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Analysis of Ternary-Valued, CIC Filter-Based OFDM Channelizers in Modern Wireless Communications Systems


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ANALYSIS OF TERNARY-VALUED, CIC FILTER-BASED, OFDM CHANNELIZERS IN MODERN WIRELESS COMMUNICATIONS SYSTEMS By WILLIAM E. LAWTON A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by William Edward Lawton

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To My Family

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ACKNOWLEDGMENTS I would like to thank Dr. Fred J. Taylor for his support and guidance during my work towards this thesis. This thesis would not have been formed without his guidance. I would also like to thank my advisors, Dr. John M. Shea and Dr. William R. Eisenstadt, for their support and contributions during this thesis. I would also like to thank my wife, Leigh, daughter, Rachael, and son, Will, for supporting me through my thesis. iv

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TABLE OF CONTENTS page ACKNOWLEDGMENTS..............................................................................................iv LIST OF TABLES........................................................................................................vii LIST OF FIGURES........................................................................................................ix ABSTRACT.................................................................................................................xiv CHAPTER 1 INTRODUCTION....................................................................................................1 Approach..................................................................................................................3 Analysis Tools..........................................................................................................4 2 OFDM BACKGROUND AND BENEFITS..............................................................6 OFDM Introduction..................................................................................................6 OFDM Background..................................................................................................9 OFDM Benefits and Drawbacks.............................................................................10 Multipath in Wireless Channels..............................................................................10 Cyclic Prefix to Mitigate Effects of Multipath.........................................................12 3 OFDM CHANNELIZER OVERVIEW AND DESCRIPTION...............................15 CIC Filter Overview...............................................................................................15 CIC Filter Optimization..........................................................................................23 OFDM Channelizer Introduction............................................................................24 OFDM Channelizer Selection.................................................................................26 48-Subcarrier OFDM Channelizer Description.......................................................27 4 OFDM CHANNELIZER PERFORMANCE IN AWGN CHANNEL.....................38 Performance of 48-Point OFDM Channelizer in AWGN Channel...........................38 Analysis of Various Constellation Schemes Utilizing Filter 1 as Reference.............44 BER Normalization of Filter Banks through Constellation Density Compensation..45 v

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5 OFDM CHANNELIZER PERFORMANCE AND LIMITATIONS IN MULTIPATH CHANNEL.....................................................................................48 OFDM Channelizer Subcarrier Separation..............................................................51 Approaches to Enhance the OFDM Channelizer for Multipath Channel Conditions...........................................................................................................57 Multipath Effects on BER Performance..................................................................62 Multipath Effects on OFDM Channelizer with Coherent Alignment.......................65 Multipath Effects on OFDM Channelizer with Cyclic Prefix..................................65 Multipath Effects on OFDM Channelizer with Coherent Alignment and Cyclic Prefix..................................................................................................................69 Summary of Performance Comparison....................................................................69 6 CONCLUSIONS AND FUTURE WORK..............................................................77 Summary of Simulation Effort................................................................................77 Lessons Learned and Future Work..........................................................................78 Summary of Simulation Performance Results.........................................................79 APPENDIX A POLYMORPHIC-BASED SPW OVERVIEW.......................................................81 B SIMULATION RESULTS RAW DATA................................................................84 C SPW BLOCK DIAGRAMS..................................................................................106 D MATLAB AND MATHEMATICA ANALYSIS COMMANDS..........................125 LIST OF REFERENCES.............................................................................................126 BIOGRAPHICAL SKETCH.......................................................................................128 vi

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LIST OF TABLES Table page 1 Measured Delay Spreads in Various Wireless Channels.......................................12 2 Explored Primary Polynomials.............................................................................28 3 Transfer Functions for 48-Subcarrier OFDM Channelizer Filter Banks.................31 4 Performance Advantage of Filters Based on Number of Subcarriers.....................44 5 Performance Advantage of Modulation Schemes..................................................45 6 Per-Filter Modulation Scheme for 48-Subcarrier OFDM Channelizer...................46 7 Filter 6 Separation Filter Subcarrier +6 Coefficient Listing...................................57 8 Filter 6 Separation Filter Subcarrier -6 Coefficient Listing...................................58 9 Filter 6 Separation Filter Subcarrier +18 Coefficient Listing.................................58 10 Filter 6 Separation Filter Subcarrier -18 Coefficient Listing..................................58 11 Rappaport Multipath Channel Tap Weights..........................................................63 12 Multipath Fading Performance Delta (@ BER = 1e-2) without Alignment (AWGN reference)...............................................................................................73 13 Multipath Fading Performance Delta (@ BER = 1e-2) with Option 1 (AWGN reference).............................................................................................................74 14 Multipath Fading Performance Delta (@ BER = 1e-2) with Option 2 (AWGN reference).............................................................................................................74 15 Multipath Fading Performance Delta (@ BER = 1e-5) without Alignment (AWGN reference)...............................................................................................75 16 Multipath Fading Performance Delta (@ BER = 1e-5) with Option 1 (AWGN reference).............................................................................................................75 17 Multipath Fading Performance Delta (@ BER = 1e-5) with Option 2 (AWGN reference).............................................................................................................76 vii

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18 Raw Data for Figure 38........................................................................................84 19 Raw Data for Figure 39........................................................................................85 20 Raw Data for Figure 40........................................................................................86 21 Raw Data for Figure 41........................................................................................87 22 Raw Data for Figure 42........................................................................................88 23 Raw Data for Figure 43........................................................................................89 24 Raw Data for Figure 44........................................................................................90 25 Raw Data for Figure 45........................................................................................91 26 Raw Data for Figure 46........................................................................................93 27 Raw Data for Figure 48........................................................................................94 28 Raw Data for Figure 65........................................................................................94 29 Raw Data for Figure 66........................................................................................95 30 Raw Data for Figure 67........................................................................................97 31 Raw Data for Figure 68........................................................................................98 32 Raw Data for Figure 69........................................................................................99 33 Raw Data for Figure 70......................................................................................101 34 Raw Data for Figure 71......................................................................................102 35 Raw Data for Figure 72......................................................................................102 36 Raw Data for Figure 73......................................................................................103 37 Raw Data for Figure 74......................................................................................104 38 Raw Data for Figure 75......................................................................................104 39 Raw Data for Figure 76......................................................................................105 viii

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LIST OF FIGURES Figure page 1 Block Diagram of Typical OFDM Transmitter.......................................................8 2 Block Diagram of Typical OFDM Receiver............................................................9 3 Common Source of Multipath..............................................................................11 4 Example of Cyclic Prefix Extension.....................................................................14 5 CIC Filter-Based Interpolator...............................................................................15 6 CIC Filter-Based Decimator.................................................................................15 7 Integrator Block Diagram.....................................................................................16 8 Magnitude Frequency Response of Integrator Filter..............................................17 9 Comb Filter Block Diagram.................................................................................17 10 Magnitude Frequency Response of Comb Filter, R*M = 1...................................18 11 Magnitude Frequency Response of Comb Filter, R*M = 2...................................18 12 Magnitude Frequency Response of Comb Filter, R*M = 4...................................19 13 Magnitude Frequency Response of Comb Filter, R*M = 8...................................19 14 Magnitude Frequency Response of Comb Filter, R*M = 16..................................20 15 Magnitude Frequency Response of CIC Filter, R*M = 1......................................21 16 Magnitude Frequency Response of CIC Filter, R*M = 2......................................21 17 Magnitude Frequency Response of CIC Filter, R*M = 4......................................22 18 Magnitude Frequency Response of CIC Filter, R*M = 8......................................22 19 Magnitude Frequency Response of CIC Filter, R*M = 16.....................................23 20 Optimized Comb Filter Block Diagram................................................................23 ix

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21 Optimized CIC Filter-Based Interpolator..............................................................24 22 Optimized CIC Filter-Based Decimator................................................................24 23 Optimized OFDM Channelizer Interpolator..........................................................24 24 Optimized OFDM Channelizer Decimator............................................................24 25 Block Diagram of an OFDM Channelizer-Based Transmitter...............................25 26 Block Diagram of an OFDM Channelizer-Based Receiver...................................25 27 Magnitude Frequency Response of Filter 1...........................................................32 28 Magnitude Frequency Response of Filter 2...........................................................32 29 Magnitude Frequency Response of Filter 3...........................................................33 30 Magnitude Frequency Response of Filter 4...........................................................33 31 Magnitude Frequency Response of Filter 5...........................................................34 32 Magnitude Frequency Response of Filter 6...........................................................34 33 Magnitude Frequency Response of Filter 7...........................................................35 34 Magnitude Frequency Response of Filter 8...........................................................35 35 Magnitude Frequency Response of Filter 9...........................................................36 36 Magnitude Frequency Response of Filter 10.........................................................36 37 Magnitude Frequency Response of All Filters......................................................37 38 BER of 48-Subcarrier OFDM Channelizer in AWGN with BPSK Modulation.....39 39 BER of 48-Subcarrier OFDM Channelizer in AWGN with QPSK Modulation.....40 40 BER of 48-Subcarrier OFDM Channelizer in AWGN with 8-PSK Modulation.....40 41 BER of 48-Subcarrier OFDM Channelizer in AWGN with 8-QAM Modulation...41 42 BER of 48-Subcarrier OFDM Channelizer in AWGN with 16-QAM Modulation.41 43 BER of 48-Subcarrier OFDM Channelizer in AWGN with 32-QAM Modulation.42 44 BER of 48-Subcarrier OFDM Channelizer in AWGN with 64-QAM Modulation.42 45 BER of 48-Subcarrier OFDM Channelizer in AWGN with 128-QAM Modulation...........................................................................................................43 x

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46 BER of 48-Subcarrier OFDM Channelizer in AWGN with 256-QAM Modulation...........................................................................................................43 47 Filter 1 OFDM Channelizer Multi-Modulation Scheme Performance...................45 48 BER of 48-Subcarrier OFDM Channelizer in AWGN with Mixed Modulation.....47 49 Multipath Constellation Scatter without Alignment..............................................50 50 Multipath Constellation Scatter with Option 1 Alignment.....................................50 51 Multipath Constellation Scatter with Option 2 Alignment.....................................51 52 Filter 3 Separation Filter Design Parameters.........................................................52 53 Filter 4 Separation Filter Design Parameters.........................................................53 54 Filter 5 Separation Filter Design Parameters.........................................................53 55 Filter 6 Separation Filter Design Parameters.........................................................54 56 Filter 7 Separation Filter Design Parameters.........................................................54 57 Filter 8 Separation Filter Design Parameters.........................................................55 58 Filter 9 Separation Filter Design Parameters.........................................................55 59 Filter 10 Separation Filter Design Parameters.......................................................56 60 Filter 6 Separation Filter Frequency Responses....................................................59 61 Block Diagram of OFDM Channelizer Receiver for Option 1...............................60 62 Block Diagram of OFDM Channelizer Transmitter for Option 2..........................61 63 Block Diagram of OFDM Channelizer Receiver for Option 2...............................61 64 Rappaport Multipath Channel Frequency Response..............................................64 65 OFDM Channelizer Filter 6 BER Results.............................................................66 66 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1............66 67 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2............67 68 BER for Filter 6 OFDM Channelizer in Multipath Channel and 4 Sample Cyclic Prefix...................................................................................................................67 xi

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69 BER for Filter 6 OFDM Channelizer in Multipath Channel and 7 Sample Cyclic Prefix...................................................................................................................68 70 BER for Filter 6 OFDM Channelizer in Multipath Channel and 10 Sample Cyclic Prefix........................................................................................................68 71 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 4 Sample Cyclic Prefix............................................................................................70 72 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 7 Sample Cyclic Prefix............................................................................................71 73 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 10 Sample Cyclic Prefix............................................................................................71 74 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 4 Sample Cyclic Prefix............................................................................................72 75 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 7 Sample Cyclic Prefix............................................................................................72 76 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 10 Sample Cyclic Prefix............................................................................................73 77 Polymorphic Block Type Illustration....................................................................82 78 Polymorphic Default Value Illustration................................................................83 79 Block Diagram of OFDM Channelizer Transmitter............................................107 80 Internal Block Diagram of OFDM Channelizer Transmitter Top Red Box.......107 81 Internal Block Diagram of OFDM Channelizer Transmitter Middle Red Box..108 82 Internal Block Diagram of OFDM Channelizer Transmitter Bottom Red Box..108 83 Block Diagram of First-Order OFDM Channelizer Transmission Filter..............109 84 Block Diagram of Second-Order OFDM Channelizer Transmission Filter..........109 85 Block Diagram of OFDM Channelizer Receiver.................................................111 86 Internal Block Diagram of OFDM Channelizer Transmitter Top Red Box.......112 87 Internal Block Diagram of OFDM Channelizer Transmitter Middle Red Box..113 88 Internal Block Diagram of OFDM Channelizer Transmitter Bottom Red Box..114 89 Block Diagram of First-Order OFDM Channelizer Receive Filter......................115 xii

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90 Block Diagram of Second-Order OFDM Channelizer Transmission Filter..........115 91 Block Diagram of AWGN Communication System............................................116 92 Blow Up Block Diagram of AWGN Communication System Top Left Red Box....................................................................................................................117 93 Blow Up Block Diagram of AWGN Communication System Middle Red Box117 94 Blow Up Block Diagram of AWGN Communication System Top Red Box, Second from Right.............................................................................................118 95 Blow Up Block Diagram of AWGN Communication System Right Red Box..118 96 Block Diagram of Multipath Communication System Using Option 1................119 97 Blow Up Block Diagram of Multipath Communication System Left Red Box..119 98 Blow Up Block Diagram of Multipath Communication System Red Box, Second from the Left..........................................................................................120 99 Blow Up Block Diagram of Multipath Communication System Red Box, Second from the Right........................................................................................120 100 Blow Up Block Diagram of Multipath Communication System Right Red Box121 101 Block Diagram of Multipath Communication System Using Option 2................121 102 Blow Up Block Diagram of Multipath Communication System Top Left Red Box....................................................................................................................122 103 Blow Up Block Diagram of Multipath Communication System Red Box, Second from the Left..........................................................................................122 104 Blow Up Block Diagram of Multipath Communication System Middle Red Box....................................................................................................................123 105 Blow Up Block Diagram of Multipath Communication System Red Box, Second from the Right........................................................................................123 106 Blow Up Block Diagram of Multipath Communication System Right Red Box124 xiii

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ANALYSIS OF TERNARY-VALUED, CIC FILTER-BASED, OFDM CHANNELIZERS IN MODERN WIRELESS COMMUNICATIONS SYSTEMS By William E. Lawton December 2005 Chair: Dr. Fred J. Taylor Major Department: Electrical and Computer Engineering Mobile wireless communications has been increasing in use and popularity over the last five to ten years. Examples of modern wireless communications systems include 2G and 3G mobile wireless as well as 802.11 a/b/g systems. The requirement for these systems to achieve greater performance and lower cost is continuing to grow. Typically system design complexity increases simultaneously with system performance. Orthogonal Frequency Division Multiplexing (OFDM) is currently being designed into an increasing number of wireless system standards in order to meet the increasing performance requirements. While OFDM is well suited to the multipath channel conditions common in wireless applications, it does require complex-valued multiplications in order to realize the Fast-Fourier Transform (FFT) central to most OFDM-based systems. This thesis explores the feasibility of using OFDM channelizers to realize the benefits inherent in FFT-based OFDM systems while simultaneously decreasing the required design complexity necessary to implement the system. xiv

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This thesis presents a study to analyze the use of OFDM channelizers as suitable alternatives to their FFT-based counterparts. OFDM channelizers consist of ternary-valued filters similar to cascaded integrator comb (CIC) filters. Number theory is leveraged in order to realize filters capable of generating OFDM symbols from a bank of filters consisting entirely of additions and subtractions. The OFDM channelizers similarity to an FFT-based OFDM system suggests that it possesses the same ability to mitigate the effect of multipath interference in a wireless channel. Simulation results created using Cowares Signal Processing Worksystem (SPW) are presented and analyzed to determine the ability for OFDM channelizers to operate effectively in multipath channel conditions. Since different systems can have widely varying requirements and operating conditions, it is not possible to present an optimal OFDM channelizer adapted for all conceivable system design parameters. Instead, the analyses are presented along with different options for implementing channel adaptation schemes based on an OFDM channelizer. Additionally, advantages and disadvantages of OFDM channelizers relative to an FFT-based OFDM system are given. These tradeoffs presented in this thesis are intended to serve as guidance for anyone interested in incorporating the benefits of an OFDM channelizer into a system design. xv

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CHAPTER 1 INTRODUCTION Wireless communication systems have been increasing in use and popularity in the past five to ten years. Examples of such systems are 802.11 a/b/g fixed wireless, 2 nd generation (2G) and 3 rd generation (3G) mobile wireless systems. Along with increased use and popularity, the demand for wireless systems to deliver increased data rates and quality of service (QoS) is also growing. In order to meet this demand, many aspects of the systems must scale to achieve the increasing target performance requirements. These include increasing the available channel capacity as well as improving receiver performance to attain performance closer to the theoretical channel capacity limit. A simultaneous design challenge to reduce the cost of equipment is created both by consumer demand and competition in the marketplace. The increased channel capacity is being realized with multiple approaches currently. Technologies such as 802.11n are employing MIMO-based (multiple-input, multiple-output) communication in order to increase channel capacity by taking advantage of multipath channel conditions. In general, this approach tends to increase overall system cost. This contradicts the market demand for lower cost devices. In order to keep the cost of these new solutions low, the digital domain is migrating closer to the antenna in order to replace some of the relatively expensive analog components with inexpensive digital equivalents. This means that digital solutions are required that can operate at very high speeds. Additionally, many wireless devices are powered with 1

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2 batteries. Therefore these high speed digital designs need to be optimized for low power consumption. The improvement of receiver performance necessary to approach channel capacity limits includes utilizing more robust error correcting codes, such as turbo codes, that can achieve performance approaching Shannons limit [1]. Furthermore, advanced modulation and demodulation techniques are being utilized in order to achieve necessary increased performance and mitigate the effects introduced by wireless communication channels. One type of advanced modulation technique being utilized in modern communication systems is Orthogonal Frequency Division Multiplexing (OFDM). OFDM utilizes multiple subcarriers, each with longer symbol lengths than single-carrier modulation symbols, in order to mitigate the effects of inter-symbol interference (ISI) introduced by multipath channels. Other benefits attributable to OFDM modulation will be explored in this thesis as well. Typically, these advanced modulation and demodulation techniques require more processing power in order to realize these advanced receivers. This increased processing power is undesirable because it increases the power utilization, thus reducing the battery life in mobile wireless devices. OFDM is typically implemented with a Fast Fourier Transform (FFT) architecture due to the relatively low complexity of the FFT compared to a mathematically equivalent DFT (Discrete Fourier Transform). Although it is typical to utilize an FFT to implement an OFDM communication system, it is not necessary. Other mechanisms can be employed to develop an OFDM-based communication system. An alternative in the form of a ternary-valued, CIC (concatenated integrator comb) filter

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3 based OFDM channelizer, hereafter referred to as OFDM channelizer, will be investigated in this thesis. Approach Chapter 2 presents a brief history of OFDM as well as certain benefits of utilizing OFDM as a modulation scheme for wireless communications. Additionally, this chapter presents the primary challenges introduced by a multipath channel and how the OFDM characteristics allow for robust communication in such an environment. This chapter defines the advantages and disadvantages of OFDM since most of them are shared by the OFDM channelizer. Chapter 3 presents the concept and structure of an OFDM channelizer. Furthermore, it describes the transformation of a typical CIC filter into an OFDM channelizer capable of substituting an FFT engine in an OFDM system. Chapter 4 presents a study to analyze the viability and usefulness of a ternary valued, CIC filter-based OFDM channelizer to replace the more common FFT to perform the modulation of the transmitted signal into multiple narrowband carriers. In addition, the primary differences between the CIC filter-based channelizer and an OFDM system are explained. This chapter also presents performance simulation results of an OFDM channelizer in an additive white Gaussian noise (AWGN) channel. Furthermore, a proposal to leverage a benefit of the OFDM channelizer is given. Chapter 5 presents concepts to add robustness to the OFDM channelizer in a multipath channel. Additionally, it presents some of the challenges and limitations introduced by the OFDM channelizer when compared to an FFT-based OFDM modulation scheme in multipath channels as well as performance simulation results in a multipath channel with various assumptions.

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4 Chapter 6 summarizes the work and results provided in this thesis. Additionally, areas of future work to be considered are provided. Appendix A provides a brief overview of SPW block diagram interpretation necessary for readers not already familiar with SPW. Appendix B presents the raw simulation data results presented elsewhere in the thesis in graph form. Appendix C presents screen captures of the SPW block diagrams developed during this thesis creation. Appendix D presents both Matlab and Mathematica commands used to perform analysis in this thesis. Analysis Tools The work presented in this thesis is supported through simulation-based analysis. The primary tool for performing this analysis is Cowares Signal Processing Worksystem (SPW) version 4.9. SPW is a system-level design tool based on a hierarchal block diagram design approach. In Appendix C, native block diagrams captured from within SPW are used as figures to describe the implementation behind the simulation results. Specifically, polymorphic designs are used to capture the designs. Polymorphic-based designs are desirable as they allow simultaneous capture and representation of a design for both floating-point and fixed-point implementations. These block diagrams are given as an alternative to code listings common in publications. Appendix A presents a high-level introduction to SPW block diagrams and polymorphic extensions in order to facilitate an understanding required of the reader in order to interpret the block diagrams.

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5 Additionally, Microsoft Excel spreadsheet processing software is used to format and plot results obtained from various SPW simulations as well as generate frequency response plots of many aspects of the simulation systems included in this thesis.

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CHAPTER 2 OFDM BACKGROUND AND BENEFITS This chapter focuses on the background of OFDM modulation as well as its usefulness in modern wireless communication systems. It begins with an overview of various techniques common within OFDM systems as well as reasons and explanations of the techniques. It continues by describing characteristics of multipath channels along with a description of how OFDM modulation schemes can be designed to mitigate the effects of multipath channel propogation. OFDM Introduction OFDM is a special form of multicarrier transmission. It can be viewed as either a data modulation technique or a data multiplexing technique. This chapter will focus on OFDM as a data modulation technique. The main benefit of OFDM is its robustness against frequency selective fading created by multipath. In a single carrier system, a fade in a portion of the signal band can cause severe degradation to the overall link. However, in an OFDM system, only a small percentage of the total carriers will exist in any one fade. Therefore, the link will continue to persist because even a simple error correcting coding scheme will resolve the errors. Figure 1 illustrates a block diagram of a typical OFDM transmitter. The basic makeup of an OFDM transmitter consists of the following components and their respective descriptions: Coding insertion of parity and cyclic redundancy check (CRC) into the data stream in order to correct and detect errors at the receiver 6

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7 Interleaver reordering of the data stream in order to more evenly distribute errors generated in the channel that are correlated either in time or in frequency. Allows for more effective and efficient error correction. Bit-to-Symbol Mapping transforms a bit stream into complex-valued symbols in the signal space domain. These symbols are the Fourier coefficients input to the Inverse Fast Fourier Transform (IFFT). Pilot Insertion pilot or synchronization symbols are common in OFDM systems in order for the receiver to accurately measure the channel response as well as the frequency of the transmitter. These estimates help ensure accurate reception of the transmitted signal. Serial-to-Parallel groups N symbols before computing the IFFT to generate an OFDM symbol. IFFT performs the inverse fast Fourier transform converting the frequency domain signal into the time domain for transmission. Parallel-to-Serial serializes the time-domain signal for transmission. Cyclic Extension Insertion transmits the tail of each OFDM symbol before wrapping around and transmitting the entire OFDM symbol from beginning to end. Windowing applies a window to a percentage of the cyclically extended OFDM symbol at both the head and tail of the symbol. This step typically also includes overlapping the windowed tail of symbol M and the windowed head of symbol M+1. This improves the spectral properties of the transmitted signal. DAC Digital-to-Analog Converter that transforms the digital signal to analog. LPF low-pass filter to eliminate spectral copies caused by conversion from digital to analog signal. RF TX optional RF transmission circuitry. Some OFDM implementations transmit the baseband OFDM signal directly.

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8 Coding:Forward Error Correction / CRC Time / Frequency Interleaver Bit-to-Symbol Mapping Optional Pilot Insertion Serial-to-Parallel IFFT Parallel-to-Serial Cyclic Extension Insertion Windowing DAC LPF / RF TX Figure 1. Block Diagram of Typical OFDM Transmitter Figure 2 illustrates a block diagram of a typical OFDM receiver. The basic makeup of an OFDM receiver consists of the following components and their respective descriptions: RF RX optional RF reception circuitry. LPF also referred to as an anti-aliasing filter. Attenuates higher frequency components prior to sampling to reduce effects of aliasing caused by sampling. ADC Analog-to-Digital Converter that transforms the analog signal to digital. Time / Frequency Synchronization Detects starting time of received frame as well as estimating sampling frequency of transmitted signal in order to minimize effects of frequency error of the receivers sampling clock relative to the transmitter clock. Cyclic Extension Removal removes the cyclic extention of the OFDM symbols in the receiver. Most of the ISI is contained within this time and therefore not processed in the receiver. Serial-to-Parallel groups N samples before computing the FFT to convert the received signal back into the frequency domain. FFT performs the fast Fourier transform converting from the time domain into the frequency domain for demodulation. Parallel-to-Serial serializes the frequency-domain signal for further processing. Channel Correction estimates the channel response on a per-carrier basis. The channel estimate is used to normalize the channel response in the received signal.

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9 Symbol-to-Bit Demapping maps the received symbols to soft bits used for decoding. Soft bits are typically generated as input to the decoder due to increased performance of the decoder versus hard-decision (antipodal) bits. Time / Frequency Deinterleaver restores original order of transmitted bit stream into the decoder. This deinterleaving increases the average distance between burst errors introduced by the channel in either the time or frequency domain. Decoder corrects bit errors introduced either in the channel or in the noise present in the receiver front end. Additionally, uncorrectable errors are typically detected by comparing the received CRC against another CRC that is computed in the receiver. RF RX / LPF ADC Time / Frequency Synchronization Cyclic Extension Removal Serial-to-Parallel FFT Parallel-to-Serial Channel Correction Symbol-to-Bit Demapping Time / Frequency Deinterleaver Coding:Forward Error Correction / CRC Figure 2. Block Diagram of Typical OFDM Receiver OFDM Background OFDM studies date back to the 1960s [2, 3]. In the 1960s, OFDM was incorporated into several military systems such as KINEPLEX [4], ANDEFT [5] and KATHRYN [6]. The first OFDM patent was filed and issued in 1971 [7]. In the 1980s, OFDM usage studies began to branch into areas including high-speed modems, digital mobile communications and high-density recording. The first commercial applications of OFDM can be traced back to Discrete Multi-Tone (DMT). DMT was developed to transmit video over twisted pair copper wires for Digital Subscriber Line (DSL). Then, during the 1990s, OFDM became integral for wideband data communications over mobile radio FM channels, various forms of high

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10 data rate digital subscriber lines (xDSL), digital audio broadcasting (DAB) and high-definition television (HDTV) terrestrial broadcasting. OFDM Benefits and Drawbacks OFDM has been characterized well over the past few decades. This section highlights the well known advantages and drawbacks of OFDM modulation. These are listed here to be explained in more detail in the following sections. OFDM has several main advantages described below: OFDM is robust against multipath. The implementation complexity is significantly lower for an OFDM-based system versus a single-carrier system with equalization for a given delay spread. OFDM has the potential to exploit distinct signal-to-noise ratios (SNR) per carrier by modulating a different number of bits per carrier. This has the net effect of increasing the capacity of the system. OFDM tolerates narrowband interference effectively. This is due to the inherent frequency division in the signaling. Any narrowband interference interferes with a relatively small number of carriers in the system. OFDM also has a couple of primary disadvantages described below: OFDM signals are usually characterized by relatively high peak-to-average power ratios (PAPR). This can lead to reduced power efficiency in the power amplifier in the system. OFDM is more sensitive to frequency offset and phase noise than a typical single-carrier system. Multipath in Wireless Channels Multipath is a common phenomenon occurring in wireless channels. Figure 3 demonstrates a typical source of the multipath phenomenon in a wireless channel. As shown, reflections from multiple objects near either the transmitter or receiver combine to contribute parts of the received signal occurring at different time offsets. These time-offset reflections combine to create a frequency selective channel response. The

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11 degradation caused by multipath is most pronounced when there is no line-of-sight (LOS) path between the transmitter and receiver. Transmitter Receiver Figure 3. Common Source of Multipath Various studies and characterizations of multipath channel profiles in different environments exist in literature. This thesis does not attempt to describe the details of these various channel characteristics. However, the most important design parameter relating to an OFDM system is the maximum delay spread introduced by a channel. This maximum delay spread defines the duration of time from the first path response to the last path response when measured from the receiver. In practice, the duration of time where a certain percentage of the received energy is localized is used to define the delay spread. This is typically more practical to design for rather than the delay spread to contain all of the received signal energy.

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12 Cyclic Prefix to Mitigate Effects of Multipath One of the primary benefits of OFDM modulation is its ability to effectively mitigate multipath delay spread. Fundamentally, this is achieved because the symbol duration of an OFDM symbol is substantially longer than the symbol time of an equivalent bit-rate, single-carrier system. Since OFDM transmits data in parallel using multiple carriers, the resulting OFDM symbol duration is proportionally longer based on the number of carriers. Therefore, the channel delay spread, relative to the symbol duration, is reduced. Many empirical studies have been performed in order to characterize wireless channel characteristics in many different environments [8, 9]. Typical delay spreads seen in various scenarios are listed in Table 1 below. Depending on the baseband sampling rate of the system under study, the various delay spreads identified in Table 1 translate into a certain number of baseband samples. This duration defines the amount of ISI introduced into the received signal. Table 1. Measured Delay Spreads in Various Wireless Channels Environment Description Median Delay Spread [ns] Maximum Delay Spread [ns] Frequency Range [GHz] Large building 40 120 4 6 Office building #1 50 60 4 6 Meeting room (metal walls) 35 55 4 6 Single room (stone walls) 10 35 4 6 Office building #2 40 130 4 6 Indoor sports arena 40 120 4 6 Factory #1 65 125 4 6 Office building #3 25 65 4 6 Office building #4 (single room) 20 30 4 6 Office building #5 -1000 0.815 Office #6 90 8000 0.915/1.9 Urban 136/258 -1.9

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13 Typical OFDM systems use a well-known approach to mitigating the delay spread introduced by a multipath channel. They reduce or eliminate the inter-symbol interference (ISI) introduced by the channel by adding a guard period to the OFDM symbols. An effective guard period should be at least as long as the maximum expected delay spread occurring over the worst-case potential channel condition. However, guard periods of excessive length add overhead to the transmission scheme, ultimately reducing system throughput. Therefore, there is a design tradeoff relative to the length of the guard interval. Guard intervals are implemented using a cyclic extension of the OFDM symbol, extending a portion of the tail of the OFDM symbol to the beginning of the new cyclically extended OFDM symbol. This is illustrated in Figure 4 below. As noted in Figure 1, the cyclic extension occurs after the IFFT. The IFFT produces a time-domain signal consisting of all carriers superimposed with one another. Figure 4 illustrates how each carrier is cyclically extended. In an actual implementation, the cyclic extension is achieved by extending a single time-domain signal consisting of all of the subcarriers superimposed into a single data stream. As an alternative to a cyclic extension, the guard period could be implemented with a zero-valued insertion. However, a zero-valued guard period would have the negative side-effect of introducing inter-carrier interference (ICI) since integer numbers of carrier cycles are no longer guaranteed to exist within a single non-extended OFDM symbol duration. A cyclic extended guard period rather than a zero-filled guard period of the transmitted symbols prevents this ICI.

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14 Cyclic Prefix Cyclically Extended OFDM Symbol Original OFDM Symbol Figure 4. Example of Cyclic Prefix Extension

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CHAPTER 3 OFDM CHANNELIZER OVERVIEW AND DESCRIPTION As described earlier, typical OFDM-based systems implement modulation and demodulation utilizing an FFT structure. The benefits of utilizing an FFT for this function have been discussed previously. This chapter will investigate utilizing an OFDM channelizer as an alternative to FFT-based OFDM. CIC Filter Overview The OFDM channelizer explored in this thesis is based on a Cascaded Integrator-Comb (CIC) filter. The CIC filter is a well-known, multiplier-less finite impulse response (FIR) filter having a wide variety of applications. One of the more common applications is as a digital sample-rate converter where a highly sampled signal contains a relatively narrowband baseband signal of interest. A CIC filter can be used in either interpolation or decimation schemes to either increase or decrease the sample rates of a signal, respectively. Top-level block diagrams of these sample-rate converters are shown below in Figure 5 and Figure 6. Comb Comb Comb RUpsample Integrator Integrator Integrator Figure 5. CIC Filter-Based Interpolator Integrator Integrator Integrator RDownsample Comb Comb Comb Figure 6. CIC Filter-Based Decimator 15

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16 A CIC filter, as its name suggests, consists of two primary stages cascaded serially. The first stage consists of one or more integrators. The second stage consists of an equal number of comb filters. A single integrator has a transfer function given below. 111)(= z zH (1) As the transfer function indicates, the integrator has a single pole located directly at unity on the real axis on the unit circle in the z-domain. A block diagram of an integrator is given in Figure 7. Z-1 + Figure 7. Integrator Block Diagram The magnitude frequency response of the integrator filter is shown in Figure 8 below. A comb filter has a transfer function given by the equation below. RMzzH=1)( (2) In the above transfer function, R is typically referred to as the integer rate change factor and M is typically referred to as the differential delay [10]. A block diagram of a comb filter is given in Figure 9.

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17 -100102030405060-0.5-0.4-0.29-0.19-0.080.020.120.230.330.44Frequency (fs = 1 Hz)Magnitude (dB) Integrator Figure 8. Magnitude Frequency Response of Integrator Filter Z-R*M + Figure 9. Comb Filter Block Diagram The magnitude frequency responses of the comb filter when R*M = 1, 2, 4, 8 and 16 are shown in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, respectively. The CIC filter is constructed by concatenating the integration and comb filter stages that are described above. A non-obvious observation regarding the CIC filter is that it is actually an FIR filter. This is not obvious since the filter has feedback contained within its integrators. Typically with digital signal processing, feedback is synonymous with infinite impulse response (IIR) filters. However, a CIC filter, under closer examination,

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18 produces a finite impulse response. This finite impulse response is realized by the exact cancellation of the zeros from the comb filter and the poles in the integrator located at unity on the real axis of the unit circle. This finite impulse response is demonstrated in the transfer response equation below. -120-100-80-60-40-20020-0.5-0.4-0.29-0.19-0.080.0210.1250.2290.3330.437Frequency (fs = 1 Hz)Magnitude (dB) Comb Figure 10. Magnitude Frequency Response of Comb Filter, R*M = 1 -120-100-80-60-40-20020-0.5-0.4-0.29-0.19-0.080.0210.1250.2290.3330.437Frequency (fs = 1 Hz)Magnitude (dB) Comb Figure 11. Magnitude Frequency Response of Comb Filter, R*M = 2

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19 -120-100-80-60-40-20020-0.5-0.4-0.29-0.19-0.080.0210.1250.2290.3330.437Frequency (fs = 1 Hz)Magnitude (dB) Comb Figure 12. Magnitude Frequency Response of Comb Filter, R*M = 4 -120-100-80-60-40-20020-0.5-0.4-0.29-0.19-0.080.0210.1250.2290.3330.437Frequency (fs = 1 Hz)Magnitude (dB) Comb Figure 13. Magnitude Frequency Response of Comb Filter, R*M = 8

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20 -120-100-80-60-40-20020-0.5-0.4-0.29-0.19-0.080.0210.1250.2290.3330.437Frequency (fs = 1 Hz)Magnitude (dB) Comb Figure 14. Magnitude Frequency Response of Comb Filter, R*M = 16 ()NMRkkNMRNzzzzH===101111)( (3) When the integrator and comb filters are cascaded into a single CIC filter, there is typically a rate conversion inserted between the integrator and comb sections of the filter. In this form, the CIC filter becomes a multirate filter. The magnitude frequency response of a CIC filter with R*M equal to one is shown in Figure 15 below. It can be seen that the response of a CIC filter with R*M equal to one is flat. This can be seen by the transfer function given in Equation 3 when R*M equals one. In this case the single zero in the comb filter exactly cancels the pole in the integrator filter. In order to develop filters of more interest, R*M needs to be something other than 1 [11]. Examples of CIC filter magnitude frequency responses with R*M equal to 2, 4, 8 and 16 are shown in Figure 16, Figure 17, Figure 18 and Figure 19, respectively.

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21 -200-1000100200300400-0.5-0.4-0.29-0.19-0.080.020.120.230.330.44Frequency (fs = 1 Hz)Magnitude (dB) Integrator Comb CIC Figure 15. Magnitude Frequency Response of CIC Filter, R*M = 1 -200-1000100200300400-0.5-0.4-0.29-0.19-0.080.020.120.230.330.44Frequency (fs = 1 Hz)Magnitude (dB) Integrator Comb CIC Figure 16. Magnitude Frequency Response of CIC Filter, R*M = 2

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22 -200-1000100200300400-0.5-0.4-0.29-0.19-0.080.020.120.230.330.44Frequency (fs = 1 Hz)Magnitude (dB) Integrator Comb CIC Figure 17. Magnitude Frequency Response of CIC Filter, R*M = 4 -200-1000100200300400-0.5-0.4-0.29-0.19-0.080.020.120.230.330.44Frequency (fs = 1 Hz)Magnitude (dB) Integrator Comb CIC Figure 18. Magnitude Frequency Response of CIC Filter, R*M = 8

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23 -200-1000100200300400-0.5-0.4-0.29-0.19-0.080.020.120.230.330.44Frequency (fs = 1 Hz)Magnitude (dB) Integrator Comb CIC Figure 19. Magnitude Frequency Response of CIC Filter, R*M = 16 CIC Filter Optimization A simple modification can be made to the typical CIC-based FIR filter presented above in order to optimize the architecture. The optimization is realized by pushing the comb filter section of the CIC FIR through the rate change operation. In order to maintain an equivalent filter response with this architecture change, the comb filter must be altered accordingly. The simple modification to the comb filter involves reducing the delay operator from R*M delays to M delays as shown in Figure 20. Z-M + Figure 20. Optimized Comb Filter Block Diagram

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24 The corresponding optimized CIC interpolation and decimation filters are shown in Figure 21 and Figure 22, respectively. Comb Comb Comb RUpsample Integrator Integrator Integrator Figure 21. Optimized CIC Filter-Based Interpolator Integrator Integrator Integrator RDownsample Comb Comb Comb Figure 22. Optimized CIC Filter-Based Decimator OFDM Channelizer Introduction The OFDM channelizer utilizes number theory in order to enhance the capability of the more common CIC filter. In order to transform a CIC filter into an OFDM channelizer, a reduced polynomial based filter is chosen to replace the interpolator stage of the CIC filter [12, 13]. The corresponding optimized OFDM channelizer interpolation and decimation filters are shown in Figure 23 and Figure 24, respectively. Although the figures show multiple stages of comb and reduced-polynomial based filters (N > 1), this thesis will concentrate on analyzing the OFDM channelizer when N=1. In this case, the OFDM channelizer produces a spectrum equivalent to the same size FFT-based OFDM transmitter. Increasing N narrows the spectrum of each harmonic produced by the filter. Comb Comb Comb RUpsample Reduced Polynomial Based Filter Reduced Polynomial Based Filter Reduced Polynomial Based Filter Figure 23. Optimized OFDM Channelizer Interpolator Reduced Polynomial Based Filter Reduced Polynomial Based Filter Reduced Polynomial Based Filter RDownsample Comb Comb Comb Figure 24. Optimized OFDM Channelizer Decimator

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25 System-level block diagrams of a transmitter and receiver as part of an OFDM channelizer-based system are shown below in Figure 25 and Figure 26, respectively. It should be noted that the system-level block diagrams of the OFDM channelizer are similar to the original FFT-based OFDM diagrams presented earlier. The primary difference is the replacement of the FFT / IFFT by the equivalent Channelizer filter bank. Coding:Forward Error Correction / CRC Time / Frequency Interleaver Bit-to-Symbol Mapping Optional Pilot Insertion Serial-to-Parallel Channelizer Filter Bank Parallel-to-Serial Cyclic Extension Insertion Windowing DAC LPF / RF TX Figure 25. Block Diagram of an OFDM Channelizer-Based Transmitter RF RX / LPF ADC Time / Frequency Synchronization Cyclic Extension Removal Serial-to-Parallel Channelizer Filter Bank Parallel-to-Serial Channel Correction Symbol-to-Bit Demapping Time / Frequency Deinterleaver Coding:Forward Error Correction / CRC Carrier Combining (optional) Figure 26. Block Diagram of an OFDM Channelizer-Based Receiver The transformation of a typical CIC filter into an OFDM channelizer begins with selecting the number of subcarriers created by the OFDM channelizer. This is synonymous with selecting the FFT size in a typical OFDM system. However, there are additional design tradeoffs that must be taken into account with the OFDM channelizer that do not need to be taken into account for an FFT-based OFDM system. The details of selecting the number of subcarriers are described in the following section.

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26 Once the number of subcarriers is chosen (N), a corresponding primary polynomial can be factored into a set of irreducible primitive polynomials. The coefficient set of the primitive polynomials is limited to the terary values, {-1, 0, 1}. The primary polynomial is of the form shown in Equation 4. 1)(=Nzzg (4) Factoring the primary polynomials into irreducible primitive polynomials yields the transfer functions that are used to replace the integrators in the typical CIC filter. OFDM Channelizer Selection A primary polynomial of form z N -1 is chosen based on two relevant criteria for the reduction of the polynomial over the ternary-valued coefficient set {-1, 0, 1}. First, the polynomial should reduce into a relatively large number of independent polynomials. Second, each polynomial resulting from the reduction of the primary polynomial should consist of a small number of terms. Table 2 shows a list of various primary polynomials and the corresponding number of irreducible polynomials resulting from the reduction against the possible coefficient set as well as the maximum number of terms in the reduced polynomials. It is interesting to note that the number of polynomials resulting in the reduction of the primary polynomial with only ternary-valued coefficients is equal to the number of divisors of the polynomial order, N. For example, z 12 -1 results in 6 ternary-valued coefficients. The integer, 12, has 6 divisors: 1, 2, 3, 4, 6 and 12. This holds true for all of the polynomials analyzed in Table 2 below. Given Table 2, N equal to 48 was chosen for further analysis in this thesis due to a relatively high number of reduced polynomials, each having a low number of terms. The

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27 number of terms per polynomial is an important design factor since the number of terms in the reduced polynomials directly maps into the number of additions required per sample per filter. A higher number of additions per sample per filter has a direct affect on the speed at which the filters can be operated. The number of irreducible polynomials defines the number of independent filter banks that can be realized. This design criterion will become more important as the characteristics of an OFDM channelizer are explored under wireless channel propagation conditions. It should be noted that in an FFT-based OFDM system, every carrier can be independently modulated and demodulated. 48-Subcarrier OFDM Channelizer Description The detailed simulation analysis in this thesis will be performed for the 48-subcarrier system design. The factorization of the primary polynomial, z -48 -1 is shown in Table 3 along with various other filter characteristics. Table 3 highlights the primary difference between an OFDM channelizer and a similar FFT-based OFDM system. Where an FFT contains N distinct carriers, each of which can contain independent data streams, an OFDM channelizer only has as many distinct channels as filters. For the case of the 48-subcarrier OFDM channelizer, ten independent data streams are supported. The number of frequency bins column shows how many subcarriers (harmonics) are generated through each filter. The total number of subcarriers across all filters is equal to 48 and the resulting total combined spectrum is equivalent to that of a 48-point FFT. The magnitude frequency responses of the ten distinct filters for the 48-subcarrier OFDM channelizer are shown below in Figure 27, Figure 28, Figure 29, Figure 30, Figure 31, Figure 32, Figure 33, Figure 34, Figure 35 and Figure 36. The combined frequency response of all filters superimposed on a single graph is shown in Figure 37.

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28 Figure 37 shows the frequency response of the entire 48-subcarrier OFDM channelizer after the output of each filter has been normalized. The normalization factors on each filter are shown in Table 3. For an optimal implementation, the scaling of each filter output should be applied at the low end rate of the multirate filter. Therefore, although this normalization factor is a multiplication necessary to flatten the power spectral density, it operates at a relatively slow rate and therefore does not limit the speed at which the implementation can be executed. Table 2. Explored Primary Polynomials N Number of Reduced Polynomials Largest Number of Terms in Reduced Polynomials Highest Reduced Polynomial Order Possible Use 12 6 3 4 Yes 13 2 13 12 No 1,2 14 4 7 6 No 2 15 4 7 8 No 2 16 5 2 8 Yes 17 2 17 16 No 1,2 18 6 3 6 Yes 19 2 19 18 No 1,2 20 6 5 8 No 2 21 4 9 12 No 2 22 4 11 10 No 2 23 2 23 22 No 1,2 24 8 3 8 Yes 25 3 5 20 No 1 26 4 13 12 No 2 27 4 3 18 No 1 28 6 7 12 No 2 29 2 29 28 No 1,2,3 30 8 7 8 No 2 31 2 31 30 No 1,2,3 32 6 2 16 Yes

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29 Table 2. Continued. N Number of Reduced Polynomials Largest Number of Terms in Reduced Polynomials Highest Reduced Polynomial Order Possible Use 33 4 15 20 No 2 34 4 17 16 No 2 35 4 17 24 No 2 36 9 3 12 Yes 37 2 37 36 No 1,2,3 38 4 19 18 No 2 39 4 17 24 No 2 40 8 5 16 Yes 41 2 41 40 No 1,2,3 42 8 9 12 No 2 43 2 43 42 No 1,2,3 44 6 11 20 No 2 45 6 7 24 No 2 46 4 23 22 No 2 47 2 47 46 No 1,2,3 48 10 3 16 Yes 49 3 7 42 No 2,3 50 6 5 20 No 1 51 4 23 32 No 2,3 52 6 13 24 No 2 53 2 53 52 No 1,2,3 54 8 3 18 Yes 55 4 17 40 No 2,3 56 8 7 24 No 2 57 4 25 36 No 2,3 58 4 29 28 No 2,3 59 2 59 58 No 1,2,3 60 12 7 16 No 2 61 2 61 60 No 1,2,3 62 4 31 30 No 2,3 63 6 9 32 No 2,3 64 7 2 32 No 1,3 65 4 31 48 No 2,3 66 8 15 20 No 2 67 2 67 66 No 1,2,3 68 6 17 32 No 2,3 69 4 31 44 No 2,3 70 8 17 24 No 2 71 2 71 70 No 1,2,3

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30 Table 2. Continued. N Number of Reduced Polynomials Largest Number of Terms in Reduced Polynomials Highest Reduced Polynomial Order Possible Use 72 12 3 24 Yes 73 2 73 72 No 1,2,3 74 4 37 36 No 2,3 75 6 7 40 No 2,3 76 6 19 36 No 2,3 77 4 31 60 No 2,3 78 8 17 24 No 2 79 2 79 78 No 1,2,3 80 10 5 32 No 3 81 5 3 54 No 1,3 82 4 41 40 No 2,3 83 2 83 82 No 1,2,3 84 12 9 24 No 2 85 4 41 64 No 2,3 86 4 43 42 No 2,3 87 4 39 56 No 2,3 88 8 11 40 No 2,3 89 2 89 88 No 1,2,3 90 12 7 24 No 2 91 4 23 72 No 2,3 92 6 23 44 No 2,3 93 4 41 60 No 2,3 94 4 47 46 No 2,3 Note 1: Undesirable number of irreducible polynomials. Too few. Note 2: Undesirable number of terms in irreducible polynomials. Too many. Note 3: Undesirable highest order of reduced polynomial. Too large.

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31 Table 3. Transfer Functions for 48-Subcarrier OFDM Channelizer Filter Banks Filter Number Reduced Polynomial for Interpolator Replacement Filter Transfer Response Number of Harmonics Filter Resonant Harmonic Frequencies (f s =48) Filter Normalization Factor 1 11 z 14811 z z 1 0 1 2 11 + z 14811 + z z 1 -24 1 3 21 + z 24811 + z z 2 +/-12 2 4 211 + z z 214811 + z z z 2 +/-8 3 5 211 + + z z 214811 + + z z z 2 +/-16 3 6 41 + z 44811 + z z 4 +/-6, +/-18 4 7 421 + z z 424811 + z z z 4 +/-4, +/-20 32 8 81 + z 84811 + z z 8 +/-3, +/-9, +/-15, +/-21 8 9 841 + z z 844811 + z z z 8 +/-2, +/-10, +/-14, +/-22 32 10 1681 + z z 1684811 + z z z 16 +/-1, +/-5, +/-7, +/-11, +/-13, +/-17, +/-19, +/-23 38

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32 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 1 Figure 27. Magnitude Frequency Response of Filter 1 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 2 Figure 28. Magnitude Frequency Response of Filter 2

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33 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 3 Figure 29. Magnitude Frequency Response of Filter 3 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 4 Figure 30. Magnitude Frequency Response of Filter 4

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34 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 5 Figure 31. Magnitude Frequency Response of Filter 5 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 6 Figure 32. Magnitude Frequency Response of Filter 6

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35 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 7 Figure 33. Magnitude Frequency Response of Filter 7 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 8 Figure 34. Magnitude Frequency Response of Filter 8

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36 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 9 Figure 35. Magnitude Frequency Response of Filter 9 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 10 Figure 36. Magnitude Frequency Response of Filter 10

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37 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 A Figure 37. Magnitude Frequency Response of All Filters

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CHAPTER 4 OFDM CHANNELIZER PERFORMANCE IN AWGN CHANNEL This chapter presents the performance of the 48-point OFDM channelizer in an AWGN channel. It goes on to describe techniques for overcoming apparent limitations associated with the OFDM channelizer when compared to an FFT-based OFDM modulation scheme under these channel conditions. An assumption for this analysis is that the energy per subcarrier is constant across all subcarriers. Therefore, bits modulated through filters generating a greater number of subcarriers produce proportionally more energy than filters generating a smaller number of subcarriers. A representative power spectral density has been shown previously in Figure 37. Performance of 48-Point OFDM Channelizer in AWGN Channel In an AWGN channel, the OFDM channelizer provides the same performance as a matched-filter receiver. This section presents simulation results for the 48-Point Channelizer and derives the process gains achieved across the various filter banks. The calculation for the probability of error in an uncoded, antipodal, BPSK-modulated system with a matched-filter receiver is given below, where the energy per bit is given by E b and the noise power is given by N o /2 [14]. =obbNEQE2}Pr{ (5) With an OFDM channelizer, the probability of bit error varies across each of the filter banks. This is due to the fact that each filter can generate a different number of subcarriers. Given a uniform power spectral density across all subcarriers, the greater the 38

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39 number of subcarriers a filter occupies, the greater the transmitted energy per bit with respect to the information modulated through the filter and the lower the probability of bit error after the energy from the subcarriers is coherently combined at the receiver. The measured performance of each filter bank in an AWGN channel with various modulation schemes are shown in the following figures below. The modulation schemes considered include BPSK, QPSK, 8-PSK, 8-QAM, 16-QAM, 32-QAM, 64-QAM, 128-QAM and 256-QAM [14]. The modulation schemes consist of common square or rectangular constellations with Gray code mapping of bits to symbols in the signal space domain. The theoretical probability of bit error for antipodal BPSK signaling is given as a common reference in all of the figures below. 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-21471013161922SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 38. BER of 48-Subcarrier OFDM Channelizer in AWGN with BPSK Modulation

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40 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-21471013161922SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 39. BER of 48-Subcarrier OFDM Channelizer in AWGN with QPSK Modulation 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-21471013161922SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 40. BER of 48-Subcarrier OFDM Channelizer in AWGN with 8-PSK Modulation

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41 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-21471013161922SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 41. BER of 48-Subcarrier OFDM Channelizer in AWGN with 8-QAM Modulation 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-21471013161922SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 42. BER of 48-Subcarrier OFDM Channelizer in AWGN with 16-QAM Modulation

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42 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-2147101316192225SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 43. BER of 48-Subcarrier OFDM Channelizer in AWGN with 32-QAM Modulation 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-214710131619222528SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 44. BER of 48-Subcarrier OFDM Channelizer in AWGN with 64-QAM Modulation

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43 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-214710131619222528SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 45. BER of 48-Subcarrier OFDM Channelizer in AWGN with 128-QAM Modulation 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-23-20-17-14-11-8-5-214710131619222528SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 BPSK Theoretical Figure 46. BER of 48-Subcarrier OFDM Channelizer in AWGN with 256-QAM Modulation

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44 As shown in the above figures, independent of modulation scheme, the groups of filters within the OFDM channelizer yield widely varying performance. Essentially, the relative performance difference between any two filters is defined by the ratio of the number of subcarriers produced by the filters. Note that as shown in the above figures, all carriers generating equal number of subcarriers yield the same performance. The relative performance of each filter is given below in Table 4. The performance advantage of any given filter can be calculated by the following equation given that the per-subcarrier energy is constant independent of the number of subcarriers produced by the filter. (ssubcarriernumG_log1010= ) (6) Table 4. Performance Advantage of Filters Based on Number of Subcarriers Filter Number Number of Subcarriers Performance Advantage of Filter, G (dB) 1 1 0 2 1 0 3 2 3.01 4 2 3.01 5 2 3.01 6 4 6.02 7 4 6.02 8 8 9.03 9 8 9.03 10 16 12.04 Analysis of Various Constellation Schemes Utilizing Filter 1 as Reference This section presents the performance analysis of various modulation schemes using the same filter. This is the same data included in the previous graphs but grouped together specifically to measure the BER performance delta across the modulations schemes. The performance loss between modulation schemes is shown below in Table 5.

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45 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00024681012141618202224Eb/No (dB)Pr{e} bpsk ber qpsk ber 8psk ber 16qam ber 64qam ber 256qam ber 8qam ber 32qam ber 128qam ber Figure 47. Filter 1 OFDM Channelizer Multi-Modulation Scheme Performance Table 5. Performance Advantage of Modulation Schemes Modulation Scheme Eb/No Performance Advantage @ BER = 10e-2 (dB) SNR Performance Advantage @ BER = 10e-2 (dB) Eb/No Performance Advantage @ BER = 10e-5 (dB) SNR Performance Advantage @ BER = 10e-5 (dB) BPSK (ref) 0 0 0 0 QPSK 0 -3.01 0 -3.01 8-PSK -2.25 -7.02 -2.2 -6.97 8-QAM -3 -7.77 -3.3 -8.07 16-QAM -3.55 -9.57 -3.9 -9.92 32-QAM -6.5 -13.49 -7 -13.99 64-QAM -7.65 -15.43 -7.8 -15.58 128-QAM -10.85 -19.3 -11.2 -19.65 256-QAM -12.05 -21.08 -13 -22.03 BER Normalization of Filter Banks through Constellation Density Compensation It is desirable to take advantage of the process gain inherent across various filters within the OFDM channelizer by normalizing the BER across each filter in the 48-subcarrier OFDM channelizer. Given the measured BER across various modulation

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46 schemes in Table 5 as well as the performance advantage of each filter, G, in Table 4, an attempt is made to normalize the probability of bit error across each filter by leveraging the inherent process gain per filter to signal denser constellations while achieving similar probabilities of bit error. The selected modulation scheme per filter is given in Table 6 below along with the expected residual performance advantage of each filter with respect to the first filter. Note that the negative advantage denotes a loss in BER performance relative to the first filter. Table 6. Per-Filter Modulation Scheme for 48-Subcarrier OFDM Channelizer Filter Number Modulation Scheme Expected SNR Performance Advantage @ BER = 10e-2 (dB) Expected SNR Performance Advantage @ BER = 10e-5 (dB) 1 QPSK 0 0 2 QPSK 0 0 3 8-QAM -1.01 -0.96 4 8-QAM -1.01 -0.96 5 8-QAM -1.01 -0.96 6 16-QAM -0.56 -0.91 7 16-QAM -0.56 -0.91 8 32-QAM -1.48 -1.98 9 32-QAM -1.48 -1.98 10 64-QAM -0.42 -0.57 The measured BER performance of the above modulation scheme to filter mapping is summarized in Figure 48 below. It is observed that the measured BER performance correlates against the expected performance advantage based on the previous simulation results. The simulation results shown in Figure 48 demonstrate how the OFDM channelizer can be adapted to utilize apparent limitations in order to compensate for the inherent properties of the filters making up the OFDM channelizer. This residual performance

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47 advantage delta of less than 2 dB can further be reduced by making certain adjustments, such as puncturing, to the error control coding scheme per filter. Although this would not have any net effect on the uncoded BER as measured in this chapter, the post-correction BER would converge. Error control coding is not investigated in this thesis. 1.0E-091.0E-081.0E-071.0E-061.0E-051.0E-041.0E-031.0E-021.0E-011.0E+00024681012141618SNRPr{e} Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 Figure 48. BER of 48-Subcarrier OFDM Channelizer in AWGN with Mixed Modulation

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CHAPTER 5 OFDM CHANNELIZER PERFORMANCE AND LIMITATIONS IN MULTIPATH CHANNEL This chapter adapts an OFDM channelizer to allow for robust communications in a typical multipath channel common in wireless communications. Various wireless channel characteristics were presented in Chapter 2 along with inherent properties of typical FFT-based OFDM modulation schemes capable of mitigating the effects of multipath. This analysis will leverage a cyclic prefix extension in order to mitigate the effects of multipath channel propagation conditions. While considering the effect of multipath conditions on an OFDM channelizer, it is important to note that there is a fundamental challenge with the OFDM channelizer that must be solved. This problem is created by the coherent combining of multiple subcarriers inherent in an OFDM channelizer receiver. Multipath channels have frequency-varying phase and amplitude responses. This generally means that any two subcarriers will experience different phase and amplitude responses through the channel. Since an OFDM channelizer receiver coherently combines multiple subcarriers, some amount of destructive interference will be observed at the receiver. To mitigate this interference, an OFDM channelizer must provide some form of alignment across the subcarriers that are common to each filter. In this manner, the subcarriers will be coherently aligned in magnitude and / or phase prior to combining and the transmitted symbols can be recovered. With phase-shift keying (PSK) modulation schemes, it is sufficient for only the phases to be aligned between the subcarriers in order 48

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49 to prevent destructive combining through the OFDM channelizer receiver. In quadrature amplitude modulation (QAM) modulation schemes, both the amplitude and phase must be aligned prior to coherent subcarrier combining at the receiver. Various weighted combining schemes, such as maximal-ratio combining (MRC), can be used to achieve optimal signal-to-noise ratio enhancements. However, these combining schemes are outside of the scope of this thesis. An example of this phenomenon is shown in the constellation scatter diagram figures below. The below figures demonstrate the effects of multiple cyclic prefix lengths on filter 6 subcarriers using QPSK modulation. A multipath channel described later in this chapter was used to induce the frequency-varying phase and amplitude response into the transmitted signal. All of the scatter diagrams below were generated with an SNR of 30 dB. Figure 49 demonstrates the scattering resulting from the ISI in the multipath channel combined with the additive noise. Figure 50 and Figure 51 demonstrate similar scattering except that two alternative alignment schemes have been used to coherently combine the subcarriers associated with filter 6. The alignment schemes are further discussed later in this Chapter. Note that the ranges of the axes in Figure 49 is larger than the axes in the other two figures. This should be accounted for when comparing the amount of scatter among the different scenarios. The constellation diagrams above show a significant increase in effective SNR when using subcarrier alignment versus non-aligned combining.

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50 Figure 49. Multipath Constellation Scatter without Alignment Figure 50. Multipath Constellation Scatter with Option 1 Alignment

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51 Figure 51. Multipath Constellation Scatter with Option 2 Alignment OFDM Channelizer Subcarrier Separation Recall one advantage of FFT-based OFDM modulation is the potential to eliminate the need for equalization by using differential modulation schemes [8]. While this is true for OFDM channelizers as well, in multipath channel conditions, OFDM channelizers still must separately filter the individual subcarriers in order to differentially demodulate them individually prior to combining the subcarriers containing the same information. This phenomenon is the largest potential disadvantage of the OFDM channelizer versus a typical FFT-based OFDM system. Separation filters are required to accomplish this separation for subcarriers common to a single OFDM channelizer filter. The OFDM channelizer separation filter design parameters are unique per filter because the frequency separation between multiple subcarriers common to a filter varies per filter. The per-filter design parameters for the 48-point OFDM channelizer filters are

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52 shown in the figures below. Note that both filters 1 and 2 generate a single harmonic and therefore do not need any separation filtering and alignment to prevent destructive combining. The specified separation design parameters are superimposed along with the frequency response of the respective OFDM channelizer filter in the figures below. Each separation filter is shown in a different color. In general, one separation filter is needed per subcarrier that the OFDM channelizer filter produces. Note from the filter prototypes in the figures above that the separation filters only need to suppress energy from subcarriers belonging to the same OFDM channelizer filter. Rejection of subcarriers from other OFDM channelizer filters is provided by the OFDM channelizer filters themselves. This increases the available frequency span for a particular filters transition band to occupy. Also note that the filter prototypes shown in the figures above are not symmetric about DC. Therefore, complex-valued filter -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 3 Figure 52. Filter 3 Separation Filter Design Parameters

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53 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 4 Figure 53. Filter 4 Separation Filter Design Parameters -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 5 Figure 54. Filter 5 Separation Filter Design Parameters

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54 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 6 Figure 55. Filter 6 Separation Filter Design Parameters -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 7 Figure 56. Filter 7 Separation Filter Design Parameters

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55 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 8 Figure 57. Filter 8 Separation Filter Design Parameters -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 9 Figure 58. Filter 9 Separation Filter Design Parameters

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56 -30-20-10010203040-0.5-0.4-0.3-0.2-0.1-00.10.20.30.40.5Frequency (fs = 1 Hz)Magnitude (dB) Filter 10 Figure 59. Filter 10 Separation Filter Design Parameters coefficients are required to provide this frequency response. It should also be noted that the duration of the filters transition bands varies across the OFDM channelizers proportionally to the number of subcarriers produced by each OFDM channelizer. This tends to tighten the design constraints for the separation filter prototype as the number of subcarriers in the respective OFDM channelizer filter increases. In general, the separation filter prototypes for the OFDM channelizer filters generating a relatively low number of subcarriers can be realized using multiplier-less FIR filters. The drawback of this required frequency separation filtering is that high-rate multiplications may be necessary in order to realize separation filters that provide sufficient subcarrier rejection for OFDM channelizer filters producing a large number of subcarriers. The analysis that follows focuses on filter 6 as a nominal case to analyze the effects of multipath on the OFDM channelizer. This thesis does not focus on the optimal

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57 separation filter design techniques that may be available to reduce or eliminate the potential multiplications necessary to design the separation filters. Symmetric FIR filters are used as the basis for the separation filters in this thesis. The separation FIR filter coefficient sets designed for this analysis are shown in Table 7, Table 8, Table 9 and Table 10 below while the corresponding frequency responses are shown in Figure 60. The above filters are created by modifying a traditional rectangular windowed sinc bandpass filter [15]. To accomplish this, an FFT of the real-valued coefficients is taken. Next, either the positive or negative frequency Fourier coefficients are forced to zero. Finally, an inverse FFT produces the resulting complex-valued coefficients. An FFT larger than the number of filter taps is used for this purpose. The resulting complex-valued coefficients are truncated in time to yield a filter impulse response equal in length to the initial impulse response. Approaches to Enhance the OFDM Channelizer for Multipath Channel Conditions In order to determine the feasibility of providing sufficient channel separation, coherent phase and amplitude alignment between subcarriers, and combining of the appropriate subcarriers, two approaches will be considered. The first approach uses an Table 7. Filter 6 Separation Filter Subcarrier +6 Coefficient Listing Filter Tap Real Imaginary 1 -2.37335 1.39253 2 -3.56803 -1.49313 3 -1.91729 -4.48337 4 2.2473 -5.47087 5 5.84983 -2.3415 6 5.84983 2.3415 7 2.2473 5.47087 8 -1.91729 4.48337 9 -3.56803 1.49314 10 -2.37335 -1.39253

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58 Table 8. Filter 6 Separation Filter Subcarrier -6 Coefficient Listing Filter Tap Real Imaginary 1 -2.37428 -1.39253 2 -3.56895 1.49314 3 -1.91822 4.48337 4 2.24637 5.47087 5 5.84891 2.3415 6 5.84891 -2.3415 7 2.24637 -5.47087 8 -1.91822 -4.48337 9 -3.56895 -1.49314 10 -2.37428 1.39253 Table 9. Filter 6 Separation Filter Subcarrier +18 Coefficient Listing Filter Tap Real Imaginary 1 -1.10227 1.60279 2 -1.6561 -3.65139 3 5.17983 1.6766 4 -6.07262 2.83778 5 2.70982 -6.90155 6 2.70982 6.90155 7 -6.07262 -2.83778 8 5.17983 -1.6766 9 -1.6561 3.65139 10 -1.10227 -1.60279 Table 10. Filter 6 Separation Filter Subcarrier -18 Coefficient Listing Filter Tap Real Imaginary 1 -1.09863 -1.60279 2 -1.65246 3.65139 3 5.18347 -1.6766 4 -6.06898 -2.83778 5 2.71346 6.90155 6 2.71346 -6.90155 7 -6.06898 2.83778 8 5.18347 1.6766 9 -1.65246 -3.65139 10 -1.09863 1.60279

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59 -30-20-10010203040-0.5-0.4-0.3-0.3-0.2-0.1-00.070.150.230.310.390.47Frequency (fs = 1 Hz)Magnitude (dB) Positive 6 Negative 6 Positive 18 Negative 18 Filter 6 FreqencyResponse Figure 60. Filter 6 Separation Filter Frequency Responses open-loop equalization scheme directly at the receiver. This approach assumes ideal channel estimation and applies the compensation at the receiver after the subcarrier separation filtering and prior to the coherent combining of the subcarriers and making hard decisions of the received symbols. The second approach uses a closed-loop equalization scheme by informing the transmitter of the per-subcarrier channel response and having the transmitter pre-distort the subcarriers in order to allow for simple coherent combining in an unmodified OFDM channelizer at the receiver. The block diagram of option 1 is in Figure 61 below. This approach allows for equalization at the receiver independent of any support from the transmitter. The disadvantage of this approach is that the receiver must separate, channel estimate and combine real-time. This has the negative effect of increasing the complexity and limiting the maximum baseband operating frequency of the system, which is contrary to the

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60 primary benefit of an OFDM channelizer. Note that all subcarriers must be processed through a separate channelization filter. For the 48 subcarrier case, it increases the number of channelizer filters from ten to 48. The transmitter for this approach remains unchanged from the OFDM channelizer transmitter presented in Chapter 3. RF RX / LPF ADC Time / Frequency Synchronization Cyclic Extension Removal Serial-to-Parallel Supplemented Channelizer Filter Bank Parallel-to-Serial Channel Correction Symbol-to-Bit Demapping Time / Frequency Deinterleaver Coding:Forward Error Correction / CRC Carrier Combining Pre-Channelizer Separation Filters Figure 61. Block Diagram of OFDM Channelizer Receiver for Option 1 The block diagrams of option 2 are shown in Figure 62 and Figure 63 below. This approach allows for pre-compensation to be applied at the transmitter in order to coherently align the phase and magnitude of each subcarrier belonging to a single filter at the channel output. In this manner, the receive filters making up the OFDM channelizer can operate as usual and combine the received subcarriers as demonstrated in Chapter 4. Note that no additional channelizer filters are required at the receiver. This method requires the transmitter to have knowledge of the per-subcarrier channel characteristics in order to apply the pre-distortion to the transmitted signal and inform the transmitter of the per-subcarrier channel response. This can be accomplished by having the receiver compute a per-subcarrier estimation of the distortion introduced by the channel. This estimation can potentially be simplified by establishing a channel estimation procedure during which time the transmitter sequences through each subcarrier in a filter one at a time. In this manner the receive filters would not suffer the loss associated with destructive combining of multiple subcarriers while performing this

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61 Coding:Forward Error Correction / CRC Time / Frequency Interleaver Bit-to-Symbol Mapping Optional Pilot Insertion Serial-to-Parallel IFFT Parallel-to-Serial Cyclic Extension Insertion Windowing DAC LPF / RF TX Complex-Valued, Per-Subcarrier Weight Generation Figure 62. Block Diagram of OFDM Channelizer Transmitter for Option 2 RF RX / LPF ADC Time / Frequency Synchronization Cyclic Extension Removal Serial-to-Parallel Supplemented Channelizer Filter Bank Parallel-to-Serial Channel Correction Symbol-to-Bit Demapping Time / Frequency Deinterleaver Coding:Forward Error Correction / CRC Carrier Combining Pre-Channelizer Separation Filters Complex-Valued, Per Subcarrier Weights Figure 63. Block Diagram of OFDM Channelizer Receiver for Option 2 channel estimation. The transmitter and receiver would synchronously cycle through patterns of single subcarriers being transmitted in a particular filter. Each filter can perform the estimation sequence disjoint from the other filters. This method has the disadvantage of sacrificing system capacity to handle these channel estimation sequencing scenarios. A procedure to mitigate this channel estimation overhead is to allow the receiver to detect and signal a potential breakdown of performance on a per-filter basis. When a non-optimal condition is detected and signaled to the transmitter, the subcarriers for that filter would initiate a channel estimation transmission sequence for the subcarriers

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62 contained within that filter. There are various approaches that could be chosen. The most efficient method would strongly depend on the channel characteristics. The primary application for this second approach is for a one-to-many type network where all communication in the network goes through a single access point. Examples of this sort of network topology exist in various wireless protocols including 802.11a/b/g as well as mobile wireless standards. The requirement for this second option is that the access point would be capable of applying a per-subcarrier pre-distortion to each sub-carrier individually. This is a viable assumption since many on-to-many networks allow for relatively more complex and costly access points to be able to handle specific network management tasks. Therefore, it is reasonable to assume that the access point in this network could have an FFT-based OFDM transmission scheme while the numerous end-points could have lower-cost OFDM channelizers. In this manner, the transmitter could handle the pre-distortion on a per-subcarrier basis and allow the receiver to have a much simpler and therefore cheaper OFDM channelizer based receiver. The network would benefit by having the large majority of its nodes have relatively lower cost. Multipath Effects on BER Performance As described above, multipath in a channel has the potential to distort multiple subcarriers associated with the same filter in a manner such that the combining in the OFDM channelizer receiver combines the subcarriers in a non-optimal fashion. This section will demonstrate this phenomenon through figures captured in simulation. A Rappaport channel is used for this demonstration [9]. The Non-Line-of-Sight model is given in Equation (7) below.

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63 () nBddLdL0 (7) In order to simplify the analysis, only filter 6 is included in the analysis. Recall that filter 6 generates frequency bins 6 and 18. The characteristic parameters for the Rappaport channel model used for this analysis are given below. n = Path loss exponent; typical range of n is 3.5 n 5 d = Distance (separation) between transmit and receive antennas d 0 = Reference distance or free space propagation corner distance L B = Propagation loss of the LOS path for d 0 [m] L = Loss (propagation loss) of the combined NLOS and LOS signal path The Rappaport channel is based on empirically gathered field data and the model is statistical in nature with randomly generated multipath weights. The actual path weights used for this analysis are given in Table 11 below. The frequency response of the Rappaport channel described above is shown in Figure 64 below. Table 11. Rappaport Multipath Channel Tap Weights Multipath Channel Tap Real Imaginary 1 0 0 2 -0.48967 0.39845 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0.02935 -0.35591 10 -0.15027 0.35342 11 -0.19401 0.53741

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64 -30-20-1001020304050-0.5-0.4-0.3-0.3-0.2-0.1-00.080.170.250.330.420.5Frequency (fs = 1 Hz)Magnitude (dB) Multipath ChannelResponse Filter 6 FreqencyResponse Figure 64. Rappaport Multipath Channel Frequency Response The performance simulation results of the OFDM channelizer filter 6 are shown in Figure 65 below. The simulations do not contain any alignment of the subcarriers prior to subcarrier combining. They are intended as a reference to measure gains of the two subcarrier alignment options. It should be noted that there is an inherent noise floor introduced by the multipath channel. This noise floor prevents any modulation scheme with order greater than 16 from achieving BERs lower than approximately 0.1. Additionally, all modulation schemes show significant degradation when compared to the AWGN performance simulations explored in Chapter 4. The noise floor introduced by the multipath channel has two main components. First, the destructive subcarrier combining reduces the effective per-filter signal level at the receiver. Second, the ISI introduced by the multipath channel causes a scattering of

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65 the received constellation absent any actual noise contribution. The two primary components of the degradation shown above will be mitigated in the following analysis. Multipath Effects on OFDM Channelizer with Coherent Alignment Simulation results are shown below to demonstrate the performance gains achievable through coherent alignment and combining of an OFDM channelizer signal as described previously. This analysis includes both options 1 and 2 presented previously. This analysis again utilizes filter 6 to perform the performance analysis. This analysis assumes ideal channel estimation in the coherent alignment of the multiple subcarriers belonging to filter 6. Figure 66 illustrates the performance achieved through option 1 while Figure 67 illustrates the performance achieved through option 2. One can observe that the performance is significantly degraded for both option 1 and option 2 above when compared to the AWGN simulations presented in Chapter 4. Additionally, there is an inherent noise floor visible in the figures above similar to the noise floor in the performance results without coherent combining compensation, however not as pronounced. Multipath Effects on OFDM Channelizer with Cyclic Prefix As previously described, FFT-based OFDM modulation schemes benefit from the presence of a guard interval typically based on a cyclic prefix extension. The simulation results provided below demonstrate a similar performance gain for an OFDM channelizer. Simulation results with various cyclic prefix lengths are shown in Figure 68, Figure 69 and Figure 70 below. One can observe that the performance of an OFDM channelizer benefits from a cyclic prefix extension in multipath channel conditions. It should be noted that, in general, that while the noise floor of the higher order constellations is removed as the

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66 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-7-4-1258111417202326293235SNRPr{e} BPSK QPSK 8-QAM 8-PSK 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 65. OFDM Channelizer Filter 6 BER Results 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-7-4-1258111417202326293235SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 66. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1

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67 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-7-4-1258111417202326293235SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 67. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 1.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-7-4-1258111417202326293235SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 68. BER for Filter 6 OFDM Channelizer in Multipath Channel and 4 Sample Cyclic Prefix

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68 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-7-4-1258111417202326293235SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 69. BER for Filter 6 OFDM Channelizer in Multipath Channel and 7 Sample Cyclic Prefix 1.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-7-4-1258111417202326293235SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 70. BER for Filter 6 OFDM Channelizer in Multipath Channel and 10 Sample Cyclic Prefix

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69 cyclic prefix extension approaches the length of the memory in the channel, the performance of the lower order modulation schemes is relatively unchanged with the increase in length of the cyclic prefix extension. Multipath Effects on OFDM Channelizer with Coherent Alignment and Cyclic Prefix The final simulation analysis combines the two performance enhancements described above, namely coherent alignment of multiple subcarriers as well as the use of a cyclic prefix extension. Both coherent alignment options 1 and 2 are explored in the simulation results below. Two notable trends can be observed in the figures above when comparing option 1 and option 2. First, the separation filters necessary for option 1 introduce additional multipath delay spread. This increases the required cyclic prefix length necessary to mitigate the effects of multipath. Second, the performance with option 2 exceeds the performance of option 1. Summary of Performance Comparison A performance comparison is shown in the tables below. These tables compare performance against the various simulation scenarios presented above. The AWGN performance is used as a reference and the values in the tables are the losses, in dB, of each scenario relative to the AWGN simulation with the same effective SNR. As a general trend, the performance of all scenarios increase (e.g. the loss relative to AWGN decreases) as the cyclic prefix is extended to a point equal to the ISI present in the system. It can be seen that the performance of the system without subcarrier alignment never gets smaller than 10 dB degradation relative to AWGN performance even with a cyclic prefix extension. By comparison, the performance of option 1 gets

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70 within 2 dB of AWGN performance while option 2 gets within 1 dB of AWGN performance. This demonstrates the significant gains achievable through coherent alignment in a system based on an OFDM channelizer scheme. It should be noted that in order to achieve optimal performance with option 1 a cyclic prefix approximately 50% longer is necessary. This additional cyclic prefix extension directly reduces the available system capacity since any time allocated to a cyclic prefix extension is not available for information transmission. In this example with a 48-subcarrier system, the data rate reduction due to a 10 sample cyclic prefix is 82.8%, while the data rate reduction due to a 16 sample cyclic prefix is 75%. 1.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-42814202632SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 71. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 4 Sample Cyclic Prefix

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71 1.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-42814202632SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 72. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 7 Sample Cyclic Prefix 1.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-42814202632SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 73. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 10 Sample Cyclic Prefix

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72 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-42814202632SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 74. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 4 Sample Cyclic Prefix 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-42814202632SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 75. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 7 Sample Cyclic Prefix

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73 1.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00-10-42814202632SNRPr{e} BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM Figure 76. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 10 Sample Cyclic Prefix Table 12. Multipath Fading Performance Delta (@ BER = 1e-2) without Alignment (AWGN reference) Constellation Type No Alignment / No Cyclic Prefix No Alignment / 4 Sample Cyclic Prefix No Alignment / 7 Sample Cyclic Prefix No Alignment / 10 Sample Cyclic Prefix BPSK 12.8 11.2 11.1 10.2 QPSK 12.8 11.2 11 10.2 8-PSK 15 11.5 11 10.2 8-QAM 14.6 11.4 11 10.1 16-QAM 17 11.7 10.9 10.1 32-QAM NA 17.2 11 10.2 64-QAM NA NA 10.8 10.2 128-QAM NA NA 11 10.2 256-QAM NA NA 11 10.2

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74 Table 13. Multipath Fading Performance Delta (@ BER = 1e-2) with Option 1 (AWGN reference) Constellation Type Option 2 Alignment / No Cyclic Prefix Option 2 Alignment / 4 Sample Cyclic Prefix Option 2 Alignment / 10 Sample Cyclic Prefix Option 2 Alignment / 16 Sample Cyclic Prefix BPSK 5.6 3.8 2.9 2.6 QPSK 5.8 3.8 2.9 2.8 8-PSK 8.9 5 3 2.6 8-QAM 10.1 4.7 3 2.7 16-QAM 22.5 5.9 2.9 2.5 32-QAM NA NA 3 2.6 64-QAM NA NA 3.2 1.6 128-QAM NA NA 3.8 2.7 256-QAM NA NA 4.3 2.7 Table 14. Multipath Fading Performance Delta (@ BER = 1e-2) with Option 2 (AWGN reference) Constellation Type Option 1 Alignment / No Cyclic Prefix Option 1 Alignment / 4 Sample Cyclic Prefix Option 1 Alignment / 7 Sample Cyclic Prefix Option 1 Alignment / 10 Sample Cyclic Prefix BPSK 2.3 1.7 1.4 0.8 QPSK 2.5 1.6 1.1 0.8 8-PSK 3.8 2.1 1.4 0.9 8-QAM 5 2.2 1.2 0.8 16-QAM 8 2.8 1.3 0.7 32-QAM NA 8 1.8 0.7 64-QAM NA NA 2.2 0.8 128-QAM NA NA 4.4 0.8 256-QAM NA NA 7.6 0.8

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75 Table 15. Multipath Fading Performance Delta (@ BER = 1e-5) without Alignment (AWGN reference) Constellation Type No Alignment / No Cyclic Prefix No Alignment / 4 Sample Cyclic Prefix No Alignment / 7 Sample Cyclic Prefix No Alignment / 10 Sample Cyclic Prefix BPSK 13.5 11.6 11 10.3 QPSK 13.5 11.6 11 10.3 8-PSK 17.4 13 11.5 10.8 8-QAM 17.4 12.4 11.1 10.3 16-QAM 19.3 13.1 11 10.3 32-QAM NA 18.3 11 10.3 64-QAM NA 19.1 11 10.3 128-QAM NA NA 11 10.3 256-QAM NA NA 10.9 10.2 Table 16. Multipath Fading Performance Delta (@ BER = 1e-5) with Option 1 (AWGN reference) BPSK 6.4 4.3 3 2.6 QPSK 7 4.6 3.1 2.6 8-PSK 11.8 6.9 3.6 3 8-QAM 13.8 6 3 2.6 16-QAM NA 9.2 3.2 2.7 32-QAM NA NA 3.6 2.7 64-QAM NA NA 4 2.7 128-QAM NA NA 5.8 2.6 256-QAM NA NA 6.3 2.6 BPSK 6.4 4.3 3 2.6

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76 Table 17. Multipath Fading Performance Delta (@ BER = 1e-5) with Option 2 (AWGN reference) Constellation Type Option 1 Alignment / No Cyclic Prefix Option 1 Alignment / 4 Sample Cyclic Prefix Option 1 Alignment / 7 Sample Cyclic Prefix Option 1 Alignment / 10 Sample Cyclic Prefix BPSK 3 1.9 1.1 0.8 QPSK 3.4 2.2 1.4 1 8-PSK 6.7 4.1 2.1 1.2 8-QAM 7.8 3.7 1.5 0.8 16-QAM 18.2 6.5 1.9 0.9 32-QAM NA NA 3.3 0.9 64-QAM NA NA 4.7 0.9 128-QAM NA NA NA 0.9 256-QAM NA NA NA 0.9

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CHAPTER 6 CONCLUSIONS AND FUTURE WORK This thesis explores the usefulness of an OFDM channelizer in both AWGN and multipath channel conditions in order to achieve the advantages of an FFT-based OFDM modulation scheme while reducing the overall design complexities. The desired goal is to leverage the reduced design complexity to realize a system capable of increased baseband operating frequencies simultaneous with decreased cost. CIC filter-based OFDM channelizers are interesting from an implementation complexity and operating clock frequency perspective. They can potentially provide a low-complexity alternative to implementing an FFT-based OFDM system that has the potential to operate at very high clock rates due to the multiplier-less structure from which they are derived. Summary of Simulation Effort The simulation effort consisted of multiple phases. The first phase involved creating a functional model of the OFDM channelizer and simulation system capable of analyzing the performance of the OFDM channelizer. The second phase involved calibrating and capturing the performance of the OFDM channelizer in an AWGN channel. The third phase consisted of measuring the potential performance loss of an OFDM channelizer in a multipath channel. The fourth phase included development of two different methods for overcoming the potential destructive combining of multiple subcarriers belonging to a common OFDM channelizer filter. The final phase included a performance comparison of the two methods assuming various cyclic prefix lengths. 77

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78 Lessons Learned and Future Work A number of lessons were learned during the analysis contributing to this thesis. The results of these lessons were presented in previous chapters. A few notable lessons learned are noted below. Typical E b /N 0 BER curves presented in textbooks and other reference material measure complex-valued noise power even if the signaling is real-valued BPSK. Therefore the noise power relative to the signal power must be calibrated assuming complex valued noise. Without taking this into account, this can introduce a 3 dB error in the system calibration. Certain wireless channel models (i.e. Rummler [9]) are represented with mathematical models that result in symmetric frequency responses. Given the symmetric frequency response of all OFDM channelizer filters, this likely leads to channel phase responses that cancel each other at the receiver. In reality, channel responses are not symmetric and the analysis could falsely conclude that destructive interference at the receiver is not a limitation. Given the impulse responses of each OFDM channelizer filter, each filter can have an optimal sampling point. These ideal sampling points were found to be located within one or two samples of the end of the received symbols. Furthermore, given the ideal sampling point of each filter, a negation of the received complex-valued samples is necessary at the receiver for correct interpretation of the received bit stream. Many interesting aspects of OFDM channelizers were explored in this thesis. Still many more aspects can be considered. The list below describes further analysis that should be investigated in order to further refine the advantages of an OFDM channelizer beyond those of a typical FFT-based OFDM system. Investigate multiplier-less filters for OFDM channelizer subcarrier separation in multipath channels. Investigate fixed-point requirements for an OFDM channelizer and how this fixed-point design might compare to an FFT-based OFDM channelizer. Implement and synthesize an OFDM channelizer targeting current state-of-the-art technology to determine actual realizable operating speed. The realizable operating speed of an OFDM channelizer should be contrasted against an FFT-based OFDM channelizer.

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79 Investigate utilizing variable-rate error correction coding in order to achieve more uniform parity of BER across various filter banks with different process gains. Investigate benefit of maximal ratio combining. Theoretically, maximal ratio combining provides optimal reception when combining diversity branches through a channel providing uncorrelated diversity through the branches. Investigate performance of OFDM channelizer in a fading channel similar to mobile wireless channel models. Consideration should be taken into account for the channel coherence time relative to the time necessary to estimate the channel conditions. Summary of Simulation Performance Results It has been shown that the performance of an OFDM channelizer meets the expected performance for a more general matched filter receiver in an AWGN channel. It has also been shown that the performance of an OFDM channelizer in a multipath channel can approach the performance of a matched filter in an AWGN channel. This can be accomplished through the use of coherent subcarrier combining either provided either by pre-compensation the transmitter or by separation filters at the receiver as well as through the use of a cyclic prefix extension scheme. Given system design constraints, the transmitter-based pre-compensation scheme has a limitation in a fading channel where the channel estimate changes over time. This is due to the time necessary to both make the estimate and deliver the estimate to the transmitter before it can start applying the pre-distortion to its transmission. During this elapsed time, the channel conditions will change and might not be well correlated to the channel conditions from when the channel estimate was measured. The receiver-based separation filter scheme has a limitation of requiring additional filters to perform subcarrier separation in order to compute per-subcarrier channel estimations. This must be done prior to coherent subcarrier combining. These separation filters potentially introduce high-rate multipliers into the system architecture. Design

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80 consideration needs to be made to ensure the order of the OFDM channelizer is kept sufficiently small so as not to require extremely steep skirt filter design parameters for the separation filters. Finally, the separation filters can increase the delay spread observed at the receiver and effectively increase the required cyclic prefix length. This longer cyclic prefix reduces the effective bit rate achievable across the channel. All other design constraints being equal, option 1 subcarrier alignment is shown to yield superior performance compared to option 2. The two primary factors contributing to this are the 1-2 dB performance advantage that option 1 gives over option 2 as well as the additional cyclic prefix extension necessary with option 2 to achieve optimal performance. This additional cyclic prefix is necessary to eliminate the additional ISI introduced by the separation filters themselves.

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APPENDIX A POLYMORPHIC-BASED SPW OVERVIEW As mentioned in the introduction, SPW is a hierarchal, block-based modeling and simulation environment useful for performing system analysis. SPW is capable of executing both small and large-scale system simulations. The polymorphic technology within SPW extends the flexibility and capability of the tool by allowing a single representation of a system capture both the floating-point and fixed-point design. The SPW tool and accompanying polymorphic feature set are powerful but also complex. A brief introduction to introduce the reader to this tool is provided in this appendix. Polymorphic models available within SPW can be configured in a wide-variety of block types (54 total). A block type consists of two sub-types: element type and composite type. There are six element types and nine composite types that combine to produce the 54 total block types (6 element types 9 composite types = 54 block types). The element type defines the type of each element within the signal operated on by the block. Examples of element types are: Double, ComplexDouble, Fixed-Point and ComplexFixed-Point. The composite type defines the composite structure type on which the block operates. Examples of composite types are: Scalar (none), Vector and Matrix. SPW polymorphic block types also support various video signal formats that will not be utilized in this thesis. Figure 77 shows the presentation of the block type information of a polymorphic block within SPW. 81

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82 Figure 77. Polymorphic Block Type Illustration In addition to the block type information, it can be seen in Figure 77 that additional configuration information is listed on the symbol of a block. This information is referred to as the default value information of the block. In Figure 78, two of the fields composing a default value field are listed. The 0.5 is the initial value (constant value in this example) that the block produces at its output. The <8,0,t> are the fixed-point attributes that are defined for the block if the blocks type is set for a fixed-point type. In order to further define these fixed-point attributes: 8 is the total number of bits (including optional sign bit), 0 is the bit position of the most significant bit (MSB), not including any sign bit, and t denotes twos complement signal representation as opposed to u for unsigned signal representation. This parameter in the default value field is ignored if the blocks type is not set to a fixed-point type. Two other optional parameters in the default value field are the composite type size (i.e. vector size) and the fixed-point modes of operation. These two parameters are not shown in Figure 78. The composite type size, when present, is enclosed within square brackets []. The fixed-point modes-of-operation parameter is composed of two parameters: loss-of-precision mode and overflow mode. These two parameters are specified within

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83 parentheses and separated by a comma. The first parameter is the loss-of-precision mode and a couple of examples are truncation and round. The second parameter is the overflow mode and the two possible settings are clip and wrap. Figure 78. Polymorphic Default Value Illustration

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APPENDIX B SIMULATION RESULTS RAW DATA Table 18. Raw Data for Figure 38 SNR (dB) Filter 1 Filter 2 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.60E-1 4.60E-1 4.44E-1 4.44E-1 4.21E-1 4.21E-1 3.88E-1 3.89E-1 3.44E-1 -22 4.55E-1 4.55E-1 4.37E-1 4.37E-1 4.11E-1 4.11E-1 3.75E-1 3.75E-1 3.27E-1 -21 4.50E-1 4.50E-1 4.29E-1 4.29E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.07E-1 -20 4.44E-1 4.44E-1 Filter 3 4.44E-1 4.37E-1 4.29E-1 4.21E-1 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1 -19 4.37E-1 4.37E-1 4.11E-1 4.11E-1 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1 -18 4.29E-1 4.29E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.07E-1 3.07E-1 2.38E-1 -17 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1 2.86E-1 2.12E-1 -16 4.11E-1 4.11E-1 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1 2.63E-1 1.85E-1 -15 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.08E-1 3.08E-1 2.38E-1 2.39E-1 1.57E-1 -14 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1 2.86E-1 2.12E-1 2.12E-1 1.30E-1 -13 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1 2.63E-1 1.85E-1 1.85E-1 1.03E-1 -12 3.61E-1 3.61E-1 3.08E-1 3.08E-1 2.39E-1 2.39E-1 1.57E-1 1.58E-1 7.77E-2 -11 3.45E-1 3.45E-1 2.86E-1 2.87E-1 4.11E-1 4.01E-1 3.89E-1 3.76E-1 3.61E-1 3.45E-1 3.27E-1 3.08E-1 2.86E-1 2.13E-1 2.13E-1 1.30E-1 1.30E-1 5.54E-2 -10 3.27E-1 3.27E-1 2.64E-1 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.03E-1 1.03E-1 3.68E-2 -9 3.08E-1 3.08E-1 2.39E-1 2.39E-1 2.39E-1 1.58E-1 1.58E-1 7.79E-2 7.79E-2 2.24E-2 -8 2.87E-1 2.87E-1 2.13E-1 2.13E-1 2.13E-1 1.30E-1 1.30E-1 5.56E-2 5.56E-2 1.22E-2 -7 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.86E-1 1.03E-1 1.03E-1 3.70E-2 3.70E-2 5.76E-3 -6 2.39E-1 2.39E-1 1.58E-1 1.58E-1 1.58E-1 7.82E-2 7.82E-2 2.25E-2 2.25E-2 2.29E-3 -5 2.13E-1 2.13E-1 1.30E-1 1.30E-1 1.30E-1 5.59E-2 5.59E-2 1.22E-2 1.22E-2 7.33E-4 -4 1.86E-1 1.86E-1 1.03E-1 1.03E-1 1.04E-1 3.72E-2 3.72E-2 5.79E-3 5.80E-3 1.80E-4 -3 1.58E-1 1.58E-1 7.84E-2 7.84E-2 7.84E-2 2.26E-2 2.26E-2 2.31E-3 2.31E-3 3.08E-5 -2 1.31E-1 1.31E-1 5.61E-2 5.61E-2 5.61E-2 1.23E-2 1.23E-2 7.46E-4 7.42E-4 3.20E-6 -1 1.04E-1 1.04E-1 3.74E-2 3.74E-2 3.73E-2 5.85E-3 5.86E-3 1.83E-4 1.82E-4 1.10E-7 0 7.86E-2 7.86E-2 2.28E-2 2.28E-2 2.28E-2 2.33E-3 2.34E-3 3.34E-5 3.19E-5 2.00E-8 1 5.63E-2 5.62E-2 1.24E-2 1.24E-2 1.24E-2 7.53E-4 7.52E-4 3.68E-6 3.62E-6 2 3.75E-2 3.75E-2 5.90E-3 5.91E-3 5.90E-3 1.82E-4 1.85E-4 2.40E-7 2.80E-7 4 1.25E-2 1.25E-2 7.62E-4 7.62E-4 7.63E-4 3.00E-6 4.05E-6 5 5.94E-3 5.96E-3 1.89E-4 1.88E-4 1.89E-4 1.90E-7 1.80E-7 6 2.38E-3 2.39E-3 3.28E-5 3.30E-5 3.28E-5 1.00E-8 1.00E-8 7 7.67E-4 7.74E-4 3.85E-6 3.90E-6 3.62E-6 8 1.88E-4 1.91E-4 2.00E-7 3.30E-7 2.70E-7 9 3.28E-5 3.34E-5 2.00E-8 1.00E-8 10 3.48E-6 4.01E-6 11 2.20E-7 3.30E-7 12 2.00E-8 1.00E-8 84

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85 Table 19. Raw Data for Figure 39 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.72E-1 4.72E-1 4.60E-1 4.60E-1 4.60E-1 4.44E-1 4.44E-1 4.21E-1 4.21E-1 3.89E-1 -22 4.68E-1 4.68E-1 4.55E-1 4.55E-1 4.55E-1 4.37E-1 4.37E-1 4.11E-1 4.11E-1 3.75E-1 -21 4.64E-1 4.65E-1 4.50E-1 4.50E-1 4.50E-1 4.29E-1 4.29E-1 4.00E-1 4.00E-1 3.61E-1 -20 4.60E-1 4.60E-1 4.44E-1 4.44E-1 4.44E-1 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.45E-1 -19 4.55E-1 4.55E-1 4.37E-1 4.37E-1 4.37E-1 4.11E-1 4.11E-1 3.75E-1 3.75E-1 3.27E-1 -18 4.50E-1 4.50E-1 4.29E-1 4.29E-1 4.29E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.07E-1 -17 4.44E-1 4.44E-1 4.21E-1 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1 -16 4.37E-1 4.37E-1 4.11E-1 4.11E-1 4.11E-1 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1 -15 4.29E-1 4.29E-1 4.01E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.07E-1 3.07E-1 2.38E-1 -14 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1 2.86E-1 2.12E-1 -13 4.11E-1 4.11E-1 3.76E-1 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1 2.63E-1 1.85E-1 -12 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.61E-1 3.08E-1 3.08E-1 2.39E-1 2.39E-1 1.58E-1 -11 3.89E-1 3.89E-1 3.45E-1 3.45E-1 3.45E-1 2.87E-1 2.87E-1 2.13E-1 2.13E-1 1.30E-1 -10 3.76E-1 3.76E-1 3.27E-1 3.27E-1 3.27E-1 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.03E-1 -9 3.61E-1 3.61E-1 3.08E-1 3.08E-1 3.08E-1 2.39E-1 2.39E-1 1.58E-1 1.58E-1 7.79E-2 -8 3.45E-1 3.45E-1 2.87E-1 2.87E-1 2.87E-1 2.13E-1 2.13E-1 1.30E-1 1.30E-1 5.57E-2 -7 3.28E-1 3.28E-1 2.64E-1 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.03E-1 1.03E-1 3.70E-2 -6 3.08E-1 3.08E-1 2.39E-1 2.39E-1 2.39E-1 1.58E-1 1.58E-1 7.81E-2 7.81E-2 2.25E-2 -5 2.87E-1 2.87E-1 2.13E-1 2.13E-1 2.13E-1 1.30E-1 1.30E-1 5.58E-2 5.58E-2 1.23E-2 -4 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.86E-1 1.03E-1 1.04E-1 3.71E-2 3.71E-2 5.81E-3 -3 2.39E-1 2.40E-1 1.58E-1 1.58E-1 1.58E-1 7.84E-2 7.85E-2 2.26E-2 2.26E-2 2.32E-3 -2 2.13E-1 2.14E-1 1.31E-1 1.31E-1 1.31E-1 5.61E-2 5.61E-2 1.23E-2 1.23E-2 7.40E-4 -1 1.86E-1 1.86E-1 1.04E-1 1.04E-1 1.04E-1 3.73E-2 3.74E-2 5.85E-3 5.85E-3 1.80E-4 0 1.59E-1 1.59E-1 7.87E-2 7.87E-2 7.86E-2 2.27E-2 2.28E-2 2.34E-3 2.34E-3 3.10E-5 1 1.31E-1 1.31E-1 5.63E-2 5.63E-2 5.63E-2 1.24E-2 1.24E-2 7.51E-4 7.51E-4 3.64E-6 2 1.04E-1 1.04E-1 3.75E-2 3.75E-2 3.75E-2 5.90E-3 5.91E-3 1.85E-4 1.84E-4 2.20E-7 3 7.89E-2 7.89E-2 2.29E-2 2.29E-2 2.29E-2 2.36E-3 2.37E-3 3.28E-5 3.24E-5 1.50E-8 4 5.65E-2 5.65E-2 1.25E-2 1.25E-2 1.25E-2 7.63E-4 7.62E-4 3.52E-6 3.77E-6 5 3.77E-2 3.77E-2 5.96E-3 5.95E-3 5.95E-3 1.86E-4 1.88E-4 2.35E-7 2.90E-7 6 2.30E-2 2.30E-2 2.39E-3 2.38E-3 2.39E-3 3.31E-5 3.31E-5 1.00E-8 1.50E-8 7 1.26E-2 1.26E-2 7.73E-4 7.69E-4 7.74E-4 3.70E-6 3.91E-6 8 6.00E-3 6.01E-3 1.92E-4 1.91E-4 1.92E-4 2.70E-7 2.70E-7 9 2.41E-3 2.41E-3 3.39E-5 3.36E-5 3.33E-5 5.00E-9 2.00E-8 10 7.79E-4 7.83E-4 3.95E-6 3.85E-6 4.06E-6 11 1.93E-4 1.94E-4 2.15E-7 3.15E-7 2.55E-7 12 3.41E-5 3.41E-5 1.00E-8 1.50E-8 13 3.79E-6 4.11E-6 14 2.75E-7 2.55E-7 15 1.50E-8 5.00E-9

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86 Table 20. Raw Data for Figure 40 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.82E-1 4.82E-1 4.75E-1 4.75E-1 4.75E-1 4.64E-1 4.64E-1 4.48E-1 4.48E-1 4.26E-1 -22 4.80E-1 4.80E-1 4.72E-1 4.72E-1 4.72E-1 4.59E-1 4.59E-1 4.42E-1 4.42E-1 4.16E-1 -21 4.78E-1 4.78E-1 4.68E-1 4.68E-1 4.68E-1 4.54E-1 4.54E-1 4.34E-1 4.34E-1 4.06E-1 -20 4.75E-1 4.75E-1 4.64E-1 4.64E-1 4.64E-1 4.48E-1 4.48E-1 4.26E-1 4.26E-1 3.94E-1 -19 4.72E-1 4.72E-1 4.59E-1 4.59E-1 4.59E-1 4.42E-1 4.42E-1 4.16E-1 4.16E-1 3.80E-1 -18 4.68E-1 4.68E-1 4.54E-1 4.54E-1 4.54E-1 4.34E-1 4.34E-1 4.06E-1 4.06E-1 3.65E-1 -17 4.64E-1 4.64E-1 4.48E-1 4.48E-1 4.48E-1 4.26E-1 4.26E-1 3.94E-1 3.94E-1 3.48E-1 -16 4.59E-1 4.59E-1 4.42E-1 4.42E-1 4.42E-1 4.17E-1 4.16E-1 3.80E-1 3.80E-1 3.30E-1 -15 4.54E-1 4.54E-1 4.34E-1 4.34E-1 4.34E-1 4.06E-1 4.06E-1 3.65E-1 3.65E-1 3.10E-1 -14 4.48E-1 4.48E-1 4.26E-1 4.26E-1 4.26E-1 3.94E-1 3.94E-1 3.49E-1 3.49E-1 2.88E-1 -13 4.42E-1 4.42E-1 4.17E-1 4.17E-1 4.17E-1 3.80E-1 3.80E-1 3.30E-1 3.30E-1 2.65E-1 -12 4.34E-1 4.34E-1 4.06E-1 4.06E-1 4.06E-1 3.65E-1 3.65E-1 3.10E-1 3.10E-1 2.40E-1 -11 4.26E-1 4.26E-1 3.94E-1 3.94E-1 3.94E-1 3.49E-1 3.49E-1 2.88E-1 2.88E-1 2.15E-1 -10 4.17E-1 4.17E-1 3.81E-1 3.81E-1 3.81E-1 3.30E-1 3.30E-1 2.65E-1 2.65E-1 1.89E-1 -9 4.06E-1 4.06E-1 3.66E-1 3.66E-1 3.66E-1 3.10E-1 3.10E-1 2.40E-1 2.40E-1 1.64E-1 -8 3.94E-1 3.94E-1 3.49E-1 3.49E-1 3.49E-1 2.88E-1 2.89E-1 2.15E-1 2.15E-1 1.40E-1 -7 3.81E-1 3.81E-1 3.31E-1 3.31E-1 3.31E-1 2.65E-1 2.65E-1 1.89E-1 1.89E-1 1.17E-1 -6 3.66E-1 3.66E-1 3.10E-1 3.10E-1 3.10E-1 2.41E-1 2.41E-1 1.64E-1 1.64E-1 9.52E-2 -5 3.49E-1 3.49E-1 2.89E-1 2.89E-1 2.89E-1 2.15E-1 2.15E-1 1.40E-1 1.40E-1 7.55E-2 -4 3.31E-1 3.31E-1 2.65E-1 2.65E-1 2.65E-1 1.90E-1 1.90E-1 1.17E-1 1.17E-1 5.76E-2 -3 3.11E-1 3.11E-1 2.41E-1 2.41E-1 2.41E-1 1.65E-1 1.65E-1 9.54E-2 9.54E-2 4.19E-2 -2 2.89E-1 2.89E-1 2.16E-1 2.16E-1 2.16E-1 1.40E-1 1.40E-1 7.57E-2 7.57E-2 2.85E-2 -1 2.66E-1 2.66E-1 1.90E-1 1.90E-1 1.90E-1 1.17E-1 1.17E-1 5.78E-2 5.78E-2 1.79E-2 0 2.41E-1 2.41E-1 1.65E-1 1.65E-1 1.65E-1 9.56E-2 9.56E-2 4.20E-2 4.20E-2 1.01E-2 1 2.16E-1 2.16E-1 1.40E-1 1.40E-1 1.40E-1 7.59E-2 7.59E-2 2.86E-2 2.86E-2 5.05E-3 2 1.90E-1 1.90E-1 1.17E-1 1.17E-1 1.17E-1 5.80E-2 5.80E-2 1.80E-2 1.80E-2 2.14E-3 3 1.65E-1 1.65E-1 9.58E-2 9.58E-2 9.58E-2 4.22E-2 4.22E-2 1.02E-2 1.02E-2 7.41E-4 4 1.41E-1 1.41E-1 7.60E-2 7.60E-2 7.60E-2 2.88E-2 2.88E-2 5.09E-3 5.08E-3 1.99E-4 5 1.18E-1 1.18E-1 5.81E-2 5.81E-2 5.81E-2 1.81E-2 1.81E-2 2.16E-3 2.16E-3 3.89E-5 6 9.60E-2 9.60E-2 4.23E-2 4.23E-2 4.23E-2 1.03E-2 1.03E-2 7.50E-4 7.54E-4 5.15E-6 7 7.62E-2 7.62E-2 2.89E-2 2.89E-2 2.89E-2 5.13E-3 5.14E-3 2.03E-4 2.03E-4 4.60E-7 8 5.83E-2 5.83E-2 1.82E-2 1.82E-2 1.82E-2 2.19E-3 2.19E-3 4.05E-5 4.01E-5 3.33E-8 9 4.25E-2 4.25E-2 1.03E-2 1.03E-2 1.03E-2 7.63E-4 7.63E-4 5.28E-6 5.39E-6 10 2.90E-2 2.90E-2 5.17E-3 5.17E-3 5.17E-3 2.06E-4 2.07E-4 4.33E-7 4.60E-7 11 1.83E-2 1.83E-2 2.20E-3 2.20E-3 2.20E-3 4.06E-5 4.13E-5 2.33E-8 3.00E-8 12 1.04E-2 1.04E-2 7.70E-4 7.69E-4 7.70E-4 5.24E-6 5.47E-6 13 5.20E-3 5.21E-3 2.11E-4 2.09E-4 2.09E-4 4.37E-7 4.17E-7 14 2.22E-3 2.23E-3 4.24E-5 4.17E-5 4.19E-5 3.00E-8 2.00E-8 15 7.76E-4 7.80E-4 5.69E-6 5.83E-6 5.62E-6 16 2.13E-4 2.12E-4 4.43E-7 4.13E-7 4.90E-7 17 4.26E-5 4.24E-5 1.33E-8 1.67E-8 3.33E-9 18 5.82E-6 5.76E-6 3.33E-9 19 4.73E-7 4.87E-7 20 3.00E-8 3.00E-8

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87 Table 21. Raw Data for Figure 41 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.83E-1 4.83E-1 4.76E-1 4.76E-1 4.76E-1 4.66E-1 4.66E-1 4.51E-1 4.51E-1 4.30E-1 -22 4.81E-1 4.81E-1 4.73E-1 4.73E-1 4.73E-1 4.62E-1 4.62E-1 4.45E-1 4.45E-1 4.21E-1 -21 4.79E-1 4.79E-1 4.70E-1 4.70E-1 4.70E-1 4.57E-1 4.57E-1 4.38E-1 4.38E-1 4.11E-1 -20 4.76E-1 4.76E-1 4.66E-1 4.66E-1 4.66E-1 4.52E-1 4.51E-1 4.30E-1 4.30E-1 3.99E-1 -19 4.73E-1 4.73E-1 4.62E-1 4.62E-1 4.62E-1 4.45E-1 4.45E-1 4.21E-1 4.21E-1 3.86E-1 -18 4.70E-1 4.70E-1 4.57E-1 4.57E-1 4.57E-1 4.38E-1 4.38E-1 4.11E-1 4.11E-1 3.72E-1 -17 4.66E-1 4.66E-1 4.51E-1 4.52E-1 4.52E-1 4.30E-1 4.30E-1 3.99E-1 3.99E-1 3.56E-1 -16 4.62E-1 4.62E-1 4.45E-1 4.45E-1 4.45E-1 4.21E-1 4.21E-1 3.86E-1 3.86E-1 3.38E-1 -15 4.57E-1 4.57E-1 4.38E-1 4.38E-1 4.38E-1 4.11E-1 4.11E-1 3.72E-1 3.72E-1 3.19E-1 -14 4.52E-1 4.52E-1 4.30E-1 4.30E-1 4.30E-1 3.99E-1 3.99E-1 3.56E-1 3.56E-1 2.98E-1 -13 4.45E-1 4.45E-1 4.21E-1 4.21E-1 4.21E-1 3.87E-1 3.86E-1 3.38E-1 3.38E-1 2.76E-1 -12 4.38E-1 4.38E-1 4.11E-1 4.11E-1 4.11E-1 3.72E-1 3.72E-1 3.19E-1 3.19E-1 2.53E-1 -11 4.30E-1 4.30E-1 3.99E-1 4.00E-1 3.99E-1 3.56E-1 3.56E-1 2.98E-1 2.98E-1 2.29E-1 -10 4.21E-1 4.21E-1 3.87E-1 3.87E-1 3.87E-1 3.38E-1 3.38E-1 2.76E-1 2.76E-1 2.05E-1 -9 4.11E-1 4.11E-1 3.72E-1 3.72E-1 3.72E-1 3.19E-1 3.19E-1 2.53E-1 2.53E-1 1.80E-1 -8 4.00E-1 4.00E-1 3.56E-1 3.56E-1 3.56E-1 2.98E-1 2.98E-1 2.29E-1 2.29E-1 1.55E-1 -7 3.87E-1 3.87E-1 3.38E-1 3.38E-1 3.38E-1 2.76E-1 2.76E-1 2.05E-1 2.05E-1 1.30E-1 -6 3.72E-1 3.72E-1 3.19E-1 3.19E-1 3.19E-1 2.53E-1 2.53E-1 1.81E-1 1.81E-1 1.05E-1 -5 3.56E-1 3.56E-1 2.98E-1 2.99E-1 2.98E-1 2.30E-1 2.30E-1 1.56E-1 1.56E-1 8.05E-2 -4 3.39E-1 3.39E-1 2.77E-1 2.77E-1 2.77E-1 2.06E-1 2.06E-1 1.30E-1 1.30E-1 5.82E-2 -3 3.19E-1 3.19E-1 2.54E-1 2.54E-1 2.54E-1 1.81E-1 1.81E-1 1.05E-1 1.05E-1 3.89E-2 -2 2.99E-1 2.99E-1 2.30E-1 2.30E-1 2.30E-1 1.56E-1 1.56E-1 8.07E-2 8.08E-2 2.36E-2 -1 2.77E-1 2.77E-1 2.06E-1 2.06E-1 2.06E-1 1.31E-1 1.31E-1 5.83E-2 5.84E-2 1.27E-2 0 2.54E-1 2.54E-1 1.81E-1 1.81E-1 1.81E-1 1.05E-1 1.05E-1 3.91E-2 3.90E-2 5.90E-3 1 2.30E-1 2.30E-1 1.56E-1 1.56E-1 1.56E-1 8.10E-2 8.11E-2 2.37E-2 2.37E-2 2.27E-3 2 2.06E-1 2.06E-1 1.31E-1 1.31E-1 1.31E-1 5.86E-2 5.86E-2 1.28E-2 1.28E-2 6.90E-4 3 1.81E-1 1.81E-1 1.06E-1 1.06E-1 1.06E-1 3.92E-2 3.93E-2 5.94E-3 5.95E-3 1.57E-4 4 1.56E-1 1.56E-1 8.13E-2 8.13E-2 8.12E-2 2.39E-2 2.39E-2 2.29E-3 2.30E-3 2.48E-5 5 1.31E-1 1.31E-1 5.88E-2 5.88E-2 5.88E-2 1.29E-2 1.29E-2 6.98E-4 7.01E-4 2.57E-6 6 1.06E-1 1.06E-1 3.95E-2 3.94E-2 3.94E-2 6.00E-3 6.01E-3 1.60E-4 1.60E-4 1.30E-7 7 8.15E-2 8.15E-2 2.40E-2 2.40E-2 2.40E-2 2.32E-3 2.33E-3 2.56E-5 2.55E-5 8 5.90E-2 5.90E-2 1.30E-2 1.30E-2 1.30E-2 7.11E-4 7.13E-4 2.54E-6 2.36E-6 9 3.96E-2 3.96E-2 6.06E-3 6.06E-3 6.05E-3 1.62E-4 1.63E-4 1.60E-7 1.30E-7 10 2.41E-2 2.42E-2 2.35E-3 2.34E-3 2.34E-3 2.62E-5 2.60E-5 1.33E-8 1.33E-8 11 1.31E-2 1.31E-2 7.22E-4 7.19E-4 7.19E-4 2.54E-6 2.48E-6 12 6.10E-3 6.10E-3 1.68E-4 1.66E-4 1.65E-4 1.17E-7 1.60E-7 13 2.37E-3 2.37E-3 2.68E-5 2.70E-5 2.67E-5 6.67E-9 6.67E-9 14 7.29E-4 7.28E-4 2.71E-6 2.80E-6 2.76E-6 15 1.69E-4 1.68E-4 1.23E-7 2.17E-7 1.53E-7 16 2.72E-5 2.68E-5 1.33E-8 17 2.55E-6 2.85E-6 18 1.27E-7 1.53E-7 19 3.33E-9 3.33E-9

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88 Table 22. Raw Data for Figure 42 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.87E-1 4.87E-1 4.82E-1 4.82E-1 4.82E-1 4.74E-1 4.74E-1 4.63E-1 4.63E-1 4.47E-1 -22 4.86E-1 4.86E-1 4.80E-1 4.80E-1 4.80E-1 4.71E-1 4.71E-1 4.59E-1 4.59E-1 4.40E-1 -21 4.84E-1 4.84E-1 4.77E-1 4.77E-1 4.77E-1 4.68E-1 4.68E-1 4.53E-1 4.54E-1 4.32E-1 -20 4.82E-1 4.82E-1 4.74E-1 4.75E-1 4.74E-1 4.64E-1 4.64E-1 4.47E-1 4.47E-1 4.23E-1 -19 4.80E-1 4.80E-1 4.71E-1 4.71E-1 4.71E-1 4.59E-1 4.59E-1 4.40E-1 4.41E-1 4.13E-1 -18 4.77E-1 4.77E-1 4.68E-1 4.68E-1 4.68E-1 4.54E-1 4.54E-1 4.32E-1 4.33E-1 4.00E-1 -17 4.74E-1 4.75E-1 4.64E-1 4.64E-1 4.64E-1 4.48E-1 4.48E-1 4.23E-1 4.23E-1 3.86E-1 -16 4.71E-1 4.71E-1 4.59E-1 4.59E-1 4.59E-1 4.41E-1 4.41E-1 4.13E-1 4.13E-1 3.70E-1 -15 4.68E-1 4.68E-1 4.54E-1 4.54E-1 4.54E-1 4.33E-1 4.33E-1 4.01E-1 4.01E-1 3.52E-1 -14 4.64E-1 4.64E-1 4.48E-1 4.48E-1 4.48E-1 4.23E-1 4.24E-1 3.87E-1 3.87E-1 3.32E-1 -13 4.59E-1 4.59E-1 4.41E-1 4.41E-1 4.41E-1 4.13E-1 4.13E-1 3.71E-1 3.71E-1 3.10E-1 -12 4.54E-1 4.54E-1 4.33E-1 4.33E-1 4.33E-1 4.01E-1 4.01E-1 3.53E-1 3.53E-1 2.86E-1 -11 4.48E-1 4.48E-1 4.24E-1 4.24E-1 4.24E-1 3.87E-1 3.87E-1 3.32E-1 3.32E-1 2.61E-1 -10 4.41E-1 4.41E-1 4.13E-1 4.13E-1 4.13E-1 3.71E-1 3.71E-1 3.10E-1 3.10E-1 2.36E-1 -9 4.33E-1 4.33E-1 4.01E-1 4.01E-1 4.01E-1 3.53E-1 3.53E-1 2.86E-1 2.87E-1 2.11E-1 -8 4.24E-1 4.24E-1 3.87E-1 3.87E-1 3.87E-1 3.33E-1 3.33E-1 2.62E-1 2.62E-1 1.87E-1 -7 4.13E-1 4.13E-1 3.71E-1 3.71E-1 3.71E-1 3.11E-1 3.11E-1 2.36E-1 2.36E-1 1.63E-1 -6 4.01E-1 4.01E-1 3.53E-1 3.53E-1 3.53E-1 2.87E-1 2.87E-1 2.11E-1 2.11E-1 1.41E-1 -5 3.87E-1 3.87E-1 3.33E-1 3.33E-1 3.33E-1 2.62E-1 2.62E-1 1.87E-1 1.87E-1 1.19E-1 -4 3.71E-1 3.71E-1 3.11E-1 3.11E-1 3.11E-1 2.37E-1 2.37E-1 1.63E-1 1.63E-1 9.73E-2 -3 3.53E-1 3.53E-1 2.87E-1 2.87E-1 2.87E-1 2.12E-1 2.12E-1 1.41E-1 1.41E-1 7.70E-2 -2 3.33E-1 3.33E-1 2.62E-1 2.62E-1 2.62E-1 1.87E-1 1.87E-1 1.19E-1 1.19E-1 5.82E-2 -1 3.11E-1 3.11E-1 2.37E-1 2.37E-1 2.37E-1 1.64E-1 1.64E-1 9.75E-2 9.75E-2 4.16E-2 0 2.87E-1 2.87E-1 2.12E-1 2.12E-1 2.12E-1 1.41E-1 1.41E-1 7.72E-2 7.72E-2 2.76E-2 1 2.62E-1 2.62E-1 1.88E-1 1.87E-1 1.87E-1 1.19E-1 1.19E-1 5.84E-2 5.84E-2 1.68E-2 2 2.37E-1 2.37E-1 1.64E-1 1.64E-1 1.64E-1 9.77E-2 9.78E-2 4.17E-2 4.17E-2 9.12E-3 3 2.12E-1 2.12E-1 1.41E-1 1.41E-1 1.41E-1 7.75E-2 7.75E-2 2.77E-2 2.77E-2 4.32E-3 4 1.88E-1 1.88E-1 1.19E-1 1.19E-1 1.19E-1 5.86E-2 5.87E-2 1.69E-2 1.69E-2 1.72E-3 5 1.64E-1 1.64E-1 9.80E-2 9.79E-2 9.80E-2 4.19E-2 4.19E-2 9.18E-3 9.18E-3 5.48E-4 6 1.41E-1 1.41E-1 7.77E-2 7.76E-2 7.76E-2 2.79E-2 2.79E-2 4.35E-3 4.35E-3 1.33E-4 7 1.19E-1 1.19E-1 5.88E-2 5.88E-2 5.88E-2 1.70E-2 1.70E-2 1.73E-3 1.74E-3 2.31E-5 8 9.82E-2 9.82E-2 4.21E-2 4.21E-2 4.21E-2 9.24E-3 9.25E-3 5.56E-4 5.57E-4 2.71E-6 9 7.79E-2 7.79E-2 2.80E-2 2.80E-2 2.80E-2 4.39E-3 4.39E-3 1.36E-4 1.36E-4 1.28E-7 10 5.90E-2 5.90E-2 1.71E-2 1.71E-2 1.71E-2 1.75E-3 1.75E-3 2.39E-5 2.38E-5 7.50E-9 11 4.22E-2 4.22E-2 9.32E-3 9.32E-3 9.31E-3 5.65E-4 5.65E-4 2.62E-6 2.75E-6 12 2.81E-2 2.81E-2 4.43E-3 4.43E-3 4.43E-3 1.38E-4 1.39E-4 1.80E-7 2.10E-7 13 1.72E-2 1.72E-2 1.77E-3 1.77E-3 1.77E-3 2.40E-5 2.41E-5 1.00E-8 1.50E-8 14 9.37E-3 9.38E-3 5.74E-4 5.73E-4 5.72E-4 2.61E-6 2.79E-6 15 4.46E-3 4.47E-3 1.42E-4 1.42E-4 1.41E-4 1.68E-7 1.45E-7 16 1.79E-3 1.79E-3 2.48E-5 2.49E-5 2.46E-5 2.50E-9 1.00E-8 17 5.78E-4 5.80E-4 2.88E-6 2.88E-6 2.97E-6 18 1.42E-4 1.43E-4 1.53E-7 2.30E-7 2.00E-7 19 2.52E-5 2.53E-5 5.00E-9 2.50E-9 2.50E-9 20 2.88E-6 2.90E-6 21 1.90E-7 1.95E-7 22 1.00E-8 2.50E-9

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89 Table 23. Raw Data for Figure 43 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.91E-1 4.91E-1 4.87E-1 4.87E-1 4.87E-1 4.81E-1 4.81E-1 4.73E-1 4.73E-1 4.60E-1 -22 4.89E-1 4.89E-1 4.85E-1 4.85E-1 4.85E-1 4.78E-1 4.79E-1 4.69E-1 4.69E-1 4.55E-1 -21 4.88E-1 4.88E-1 4.83E-1 4.83E-1 4.83E-1 4.76E-1 4.76E-1 4.65E-1 4.65E-1 4.49E-1 -20 4.87E-1 4.87E-1 4.81E-1 4.81E-1 4.81E-1 4.73E-1 4.73E-1 4.60E-1 4.60E-1 4.42E-1 -19 4.85E-1 4.85E-1 4.79E-1 4.79E-1 4.79E-1 4.69E-1 4.69E-1 4.55E-1 4.55E-1 4.34E-1 -18 4.83E-1 4.83E-1 4.76E-1 4.76E-1 4.76E-1 4.65E-1 4.65E-1 4.49E-1 4.49E-1 4.25E-1 -17 4.81E-1 4.81E-1 4.73E-1 4.73E-1 4.73E-1 4.60E-1 4.60E-1 4.42E-1 4.42E-1 4.14E-1 -16 4.79E-1 4.79E-1 4.69E-1 4.69E-1 4.69E-1 4.55E-1 4.55E-1 4.34E-1 4.34E-1 4.03E-1 -15 4.76E-1 4.76E-1 4.65E-1 4.65E-1 4.65E-1 4.49E-1 4.49E-1 4.25E-1 4.25E-1 3.90E-1 -14 4.73E-1 4.73E-1 4.60E-1 4.60E-1 4.60E-1 4.42E-1 4.42E-1 4.15E-1 4.15E-1 3.76E-1 -13 4.69E-1 4.69E-1 4.55E-1 4.55E-1 4.34E-1 4.34E-1 4.03E-1 4.03E-1 3.60E-1 -12 4.65E-1 4.65E-1 4.49E-1 4.49E-1 4.49E-1 4.25E-1 4.25E-1 3.90E-1 3.90E-1 3.42E-1 -11 4.60E-1 4.61E-1 4.42E-1 4.42E-1 4.42E-1 4.15E-1 4.15E-1 3.76E-1 3.76E-1 3.24E-1 -10 4.55E-1 4.55E-1 4.34E-1 4.34E-1 4.34E-1 4.03E-1 4.03E-1 3.60E-1 3.60E-1 3.03E-1 -9 4.49E-1 4.49E-1 4.25E-1 4.25E-1 4.25E-1 3.90E-1 3.90E-1 3.43E-1 3.43E-1 2.81E-1 -8 4.42E-1 4.42E-1 4.15E-1 4.15E-1 4.15E-1 3.76E-1 3.76E-1 3.24E-1 3.24E-1 2.58E-1 -7 4.34E-1 4.34E-1 4.03E-1 4.03E-1 4.03E-1 3.60E-1 3.60E-1 3.03E-1 3.03E-1 2.35E-1 -6 4.25E-1 4.25E-1 3.90E-1 3.90E-1 3.90E-1 3.43E-1 3.43E-1 2.81E-1 2.81E-1 2.11E-1 -5 4.15E-1 4.15E-1 3.76E-1 3.76E-1 3.76E-1 3.24E-1 3.24E-1 2.59E-1 2.59E-1 1.88E-1 -4 4.03E-1 4.03E-1 3.60E-1 3.60E-1 3.60E-1 3.04E-1 3.04E-1 2.35E-1 2.35E-1 1.66E-1 -3 3.91E-1 3.91E-1 3.43E-1 3.43E-1 3.43E-1 2.82E-1 2.82E-1 2.12E-1 2.12E-1 1.45E-1 -2 3.76E-1 3.76E-1 3.24E-1 3.24E-1 3.24E-1 2.59E-1 2.59E-1 1.89E-1 1.89E-1 1.25E-1 -1 3.61E-1 3.61E-1 3.04E-1 3.04E-1 3.04E-1 2.35E-1 2.35E-1 1.66E-1 1.66E-1 1.06E-1 0 3.43E-1 3.43E-1 2.82E-1 2.82E-1 2.82E-1 2.12E-1 2.12E-1 1.45E-1 1.45E-1 8.71E-2 1 3.24E-1 3.24E-1 2.59E-1 2.59E-1 2.59E-1 1.89E-1 1.89E-1 1.25E-1 1.25E-1 6.93E-2 2 3.04E-1 3.04E-1 2.36E-1 2.36E-1 2.36E-1 1.67E-1 1.67E-1 1.06E-1 1.06E-1 5.28E-2 3 2.82E-1 2.82E-1 2.12E-1 2.12E-1 2.12E-1 1.46E-1 1.46E-1 8.72E-2 8.73E-2 3.81E-2 4 2.59E-1 2.59E-1 1.89E-1 1.89E-1 1.89E-1 1.25E-1 1.25E-1 6.95E-2 6.95E-2 2.56E-2 5 2.36E-1 2.36E-1 1.67E-1 1.67E-1 1.67E-1 1.06E-1 1.06E-1 5.30E-2 5.30E-2 1.58E-2 6 2.12E-1 2.12E-1 1.46E-1 1.46E-1 1.46E-1 8.75E-2 8.75E-2 3.82E-2 3.82E-2 8.73E-3 7 1.89E-1 1.89E-1 1.26E-1 1.26E-1 1.26E-1 6.97E-2 6.97E-2 2.57E-2 2.57E-2 4.23E-3 8 1.67E-1 1.67E-1 1.06E-1 1.06E-1 1.06E-1 5.31E-2 5.32E-2 1.58E-2 1.59E-2 1.73E-3 9 1.46E-1 1.46E-1 8.76E-2 8.76E-2 8.76E-2 3.83E-2 3.84E-2 8.79E-3 8.79E-3 5.73E-4 10 1.26E-1 1.26E-1 6.99E-2 6.99E-2 6.99E-2 2.58E-2 2.58E-2 4.26E-3 4.26E-3 1.46E-4 11 1.06E-1 1.06E-1 5.33E-2 5.33E-2 5.33E-2 1.59E-2 1.59E-2 1.75E-3 1.75E-3 2.65E-5 12 8.78E-2 8.78E-2 3.85E-2 3.85E-2 3.85E-2 8.84E-3 8.85E-3 5.81E-4 5.82E-4 3.22E-6 13 7.01E-2 7.00E-2 2.59E-2 2.59E-2 2.59E-2 4.30E-3 4.30E-3 1.48E-4 1.49E-4 1.90E-7 14 5.35E-2 5.35E-2 1.60E-2 1.60E-2 1.60E-2 1.76E-3 1.77E-3 2.72E-5 2.75E-5 1.40E-8 15 3.86E-2 3.86E-2 8.91E-3 8.91E-3 8.91E-3 5.89E-4 5.89E-4 3.22E-6 3.35E-6 16 2.60E-2 2.60E-2 4.33E-3 4.33E-3 4.33E-3 1.51E-4 1.51E-4 2.42E-7 2.76E-7 17 1.61E-2 1.61E-2 1.79E-3 1.78E-3 1.78E-3 2.76E-5 2.79E-5 8.00E-9 2.00E-8 18 8.96E-3 8.97E-3 5.97E-4 5.98E-4 5.96E-4 3.24E-6 3.50E-6 4.55E-1

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90 Table 23. Continued. SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 19 4.36E-3 4.37E-3 1.54E-4 1.54E-4 1.54E-4 2.34E-7 2.34E-7 20 1.80E-3 1.80E-3 2.85E-5 2.87E-5 2.84E-5 8.00E-9 1.00E-8 21 6.02E-4 6.04E-4 3.54E-6 3.60E-6 3.50E-6 22 1.55E-4 1.56E-4 2.18E-7 3.00E-7 2.64E-7 23 2.91E-5 2.92E-5 6.00E-9 1.60E-8 6.00E-9 24 3.56E-6 3.59E-6 25 2.76E-7 2.62E-7 26 1.20E-8 6.00E-9 Table 24. Raw Data for Figure 44 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.92E-1 4.92E-1 4.88E-1 4.88E-1 4.88E-1 4.83E-1 4.83E-1 4.76E-1 4.76E-1 4.65E-1 -22 4.91E-1 4.91E-1 4.87E-1 4.85E-1 4.66E-1 -11 4.61E-1 4.35E-1 3.04E-1 -7 3.75E-1 4.35E-1 3.04E-1 3.90E-1 3.76E-1 3.05E-1 2.17E-1 4.87E-1 4.87E-1 4.81E-1 4.81E-1 4.73E-1 4.73E-1 4.61E-1 -21 4.90E-1 4.90E-1 4.85E-1 4.85E-1 4.79E-1 4.79E-1 4.70E-1 4.70E-1 4.56E-1 -20 4.88E-1 4.88E-1 4.83E-1 4.83E-1 4.83E-1 4.76E-1 4.76E-1 4.66E-1 4.66E-1 4.49E-1 -19 4.87E-1 4.87E-1 4.81E-1 4.81E-1 4.81E-1 4.73E-1 4.73E-1 4.61E-1 4.61E-1 4.42E-1 -18 4.85E-1 4.85E-1 4.79E-1 4.79E-1 4.79E-1 4.70E-1 4.70E-1 4.56E-1 4.56E-1 4.34E-1 -17 4.83E-1 4.83E-1 4.76E-1 4.76E-1 4.76E-1 4.66E-1 4.66E-1 4.50E-1 4.50E-1 4.25E-1 -16 4.81E-1 4.81E-1 4.73E-1 4.73E-1 4.73E-1 4.61E-1 4.61E-1 4.43E-1 4.43E-1 4.15E-1 -15 4.79E-1 4.79E-1 4.70E-1 4.70E-1 4.70E-1 4.56E-1 4.56E-1 4.34E-1 4.34E-1 4.03E-1 -14 4.76E-1 4.76E-1 4.66E-1 4.66E-1 4.50E-1 4.50E-1 4.25E-1 4.25E-1 3.90E-1 -13 4.73E-1 4.73E-1 4.61E-1 4.61E-1 4.61E-1 4.43E-1 4.43E-1 4.15E-1 4.15E-1 3.75E-1 -12 4.70E-1 4.70E-1 4.56E-1 4.56E-1 4.56E-1 4.35E-1 4.35E-1 4.03E-1 4.03E-1 3.59E-1 4.66E-1 4.66E-1 4.50E-1 4.50E-1 4.50E-1 4.25E-1 4.25E-1 3.90E-1 3.90E-1 3.42E-1 -10 4.61E-1 4.43E-1 4.43E-1 4.43E-1 4.15E-1 4.15E-1 3.75E-1 3.75E-1 3.24E-1 -9 4.56E-1 4.56E-1 4.35E-1 4.35E-1 4.03E-1 4.03E-1 3.59E-1 3.59E-1 -8 4.50E-1 4.50E-1 4.25E-1 4.25E-1 4.25E-1 3.90E-1 3.90E-1 3.42E-1 3.42E-1 2.83E-1 4.43E-1 4.43E-1 4.15E-1 4.15E-1 4.15E-1 3.75E-1 3.24E-1 3.24E-1 2.62E-1 -6 4.35E-1 4.03E-1 4.03E-1 4.03E-1 3.60E-1 3.60E-1 3.04E-1 2.39E-1 -5 4.25E-1 4.26E-1 3.90E-1 3.90E-1 3.42E-1 3.42E-1 2.84E-1 2.84E-1 2.17E-1 -4 4.15E-1 4.15E-1 3.76E-1 3.75E-1 3.24E-1 3.24E-1 2.62E-1 2.62E-1 1.94E-1 -3 4.03E-1 4.03E-1 3.60E-1 3.60E-1 3.60E-1 3.05E-1 3.05E-1 2.39E-1 2.39E-1 1.72E-1 -2 3.90E-1 3.90E-1 3.43E-1 3.43E-1 3.43E-1 2.84E-1 2.84E-1 2.17E-1 2.17E-1 1.52E-1 -1 3.76E-1 3.76E-1 3.24E-1 3.24E-1 3.24E-1 2.62E-1 2.62E-1 1.94E-1 1.94E-1 1.32E-1 0 3.60E-1 3.60E-1 3.05E-1 3.05E-1 2.40E-1 2.40E-1 1.73E-1 1.73E-1 1.14E-1 1 3.43E-1 3.43E-1 2.84E-1 2.84E-1 2.84E-1 2.17E-1 2.17E-1 1.52E-1 1.52E-1 9.63E-2 2 3.24E-1 3.24E-1 2.62E-1 2.62E-1 2.62E-1 1.95E-1 1.95E-1 1.32E-1 1.32E-1 7.95E-2 3 3.05E-1 3.05E-1 2.40E-1 2.40E-1 2.40E-1 1.73E-1 1.73E-1 1.14E-1 1.14E-1 6.35E-2 4 2.84E-1 2.84E-1 2.17E-1 2.17E-1 1.52E-1 1.52E-1 9.65E-2 9.65E-2 4.86E-2 5 2.63E-1 2.63E-1 1.95E-1 1.95E-1 1.95E-1 1.33E-1 1.33E-1 7.97E-2 7.97E-2 3.52E-2 6 2.40E-1 2.40E-1 1.73E-1 1.73E-1 1.73E-1 1.14E-1 1.14E-1 6.37E-2 6.37E-2 2.38E-2

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91 Table 24. Continued. SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 7 2.17E-1 2.17E-1 1.52E-1 1.52E-1 1.52E-1 9.67E-2 9.67E-2 4.87E-2 4.87E-2 1.48E-2 8 1.95E-1 1.95E-1 1.33E-1 1.33E-1 1.33E-1 7.99E-2 7.99E-2 3.53E-2 3.53E-2 8.27E-3 9 1.73E-1 1.73E-1 1.14E-1 1.14E-1 1.14E-1 6.38E-2 6.39E-2 2.39E-2 2.39E-2 4.05E-3 10 1.53E-1 1.53E-1 9.69E-2 9.68E-2 9.68E-2 4.89E-2 4.89E-2 1.49E-2 1.49E-2 1.69E-3 11 1.33E-1 1.33E-1 8.00E-2 4.15E-3 2.97E-5 8.00E-2 8.00E-2 3.54E-2 3.55E-2 8.32E-3 8.32E-3 5.70E-4 12 1.15E-1 1.15E-1 6.40E-2 6.40E-2 6.40E-2 2.40E-2 2.40E-2 4.08E-3 4.08E-3 1.49E-4 13 9.70E-2 9.70E-2 4.90E-2 4.90E-2 4.90E-2 1.49E-2 1.50E-2 1.70E-3 1.70E-3 2.78E-5 14 8.02E-2 8.02E-2 3.56E-2 3.56E-2 3.56E-2 8.37E-3 8.38E-3 5.75E-4 5.77E-4 3.51E-6 15 6.42E-2 6.42E-2 2.41E-2 2.41E-2 2.41E-2 4.12E-3 4.12E-3 1.51E-4 1.51E-4 2.38E-7 16 4.92E-2 4.92E-2 1.50E-2 1.50E-2 1.50E-2 1.72E-3 1.72E-3 2.86E-5 2.89E-5 1.33E-8 17 3.57E-2 3.57E-2 8.43E-3 8.44E-3 8.43E-3 5.84E-4 5.85E-4 3.56E-6 3.62E-6 18 2.42E-2 2.42E-2 4.15E-3 4.15E-3 1.53E-4 1.54E-4 2.78E-7 2.98E-7 19 1.51E-2 1.51E-2 1.73E-3 1.73E-3 1.74E-3 2.88E-5 2.93E-5 1.50E-8 1.83E-8 20 8.49E-3 8.49E-3 5.92E-4 5.92E-4 5.92E-4 3.59E-6 3.74E-6 21 4.18E-3 4.19E-3 1.56E-4 1.57E-4 1.56E-4 2.62E-7 2.73E-7 22 1.75E-3 1.75E-3 2.99E-5 2.98E-5 8.33E-9 1.00E-8 23 5.97E-4 5.98E-4 3.78E-6 3.94E-6 3.86E-6 24 1.58E-4 1.58E-4 2.68E-7 3.13E-7 2.80E-7 25 3.04E-5 3.06E-5 6.67E-9 1.83E-8 8.33E-9 26 3.87E-6 3.91E-6 1.67E-9 27 3.00E-7 3.28E-7 28 1.83E-8 1.17E-8 Table 25. Raw Data for Figure 45 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.93E-1 4.93E-1 4.91E-1 4.91E-1 4.91E-1 4.86E-1 4.86E-1 4.81E-1 4.81E-1 4.72E-1 -22 4.93E-1 4.92E-1 4.89E-1 4.89E-1 4.89E-1 4.85E-1 4.85E-1 4.78E-1 4.78E-1 4.68E-1 -21 4.92E-1 4.92E-1 4.88E-1 4.88E-1 4.88E-1 4.83E-1 4.83E-1 4.75E-1 4.75E-1 4.64E-1 -20 4.91E-1 4.72E-1 4.53E-1 -17 4.72E-1 4.64E-1 4.64E-1 4.47E-1 -14 4.68E-1 4.64E-1 4.39E-1 4.91E-1 4.86E-1 4.86E-1 4.86E-1 4.81E-1 4.81E-1 4.72E-1 4.59E-1 -19 4.89E-1 4.89E-1 4.85E-1 4.85E-1 4.85E-1 4.78E-1 4.78E-1 4.68E-1 4.68E-1 -18 4.88E-1 4.88E-1 4.83E-1 4.83E-1 4.83E-1 4.75E-1 4.75E-1 4.64E-1 4.64E-1 4.47E-1 4.87E-1 4.87E-1 4.81E-1 4.81E-1 4.81E-1 4.72E-1 4.59E-1 4.59E-1 4.39E-1 -16 4.85E-1 4.85E-1 4.78E-1 4.78E-1 4.78E-1 4.68E-1 4.68E-1 4.53E-1 4.53E-1 4.31E-1 -15 4.83E-1 4.83E-1 4.75E-1 4.75E-1 4.75E-1 4.47E-1 4.22E-1 4.81E-1 4.81E-1 4.72E-1 4.72E-1 4.72E-1 4.59E-1 4.59E-1 4.39E-1 4.39E-1 4.12E-1 -13 4.78E-1 4.78E-1 4.68E-1 4.68E-1 4.53E-1 4.53E-1 4.31E-1 4.31E-1 4.00E-1 -12 4.75E-1 4.75E-1 4.64E-1 4.64E-1 4.47E-1 4.47E-1 4.22E-1 4.22E-1 3.88E-1 -11 4.72E-1 4.72E-1 4.59E-1 4.59E-1 4.59E-1 4.39E-1 4.39E-1 4.12E-1 4.12E-1 3.75E-1 -10 4.68E-1 4.68E-1 4.53E-1 4.53E-1 4.53E-1 4.31E-1 4.31E-1 4.00E-1 4.00E-1 3.60E-1 -9 4.64E-1 4.64E-1 4.47E-1 4.47E-1 4.47E-1 4.22E-1 4.22E-1 3.88E-1 3.88E-1 3.45E-1 -8 4.59E-1 4.59E-1 4.39E-1 4.39E-1 4.12E-1 4.12E-1 3.75E-1 3.75E-1 3.29E-1

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92 Table 25. Continued. SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -7 4.53E-1 4.53E-1 4.31E-1 4.31E-1 4.31E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.12E-1 -6 4.47E-1 4.47E-1 3.89E-1 3.61E-1 3.13E-1 2.76E-1 7.15E-2 5.72E-2 4.39E-2 3.19E-2 1.00E-8 1.45E-4 29 4.22E-1 4.22E-1 4.22E-1 3.88E-1 3.88E-1 3.45E-1 3.45E-1 2.94E-1 -5 4.40E-1 4.40E-1 4.12E-1 4.12E-1 4.12E-1 3.75E-1 3.75E-1 3.29E-1 3.29E-1 2.75E-1 -4 4.31E-1 4.31E-1 4.01E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.12E-1 3.12E-1 2.56E-1 -3 4.22E-1 4.22E-1 3.88E-1 3.88E-1 3.88E-1 3.45E-1 3.45E-1 2.94E-1 2.94E-1 2.36E-1 -2 4.12E-1 4.12E-1 3.75E-1 3.75E-1 3.75E-1 3.29E-1 3.29E-1 2.76E-1 2.76E-1 2.15E-1 -1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.61E-1 3.12E-1 3.12E-1 2.56E-1 2.56E-1 1.94E-1 0 3.89E-1 3.46E-1 3.46E-1 3.46E-1 2.94E-1 2.94E-1 2.36E-1 2.36E-1 1.74E-1 1 3.75E-1 3.75E-1 3.29E-1 3.29E-1 3.29E-1 2.76E-1 2.76E-1 2.15E-1 2.15E-1 1.54E-1 2 3.61E-1 3.12E-1 3.12E-1 3.12E-1 2.56E-1 2.56E-1 1.94E-1 1.94E-1 1.35E-1 3 3.46E-1 3.46E-1 2.95E-1 2.95E-1 2.95E-1 2.36E-1 2.36E-1 1.74E-1 1.74E-1 1.18E-1 4 3.30E-1 3.30E-1 2.76E-1 2.76E-1 2.76E-1 2.15E-1 2.15E-1 1.54E-1 1.54E-1 1.01E-1 5 3.13E-1 2.56E-1 2.56E-1 2.56E-1 1.95E-1 1.95E-1 1.36E-1 1.36E-1 8.59E-2 6 2.95E-1 2.95E-1 2.36E-1 2.36E-1 2.36E-1 1.74E-1 1.74E-1 1.18E-1 1.18E-1 7.09E-2 7 2.76E-1 2.16E-1 2.16E-1 2.16E-1 1.54E-1 1.55E-1 1.02E-1 1.02E-1 5.67E-2 8 2.57E-1 2.57E-1 1.95E-1 1.95E-1 1.95E-1 1.36E-1 1.36E-1 8.60E-2 8.60E-2 4.34E-2 9 2.37E-1 2.36E-1 1.74E-1 1.74E-1 1.74E-1 1.18E-1 1.18E-1 7.10E-2 7.11E-2 3.15E-2 10 2.16E-1 2.16E-1 1.55E-1 1.55E-1 1.55E-1 1.02E-1 1.02E-1 5.68E-2 5.68E-2 2.13E-2 11 1.95E-1 1.95E-1 1.36E-1 1.36E-1 1.36E-1 8.62E-2 8.62E-2 4.35E-2 4.35E-2 1.33E-2 12 1.75E-1 1.75E-1 1.18E-1 1.18E-1 1.18E-1 7.12E-2 7.12E-2 3.16E-2 3.16E-2 7.44E-3 13 1.55E-1 1.55E-1 1.02E-1 1.02E-1 1.02E-1 5.70E-2 5.70E-2 2.14E-2 2.14E-2 3.66E-3 14 1.36E-1 1.36E-1 8.63E-2 8.63E-2 8.63E-2 4.36E-2 4.37E-2 1.33E-2 1.33E-2 1.53E-3 15 1.19E-1 1.19E-1 7.14E-2 7.13E-2 7.14E-2 3.17E-2 3.17E-2 7.48E-3 7.48E-3 5.19E-4 16 1.02E-1 1.02E-1 5.71E-2 5.71E-2 5.71E-2 2.15E-2 2.15E-2 3.68E-3 3.68E-3 1.36E-4 17 8.65E-2 8.65E-2 4.38E-2 4.38E-2 4.38E-2 1.34E-2 1.34E-2 1.54E-3 1.54E-3 2.57E-5 18 7.15E-2 3.18E-2 3.18E-2 3.18E-2 7.53E-3 7.54E-3 5.24E-4 5.25E-4 3.28E-6 19 5.72E-2 2.16E-2 2.16E-2 2.16E-2 3.71E-3 3.72E-3 1.38E-4 1.38E-4 2.33E-7 20 4.39E-2 1.35E-2 1.35E-2 1.35E-2 1.55E-3 1.56E-3 2.64E-5 2.66E-5 1.14E-8 21 3.19E-2 7.59E-3 7.59E-3 7.58E-3 5.32E-4 5.32E-4 3.35E-6 3.37E-6 22 2.17E-2 2.17E-2 3.74E-3 3.75E-3 3.74E-3 1.40E-4 1.41E-4 2.61E-7 2.87E-7 23 1.36E-2 1.36E-2 1.57E-3 1.57E-3 1.57E-3 2.66E-5 2.71E-5 1.43E-8 1.71E-8 24 7.63E-3 7.64E-3 5.39E-4 5.39E-4 5.38E-4 3.36E-6 3.46E-6 25 3.77E-3 3.78E-3 1.43E-4 1.43E-4 1.43E-4 2.44E-7 2.59E-7 26 1.59E-3 1.59E-3 2.75E-5 2.76E-5 2.74E-5 8.57E-9 27 5.43E-4 5.44E-4 3.48E-6 3.67E-6 3.62E-6 28 1.45E-4 2.60E-7 2.94E-7 2.59E-7 2.80E-5 2.81E-5 7.14E-9 1.71E-8 1.00E-8 30 3.64E-6 3.63E-6 1.43E-9

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93 Table 26. Raw Data for Figure 46 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 -23 4.94E-1 4.94E-1 4.91E-1 4.91E-1 4.91E-1 4.88E-1 4.88E-1 4.82E-1 4.82E-1 4.74E-1 -22 4.93E-1 4.93E-1 4.90E-1 4.90E-1 4.90E-1 4.86E-1 4.86E-1 4.80E-1 4.80E-1 4.71E-1 -21 4.92E-1 4.92E-1 4.77E-1 4.74E-1 4.71E-1 4.67E-1 4.62E-1 3.55E-1 3.40E-1 3.24E-1 3.07E-1 2.90E-1 1.79E-1 1.42E-1 3.45E-3 4.99E-4 1.33E-4 4.89E-1 4.89E-1 4.89E-1 4.84E-1 4.84E-1 4.77E-1 4.77E-1 4.67E-1 -20 4.91E-1 4.91E-1 4.88E-1 4.88E-1 4.88E-1 4.82E-1 4.82E-1 4.74E-1 4.74E-1 4.62E-1 -19 4.90E-1 4.90E-1 4.86E-1 4.86E-1 4.86E-1 4.80E-1 4.80E-1 4.71E-1 4.71E-1 4.57E-1 -18 4.89E-1 4.89E-1 4.84E-1 4.84E-1 4.84E-1 4.77E-1 4.77E-1 4.67E-1 4.67E-1 4.51E-1 -17 4.88E-1 4.88E-1 4.82E-1 4.82E-1 4.82E-1 4.74E-1 4.74E-1 4.62E-1 4.62E-1 4.44E-1 -16 4.86E-1 4.86E-1 4.80E-1 4.80E-1 4.80E-1 4.71E-1 4.71E-1 4.57E-1 4.57E-1 4.36E-1 -15 4.84E-1 4.84E-1 4.77E-1 4.77E-1 4.77E-1 4.67E-1 4.67E-1 4.51E-1 4.51E-1 4.27E-1 -14 4.82E-1 4.82E-1 4.74E-1 4.74E-1 4.74E-1 4.62E-1 4.62E-1 4.44E-1 4.44E-1 4.18E-1 -13 4.80E-1 4.80E-1 4.71E-1 4.71E-1 4.71E-1 4.57E-1 4.57E-1 4.36E-1 4.36E-1 4.07E-1 -12 4.77E-1 4.67E-1 4.67E-1 4.67E-1 4.51E-1 4.51E-1 4.27E-1 4.27E-1 3.95E-1 -11 4.74E-1 4.62E-1 4.62E-1 4.62E-1 4.44E-1 4.44E-1 4.18E-1 4.18E-1 3.82E-1 -10 4.71E-1 4.57E-1 4.57E-1 4.57E-1 4.36E-1 4.36E-1 4.07E-1 4.07E-1 3.69E-1 -9 4.67E-1 4.51E-1 4.51E-1 4.51E-1 4.28E-1 4.28E-1 3.95E-1 3.95E-1 3.54E-1 -8 4.62E-1 4.44E-1 4.44E-1 4.44E-1 4.18E-1 4.18E-1 3.82E-1 3.82E-1 3.39E-1 -7 4.57E-1 4.57E-1 4.36E-1 4.36E-1 4.36E-1 4.07E-1 4.07E-1 3.69E-1 3.69E-1 3.23E-1 -6 4.51E-1 4.51E-1 4.28E-1 4.28E-1 4.28E-1 3.95E-1 3.95E-1 3.54E-1 3.54E-1 3.07E-1 -5 4.44E-1 4.44E-1 4.18E-1 4.18E-1 4.18E-1 3.83E-1 3.83E-1 3.39E-1 3.39E-1 2.90E-1 -4 4.36E-1 4.36E-1 4.07E-1 4.07E-1 4.07E-1 3.69E-1 3.69E-1 3.23E-1 3.23E-1 2.72E-1 -3 4.28E-1 4.28E-1 3.95E-1 3.95E-1 3.95E-1 3.55E-1 3.55E-1 3.07E-1 3.07E-1 2.54E-1 -2 4.18E-1 4.18E-1 3.83E-1 3.83E-1 3.83E-1 3.39E-1 3.39E-1 2.90E-1 2.90E-1 2.36E-1 -1 4.07E-1 4.07E-1 3.69E-1 3.69E-1 3.69E-1 3.24E-1 3.24E-1 2.73E-1 2.73E-1 2.17E-1 0 3.95E-1 3.95E-1 3.55E-1 3.55E-1 3.55E-1 3.07E-1 3.07E-1 2.55E-1 2.55E-1 1.97E-1 1 3.83E-1 3.83E-1 3.40E-1 3.40E-1 3.40E-1 2.90E-1 2.90E-1 2.36E-1 2.36E-1 1.78E-1 2 3.69E-1 3.69E-1 3.24E-1 3.24E-1 3.24E-1 2.73E-1 2.73E-1 2.17E-1 2.17E-1 1.59E-1 3 3.55E-1 3.07E-1 3.07E-1 3.07E-1 2.55E-1 2.55E-1 1.98E-1 1.98E-1 1.41E-1 4 3.40E-1 2.90E-1 2.90E-1 2.90E-1 2.36E-1 2.36E-1 1.78E-1 1.78E-1 1.24E-1 5 3.24E-1 2.73E-1 2.73E-1 2.73E-1 2.17E-1 2.17E-1 1.59E-1 1.59E-1 1.08E-1 6 3.07E-1 2.55E-1 2.55E-1 2.55E-1 1.98E-1 1.98E-1 1.41E-1 1.41E-1 9.27E-2 7 2.90E-1 2.36E-1 2.36E-1 2.36E-1 1.79E-1 1.79E-1 1.24E-1 1.24E-1 7.85E-2 8 2.73E-1 2.73E-1 2.17E-1 2.17E-1 2.17E-1 1.60E-1 1.60E-1 1.08E-1 1.08E-1 6.49E-2 9 2.55E-1 2.55E-1 1.98E-1 1.98E-1 1.98E-1 1.41E-1 1.41E-1 9.28E-2 9.28E-2 5.19E-2 10 2.37E-1 2.37E-1 1.79E-1 1.79E-1 1.79E-1 1.24E-1 1.24E-1 7.86E-2 7.86E-2 3.99E-2 11 2.18E-1 2.18E-1 1.60E-1 1.60E-1 1.60E-1 1.08E-1 1.08E-1 6.50E-2 6.50E-2 2.90E-2 12 1.98E-1 1.98E-1 1.42E-1 1.42E-1 1.42E-1 9.30E-2 9.30E-2 5.21E-2 5.21E-2 1.97E-2 13 1.79E-1 1.24E-1 1.24E-1 1.24E-1 7.87E-2 7.88E-2 4.00E-2 4.00E-2 1.23E-2 14 1.60E-1 1.60E-1 1.08E-1 1.08E-1 1.08E-1 6.52E-2 6.52E-2 2.91E-2 2.91E-2 6.96E-3 15 1.42E-1 9.31E-2 9.31E-2 9.31E-2 5.22E-2 5.22E-2 1.98E-2 1.98E-2 16 1.24E-1 1.24E-1 7.89E-2 7.89E-2 7.89E-2 4.01E-2 4.01E-2 1.24E-2 1.24E-2 1.45E-3 17 1.08E-1 1.08E-1 6.53E-2 6.53E-2 6.53E-2 2.92E-2 2.92E-2 7.00E-3 7.00E-3 18 9.33E-2 9.33E-2 5.23E-2 5.23E-2 5.23E-2 1.99E-2 1.99E-2 3.47E-3 3.47E-3

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94 Table 26. Continued. SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter 7 Filter 8 Filter 9 Filter 10 19 7.90E-2 7.90E-2 4.02E-2 4.02E-2 4.02E-2 1.25E-2 1.25E-2 1.46E-3 1.46E-3 2.54E-5 20 6.54E-2 6.54E-2 2.93E-2 2.93E-2 2.93E-2 7.04E-3 7.05E-3 5.04E-4 5.06E-4 3.34E-6 21 5.25E-2 5.25E-2 2.00E-2 2.00E-2 2.00E-2 3.50E-3 3.50E-3 1.35E-4 1.35E-4 2.45E-7 22 4.03E-2 4.03E-2 1.25E-2 1.25E-2 1.25E-2 1.48E-3 1.48E-3 2.63E-5 2.65E-5 1.00E-8 23 2.94E-2 2.94E-2 7.09E-3 7.10E-3 7.09E-3 5.12E-4 5.12E-4 3.40E-6 3.48E-6 24 2.01E-2 2.01E-2 3.53E-3 3.53E-3 3.53E-3 1.36E-4 1.38E-4 2.75E-7 3.03E-7 25 1.26E-2 1.26E-2 1.49E-3 1.49E-3 1.49E-3 2.65E-5 2.68E-5 1.50E-8 2.00E-8 26 7.13E-3 7.14E-3 5.18E-4 5.18E-4 5.18E-4 3.40E-6 3.52E-6 27 3.55E-3 3.56E-3 1.40E-4 1.40E-4 1.40E-4 2.53E-7 2.70E-7 28 1.51E-3 1.51E-3 2.74E-5 2.75E-5 2.74E-5 8.75E-9 1.13E-8 29 5.23E-4 5.23E-4 3.53E-6 3.69E-6 3.69E-6 30 1.41E-4 1.41E-4 2.80E-7 2.89E-7 2.88E-7 Table 27. Raw Data for Figure 48 SNR (dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 7 Filter 8 Filter 9 Filter 10 0 1.59E-1 1.59E-1 1.81E-1 1.81E-1 1.81E-1 1.41E-1 1.41E-1 1.45E-1 1.45E-1 1.14E-1 1 1.31E-1 1.31E-1 1.56E-1 1.56E-1 1.56E-1 1.19E-1 1.19E-1 1.25E-1 1.25E-1 9.63E-2 2 1.04E-1 1.04E-1 1.31E-1 1.31E-1 1.31E-1 9.77E-2 9.77E-2 1.06E-1 1.06E-1 7.95E-2 3 7.89E-2 7.89E-2 1.06E-1 1.06E-1 1.06E-1 7.75E-2 7.75E-2 8.72E-2 8.73E-2 6.35E-2 4 5.65E-2 5.65E-2 8.13E-2 8.12E-2 8.13E-2 5.86E-2 5.86E-2 6.95E-2 6.95E-2 4.86E-2 5 3.77E-2 3.77E-2 5.88E-2 5.88E-2 5.88E-2 4.19E-2 4.19E-2 5.30E-2 5.30E-2 3.52E-2 6 2.30E-2 2.30E-2 3.94E-2 3.94E-2 3.94E-2 2.79E-2 2.79E-2 3.82E-2 3.82E-2 2.38E-2 7 1.26E-2 1.26E-2 2.40E-2 2.40E-2 2.40E-2 1.70E-2 1.70E-2 2.57E-2 2.57E-2 1.48E-2 8 6.01E-3 6.00E-3 1.30E-2 1.30E-2 1.30E-2 9.25E-3 9.26E-3 1.59E-2 1.59E-2 8.27E-3 9 2.41E-3 2.41E-3 6.06E-3 6.05E-3 6.06E-3 4.39E-3 4.40E-3 8.79E-3 8.79E-3 4.05E-3 10 7.84E-4 7.82E-4 2.35E-3 2.34E-3 2.35E-3 1.75E-3 1.76E-3 4.26E-3 4.26E-3 1.68E-3 11 1.94E-4 1.94E-4 7.24E-4 7.20E-4 7.22E-4 5.65E-4 5.65E-4 1.75E-3 1.75E-3 5.70E-4 12 3.47E-5 3.44E-5 1.68E-4 1.66E-4 1.66E-4 1.39E-4 1.39E-4 5.82E-4 5.80E-4 1.49E-4 13 3.96E-6 3.99E-6 2.68E-5 2.64E-5 2.67E-5 2.40E-5 2.44E-5 1.49E-4 1.48E-4 2.81E-5 14 2.85E-7 2.63E-7 2.65E-6 2.98E-6 2.65E-6 2.67E-6 2.81E-6 2.72E-5 2.74E-5 3.60E-6 15 1.33E-8 6.67E-9 1.58E-7 1.88E-7 1.74E-7 1.81E-7 1.68E-7 3.29E-6 3.23E-6 2.52E-7 16 5.56E-9 5.56E-9 2.22E-9 5.00E-9 5.00E-9 2.50E-7 2.39E-7 1.11E-8 17 7.33E-9 1.40E-8 Filter 6 Table 28. Raw Data for Figure 65 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 4.10E-1 4.36E-1 4.61E-1 4.59E-1 4.71E-1 4.78E-1 4.81E-1 4.85E-1 4.86E-1 -9 3.99E-1 4.28E-1 4.56E-1 4.53E-1 4.67E-1 4.75E-1 4.79E-1 4.83E-1 4.84E-1 -8 3.87E-1 4.19E-1 4.51E-1 4.47E-1 4.63E-1 4.72E-1 4.76E-1 4.80E-1 4.82E-1 -7 3.74E-1 4.10E-1 4.44E-1 4.41E-1 4.58E-1 4.69E-1 4.73E-1 4.78E-1 4.80E-1 -6 3.59E-1 3.99E-1 4.37E-1 4.33E-1 4.53E-1 4.64E-1 4.69E-1 4.75E-1 4.77E-1

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95 Table 28. Continued. SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -5 3.43E-1 3.87E-1 4.29E-1 4.25E-1 4.47E-1 4.60E-1 4.65E-1 4.71E-1 4.74E-1 -4 3.25E-1 3.74E-1 4.20E-1 4.15E-1 4.40E-1 4.54E-1 4.60E-1 4.68E-1 4.70E-1 -3 3.05E-1 3.59E-1 4.10E-1 4.04E-1 4.32E-1 4.48E-1 4.55E-1 4.63E-1 4.66E-1 -2 2.84E-1 3.43E-1 3.98E-1 3.92E-1 4.22E-1 4.41E-1 4.49E-1 4.58E-1 4.62E-1 -1 2.61E-1 3.25E-1 3.85E-1 3.79E-1 4.12E-1 4.33E-1 4.42E-1 4.53E-1 4.56E-1 0 2.37E-1 3.05E-1 3.70E-1 3.64E-1 3.99E-1 4.24E-1 4.34E-1 4.46E-1 4.50E-1 1 2.12E-1 2.84E-1 3.54E-1 3.47E-1 3.85E-1 4.14E-1 4.24E-1 4.39E-1 4.43E-1 2 1.85E-1 2.62E-1 3.37E-1 3.29E-1 3.70E-1 4.02E-1 4.14E-1 4.31E-1 4.36E-1 3 1.58E-1 2.37E-1 3.18E-1 3.09E-1 3.52E-1 3.90E-1 4.02E-1 4.22E-1 4.27E-1 4 1.32E-1 2.12E-1 2.97E-1 2.87E-1 3.32E-1 3.76E-1 3.89E-1 4.12E-1 4.17E-1 5 1.07E-1 1.85E-1 2.76E-1 2.65E-1 3.10E-1 3.61E-1 3.75E-1 4.01E-1 4.07E-1 6 8.29E-2 1.59E-1 2.54E-1 2.41E-1 2.88E-1 3.44E-1 3.60E-1 3.89E-1 3.96E-1 7 6.18E-2 1.32E-1 2.31E-1 2.17E-1 2.64E-1 3.26E-1 3.44E-1 3.77E-1 3.84E-1 8 4.38E-2 1.07E-1 2.08E-1 1.93E-1 2.40E-1 3.07E-1 3.27E-1 3.63E-1 3.71E-1 9 2.93E-2 8.31E-2 1.86E-1 1.69E-1 2.17E-1 2.87E-1 3.09E-1 3.49E-1 3.58E-1 10 1.83E-2 6.20E-2 1.63E-1 1.26E-1 2.06E-1 2.15E-1 2.60E-1 3.78E-3 3.69E-4 1.11E-1 1.89E-1 9.91E-4 1.52E-1 6.33E-8 32 1.09E-1 7.38E-7 1.47E-1 1.94E-1 2.67E-1 2.90E-1 3.35E-1 3.44E-1 11 1.05E-2 4.40E-2 1.40E-1 1.73E-1 2.46E-1 2.71E-1 3.20E-1 3.30E-1 12 5.44E-3 2.95E-2 1.18E-1 1.07E-1 1.53E-1 2.26E-1 2.52E-1 3.05E-1 3.16E-1 13 2.47E-3 1.84E-2 9.76E-2 8.93E-2 1.34E-1 2.33E-1 2.90E-1 3.02E-1 14 9.38E-4 1.06E-2 7.82E-2 7.37E-2 1.17E-1 1.88E-1 2.75E-1 2.89E-1 15 2.85E-4 5.49E-3 6.09E-2 6.00E-2 1.01E-1 1.71E-1 1.98E-1 2.76E-1 16 6.47E-5 2.49E-3 4.60E-2 4.80E-2 8.60E-2 1.55E-1 1.82E-1 2.46E-1 2.64E-1 17 1.03E-5 9.49E-4 3.36E-2 3.76E-2 7.26E-2 1.42E-1 1.68E-1 2.32E-1 2.52E-1 18 1.09E-6 2.88E-4 2.38E-2 2.87E-2 6.06E-2 1.30E-1 1.56E-1 2.20E-1 2.42E-1 19 6.00E-8 6.65E-5 1.62E-2 2.12E-2 5.01E-2 1.19E-1 1.46E-1 2.08E-1 2.32E-1 20 1.00E-8 1.04E-5 1.05E-2 1.50E-2 4.08E-2 1.10E-1 1.37E-1 1.98E-1 2.24E-1 21 1.08E-6 6.51E-3 1.02E-2 3.26E-2 1.02E-1 1.30E-1 1.89E-1 2.16E-1 22 6.50E-8 3.76E-3 6.44E-3 2.55E-2 9.49E-2 1.24E-1 1.81E-1 2.10E-1 23 1.99E-3 1.94E-2 8.91E-2 1.20E-1 1.75E-1 2.05E-1 24 9.44E-4 2.00E-3 1.42E-2 8.43E-2 1.17E-1 1.69E-1 2.00E-1 25 3.86E-4 9.33E-4 9.94E-3 8.02E-2 1.14E-1 1.65E-1 1.97E-1 26 1.31E-4 6.50E-3 7.69E-2 1.12E-1 1.61E-1 1.94E-1 27 3.51E-5 1.18E-4 3.91E-3 7.41E-2 1.58E-1 1.91E-1 28 7.10E-6 2.90E-5 2.10E-3 7.18E-2 1.10E-1 1.56E-1 29 8.50E-7 5.02E-6 7.00E-2 1.10E-1 1.54E-1 1.87E-1 30 3.33E-8 6.17E-7 3.95E-4 6.85E-2 1.10E-1 1.85E-1 31 1.28E-4 6.74E-2 1.09E-1 1.51E-1 1.83E-1 3.18E-5 6.66E-2 1.50E-1 1.82E-1 33 5.71E-6 6.61E-2 1.09E-1 1.49E-1 1.81E-1 34 6.58E-2 1.09E-1 1.49E-1 1.80E-1 35 3.50E-8 6.57E-2 1.09E-1 1.48E-1 1.79E-1 36 5.00E-9 6.57E-2 1.09E-1 1.48E-1 1.78E-1

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96 Table 29. Raw Data for Figure 66 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 2.93E-1 3.49E-1 4.03E-1 3.97E-1 4.26E-1 4.44E-1 4.51E-1 4.60E-1 4.64E-1 -9 2.71E-1 3.32E-1 3.84E-1 3.90E-1 4.16E-1 4.37E-1 4.45E-1 4.55E-1 4.59E-1 -8 2.48E-1 3.14E-1 3.76E-1 3.70E-1 4.05E-1 4.28E-1 4.37E-1 4.49E-1 4.53E-1 -7 2.23E-1 2.93E-1 3.61E-1 3.54E-1 3.91E-1 4.18E-1 4.28E-1 4.42E-1 4.46E-1 -6 1.97E-1 2.71E-1 3.44E-1 3.37E-1 3.77E-1 4.08E-1 4.19E-1 4.34E-1 4.39E-1 -5 1.71E-1 2.76E-1 3.40E-1 7.43E-2 2.38E-1 1.99E-2 2.22E-6 2.25E-1 1.27E-2 1.90E-1 31 1.41E-1 2.48E-1 3.26E-1 3.18E-1 3.60E-1 3.96E-1 4.08E-1 4.26E-1 4.31E-1 -4 1.45E-1 2.23E-1 3.07E-1 2.97E-1 3.41E-1 3.82E-1 3.95E-1 4.16E-1 4.22E-1 -3 1.19E-1 1.98E-1 2.86E-1 2.75E-1 3.21E-1 3.68E-1 3.82E-1 4.06E-1 4.12E-1 -2 9.53E-2 1.72E-1 2.65E-1 2.53E-1 2.99E-1 3.52E-1 3.68E-1 3.95E-1 4.01E-1 -1 7.35E-2 1.46E-1 2.43E-1 2.29E-1 3.36E-1 3.52E-1 3.83E-1 3.90E-1 0 5.45E-2 1.20E-1 2.21E-1 2.06E-1 2.53E-1 3.18E-1 3.36E-1 3.71E-1 3.78E-1 1 3.86E-2 9.66E-2 1.99E-1 1.83E-1 2.30E-1 2.99E-1 3.19E-1 3.58E-1 3.66E-1 2 2.59E-2 7.50E-2 1.77E-1 1.61E-1 2.08E-1 2.80E-1 3.02E-1 3.44E-1 3.53E-1 3 1.62E-2 5.60E-2 1.56E-1 1.41E-1 1.88E-1 2.61E-1 2.85E-1 3.31E-1 4 9.30E-3 4.01E-2 1.36E-1 1.22E-1 1.68E-1 2.42E-1 2.67E-1 3.17E-1 3.27E-1 5 4.77E-3 2.73E-2 1.16E-1 1.05E-1 1.51E-1 2.24E-1 2.50E-1 3.03E-1 3.15E-1 6 2.13E-3 1.75E-2 9.76E-2 8.95E-2 1.35E-1 2.07E-1 2.33E-1 2.90E-1 3.03E-1 7 7.87E-4 1.04E-2 8.08E-2 7.62E-2 1.20E-1 1.91E-1 2.18E-1 2.78E-1 2.91E-1 8 2.33E-4 5.68E-3 6.59E-2 6.44E-2 1.06E-1 1.77E-1 2.04E-1 2.66E-1 2.81E-1 9 5.03E-5 2.75E-3 5.28E-2 5.40E-2 9.45E-2 1.64E-1 1.92E-1 2.55E-1 2.72E-1 10 7.66E-6 1.15E-3 4.16E-2 4.48E-2 8.38E-2 1.54E-1 1.81E-1 2.45E-1 2.63E-1 11 7.70E-7 3.99E-4 3.21E-2 3.66E-2 1.45E-1 1.71E-1 2.36E-1 2.56E-1 12 1.11E-4 2.44E-2 2.93E-2 6.59E-2 1.37E-1 1.64E-1 2.28E-1 2.50E-1 13 2.24E-5 1.81E-2 2.28E-2 5.84E-2 1.30E-1 1.58E-1 2.21E-1 2.45E-1 14 3.30E-6 1.31E-2 1.71E-2 5.19E-2 1.25E-1 1.53E-1 2.15E-1 2.41E-1 15 2.75E-7 9.12E-3 1.24E-2 4.61E-2 1.21E-1 1.49E-1 2.10E-1 16 1.00E-8 6.04E-3 8.57E-3 4.10E-2 1.18E-1 1.46E-1 2.06E-1 2.35E-1 17 5.00E-9 3.74E-3 5.60E-3 3.66E-2 1.15E-1 1.44E-1 2.03E-1 2.33E-1 18 2.12E-3 3.44E-3 3.28E-2 1.13E-1 1.43E-1 2.00E-1 2.31E-1 19 1.07E-3 1.96E-3 2.95E-2 1.11E-1 1.42E-1 1.98E-1 2.30E-1 20 4.66E-4 1.01E-3 2.66E-2 1.10E-1 1.41E-1 1.96E-1 2.29E-1 21 1.68E-4 4.62E-4 2.41E-2 1.08E-1 1.41E-1 1.95E-1 2.28E-1 22 4.87E-5 1.79E-4 2.19E-2 1.08E-1 1.41E-1 1.93E-1 2.27E-1 23 1.02E-5 5.65E-5 1.07E-1 1.40E-1 1.92E-1 2.27E-1 24 1.60E-6 1.32E-5 1.82E-2 1.06E-1 1.40E-1 1.92E-1 2.26E-1 25 1.03E-7 1.66E-2 1.06E-1 1.40E-1 1.91E-1 2.26E-1 26 3.33E-9 2.37E-7 1.52E-2 1.05E-1 1.40E-1 1.91E-1 2.25E-1 27 1.00E-8 1.39E-2 1.05E-1 1.40E-1 1.90E-1 28 1.05E-1 1.41E-1 1.90E-1 2.25E-1 29 1.15E-2 1.04E-1 1.41E-1 2.25E-1 30 1.04E-2 1.04E-1 1.41E-1 1.89E-1 2.25E-1 9.25E-3 1.04E-1 1.89E-1 2.25E-1

PAGE 112

97 Table 29. Continued. SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM 32 8.13E-3 1.04E-1 1.41E-1 2.24E-1 1.89E-1 33 6.01E-3 1.03E-1 7.04E-3 1.04E-1 1.41E-1 1.89E-1 2.24E-1 34 1.04E-1 1.41E-1 1.89E-1 2.24E-1 35 5.04E-3 1.41E-1 1.89E-1 2.24E-1 36 4.16E-3 1.03E-1 1.41E-1 1.89E-1 2.24E-1 37 3.38E-3 1.03E-1 1.41E-1 1.89E-1 2.24E-1 Table 30. Raw Data for Figure 67 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 2.32E-1 3.02E-1 3.67E-1 3.61E-1 3.96E-1 4.21E-1 4.31E-1 4.44E-1 4.49E-1 -9 2.06E-1 2.80E-1 3.51E-1 3.44E-1 3.81E-1 4.11E-1 4.22E-1 4.37E-1 4.41E-1 -6 2.59E-1 2.99E-1 6 2.16E-4 -8 1.80E-1 2.57E-1 3.33E-1 3.25E-1 3.65E-1 3.99E-1 4.11E-1 4.28E-1 4.33E-1 -7 1.53E-1 2.33E-1 3.14E-1 3.05E-1 3.46E-1 3.86E-1 3.99E-1 4.19E-1 4.24E-1 1.26E-1 2.07E-1 2.94E-1 2.83E-1 3.26E-1 3.71E-1 3.85E-1 4.08E-1 4.14E-1 -5 1.01E-1 1.80E-1 2.72E-1 2.60E-1 3.04E-1 3.55E-1 3.71E-1 3.97E-1 4.03E-1 -4 7.71E-2 1.53E-1 2.50E-1 2.36E-1 2.80E-1 3.38E-1 3.55E-1 3.85E-1 3.92E-1 -3 5.64E-2 1.27E-1 2.28E-1 2.12E-1 2.57E-1 3.20E-1 3.38E-1 3.72E-1 3.79E-1 -2 3.91E-2 1.01E-1 2.05E-1 1.88E-1 2.33E-1 3.00E-1 3.21E-1 3.59E-1 3.66E-1 -1 2.54E-2 7.80E-2 1.83E-1 1.64E-1 2.10E-1 2.80E-1 3.02E-1 3.44E-1 3.53E-1 0 1.52E-2 5.73E-2 1.60E-1 1.42E-1 1.88E-1 2.83E-1 3.29E-1 3.39E-1 1 8.33E-3 3.99E-2 1.37E-1 1.21E-1 1.68E-1 2.39E-1 2.64E-1 3.14E-1 3.25E-1 2 4.05E-3 2.61E-2 1.15E-1 1.02E-1 1.48E-1 2.19E-1 2.45E-1 3.11E-1 3 1.70E-3 1.59E-2 9.40E-2 8.42E-2 1.30E-1 2.00E-1 2.26E-1 2.84E-1 2.97E-1 4 5.91E-4 8.87E-3 7.44E-2 6.87E-2 1.13E-1 1.82E-1 2.08E-1 2.69E-1 2.83E-1 5 1.61E-4 4.45E-3 5.72E-2 5.51E-2 9.68E-2 1.66E-1 1.92E-1 2.54E-1 2.70E-1 3.21E-5 1.95E-3 4.24E-2 4.32E-2 8.22E-2 1.51E-1 1.77E-1 2.40E-1 2.58E-1 7 4.12E-6 7.24E-4 3.04E-2 3.30E-2 6.90E-2 1.38E-1 1.63E-1 2.26E-1 2.46E-1 8 2.80E-7 2.10E-2 2.45E-2 5.74E-2 1.26E-1 1.52E-1 2.14E-1 2.35E-1 9 1.00E-8 4.96E-5 1.39E-2 1.75E-2 4.73E-2 1.15E-1 1.41E-1 2.03E-1 2.26E-1 10 8.10E-6 8.85E-3 1.19E-2 3.87E-2 1.06E-1 1.33E-1 1.92E-1 2.17E-1 11 8.25E-7 5.33E-3 7.61E-3 3.13E-2 9.83E-2 1.26E-1 1.84E-1 2.09E-1 12 3.00E-8 3.00E-3 4.53E-3 2.52E-2 9.15E-2 1.20E-1 1.76E-1 2.02E-1 13 5.00E-9 1.56E-3 2.46E-3 2.02E-2 8.57E-2 1.15E-1 1.70E-1 1.97E-1 14 7.35E-4 1.20E-3 1.60E-2 8.07E-2 1.11E-1 1.64E-1 1.92E-1 15 3.05E-4 5.09E-4 1.27E-2 7.66E-2 1.08E-1 1.60E-1 1.88E-1 16 1.09E-4 1.82E-4 1.00E-2 7.32E-2 1.05E-1 1.56E-1 1.85E-1 17 3.40E-5 5.38E-5 7.87E-3 7.04E-2 1.03E-1 1.53E-1 1.82E-1 18 8.54E-6 1.24E-5 6.18E-3 6.81E-2 1.01E-1 1.51E-1 1.80E-1 19 1.75E-6 2.02E-6 4.83E-3 6.62E-2 9.97E-2 1.49E-1 1.78E-1 20 2.07E-7 2.57E-7 3.76E-3 6.46E-2 9.85E-2 1.47E-1 1.77E-1 21 1.33E-8 3.33E-9 2.90E-3 6.34E-2 9.76E-2 1.46E-1 1.76E-1

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98 Table 30. Continued. SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM 22 2.21E-3 6.23E-2 9.69E-2 1.45E-1 1.75E-1 23 3.51E-4 1.72E-1 4.90E-5 1.72E-1 3.45E-7 9.43E-2 1.65E-3 6.15E-2 9.63E-2 1.44E-1 1.74E-1 24 1.20E-3 6.08E-2 9.58E-2 1.44E-1 1.74E-1 25 8.45E-4 6.03E-2 9.54E-2 1.43E-1 1.73E-1 26 5.62E-4 5.99E-2 9.51E-2 1.43E-1 1.73E-1 27 5.95E-2 9.49E-2 1.43E-1 1.73E-1 28 2.02E-4 5.92E-2 9.47E-2 1.42E-1 1.72E-1 29 1.05E-4 5.90E-2 9.46E-2 1.42E-1 30 5.89E-2 9.45E-2 1.42E-1 31 1.96E-5 5.88E-2 9.44E-2 1.42E-1 1.72E-1 32 6.88E-6 5.87E-2 9.44E-2 1.42E-1 1.72E-1 33 1.83E-6 5.87E-2 9.43E-2 1.42E-1 1.72E-1 34 5.87E-2 9.43E-2 1.42E-1 1.72E-1 35 4.75E-8 5.87E-2 9.43E-2 1.42E-1 1.72E-1 36 2.50E-9 5.87E-2 1.42E-1 1.72E-1 37 5.87E-2 9.43E-2 1.41E-1 1.71E-1 Table 31. Raw Data for Figure 68 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 4.01E-1 4.29E-1 4.54E-1 4.57E-1 4.68E-1 4.76E-1 4.79E-1 4.83E-1 4.84E-1 -9 3.89E-1 4.21E-1 4.48E-1 4.52E-1 4.64E-1 4.73E-1 4.76E-1 4.81E-1 4.82E-1 -8 3.76E-1 4.11E-1 4.42E-1 4.45E-1 4.59E-1 4.69E-1 4.73E-1 4.78E-1 4.80E-1 -7 3.61E-1 4.01E-1 4.34E-1 4.38E-1 4.54E-1 4.65E-1 4.70E-1 4.75E-1 4.77E-1 -6 3.45E-1 3.89E-1 4.26E-1 4.30E-1 4.48E-1 4.61E-1 4.66E-1 4.72E-1 4.74E-1 -5 3.27E-1 3.76E-1 4.17E-1 4.21E-1 4.41E-1 4.55E-1 4.61E-1 4.68E-1 4.71E-1 -4 3.08E-1 3.61E-1 4.06E-1 4.11E-1 4.33E-1 4.49E-1 4.56E-1 4.64E-1 4.67E-1 -3 2.87E-1 3.45E-1 3.94E-1 3.87E-1 4.01E-1 4.15E-1 4.15E-1 4.00E-1 4.24E-1 4.42E-1 4.50E-1 4.59E-1 4.62E-1 -2 2.64E-1 3.28E-1 3.81E-1 4.13E-1 4.34E-1 4.43E-1 4.53E-1 4.57E-1 -1 2.39E-1 3.08E-1 3.66E-1 3.72E-1 4.25E-1 4.35E-1 4.47E-1 4.51E-1 0 2.14E-1 2.87E-1 3.49E-1 3.56E-1 3.87E-1 4.26E-1 4.40E-1 4.44E-1 1 1.87E-1 2.64E-1 3.31E-1 3.39E-1 3.71E-1 4.04E-1 4.31E-1 4.37E-1 2 1.59E-1 2.40E-1 3.11E-1 3.20E-1 3.53E-1 3.91E-1 4.03E-1 4.22E-1 4.28E-1 3 1.32E-1 2.14E-1 2.89E-1 2.99E-1 3.33E-1 3.77E-1 3.90E-1 4.12E-1 4.18E-1 4 1.05E-1 1.87E-1 2.66E-1 2.77E-1 3.11E-1 3.61E-1 3.76E-1 4.01E-1 4.07E-1 5 8.02E-2 1.59E-1 2.42E-1 2.55E-1 2.88E-1 3.44E-1 3.60E-1 3.89E-1 3.96E-1 6 5.80E-2 1.32E-1 2.17E-1 2.31E-1 2.63E-1 3.25E-1 3.43E-1 3.76E-1 3.83E-1 7 3.93E-2 1.05E-1 1.92E-1 2.07E-1 2.39E-1 3.05E-1 3.25E-1 3.62E-1 3.70E-1 8 2.46E-2 8.05E-2 1.67E-1 1.83E-1 2.14E-1 2.84E-1 3.06E-1 3.47E-1 3.56E-1 9 1.40E-2 5.82E-2 1.43E-1 1.59E-1 1.90E-1 2.61E-1 2.86E-1 3.31E-1 3.41E-1 10 7.05E-3 3.95E-2 1.20E-1 1.34E-1 1.67E-1 2.39E-1 2.65E-1 3.15E-1 3.26E-1 11 3.09E-3 2.47E-2 9.88E-2 1.09E-1 1.44E-1 2.16E-1 2.43E-1 2.97E-1 3.10E-1

PAGE 114

99 Table 31. Continued. SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM 12 1.13E-3 1.41E-2 7.95E-2 8.56E-2 1.23E-1 1.93E-1 2.21E-1 2.80E-1 2.93E-1 13 3.38E-4 7.10E-3 6.20E-2 6.36E-2 1.03E-1 1.72E-1 1.79E-1 2.23E-1 2.26E-1 7.40E-5 1.31E-2 1.60E-7 4.05E-7 2.85E-3 7.19E-3 5.16E-2 2.00E-1 2.61E-1 2.77E-1 14 7.70E-5 3.12E-3 4.65E-2 4.45E-2 8.33E-2 1.52E-1 2.42E-1 2.60E-1 15 1.19E-5 1.15E-3 3.33E-2 2.89E-2 6.52E-2 1.33E-1 1.60E-1 2.43E-1 16 1.36E-6 3.41E-4 2.25E-2 1.72E-2 4.91E-2 1.15E-1 1.41E-1 2.04E-1 17 6.00E-8 7.68E-5 1.41E-2 9.26E-3 3.52E-2 9.81E-2 1.24E-1 1.86E-1 2.09E-1 18 1.18E-5 8.20E-3 4.42E-3 2.39E-2 8.23E-2 1.09E-1 1.68E-1 1.92E-1 19 1.27E-6 4.32E-3 1.84E-3 1.53E-2 6.76E-2 9.43E-2 1.52E-1 1.76E-1 20 7.00E-8 2.02E-3 6.50E-4 9.07E-3 5.44E-2 8.08E-2 1.37E-1 1.61E-1 21 8.16E-4 1.89E-4 4.94E-3 4.27E-2 6.85E-2 1.23E-1 1.47E-1 22 2.74E-4 4.33E-5 2.42E-3 3.28E-2 5.73E-2 1.11E-1 1.34E-1 23 7.44E-6 1.04E-3 2.47E-2 4.73E-2 1.00E-1 1.23E-1 24 1.49E-5 7.73E-7 3.78E-4 1.82E-2 3.84E-2 9.09E-2 1.13E-1 25 2.02E-6 5.00E-8 1.11E-4 3.08E-2 8.25E-2 1.05E-1 26 2.51E-5 9.31E-3 2.42E-2 7.51E-2 9.85E-2 27 1.00E-8 3.87E-6 6.46E-3 1.87E-2 6.87E-2 9.30E-2 28 4.36E-3 1.41E-2 6.33E-2 8.85E-2 29 2.75E-8 1.02E-2 5.86E-2 8.48E-2 30 1.77E-3 5.48E-2 8.19E-2 31 1.02E-3 4.82E-3 7.95E-2 32 5.38E-4 3.05E-3 4.90E-2 7.76E-2 33 2.49E-4 1.78E-3 4.69E-2 7.62E-2 34 9.75E-5 9.48E-4 4.53E-2 7.51E-2 35 3.11E-5 4.46E-4 4.41E-2 7.43E-2 36 7.52E-6 1.78E-4 4.32E-2 7.38E-2 37 1.24E-6 5.78E-5 4.26E-2 7.35E-2 Table 32. Raw Data for Figure 69 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 4.01E-1 4.29E-1 4.54E-1 4.57E-1 4.68E-1 4.76E-1 4.79E-1 4.83E-1 4.84E-1 -9 3.89E-1 4.21E-1 4.48E-1 4.52E-1 4.64E-1 4.73E-1 4.76E-1 4.81E-1 4.82E-1 -8 3.76E-1 4.11E-1 4.42E-1 4.45E-1 4.59E-1 4.69E-1 4.73E-1 4.78E-1 4.80E-1 -7 3.61E-1 4.01E-1 4.34E-1 4.38E-1 4.54E-1 4.65E-1 4.70E-1 4.75E-1 4.77E-1 -6 3.45E-1 3.89E-1 4.26E-1 4.30E-1 4.48E-1 4.60E-1 4.66E-1 4.72E-1 4.74E-1 -5 3.27E-1 3.76E-1 4.17E-1 4.21E-1 4.41E-1 4.55E-1 4.68E-1 4.71E-1 -4 3.08E-1 3.61E-1 4.06E-1 4.11E-1 4.33E-1 4.49E-1 4.56E-1 4.64E-1 4.67E-1 -3 2.87E-1 3.45E-1 3.94E-1 4.00E-1 4.24E-1 4.42E-1 4.50E-1 4.59E-1 4.62E-1 -2 2.64E-1 3.28E-1 3.81E-1 3.87E-1 4.13E-1 4.34E-1 4.43E-1 4.53E-1 4.57E-1 -1 2.39E-1 3.08E-1 3.66E-1 3.72E-1 4.01E-1 4.25E-1 4.35E-1 4.47E-1 4.51E-1 0 2.13E-1 2.87E-1 3.49E-1 3.56E-1 3.87E-1 4.15E-1 4.25E-1 4.39E-1 4.44E-1 1 1.86E-1 2.64E-1 3.31E-1 3.39E-1 3.71E-1 4.03E-1 4.15E-1 4.31E-1 4.36E-1 4.61E-1

PAGE 115

100 Table 32. Continued. SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM 2 1.58E-1 2.39E-1 3.11E-1 3.19E-1 3.53E-1 3.91E-1 4.03E-1 4.22E-1 4.28E-1 3 1.30E-1 2.13E-1 2.89E-1 2.99E-1 3.33E-1 3.76E-1 3.90E-1 4.12E-1 4.18E-1 4 1.04E-1 1.86E-1 2.66E-1 2.77E-1 3.11E-1 3.60E-1 3.76E-1 4.01E-1 4.07E-1 5 7.85E-2 1.59E-1 2.41E-1 2.54E-1 2.87E-1 3.43E-1 3.60E-1 3.89E-1 3.95E-1 6 5.62E-2 1.31E-1 2.16E-1 2.30E-1 2.62E-1 3.24E-1 3.43E-1 3.75E-1 3.83E-1 7 3.74E-2 1.04E-1 1.90E-1 2.06E-1 2.37E-1 3.04E-1 3.24E-1 3.61E-1 3.69E-1 8 2.28E-2 7.88E-2 1.65E-1 1.81E-1 2.12E-1 2.82E-1 3.05E-1 3.46E-1 3.55E-1 9 1.24E-2 5.64E-2 1.41E-1 1.56E-1 1.88E-1 2.59E-1 2.84E-1 3.29E-1 3.40E-1 10 5.92E-3 3.76E-2 1.17E-1 1.31E-1 1.64E-1 2.36E-1 2.62E-1 3.12E-1 3.24E-1 11 2.37E-3 2.29E-2 9.59E-2 1.06E-1 1.41E-1 2.12E-1 2.40E-1 2.95E-1 3.07E-1 12 7.64E-4 1.25E-2 7.61E-2 8.14E-2 1.19E-1 1.89E-1 2.17E-1 2.76E-1 2.90E-1 13 1.89E-4 5.97E-3 5.82E-2 5.89E-2 9.81E-2 1.67E-1 1.95E-1 2.57E-1 2.73E-1 14 3.34E-5 2.40E-3 4.24E-2 3.95E-2 7.77E-2 1.46E-1 1.73E-1 2.36E-1 2.55E-1 15 3.61E-6 7.77E-4 2.89E-2 2.41E-2 5.89E-2 1.26E-1 1.52E-1 2.16E-1 2.37E-1 16 2.10E-7 1.91E-4 1.82E-2 1.30E-2 4.21E-2 1.06E-1 1.33E-1 1.95E-1 2.18E-1 17 1.00E-8 3.30E-5 1.04E-2 6.08E-3 2.81E-2 8.77E-2 1.14E-1 1.74E-1 1.98E-1 18 3.75E-6 5.18E-3 2.36E-3 1.71E-2 7.00E-2 9.69E-2 1.55E-1 1.79E-1 19 2.15E-7 2.22E-3 7.25E-4 9.33E-3 5.34E-2 8.01E-2 1.36E-1 1.60E-1 20 1.00E-8 7.72E-4 1.65E-4 4.44E-3 3.85E-2 6.41E-2 1.19E-1 1.42E-1 21 2.09E-4 2.61E-5 1.78E-3 2.60E-2 4.91E-2 1.02E-1 1.24E-1 22 4.15E-5 2.79E-6 5.73E-4 1.61E-2 3.56E-2 8.64E-2 1.08E-1 23 5.57E-6 1.27E-7 1.41E-4 8.93E-3 2.42E-2 7.14E-2 9.32E-2 24 4.53E-7 1.00E-8 2.42E-5 4.35E-3 1.51E-2 5.72E-2 7.89E-2 25 6.67E-9 2.66E-6 1.79E-3 8.45E-3 4.38E-2 6.53E-2 26 1.63E-7 5.98E-4 4.16E-3 3.19E-2 5.24E-2 27 7.50E-9 1.53E-4 1.74E-3 2.16E-2 4.03E-2 28 2.80E-5 5.94E-4 1.35E-2 2.94E-2 29 3.35E-6 1.56E-4 7.60E-3 2.00E-2 30 2.10E-7 2.93E-5 3.76E-3 1.26E-2 31 8.00E-9 3.76E-6 1.58E-3 7.11E-3 32 2.45E-7 5.40E-4 3.54E-3 33 8.33E-9 1.43E-4 1.50E-3 34 2.70E-5 5.19E-4 35 3.51E-6 1.40E-4 36 2.31E-7 2.70E-5 37 7.14E-9 3.55E-6

PAGE 116

101 Table 33. Raw Data for Figure 70 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 3.92E-1 4.23E-1 4.50E-1 4.53E-1 4.65E-1 4.74E-1 4.77E-1 4.81E-1 4.83E-1 -9 3.79E-1 4.14E-1 4.44E-1 4.40E-1 4.49E-1 1.65E-1 8.44E-2 2.36E-1 3.09E-1 8.47E-2 1.93E-1 7.20E-3 3.17E-1 1.12E-1 11 7.27E-3 9.22E-2 4.60E-5 4.47E-1 4.60E-1 4.70E-1 4.74E-1 4.79E-1 4.81E-1 -8 3.65E-1 4.04E-1 4.36E-1 4.55E-1 4.66E-1 4.71E-1 4.76E-1 4.78E-1 -7 3.49E-1 3.92E-1 4.28E-1 4.32E-1 4.62E-1 4.67E-1 4.73E-1 4.75E-1 -6 3.32E-1 3.79E-1 4.19E-1 4.24E-1 4.42E-1 4.57E-1 4.62E-1 4.69E-1 4.72E-1 -5 3.13E-1 3.65E-1 4.09E-1 4.14E-1 4.35E-1 4.51E-1 4.57E-1 4.65E-1 4.68E-1 -4 2.92E-1 3.49E-1 3.97E-1 4.02E-1 4.26E-1 4.44E-1 4.51E-1 4.60E-1 4.64E-1 -3 2.69E-1 3.32E-1 3.84E-1 3.90E-1 4.16E-1 4.36E-1 4.44E-1 4.55E-1 4.58E-1 -2 2.45E-1 3.13E-1 3.69E-1 3.76E-1 4.04E-1 4.27E-1 4.37E-1 4.48E-1 4.53E-1 -1 2.20E-1 2.92E-1 3.53E-1 3.60E-1 3.91E-1 4.18E-1 4.28E-1 4.41E-1 4.46E-1 0 1.93E-1 2.70E-1 3.35E-1 3.43E-1 3.75E-1 4.06E-1 4.18E-1 4.33E-1 4.38E-1 1 2.45E-1 3.16E-1 3.24E-1 3.58E-1 3.94E-1 4.06E-1 4.24E-1 4.30E-1 2 1.37E-1 2.20E-1 2.94E-1 3.04E-1 3.38E-1 3.80E-1 3.93E-1 4.14E-1 4.20E-1 3 1.10E-1 1.93E-1 2.71E-1 2.82E-1 3.16E-1 3.64E-1 3.79E-1 4.04E-1 4.10E-1 4 1.65E-1 2.47E-1 2.59E-1 2.93E-1 3.48E-1 3.64E-1 3.92E-1 3.98E-1 5 6.13E-2 1.37E-1 2.22E-1 2.68E-1 3.29E-1 3.47E-1 3.79E-1 3.86E-1 6 4.16E-2 1.10E-1 1.96E-1 2.12E-1 2.43E-1 3.29E-1 3.65E-1 3.73E-1 7 2.60E-2 1.71E-1 1.87E-1 2.18E-1 2.87E-1 3.10E-1 3.49E-1 3.58E-1 8 1.46E-2 6.15E-2 1.46E-1 1.62E-1 2.65E-1 2.89E-1 3.33E-1 3.43E-1 9 4.18E-2 1.23E-1 1.37E-1 1.70E-1 2.41E-1 2.68E-1 3.28E-1 10 3.02E-3 2.61E-2 1.01E-1 1.47E-1 2.18E-1 2.45E-1 2.99E-1 3.11E-1 1.03E-3 1.47E-2 8.08E-2 8.72E-2 1.25E-1 1.95E-1 2.23E-1 2.81E-1 2.95E-1 12 2.72E-4 6.24E-2 6.41E-2 1.03E-1 1.72E-1 2.00E-1 2.61E-1 2.77E-1 13 5.17E-5 3.05E-3 4.60E-2 4.39E-2 8.26E-2 1.51E-1 1.78E-1 2.41E-1 2.59E-1 14 6.66E-6 1.05E-3 3.20E-2 2.74E-2 6.33E-2 1.30E-1 1.57E-1 2.21E-1 2.41E-1 15 5.10E-7 2.77E-4 2.06E-2 1.53E-2 4.60E-2 1.11E-1 1.38E-1 2.00E-1 2.22E-1 16 2.00E-8 5.33E-5 1.20E-2 7.44E-3 3.12E-2 1.19E-1 1.79E-1 2.03E-1 17 6.82E-6 6.23E-3 3.03E-3 1.95E-2 7.42E-2 1.01E-1 1.59E-1 1.83E-1 18 4.10E-7 2.77E-3 9.90E-4 1.09E-2 5.73E-2 8.41E-2 1.41E-1 1.64E-1 19 1.00E-8 1.02E-3 2.46E-4 5.40E-3 4.20E-2 6.79E-2 1.23E-1 1.46E-1 20 2.98E-4 4.35E-5 2.27E-3 2.88E-2 5.26E-2 1.06E-1 1.28E-1 21 6.43E-5 4.84E-6 7.75E-4 2.00E-2 3.87E-2 9.01E-2 1.10E-1 22 9.73E-6 2.60E-7 2.04E-4 1.04E-2 2.67E-2 7.50E-2 9.68E-2 23 8.50E-7 1.00E-8 3.91E-5 5.26E-3 1.70E-2 6.06E-2 8.24E-2 24 3.67E-8 4.92E-6 2.26E-3 9.84E-3 4.70E-2 6.86E-2 25 3.60E-7 8.00E-4 5.02E-3 3.46E-2 5.55E-2 26 1.00E-8 2.21E-4 2.19E-3 2.39E-2 4.31E-2 27 4.44E-5 7.89E-4 1.53E-2 3.19E-2 28 6.00E-6 2.23E-4 8.85E-3 2.21E-2 29 5.00E-7 4.52E-3 1.42E-2 30 2.80E-8 6.57E-6 1.98E-3 8.26E-3 31 5.73E-7 7.17E-4 4.25E-3

PAGE 117

102 Table 33. Continued. SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM 32 3.00E-8 2.04E-4 1.88E-3 33 4.24E-5 6.88E-4 34 6.09E-6 1.98E-4 35 5.20E-7 4.19E-5 36 2.71E-8 6.16E-6 37 5.55E-7 Table 34. Raw Data for Figure 71 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 2.74E-1 3.36E-1 3.87E-1 3.93E-1 4.18E-1 4.38E-1 4.46E-1 4.56E-1 4.60E-1 -8 2.25E-1 2.97E-1 3.57E-1 2.14E-1 2.10E-4 2.58E-5 1.45E-2 1.01E-1 3.64E-1 3.94E-1 4.20E-1 4.30E-1 4.43E-1 4.48E-1 -6 1.72E-1 2.51E-1 3.20E-1 3.29E-1 3.62E-1 3.97E-1 4.09E-1 4.27E-1 4.32E-1 -4 1.18E-1 2.00E-1 2.77E-1 2.88E-1 3.22E-1 3.68E-1 3.83E-1 4.06E-1 4.13E-1 -2 6.88E-2 1.45E-1 2.29E-1 2.43E-1 2.76E-1 3.35E-1 3.52E-1 3.83E-1 3.90E-1 0 3.20E-2 9.31E-2 1.80E-1 1.96E-1 2.27E-1 2.95E-1 3.16E-1 3.55E-1 3.63E-1 2 1.08E-2 4.97E-2 1.33E-1 1.48E-1 1.80E-1 2.52E-1 2.77E-1 3.24E-1 3.35E-1 4 2.25E-3 2.05E-2 9.20E-2 1.01E-1 1.37E-1 2.08E-1 2.36E-1 2.92E-1 3.04E-1 6 2.25E-4 5.86E-3 5.85E-2 5.91E-2 9.84E-2 1.68E-1 1.95E-1 2.57E-1 2.74E-1 8 7.10E-6 1.00E-3 3.29E-2 2.83E-2 6.49E-2 1.33E-1 1.59E-1 2.23E-1 2.43E-1 10 8.31E-5 1.56E-2 1.04E-2 3.85E-2 1.03E-1 1.29E-1 1.92E-1 12 1.65E-6 5.79E-3 2.69E-3 2.04E-2 7.86E-2 1.05E-1 1.64E-1 1.88E-1 14 1.48E-3 4.22E-4 9.44E-3 5.91E-2 8.61E-2 1.43E-1 1.67E-1 16 3.05E-5 3.72E-3 4.44E-2 7.15E-2 1.27E-1 1.50E-1 18 1.29E-5 4.67E-7 1.15E-3 3.37E-2 6.06E-2 1.16E-1 1.39E-1 20 1.00E-7 2.42E-4 2.61E-2 5.27E-2 1.10E-1 1.31E-1 22 2.08E-2 4.71E-2 1.05E-1 1.26E-1 24 8.75E-7 1.70E-2 4.34E-2 1.03E-1 1.23E-1 26 4.10E-2 1.01E-1 1.21E-1 28 1.29E-2 3.94E-2 1.20E-1 30 1.20E-2 3.85E-2 1.00E-1 1.19E-1 32 1.16E-2 3.80E-2 9.97E-2 1.18E-1 34 1.14E-2 3.77E-2 9.95E-2 1.18E-1 36 1.14E-2 3.76E-2 9.94E-2 1.18E-1 Table 35. Raw Data for Figure 72 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 2.63E-1 3.27E-1 3.80E-1 3.86E-1 4.13E-1 4.34E-1 4.42E-1 4.53E-1 4.57E-1 -8 2.12E-1 2.86E-1 3.48E-1 3.56E-1 3.52E-1 3.60E-1 3.86E-1 4.15E-1 4.25E-1 4.39E-1 4.44E-1 -6 1.57E-1 2.38E-1 3.10E-1 3.19E-1 3.90E-1 4.03E-1 4.22E-1 4.27E-1 -4 1.03E-1 1.85E-1 2.65E-1 2.76E-1 3.10E-1 3.75E-1 4.00E-1 4.07E-1

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103 Table 35. Continued. SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -2 5.55E-2 1.30E-1 2.15E-1 2.29E-1 2.62E-1 3.24E-1 3.42E-1 3.75E-1 3.82E-1 0 2.24E-2 7.80E-2 1.64E-1 1.81E-1 2.11E-1 2.81E-1 3.04E-1 3.45E-1 3.54E-1 2 5.76E-3 3.71E-2 1.17E-1 1.30E-1 1.63E-1 2.35E-1 2.62E-1 3.12E-1 3.23E-1 4 7.42E-4 1.24E-2 7.58E-2 8.10E-2 1.19E-1 1.89E-1 2.17E-1 2.76E-1 2.90E-1 6 3.42E-5 2.38E-3 4.23E-2 3.93E-2 7.76E-2 1.46E-1 1.73E-1 2.36E-1 2.55E-1 8 2.00E-7 1.96E-4 1.82E-2 1.31E-2 4.22E-2 1.06E-1 1.33E-1 1.95E-1 2.18E-1 10 5.36E-5 7.44E-5 2.10E-3 3.80E-6 5.30E-3 2.44E-3 1.74E-2 7.04E-2 9.74E-2 1.55E-1 1.79E-1 12 8.36E-4 1.83E-4 4.69E-3 3.95E-2 6.52E-2 1.20E-1 1.43E-1 14 3.50E-6 6.71E-4 1.71E-2 3.73E-2 8.83E-2 1.10E-1 16 8.67E-7 3.73E-5 5.13E-3 1.69E-2 6.02E-2 8.21E-2 18 4.25E-7 8.99E-4 5.46E-3 3.60E-2 5.72E-2 20 1.10E-3 1.79E-2 3.57E-2 22 1.80E-6 1.14E-4 7.08E-3 1.93E-2 24 3.37E-6 8.71E-3 26 4.42E-4 3.09E-3 28 5.79E-5 7.72E-4 30 3.33E-6 1.15E-4 32 2.86E-8 7.81E-6 34 1.63E-7 Table 36. Raw Data for Figure 73 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 2.56E-1 3.21E-1 3.76E-1 3.82E-1 4.09E-1 4.31E-1 4.40E-1 4.51E-1 4.55E-1 -8 2.05E-1 2.80E-1 3.43E-1 3.51E-1 3.82E-1 4.11E-1 4.22E-1 4.37E-1 4.42E-1 -6 1.49E-1 2.31E-1 3.04E-1 3.13E-1 3.47E-1 3.86E-1 3.99E-1 4.19E-1 4.25E-1 -4 9.55E-2 1.78E-1 2.58E-1 2.69E-1 3.04E-1 3.55E-1 3.71E-1 3.97E-1 4.04E-1 -2 4.98E-2 1.22E-1 2.08E-1 2.22E-1 2.54E-1 3.18E-1 3.37E-1 3.71E-1 3.79E-1 0 1.91E-2 7.14E-2 1.57E-1 1.73E-1 2.04E-1 2.28E-1 2.10E-1 2.30E-1 1.27E-1 1.49E-1 1.36E-1 18 2.85E-2 24 3.58E-4 2.75E-1 2.98E-1 3.41E-1 3.50E-1 2 4.56E-3 3.25E-2 1.11E-1 1.23E-1 1.57E-1 2.55E-1 3.07E-1 3.19E-1 4 5.16E-4 1.01E-2 7.04E-2 7.40E-2 1.13E-1 1.82E-1 2.70E-1 2.85E-1 6 2.00E-5 1.73E-3 3.80E-2 3.42E-2 7.17E-2 1.39E-1 1.66E-1 2.49E-1 8 1.16E-4 1.55E-2 1.04E-2 3.74E-2 1.00E-1 1.88E-1 2.11E-1 10 1.60E-6 4.06E-3 1.68E-3 1.43E-2 6.47E-2 9.16E-2 1.73E-1 12 5.28E-4 9.82E-5 3.41E-3 3.43E-2 5.93E-2 1.13E-1 14 2.23E-5 1.17E-6 3.85E-4 1.35E-2 3.18E-2 8.16E-2 1.03E-1 16 6.67E-8 1.34E-5 3.37E-3 1.27E-2 5.29E-2 7.46E-2 4.06E-4 3.24E-3 4.85E-2 20 1.56E-5 4.07E-4 1.14E-2 2.63E-2 22 6.00E-8 1.67E-5 2.93E-3 1.06E-2 6.67E-8 3.71E-4 2.76E-3 26 1.56E-5

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104 Table 37. Raw Data for Figure 74 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 2.28E-1 2.99E-1 3.58E-1 3.66E-1 3.94E-1 4.20E-1 4.30E-1 4.43E-1 4.48E-1 -8 1.74E-1 2.53E-1 3.22E-1 3.30E-1 3.62E-1 3.97E-1 4.09E-1 4.27E-1 4.32E-1 -6 1.19E-1 2.02E-1 2.79E-1 2.90E-1 3.22E-1 3.68E-1 3.83E-1 4.06E-1 4.12E-1 -4 6.89E-2 18 1.47E-1 2.31E-1 2.45E-1 2.75E-1 3.34E-1 3.51E-1 3.82E-1 3.89E-1 -2 3.14E-2 9.37E-2 1.80E-1 1.98E-1 2.26E-1 2.94E-1 3.15E-1 3.54E-1 3.62E-1 0 9.89E-3 4.92E-2 1.32E-1 1.49E-1 1.79E-1 2.50E-1 2.75E-1 3.22E-1 3.33E-1 2 1.80E-3 1.94E-2 9.02E-2 1.00E-1 1.36E-1 2.05E-1 2.32E-1 2.89E-1 3.01E-1 4 1.45E-4 5.03E-3 5.54E-2 5.61E-2 9.55E-2 1.64E-1 1.91E-1 2.53E-1 2.69E-1 6 3.19E-6 7.07E-4 2.88E-2 2.44E-2 6.01E-2 1.28E-1 1.53E-1 2.16E-1 2.37E-1 8 2.00E-8 4.04E-5 1.19E-2 7.48E-3 3.26E-2 9.56E-2 1.22E-1 1.82E-1 2.05E-1 10 6.60E-7 3.58E-3 1.48E-3 1.48E-2 6.80E-2 9.49E-2 1.52E-1 1.75E-1 12 7.04E-4 1.73E-4 5.58E-3 4.60E-2 7.24E-2 1.28E-1 1.50E-1 14 7.39E-5 1.10E-5 1.71E-3 3.02E-2 5.46E-2 1.09E-1 1.31E-1 16 2.83E-6 2.70E-7 4.21E-4 1.98E-2 4.13E-2 9.48E-2 1.17E-1 2.00E-8 7.82E-5 1.34E-2 3.21E-2 8.45E-2 1.06E-1 20 9.51E-6 9.54E-3 2.59E-2 7.73E-2 9.95E-2 22 5.88E-7 7.20E-3 2.18E-2 7.24E-2 9.50E-2 24 7.50E-9 5.77E-3 1.92E-2 6.92E-2 9.20E-2 26 4.88E-3 1.76E-2 6.71E-2 9.00E-2 28 4.31E-3 1.65E-2 6.58E-2 8.88E-2 30 3.96E-3 1.58E-2 6.49E-2 8.80E-2 32 3.73E-3 1.54E-2 6.44E-2 8.75E-2 34 3.59E-3 1.51E-2 6.40E-2 8.72E-2 36 3.48E-3 1.49E-2 6.38E-2 8.70E-2 Table 38. Raw Data for Figure 75 SNR (dB) BPSK QPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 2.63E-1 3.27E-1 3.80E-1 3.86E-1 4.13E-1 4.34E-1 4.42E-1 4.53E-1 4.57E-1 -8 2.12E-1 2.86E-1 3.48E-1 3.56E-1 3.86E-1 4.15E-1 4.25E-1 4.39E-1 4.44E-1 -6 1.57E-1 2.38E-1 3.10E-1 3.19E-1 3.52E-1 3.90E-1 3.60E-1 3.24E-1 2.81E-1 2.35E-1 1.89E-1 1.46E-1 1.06E-1 7.04E-2 3.95E-2 1.71E-2 5.13E-3 4.03E-1 4.22E-1 4.27E-1 -4 1.03E-1 1.85E-1 2.65E-1 2.76E-1 3.10E-1 3.75E-1 4.00E-1 4.07E-1 -2 5.55E-2 7.80E-2 1.17E-1 8.10E-2 7.76E-2 1.30E-1 2.15E-1 2.29E-1 2.62E-1 3.42E-1 3.75E-1 3.82E-1 0 2.24E-2 1.64E-1 1.81E-1 2.11E-1 3.04E-1 3.45E-1 3.54E-1 2 5.76E-3 3.71E-2 1.30E-1 1.63E-1 2.62E-1 3.12E-1 3.23E-1 4 7.42E-4 1.24E-2 7.58E-2 1.19E-1 2.17E-1 2.76E-1 2.90E-1 6 3.42E-5 2.38E-3 4.23E-2 3.93E-2 1.73E-1 2.36E-1 2.55E-1 8 2.00E-7 1.96E-4 1.82E-2 1.31E-2 4.22E-2 1.33E-1 1.95E-1 2.18E-1 10 3.80E-6 5.30E-3 2.44E-3 1.74E-2 9.74E-2 1.55E-1 1.79E-1 12 8.36E-4 1.83E-4 4.69E-3 6.52E-2 1.20E-1 1.43E-1 14 5.36E-5 3.50E-6 6.71E-4 3.73E-2 8.83E-2 1.10E-1 16 8.67E-7 3.73E-5 1.69E-2 6.02E-2 8.21E-2

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105 Table 38. Continued. SNR (dB) BPSK QPSK 8-PSK 16-QAM 32-QAM 128-QAM 64-QAM 256-QAM 18 4.25E-7 8.99E-4 5.46E-3 3.60E-2 5.72E-2 20 7.44E-5 1.10E-3 1.79E-2 3.57E-2 22 7.08E-3 30 1.80E-6 1.14E-4 1.93E-2 24 3.37E-6 2.10E-3 8.71E-3 26 4.42E-4 3.09E-3 28 5.79E-5 7.72E-4 3.33E-6 1.15E-4 32 2.86E-8 7.81E-6 34 1.63E-7 Table 39. Raw Data for Figure 76 SNR (dB) BPSK 8-PSK 8-QAM 16-QAM 32-QAM 64-QAM 128-QAM 256-QAM -10 2.56E-1 3.76E-1 3.82E-1 4.09E-1 4.31E-1 4.40E-1 4.51E-1 4.55E-1 -8 2.05E-1 3.43E-1 3.51E-1 3.82E-1 4.11E-1 4.22E-1 4.37E-1 4.42E-1 -6 1.49E-1 3.04E-1 3.13E-1 3.47E-1 3.86E-1 3.99E-1 4.19E-1 4.25E-1 -4 9.55E-2 2.58E-1 2.69E-1 3.04E-1 3.55E-1 3.71E-1 3.97E-1 4.04E-1 -2 4.98E-2 8-QAM QPSK 3.21E-1 2.80E-1 2.31E-1 1.78E-1 1.22E-1 2.08E-1 2.22E-1 1.73E-1 1.23E-1 7.40E-2 3.42E-2 7.17E-2 1.39E-1 1.66E-1 2.30E-1 2.49E-1 8 1.16E-4 1.55E-2 1.04E-2 3.74E-2 1.00E-1 1.27E-1 1.88E-1 2.11E-1 10 1.60E-6 4.06E-3 1.68E-3 1.43E-2 6.47E-2 9.16E-2 1.49E-1 1.73E-1 12 5.28E-4 9.82E-5 3.41E-3 3.43E-2 5.93E-2 1.13E-1 1.36E-1 14 2.23E-5 1.17E-6 3.85E-4 1.35E-2 3.18E-2 8.16E-2 1.03E-1 16 6.67E-8 1.34E-5 3.37E-3 1.27E-2 5.29E-2 7.46E-2 18 4.06E-4 3.24E-3 2.85E-2 4.85E-2 20 1.56E-5 4.07E-4 1.14E-2 2.63E-2 22 6.00E-8 1.67E-5 2.93E-3 1.06E-2 24 6.67E-8 3.71E-4 2.76E-3 26 1.56E-5 3.58E-4 28 8.57E-8 1.58E-5 30 8.75E-8 2.54E-1 3.18E-1 3.37E-1 3.71E-1 3.79E-1 0 1.91E-2 7.14E-2 1.57E-1 2.04E-1 2.75E-1 2.98E-1 3.41E-1 3.50E-1 2 4.56E-3 3.25E-2 1.11E-1 1.57E-1 2.28E-1 2.55E-1 3.07E-1 3.19E-1 4 5.16E-4 1.01E-2 7.04E-2 1.13E-1 1.82E-1 2.10E-1 2.70E-1 2.85E-1 6 2.00E-5 1.73E-3 3.80E-2

PAGE 121

APPENDIX C SPW BLOCK DIAGRAMS This appendix provides illustrations of the various block based diagrams developed during the development of this thesis. These figures are provided as an alternative to more typical pseudo code that is usually provided in this type of thesis. The reader should refer to Appendix A for an introduction regarding polymorphic modeling within SPW. The following figures are detailed diagrams of the transmitter and receiver models used in all of the relevant systems to characterize various performance metrics. The transmitter models are shown in Figure 79, Figure 80, Figure 81, Figure 82, Figure 83 and Figure 84. Figure 79 shows the top level hierarchal model of the OFDM channelizer transmitter. At a high-level, the transmitter consists of ten filters each driven by separate data input ports since each filter transmits data in parallel. The outputs of all ten filters are summed together at the output. The diagram contains three red boxes denoting the blown up representation of the transmitter contained in Figure 80, Figure 81 and Figure 82. Figure 83 and Figure 84 represent the first and second order models of the actual filters that make up the OFDM channelizer transmitter. The 1_tap and 2_tap denotation on the block symbol in Figure 79 denotes whether that particular filter is first-order or second-order, respectively. The parameter configuration contained within is representative of a nominal filter and these parameter values are different for each of the ten filters within the 48-point OFDM channelizer. 106

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107 Figure 79. Block Diagram of OFDM Channelizer Transmitter Figure 80. Internal Block Diagram of OFDM Channelizer Transmitter Top Red Box

PAGE 123

108 Figure 81. Internal Block Diagram of OFDM Channelizer Transmitter Middle Red Box Figure 82. Internal Block Diagram of OFDM Channelizer Transmitter Bottom Red Box

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109 As described above, Figure 83 and Figure 84 below include the detailed models of the first-order and second-order OFDM channelizer filters. They include the polynomial filter, rate conversion as well as the comb filter. Figure 83. Block Diagram of First-Order OFDM Channelizer Transmission Filter Figure 84. Block Diagram of Second-Order OFDM Channelizer Transmission Filter

PAGE 125

110 The receiver models are shown in Figure 85, Figure 86, Figure 87, Figure 88, Figure 89 and Figure 90. Figure 85 shows the top level hierarchal model of the OFDM channelizer receiver. Similar to the transmitter, the receiver consists of ten filters each driven by a common signal input port since each filter extracts data from separate subcarriers contained within the received signal. The outputs of all ten filters are output in parallel since the data sent across all ten filters is independent. The diagram contains three red boxes denoting the blown up representation of the receiver contained in Figure 86, Figure 87 and Figure 88. Figure 89 and Figure 90 represent the first and second order models of the actual filters that make up the OFDM channelizer receiver. The 1_tap and 2_tap denotation on the block symbol in Figure denotes whether that particular filter is first-order or second-order, respectively. The parameter configuration contained within is representative of a nominal filter and these parameter values are different for each of the ten filters within the 48-point OFDM channelizer. 85 As described above, Figure 89 and Figure 90 below include the detailed models of the first-order and second-order OFDM channelizer filters. They include the polynomial filter, rate conversion as well as the comb filter. Three main system diagrams are provided below that were created to perform certain simulation based analysis: an AWGN based communications system, a multipath based communications system with Option 1 and a multipath communications system with Option 2. Each of these communication system diagrams includes blown up diagrams for better visibility. The blown up regions of the diagrams are denoted in the

PAGE 126

111 main diagrams surrounded by red boxes similar to the transmitter and receiver models above. Figure 85. Block Diagram of OFDM Channelizer Receiver

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112 Figure 86. Internal Block Diagram of OFDM Channelizer Transmitter Top Red Box

PAGE 128

113 Figure 87. Internal Block Diagram of OFDM Channelizer Transmitter Middle Red Box

PAGE 129

114 Figure 88. Internal Block Diagram of OFDM Channelizer Transmitter Bottom Red Box

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115 Figure 89. Block Diagram of First-Order OFDM Channelizer Receive Filter Figure 90. Block Diagram of Second-Order OFDM Channelizer Transmission Filter

PAGE 131

116 Figure 91 represents a block diagram of the AWGN system. It consists of ten symbol modulators the provide input complex-valued symbols to each of the ten OFDM channelizer filters. To support specific analysis contained within this thesis, each of the ten filters can independently be provided with unique modulation schemes including a varying number of bits per symbol. The center of the diagram shows the OFDM channelizer transmitter, channel including AWGN and OFDM channelizer receiver. The outputs of each of the OFDM channelizer receive filters goes into a hard-decision symbol demapping block to detect and report the symbol estimated to be transmitted. The per-bit and per-symbol decisions are then input to error rate estimators which average the bit errors and symbol errors during the course of the simulation. Figure 92, Figure 93, Figure 94 and Figure 95 show blown up diagrams of the regions of Figure 91 surrounded by red boxes for additional clarity. Figure 91. Block Diagram of AWGN Communication System

PAGE 132

117 Figure 92. Blow Up Block Diagram of AWGN Communication System Top Left Red Box Figure 93. Blow Up Block Diagram of AWGN Communication System Middle Red Box

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118 Figure 94. Blow Up Block Diagram of AWGN Communication System Top Red Box, Second from Right Figure 95. Blow Up Block Diagram of AWGN Communication System Right Red Box Figure 101 illustrates the system used to capture the multipath performance of an OFDM channelizer using Option 1 described in Chapter 5. This system differs from the

PAGE 134

119 AWGN system by the inclusion of a multipath channel model as well the separation filters and multiple OFDM channelizer receive filters for filter 6. Additionally, the transmitter includes optional cyclic prefix extension insertion and associated removal at the receiver. The blown up portions denoted in red boxes are shown in Figure 97, Figure 98, Figure 99 and Figure 100 that follow. Note that the separation filters consist of a common tapped-delay line and separate pointwise multiplications with unique filter coefficient constants followed by summations per output sample. Figure 96. Block Diagram of Multipath Communication System Using Option 1 Figure 97. Blow Up Block Diagram of Multipath Communication System Left Red Box

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120 Figure 98. Blow Up Block Diagram of Multipath Communication System Red Box, Second from the Left Figure 99. Blow Up Block Diagram of Multipath Communication System Red Box, Second from the Right

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121 Figure 100. Blow Up Block Diagram of Multipath Communication System Right Red Box Figure 101 illustrates the system used to capture the multipath performance of an OFDM channelizer using Option 2 described in Chapter 5. This system differs from the AWGN system by the inclusion of a multipath channel model as well as an FFT-based transmitter. The FFT-based transmitter modulates random data for the subcarriers not associated with filter 6 used in the performance analysis. Figure 102, Figure 103, Figure 104, Figure 105 and Figure 106 show the blown up portions of the system figure for added clarity. Figure 101. Block Diagram of Multipath Communication System Using Option 2

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122 Figure 102. Blow Up Block Diagram of Multipath Communication System Top Left Red Box Figure 103. Blow Up Block Diagram of Multipath Communication System Red Box, Second from the Left

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123 Figure 104. Blow Up Block Diagram of Multipath Communication System Middle Red Box Figure 105. Blow Up Block Diagram of Multipath Communication System Red Box, Second from the Right

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124 Figure 106. Blow Up Block Diagram of Multipath Communication System Right Red Box

PAGE 140

APPENDIX D MATLAB AND MATHEMATICA ANALYSIS COMMANDS This appendix presents both the Matlab and Mathematica commands used for limited analysis throughout this thesis. Matlab was used to produce theoretical BPSK BER statistics for comparisons to the AWGN simulation results presented in Chapter 4. The following Matlab commands were used to generate the reference theoretical BPSK BER plots. ;12:23_=dbx );10/(.^10xx= ));((*5.0xsqrterfcy= The following mapping between the more traditional Q(x) function and Matlabs erfc() function is given in the following equation. ));2(/(*5.0)(sqrtxerfcxQ= (8) Mathematica was used to produce Table 2. The command for factoring the root polynomial of form z N -1 into ternary-valued coefficients is given below. Factor[z^N-1,Extension->{0,1,-1}] The command for determining the divisors of integers a given integer, N, is given below. Divisors[N] 125

PAGE 141

LIST OF REFERENCES [1] S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications, Prentice Hall, New Jersey, 1983. [2] S. B. Weinstein and P. M. Ebert, Data Transmission by Frequency-Division Multiplexing Using the Discrete Fourier Transform, IEEE Transactions on Communication Technology, Vol. 19, No. 5, pp. 628-634, Oct. 1971. [3] R. W. Chang and R. A. Gibbey, A Theoretical Study of Performance of an Orthogonal Multiplexing Data Transmission Scheme, IEEE Transactions on Communication Technology, Vol. 16, No. 4, pp. 529-540, Aug. 1968. [4] R. R. Mosier and R. G. Clabaugh, Kineplex, a Bandwidth Efficient Binary Transmission System, AEII Trans., Vol. 76, pp. 723-728, Jan. 1958. [5] G. C. Porter, Error Distribution and Diversity Performance of a Frequency Differential PSK HF modem, IEEE Trans. Comm., Vol. COM-16, pp. 567-575, Aug. 1968. [6] M. S. Zimmerman and A. L. Kirsch, The AN/GSC-10 (KATHRYN) variable rate data modem for HF radio, IEEE Transactions on Communications, Vol. COM-15, pp. 197-205, Apr. 1967. [7] Orthogonal Frequency Division Multiplexing, U.S. Patent No. 3,488,455, filed Nov. 14, 1966, issued Jan. 6, 1970. [8] R. Van Nee and R. Prasad, OFDM for Wireless Multimedia Communications, Artech House Publishers, Boston, 2000. [9] K. Feher, Wireless Digital Communications: Modulation and Spread Spectrum Applications, Prentice Hall Digital and Wireless Communication Series, New Jersey, 1995. [10] E. B. Hogenauer, An Economical Class of Digital Filters for Decimation and Interpolation, IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 29, No.2, pp.155-162, Apr. 1981. [11] A. Y. Kwentus, Z. Jiang and A. N. Willson, Application of Filter Sharpening to Cascaded Integrator-Comb Decimation Filters, IEEE Transactions on Signal Processing, Vol. 45, No. 2, pp. 457-467, Feb. 1997. 126

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127 [12] A. Santraine, S. Leprince and F. Taylor, Multiplier-Free Bandpass Channelizer for Undersampled Applications, IEEE Signal Processing Letters, Vol. 11, No. 11, pp. 904-907, Nov. 2004. [13] A. Santraine, S. Leprince and F. Taylor, Multiplier-Free Band-Selectable Digital Filters, IEEE International Conference on Acoustics, Speech and Signal Processing 2004 Proceedings, Vol. 2, pp. 17-21, May 2004. [14] J. G. Proakis, Digital Communications, McGraw-Hill Series in Electrical and Computer Engineering, New York, 1995. [15] F. Taylor and J. Mellot, Hands-On Digital Signal Processing, McGraw-Hill, New York, 1998.

PAGE 143

BIOGRAPHICAL SKETCH William Edward Lawton was born on March 23, 1975, in Tallahassee, Florida. He attended the University of Florida in Gainesville, Florida, and graduated with a Bachelor of Science in electrical and computer engineering in May 1998. He married his wife, Leigh Jerkins Lawton, on February 3, 2001. They relocated in the Fall of 2002 to Pennsylvania where his wife gave birth to their daughter, Rachael Elizabeth Lawton, on May 20, 2004. His son, William Evan Lawton, was born on September 23, 2005. 128


Permanent Link: http://ufdc.ufl.edu/UFE0013271/00001

Material Information

Title: Analysis of Ternary-Valued, CIC Filter-Based OFDM Channelizers in Modern Wireless Communications Systems
Physical Description: Mixed Material
Copyright Date: 2008

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Holding Location: University of Florida
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Permanent Link: http://ufdc.ufl.edu/UFE0013271/00001

Material Information

Title: Analysis of Ternary-Valued, CIC Filter-Based OFDM Channelizers in Modern Wireless Communications Systems
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013271:00001


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ANALYSIS OF TERNARY-VALUED, CIC FILTER-BASED, OFDM
CHANNELIZERS IN MODERN WIRELESS COMMUNICATIONS SYSTEMS















By

WILLIAM E. LAWTON


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2005





























Copyright 2005

by

William Edward Lawton


































To My Family















ACKNOWLEDGMENTS

I would like to thank Dr. Fred J. Taylor for his support and guidance during my

work towards this thesis. This thesis would not have been formed without his guidance.

I would also like to thank my advisors, Dr. John M. Shea and Dr. William R.

Eisenstadt, for their support and contributions during this thesis.

I would also like to thank my wife, Leigh, daughter, Rachael, and son, Will, for

supporting me through my thesis.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ...................................................................... ...................iv

L IST O F T A B L E S ...........................................................................vii

LIST O F FIG U RE S ......................................................................................... ... ........ix

A B STR A C T .................................................................................................... xiv

CHAPTER

1 IN TROD U CTION .............................................. ........... ........ .. .............. 1

A p p ro a c h ................................................................................ 3
A analysis T ools ...................................................................................................

2 OFDM BACKGROUND AND BENEFITS ................................. ..............6

OFDM Introduction .................................. ............................ ... .............
OFDM Background ................................... ............................9
OFDM Benefits and Drawbacks .................................................................... 10
M ultipath in W wireless Channels .................................................................... 10
Cyclic Prefix to Mitigate Effects of Multipath.................................... 12

3 OFDM CHANNELIZER OVERVIEW AND DESCRIPTION .............. ..............15

C IC F ilter O v erv iew ............................................................................................... 15
CIC Filter Optimization ..... ............ ......... ............... 23
OFDM Channelizer Introduction ................. ......... .......... 24
OFDM Channelizer Selection ....... .. ...................................................... ..........26
48-Subcarrier OFDM Channelizer Description ............. ................... .................27

4 OFDM CHANNELIZER PERFORMANCE IN AWGN CHANNEL .....................38

Performance of 48-Point OFDM Channelizer in AWGN Channel ...........................38
Analysis of Various Constellation Schemes Utilizing Filter 1 as Reference.............44
BER Normalization of Filter Banks through Constellation Density Compensation..45









5 OFDM CHANNELIZER PERFORMANCE AND LIMITATIONS IN
M U L TIPA TH CH A N N E L ........................................................... ....................48

OFDM Channelizer Subcarrier Separation ........... ........... ........ .......... ...........51
Approaches to Enhance the OFDM Channelizer for Multipath Channel
Conditions .......................................................................... ........ ......... ........ ........ 57
M ultipath Effects on BER Performance ................................... ......... .. .............. 62
Multipath Effects on OFDM Channelizer with Coherent Alignment ................ 65
Multipath Effects on OFDM Channelizer with Cyclic Prefix ......... .......... 65
Multipath Effects on OFDM Channelizer with Coherent Alignment and Cyclic
Prefix ............. ........................................... ................ .............. 69
Summary of Performance Comparison..... .................... ...............69

6 CONCLUSIONS AND FUTURE WORK .......................................................77

Summary of Simulation Effort.. ... ............................................................ 77
Lessons Learned and Future W ork................................................. .............. 78
Summary of Simulation Performance Results ................ ................................ 79

APPENDIX

A POLYMORPHIC-BASED SPW OVERVIEW.........................81

B SIMULATION RESULTS RAW DATA ............... ...... .............. ............ 84

C SPW B LO CK D IA G R A M S .......................... .................................................. 106

D MATLAB AND MATHEMATICS ANALYSIS COMMANDS ....................... 125

L IST O F R E FE R E N C E S ............. .......................................................................... 126

BIOGRAPHICAL SKETCH ............ ..... ............................. 128















LIST OF TABLES


Table page

1 Measured Delay Spreads in Various Wireless Channels .................................... 12

2 Explored Prim ary Polynom ials ........................................ ........................ 28

3 Transfer Functions for 48-Subcarrier OFDM Channelizer Filter Banks.................31

4 Performance Advantage of Filters Based on Number of Subcarriers ...................44

5 Performance Advantage of Modulation Schemes.................. ........................... 45

6 Per-Filter Modulation Scheme for 48-Subcarrier OFDM Channelizer...................46

7 Filter 6 Separation Filter Subcarrier +6 Coefficient Listing ...............................57

8 Filter 6 Separation Filter Subcarrier -6 Coefficient Listing ..................................58

9 Filter 6 Separation Filter Subcarrier +18 Coefficient Listing .............. ............. 58

10 Filter 6 Separation Filter Subcarrier -18 Coefficient Listing............................ 58

11 Rappaport Multipath Channel Tap Weights........................................... 63

12 Multipath Fading Performance Delta (@ BER = le-2) without Alignment
(AW GN reference) .................................... ...................................... 73

13 Multipath Fading Performance Delta (@ BER = le-2) with Option 1 (AWGN
reference) ..................................................... ................... ..... ........ 74

14 Multipath Fading Performance Delta (@ BER = le-2) with Option 2 (AWGN
reference) ..................................................... ................... ..... ........ 74

15 Multipath Fading Performance Delta (@ BER = 1e-5) without Alignment
(AW GN reference) .................................... ...................................... 75

16 Multipath Fading Performance Delta (@ BER = le-5) with Option 1 (AWGN
reference) ..................................................... ................... ..... ........ 75

17 Multipath Fading Performance Delta (@ BER = le-5) with Option 2 (AWGN
referen ce) ...................................... ............................... ............... 7 6










18 Raw D ata for Figure 38 ...................................................................... 84

19 R aw D ata for Figure 39 ............................................... ............................. 85

20 Raw Data for Figure 40 ...................................................................... 86

21 Raw D ata for Figure 41 ...................................................................... 87

22 R aw D ata for Figure 42 .............................................. .............................. 88

23 Raw Data for Figure 43 ...................................................................... 89

24 Raw D ata for Figure 44 ...................................................................... 90

25 R aw D ata for Figure 45 ............................................... ............................. 91

26 Raw D ata for Figure 46 ...................................................................... 93

27 Raw D ata for Figure 48 ...................................................................... 94

28 Raw Data for Figure 65 ...................................................................... 94

29 Raw D ata for Figure 66 ...................................................................... 95

30 Raw D ata for Figure 67 ...................................................................... 97

31 Raw D ata for Figure 68 ...................................................................... 98

32 Raw D ata for Figure 69 ...................................................................... 99

33 Raw D ata for Figure 70 ..................................................................... 101

34 Raw D ata for Figure 71 ..................................................................... 102

35 Raw D ata for Figure 72 ..................................................................... 102

36 Raw D ata for Figure 73 ..................................................................... 103

37 Raw D ata for Figure 74 ..................................................................... 104

38 Raw D ata for Figure 75 ..................................................................... 104

39 Raw D ata for Figure 76 ..................................................................... 105










viii
















LIST OF FIGURES


Figure page

1 Block Diagram of Typical OFDM Transmitter ................................8

2 Block Diagram of Typical OFDM Receiver ...............................

3 Com m on Source of M ultipath .................................... .......................... .......... 11

4 Example of Cyclic Prefix Extension .............................................. .............. 14

5 C IC F ilter-B ased Interpolator ................................................................. .. ..... 15

6 C IC Filter-B ased D ecim ator ................................................................... .. .. .... 15

7 Integrator B lock D iagram .......................................................................... ..... 16

8 Magnitude Frequency Response of Integrator Filter............................................. 17

9 C om b Filter B lock D iagram ......................................................................... .... 17

10 Magnitude Frequency Response of Comb Filter, R*M = 1 ............. ................18

11 Magnitude Frequency Response of Comb Filter, R*M = 2 ............. ................18

12 Magnitude Frequency Response of Comb Filter, R*M = 4 ............. ................ 19

13 Magnitude Frequency Response of Comb Filter, R*M = 8 ............. ................ 19

14 Magnitude Frequency Response of Comb Filter, R*M = 16..............................20

15 Magnitude Frequency Response of CIC Filter, R*M = 1 ............. .................21

16 Magnitude Frequency Response of CIC Filter, R*M = 2 ............. .................21

17 Magnitude Frequency Response of CIC Filter, R*M = 4 ............. .................22

18 Magnitude Frequency Response of CIC Filter, R*M = 8 ............. .................22

19 Magnitude Frequency Response of CIC Filter, R*M = 16.............. .................23

20 Optimized Comb Filter Block Diagram .......................................................23









21 Optimized CIC Filter-Based Interpolator.... ........... ....................................24

22 Optimized CIC Filter-Based Decimator ............ ...........................................24

23 Optimized OFDM Channelizer Interpolator................................ ................. 24

24 Optimized OFDM Channelizer Decimator.................................. .............. 24

25 Block Diagram of an OFDM Channelizer-Based Transmitter ........................... 25

26 Block Diagram of an OFDM Channelizer-Based Receiver ........................... 25

27 Magnitude Frequency Response of Filter 1................................. .................32

28 Magnitude Frequency Response of Filter 2................................. .................32

29 Magnitude Frequency Response of Filter 3................................. .................33

30 Magnitude Frequency Response of Filter 4................................. .................33

31 Magnitude Frequency Response of Filter 5................................. .................34

32 Magnitude Frequency Response of Filter 6................................. .................34

33 Magnitude Frequency Response of Filter 7................................. .................35

34 Magnitude Frequency Response of Filter 8................................. .................35

35 Magnitude Frequency Response of Filter 9................................. .................36

36 M agnitude Frequency Response of Filter 10..................................... ............... 36

37 M agnitude Frequency Response of All Filters ......... ....................................... 37

38 BER of 48-Subcarrier OFDM Channelizer in AWGN with BPSK Modulation .....39

39 BER of 48-Subcarrier OFDM Channelizer in AWGN with QPSK Modulation.....40

40 BER of 48-Subcarrier OFDM Channelizer in AWGN with 8-PSK Modulation.....40

41 BER of 48-Subcarrier OFDM Channelizer in AWGN with 8-QAM Modulation...41

42 BER of 48-Subcarrier OFDM Channelizer in AWGN with 16-QAM Modulation.41

43 BER of 48-Subcarrier OFDM Channelizer in AWGN with 32-QAM Modulation.42

44 BER of 48-Subcarrier OFDM Channelizer in AWGN with 64-QAM Modulation.42

45 BER of48-Subcarrier OFDM Channelizer in AWGN with 128-QAM
M o du latio n ...................................... .............................. ............... 4 3









46 BER of 48-Subcarrier OFDM Channelizer in AWGN with 256-QAM
M o du latio n ...................................... .............................. ............... 4 3

47 Filter 1 OFDM Channelizer Multi-Modulation Scheme Performance ...................45

48 BER of 48-Subcarrier OFDM Channelizer in AWGN with Mixed Modulation.....47

49 Multipath Constellation Scatter without Alignment ...........................................50

50 Multipath Constellation Scatter with Option 1 Alignment............................... 50

51 Multipath Constellation Scatter with Option 2 Alignment............................ 51

52 Filter 3 Separation Filter Design Parameters ........... ......................................... 52

53 Filter 4 Separation Filter Design Parameters ........... ......................................... 53

54 Filter 5 Separation Filter Design Parameters ........... ......................................... 53

55 Filter 6 Separation Filter Design Parameters ........... ......................................... 54

56 Filter 7 Separation Filter Design Parameters ........... ......................................... 54

57 Filter 8 Separation Filter Design Parameters ........... ......................................... 55

58 Filter 9 Separation Filter Design Parameters ........... ......................................... 55

59 Filter 10 Separation Filter Design Parameters.......................................................56

60 Filter 6 Separation Filter Frequency Responses ................... ................. ......... 59

61 Block Diagram of OFDM Channelizer Receiver for Option 1.......................... 60

62 Block Diagram of OFDM Channelizer Transmitter for Option 2 ................... 61

63 Block Diagram of OFDM Channelizer Receiver for Option 2....................... 61

64 Rappaport Multipath Channel Frequency Response............................................64

65 OFDM Channelizer Filter 6 BER Results............................ 66

66 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1............66

67 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2............67

68 BER for Filter 6 OFDM Channelizer in Multipath Channel and 4 Sample Cyclic
P re fi x ............ ..... .......... ..................... ............................... 6 7









69 BER for Filter 6 OFDM Channelizer in Multipath Channel and 7 Sample Cyclic
P re fi x ...................... .. ............... ... ...................... ............... 6 8

70 BER for Filter 6 OFDM Channelizer in Multipath Channel and 10 Sample
Cyclic Prefix .............. .............................................................. ..... 68

71 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 4
Sample Cyclic Prefix............................................. ............... 70

72 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 7
Sam ple Cyclic Prefix .............................................. .............. .............. 71

73 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and 10
Sam ple Cyclic Prefix .............................................. .............. .............. 71

74 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 4
Sam ple Cyclic Prefix................................................................. .............. 72

75 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 7
Sam ple Cyclic Prefix................................................................. .............. 72

76 BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and 10
Sam ple Cyclic Prefix .............................................. .............. .............. 73

77 Polymorphic Block Type Illustration...................... ................... ............. 82

78 Polymorphic Default Value Illustration .............................. ..... .............. 83

79 Block Diagram of OFDM Channelizer Transmitter .......... .............. 107

80 Internal Block Diagram of OFDM Channelizer Transmitter Top Red Box....... 107

81 Internal Block Diagram of OFDM Channelizer Transmitter Middle Red Box.. 108

82 Internal Block Diagram of OFDM Channelizer Transmitter Bottom Red Box.. 108

83 Block Diagram of First-Order OFDM Channelizer Transmission Filter.............. 109

84 Block Diagram of Second-Order OFDM Channelizer Transmission Filter......... 109

85 Block Diagram of OFDM Channelizer Receiver.............. ........... ... 111

86 Internal Block Diagram of OFDM Channelizer Transmitter Top Red Box....... 112

87 Internal Block Diagram of OFDM Channelizer Transmitter Middle Red Box.. 113

88 Internal Block Diagram of OFDM Channelizer Transmitter Bottom Red Box. 114

89 Block Diagram of First-Order OFDM Channelizer Receive Filter ............. 115









90 Block Diagram of Second-Order OFDM Channelizer Transmission Filter ........115

91 Block Diagram of AWGN Communication System........................................... 116

92 Blow Up Block Diagram of AWGN Communication System Top Left Red
B ox .................................... .................................. ......... 117

93 Blow Up Block Diagram of AWGN Communication System Middle Red Box 117

94 Blow Up Block Diagram of AWGN Communication System Top Red Box,
Second from Right ................................... ........................... .. ........ 118

95 Blow Up Block Diagram of AWGN Communication System Right Red Box.. 118

96 Block Diagram of Multipath Communication System Using Option 1 ............ 119

97 Blow Up Block Diagram of Multipath Communication System -Left Red Box.. 119

98 Blow Up Block Diagram of Multipath Communication System -Red Box,
Second from the L eft .. .... .......................................................... .............. 120

99 Blow Up Block Diagram of Multipath Communication System -Red Box,
Second from the Right ............. ............. ........................ 120

100 Blow Up Block Diagram of Multipath Communication System -Right Red Box 121

101 Block Diagram of Multipath Communication System Using Option 2................ 121

102 Blow Up Block Diagram of Multipath Communication System Top Left Red
B ox ............. ......................................................................... 122

103 Blow Up Block Diagram of Multipath Communication System Red Box,
Second from the L eft ....................................... .. ....................... .............. 122

104 Blow Up Block Diagram of Multipath Communication System Middle Red
B ox .............. ..................................................... ... .. ....... 123

105 Blow Up Block Diagram of Multipath Communication System Red Box,
Second from the R ight...................... ...................... .................. .............. 123

106 Blow Up Block Diagram of Multipath Communication System Right Red Boxl24















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

ANALYSIS OF TERNARY-VALUED, CIC FILTER-BASED, OFDM
CHANNELIZERS IN MODERN WIRELESS COMMUNICATIONS SYSTEMS

By

William E. Lawton

December 2005

Chair: Dr. Fred J. Taylor
Major Department: Electrical and Computer Engineering

Mobile wireless communications has been increasing in use and popularity over the

last five to ten years. Examples of modern wireless communications systems include 2G

and 3G mobile wireless as well as 802.11 a/b/g systems. The requirement for these

systems to achieve greater performance and lower cost is continuing to grow. Typically

system design complexity increases simultaneously with system performance.

Orthogonal Frequency Division Multiplexing (OFDM) is currently being designed into

an increasing number of wireless system standards in order to meet the increasing

performance requirements. While OFDM is well suited to the multipath channel

conditions common in wireless applications, it does require complex-valued

multiplications in order to realize the Fast-Fourier Transform (FFT) central to most

OFDM-based systems. This thesis explores the feasibility of using OFDM channelizers

to realize the benefits inherent in FFT-based OFDM systems while simultaneously

decreasing the required design complexity necessary to implement the system.









This thesis presents a study to analyze the use of OFDM channelizers as suitable

alternatives to their FFT-based counterparts. OFDM channelizers consist of ternary-

valued filters similar to cascaded integrator comb (CIC) filters. Number theory is

leveraged in order to realize filters capable of generating OFDM symbols from a bank of

filters consisting entirely of additions and subtractions. The OFDM channelizer's

similarity to an FFT-based OFDM system suggests that it possesses the same ability to

mitigate the effect of multipath interference in a wireless channel.

Simulation results created using Coware's Signal Processing Worksystem (SPW)

are presented and analyzed to determine the ability for OFDM channelizers to operate

effectively in multipath channel conditions. Since different systems can have widely

varying requirements and operating conditions, it is not possible to present an optimal

OFDM channelizer adapted for all conceivable system design parameters. Instead, the

analyses are presented along with different options for implementing channel adaptation

schemes based on an OFDM channelizer. Additionally, advantages and disadvantages of

OFDM channelizers relative to an FFT-based OFDM system are given. These tradeoffs

presented in this thesis are intended to serve as guidance for anyone interested in

incorporating the benefits of an OFDM channelizer into a system design.














CHAPTER 1
INTRODUCTION

Wireless communication systems have been increasing in use and popularity in the

past five to ten years. Examples of such systems are 802.11 a/b/g fixed wireless, 2nd

generation (2G) and 3rd generation (3G) mobile wireless systems. Along with increased

use and popularity, the demand for wireless systems to deliver increased data rates and

quality of service (QoS) is also growing. In order to meet this demand, many aspects of

the systems must scale to achieve the increasing target performance requirements. These

include increasing the available channel capacity as well as improving receiver

performance to attain performance closer to the theoretical channel capacity limit. A

simultaneous design challenge to reduce the cost of equipment is created both by

consumer demand and competition in the marketplace.

The increased channel capacity is being realized with multiple approaches

currently. Technologies such as 802.1 In are employing MIMO-based (multiple-input,

multiple-output) communication in order to increase channel capacity by taking

advantage of multipath channel conditions. In general, this approach tends to increase

overall system cost. This contradicts the market demand for lower cost devices. In order

to keep the cost of these new solutions low, the digital domain is migrating closer to the

antenna in order to replace some of the relatively expensive analog components with

inexpensive digital equivalents. This means that digital solutions are required that can

operate at very high speeds. Additionally, many wireless devices are powered with









batteries. Therefore these high speed digital designs need to be optimized for low power

consumption.

The improvement of receiver performance necessary to approach channel capacity

limits includes utilizing more robust error correcting codes, such as turbo codes, that can

achieve performance approaching Shannon's limit [1]. Furthermore, advanced

modulation and demodulation techniques are being utilized in order to achieve necessary

increased performance and mitigate the effects introduced by wireless communication

channels. One type of advanced modulation technique being utilized in modern

communication systems is Orthogonal Frequency Division Multiplexing (OFDM).

OFDM utilizes multiple subcarriers, each with longer symbol lengths than single-carrier

modulation symbols, in order to mitigate the effects of inter-symbol interference (ISI)

introduced by multipath channels. Other benefits attributable to OFDM modulation will

be explored in this thesis as well.

Typically, these advanced modulation and demodulation techniques require more

processing power in order to realize these advanced receivers. This increased processing

power is undesirable because it increases the power utilization, thus reducing the battery

life in mobile wireless devices. OFDM is typically implemented with a Fast Fourier

Transform (FFT) architecture due to the relatively low complexity of the FFT compared

to a mathematically equivalent DFT (Discrete Fourier Transform). Although it is typical

to utilize an FFT to implement an OFDM communication system, it is not necessary.

Other mechanisms can be employed to develop an OFDM-based communication system.

An alternative in the form of a ternary-valued, CIC concatenatedd integrator comb) filter-









based OFDM channelizer, hereafter referred to as OFDM channelizer, will be

investigated in this thesis.

Approach

Chapter 2 presents a brief history of OFDM as well as certain benefits of utilizing

OFDM as a modulation scheme for wireless communications. Additionally, this chapter

presents the primary challenges introduced by a multipath channel and how the OFDM

characteristics allow for robust communication in such an environment. This chapter

defines the advantages and disadvantages of OFDM since most of them are shared by the

OFDM channelizer.

Chapter 3 presents the concept and structure of an OFDM channelizer.

Furthermore, it describes the transformation of a typical CIC filter into an OFDM

channelizer capable of substituting an FFT engine in an OFDM system.

Chapter 4 presents a study to analyze the viability and usefulness of a ternary

valued, CIC filter-based OFDM channelizer to replace the more common FFT to perform

the modulation of the transmitted signal into multiple narrowband carriers. In addition,

the primary differences between the CIC filter-based channelizer and an OFDM system

are explained. This chapter also presents performance simulation results of an OFDM

channelizer in an additive white Gaussian noise (AWGN) channel. Furthermore, a

proposal to leverage a benefit of the OFDM channelizer is given.

Chapter 5 presents concepts to add robustness to the OFDM channelizer in a

multipath channel. Additionally, it presents some of the challenges and limitations

introduced by the OFDM channelizer when compared to an FFT-based OFDM

modulation scheme in multipath channels as well as performance simulation results in a

multipath channel with various assumptions.









Chapter 6 summarizes the work and results provided in this thesis. Additionally,

areas of future work to be considered are provided.

Appendix A provides a brief overview of SPW block diagram interpretation

necessary for readers not already familiar with SPW.

Appendix B presents the raw simulation data results presented elsewhere in the

thesis in graph form.

Appendix C presents screen captures of the SPW block diagrams developed during

this thesis creation.

Appendix D presents both Matlab and Mathematica commands used to perform

analysis in this thesis.

Analysis Tools

The work presented in this thesis is supported through simulation-based analysis.

The primary tool for performing this analysis is Coware's Signal Processing Worksystem

(SPW) version 4.9. SPW is a system-level design tool based on a hierarchal block

diagram design approach. In Appendix C, native block diagrams captured from within

SPW are used as figures to describe the implementation behind the simulation results.

Specifically, polymorphic designs are used to capture the designs. Polymorphic-based

designs are desirable as they allow simultaneous capture and representation of a design

for both floating-point and fixed-point implementations.

These block diagrams are given as an alternative to code listings common in

publications. Appendix A presents a high-level introduction to SPW block diagrams and

polymorphic extensions in order to facilitate an understanding required of the reader in

order to interpret the block diagrams.






5


Additionally, Microsoft Excel spreadsheet processing software is used to format

and plot results obtained from various SPW simulations as well as generate frequency

response plots of many aspects of the simulation systems included in this thesis.














CHAPTER 2
OFDM BACKGROUND AND BENEFITS

This chapter focuses on the background of OFDM modulation as well as its

usefulness in modern wireless communication systems. It begins with an overview of

various techniques common within OFDM systems as well as reasons and explanations

of the techniques. It continues by describing characteristics of multipath channels along

with a description of how OFDM modulation schemes can be designed to mitigate the

effects of multipath channel propogation.

OFDM Introduction

OFDM is a special form of multicarrier transmission. It can be viewed as either a

data modulation technique or a data multiplexing technique. This chapter will focus on

OFDM as a data modulation technique. The main benefit of OFDM is its robustness

against frequency selective fading created by multipath. In a single carrier system, a fade

in a portion of the signal band can cause severe degradation to the overall link. However,

in an OFDM system, only a small percentage of the total carriers will exist in any one

fade. Therefore, the link will continue to persist because even a simple error correcting

coding scheme will resolve the errors.

Figure 1 illustrates a block diagram of a typical OFDM transmitter. The basic

makeup of an OFDM transmitter consists of the following components and their

respective descriptions:

S Coding insertion of parity and cyclic redundancy check (CRC) into the data
stream in order to correct and detect errors at the receiver









* Interleaver reordering of the data stream in order to more evenly distribute
errors generated in the channel that are correlated either in time or in
frequency. Allows for more effective and efficient error correction.

* Bit-to-Symbol Mapping transforms a bit stream into complex-valued
symbols in the signal space domain. These symbols are the Fourier
coefficients input to the Inverse Fast Fourier Transform (IFFT).

* Pilot Insertion pilot or synchronization symbols are common in OFDM
systems in order for the receiver to accurately measure the channel response
as well as the frequency of the transmitter. These estimates help ensure
accurate reception of the transmitted signal.

* Serial-to-Parallel groups N symbols before computing the IFFT to generate
an OFDM symbol.

* IFFT performs the inverse fast Fourier transform converting the frequency
domain signal into the time domain for transmission.

* Parallel-to-Serial serializes the time-domain signal for transmission.

* Cyclic Extension Insertion transmits the tail of each OFDM symbol before
wrapping around and transmitting the entire OFDM symbol from beginning
to end.

* Windowing applies a window to a percentage of the cyclically extended
OFDM symbol at both the head and tail of the symbol. This step typically
also includes overlapping the windowed tail of symbol M and the windowed
head of symbol M+1. This improves the spectral properties of the
transmitted signal.

* DAC Digital-to-Analog Converter that transforms the digital signal to
analog.

* LPF low-pass filter to eliminate spectral copies caused by conversion from
digital to analog signal.

* RF TX optional RF transmission circuitry. Some OFDM implementations
transmit the baseband OFDM signal directly.



















Figure 1. Block Diagram of Typical OFDM Transmitter


Figure 2 illustrates a block diagram of a typical OFDM receiver. The basic makeup

of an OFDM receiver consists of the following components and their respective

descriptions:

RF RX optional RF reception circuitry.

LPF also referred to as an anti-aliasing filter. Attenuates higher frequency
components prior to sampling to reduce effects of aliasing caused by
sampling.

ADC Analog-to-Digital Converter that transforms the analog signal to
digital.

Time / Frequency Synchronization Detects starting time of received frame
as well as estimating sampling frequency of transmitted signal in order to
minimize effects of frequency error of the receivers sampling clock relative
to the transmitter clock.

Cyclic Extension Removal removes the cyclic extention of the OFDM
symbols in the receiver. Most of the ISI is contained within this time and
therefore not processed in the receiver.

Serial-to-Parallel groups N samples before computing the FFT to convert
the received signal back into the frequency domain.

FFT performs the fast Fourier transform converting from the time domain
into the frequency domain for demodulation.

Parallel-to-Serial serializes the frequency-domain signal for further
processing.

Channel Correction estimates the channel response on a per-carrier basis.
The channel estimate is used to normalize the channel response in the
received signal.










Symbol-to-Bit Demapping maps the received symbols to soft bits used for
decoding. Soft bits are typically generated as input to the decoder due to
increased performance of the decoder versus hard-decision (antipodal) bits.

Time / Frequency Deinterleaver restores original order of transmitted bit
stream into the decoder. This deinterleaving increases the average distance
between burst errors introduced by the channel in either the time or frequency
domain.

Decoder corrects bit errors introduced either in the channel or in the noise
present in the receiver front end. Additionally, uncorrectable errors are
typically detected by comparing the received CRC against another CRC that
is computed in the receiver.



STime /Frequency Cyclic Serial-to-
RF RX/ LPF ADC z Extension Parallel FFT
yRemoval




Time Coding
Parallel-to- Channel Symbol-to-Bit T me! Forward Error
Frequency
Serial Correction Demapping Deinterleaver Correction
/CRC


Figure 2. Block Diagram of Typical OFDM Receiver

OFDM Background

OFDM studies date back to the 1960s [2, 3]. In the 1960s, OFDM was

incorporated into several military systems such as KINEPLEX [4], ANDEFT [5] and

KATHRYN [6]. The first OFDM patent was filed and issued in 1971 [7]. In the 1980s,

OFDM usage studies began to branch into areas including high-speed modems, digital

mobile communications and high-density recording.

The first commercial applications of OFDM can be traced back to Discrete Multi-

Tone (DMT). DMT was developed to transmit video over twisted pair copper wires for

Digital Subscriber Line (DSL). Then, during the 1990s, OFDM became integral for

wideband data communications over mobile radio FM channels, various forms of high









data rate digital subscriber lines (xDSL), digital audio broadcasting (DAB) and high-

definition television (HDTV) terrestrial broadcasting.

OFDM Benefits and Drawbacks

OFDM has been characterized well over the past few decades. This section

highlights the well known advantages and drawbacks of OFDM modulation. These are

listed here to be explained in more detail in the following sections. OFDM has several

main advantages described below:

OFDM is robust against multipath. The implementation complexity is
significantly lower for an OFDM-based system versus a single-carrier system
with equalization for a given delay spread.

OFDM has the potential to exploit distinct signal-to-noise ratios (SNR) per
carrier by modulating a different number of bits per carrier. This has the net
effect of increasing the capacity of the system.

OFDM tolerates narrowband interference effectively. This is due to the
inherent frequency division in the signaling. Any narrowband interference
interferes with a relatively small number of carriers in the system.

OFDM also has a couple of primary disadvantages described below:

OFDM signals are usually characterized by relatively high peak-to-average
power ratios (PAPR). This can lead to reduced power efficiency in the power
amplifier in the system.

OFDM is more sensitive to frequency offset and phase noise than a typical
single-carrier system.

Multipath in Wireless Channels

Multipath is a common phenomenon occurring in wireless channels. Figure 3

demonstrates a typical source of the multipath phenomenon in a wireless channel. As

shown, reflections from multiple objects near either the transmitter or receiver combine

to contribute parts of the received signal occurring at different time offsets. These time-

offset reflections combine to create a frequency selective channel response. The










degradation caused by multipath is most pronounced when there is no line-of-sight (LOS)

path between the transmitter and receiver.



HI HI II I
II EI II EI






1I I 1I I -

Transmitter Receiver
\ /
\ % /












Figure 3. Common Source of Multipath

Various studies and characterizations of multipath channel profiles in different

environments exist in literature. This thesis does not attempt to describe the details of

these various channel characteristics. However, the most important design parameter

relating to an OFDM system is the maximum delay spread introduced by a channel. This

maximum delay spread defines the duration of time from the first path response to the

last path response when measured from the receiver. In practice, the duration of time

where a certain percentage of the received energy is localized is used to define the delay

spread. This is typically more practical to design for rather than the delay spread to

contain all of the received signal energy.









Cyclic Prefix to Mitigate Effects of Multipath

One of the primary benefits of OFDM modulation is its ability to effectively

mitigate multipath delay spread. Fundamentally, this is achieved because the symbol

duration of an OFDM symbol is substantially longer than the symbol time of an

equivalent bit-rate, single-carrier system. Since OFDM transmits data in parallel using

multiple carriers, the resulting OFDM symbol duration is proportionally longer based on

the number of carriers. Therefore, the channel delay spread, relative to the symbol

duration, is reduced.

Many empirical studies have been performed in order to characterize wireless

channel characteristics in many different environments [8, 9]. Typical delay spreads seen

in various scenarios are listed in Table 1 below.

Depending on the baseband sampling rate of the system under study, the various

delay spreads identified in Table 1 translate into a certain number of baseband samples.

This duration defines the amount ofISI introduced into the received signal.

Table 1. Measured Delay Spreads in Various Wireless Channels
Median Maximum
Frequency
Delay Delay
Environment Description eay Deay Range
Spread Spread GHz]
[ns] [ns]
Large building 40 120 4- 6
Office building #1 50 60 4- 6
Meeting room (metal walls) 35 55 4-6
Single room (stone walls) 10 35 4- 6
Office building #2 40 130 4- 6
Indoor sports arena 40 120 4- 6
Factory #1 65 125 4- 6
Office building #3 25 65 4- 6
Office building #4 (single room) 20 30 4- 6
Office building #5 -- 1000 0.815
Office #6 90 8000 0.915/1.9
Urban 136/258 -- 1.9









Typical OFDM systems use a well-known approach to mitigating the delay spread

introduced by a multipath channel. They reduce or eliminate the inter-symbol

interference (ISI) introduced by the channel by adding a guard period to the OFDM

symbols. An effective guard period should be at least as long as the maximum expected

delay spread occurring over the worst-case potential channel condition. However, guard

periods of excessive length add overhead to the transmission scheme, ultimately reducing

system throughput. Therefore, there is a design tradeoff relative to the length of the

guard interval. Guard intervals are implemented using a cyclic extension of the OFDM

symbol, extending a portion of the tail of the OFDM symbol to the beginning of the new

cyclically extended OFDM symbol. This is illustrated in Figure 4 below.

As noted in Figure 1, the cyclic extension occurs after the IFFT. The IFFT

produces a time-domain signal consisting of all carriers superimposed with one another.

Figure 4 illustrates how each carrier is cyclically extended. In an actual implementation,

the cyclic extension is achieved by extending a single time-domain signal consisting of

all of the subcarriers superimposed into a single data stream.

As an alternative to a cyclic extension, the guard period could be implemented with

a zero-valued insertion. However, a zero-valued guard period would have the negative

side-effect of introducing inter-carrier interference (ICI) since integer numbers of carrier

cycles are no longer guaranteed to exist within a single non-extended OFDM symbol

duration. A cyclic extended guard period rather than a zero-filled guard period of the

transmitted symbols prevents this ICI.









































I I I I
Cyclic Prefix Original OFDM Symbol


Figure 4. Example of Cyclic Prefix Extension


Cyclically Extended OFDM Symbol















CHAPTER 3
OFDM CHANNELIZER OVERVIEW AND DESCRIPTION

As described earlier, typical OFDM-based systems implement modulation and

demodulation utilizing an FFT structure. The benefits of utilizing an FFT for this

function have been discussed previously. This chapter will investigate utilizing an

OFDM channelizer as an alternative to FFT-based OFDM.

CIC Filter Overview

The OFDM channelizer explored in this thesis is based on a Cascaded Integrator-

Comb (CIC) filter. The CIC filter is a well-known, multiplier-less finite impulse

response (FIR) filter having a wide variety of applications. One of the more common

applications is as a digital sample-rate converter where a highly sampled signal contains a

relatively narrowband baseband signal of interest. A CIC filter can be used in either

interpolation or decimation schemes to either increase or decrease the sample rates of a

signal, respectively. Top-level block diagrams of these sample-rate converters are shown

below in Figure 5 and Figure 6.


R Comb Comb Comb Integrator Integrator Integrator
Upsample


Figure 5. CIC Filter-Based Interpolator


-- Integrator Integrator e Integrator --- Comb -- Comb -w Comb --w IR
Downsample


Figure 6. CIC Filter-Based Decimator









A CIC filter, as its name suggests, consists of two primary stages cascaded serially.

The first stage consists of one or more integrators. The second stage consists of an equal

number of comb filters. A single integrator has a transfer function given below.


H(z) = (1)
1-z1

As the transfer function indicates, the integrator has a single pole located directly at

unity on the real axis on the unit circle in the z-domain. A block diagram of an integrator

is given in Figure 7.




Z-1






---- -----------------


Figure 7. Integrator Block Diagram

The magnitude frequency response of the integrator filter is shown in Figure 8

below.

A comb filter has a transfer function given by the equation below.

H(z) = l+ z-R (2)

In the above transfer function, R is typically referred to as the integer rate change

factor and M is typically referred to as the differential delay [10]. A block diagram of a

comb filter is given in Figure 9.






17



60
50
40
w 30
0 -- Integrator

10-
0
-10 -
-0.5 -0.4 -0.29-0.19-0.08 0.02 0.12 0.23 0.33 0.44
Frequency (fs= 1 Hz)


Figure 8. Magnitude Frequency Response of Integrator Filter




z-R*M





-----------------------
+




Figure 9. Comb Filter Block Diagram

The magnitude frequency responses of the comb filter when R*M = 1, 2, 4, 8 and

16 are shown in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, respectively.

The CIC filter is constructed by concatenating the integration and comb filter stages that

are described above. A non-obvious observation regarding the CIC filter is that it is

actually an FIR filter. This is not obvious since the filter has feedback contained within

its integrators. Typically with digital signal processing, feedback is synonymous with

infinite impulse response (IIR) filters. However, a CIC filter, under closer examination,







18


produces a finite impulse response. This finite impulse response is realized by the exact

cancellation of the zeros from the comb filter and the poles in the integrator located at

unity on the real axis of the unit circle. This finite impulse response is demonstrated in

the transfer response equation below.


Figure 10. Magnitude Frequency Response of Comb Filter, R*M = 1


Figure 11. Magnitude Frequency Response of Comb Filter, R*M = 2


20

0

-20

-40

-60

-80

-100

-120 ..
-0.5 -0.4 -0.29 -0.19 -0.08 0.021 0.1250.2290.3330.437
Frequency (fs = 1 Hz)


__Comb


20

0

-20

-40

-60

-80

-100

-120
-0.5 -0.4 -0.29 -0.19 -0.08 0.021 0.1250.2290.3330.437
Frequency (fs = 1 Hz)


__Comb


































Figure 12. Magnitude Frequency Response of Comb Filter, R*M = 4


Figure 13. Magnitude Frequency Response of Comb Filter, R*M = 8


20

0

-20

-40

-60

-80

-100

-120 ..
-0.5 -0.4 -0.29 -0.19 -0.08 0.021 0.1250.2290.3330.437

Frequency (fs = 1 Hz)


__Comb


20

0

-20

-40 -

-60 -

-80

-100

-120. ....
-0.5 -0.4 -0.29 -0.19 -0.08 0.021 0.1250.2290.3330.437

Frequency (fs = 1 Hz)


__Comb
































Figure 14. Magnitude Frequency Response of Comb Filter, R*M = 16

1 R M-1 NI
H(z)= -Z +z-RM ,= 2-Z-k (3)
I- k=0 j


When the integrator and comb filters are cascaded into a single CIC filter, there is

typically a rate conversion inserted between the integrator and comb sections of the filter.

In this form, the CIC filter becomes a multirate filter. The magnitude frequency response

of a CIC filter with R*M equal to one is shown in Figure 15 below. It can be seen that

the response of a CIC filter with R*M equal to one is flat. This can be seen by the

transfer function given in Equation 3 when R*M equals one. In this case the single zero

in the comb filter exactly cancels the pole in the integrator filter.

In order to develop filters of more interest, R*M needs to be something other than 1

[11]. Examples of CIC filter magnitude frequency responses with R*M equal to 2, 4, 8

and 16 are shown in Figure 16, Figure 17, Figure 18 and Figure 19, respectively.


20 -
20



-20--- --- --

-40

-60

-80--- ---------- --

-100

-120 ..
-0.5 -0.4 -0.29 -0.19 -0.08 0.021 0.1250.2290.3330.437
Frequency (fs = 1 Hz)


__Comb











400

300

200

100

0

-100

-200
-0.5 -0.4 -0.29-0.19-0.08 0.02 0.12 0.23 0.33 0.44
Frequency (fs = 1 Hz)


- Integrator
--Comb
CIC


Figure 15. Magnitude Frequency Response of CIC Filter, R*M = 1


400

300

200

100

0

-100

-200 ..
-0.5 -0.4 -0.29-0.19-0.08 0.02 0.12 0.23 0.33 0.44
Frequency (fs = 1 Hz)


- Integrator
--Comb
CIC


Figure 16. Magnitude Frequency Response of CIC Filter, R*M = 2











400

300

200

100 -

0

-100

-200
-0.5 -0.4 -0.29-0.19-0.08 0.02 0.12 0.23 0.33 0.44
Frequency (fs = 1 Hz)


- Integrator
--Comb
CIC


Figure 17. Magnitude Frequency Response of CIC Filter, R*M = 4


400

300

200

100



-100

-200
-0.5 -0.4 -0.29-0.19-0.08 0.02 0.12 0.23 0.33 0.44
Frequency (fs = 1 Hz)


- Integrator
--Comb
CIC


Figure 18. Magnitude Frequency Response of CIC Filter, R*M = 8











400

300

200

100



-100
-100 ---------------


-0.5 -0.4 -0.29-0.19-0.08 0.02 0.12 0.23 0.33 0.44
Frequency (fs = 1 Hz)


- Integrator
--Comb
CIC


Figure 19. Magnitude Frequency Response of CIC Filter, R*M = 16

CIC Filter Optimization

A simple modification can be made to the typical CIC-based FIR filter presented

above in order to optimize the architecture. The optimization is realized by pushing the

comb filter section of the CIC FIR through the rate change operation. In order to

maintain an equivalent filter response with this architecture change, the comb filter must

be altered accordingly. The simple modification to the comb filter involves reducing the

delay operator from R*M delays to M delays as shown in Figure 20.

r----------------I






r-----------------
z-M i





I + I

I I
I I


Figure 20. Optimized Comb Filter Block Diagram










The corresponding optimized CIC interpolation and decimation filters are shown in

Figure 21 and Figure 22, respectively.


- Comb -- Comb -- Comb TfR Integrator -- Integrator -- Integrator -
Upsample


Figure 21. Optimized CIC Filter-Based Interpolator


-- Integrator -- Integrator -- Integrator --w mJR -W Comb -- Comb -- Comb
Downsample


Figure 22. Optimized CIC Filter-Based Decimator

OFDM Channelizer Introduction

The OFDM channelizer utilizes number theory in order to enhance the capability of

the more common CIC filter. In order to transform a CIC filter into an OFDM

channelizer, a reduced polynomial based filter is chosen to replace the interpolator stage

of the CIC filter [12, 13]. The corresponding optimized OFDM channelizer interpolation

and decimation filters are shown in Figure 23 and Figure 24, respectively. Although the

figures show multiple stages of comb and reduced-polynomial based filters (N > 1), this

thesis will concentrate on analyzing the OFDM channelizer when N=1. In this case, the

OFDM channelizer produces a spectrum equivalent to the same size FFT-based OFDM

transmitter. Increasing N narrows the spectrum of each harmonic produced by the filter.

Reduced Reduced Reduced
Comb Comb w Comb p R p Polynomial Polynomial _, Polynomial
Upsample Based Based Based
Filter Filter Filter


Figure 23. Optimized OFDM Channelizer Interpolator

Reduced Reduced Reduced
Polynomial Polynomial Polynomial JR spComb -w Comb Comb
Based Based Based Downsample
Filter Filter Filter


Figure 24. Optimized OFDM Channelizer Decimator









System-level block diagrams of a transmitter and receiver as part of an OFDM

channelizer-based system are shown below in Figure 25 and Figure 26, respectively. It

should be noted that the system-level block diagrams of the OFDM channelizer are

similar to the original FFT-based OFDM diagrams presented earlier. The primary

difference is the replacement of the FFT / IFFT by the equivalent Channelizer filter bank.


Figure 25. Block Diagram of an OFDM Channelizer-Based Transmitter


Figure 26. Block Diagram of an OFDM Channelizer-Based Receiver

The transformation of a typical CIC filter into an OFDM channelizer begins with

selecting the number of subcarriers created by the OFDM channelizer. This is

synonymous with selecting the FFT size in a typical OFDM system. However, there are

additional design tradeoffs that must be taken into account with the OFDM channelizer

that do not need to be taken into account for an FFT-based OFDM system. The details of

selecting the number of subcarriers are described in the following section.









Once the number of subcarriers is chosen (N), a corresponding primary polynomial

can be factored into a set of irreducible primitive polynomials. The coefficient set of the

primitive polynomials is limited to the terary values, {-1, 0, 1}. The primary polynomial

is of the form shown in Equation 4.

g(z) = z-N -1. (4)

Factoring the primary polynomials into irreducible primitive polynomials yields the

transfer functions that are used to replace the integrators in the typical CIC filter.

OFDM Channelizer Selection

A primary polynomial of form zN-1 is chosen based on two relevant criteria for the

reduction of the polynomial over the ternary-valued coefficient set {-1, 0, 1}. First, the

polynomial should reduce into a relatively large number of independent polynomials.

Second, each polynomial resulting from the reduction of the primary polynomial should

consist of a small number of terms. Table 2 shows a list of various primary polynomials

and the corresponding number of irreducible polynomials resulting from the reduction

against the possible coefficient set as well as the maximum number of terms in the

reduced polynomials.

It is interesting to note that the number of polynomials resulting in the reduction of

the primary polynomial with only ternary-valued coefficients is equal to the number of

divisors of the polynomial order, N. For example, z12-1 results in 6 ternary-valued

coefficients. The integer, 12, has 6 divisors: 1, 2, 3, 4, 6 and 12. This holds true for all of

the polynomials analyzed in Table 2 below.

Given Table 2, N equal to 48 was chosen for further analysis in this thesis due to a

relatively high number of reduced polynomials, each having a low number of terms. The









number of terms per polynomial is an important design factor since the number of terms

in the reduced polynomials directly maps into the number of additions required per

sample per filter. A higher number of additions per sample per filter has a direct affect

on the speed at which the filters can be operated.

The number of irreducible polynomials defines the number of independent filter

banks that can be realized. This design criterion will become more important as the

characteristics of an OFDM channelizer are explored under wireless channel propagation

conditions. It should be noted that in an FFT-based OFDM system, every carrier can be

independently modulated and demodulated.

48-Subcarrier OFDM Channelizer Description

The detailed simulation analysis in this thesis will be performed for the 48-

subcarrier system design. The factorization of the primary polynomial, z-48-1 is shown in

Table 3 along with various other filter characteristics. Table 3 highlights the primary

difference between an OFDM channelizer and a similar FFT-based OFDM system.

Where an FFT contains N distinct carriers, each of which can contain independent data

streams, an OFDM channelizer only has as many distinct channels as filters. For the case

of the 48-subcarrier OFDM channelizer, ten independent data streams are supported. The

number of frequency bins column shows how many subcarriers (harmonics) are

generated through each filter. The total number of subcarriers across all filters is equal to

48 and the resulting total combined spectrum is equivalent to that of a 48-point FFT.

The magnitude frequency responses of the ten distinct filters for the 48-subcarrier

OFDM channelizer are shown below in Figure 27, Figure 28, Figure 29, Figure 30,

Figure 31, Figure 32, Figure 33, Figure 34, Figure 35 and Figure 36. The combined

frequency response of all filters superimposed on a single graph is shown in Figure 37.









Figure 37 shows the frequency response of the entire 48-subcarrier OFDM

channelizer after the output of each filter has been normalized. The normalization factors

on each filter are shown in Table 3. For an optimal implementation, the scaling of each

filter output should be applied at the low end rate of the multirate filter. Therefore,

although this normalization factor is a multiplication necessary to flatten the power

spectral density, it operates at a relatively slow rate and therefore does not limit the speed

at which the implementation can be executed.

Table 2. Explored Primary Polynomials
N Number of Largest Highest Possible
Reduced Number of Reduced Use
Polynomials Terms in Polynomial
Reduced Order
Polynomials
12 6 3 4 Yes
13 2 13 12 No1'2
14 4 7 6 No2
15 4 7 8 No2
16 5 2 8 Yes
17 2 17 16 No1,2
18 6 3 6 Yes
19 2 19 18 No1,2
20 6 5 8 No2
21 4 9 12 No2
22 4 11 10 No2
23 2 23 22 No1,2
24 8 3 8 Yes
25 3 5 20 No1
26 4 13 12 No2
27 4 3 18 No1
28 6 7 12 No2
29 2 29 28 No1'2'3
30 8 7 8 No2
31 2 31 30 No1'2'3
32 6 2 16 Yes





Highest
Reduced
Polynomial
Order


Possible
Use


Table 2. Continued.
N Number of Largest
Reduced Number of
Polynomials Terms in
Reduced
Polynomials
33 4 15
34 4 17
35 4 17
36 9 3
37 2 37
38 4 19
39 4 17
40 8 5
41 2 41
42 8 9
43 2 43
44 6 11
45 6 7
46 4 23
47 2 47
48 10 3
49 3 7
50 6 5
51 4 23
52 6 13
53 2 53
54 8 3
55 4 17
56 8 7
57 4 25
58 4 29
59 2 59
60 12 7
61 2 61
62 4 31
63 6 9
64 7 2
65 4 31
66 8 15
67 2 67
68 6 17
69 4 31
70 8 17
71 2 71


20 No
16 No2
24 No2
12 Yes
36 No1'2'3
18 No2
24 No2
16 Yes
40 No1'2'3
12 No2
42 No1'2'3
20 No2
24 No2
22 No2
46 No1'2'3
16 Yes
42 No2,3
20 No1
32 No2,3
24 No2
52 No1'2'3
18 Yes
40 No2,3
24 No2
36 No2,3
28 No2,3
58 No1'2'3
16 No2
60 No1'2'3
30 No2,3
32 No2,3
32 No1,3
48 No2,3
20 No2
66 No1'2'3
32 No2',3
44 No2,3
24 No2
70 No1'2'3









Table 2. Continued.
N Number of Largest Highest Possible
Reduced Number of Reduced Use
Polynomials Terms in Polynomial
Reduced Order
Polynomials
72 12 3 24 Yes
73 2 73 72 No1'2'3
74 4 37 36 No2,3
75 6 7 40 No2,3
76 6 19 36 No2,3
77 4 31 60 No2'3
78 8 17 24 No2
79 2 79 78 No1'2'3
80 10 5 32 No3
81 5 3 54 No1,3
82 4 41 40 No2,3
83 2 83 82 No1'2'3
84 12 9 24 No2
85 4 41 64 No2,3
86 4 43 42 No2,3
87 4 39 56 No2,3
88 8 11 40 No2,3
89 2 89 88 No1'2'3
90 12 7 24 No2
91 4 23 72 No2,3
92 6 23 44 No2,3
93 4 41 60 No2,3
94 4 47 46 No2,3
Note 1: Undesirable number of irreducible polynomials. Too few.
Note 2: Undesirable number of terms in irreducible polynomials. Too many.
Note 3: Undesirable highest order of reduced polynomial. Too large.










Table 3. Transfer Functions for 48-Subcarrier OFDM Channelizer Filter Banks
Filter Reduced Filter Number of Filter Filter
Number Polynomial Transfer Harmonics Resonant Normalization
for Response Harmonic Factor
Interpolator Frequencies
Replacement (fs=48)
1 1-z-' 1-z-48 1 0 1


1+z-1


1+z-2


-z-1 +z-2


1+z- + -2


1+z-4


1-z- +z-4


1+z-8


1-z-4 + -8


10 1-z- + -16


1-z-48

1+z-'
1-z-48

1+z-2
1-z-48
1-+z-2
1-z-48
1+z- +z-2
1- -48

1+z-4
1- -48

l-z-2 +Z-4
1-z-48
1-z-48


1 z-+ -8

1-z-48
1- z + z-16


+/-12


+/-8


+/-16


+/-6, +/-18


+/-4, +/-20


+/-3, +/-9,
+/-15, +/-21

+/-2, +/-10,
+/-14, +/-22

+/-1, +/-5,
+/-7, +/-11,
+/-13, +/-
17, +/-19,
+/-23


1


2








4


2.-v

8


2.V-


8-.,






































Figure 27. Magnitude Frequency Response of Filter 1


Figure 28. Magnitude Frequency Response of Filter 2


40


30


20


10
S-- Filter 1
04
Co
cc

-10


-20


-30
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)


40


30


20 -


10
- Filter2
0


-10


-20


-30 I I I I I
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)












40


30


20 ----


S10 -
[ Filter
-10 -----------------------------------------------
cc

-10 ----------------------------------- ----



-30
-30 --------------- -----

-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)



Figure 29. Magnitude Frequency Response of Filter 3


40


30


20


10 -- ---------- -----
[ Filter4
0 -----------------------------------------------


-10 -----------------------------------



-30
-30 --- ---- ---- ----------

-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)


Figure 30. Magnitude Frequency Response of Filter 4





































Figure 31. Magnitude Frequency Response of Filter 5


Figure 32. Magnitude Frequency Response of Filter 6


40

30

20




-10 -
[2- Filter5





-20


-30

-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)


40

30

20


10
0 --Filter 6
cc


-10

-20


-30 ....
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)


































Figure 33. Magnitude Frequency Response of Filter 7


Figure 34. Magnitude Frequency Response of Filter 8


40

30

20 -

10 -
3 Filter7

cc
Co


-10

-20

-30 ....
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)


40 --------------------

30 J 1A Aln nA Al II. n Al nAt1t )1 A An
40





-10 I
S--Filter 8]

cn



-20

-30 -------
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)


































-3 0 + I I I I I I I I I
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5

Frequency (fs = 1 Hz)



Figure 35. Magnitude Frequency Response of Filter 9


40


30


20


10
-o
S10 -------- -- ----- -------- -------- --___
[ Filter 10
"E 0 ------- -



-10


-20


-3 0 .
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5

Frequency (fs = 1 Hz)


Figure 36. Magnitude Frequency Response of Filter 10


40


30


20
S"l

10
-20

C


-10


-20


----- -Filer
---- ---- --- ------ --- --------- ---- -- -


Filter 9










40

30 A A- --- Filter 1--
20- Filter 1
30 Filter 2
Filter 3
10 filter
Filter 5
-- Filter 6
1 0 A -, "1 -- Filter 7ilter
li Filter 8
-30 ----------- ------
-20 Filter 10

-30
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)


Figure 37. Magnitude Frequency Response of All Filters














CHAPTER 4
OFDM CHANNELIZER PERFORMANCE IN AWGN CHANNEL

This chapter presents the performance of the 48-point OFDM channelizer in an

AWGN channel. It goes on to describe techniques for overcoming apparent limitations

associated with the OFDM channelizer when compared to an FFT-based OFDM

modulation scheme under these channel conditions. An assumption for this analysis is

that the energy per subcarrier is constant across all subcarriers. Therefore, bits modulated

through filters generating a greater number of subcarriers produce proportionally more

energy than filters generating a smaller number of subcarriers. A representative power

spectral density has been shown previously in Figure 37.

Performance of 48-Point OFDM Channelizer in AWGN Channel

In an AWGN channel, the OFDM channelizer provides the same performance as a

matched-filter receiver. This section presents simulation results for the 48-Point

Channelizer and derives the process gains achieved across the various filter banks. The

calculation for the probability of error in an uncoded, antipodal, BPSK-modulated system

with a matched-filter receiver is given below, where the energy per bit is given by Eb and

the noise power is given by No/2 [14].


Pr{Eb} = Eb (5)


With an OFDM channelizer, the probability of bit error varies across each of the

filter banks. This is due to the fact that each filter can generate a different number of

subcarriers. Given a uniform power spectral density across all subcarriers, the greater the










number of subcarriers a filter occupies, the greater the transmitted energy per bit with

respect to the information modulated through the filter and the lower the probability of bit

error after the energy from the subcarriers is coherently combined at the receiver.

The measured performance of each filter bank in an AWGN channel with various

modulation schemes are shown in the following figures below. The modulation schemes

considered include BPSK, QPSK, 8-PSK, 8-QAM, 16-QAM, 32-QAM, 64-QAM, 128-

QAM and 256-QAM [14]. The modulation schemes consist of common square or

rectangular constellations with Gray code mapping of bits to symbols in the signal space

domain. The theoretical probability of bit error for antipodal BPSK signaling is given as

a common reference in all of the figures below.


1.E+00

1.E-01

1.E-02

1.E-03

1.E-04

1.E-05

1.E-06

1.E-07

1.E-08

1.E-09


I ri


_s Filter

Filter


r 1
r 2
r 3
r 4


ME 9 % CJ IT W LO C)M D

SNR


I


-- Filter 5
-- Filter 6
- Filter 7
- Filter 8
Filter 9
Filter 10
BPSK Theoretical


Figure 38. BER of 48-Subcarrier OFDM Channelizer in AWGN with BPSK Modulation


r


- ~-------

-- .~-------


-*- Filtel







40



1.E+00

1.E-01 Filter 1
Filter 2
1.E-02
S\ \Filter 3
1.E-03 Filter 4
1E-04 --- Filter 5
1.E-04
Filter 6
S1.E-05 Filter 7

1.E-06 Filter 8
Filter 9
1.E-07 Filter 10

1.E-08 BPSK Theoretical

1.E-09
c r- T LO r- C cD M c.

SNR


Figure 39. BER of 48-Subcarrier OFDM Channelizer in AWGN with QPSK Modulation


1.E+00


1.E-02

1.E-03 F,,-. 2
--Firi 5
1.E-04
.-- IIr 6
.. 1.E-05 -F

1.E-06

1.E-07 ,

1.E-08 .. .,,,

1.E-09
M Co r- T- W LO r- cD M.

SNR


Figure 40. BER of 48-Subcarrier OFDM Channelizer in AWGN with 8-PSK Modulation











1.E+00

1.E-01 --Filter 1
--Filter 2
1.E-02 *=
1.E-0-- Filter 3

1.E-03 Filter 4
x Filter 5
1.E-04
-*- Filter 6
0. 1.E-05 -- Filter 7

1.E-06 --Filter 8
Filter 9
1.E-07 Filter 10

1.E-08 BPSK Theoretical

1.E-09
Coo r-'- T- W LO r--'r- C oM cD 0

SNR


Figure 41. BER of 48-Subcarrier OFDM Channelizer in AWGN with 8-QAM
Modulation


1.E+00
Filter 1
1.E-01 __ -__ __"
--_ _-__ -Filter2
1.E-02 Filter 3
Filter4
1.E-03 -. Filter
-x-Filter 5

a 1.E-05 sFilter7
Filter 8
1.E-06 =^
1. 06 Filter 9

1.E-07 Filter 10
1E BPSK Theoretical
1.E-08 ---

1.E-09
C'.J IC'. I I r r'.r

SNR


Figure 42. BER of 48-Subcarrier OFDM Channelizer in AWGN with 16-QAM
Modulation










1.E+00

1.E-01 F,,,- I
-1.E-02 ---
1.E-02


1.E-04 =6

a 1.E-05 -- ,,, -

1.E-06 _
1.E-07 \=r F\,, .,

1.E-08 PS ,,

1.E-09
C\j C\j Ir i I I r- r- r- r- c\j c\j

SNR


Figure 43. BER of 48-Subcarrier OFDM Channelizer in AWGN with 32-QAM
Modulation


1.E+00

1.E-01 -- Filter 1
1.E-02 -Filter2
1.E-02
Filter 3
1.E-03 Filter 4
A- 1E-0 -- Filter 5
FieFilter 6
S1.E-05
-- Filter 7
1.E-06 Filter 8
Filter 9
1.E-07
Filter 10
1.E-08 BPSK Theoretical

1.E-09
C'J C'J Ir i- r i CIrJ C\J C\J

SNR


Figure 44. BER of 48-Subcarrier OFDM Channelizer in AWGN with 64-QAM
Modulation











1.E+00

1.E-01 -_ F-Ir r I

1.E-02 _
F"l ,.
1.E-03 F,,,r A

'- 1.E-04 -- -- -.- r 6
1.E-05

1.E-06 -F-

1.E-07
1.E-08 6PS Tri ,i .r.IC ,

1.E-09
C\J C\J Ir i C'.J C\J C\J

SNR


Figure 45. BER of 48-Subcarrier OFDM Channelizer in AWGN with 128-QAM
Modulation


1.E+00

1.E-01 s..'_ -- Filter 1
-Filter2
1.E-02
Filter 3
1.E-03 Filter 4

1.E-04 4 Filter 5
-- \ -*o-Filter6
--Filter 7
1.E-06 Filter 8
Filter 9
1.E-07
Filter 10
1.E-08 BPSK Theoretical

1.E-09
C'J C'.J i I- I I r-- r- C\J C\J C\J
I I I I I

SNR


Figure 46. BER of 48-Subcarrier OFDM Channelizer in AWGN with 256-QAM
Modulation









As shown in the above figures, independent of modulation scheme, the groups of

filters within the OFDM channelizer yield widely varying performance. Essentially, the

relative performance difference between any two filters is defined by the ratio of the

number of subcarriers produced by the filters. Note that as shown in the above figures,

all carriers generating equal number of subcarriers yield the same performance. The

relative performance of each filter is given below in Table 4. The performance advantage

of any given filter can be calculated by the following equation given that the per-

subcarrier energy is constant independent of the number of subcarriers produced by the

filter.

G = 10 logo1 (num subcarriers) (6)

Table 4. Performance Advantage of Filters Based on Number of Subcarriers
Filter Number of Performance
Number Subcarriers Advantage of
Filter, G (dB)
1 1 0
2 1 0
3 2 3.01
4 2 3.01
5 2 3.01
6 4 6.02
7 4 6.02
8 8 9.03
9 8 9.03
10 16 12.04


Analysis of Various Constellation Schemes Utilizing Filter 1 as Reference

This section presents the performance analysis of various modulation schemes

using the same filter. This is the same data included in the previous graphs but grouped

together specifically to measure the BER performance delta across the modulations

schemes. The performance loss between modulation schemes is shown below in Table 5.

































Figure 47. Filter 1 OFDM Channelizer Multi-Modulation Scheme Performance

Table 5. Performance Advantage of Modulation Schemes
Modulation Eb/No Performance SNR Performance Eb/No Performance SNR Performance
Scheme Advantage @ BER = Advantage @ BER Advantage @ BER Advantage @
10e-2 (dB) = 10e-2 (dB) = 10e-5 (dB) BER = 10e-5 (dB)
BPSK (ref) 0 0 0 0
QPSK 0 -3.01 0 -3.01
8-PSK -2.25 -7.02 -2.2 -6.97
8-QAM -3 -7.77 -3.3 -8.07
16-QAM -3.55 -9.57 -3.9 -9.92
32-QAM -6.5 -13.49 -7 -13.99
64-QAM -7.65 -15.43 -7.8 -15.58
128-QAM -10.85 -19.3 -11.2 -19.65
256-QAM -12.05 -21.08 -13 -22.03


BER Normalization of Filter Banks through Constellation Density Compensation

It is desirable to take advantage of the process gain inherent across various filters

within the OFDM channelizer by normalizing the BER across each filter in the 48-

subcarrier OFDM channelizer. Given the measured BER across various modulation


1.E+00

1.E-01

1.E-02

1.E-03

1.E-04

. 1.E-05

1.E-06

1.E-07

1.E-08

1.E-09


- bpsk ber
- qpsk ber
8psk ber
16qam ber
- 64qam ber
- 256qam ber
- 8qam ber
-32qam ber
128qam ber


o 1 EbNob t, (dB) rrr
Eb/No (dB)









schemes in Table 5 as well as the performance advantage of each filter, G, in Table 4, an

attempt is made to normalize the probability of bit error across each filter by leveraging

the inherent process gain per filter to signal denser constellations while achieving similar

probabilities of bit error.

The selected modulation scheme per filter is given in Table 6 below along with the

expected residual performance advantage of each filter with respect to the first filter.

Note that the negative advantage denotes a loss in BER performance relative to the first

filter.

Table 6. Per-Filter Modulation Scheme for 48-Subcarrier OFDM Channelizer
Expected SNR Expected SNR
Filter Modulation Performance Performance
Number Scheme Advantage @ BER Advantage @ BER
= 10e-2 (dB) = 10e-5 (dB)
1 QPSK 0 0
2 QPSK 0 0
3 8-QAM -1.01 -0.96
4 8-QAM -1.01 -0.96
5 8-QAM -1.01 -0.96
6 16-QAM -0.56 -0.91
7 16-QAM -0.56 -0.91
8 32-QAM -1.48 -1.98
9 32-QAM -1.48 -1.98
10 64-QAM -0.42 -0.57

The measured BER performance of the above modulation scheme to filter mapping

is summarized in Figure 48 below. It is observed that the measured BER performance

correlates against the expected performance advantage based on the previous simulation

results.

The simulation results shown in Figure 48 demonstrate how the OFDM channelizer

can be adapted to utilize apparent limitations in order to compensate for the inherent

properties of the filters making up the OFDM channelizer. This residual performance









advantage delta of less than 2 dB can further be reduced by making certain adjustments,

such as puncturing, to the error control coding scheme per filter. Although this would not

have any net effect on the uncoded BER as measured in this chapter, the post-correction

BER would converge. Error control coding is not investigated in this thesis.


-- Filter 1
- Filter 2
Filter 3
Filter 4
-- Filter 5
--Filter 6
- Filter 7
- Filter 8
Filter 9
Filter 10


Sb SNR b 6
SNR


Figure 48. BER of 48-Subcarrier OFDM Channelizer in AWGN with Mixed Modulation


1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06
1.0E-07
1.0E-08
1.0E-09


rw














CHAPTER 5
OFDM CHANNELIZER PERFORMANCE AND LIMITATIONS IN MULTIPATH
CHANNEL

This chapter adapts an OFDM channelizer to allow for robust communications in a

typical multipath channel common in wireless communications. Various wireless

channel characteristics were presented in Chapter 2 along with inherent properties of

typical FFT-based OFDM modulation schemes capable of mitigating the effects of

multipath. This analysis will leverage a cyclic prefix extension in order to mitigate the

effects of multipath channel propagation conditions.

While considering the effect of multipath conditions on an OFDM channelizer, it is

important to note that there is a fundamental challenge with the OFDM channelizer that

must be solved. This problem is created by the coherent combining of multiple

subcarriers inherent in an OFDM channelizer receiver. Multipath channels have

frequency-varying phase and amplitude responses. This generally means that any two

subcarriers will experience different phase and amplitude responses through the channel.

Since an OFDM channelizer receiver coherently combines multiple subcarriers, some

amount of destructive interference will be observed at the receiver.

To mitigate this interference, an OFDM channelizer must provide some form of

alignment across the subcarriers that are common to each filter. In this manner, the

subcarriers will be coherently aligned in magnitude and / or phase prior to combining and

the transmitted symbols can be recovered. With phase-shift keying (PSK) modulation

schemes, it is sufficient for only the phases to be aligned between the subcarriers in order









to prevent destructive combining through the OFDM channelizer receiver. In quadrature

amplitude modulation (QAM) modulation schemes, both the amplitude and phase must

be aligned prior to coherent subcarrier combining at the receiver. Various weighted

combining schemes, such as maximal-ratio combining (MRC), can be used to achieve

optimal signal-to-noise ratio enhancements. However, these combining schemes are

outside of the scope of this thesis.

An example of this phenomenon is shown in the constellation scatter diagram

figures below. The below figures demonstrate the effects of multiple cyclic prefix

lengths on filter 6 subcarriers using QPSK modulation. A multipath channel described

later in this chapter was used to induce the frequency-varying phase and amplitude

response into the transmitted signal. All of the scatter diagrams below were generated

with an SNR of 30 dB.

Figure 49 demonstrates the scattering resulting from the ISI in the multipath

channel combined with the additive noise. Figure 50 and Figure 51 demonstrate similar

scattering except that two alternative alignment schemes have been used to coherently

combine the subcarriers associated with filter 6. The alignment schemes are further

discussed later in this Chapter. Note that the ranges of the axes in Figure 49 is larger than

the axes in the other two figures. This should be accounted for when comparing the

amount of scatter among the different scenarios.

The constellation diagrams above show a significant increase in effective SNR

when using subcarrier alignment versus non-aligned combining.























+ ':I.': Pi eii. = 1



+ : Pet = +



Figure 49. Multipath Constellation Scatter without Alignment

-2

+




Figure 49. Multipath Constellation Scatter without Alignment


':i .i: Prefl. = 1
*, I.: P[eD, =
.7 i,:hi: Pi el,. = 7
j' /:lI: F'I. -l 1 11


-2 ['


Figure 50. Multipath Constellation Scatter with Option 1 Alignment


-2 0 -.1


L? C










2']-


C, I,



.. + C .:I: PP.ii. = 1
I' I I C Pr [i =
'-' .'' C- .I, P rel. = 1 C'
C,- Preli, = IC
C-B.-:I -e -

.. i if
-






Figure 51. Multipath Constellation Scatter with Option 2 Alignment

OFDM Channelizer Subcarrier Separation

Recall one advantage ofFFT-based OFDM modulation is the potential to eliminate

the need for equalization by using differential modulation schemes [8]. While this is true

for OFDM channelizers as well, in multipath channel conditions, OFDM channelizers

still must separately filter the individual subcarriers in order to differentially demodulate

them individually prior to combining the subcarriers containing the same information.

This phenomenon is the largest potential disadvantage of the OFDM channelizer versus a

typical FFT-based OFDM system. Separation filters are required to accomplish this

separation for subcarriers common to a single OFDM channelizer filter.

The OFDM channelizer separation filter design parameters are unique per filter

because the frequency separation between multiple subcarriers common to a filter varies

per filter. The per-filter design parameters for the 48-point OFDM channelizer filters are







52


shown in the figures below. Note that both filters 1 and 2 generate a single harmonic and

therefore do not need any separation filtering and alignment to prevent destructive

combining. The specified separation design parameters are superimposed along with the

frequency response of the respective OFDM channelizer filter in the figures below. Each

separation filter is shown in a different color. In general, one separation filter is needed

per subcarrier that the OFDM channelizer filter produces.

Note from the filter prototypes in the figures above that the separation filters only

need to suppress energy from subcarriers belonging to the same OFDM channelizer filter.

Rejection of subcarriers from other OFDM channelizer filters is provided by the OFDM

channelizer filters themselves. This increases the available frequency span for a

particular filter's transition band to occupy. Also note that the filter prototypes shown in

the figures above are not symmetric about DC. Therefore, complex-valued filter


40

30

20

10




10 ---- -------------------------
-210 --/---------------- -- ----- ------------ -----_
I Filter 3



-130
-20
S-10 --- ------- --------- ---

-2 0 - -

3 0 ,-,-,- -- -- -- ----

-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5

Frequency (fs = 1 Hz)


Figure 52. Filter 3 Separation Filter Design Parameters




































Figure 53. Filter 4 Separation Filter Design Parameters


Figure 54. Filter 5 Separation Filter Design Parameters


40

30

20

3 10 ------------ ---- ---------
S10
Filter 4

0)

-10 -------------------------

-20 ---------- ------ -

-30 1 1 1 1 1
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5

Frequency (fs = 1 Hz)


40

30 -





Filter 5





-20 -

-30
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5

Frequency (fs = 1 Hz)





























-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2
Frequency (fs = 1 Hz)


Figure 55. Filter 6 Separation Filter Design Parameters


Figure 56. Filter 7 Separation Filter Design Parameters


Filter 6


0.3 0.4 0.5


40_

30

20

S10
Filter 7
c 0

-10i

-20 -

-30 -
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)






























-0.5 -0.4 -0.3 -0.2 -0.1


-Filter 8


-0 0.1 0.2 0.3 0.4 0.5


Frequency (fs = 1 Hz)


Figure 57. Filter 8 Separation Filter Design Parameters


Figure 58. Filter 9 Separation Filter Design Parameters


40

30







20
CM



-20


-30
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5
Frequency (fs = 1 Hz)










40

30

20

10

1Filter 10

0)

-10 -------

-20 ---- --- ----

-30
-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5

Frequency (fs = 1 Hz)


Figure 59. Filter 10 Separation Filter Design Parameters

coefficients are required to provide this frequency response. It should also be noted that

the duration of the filter's transition bands varies across the OFDM channelizers

proportionally to the number of subcarriers produced by each OFDM channelizer. This

tends to tighten the design constraints for the separation filter prototype as the number of

subcarriers in the respective OFDM channelizer filter increases. In general, the

separation filter prototypes for the OFDM channelizer filters generating a relatively low

number of subcarriers can be realized using multiplier-less FIR filters. The drawback of

this required frequency separation filtering is that high-rate multiplications may be

necessary in order to realize separation filters that provide sufficient subcarrier rejection

for OFDM channelizer filters producing a large number of subcarriers.

The analysis that follows focuses on filter 6 as a nominal case to analyze the effects

of multipath on the OFDM channelizer. This thesis does not focus on the optimal









separation filter design techniques that may be available to reduce or eliminate the

potential multiplications necessary to design the separation filters. Symmetric FIR filters

are used as the basis for the separation filters in this thesis. The separation FIR filter

coefficient sets designed for this analysis are shown in Table 7, Table 8, Table 9 and

Table 10 below while the corresponding frequency responses are shown in Figure 60.

The above filters are created by modifying a traditional rectangular windowed since

bandpass filter [15]. To accomplish this, an FFT of the real-valued coefficients is taken.

Next, either the positive or negative frequency Fourier coefficients are forced to zero.

Finally, an inverse FFT produces the resulting complex-valued coefficients. An FFT

larger than the number of filter taps is used for this purpose. The resulting complex-

valued coefficients are truncated in time to yield a filter impulse response equal in length

to the initial impulse response.

Approaches to Enhance the OFDM Channelizer for Multipath Channel Conditions

In order to determine the feasibility of providing sufficient channel separation,

coherent phase and amplitude alignment between subcarriers, and combining of the

appropriate subcarriers, two approaches will be considered. The first approach uses an

Table 7. Filter 6 Separation Filter Subcarrier +6 Coefficient Listing
Filter
Real Imaginary
Tap
1 -2.37335 1.39253
2 -3.56803 -1.49313
3 -1.91729 -4.48337
4 2.2473 -5.47087
5 5.84983 -2.3415
6 5.84983 2.3415
7 2.2473 5.47087
8 -1.91729 4.48337
9 -3.56803 1.49314
10 -2.37335 -1.39253










Table 8. Filter 6 Separation Filter Subcarrier -6 Coefficient Listing
Filter
r Real Imaginary
Tap
1 -2.37428 -1.39253
2 -3.56895 1.49314
3 -1.91822 4.48337
4 2.24637 5.47087
5 5.84891 2.3415
6 5.84891 -2.3415
7 2.24637 -5.47087
8 -1.91822 -4.48337
9 -3.56895 -1.49314
10 -2.37428 1.39253


Table 9. Filter 6 Separation Filter


Filter
Tap
1
2
3
4
5
6
7
8
9
10


Real


-1.10227
-1.6561
5.17983
-6.07262
2.70982
2.70982
-6.07262
5.17983
-1.6561
-1.10227


Subcarrier +18 Coefficient Listing


Imaginary

1.60279
-3.65139
1.6766
2.83778
-6.90155
6.90155
-2.83778
-1.6766
3.65139
-1.60279


Table 10. Filter 6 Separation Filter


Filter
Tap
1
2
3
4
5
6
7
8
9
10


Real


-1.09863
-1.65246
5.18347
-6.06898
2.71346
2.71346
-6.06898
5.18347
-1.65246
-1.09863


Subcarrier -18 Coefficient Listing


Imaginary

-1.60279
3.65139
-1.6766
-2.83778
6.90155
-6.90155
2.83778
1.6766
-3.65139
1.60279










40

30 Positive 6

20 l Negative 6

Positive 18
S0
M INegative 18
l -10
Filter 6 Freqency
-20 Response



0 0 0 0 0 0 0 0
Frequency (fs = 1 Hz)


Figure 60. Filter 6 Separation Filter Frequency Responses

open-loop equalization scheme directly at the receiver. This approach assumes ideal

channel estimation and applies the compensation at the receiver after the subcarrier

separation filtering and prior to the coherent combining of the subcarriers and making

hard decisions of the received symbols. The second approach uses a closed-loop

equalization scheme by informing the transmitter of the per-subcarrier channel response

and having the transmitter pre-distort the subcarriers in order to allow for simple coherent

combining in an unmodified OFDM channelizer at the receiver.

The block diagram of option 1 is in Figure 61 below. This approach allows for

equalization at the receiver independent of any support from the transmitter. The

disadvantage of this approach is that the receiver must separate, channel estimate and

combine real-time. This has the negative effect of increasing the complexity and limiting

the maximum baseband operating frequency of the system, which is contrary to the










primary benefit of an OFDM channelizer. Note that all subcarriers must be processed

through a separate channelization filter. For the 48 subcarrier case, it increases the

number of channelizer filters from ten to 48. The transmitter for this approach remains

unchanged from the OFDM channelizer transmitter presented in Chapter 3.


CyclicPre Supplemented
x/ FTime / Frequency Cycc Senal-to- Channezer
RF RXI LPF ADC oExC ionSe r Channelizer
mSynch ronization Exteso Parallel Separation F Bn
RemovalFilters Filter Bank




Time/ Coding
Parallel-to- Channel Carrier Symbol-to-Bit __ Frequency Forward Error
Senal Colrrection Combining Demapping D ntel. Correction
Deinte/eaver CRC


Figure 61. Block Diagram of OFDM Channelizer Receiver for Option 1

The block diagrams of option 2 are shown in Figure 62 and Figure 63 below. This

approach allows for pre-compensation to be applied at the transmitter in order to

coherently align the phase and magnitude of each subcarrier belonging to a single filter at

the channel output. In this manner, the receive filters making up the OFDM channelizer

can operate as usual and combine the received subcarriers as demonstrated in Chapter 4.

Note that no additional channelizer filters are required at the receiver.

This method requires the transmitter to have knowledge of the per-subcarrier

channel characteristics in order to apply the pre-distortion to the transmitted signal and

inform the transmitter of the per-subcarrier channel response. This can be accomplished

by having the receiver compute a per-subcarrier estimation of the distortion introduced by

the channel. This estimation can potentially be simplified by establishing a channel

estimation procedure during which time the transmitter sequences through each

subcarrier in a filter one at a time. In this manner the receive filters would not suffer the

loss associated with destructive combining of multiple subcarriers while performing this























Figure 62. Block Diagram of OFDM Channelizer Transmitter for Option 2


Figure 63. Block Diagram of OFDM Channelizer Receiver for Option 2

channel estimation. The transmitter and receiver would synchronously cycle through

patterns of single subcarriers being transmitted in a particular filter. Each filter can

perform the estimation sequence disjoint from the other filters. This method has the

disadvantage of sacrificing system capacity to handle these channel estimation

sequencing scenarios.

A procedure to mitigate this channel estimation overhead is to allow the receiver to

detect and signal a potential breakdown of performance on a per-filter basis. When a

non-optimal condition is detected and signaled to the transmitter, the subcarriers for that

filter would initiate a channel estimation transmission sequence for the subcarriers









contained within that filter. There are various approaches that could be chosen. The

most efficient method would strongly depend on the channel characteristics.

The primary application for this second approach is for a one-to-many type

network where all communication in the network goes through a single access point.

Examples of this sort of network topology exist in various wireless protocols including

802.1 la/b/g as well as mobile wireless standards. The requirement for this second option

is that the access point would be capable of applying a per-subcarrier pre-distortion to

each sub-carrier individually. This is a viable assumption since many on-to-many

networks allow for relatively more complex and costly access points to be able to handle

specific network management tasks. Therefore, it is reasonable to assume that the access

point in this network could have an FFT-based OFDM transmission scheme while the

numerous end-points could have lower-cost OFDM channelizers.

In this manner, the transmitter could handle the pre-distortion on a per-subcarrier

basis and allow the receiver to have a much simpler and therefore cheaper OFDM

channelizer based receiver. The network would benefit by having the large majority of

its nodes have relatively lower cost.

Multipath Effects on BER Performance

As described above, multipath in a channel has the potential to distort multiple

subcarriers associated with the same filter in a manner such that the combining in the

OFDM channelizer receiver combines the subcarriers in a non-optimal fashion. This

section will demonstrate this phenomenon through figures captured in simulation. A

Rappaport channel is used for this demonstration [9]. The Non-Line-of-Sight model is

given in Equation (7) below.









( -n

L(d) (7)


In order to simplify the analysis, only filter 6 is included in the analysis. Recall

that filter 6 generates frequency bins 6 and 18. The characteristic parameters for the

Rappaport channel model used for this analysis are given below.

* n = Path loss exponent; typical range of n is 3.5 < n < 5
* d = Distance (separation) between transmit and receive antennas
* do = Reference distance or free space propagation corner distance
* LB = Propagation loss of the LOS path for do[m]
* L = Loss (propagation loss) of the combined NLOS and LOS signal path


The Rappaport channel is based on empirically gathered field data and the model is

statistical in nature with randomly generated multipath weights. The actual path weights

used for this analysis are given in Table 11 below.

The frequency response of the Rappaport channel described above is shown in

Figure 64 below.

Table 11. Rappaport Multipath Channel Tap Weights
Multipath
Multipath Real Imaginary
Channel Tap
1 0 0
2 -0.48967 0.39845
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
9 0.02935 -0.35591
10 -0.15027 0.35342
11 -0.19401 0.53741











50

40

30
S30---A ----h------ ----ft--

S 20 -- Multipath Channel
0 A AAAn fiAA AA Response
3 1 II 1
0 Filter6 Freqency
S0- Response

-10-

-20

-30
L0 q CO C CO 0 4O I0P- LO CO CN4 LO
0 0 0 0 0 o C 0
C' C- 0 0 0 0 0
I I I I I I o oo od
Frequency (fs = 1 Hz)


Figure 64. Rappaport Multipath Channel Frequency Response

The performance simulation results of the OFDM channelizer filter 6 are shown in

Figure 65 below. The simulations do not contain any alignment of the subcarriers prior

to subcarrier combining. They are intended as a reference to measure gains of the two

subcarrier alignment options.

It should be noted that there is an inherent noise floor introduced by the multipath

channel. This noise floor prevents any modulation scheme with order greater than 16

from achieving BERs lower than approximately 0.1. Additionally, all modulation

schemes show significant degradation when compared to the AWGN performance

simulations explored in Chapter 4.

The noise floor introduced by the multipath channel has two main components.

First, the destructive subcarrier combining reduces the effective per-filter signal level at

the receiver. Second, the ISI introduced by the multipath channel causes a scattering of









the received constellation absent any actual noise contribution. The two primary

components of the degradation shown above will be mitigated in the following analysis.

Multipath Effects on OFDM Channelizer with Coherent Alignment

Simulation results are shown below to demonstrate the performance gains

achievable through coherent alignment and combining of an OFDM channelizer signal as

described previously. This analysis includes both options 1 and 2 presented previously.

This analysis again utilizes filter 6 to perform the performance analysis. This analysis

assumes ideal channel estimation in the coherent alignment of the multiple subcarriers

belonging to filter 6. Figure 66 illustrates the performance achieved through option 1

while Figure 67 illustrates the performance achieved through option 2.

One can observe that the performance is significantly degraded for both option 1 and

option 2 above when compared to the AWGN simulations presented in Chapter 4.

Additionally, there is an inherent noise floor visible in the figures above similar to the

noise floor in the performance results without coherent combining compensation,

however not as pronounced.

Multipath Effects on OFDM Channelizer with Cyclic Prefix

As previously described, FFT-based OFDM modulation schemes benefit from the

presence of a guard interval typically based on a cyclic prefix extension. The simulation

results provided below demonstrate a similar performance gain for an OFDM

channelizer. Simulation results with various cyclic prefix lengths are shown in Figure 68,

Figure 69 and Figure 70 below.

One can observe that the performance of an OFDM channelizer benefits from a

cyclic prefix extension in multipath channel conditions. It should be noted that, in

general, that while the noise floor of the higher order constellations is removed as the







66



1.E+00

1.E-01
-,-QPSK
1.E-02
8-QAM
1.E-03 SK
8-PSK
1.E-04 -16-QAM

a 1.E-05 -*-32-QAM

1.E-06 --64-QAM

1.E-07 ---128-QAM
256-QAM

1.E-08
1 .E-08 256-QAM


1.E-09


SNR


Figure 65. OFDM Channelizer Filter 6 BER Results


1.E+00

1.E-01 --BPSK
S---QPSK
1.E-02
8-PSK
1.E-03 8-
8-QAM
$ 1.E-04 -- 16-QAM

a 1.E-05 _-32-QAM

1.E-06 -+-64-QAM

1.E-07 128-QAM
256-QAM
1.E-08

1.E-09
r-T- -- r- o M ID MI LOI

SNR


Figure 66. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1











1.E+00

1.E-01 -- BPSK

1.E-02 -u-QPSK
8-PSK
1.E-03
8-QAM
1.E-04
-- 16-QAM
0.. 1.E-05
--. 5 32-QAM

1.E-06 -I-64-QAM

1.E-07 --128-QAM

1.E-08 256-QAM

1.E-09
r-- IT-- -- C- M IID M I LO

SNR


Figure 67. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2


1.E+00

1.E-01 BPSK

1.E-02 ----QPSK
8-PSK
1.E-03 8-Q AM
-IL
z 1.E-04 -- 16-QAM
32-QAM
1.E-05
-64-QAM
1.E-06 --128-QAM

1.E-07 256-QAM

1.E-08
v-- -- v-- v-- 04 04 4c 0c co co

SNR


Figure 68. BER for Filter 6 OFDM Channelizer in Multipath Channel and 4 Sample
Cyclic Prefix











1.E+00

1.E-01

1.E-02
-*-BPSK
1.E-03 -QP
---QPSK
1.E-04 8-PSK

1 1.E-05 8-QAM
16-QA M
1.E-066-QAM
-.-32-QAM
1.E-07
---64-QAM
1.E-08 128QAM
--128-QAM
1.E-09 256-QAM
O ', O O CN 0 CO CO

SNR


Figure 69. BER for Filter 6 OFDM Channelizer in Multipath Channel and 7 Sample
Cyclic Prefix


1.E+00

1.E-01--
-U-)
1.E-02


;"-
1.E-03 I
-I II
1.E-04

1.E-05 -

1.E-06 -- -

1.E-07

1.E-08
I- I 0 L 00 IT P- 0 CO CO C^ CO

SNR


Figure 70. BER for Filter 6 OFDM Channelizer in Multipath Channel and 10 Sample
Cyclic Prefix









cyclic prefix extension approaches the length of the memory in the channel, the

performance of the lower order modulation schemes is relatively unchanged with the

increase in length of the cyclic prefix extension.

Multipath Effects on OFDM Channelizer with Coherent Alignment and Cyclic
Prefix

The final simulation analysis combines the two performance enhancements

described above, namely coherent alignment of multiple subcarriers as well as the use of

a cyclic prefix extension. Both coherent alignment options 1 and 2 are explored in the

simulation results below.

Two notable trends can be observed in the figures above when comparing option 1

and option 2. First, the separation filters necessary for option 1 introduce additional

multipath delay spread. This increases the required cyclic prefix length necessary to

mitigate the effects of multipath. Second, the performance with option 2 exceeds the

performance of option 1.

Summary of Performance Comparison

A performance comparison is shown in the tables below. These tables compare

performance against the various simulation scenarios presented above. The AWGN

performance is used as a reference and the values in the tables are the losses, in dB, of

each scenario relative to the AWGN simulation with the same effective SNR.

As a general trend, the performance of all scenarios increase (e.g. the loss relative to

AWGN decreases) as the cyclic prefix is extended to a point equal to the ISI present in

the system. It can be seen that the performance of the system without subcarrier

alignment never gets smaller than 10 dB degradation relative to AWGN performance

even with a cyclic prefix extension. By comparison, the performance of option 1 gets










within 2 dB of AWGN performance while option 2 gets within 1 dB of AWGN

performance. This demonstrates the significant gains achievable through coherent

alignment in a system based on an OFDM channelizer scheme.

It should be noted that in order to achieve optimal performance with option 1 a

cyclic prefix approximately 50% longer is necessary. This additional cyclic prefix

extension directly reduces the available system capacity since any time allocated to a

cyclic prefix extension is not available for information transmission. In this example

with a 48-subcarrier system, the data rate reduction due to a 10 sample cyclic prefix is

82.8%, while the data rate reduction due to a 16 sample cyclic prefix is 75%.


1.E+00

1.E-01 ~ BPSK
--QPSK
1.E-02 PSK

S 1.E-03 -QAM
-a- -16-QAM
1.E-04 --32AM
\---32-QAM

1.E-05 -- 64-QAM
--128-QAM
1.E-06 =
256-QAM

1.E-07
o ,- 4 0 1- C0 CD CN
NI CN N co
SNR


Figure 71. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and
4 Sample Cyclic Prefix











1.E+00

1.E-01 -*--BPSK
QPSK
1.E-02

1.E-03 8--AM

t' 1.E-04 -- i16-QAM

1.E-05 -*-32-QAM
~- -64-QAM
1.E-06 -128-QAM

1.E-07 256-QAM

1.E-08
0 1 0N (N CO

SNR


Figure 72. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and
7 Sample Cyclic Prefix


1.E+00

1.E-01 -_-* -

1.E-02 --

1.E-03 -
-
1.E-04 -: ,

1.E-05 --

1.E-06 -

1.E-07 -

1.E-08 -
o -- CN co -- 0 CD CN
S- CN CN CO

SNR


Figure 73. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 1 and
10 Sample Cyclic Prefix












1.E+00

1.E-01

1.E-02

1.E-03

1.E-04

1.E-05

1.E-06

1.E-07

1.E-08

1.E-09


_.E- 8* PS




1.E-0 8 QA


-*-- BPSK

--QPSK

8-PSK

8-QAM
--16-QAM

-*-32-QAM

-1-64-QAM

--128-QAM

256-QAM


=2


0 (D C%4
C%4 C%4 co


O ,1- 0C c I-

SNR


Figure 74. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and
4 Sample Cyclic Prefix


1.E+00

1.E-01 -_ BPSK

1.E-02 --QPSK


8-QAM
1.E-04
1. -- 16-QAM
a 1.E-05
--32-QAM
1.E-06 -64-QAM

1.E-07 -- 128-QAM

1. E-08 256-QAM

1.E-09
0 ,1- CN co 0 (D CN
-- CN CN CO

SNR



Figure 75. BER for Filter 6 OFDM Channelizer in Multipath Channel with Option 2 and
7 Sample Cyclic Prefix


-

































Figure 76. BER for Filter 6 OFDM
10 Sample Cyclic Prefix


Channelizer in Multipath Channel with Option 2 and


Table 12. Multipath Fading Performance Delta (@ BER = le-2) without Alignment
(AWGN reference)
No
No No Alignment / No Alignment / No Alignment / 10
Constellation Alignment / 7
Cotel n A ment 4 Sample 7 Sample Sample Cyclic
SN C ic Cyclic Prefix Cyclic Prefix Prefix
Prefix
BPSK 12.8 11.2 11.1 10.2
QPSK 12.8 11.2 11 10.2
8-PSK 15 11.5 11 10.2
8-QAM 14.6 11.4 11 10.1
16-QAM 17 11.7 10.9 10.1
32-QAM NA 17.2 11 10.2
64-QAM NA NA 10.8 10.2
128-QAM NA NA 11 10.2
256-QAM NA NA 11 10.2


1.E+00
1.E-01 --- BPSK
1.E-02 -.-QPSK
1.E-03 8-PSK

" 1.E-04 8-QAM
'' 16-QAM
. 1.E-05
-*.-32-QAM
1.E-06
--64-QAM
1.E-07 -128-QAM
1.E-08 256-QAM
1.E-09
o ,1- CN oo '- 0 (O CN
t (N (N C
SNR










Table 13. Multipath Fading Performance Delta (@ BER = le-2) with Option 1 (AWGN
reference)
Option 2 Option 2 Option 2 Option 2
Constellation Alignment / Alignment / 4 Alignment/ 10 Alignment / 16
Type No Cyclic Sample Cyclic Sample Cyclic Sample Cyclic
Prefix Prefix Prefix Prefix
BPSK 5.6 3.8 2.9 2.6
QPSK 5.8 3.8 2.9 2.8
8-PSK 8.9 5 3 2.6
8-QAM 10.1 4.7 3 2.7
16-QAM 22.5 5.9 2.9 2.5
32-QAM NA NA 3 2.6
64-QAM NA NA 3.2 1.6
128-QAM NA NA 3.8 2.7
256-QAM NA NA 4.3 2.7


Table 14. Multipath Fading Performance Delta (@ BER
reference)


1e-2) with Option 2 (AWGN


Option 1 Option 1 Option 1 Option 1
Constellation Alignment / Alignment / 4 Alignment / 7 Alignment/ 10
Type No Cyclic Sample Cyclic Sample Cyclic Sample Cyclic
Prefix Prefix Prefix Prefix
BPSK 2.3 1.7 1.4 0.8
QPSK 2.5 1.6 1.1 0.8
8-PSK 3.8 2.1 1.4 0.9
8-QAM 5 2.2 1.2 0.8
16-QAM 8 2.8 1.3 0.7
32-QAM NA 8 1.8 0.7
64-QAM NA NA 2.2 0.8
128-QAM NA NA 4.4 0.8
256-QAM NA NA 7.6 0.8










Table 15. Multipath Fading Performance Delta (@ BER = le-5) without Alignment
(AWGN reference)
No
No No Alignment / No Alignment / No Alignment / 10
Constellation Alignment / 7
Cotel n A ment 4 Sample 7 Sample Sample Cyclic
SN C ic Cyclic Prefix Cyclic Prefix Prefix
Prefix
BPSK 13.5 11.6 11 10.3
QPSK 13.5 11.6 11 10.3
8-PSK 17.4 13 11.5 10.8
8-QAM 17.4 12.4 11.1 10.3
16-QAM 19.3 13.1 11 10.3
32-QAM NA 18.3 11 10.3
64-QAM NA 19.1 11 10.3
128-QAM NA NA 11 10.3
256-QAM NA NA 10.9 10.2



Table 16. Multipath Fading Performance Delta (@ BER = le-5) with Option 1 (AWGN
reference)
BPSK 6.4 4.3 3 2.6
QPSK 7 4.6 3.1 2.6
8-PSK 11.8 6.9 3.6 3
8-QAM 13.8 6 3 2.6
16-QAM NA 9.2 3.2 2.7
32-QAM NA NA 3.6 2.7
64-QAM NA NA 4 2.7
128-QAM NA NA 5.8 2.6
256-QAM NA NA 6.3 2.6
BPSK 6.4 4.3 3 2.6










Table 17. Multipath Fading Performance Delta (@ BER = le-5) with Option 2 (AWGN
reference)
Option 1 Option 1 Option 1 Option 1
Constellation Alignment / Alignment / 4 Alignment / 7 Alignment/ 10
Type No Cyclic Sample Cyclic Sample Cyclic Sample Cyclic
Prefix Prefix Prefix Prefix
BPSK 3 1.9 1.1 0.8
QPSK 3.4 2.2 1.4 1
8-PSK 6.7 4.1 2.1 1.2
8-QAM 7.8 3.7 1.5 0.8
16-QAM 18.2 6.5 1.9 0.9
32-QAM NA NA 3.3 0.9
64-QAM NA NA 4.7 0.9
128-QAM NA NA NA 0.9
256-QAM NA NA NA 0.9














CHAPTER 6
CONCLUSIONS AND FUTURE WORK

This thesis explores the usefulness of an OFDM channelizer in both AWGN and

multipath channel conditions in order to achieve the advantages of an FFT-based OFDM

modulation scheme while reducing the overall design complexities. The desired goal is

to leverage the reduced design complexity to realize a system capable of increased

baseband operating frequencies simultaneous with decreased cost. CIC filter-based

OFDM channelizers are interesting from an implementation complexity and operating

clock frequency perspective. They can potentially provide a low-complexity alternative

to implementing an FFT-based OFDM system that has the potential to operate at very

high clock rates due to the multiplier-less structure from which they are derived.

Summary of Simulation Effort

The simulation effort consisted of multiple phases. The first phase involved

creating a functional model of the OFDM channelizer and simulation system capable of

analyzing the performance of the OFDM channelizer. The second phase involved

calibrating and capturing the performance of the OFDM channelizer in an AWGN

channel. The third phase consisted of measuring the potential performance loss of an

OFDM channelizer in a multipath channel. The fourth phase included development of

two different methods for overcoming the potential destructive combining of multiple

subcarriers belonging to a common OFDM channelizer filter. The final phase included a

performance comparison of the two methods assuming various cyclic prefix lengths.









Lessons Learned and Future Work

A number of lessons were learned during the analysis contributing to this thesis.

The results of these lessons were presented in previous chapters. A few notable lessons

learned are noted below.

* Typical Eb/No BER curves presented in textbooks and other reference material
measure complex-valued noise power even if the signaling is real-valued BPSK.
Therefore the noise power relative to the signal power must be calibrated assuming
complex valued noise. Without taking this into account, this can introduce a 3 dB
error in the system calibration.

* Certain wireless channel models (i.e. Rummler [9]) are represented with
mathematical models that result in symmetric frequency responses. Given the
symmetric frequency response of all OFDM channelizer filters, this likely leads to
channel phase responses that cancel each other at the receiver. In reality, channel
responses are not symmetric and the analysis could falsely conclude that
destructive interference at the receiver is not a limitation.

* Given the impulse responses of each OFDM channelizer filter, each filter can have
an optimal sampling point. These ideal sampling points were found to be located
within one or two samples of the end of the received symbols. Furthermore, given
the ideal sampling point of each filter, a negation of the received complex-valued
samples is necessary at the receiver for correct interpretation of the received bit
stream.

Many interesting aspects of OFDM channelizers were explored in this thesis. Still

many more aspects can be considered. The list below describes further analysis that

should be investigated in order to further refine the advantages of an OFDM channelizer

beyond those of a typical FFT-based OFDM system.

* Investigate multiplier-less filters for OFDM channelizer subcarrier separation in
multipath channels.

* Investigate fixed-point requirements for an OFDM channelizer and how this fixed-
point design might compare to an FFT-based OFDM channelizer.

* Implement and synthesize an OFDM channelizer targeting current state-of-the-art
technology to determine actual realizable operating speed. The realizable operating
speed of an OFDM channelizer should be contrasted against an FFT-based OFDM
channelizer.









* Investigate utilizing variable-rate error correction coding in order to achieve more
uniform parity of BER across various filter banks with different process gains.

* Investigate benefit of maximal ratio combining. Theoretically, maximal ratio
combining provides optimal reception when combining diversity branches through
a channel providing uncorrelated diversity through the branches.

* Investigate performance of OFDM channelizer in a fading channel similar to
mobile wireless channel models. Consideration should be taken into account for
the channel coherence time relative to the time necessary to estimate the channel
conditions.

Summary of Simulation Performance Results

It has been shown that the performance of an OFDM channelizer meets the

expected performance for a more general matched filter receiver in an AWGN channel.

It has also been shown that the performance of an OFDM channelizer in a multipath

channel can approach the performance of a matched filter in an AWGN channel. This

can be accomplished through the use of coherent subcarrier combining either provided

either by pre-compensation the transmitter or by separation filters at the receiver as well

as through the use of a cyclic prefix extension scheme.

Given system design constraints, the transmitter-based pre-compensation scheme

has a limitation in a fading channel where the channel estimate changes over time. This

is due to the time necessary to both make the estimate and deliver the estimate to the

transmitter before it can start applying the pre-distortion to its transmission. During this

elapsed time, the channel conditions will change and might not be well correlated to the

channel conditions from when the channel estimate was measured.

The receiver-based separation filter scheme has a limitation of requiring additional

filters to perform subcarrier separation in order to compute per-subcarrier channel

estimations. This must be done prior to coherent subcarrier combining. These separation

filters potentially introduce high-rate multipliers into the system architecture. Design









consideration needs to be made to ensure the order of the OFDM channelizer is kept

sufficiently small so as not to require extremely steep skirt filter design parameters for the

separation filters. Finally, the separation filters can increase the delay spread observed at

the receiver and effectively increase the required cyclic prefix length. This longer cyclic

prefix reduces the effective bit rate achievable across the channel.

All other design constraints being equal, option 1 subcarrier alignment is shown to

yield superior performance compared to option 2. The two primary factors contributing

to this are the 1-2 dB performance advantage that option 1 gives over option 2 as well as

the additional cyclic prefix extension necessary with option 2 to achieve optimal

performance. This additional cyclic prefix is necessary to eliminate the additional ISI

introduced by the separation filters themselves.














APPENDIX A
POLYMORPHIC-BASED SPW OVERVIEW

As mentioned in the introduction, SPW is a hierarchal, block-based modeling and

simulation environment useful for performing system analysis. SPW is capable of

executing both small and large-scale system simulations. The polymorphic technology

within SPW extends the flexibility and capability of the tool by allowing a single

representation of a system capture both the floating-point and fixed-point design. The

SPW tool and accompanying polymorphic feature set are powerful but also complex. A

brief introduction to introduce the reader to this tool is provided in this appendix.

Polymorphic models available within SPW can be configured in a wide-variety of

block types (54 total). A block type consists of two sub-types: element type and

composite type. There are six element types and nine composite types that combine to

produce the 54 total block types (6 element types 9 composite types = 54 block types).

The element type defines the type of each element within the signal operated on by the

block. Examples of element types are: 'Double,' 'ComplexDouble,' 'Fixed-Point' and

'ComplexFixed-Point.' The composite type defines the composite structure type on

which the block operates. Examples of composite types are: 'Scalar' (none), 'Vector'

and 'Matrix.' SPW polymorphic block types also support various video signal formats

that will not be utilized in this thesis. Figure 77 shows the presentation of the block type

information of a polymorphic block within SPW.












N I or- Composite type
RGB Long
8 --- Element type
ine

X /n out





Figure 77. Polymorphic Block Type Illustration

In addition to the block type information, it can be seen in Figure 77 that additional

configuration information is listed on the symbol of a block. This information is referred

to as the default value information of the block. In Figure 78, two of the fields

composing a default value field are listed. The '0.5' is the initial value (constant value in

this example) that the block produces at its output. The '<8,0,t>' are the fixed-point

attributes that are defined for the block if the block's type is set for a fixed-point type. In

order to further define these fixed-point attributes: '8' is the total number of bits

(including optional sign bit), '0' is the bit position of the most significant bit (MSB), not

including any sign bit, and 't' denotes two's complement signal representation as

opposed to 'u' for unsigned signal representation. This parameter in the default value

field is ignored if the block's type is not set to a fixed-point type. Two other optional

parameters in the default value field are the composite type size (i.e. vector size) and the

fixed-point modes of operation. These two parameters are not shown in Figure 78.

The composite type size, when present, is enclosed within square brackets "[]."

The fixed-point modes-of-operation parameter is composed of two parameters: loss-of-

precision mode and overflow mode. These two parameters are specified within









parentheses and separated by a comma. The first parameter is the loss-of-precision mode

and a couple of examples are 'truncation' and 'round.' The second parameter is the

overflow mode and the two possible settings are 'clip' and 'wrap.'

BLOCK TYPE DEFAULT VALUE
parameter, which specifies the fixed-point
attributes that convert the output data.

Doub le ---
0.5 <8,0,t> U I ue: 2
______ rFxp


Figure 78. Polymorphic Default Value Illustration


















APPENDIX B
SIMULATION RESULTS RAW DATA

Table 18. Raw Data for Figure 38
SNR(dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter7 Filter 8 Filter 9 Filter 10
-23 4.60E-1 4.60E-1 4.44E-1 4.44E-1 4.44E-1 4.21E-1 4.21E-1 3.88E-1 3.89E-1 3.44E-1
-22 4.55E-1 4.55E-1 4.37E-1 4.37E-1 4.37E-1 4.11E-1 4.11E-1 3.75E-1 3.75E-1 3.27E-1
-21 4.50E-1 4.50E-1 4.29E-1 4.29E-1 4.29E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.07E-1
-20 4.44E-1 4.44E-1 4.21E-1 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1
-19 4.37E-1 4.37E-1 4.11E-1 4.11E-1 4.11E-1 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1
-18 4.29E-1 4.29E-1 4.01E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.07E-1 3.07E-1 2.38E-1
-17 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1 2.86E-1 2.12E-1
-16 4.11E-1 4.11E-1 3.76E-1 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1 2.63E-1 1.85E-1
-15 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.61E-1 3.08E-1 3.08E-1 2.38E-1 2.39E-1 1.57E-1
-14 3.89E-1 3.89E-1 3.45E-1 3.45E-1 3.45E-1 2.86E-1 2.86E-1 2.12E-1 2.12E-1 1.30E-1
-13 3.76E-1 3.76E-1 3.27E-1 3.27E-1 3.27E-1 2.63E-1 2.63E-1 1.85E-1 1.85E-1 1.03E-1
-12 3.61E-1 3.61E-1 3.08E-1 3.08E-1 3.08E-1 2.39E-1 2.39E-1 1.57E-1 1.58E-1 7.77E-2
-11 3.45E-1 3.45E-1 2.86E-1 2.87E-1 2.86E-1 2.13E-1 2.13E-1 1.30E-1 1.30E-1 5.54E-2
-10 3.27E-1 3.27E-1 2.64E-1 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.03E-1 1.03E-1 3.68E-2
-9 3.08E-1 3.08E-1 2.39E-1 2.39E-1 2.39E-1 1.58E-1 1.58E-1 7.79E-2 7.79E-2 2.24E-2
-8 2.87E-1 2.87E-1 2.13E-1 2.13E-1 2.13E-1 1.30E-1 1.30E-1 5.56E-2 5.56E-2 1.22E-2
-7 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.86E-1 1.03E-1 1.03E-1 3.70E-2 3.70E-2 5.76E-3
-6 2.39E-1 2.39E-1 1.58E-1 1.58E-1 1.58E-1 7.82E-2 7.82E-2 2.25E-2 2.25E-2 2.29E-3
-5 2.13E-1 2.13E-1 1.30E-1 1.30E-1 1.30E-1 5.59E-2 5.59E-2 1.22E-2 1.22E-2 7.33E-4
-4 1.86E-1 1.86E-1 1.03E-1 1.03E-1 1.04E-1 3.72E-2 3.72E-2 5.79E-3 5.80E-3 1.80E-4
-3 1.58E-1 1.58E-1 7.84E-2 7.84E-2 7.84E-2 2.26E-2 2.26E-2 2.31E-3 2.31E-3 3.08E-5
-2 1.31E-1 1.31E-1 5.61E-2 5.61E-2 5.61E-2 1.23E-2 1.23E-2 7.46E-4 7.42E-4 3.20E-6
-1 1.04E-1 1.04E-1 3.74E-2 3.74E-2 3.73E-2 5.85E-3 5.86E-3 1.83E-4 1.82E-4 1.10E-7
0 7.86E-2 7.86E-2 2.28E-2 2.28E-2 2.28E-2 2.33E-3 2.34E-3 3.34E-5 3.19E-5 2.00E-8
1 5.63E-2 5.62E-2 1.24E-2 1.24E-2 1.24E-2 7.53E-4 7.52E-4 3.68E-6 3.62E-6
2 3.75E-2 3.75E-2 5.90E-3 5.91E-3 5.90E-3 1.82E-4 1.85E-4 2.40E-7 2.80E-7
4 1.25E-2 1.25E-2 7.62E-4 7.62E-4 7.63E-4 3.00E-6 4.05E-6
5 5.94E-3 5.96E-3 1.89E-4 1.88E-4 1.89E-4 1.90E-7 1.80E-7
6 2.38E-3 2.39E-3 3.28E-5 3.30E-5 3.28E-5 1.00E-8 1.00E-8
7 7.67E-4 7.74E-4 3.85E-6 3.90E-6 3.62E-6
8 1.88E-4 1.91E-4 2.00E-7 3.30E-7 2.70E-7
9 3.28E-5 3.34E-5 2.00E-8 1.00E-8
10 3.48E-6 4.01E-6
11 2.20E-7 3.30E-7
12 2.00E-8 1.00E-8











Table 19. Raw Data for Figure 39
SNR(dB) Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 Filter 6 Filter7 Filter 8 Filter 9 Filter 10
-23 4.72E-1 4.72E-1 4.60E-1 4.60E-1 4.60E-1 4.44E-1 4.44E-1 4.21E-1 4.21E-1 3.89E-1
-22 4.68E-1 4.68E-1 4.55E-1 4.55E-1 4.55E-1 4.37E-1 4.37E-1 4.11E-1 4.11E-1 3.75E-1
-21 4.64E-1 4.65E-1 4.50E-1 4.50E-1 4.50E-1 4.29E-1 4.29E-1 4.00E-1 4.00E-1 3.61E-1
-20 4.60E-1 4.60E-1 4.44E-1 4.44E-1 4.44E-1 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.45E-1
-19 4.55E-1 4.55E-1 4.37E-1 4.37E-1 4.37E-1 4.11E-1 4.11E-1 3.75E-1 3.75E-1 3.27E-1
-18 4.50E-1 4.50E-1 4.29E-1 4.29E-1 4.29E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.07E-1
-17 4.44E-1 4.44E-1 4.21E-1 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1
-16 4.37E-1 4.37E-1 4.11E-1 4.11E-1 4.11E-1 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1
-15 4.29E-1 4.29E-1 4.01E-1 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.07E-1 3.07E-1 2.38E-1
-14 4.21E-1 4.21E-1 3.89E-1 3.89E-1 3.89E-1 3.45E-1 3.45E-1 2.86E-1 2.86E-1 2.12E-1
-13 4.11E-1 4.11E-1 3.76E-1 3.76E-1 3.76E-1 3.27E-1 3.27E-1 2.63E-1 2.63E-1 1.85E-1
-12 4.01E-1 4.01E-1 3.61E-1 3.61E-1 3.61E-1 3.08E-1 3.08E-1 2.39E-1 2.39E-1 1.58E-1
-11 3.89E-1 3.89E-1 3.45E-1 3.45E-1 3.45E-1 2.87E-1 2.87E-1 2.13E-1 2.13E-1 1.30E-1
-10 3.76E-1 3.76E-1 3.27E-1 3.27E-1 3.27E-1 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.03E-1
-9 3.61E-1 3.61E-1 3.08E-1 3.08E-1 3.08E-1 2.39E-1 2.39E-1 1.58E-1 1.58E-1 7.79E-2
-8 3.45E-1 3.45E-1 2.87E-1 2.87E-1 2.87E-1 2.13E-1 2.13E-1 1.30E-1 1.30E-1 5.57E-2
-7 3.28E-1 3.28E-1 2.64E-1 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.03E-1 1.03E-1 3.70E-2
-6 3.08E-1 3.08E-1 2.39E-1 2.39E-1 2.39E-1 1.58E-1 1.58E-1 7.81E-2 7.81E-2 2.25E-2
-5 2.87E-1 2.87E-1 2.13E-1 2.13E-1 2.13E-1 1.30E-1 1.30E-1 5.58E-2 5.58E-2 1.23E-2
-4 2.64E-1 2.64E-1 1.86E-1 1.86E-1 1.86E-1 1.03E-1 1.04E-1 3.71E-2 3.71E-2 5.81E-3
-3 2.39E-1 2.40E-1 1.58E-1 1.58E-1 1.58E-1 7.84E-2 7.85E-2 2.26E-2 2.26E-2 2.32E-3
-2 2.13E-1 2.14E-1 1.31E-1 1.31E-1 1.31E-1 5.61E-2 5.61E-2 1.23E-2 1.23E-2 7.40E-4
-1 1.86E-1 1.86E-1 1.04E-1 1.04E-1 1.04E-1 3.73E-2 3.74E-2 5.85E-3 5.85E-3 1.80E-4
0 1.59E-1 1.59E-1 7.87E-2 7.87E-2 7.86E-2 2.27E-2 2.28E-2 2.34E-3 2.34E-3 3.10E-5
1 1.31E-1 1.31E-1 5.63E-2 5.63E-2 5.63E-2 1.24E-2 1.24E-2 7.51E-4 7.51E-4 3.64E-6
2 1.04E-1 1.04E-1 3.75E-2 3.75E-2 3.75E-2 5.90E-3 5.91E-3 1.85E-4 1.84E-4 2.20E-7
3 7.89E-2 7.89E-2 2.29E-2 2.29E-2 2.29E-2 2.36E-3 2.37E-3 3.28E-5 3.24E-5 1.50E-8
4 5.65E-2 5.65E-2 1.25E-2 1.25E-2 1.25E-2 7.63E-4 7.62E-4 3.52E-6 3.77E-6
5 3.77E-2 3.77E-2 5.96E-3 5.95E-3 5.95E-3 1.86E-4 1.88E-4 2.35E-7 2.90E-7
6 2.30E-2 2.30E-2 2.39E-3 2.38E-3 2.39E-3 3.31E-5 3.31E-5 1.00E-8 1.50E-8
7 1.26E-2 1.26E-2 7.73E-4 7.69E-4 7.74E-4 3.70E-6 3.91E-6
8 6.00E-3 6.01E-3 1.92E-4 1.91E-4 1.92E-4 2.70E-7 2.70E-7
9 2.41E-3 2.41E-3 3.39E-5 3.36E-5 3.33E-5 5.00E-9 2.00E-8
10 7.79E-4 7.83E-4 3.95E-6 3.85E-6 4.06E-6
11 1.93E-4 1.94E-4 2.15E-7 3.15E-7 2.55E-7
12 3.41E-5 3.41E-5 1.00E-8 1.50E-8
13 3.79E-6 4.11E-6
14 2.75E-7 2.55E-7
15 1.50E-8 5.00E-9