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ANALYSIS OF TERNARYVALUED, CIC FILTERBASED, 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 48Subcarrier OFDM Channelizer Description ............. ................... .................27 4 OFDM CHANNELIZER PERFORMANCE IN AWGN CHANNEL .....................38 Performance of 48Point 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 POLYMORPHICBASED 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 48Subcarrier OFDM Channelizer Filter Banks.................31 4 Performance Advantage of Filters Based on Number of Subcarriers ...................44 5 Performance Advantage of Modulation Schemes.................. ........................... 45 6 PerFilter Modulation Scheme for 48Subcarrier 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 = le2) without Alignment (AW GN reference) .................................... ...................................... 73 13 Multipath Fading Performance Delta (@ BER = le2) with Option 1 (AWGN reference) ..................................................... ................... ..... ........ 74 14 Multipath Fading Performance Delta (@ BER = le2) with Option 2 (AWGN reference) ..................................................... ................... ..... ........ 74 15 Multipath Fading Performance Delta (@ BER = 1e5) without Alignment (AW GN reference) .................................... ...................................... 75 16 Multipath Fading Performance Delta (@ BER = le5) with Option 1 (AWGN reference) ..................................................... ................... ..... ........ 75 17 Multipath Fading Performance Delta (@ BER = le5) 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 ilterB ased Interpolator ................................................................. .. ..... 15 6 C IC FilterB 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 FilterBased Interpolator.... ........... ....................................24 22 Optimized CIC FilterBased Decimator ............ ...........................................24 23 Optimized OFDM Channelizer Interpolator................................ ................. 24 24 Optimized OFDM Channelizer Decimator.................................. .............. 24 25 Block Diagram of an OFDM ChannelizerBased Transmitter ........................... 25 26 Block Diagram of an OFDM ChannelizerBased 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 48Subcarrier OFDM Channelizer in AWGN with BPSK Modulation .....39 39 BER of 48Subcarrier OFDM Channelizer in AWGN with QPSK Modulation.....40 40 BER of 48Subcarrier OFDM Channelizer in AWGN with 8PSK Modulation.....40 41 BER of 48Subcarrier OFDM Channelizer in AWGN with 8QAM Modulation...41 42 BER of 48Subcarrier OFDM Channelizer in AWGN with 16QAM Modulation.41 43 BER of 48Subcarrier OFDM Channelizer in AWGN with 32QAM Modulation.42 44 BER of 48Subcarrier OFDM Channelizer in AWGN with 64QAM Modulation.42 45 BER of48Subcarrier OFDM Channelizer in AWGN with 128QAM M o du latio n ...................................... .............................. ............... 4 3 46 BER of 48Subcarrier OFDM Channelizer in AWGN with 256QAM M o du latio n ...................................... .............................. ............... 4 3 47 Filter 1 OFDM Channelizer MultiModulation Scheme Performance ...................45 48 BER of 48Subcarrier 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 FirstOrder OFDM Channelizer Transmission Filter.............. 109 84 Block Diagram of SecondOrder 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 FirstOrder OFDM Channelizer Receive Filter ............. 115 90 Block Diagram of SecondOrder 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 TERNARYVALUED, CIC FILTERBASED, 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 complexvalued multiplications in order to realize the FastFourier Transform (FFT) central to most OFDMbased systems. This thesis explores the feasibility of using OFDM channelizers to realize the benefits inherent in FFTbased 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 FFTbased 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 FFTbased 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 FFTbased 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 MIMObased (multipleinput, multipleoutput) 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 singlecarrier modulation symbols, in order to mitigate the effects of intersymbol 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 OFDMbased communication system. An alternative in the form of a ternaryvalued, 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 filterbased 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 filterbased 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 FFTbased 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 simulationbased analysis. The primary tool for performing this analysis is Coware's Signal Processing Worksystem (SPW) version 4.9. SPW is a systemlevel 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. Polymorphicbased designs are desirable as they allow simultaneous capture and representation of a design for both floatingpoint and fixedpoint implementations. These block diagrams are given as an alternative to code listings common in publications. Appendix A presents a highlevel 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. * BittoSymbol Mapping transforms a bit stream into complexvalued 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. * SerialtoParallel 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. * ParalleltoSerial serializes the timedomain 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 DigitaltoAnalog Converter that transforms the digital signal to analog. * LPF lowpass 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 antialiasing filter. Attenuates higher frequency components prior to sampling to reduce effects of aliasing caused by sampling. ADC AnalogtoDigital 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. SerialtoParallel 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. ParalleltoSerial serializes the frequencydomain signal for further processing. Channel Correction estimates the channel response on a percarrier basis. The channel estimate is used to normalize the channel response in the received signal. SymboltoBit 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 harddecision (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 Serialto RF RX/ LPF ADC z Extension Parallel FFT yRemoval Time Coding Parallelto Channel SymboltoBit 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 highspeed modems, digital mobile communications and highdensity 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 OFDMbased system versus a singlecarrier system with equalization for a given delay spread. OFDM has the potential to exploit distinct signaltonoise 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 peaktoaverage 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 singlecarrier 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 lineofsight (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 bitrate, singlecarrier 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 46 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 wellknown approach to mitigating the delay spread introduced by a multipath channel. They reduce or eliminate the intersymbol 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 worstcase 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 timedomain 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 timedomain 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 zerovalued insertion. However, a zerovalued guard period would have the negative sideeffect of introducing intercarrier interference (ICI) since integer numbers of carrier cycles are no longer guaranteed to exist within a single nonextended OFDM symbol duration. A cyclic extended guard period rather than a zerofilled 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 OFDMbased 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 FFTbased 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 wellknown, multiplierless finite impulse response (FIR) filter having a wide variety of applications. One of the more common applications is as a digital samplerate 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. Toplevel block diagrams of these samplerate converters are shown below in Figure 5 and Figure 6. R Comb Comb Comb Integrator Integrator Integrator Upsample Figure 5. CIC FilterBased Interpolator  Integrator Integrator e Integrator  Comb  Comb w Comb w IR Downsample Figure 6. CIC FilterBased 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) 1z1 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 zdomain. A block diagram of an integrator is given in Figure 7. Z1   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+ zR (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 0.290.190.08 0.02 0.12 0.23 0.33 0.44 Frequency (fs= 1 Hz) Figure 8. Magnitude Frequency Response of Integrator Filter zR*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 nonobvious 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 M1 NI H(z)= Z +zRM ,= 2Zk (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 0.290.190.08 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 0.290.190.08 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 0.290.190.08 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 0.290.190.08 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 0.290.190.08 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 CICbased 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. rI r zM 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 FilterBased Interpolator  Integrator  Integrator  Integrator w mJR W Comb  Comb  Comb Downsample Figure 22. Optimized CIC FilterBased 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 reducedpolynomial 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 FFTbased 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 Systemlevel block diagrams of a transmitter and receiver as part of an OFDM channelizerbased system are shown below in Figure 25 and Figure 26, respectively. It should be noted that the systemlevel block diagrams of the OFDM channelizer are similar to the original FFTbased 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 ChannelizerBased Transmitter Figure 26. Block Diagram of an OFDM ChannelizerBased 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 FFTbased 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) = zN 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 zN1 is chosen based on two relevant criteria for the reduction of the polynomial over the ternaryvalued 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 ternaryvalued coefficients is equal to the number of divisors of the polynomial order, N. For example, z121 results in 6 ternaryvalued 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 FFTbased OFDM system, every carrier can be independently modulated and demodulated. 48Subcarrier 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, z481 is shown in Table 3 along with various other filter characteristics. Table 3 highlights the primary difference between an OFDM channelizer and a similar FFTbased 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 48subcarrier 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 48point FFT. The magnitude frequency responses of the ten distinct filters for the 48subcarrier 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 48subcarrier 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 48Subcarrier 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 1z' 1z48 1 0 1 1+z1 1+z2 z1 +z2 1+z + 2 1+z4 1z +z4 1+z8 1z4 + 8 10 1z + 16 1z48 1+z' 1z48 1+z2 1z48 1+z2 1z48 1+z +z2 1 48 1+z4 1 48 lz2 +Z4 1z48 1z48 1 z+ 8 1z48 1 z + z16 +/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 SFilter 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 48point 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 FFTbased 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 48Point OFDM Channelizer in AWGN Channel In an AWGN channel, the OFDM channelizer provides the same performance as a matchedfilter receiver. This section presents simulation results for the 48Point Channelizer and derives the process gains achieved across the various filter banks. The calculation for the probability of error in an uncoded, antipodal, BPSKmodulated system with a matchedfilter 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, 8PSK, 8QAM, 16QAM, 32QAM, 64QAM, 128 QAM and 256QAM [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.E01 1.E02 1.E03 1.E04 1.E05 1.E06 1.E07 1.E08 1.E09 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 48Subcarrier OFDM Channelizer in AWGN with BPSK Modulation r  ~  .~ * Filtel 40 1.E+00 1.E01 Filter 1 Filter 2 1.E02 S\ \Filter 3 1.E03 Filter 4 1E04  Filter 5 1.E04 Filter 6 S1.E05 Filter 7 1.E06 Filter 8 Filter 9 1.E07 Filter 10 1.E08 BPSK Theoretical 1.E09 c r T LO r C cD M c. SNR Figure 39. BER of 48Subcarrier OFDM Channelizer in AWGN with QPSK Modulation 1.E+00 1.E02 1.E03 F,,. 2 Firi 5 1.E04 . IIr 6 .. 1.E05 F 1.E06 1.E07 , 1.E08 .. .,,, 1.E09 M Co r T W LO r cD M. SNR Figure 40. BER of 48Subcarrier OFDM Channelizer in AWGN with 8PSK Modulation 1.E+00 1.E01 Filter 1 Filter 2 1.E02 *= 1.E0 Filter 3 1.E03 Filter 4 x Filter 5 1.E04 * Filter 6 0. 1.E05  Filter 7 1.E06 Filter 8 Filter 9 1.E07 Filter 10 1.E08 BPSK Theoretical 1.E09 Coo r' T W LO r'r C oM cD 0 SNR Figure 41. BER of 48Subcarrier OFDM Channelizer in AWGN with 8QAM Modulation 1.E+00 Filter 1 1.E01 __ __ __" _ ___ Filter2 1.E02 Filter 3 Filter4 1.E03 . Filter xFilter 5 a 1.E05 sFilter7 Filter 8 1.E06 =^ 1. 06 Filter 9 1.E07 Filter 10 1E BPSK Theoretical 1.E08  1.E09 C'.J IC'. I I r r'.r SNR Figure 42. BER of 48Subcarrier OFDM Channelizer in AWGN with 16QAM Modulation 1.E+00 1.E01 F,,, I 1.E02  1.E02 1.E04 =6 a 1.E05  ,,,  1.E06 _ 1.E07 \=r F\,, ., 1.E08 PS ,, 1.E09 C\j C\j Ir i I I r r r r c\j c\j SNR Figure 43. BER of 48Subcarrier OFDM Channelizer in AWGN with 32QAM Modulation 1.E+00 1.E01  Filter 1 1.E02 Filter2 1.E02 Filter 3 1.E03 Filter 4 A 1E0  Filter 5 FieFilter 6 S1.E05  Filter 7 1.E06 Filter 8 Filter 9 1.E07 Filter 10 1.E08 BPSK Theoretical 1.E09 C'J C'J Ir i r i CIrJ C\J C\J SNR Figure 44. BER of 48Subcarrier OFDM Channelizer in AWGN with 64QAM Modulation 1.E+00 1.E01 _ FIr r I 1.E02 _ F"l ,. 1.E03 F,,,r A ' 1.E04   . r 6 1.E05 1.E06 F 1.E07 1.E08 6PS Tri ,i .r.IC , 1.E09 C\J C\J Ir i C'.J C\J C\J SNR Figure 45. BER of 48Subcarrier OFDM Channelizer in AWGN with 128QAM Modulation 1.E+00 1.E01 s..'_  Filter 1 Filter2 1.E02 Filter 3 1.E03 Filter 4 1.E04 4 Filter 5  \ *oFilter6 Filter 7 1.E06 Filter 8 Filter 9 1.E07 Filter 10 1.E08 BPSK Theoretical 1.E09 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 48Subcarrier OFDM Channelizer in AWGN with 256QAM 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 MultiModulation 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 @ 10e2 (dB) = 10e2 (dB) = 10e5 (dB) BER = 10e5 (dB) BPSK (ref) 0 0 0 0 QPSK 0 3.01 0 3.01 8PSK 2.25 7.02 2.2 6.97 8QAM 3 7.77 3.3 8.07 16QAM 3.55 9.57 3.9 9.92 32QAM 6.5 13.49 7 13.99 64QAM 7.65 15.43 7.8 15.58 128QAM 10.85 19.3 11.2 19.65 256QAM 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.E01 1.E02 1.E03 1.E04 . 1.E05 1.E06 1.E07 1.E08 1.E09  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. PerFilter Modulation Scheme for 48Subcarrier OFDM Channelizer Expected SNR Expected SNR Filter Modulation Performance Performance Number Scheme Advantage @ BER Advantage @ BER = 10e2 (dB) = 10e5 (dB) 1 QPSK 0 0 2 QPSK 0 0 3 8QAM 1.01 0.96 4 8QAM 1.01 0.96 5 8QAM 1.01 0.96 6 16QAM 0.56 0.91 7 16QAM 0.56 0.91 8 32QAM 1.48 1.98 9 32QAM 1.48 1.98 10 64QAM 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 postcorrection 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 48Subcarrier OFDM Channelizer in AWGN with Mixed Modulation 1.0E+00 1.0E01 1.0E02 1.0E03 1.0E04 1.0E05 1.0E06 1.0E07 1.0E08 1.0E09 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 FFTbased 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 frequencyvarying 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 phaseshift 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 maximalratio combining (MRC), can be used to achieve optimal signaltonoise 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 frequencyvarying 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 nonaligned 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 CB.:I e  .. i if  Figure 51. Multipath Constellation Scatter with Option 2 Alignment OFDM Channelizer Subcarrier Separation Recall one advantage ofFFTbased 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 FFTbased 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 perfilter design parameters for the 48point 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, complexvalued filter 40 30 20 10 10   210 /    _ I Filter 3 130 20 S10     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 multiplierless FIR filters. The drawback of this required frequency separation filtering is that highrate 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 realvalued coefficients is taken. Next, either the positive or negative frequency Fourier coefficients are forced to zero. Finally, an inverse FFT produces the resulting complexvalued 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 openloop 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 closedloop equalization scheme by informing the transmitter of the persubcarrier channel response and having the transmitter predistort 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 realtime. 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 Senalto Channezer RF RXI LPF ADC oExC ionSe r Channelizer mSynch ronization Exteso Parallel Separation F Bn RemovalFilters Filter Bank Time/ Coding Parallelto Channel Carrier SymboltoBit __ 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 precompensation 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 persubcarrier channel characteristics in order to apply the predistortion to the transmitted signal and inform the transmitter of the persubcarrier channel response. This can be accomplished by having the receiver compute a persubcarrier 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 perfilter basis. When a nonoptimal 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 onetomany 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 persubcarrier predistortion to each subcarrier individually. This is a viable assumption since many ontomany 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 FFTbased OFDM transmission scheme while the numerous endpoints could have lowercost OFDM channelizers. In this manner, the transmitter could handle the predistortion on a persubcarrier 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 nonoptimal fashion. This section will demonstrate this phenomenon through figures captured in simulation. A Rappaport channel is used for this demonstration [9]. The NonLineofSight 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 S30A 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 perfilter 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, FFTbased 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.E01 ,QPSK 1.E02 8QAM 1.E03 SK 8PSK 1.E04 16QAM a 1.E05 *32QAM 1.E06 64QAM 1.E07 128QAM 256QAM 1.E08 1 .E08 256QAM 1.E09 SNR Figure 65. OFDM Channelizer Filter 6 BER Results 1.E+00 1.E01 BPSK SQPSK 1.E02 8PSK 1.E03 8 8QAM $ 1.E04  16QAM a 1.E05 _32QAM 1.E06 +64QAM 1.E07 128QAM 256QAM 1.E08 1.E09 rT  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.E01  BPSK 1.E02 uQPSK 8PSK 1.E03 8QAM 1.E04  16QAM 0.. 1.E05 . 5 32QAM 1.E06 I64QAM 1.E07 128QAM 1.E08 256QAM 1.E09 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.E01 BPSK 1.E02 QPSK 8PSK 1.E03 8Q AM IL z 1.E04  16QAM 32QAM 1.E05 64QAM 1.E06 128QAM 1.E07 256QAM 1.E08 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.E01 1.E02 *BPSK 1.E03 QP QPSK 1.E04 8PSK 1 1.E05 8QAM 16QA M 1.E066QAM .32QAM 1.E07 64QAM 1.E08 128QAM 128QAM 1.E09 256QAM 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.E01 U) 1.E02 ;" 1.E03 I I II 1.E04 1.E05  1.E06   1.E07 1.E08 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 48subcarrier 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.E01 ~ BPSK QPSK 1.E02 PSK S 1.E03 QAM a 16QAM 1.E04 32AM \32QAM 1.E05  64QAM 128QAM 1.E06 = 256QAM 1.E07 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.E01 *BPSK QPSK 1.E02 1.E03 8AM t' 1.E04  i16QAM 1.E05 *32QAM ~ 64QAM 1.E06 128QAM 1.E07 256QAM 1.E08 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.E01 _*  1.E02  1.E03   1.E04 : , 1.E05  1.E06  1.E07  1.E08  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.E01 1.E02 1.E03 1.E04 1.E05 1.E06 1.E07 1.E08 1.E09 _.E 8* PS 1.E0 8 QA * BPSK QPSK 8PSK 8QAM 16QAM *32QAM 164QAM 128QAM 256QAM =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.E01 _ BPSK 1.E02 QPSK 8QAM 1.E04 1.  16QAM a 1.E05 32QAM 1.E06 64QAM 1.E07  128QAM 1. E08 256QAM 1.E09 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 = le2) 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 8PSK 15 11.5 11 10.2 8QAM 14.6 11.4 11 10.1 16QAM 17 11.7 10.9 10.1 32QAM NA 17.2 11 10.2 64QAM NA NA 10.8 10.2 128QAM NA NA 11 10.2 256QAM NA NA 11 10.2 1.E+00 1.E01  BPSK 1.E02 .QPSK 1.E03 8PSK " 1.E04 8QAM '' 16QAM . 1.E05 *.32QAM 1.E06 64QAM 1.E07 128QAM 1.E08 256QAM 1.E09 o ,1 CN oo ' 0 (O CN t (N (N C SNR Table 13. Multipath Fading Performance Delta (@ BER = le2) 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 8PSK 8.9 5 3 2.6 8QAM 10.1 4.7 3 2.7 16QAM 22.5 5.9 2.9 2.5 32QAM NA NA 3 2.6 64QAM NA NA 3.2 1.6 128QAM NA NA 3.8 2.7 256QAM NA NA 4.3 2.7 Table 14. Multipath Fading Performance Delta (@ BER reference) 1e2) 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 8PSK 3.8 2.1 1.4 0.9 8QAM 5 2.2 1.2 0.8 16QAM 8 2.8 1.3 0.7 32QAM NA 8 1.8 0.7 64QAM NA NA 2.2 0.8 128QAM NA NA 4.4 0.8 256QAM NA NA 7.6 0.8 Table 15. Multipath Fading Performance Delta (@ BER = le5) 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 8PSK 17.4 13 11.5 10.8 8QAM 17.4 12.4 11.1 10.3 16QAM 19.3 13.1 11 10.3 32QAM NA 18.3 11 10.3 64QAM NA 19.1 11 10.3 128QAM NA NA 11 10.3 256QAM NA NA 10.9 10.2 Table 16. Multipath Fading Performance Delta (@ BER = le5) with Option 1 (AWGN reference) BPSK 6.4 4.3 3 2.6 QPSK 7 4.6 3.1 2.6 8PSK 11.8 6.9 3.6 3 8QAM 13.8 6 3 2.6 16QAM NA 9.2 3.2 2.7 32QAM NA NA 3.6 2.7 64QAM NA NA 4 2.7 128QAM NA NA 5.8 2.6 256QAM NA NA 6.3 2.6 BPSK 6.4 4.3 3 2.6 Table 17. Multipath Fading Performance Delta (@ BER = le5) 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 8PSK 6.7 4.1 2.1 1.2 8QAM 7.8 3.7 1.5 0.8 16QAM 18.2 6.5 1.9 0.9 32QAM NA NA 3.3 0.9 64QAM NA NA 4.7 0.9 128QAM NA NA NA 0.9 256QAM 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 FFTbased 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 filterbased OFDM channelizers are interesting from an implementation complexity and operating clock frequency perspective. They can potentially provide a lowcomplexity alternative to implementing an FFTbased OFDM system that has the potential to operate at very high clock rates due to the multiplierless 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 complexvalued noise power even if the signaling is realvalued 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 complexvalued 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 FFTbased OFDM system. * Investigate multiplierless filters for OFDM channelizer subcarrier separation in multipath channels. * Investigate fixedpoint requirements for an OFDM channelizer and how this fixed point design might compare to an FFTbased OFDM channelizer. * Implement and synthesize an OFDM channelizer targeting current stateoftheart technology to determine actual realizable operating speed. The realizable operating speed of an OFDM channelizer should be contrasted against an FFTbased OFDM channelizer. * Investigate utilizing variablerate 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 precompensation 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 transmitterbased precompensation 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 predistortion 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 receiverbased separation filter scheme has a limitation of requiring additional filters to perform subcarrier separation in order to compute persubcarrier channel estimations. This must be done prior to coherent subcarrier combining. These separation filters potentially introduce highrate 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 12 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 POLYMORPHICBASED SPW OVERVIEW As mentioned in the introduction, SPW is a hierarchal, blockbased modeling and simulation environment useful for performing system analysis. SPW is capable of executing both small and largescale 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 floatingpoint and fixedpoint 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 widevariety of block types (54 total). A block type consists of two subtypes: 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,' 'FixedPoint' and 'ComplexFixedPoint.' 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 fixedpoint attributes that are defined for the block if the block's type is set for a fixedpoint type. In order to further define these fixedpoint 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 fixedpoint type. Two other optional parameters in the default value field are the composite type size (i.e. vector size) and the fixedpoint modes of operation. These two parameters are not shown in Figure 78. The composite type size, when present, is enclosed within square brackets "[]." The fixedpoint modesofoperation parameter is composed of two parameters: lossof precision mode and overflow mode. These two parameters are specified within parentheses and separated by a comma. The first parameter is the lossofprecision 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 fixedpoint 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 4.60E1 4.60E1 4.44E1 4.44E1 4.44E1 4.21E1 4.21E1 3.88E1 3.89E1 3.44E1 22 4.55E1 4.55E1 4.37E1 4.37E1 4.37E1 4.11E1 4.11E1 3.75E1 3.75E1 3.27E1 21 4.50E1 4.50E1 4.29E1 4.29E1 4.29E1 4.01E1 4.01E1 3.61E1 3.61E1 3.07E1 20 4.44E1 4.44E1 4.21E1 4.21E1 4.21E1 3.89E1 3.89E1 3.45E1 3.45E1 2.86E1 19 4.37E1 4.37E1 4.11E1 4.11E1 4.11E1 3.76E1 3.76E1 3.27E1 3.27E1 2.63E1 18 4.29E1 4.29E1 4.01E1 4.01E1 4.01E1 3.61E1 3.61E1 3.07E1 3.07E1 2.38E1 17 4.21E1 4.21E1 3.89E1 3.89E1 3.89E1 3.45E1 3.45E1 2.86E1 2.86E1 2.12E1 16 4.11E1 4.11E1 3.76E1 3.76E1 3.76E1 3.27E1 3.27E1 2.63E1 2.63E1 1.85E1 15 4.01E1 4.01E1 3.61E1 3.61E1 3.61E1 3.08E1 3.08E1 2.38E1 2.39E1 1.57E1 14 3.89E1 3.89E1 3.45E1 3.45E1 3.45E1 2.86E1 2.86E1 2.12E1 2.12E1 1.30E1 13 3.76E1 3.76E1 3.27E1 3.27E1 3.27E1 2.63E1 2.63E1 1.85E1 1.85E1 1.03E1 12 3.61E1 3.61E1 3.08E1 3.08E1 3.08E1 2.39E1 2.39E1 1.57E1 1.58E1 7.77E2 11 3.45E1 3.45E1 2.86E1 2.87E1 2.86E1 2.13E1 2.13E1 1.30E1 1.30E1 5.54E2 10 3.27E1 3.27E1 2.64E1 2.64E1 2.64E1 1.86E1 1.86E1 1.03E1 1.03E1 3.68E2 9 3.08E1 3.08E1 2.39E1 2.39E1 2.39E1 1.58E1 1.58E1 7.79E2 7.79E2 2.24E2 8 2.87E1 2.87E1 2.13E1 2.13E1 2.13E1 1.30E1 1.30E1 5.56E2 5.56E2 1.22E2 7 2.64E1 2.64E1 1.86E1 1.86E1 1.86E1 1.03E1 1.03E1 3.70E2 3.70E2 5.76E3 6 2.39E1 2.39E1 1.58E1 1.58E1 1.58E1 7.82E2 7.82E2 2.25E2 2.25E2 2.29E3 5 2.13E1 2.13E1 1.30E1 1.30E1 1.30E1 5.59E2 5.59E2 1.22E2 1.22E2 7.33E4 4 1.86E1 1.86E1 1.03E1 1.03E1 1.04E1 3.72E2 3.72E2 5.79E3 5.80E3 1.80E4 3 1.58E1 1.58E1 7.84E2 7.84E2 7.84E2 2.26E2 2.26E2 2.31E3 2.31E3 3.08E5 2 1.31E1 1.31E1 5.61E2 5.61E2 5.61E2 1.23E2 1.23E2 7.46E4 7.42E4 3.20E6 1 1.04E1 1.04E1 3.74E2 3.74E2 3.73E2 5.85E3 5.86E3 1.83E4 1.82E4 1.10E7 0 7.86E2 7.86E2 2.28E2 2.28E2 2.28E2 2.33E3 2.34E3 3.34E5 3.19E5 2.00E8 1 5.63E2 5.62E2 1.24E2 1.24E2 1.24E2 7.53E4 7.52E4 3.68E6 3.62E6 2 3.75E2 3.75E2 5.90E3 5.91E3 5.90E3 1.82E4 1.85E4 2.40E7 2.80E7 4 1.25E2 1.25E2 7.62E4 7.62E4 7.63E4 3.00E6 4.05E6 5 5.94E3 5.96E3 1.89E4 1.88E4 1.89E4 1.90E7 1.80E7 6 2.38E3 2.39E3 3.28E5 3.30E5 3.28E5 1.00E8 1.00E8 7 7.67E4 7.74E4 3.85E6 3.90E6 3.62E6 8 1.88E4 1.91E4 2.00E7 3.30E7 2.70E7 9 3.28E5 3.34E5 2.00E8 1.00E8 10 3.48E6 4.01E6 11 2.20E7 3.30E7 12 2.00E8 1.00E8 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 4.72E1 4.72E1 4.60E1 4.60E1 4.60E1 4.44E1 4.44E1 4.21E1 4.21E1 3.89E1 22 4.68E1 4.68E1 4.55E1 4.55E1 4.55E1 4.37E1 4.37E1 4.11E1 4.11E1 3.75E1 21 4.64E1 4.65E1 4.50E1 4.50E1 4.50E1 4.29E1 4.29E1 4.00E1 4.00E1 3.61E1 20 4.60E1 4.60E1 4.44E1 4.44E1 4.44E1 4.21E1 4.21E1 3.89E1 3.89E1 3.45E1 19 4.55E1 4.55E1 4.37E1 4.37E1 4.37E1 4.11E1 4.11E1 3.75E1 3.75E1 3.27E1 18 4.50E1 4.50E1 4.29E1 4.29E1 4.29E1 4.01E1 4.01E1 3.61E1 3.61E1 3.07E1 17 4.44E1 4.44E1 4.21E1 4.21E1 4.21E1 3.89E1 3.89E1 3.45E1 3.45E1 2.86E1 16 4.37E1 4.37E1 4.11E1 4.11E1 4.11E1 3.76E1 3.76E1 3.27E1 3.27E1 2.63E1 15 4.29E1 4.29E1 4.01E1 4.01E1 4.01E1 3.61E1 3.61E1 3.07E1 3.07E1 2.38E1 14 4.21E1 4.21E1 3.89E1 3.89E1 3.89E1 3.45E1 3.45E1 2.86E1 2.86E1 2.12E1 13 4.11E1 4.11E1 3.76E1 3.76E1 3.76E1 3.27E1 3.27E1 2.63E1 2.63E1 1.85E1 12 4.01E1 4.01E1 3.61E1 3.61E1 3.61E1 3.08E1 3.08E1 2.39E1 2.39E1 1.58E1 11 3.89E1 3.89E1 3.45E1 3.45E1 3.45E1 2.87E1 2.87E1 2.13E1 2.13E1 1.30E1 10 3.76E1 3.76E1 3.27E1 3.27E1 3.27E1 2.64E1 2.64E1 1.86E1 1.86E1 1.03E1 9 3.61E1 3.61E1 3.08E1 3.08E1 3.08E1 2.39E1 2.39E1 1.58E1 1.58E1 7.79E2 8 3.45E1 3.45E1 2.87E1 2.87E1 2.87E1 2.13E1 2.13E1 1.30E1 1.30E1 5.57E2 7 3.28E1 3.28E1 2.64E1 2.64E1 2.64E1 1.86E1 1.86E1 1.03E1 1.03E1 3.70E2 6 3.08E1 3.08E1 2.39E1 2.39E1 2.39E1 1.58E1 1.58E1 7.81E2 7.81E2 2.25E2 5 2.87E1 2.87E1 2.13E1 2.13E1 2.13E1 1.30E1 1.30E1 5.58E2 5.58E2 1.23E2 4 2.64E1 2.64E1 1.86E1 1.86E1 1.86E1 1.03E1 1.04E1 3.71E2 3.71E2 5.81E3 3 2.39E1 2.40E1 1.58E1 1.58E1 1.58E1 7.84E2 7.85E2 2.26E2 2.26E2 2.32E3 2 2.13E1 2.14E1 1.31E1 1.31E1 1.31E1 5.61E2 5.61E2 1.23E2 1.23E2 7.40E4 1 1.86E1 1.86E1 1.04E1 1.04E1 1.04E1 3.73E2 3.74E2 5.85E3 5.85E3 1.80E4 0 1.59E1 1.59E1 7.87E2 7.87E2 7.86E2 2.27E2 2.28E2 2.34E3 2.34E3 3.10E5 1 1.31E1 1.31E1 5.63E2 5.63E2 5.63E2 1.24E2 1.24E2 7.51E4 7.51E4 3.64E6 2 1.04E1 1.04E1 3.75E2 3.75E2 3.75E2 5.90E3 5.91E3 1.85E4 1.84E4 2.20E7 3 7.89E2 7.89E2 2.29E2 2.29E2 2.29E2 2.36E3 2.37E3 3.28E5 3.24E5 1.50E8 4 5.65E2 5.65E2 1.25E2 1.25E2 1.25E2 7.63E4 7.62E4 3.52E6 3.77E6 5 3.77E2 3.77E2 5.96E3 5.95E3 5.95E3 1.86E4 1.88E4 2.35E7 2.90E7 6 2.30E2 2.30E2 2.39E3 2.38E3 2.39E3 3.31E5 3.31E5 1.00E8 1.50E8 7 1.26E2 1.26E2 7.73E4 7.69E4 7.74E4 3.70E6 3.91E6 8 6.00E3 6.01E3 1.92E4 1.91E4 1.92E4 2.70E7 2.70E7 9 2.41E3 2.41E3 3.39E5 3.36E5 3.33E5 5.00E9 2.00E8 10 7.79E4 7.83E4 3.95E6 3.85E6 4.06E6 11 1.93E4 1.94E4 2.15E7 3.15E7 2.55E7 12 3.41E5 3.41E5 1.00E8 1.50E8 13 3.79E6 4.11E6 14 2.75E7 2.55E7 15 1.50E8 5.00E9 