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- Permanent Link:
- https://ufdc.ufl.edu/UF00082210/00001
## Material Information- Title:
- Efficient communications for nonsymmetric information sources with application to picture transmission
- Creator:
- Emami, Shahriar (
*Dissertant*) Miller, Scott L. (*Thesis advisor*) Couch, Leon W. (*Reviewer*) Childers, Donald G. (*Reviewer*) Najafi, Fazil (*Reviewer*) Sigmon, Kermit (*Reviewer*) Phillips, Winfred M. (*Degree grantor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1993
- Copyright Date:
- 1993
- Language:
- English
- Physical Description:
- vii, 159 leaves : ill., photos ; 29 cm.
## Subjects- Subjects / Keywords:
- Alphabets ( jstor )
Bandwidth ( jstor ) BCH codes ( jstor ) Constellations ( jstor ) Decryption ( jstor ) Error rates ( jstor ) Polynomials ( jstor ) Signals ( jstor ) Statistics ( jstor ) Supernova remnants ( jstor ) Digital communications ( lcsh ) Digital modulation ( lcsh ) Dissertations, Academic -- Electrical Engineering -- UF ( lcsh ) Electrical Engineering thesis Ph. D ( lcsh ) Modulation (Electronics) ( lcsh ) Signal processing -- Digital techniques ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- This dissertation is concerned with issues related to nonsymmetric information sources. Signal design, alphabet size selection and decoding of information from these sources are among the topics covered in this dissertation. although the techniques presented here are applicable to any nonsymmetric source, the emphasis is placed on video sources. Initially a model for the statistics is placed on video sources. Initially a model for the statistics of DPCM (Differential Pulse Code Modulation) of video signals is derived and it is shown that DPCM of video signals results in a nonsymmetric source. The problem of signal selection for nonsymmetric sources in two dimensions is considered. Iterative methods for finding the minimum error signal (and minimum cost) constellation subject to an average (or a peak) power constraint are presented. Even though efficient techniques for source coding, channel coding and signal design exists, it is not known how the choice of alphabet size affects a communication system. Image transmission systems with various alphabet sizes are compared on the basis of equal information rate, bandwidth and average power. The systems employing various alphabet sizes are analyzed and computer simulations are performed using pictures with different amount of details. An optimum procedure for decoding Markov sequences is developed and the path metric is derived. A heuristic tree searching algorithm. Two other techniques for decoding Markov sequences, a symbol-by-symbol modified MAP (MMAP) receiver using higher order statistics and a maximum signal-to-noise ratio (MSNR) receiver, are also given. The decoding procedures were applied to image communication over noisy channels. In summary, the major contributions of this dissertation were the development of signal selection methods for nonsymmetric sources, derivation of procedures for decoding of correlated sources and application of these procedures to the picture communication in noisy situations.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1993.
- Bibliography:
- Includes bibliographical references (leaves 155-158).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Shahriar Emami.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Shahriar Emami. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 030326681 ( ALEPH )
30784206 ( OCLC ) AKA9304 ( NOTIS )
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EFFICIENT COMMUNICATIONS FOR NONSYMMETRIC INFORMATION SOURCES WITH APPLICATION TO PICTURE TRANSMISSION By SHAHRIAR EMAMI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1993 To my parents for their love, patience and support. ACKNOWLEDGEMENTS I wish to express my gratitude to my advisor, Dr. Scott L. Miller, for his encouragement, support and friendship. I am also grateful to Dr. Couch for his guidance and insight. In addition, I would like to thank Dr. Childers and Dr. Najafi for their time and interest in serving in my committee. Special thanks also go to Dr. Sigmon who has offered his valuable assistance throughout the course of this study. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS. . . . . . . . ABSTRACT.vi CHAPTER ONE REVIEW OF MODULATION TECHNIQUES, SOURCE AND CHANNEL CODING.1 1.1 Introduction.1 1.2 Two-Dimensional Modulation Formats.2 1.3 Source Coding.5 1.4 DPCM.5 1.5 Transform Coding.8 1.6 Channel Coding.10 1.7 Transmission Errors in a DPCM system . . . . 15 1.8 Optimum Prediction for Noisy Channels . . . . 16 1.9 Research Objectives.18 1.10 Description of Chapters.20 CHAPTER TWO DPCM VIDEO SIGNAL: A NONSYMMETRIC INFORMATION SOURCE.21 2.1 Introduction.21 2.2 Basics of Quantizers.21 2.3 Approaches to Quantizer Design.23 2.4 MSQE Quantizer Design.23 2.5 Analysis of DPCM Encoder.26 2.6 Results.29 2.7 Discussion.31 CHAPTER THREE SIGNAL DESIGN FOR NONSYMMETRIC SOURCES .35 3.1 Introduction.35 3.2 Maximum Likelihood Signal Design for Three Signals.37 3.3 A Numerical Approach Based on Lagrange Multipliers Method.48 3.4 Minimum Error Signal Selection.51 3.5 Minimum Average Cost Signal Selection . . . 58 3.6 Results.66 CHAPTER FOUR ALPHABET SIZE SELECTION FOR VIDEO SIGNAL CODING. . . . . 4.1 Introduction . 4.2 Preliminaries. 4.3 Analysis . . . . . . . . . . . . . . . . . . 4.4 Implementation Issues. . . . . . . 4.5 Nonbinary BCH Codes. . 4.6 Results. 4.7 Summary. CHAPTER FIVE EFFICIENT DECODING OF CORRELATED SEQUENCES 5.1 5.2 5.3 5.4 5.5 Introduction . Optimum Decoding of Markov Sequences A Modified MAP (MMAP) Receiver . A Minimum Cost Decoder . A Maximum Signal-To-Noise Ratio (MSNR) Receiver . 5.6 Redundancy in the Encoded signals 5.7 Picture Transmission over noisy Channels . 5.8 Side Information . 59 Summary. . CHAPTER SIX CONCLUSIONS AND SUMMARY . . 6.1 Summary of the Work. . . . 6.2 Directions of Future Research. APPENDIX A. EVALUATION OF AN INTEGRAL . APPENDIX B. EVALUATION OF THE DERIVATIVES REFERENCES . BIOGRAPHICAL SKETCH. . . . . 79 79 80 83 86 94 99 104 116 116 119 122 123 * . . . 124 * . . . 127 * . . . 130 * . . . 144 * . . . . 145 * . . . 147 * . . . 147 * . . . 150 151 153 155 159 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFICIENT COMMUNICATIONS FOR NONSYMMETRIC INFORMATION SOURCES WITH APPLICATION TO PICTURE TRANSMISSION By Shahriar Emami August 1993 Chairperson: Dr. Scott L. Miller Major Department: Electrical Engineering This dissertation is concerned with issues related to nonsymmetric information sources. Signal design, alphabet size selection and decoding of information from these sources are among the topics covered in this dissertation. Although the techniques presented here are applicable to any nonsymmetric source, the emphasis is placed on video sources. Initially a model for the statistics of DPCM (Differential Pulse Code Modulation) of video signals is derived and it is shown that DPCM of video signals results in a nonsymmetric source. The problem of signal selection for nonsymmetric sources in two dimensions is considered. Iterative methods for finding the minimum error signal (and minimum cost) constellation subject to an average (or a peak) power constraint are presented. Even though efficient techniques for source coding, channel coding and signal design exists, it is not known how the choice of alphabet size affects a communication system. Image transmission systems with various alphabet sizes are compared on the basis of equal information rate, bandwidth and average power. The systems employing various alphabet sizes are analyzed and computer simulations are performed using pictures with different amount of details. An optimum procedure for decoding Markov sequences is developed and the path metric is derived. A heuristic tree searching algorithm is employed to obtain a suboptimum solution. Two other techniques for decoding Markov sequences, a symbol-by-symbol modified MAP (MMAP) receiver using higher order statistics and a maximum signal-to-noise ratio (MSNR) receiver, are also given. The decoding procedures were applied to image communication over noisy channels. In summary, the major contributions of this dissertation were the development of signal selection methods for nonsymmetric sources, derivation of procedures for decoding of correlated sources and application of these procedures to the picture communication in noisy situations. vii CHAPTER ONE REVIEW OF MODULATION TECHNIQUES, SOURCE AND CHANNEL CODING IN DIGITAL COMMUNICATIONS 1.1 Introduction The goal of this chapter is to present the background necessary to follow the work presented in this dissertation, introduce the research objectives and give a brief description of the chapters. Since two-dimensional modulation has been utilized in this work, these formats are reviewed, their spectral efficiency are calculated and the upper bound on their spectral efficiency is given. Picture transmission is one of the applications presented here. To familiarize the reader with the field, a number of source coding techniques have been described. Error correction over noisy channels is an important topic and has been utilized in this thesis. The gain from channel coding has been explained and different methods of error correction have been discussed. The effects of channel errors in a DPCM system and optimum prediction in noisy channels was presented, because the enhancement of picture quality in DPCM systems has been addressed in this dissertation. 1.2 Two-Dimensional Modulation Formats M-ary phase-shift keying (MPSK) and quadrature amplitude modulation (QAM) are among the most popular two dimensional formats. In MPSK the transmitted signal is given by s(t)=Re{g(t)ect} (i.i) where g(t) =A ejO(c) , (1.2) 8( )-(i i)M i=1,2, . .,M. In other words, in MPSK while the amplitude is maintained constant the phase of signal can take on one of the M values in a symbol interval. The MPSK for M=4 is called quadrature phase-shift keying (QPSK). In MPSK signal points are confined to the circumference of a circle. But in QAM the transmitted signal is s(t)=Re{g(t) e jct} (1.3) where g(t)=x(t)+jy(t), (1.4) s(t)=x(t) cos( Ct)-y(t) sin(act). (1.5) The waveforms x(t) and y(t) are x(t) =xi h(t-iT) (1.6) i and y(t)=Eyi h(t-iT), (1.7) i where T is the symbol interval in seconds and h(t) is the pulse shape. Let us find the spectral efficiency of MPSK and QAM with rectangular pulses. The null-to-null transmission bandwidth of MPSK and QAM is 2R (1.8) where M=21 is the number of points in the signal constellation. The spectral efficiency is therefore given by R - 1 bits/sec (1.9) BT2 Hz When operating over a bandlimited channel and the overall pulse shape satisfies the raised cosine rolloff filter characteristics, the bandwidth of the modulating signal is B- (I1+r) D (1.10) 2 where D=R/l and r is the rolloff factor. Since BT =2B, the transmiss ion bandwidth of QAM is BT=. (1 R (1.11) and the spectral efficiency with raised cosine filtering is given by 1og2M bits/sec (1.12) (1+r) Hz The spectral efficiency increases with the number of the points- in the constellation. However, one can not increase the spectral efficiency by increasing the number of points in the signal constellation too much, because as you place more signals in the constellation the error rate increases. For reliable communications, the information rate must be kept below the channel capacity. Therefore, the spectral efficiency is upper bounded by q <10g2(1+-S) (1.13) N where SIN is the signal-to-noise power ratio. Two dimensional formats are well suited for high speed data transmission because of their efficient use of bandwidth. However, they require coherent detection that implies the need for synchronization circuits. 1.3 Source Coding The purpose of source coding is to remove as much redundancy as possible from the message. Efficient coding of messages provides the opportunity for significantly decreasing the transmission costs. Two main approaches to picture coding predictive coding and transform coding will be addressed here. 1.4 DPCM There is considerable correlation between adjacent samples of speech or image data, and indeed the correlation is significant even between samples that are several sampling intervals apart. The meaning of this high correlation is that, in an average sense, the signal does not change rapidly from sample to sample so that the difference between adjacent samples should have a lower variance than the variance of the signal itself. The predicted value is the output of the predictor system, whose input is a quantized version of the input signal. The difference signal may also be -called the prediction error signal, since it is the amount by which the predictor fails to exactly predict the input. Since the variance of error signal is smaller than the variance of signal, a quantizer with a given number of levels 6 can be adjusted to give a smaller quantization error than would be possible when quantizing the input directly. 1.4.1 Optimum Prediction We are interested in linear prediction of the form N R(n) =E a. x(n-j) (1.14) which is the weighted sum of N previous samples. The weights aj are linear prediction coefficients. The filter is optimized by finding the weights that minimize prediction error in a mean squared sense Ge2=E[ (x(n) -k(n) ) 2] (1.15) Since mean squared error is a function of aj and e2 -- =0 ie, . . (1.16) 3ai is a necessary condition for minimum MSE (mean-squared error). Evaluating the derivative gives E[-2x(n) - (n) 8 (n) ] (1.17) ai Equating this to zero yields, E[tx(n)-k(n)x(n-i) =0; i=1,2,.,N. (1.18) This is called the orthogonality principles which states that minimum error must be orthogonal to all data used in the 7 prediction. The expansion of this equation gives the following condition for optimum a, N IaRi(k-j)-=Ri(k); k=l,2,.,N j.1. (1.19) *Rx,(N-1) ' (al IR.,(1) * R,(N-2) a2 = R_ (2) ï¿½ (0Io) N) Rx .,N) (1.20) in matrix notation, (1.21) where R.=Q (i-j[)); r==R,(i)i: i,j=l,2, . ,N (1.22) The equations are called normal equations, Yule-Walker prediction equations or Wiener-Hopf equations. The mean squared error decreases significantly by using up to three elements in predictive coding. However, if the coefficients are not exactly matched to the statistics of a picture, the decrease in mean squared error is not significant by using three previous elements as compared to one [1]. (R,, (0) Rx (1) RXX (2) R,, ( 1) ax (0) R,, (1) ,R, ( -1) "R. (-2) ., =(N-3) 8 1.5 Transform Coding In transform coding a picture is divided into subpictures and then each of these subpictures are transformed into a set of independent coefficients. The coefficients are then quantized and coded for the transmission. An inverse transformation is applied to recover intensities of picture element. Much of the compression is a result of dropping coefficients from transmission that are small and coarsely quantizing the others as required by the picture quality. It is desirable to have a transform-which compacts most of the image energy in as few coefficients as possible. Another consideration is the ease of implementation. 1.5.1 Optimum Transform Optimum transform (K-L transform) is explicitly known, but computationally it is very demanding. This undesirable feature has prevented any hardware implementation of the optimum transform. It is mainly studied in simulations to obtain bounds. The most practical transform coding techniques are based on the DCT (discrete cosine transform), which provides a good compromise between information packing ability and computational complexity. In fact, the properties of DCT have proved of such practical value that it has become the 9 international standard for transform coding systems. In addition to that it minimizes the blocking artifact that, results when the boundaries between the subimages become visible. 1.5.2 Size of Subpictures Computer simulations on real pictures show that the mean square error-produced by transform coding improves with the size of subpicture. However, the improvement is not significant as subpicture is increased beyond blocksize of 16x16. Subjective quality of pictures, however, does not appear to improve with the size of block beyond 4x4 pixels [2]. 1.5.3 Bit Allocation One method of choosing the coefficients for transmission, is to evaluate the coefficient variances on a set of average picture, and then discard all the coefficients whose variance is lower than a certain value. Such a scheme is called zonal filtering [3)1. Having decided which coefficients to transmit, we must then design a quantizer for each of them. This could be done by dividing a given total number of bits among all the coefficients. In order to minimize the mean square error for 10 a given total number of bits f or Gaussian variables, the optimum assignment is done by making the average quantization error of each coef f icient the same. This requires that the bits be assigned to the coef f clients in proportion to the logarithm of their variance. 1.6 Channel Coding Channel coding is a method of inserting structured redundancy into the source data so that transmission errors can be identified and corrected. Block coding and convolutional coding are two important subcategories of channel coding techniques. 1.6.1 Block Codes With block coding the source data is f irst segmented into blocks of k bits; each block can represent any of M=2 k distinct messages. The encoder transforms each message into a k larger block of n digits. This set of 2 coded messages is called a code block. The (n-k) digits, which the encoder adds to each message block, are called redundant digits; they carry no new information. The ratio of data bits to total bits within a block, k/n, is called code rate. The code itself is referred to as an (nk) code. To demonstrate the performance improvement possible with 11 channel coding, let us pick a (15,11) single error correcting code. Assume a BPSK modulation, a signal-to-noise ratio of 43,776 ( S/No=43,776) and a data rate of R=4800 b/s. Let Pub and Purn represent the bit and message error rate for the uncoded system and Pcb P,m represent the bit and message error rate for the coded system, respectively. without coding Eb S =9.6dB (1.23) No RNo Pb=Q( _b ) =1.02x10-5 (1.24) u~b N0 and P um=l(l-Pub) 11=l.12xlO-4. (1.25) With coding RC=4800x (15/11) =6545 b/s Eb_ S -8.25 dB No R.No (1.26) Pc,b=Q( Eb) =1.36X10-4 N0 The bit error rate for the coded system is inferior to that of the uncoded system and the performance improvement due to coding is not apparent yet. Since the code corrects all single errors within a block of 15 bits, the message error rate for the coded system will be 15 Pcm3() Pcb(l.pb) 1-k.94x10- (1.27) It is seen by comparing the error rates that the message error rate has improved by a f actor of 58 through the use of a block code. Most of the research on block codes has been concentrated on a subclass of linear codes known as cyclic codes. A cyclic code word, after any number of cyclic shifts, has the property of remaining a valid code word from the original set of code words. Cyclic codes are attractive because they can be easily implemented with feedback shift registers. The decoding methods are simple and efficient. Examples of cyclic and related codes are BCH, ReedSolomon, Hamming, Reed-Muller, Golay,. quadratic residue and Goppa codes. The classes form overlapping sets so that a particular code may be a BCH code and also a quadratic residue code. Recent applications of these codes to digital communication include a (31,15) Reed-Solomon code for joint tactical information distribution system(JTIDS) and a (127,112) BCH code for INTELSAT V system [4]. 1.6.2 Convolutional Codes A convolutional encoder consists of some shift registers and modulo-2 summers. For the general case, k bits at a time are entered into the shift register, and the code rate is k/n. The state of the encoder is dependent upon the contents of the shift registers. Convolutional codes can be described by a code tree. It is seen that the tree contains redundant information which can be eliminated by merging, at any level, all nodes corresponding to the same encoder state. The redrawing of the tree with merging paths has been called a trellis by Forney. The problem of decoding a convolutional code can be thought of as attempting to find a path through the trellis or the tree by making use of some decoding rule. The Viterbi algorithm (5) which is shown to be a maximum likelihood decoder for convolutional codes, involves computing a metric between the received signal and the trellis path entering each state. In the event that two paths terminating on a given state are redundant, the one having the largest metric is stored (the surviving path) . This selection of survivor is performed on for all paths entering each of the other states. The decoder continues in this way to advance deeper into the trellis, making decisions by eliminating the least likely paths. The complexity of Viterbi algorithm is an exponential 14 function of the code's constraint length. For large values of constraint length (K>>10) one might consider other decoding algorithms. The complexity of sequential decoders is relatively independent of constraint length, so codes with much larger constraint length can be used. Also this technique is more suitable than Viterbi algorithm for low bit error rates. Sequential decoding (5] was first introduced by Wozencraft but the most widely used algorithm to date is due to Fano. It is an efficient method for finding the most probable code word, given the received sequence, without searching the entire tree. The explored path is probably only local; that is, the procedure is sub-optimum. The search is performed in a sequential manner, always operating on a single path, but the decoder can back up and change previous decisions. Each time the decoder moves forward, a tentative decision is made. If an incorrect decision is made, subsequent extensions of the path will be wrong. The decoder will eventually be able to recognize the situation. When this happens, a substantial amount of computation is needed to recover the correct path. Backtracking and trying alternate paths continue until it finally decodes successfully. Convolutional codes using either Viterbi or sequential decoding have the ability to utilize whatever soft-decision information might be available to the decoder. It is not surprising that they have been used widely even though their 15 theory is not as mathematically as profound as that of the block codes. Most good convolutional codes have been found by computer search rather than algebraic construction. 1.7 Transmission Errors in a DPCM System Differential PCM systems are affected differently by bit errors than PCM systems because the DPCM decoder loop causes an error propagation, while a PCM error does not propagate in time. Subjectively, DPCM is more error-robust than PCM in speech coding, but less robust than PCM for image coding. Assume a channel error changes channel input u(n) to a wrong value v(n) . Due to linearity of the decoder filter, the correct computation of output is superposed by an error output caused by input c(n)=u(n)-v(n) to the decoder loop. Since the decoder is an all-pole filter, there will -be an infinite sequence of error samples at the output, with decaying amplitudes. In the case of f irst-order prediction, the effect, on a future value, at time m, is described by (3] C(m) =C(n) a (m-n); m~tn (1.28) Transmission errors therefore propagate in the reconstructed DPCM waveform. This kind of error smearing is perceptually desirable in speech coding where a POM error spike of large magnitude is more annoying than a low amplitude error smeared over a long duration. 16 In picture coding, on the other hand, error propagation is perceptually very undesirable, taking the form of very visible streaks or blotches with one and two dimensional predictors. 1.8 Optimum Prediction for Noisy Channels One of the early approaches to system optimization under noisy conditions was presented by Chang and Donaldson [6]. Because of the importance of the result and the relevance to this dissertation a summary of the methods is given here. Let ri denote the received signal, fi denote the impulse response of the DPCM decoder. The output of the decoder is therefore given by Ri=ri *f= (s.+n) * fi (1.29) =si * f +n, * f =xi +q +ni* f i Let us define x'i as follows x/ =Aj-xi :qji ï¿½f var (Xi) =E[QI2] +2E fk E(ei1i-k] 1 k Let us assume the channel noise is uncorrelated and the difference signal samples are statistically independent, then var (X,) =E[Q2] +2E[QiNi] +E[Ni2] F fk2 (1.31) k The second term is called the mutual error and can be shown to be approximately zero if the quantizer is near optimum. The error power reduces to var (X'1) =E[Q2] +E[Ni2] fk2. (1.32) k The sum can be evaluated using an identity fk2=l+a2+a4+a6+. k (1.33) 1 -b 1-a2 The expression for the reconstruction error variance then becomes var (X'i) =var (Q.) +b var (Ni) . (1.34) We now define the following quantities to relate the quantizer and noise variances to the differential signal variance, var (Qj) =kq var (Ei), (1.35) var(N.) =Kn var(Ei) . The DPCM prediction gain is also given by Svar(Xi) =1+a 2-2a Pi, (1.36) var (E.) where P, is equal to R,,(O). Putting all this together yields var (XI) =(kq+b kn)' var (X) (1.37) Gp The second term above is the dominant term because the effect of channel noise is much more destructive to the reconstruction of the image than the effect due to quantization noise, 1-2a p1+a 2 var(X,) =K, var(X) 1-a2 (1.38) To minimize the variance of the reconstruction error we will set the derivative of this expression with respect to a to zero. The optimum value of a turns out to be a= fi-- 1 (1.39) Pi 1.9 Research Objectives This dissertation is concerned with issues related to nonsymmetric information sources. To motivate the work on nonsymmetric sources, it is shown that DPCM of digitized video signals results in a nonsymmetric information source. One of the main goals is to address the problem of signal design in 19 two dimensions for nonsymmetric sources. It is desired to find an algorithmic solution to the minimum error signal constellation for average and peak power constraints. In addition the general case where the cost function is not necessarily the error rate is discussed. Even though efficient techniques for source coding, channel coding and signal design exists, it is not known how the choice of alphabet size affects a communication system. We would like to compare communication systems with various alphabet sizes for the transmission of video signals on the basis of equal information rate, bandwidth and average power. Two realistic situations will be considered, when one is operating under tight bandwidth constraint and when the constraint is somewhat loose. System performance can be improved using standard error correction techniques at the cost of increasing the bandwidth or reducing the information rate. However, we would like to use inherent asymmetry and redundancy in the transmitted picture to improve the reception. We will model the data as a Markov source and derive the optimum method for decoding the data. We will also find a receiver that instead of minimizing the error rate maximizes the SNR (signal-to-noise) ratio. 20 1.10 Description of Chapters A review of background material relevant to this dissertation is given in Chapter one. Two dimensional modulation techniques, source coding techniques f or images, standard error correcting techniques and the ef fect of channel errors on predictive system are among the topics addressed in this chapter. In Chapter Two a DPCM system will be analyzed and a model for the statistics of the source will be derived. It will be shown theoretically and empirically that DPCM of video signals produces nonsymmetric sources. The issues of signal design are addressed in Chapter Three. Algorithmic solutions to signal design for nonsymmetric information sources under average and peak power constraints for minimizing the error rate and average cost are presented. The study on the role of alphabet size for nonsymmetric sources in a communication system is given in Chapter Four. In Chapter Five various methods for decoding Markov sequences are presented. The application to the transmission of video signals over noisy channels and a comparison of the method is also given. Chapter six contains a summary of presented approaches, conclusions and comments regarding the future research directions. CHAPTER TWO DPCM VIDEO SIGNAL: A NONSYMMETRIC INFORMATION SOURCE 2.1 Introduction The purpose of this chapter is to demonstrate that DPCM (differential pulse code modulation) of pictures results in a nonsymmetric information source. To do so, some introductory material is presented first. Since a quantizer is an important component of a DPCM system, it will be examined in some detail. Quantizers will be introduced, different criteria for the design will be mentioned and the procedure for finding an optimal quantizer (in MSQE sense) will be explained step by step. A DPCM encoder will be analyzed and a model for the resulting source statistics will be given. Eventually the model will be compared with actual picture statistics and results will be compared. 2.2 Basics of Ouantizers Quantization is the process of rounding sample values to a finite number of discrete values. This process is not an information preserving process and the reconstructed signal is 21 22 only as good as the quantized samples allow. In other words, there remains some error, the quantization error between the original and the reconstructed waveform which is related to the parameters of the quantizer. .Let the analog signal be modeled as a random waveform and let p(x) be the probability density function of the signal. The process of -quantization subdivides the range of the values of x into a number of discrete intervals. If a particular sample value of the analog signal falls anywhere in a given interval, it is assigned a single discrete value corresponding to that interval. The intervals fall between boundaries denoted by XIX21 . . .XL+I, where there are L intervals. The quantized values are denoted by 11,121 .1L and are called quantum levels or representative levels. The width of an interval is x14., -xi and is called interval's step size. If all the steps are equal and, in addition, the quantum level separations are all the same, the quantizer is said to be uniform; otherwise it is a nonuniform quantizer. It is possible to design a quantizer for a given probability density function and a given number of levels. The optimal quantizer is non-uniform unless the signal has a uniform pdf. If a uniform quantizer is used instead the mean squared quantization error will be larger than that of the optimal nonuniform quantizer. 23 2.3 Approaches to Quantizer Design The quantizers can be designed based on a mean squared error criterion. This results in overspecification of the low detailed areas of picture and consequently a small amount of granular noise but relatively poor reproduction of edges. It has been realized for sometime that for a better picture quality, quantizers should be designed on the basis of psychovisual criteria [7). One method of designing psychovisual quantizers is to minimize a weighted mean squared quantization error, where the weights are derived from subjective experiments (8]. Such optimization would be similar to mean squared error criteria, where the density function is replaced by a weighing function. 2.4 MSOE uantizer DesiQn An optimal quantizer is defined to be a quantizer with the smallest mean squared quantization error. It is desired to find the quantizer that minimizes the mean-squared quantization error for a given probability density function and number of levels. The mean-squared quantization error is given by X2 L- X+1 (x-l) 2p (X) dx. (2.1) XL Our purpose is to choose quantum levels 1i and interval boundaries xi so that eq. (2.1) is minimized. The above expression can be differentiated to obtain a set of necessary conditions that must hold for the optimum quantizer. By applying Leibniz's rule.we get (2.2) a .=-2 f i=1,2, , ali as-- =[(x-i_.1) 2-(x-1)2]p(x)dx=O, i=2,.,L, (2.3) aXi where x,=-o and XL+1= . Equation (2.3) is equivalent to i -) (2.4) which says that interval boundaries should fall midway between the adjacent quantum levels. Alternatively, I i=2xi-1i-1( (2.5) 25 Equation (2.2) is readily solved for li xl*1 f x p(x)K , ili.1 (2.6) f p (x)cbx XI The solution of the equations for the general nonuniform quantizer is difficult. However, a procedure to obtain a solution by computer iteration has been introduced by Llyod and Max [9), [10]. For a specified probability density function and a fixed value of L, 11 is first selected arbitrarily. with xi=0,we solve eq. (2.6) for X2. Next X2 and 11 are used in eq. (2.5) to obtain 12. The process is repeated to obtain X3 from eq. (2.6) and 13 from eq. (2.5). Continued iteration finally stops when 1L is obtained from eq. (2.5). If 1, has been correctly guessed, then 1L Will satisfy the equation with xL+l =00. If it does not, 11 is corrected to a new value and the process is repeated until 1L satisfies eq. (2.6). This procedure satisfies conditions eq. (2.5) and eq. (2.6) which are necessary for optimality. Max [10) used the above procedure to find the quantum levels for a zero-mean Gaussian message for quantizers up to 36 levels. Paez and Glisson [11] used the procedure to find optimum levels for signals having either a gamma density or a Laplace density for L=2,4,8,16, and 32. 2.5 Analysis of DPCM Encoder Consider an information source with alphabet {a,, a2,., aQ}. The information source is said to be nonsymmetric if the source symbols are not equally likely. It will be shown that the output of a DPCM encoder can be viewed as a nonsymmetric information source. Assume that in a DPCM encoder a predictor of order M is used. Let us model the quantizer as a additive noise source &(n)=&(n)+q(n) (2.7) where e(n) and 9(n) are the input and the output of the quantizer respectively and q(n) is the quantization noise. It has been shown that (12] 6")e~ ) (2.8) @(n)=e~n -Ebi q(n-i) i-i where e(n) is the difference signal in a DPCM without the quantizer and bi are the prediction coefficients. The distribution of 0(n) is therefore given by the M convolution of the pdf's of e(n) and Q=Zbi q(n-i) i-1 pï¿½=pe p. (0. And the probability of quantum level i is determined by PxfP(xdK (2 Xi x (n) e (n) A e (n) e(n) A x (n) (b) Figure 2.1. Block diagram of a DPCM system. decoder. (a) encoder and 2.9) .10) O'Neal [13) has shown experimentally that the pdf of 6(n) can be approximated with a Laplacian distribution p(x) =-1 exp(-6I .XI (2.11) V~co a To find the statistics of the levels an optimum quantizer must be placed in the DPCM system. If we utilize a MSQE quantizer, the statistics associated with each level will be Pi=f exp(- YTIX1) dx=f-L exp(-F2y) dy 1(2.12) =- (exp (-r a) -exp (-V2 0) 2 where a and # are tabulated in [11). We have also verified through simulations that the choice of a gamma distribution for e(n) and uniform pdf for Q results in a satisfactory approximation to the density of (n) . In other words p, (x) 8,n x)2a 2VTa) (2.13) A numerical method must be employed to find the statistics associated with each level. 29 2.6 Results In this section we will compute the statistics of the quantizers for the model developed earlier and will compare them to actual source statistics for real world pictures. Two types of quantizer were used, a MSQE quantizer and a quantizer that was found by pschovisual experiments [14]. The material used were two eight-bit pictures; a low detail picture LENNA and a high detail picture AERIAL MAP (Fig.2.2 and Fig. 2.3). Both pictures consist of 512x512 pixels. -An optimized eight-level quantizer for the, Laplacian distribution was chosen. Model parameters can be estimated by a 2 Table 2.1 contains theoretical and actual source statistics. Theoretical values are seen to be reasonably close to actual source statistics. Then a seven-level quantizer [14] that is shown to work well with different pictures was selected. The histograms for the two pictures were prepared. The Laplacian distribution and the distribution given in eq (2.13) were compared with the histograms (Figures 2.4 - 2.5). They both seem to be a fair approximation to the histograms. Table 2.2 gives the statistics given by equation eq. (2.13), Laplacian pdf and the picture. The model based statistics are close to the actual source statistics. Fig. 2.2. Lenna. Fig. 2.3. Aerial map. 31 2.7 Discussion It was shown that the pdf of input to the quantizer can be fairly approximated by equation eq. (2.13) (which is the convolution of a gamma and a uniform pdf) or simply a Laplacian distribution. The theoretical statistics derived from the models agreed well with the actual source statistics. It is seen that DPCM of video signals does in fact produce a nonsymmetric source. TABLE 2.1. A EIGHT LEVEL MSQE QUANTIZER IS USED. THE FIRST AND THE SECOND COLUMNS SHOW THE ACTUAL SOURCE STATISTICS FOR THE PICTURES. THE THIRD COLUMN SHOWS THE STATISTICS USING A LAPLACIAN DISTRIBUTION FOR SOURCE. Lenna Aerial Map Model Statistics .3324 .2800 .2549 .3574 .2795 .2549 .0946 .1416 .1510 .1060 .1390 .1510 .0337 .0578 .0744 .0331 .0573 .074 .0225 .0220 .0197 .0197 .0224 .0197 TABLE 2.2. STATISTICS FOR LENNA PICTURE. A SEVEN LEVEL QUANTIZER FOUND BY PSYCHOVISUAL EXPERIMENTS IS USED. I Laplace Eq. (13) Actual Statistics .5541 .5967 .6578 .1965 .1558 .1544 .1965 .1558 .1418 .0259 .0270 .0193 .0259 .0270 .0200 .0005 .0019 .0026 .0005 .0019 .0038 8.3 a~(a) 8.z 8.2 8.15 0. 1 8 ---- " -88 -68 -48 -28 8 26 46 68 86 (a) 8.2 8.2. Tp 8.1 8.85S 8I -88 -68 -48 -ZO 8 28 48 68 68 (b) Figure 2.2. The histogram for LENNA picture is compared with (a) Laplacian Distribution (b) Distribution given in eq. (2.13). Histogram is shown with a broken line. 8.12 8.1 0.68 // 8.84 8. 82 j-80 -60 -40 -28 0 20 40 68 8e (a) 8.12 I 0.1 8.8/ 9.86 0.0 8.ez , -88 -60 -48 -20 8 20 48 60 88 (b) Figure 2.3. The histogram for AERIAL MAP picture is compared with (a) Laplacian Distribution (b) Distribution given in equation (13). Histogram is shown with a broken line. CHAPTER THREE SIGNAL SELECTION FOR NONSYMMETRIC SOURCES 3.1 Introduction In many applications one has a bandlimited channel and has to achieve the least error rate for. a given signal-tonoise ratio. Design of high speed modems is one example where the designer is faced with the- problem of selecting an efficient set of signals with in-phase and quadrature components. The objective in signal design is to find the optimum signal constellation in presence of additive white Gaussian noise under a power constraint. Two dimensional modulation formats such as MPSK and QAM have been studied before [15). These formats confine the signal points to a certain geometry and are not optimum in the sense of minimum error rate. There has been a few attempts to solve the signal design problem under peak or average power constraint without constraining the signal points to a special geometry such as a circle or a certain lattice. Foschini et al [16] presented an iterative approach for signal selection. A gradient search procedure is given that incorporates a radial contraction technique to meet the average signal power constraint. Kernighan and Lin [17] came up with a heuristic procedure for 36 solving signal design problem under a peak power constraint. Previous investigations on signal design have focused on signal selection for equally likely signals (15]-(17]. There are some applications where the information source is nonsymmetric. A practical instance in which such a model proves rewarding is in the transmission of video signals. It was demonstrated in Chapter Two that DPCM of digitized video signals results in a nonsymmetric source. In this case signals should be mapped into a two dimensional signal constellation in an optimum manner. In other words the goal is to determine the signal constellation that minimizes the probability of error (or a given cost function) in presence of additive white Gaussian noise under an average power(or a peak power) constraint, given N signals with unequal probabilities. To illustrate the difficulty of direct approach we will design a ML receiver for a three signal constellation. We will also describe a numerical method that uses the Lagrange multipliers method for optimization. These two methods are appropriate for smaller signal sets. Then a number of iterative algorithms are developed. First a normalized conjugate gradient search algorithm and a gradient search algorithm will be presented that can be applied to signal sets of any size and with any probability distribution. The methods presented here are applicable to the design of both MAP (maximum a posteriori) and ML (maximum likelihood) receivers. These methods are generalizations and 37 modifications to the method given in (16]. Then a gradient search method for a peak power constraint is developed. Eventually a gradient search method that finds a signal constellation for an average cost function subject to a peak or average power constraint is presented. In the end, a few examples are given and conclusions are drawn. 3.2 Maximum Likelihood Signal Design for Three Signals Here a three signal constellation is designed for a 3 symbol source for transmission on a white Gaussian noise channel. The signal constellation is depicted in Figure 3.1. From geometrical considerations a1+a 2+a3+P1+P2+P3=2n, (3.1) AH cosp1=BH cosa, (3.2) AH cosa2=CH cosP2, (3.3) BH COSP3=CH cosa3. (3.4) Let r(t)=(rl,r2) be the received waveform and p(t) be a particular value of r(t). Suppose p(t)=(p,p2) is received in the symbol interval, p,(r,,r21mi) is the conditional joint probability density of random variables defining r(t) and s1, S2. s3 are the signal vectors denoted by A, B and C on the 38 constellation. A maximum likelihood receiver sets the message estimate to mk if for i=1,2,3, I pr(r1=p1Ir2=p21 Mk) > p.(r1=pl, r2=p21 mi) iok. (3.5) Assuming the noise components are Gaussian and statistically independent, P (P 1'P21 m()=-- exp_ 1 [(p1-sil) 2+ (P2-si2 ) 2] . 7N, N (3.6) The decision rule then becomes -1 [ (p1_Sk) 2+ (P2_sk2) 2] >. -__ [ (p-si,) 2+ (p2_S12) 2] N0 Ik N. itk. (3.7) Notice that the sum of squared terms on either side of the c Figure 3.1. A three signal constellation inequality are the square of the Euclidean distances between the received signal and signals si and sk. Thus the decision rule can be rewritten as: d(pSi)>- d(p,Sk) where d(x,y) is the Euclidean distance between x and y. Point H is the intersection of the decision regions and it is therefore on the boundaries of the decision regions. For the boundary points the above inequality changes to an equality, d(H,A)=d(H,B) d(H,A) =d(H, C) (3.9) (3.10) AH = BH = CH, and this results in (3.11) (3.12) C(2=P2, ( 3 = P3ï¿½ (3.13) Define P, to be the probability of word error. The probability of symbol error is one minus the probability that a symbol is correct. The probability of a correct symbol is obtained by averaging all of the conditional probabilities of a correct symbol, we have (3.14) Pw=1- P(C I Mi) pi. But P(Cjmj) is the result of point p falling in the decision i ok. (3.8) region Ii (Figure 3.2), 3 Pw=-1, PiffPr'(PIP2, mi)dp~dp2. i-1. (3.15) 2 Figure 3.2. One decision region. By substituting the expressions for probability density function eq. (3.6) into eq. (3.15) we get ___ 3 (p1-d) 2 p Pli-I --fexp[- No dp S PP2 (3.16) or equivalently - 1ptana 0PIO P2--plta 7op-.0 -nNop f ptanc1 f P2--pltanp pltand3 p2--pltanp3 L2 (p1-AH) 2 p2 exp [- -P -!L2-] dp2dpl np N0 Nï¿½ (p1-AH) 2 ,dp2dp exp [-(P- - - P2 ] dp2dpl N0 N (p 1 -AH) 2 P2 exp - -] dp 2dp1. By making the following changes of variables E=22 - dp2=dEJ-N (3.18) u=2-1 - dpj=duVIN 071 (3.19) the probability of error in terms of the new variables becomes (3.17) 3 p-i ex[-(u- AH2 - U --- f exp[-(u- AH2 u.0 - 0 , P- FI2 Ut l~fexp[-(u- ~2) f- 0 utana2 f c--utanp, tana1 f utar4 ana3 f utanp3 exp [-e2] de du exp [-C2 de du exp [-e2] de du. (3.20) Now by definition X2 erf(x,x2) =-f fexp (-t2) dt. F#CX1 (3.21) Therefore, the following is the equation for P,: =1- , f exp[-(u- - )2]erf(-u tan1'u tana2)du PP2 fexp-u- AH2 2 1xu-0 - -0 )2] erf(-u tanj3,1utana1)du - IV, -- o)2]erf(-u tanP2,u tan3) du. (3.22) 43 For a given signal to noise ratio the probability of word error depends only on a, and a2 because a,, a2 and a3 add up to 71. To find the values of aI a2 , a3 that minimize the probability of the word error, we need to differentiate P, with respect to al and a2 and set them equal to zero: a P o (Uf e22-(- )2 u2tan2 ) w=- 1-fe-- (u (l+tan a,) e- du aUa0 P2 fe x u-O (U (I+tan2a1) e u2tan2a-U (l+tan (a+a2) )e-u2tan2 (a +d2)) du u-3) 22 f_ e 1-~~a (OC1+C2) )e-u'2" ") du=O, U-0 (3.23) a" (U_- ) ---w=- p' e N./ , (u(I +tana2) e-u2tan2 2) du aa 2 7C u-0 U=O ---- 3 e d U-0 ( 1 (l+tan2 (a+a2) ) eu't a ) +u (l+tan2a2)eutan'2 du) =0. (3.24) Inspection shows that above equations consist of only one type of integral which can be expressed in a computable form, u (u-k) 2e-u du=f u e-((1+1)u2+k2-2uk) du 0 0 (3.25) By producing a perfect square in the exponent k21 - -(1+1)(u- it k du fu e-((ll)u2+k2-2uk) du= e ue (1+1) du 0 0 (3.26) and using the results obtained in Appendix A we get: k2l k2 u e-(u-k)2e-U21 du=- 1-- e + /2-n k erf( -k ) 2(1+1) 2 12_(i+i) (3.27) If we substitute for integrals from eq. (3.27) into eq. (3.23) and eq. (3.24) we will end up with the following set of nonlinear equations: --- AH2 s1er( A2co~I (Pl-P3) e + 7tA (p +p2) e NO osa1erf(- ,f2cosalI 00) -(p2+p3)e -- ICos(S . caa)lerf(- AHEcos (.,+.2)l, -)N =0 (3.28) A A2 - -(P1P2) eN +C (p1+P3) e NO UosN1 A21cos= - H2sin2 (al+OC2) A 2 (P2 +P3) e NOIcOs (CC1+2) jerf (- -A o s (C1 +a 2)1'o) =0 (3.29) Given a signal-to-noise ratio and a probability set the optimum angles can be found using eq. (3.28) and eq. (3.29). 46 We experimented with a variety of signal-to-noise ratios and probability sets. Figure 3.3 shows the optimum a, for three probability sets over a wide signal-to-noise ratio range. In all cases optimum a, approaches 60 degrees as signal to noise ratio goes up. Even when the optimum angle is somewhat different from 60 degrees from the performance point of view the two system are almost indistinguishable. 02 61 00 -10 Figure 3.3. a, (deg.) versus average power (dB) . (a) the top curves, source statistics {.9,.05,.051, (b) The middle curve, source statistics and (c) the bottom curve, source statistics {.6,.2,.2j. The presented results suggest that the three signals should be placed on the vertices of an equilateral triangle (Figure 3.4). The origin of the signal constellation can be shifted to a new location that minimizes the average energy without affecting the probability of error. Let us define 1 to be 1= AB = BC = AC in triangle ABC. Since ABC is an equilateral triangle, AH and 1 are related by: 1 =AHvf. (3.30) The point G(x,y), can be found as follows: paveP,[ (X-I)2+Yl] +p2[ (x+!)2+y2] +p3 [X2+ (y-I3)2] 2 2 2 (3.31) - (P1-P2) 1 2 - y=P31/. 2 Substituting x and y into (30) and solving for 1 we get 1= rk Pave" (3.34) where k is given by 7-- (4/3) (3.35) aP., =0 ax a, =0 ay (3.32) (3.33) P1 I (PI-P2-1) 2+3P32] +P2 [ (Pl-P2+I) 2+3P32] +P3 [ (pl-p2) 2+3 (P3-1) 2] C (o, 113/2) B(-I/2,0) A(1/2,0) Figure 3.4. The three signals form a equilateral triangle. 3.3 A Numerical Approach Based on Lagrange Multipliers Method In this approach the problem of signal selection is viewed as a constrained optimization problem. The constraints are incorporated into the optimization problem by the use of Lagrange multipliers method. Let us outline the design of a MAP receiver with three signals with unequal probabilities (Figure 3.1). A MAP receiver decision rule is m=mk if, for i=1,2,3, pr=(r1=P11r2=P2I k)Pk' p1(1P11-r2=p21 mi)pi i~k. (3.36) Upon substituting (6) into (43), we get .G (X, y) 2 2 in (Pk) -Eoj- (pj-skj) in(pi) --LE (pj-Sij)2 ij. (3.37) Assume point H is the intersection of decision regions. For point H the inequality turns into an equality and the sums on either side will be the distances between H and signals si and sj, therefore AH2 -BH2 =N' in ( 1 P2 (3.38) Similarly we obtain AH2-CH2=N in(P) . P3 (3.39) In addition to the two above MAP constraints there are four geometric constraints (eq.(3.1) through eq.(3.4)), and an additional constraint to minimize the average energy. The origin of the constellation is shifted to a location that minimizes the average power. Then the average power function evaluated at that point is set to 1/N. p a+p2b+P3 C= Pave (3.40) where a=(x-AH sinp1)2+y2 b= (X+BH sinl) 2+y2 c= (x-CH sin (3.+a2+32)) 2+ (y-AH cos31+CH cos (Pl+c2+P2)) 2 x=p1AH sin31-p2BH sina1+p3CH sin(P1+C(2+P2) y=p3AH cosP1-p3 CH COS (P1+a2+P2) . Now the problem is to minimize a function of nine variables under seven equality constraints. The method of Lagrange multipliers can be applied to this problem. We need to define a new cost function F of sixteen variables and take the partial derivatives with respect to the sixteen variables and set them equal to zero. cost function=F(a11,2,a3, j$P213,AHBH, CH, X11, 2*31I,4,15, 16,A If we eliminate Xi's among the sixteen equations we end up with a nonlinear set of equations of the other nine variables which must be solved numerically. This method becomes quite complicated as the number of the signals in the constellation grows. It was mainly used to verify the solutions obtained by the gradient based method when N was relatively small. 51 3.4 Minimum Error Signal Selection 3.4.1 Preliminaries Let us denote the signals in the constellation by SIS2,.,SN where si=(xi,yi). The average power is given by N (3.41) Pave=Y pIIsol2 n-I Where {P, ,P2 ,-. -PN} is the probability distribution of the signal set and ls.l1 is the magnitude of s,. Following the convention of [16] we choose to set P.v =1/N and define the signal-to-noise ratio to be SNR=I0 log10( No) (3.42) N0 where No is the one sided power spectral density of white noise. By definition the error rate (probability of symbol error) is N (3.43) Pei=EPn Pr(error Isn) n-1 The union bound gives the upper bound for the conditional probability of error. For a maximum likelihood receiver we have: Pr (errortsn) < E) (3.44) .1=1 ion For large values of signal-to-noise ratio the conditional probability of error will be equal to the upper bound. By using approximation for Q function when the argument is large and plugging back in (3.43) we get p N 1 exp[- Isi-s-I121 (3.45) 7En1 i1-SnII 4N, Similarly it can be shown that the symbol error rate for a MAP receiver is given by (IsiSnN2+Nln( Pn))2 Pe= N Pn Isi-Sn' exp[- 4Nï¿½1Si-_Sn|2 P n-1 ion is i-snl12 +Nln (Pn) Pi (3.46) 3.4.2 Gradient Search Algorithms for an Average Power Constraint In search of the minimum, the gradient of the probability of error is obtained analytically and an iterative gradient search algorithm which modifies the constellation at each iteration is used to find the optimum constellation. In the gradient search algorithm, the iterative rule is given by S;1 =Sk-akVFk where (3.48) denotes the signal vector at the kth step of the algorithm, ak is the step size and VF is the gradient of PC. Since the signal power may change with k, the signal vector is normalized at each step of the algorithm sk.1= . . N(pls* (k+l) 12 +P21S (k+l) 12+.2+p.sN(k+l) 12) (3.49) To speed up the convergence, instead of a conventional gradient search algorithm, the Fletcher and Reeves conjugate gradient method [18] can be utilized. In this method the information about the second derivative is used indirectly. The algorithm is described by S;+1 =.,sk+akh~k (3.50) where hk=-VFk+akk-1 (3.51) and (3.47) tk=(VFkV t vFk -(3.52) [vFk-J t: vFk-l in which ak is the step size and hk is the quantity in this algorithm that combines the information from current and previous steps to define a new direction. similar to gradient search algorithm, the power may alter with k. Therefore the signal vector needs to be normalized at each step of the algorithm. Let us summarize the procedure for iterative methods: 1. Set k=Q and select the starting points. 2. Determine the search direction by calculating hk for the conjugate gradient method and VF for the gradient method. 3. Find the improved signals coordinates and normalize them. If the improvement is smaller than a tolerance level stop, otherwise set k=k+l and go to step 2. 3.4.3 Analytical Expressions for the Gradient Vector Let us find the gradient vector for the ML receiver first. VF the gradient of Pe is a vector of 2N components in signal space, in which each signal occupies two dimensions. The gradient as a vector of N two dimensional vector components is represented by 92 vF= where (3.53) gk=g9k,g k gk is obtained by taking the derivative of P, with respect to sk gk=-E (Pk+Pi) ivk Sk-si ISj-Sil2 Ilsk-sexp[ 4No] I 1 + Ils -s-1 2N Similarly one can find the kth component of VF for a MAP receiver 2 tl pklSk~S exp [- 4NoI SkSi 12] +p Skl exp [ - 112 4N0 IISk-S-i12 t, 2 t2 t2 + t2 t 2NojSk-s1l2 2Nollsk-sII2 } (3.55) where (3.54) i~k and SkS Sk-Si ~ISk-si sk-sill t2Isk-ISki+NlD n (- ) Pi t2=lsk-si 1 -Noln (-) Pi 3.4.4 Startina Points The average signal power is equal to I/N or N 12 n-1N Thus, individual signals must satisfy Solving the inequality for sn we get 1 1 which states that starting points must be selected from inside (3.56) (3.57) (3.58) (3.59) (3.60) (3.61) . . I . a circle with radius 3.4.5 Gradient Search Algorithm for the Peak Power Constraint Even though the average power constraint is used more often, in certain applications such as space communication the peak power constraint is much more realistic. Given a peak power constraint the transmitted signal points must be placed inside the circle such that the error rate is minimum. We would like to modify the iterative procedure developed earlier to accommodate the peak power constraint. All we need to do is to further modify the modified constellation at each iteration to insure that peak signal power is bounded. Let the peak power be P=l/N. The signal set is modified using the iterative rule (gradient search or conjugate gradient search), let Mmax {Isi*k+1) 11)}. (3.62) 1:5i N We will modify the signal set again in the following manner s (k+i) 2~ (k+l) Sk.l /, j (3.63) s. (k+i) to meet the peak power constraint. 3.5 Minimum Average Cost Signal Selection The communication channels are almost never error free. The probability of error (error rate) is a measure of system performance. Various error types are usually weighted equally but in some communication system certain errors are more costly than the others. The Bayes receiver allows us to rank the different error types [19]. To utilize the Bayes receiver, one must know the source statistics and a reasonable estimate of a cost matrix must be obtained. The Bayes receiver requires more apriori knowledge about the communication system than others but it results in a superior performance if there is no mismatch between the design and operating conditions. The goal is to find a signal constellation that minimizes the average cost subject to a peak or average power constraint. First a workable expression for the average cost will be obtained. 3.5.1 Average Cost Function The Bayes receiver which minimizes the average cost is given by N N C=>3 (3P(sjIsi)pi Lji, (3.64) 1-1 j-1 where Li is cost involved when the receiver picks si when si was actually sent. The boundaries of the decision regions are no longer straight lines; instead they are two dimensional curves. For instance, the boundary between si and si is given by N N SLik P(-rlISk) Pk= F L3 k P (rlISk) Pk,3.5 k-I k-i where conditional probability of error is p(.rlsk)= 1 ex ( S-Sk12 (.6 ~No Channel noise could cause the received signal to move to an adjacent decision region and result in an error. For large values of signal-to-noise ratio; however, errors are almost entirely the result of displacement of a received signal to an adjacent decision region from the nearest point on a boundary to the transmitted signal. Let si be the transmitted signal and sji be the point with shortest distance on the boundary 60 with sj. p(sj si) can be approximated with a Q function P(sjJsj) =Q( llsi-sjil ) V(3.67) substituting back into the average cost expression we get N s- N e[SiSJ2 (3.68) 2 i id i~iPJ- ns lexp - N 2 n Lit is SJII N. 3.5.2 The Nearest Points on the Boundaries The first step in computing the average cost function is to find the set of nearest point to each and every signal point in the constellation on the boundaries. There are N signal points in the constellation and for each one there are (N-l) of nearest points on (N-i) boundaries, therefore a total of N(N-l) of nearest points must be found. The problem of finding sji which is the nearest point to si on the boundary between si and si can be formulated like a Lagrange multiplier problem. We would like to minimize lIs-si1 subject to a constraint (eq. (3.65)). Let us form the auxiliary function N N 4=:ls-si2+1(E Lik p(rIsk)pk-ELik p(rIsk)pk), (3.69) k-i k-i and differentiate it with respect to x,y and X - =0, ay ;aY=0. al Eliminating equations (3.70) X we will come up with the following set of N (XXi) E (Lik-Ljk) (x-xk) exp(- ) )Pk i - k=1 No (Y-Yi) Y (Lik-Ljk) (Y-Yk) exp (.- 112 k-I N. (3.71) NIS-Sk112 NS Ski J2 E Lik exp(- S-k)p N exp(- )pk=0 (3.72) k-1 No k-1 o where si=(xi,yi) and s=(x,y)=sji. 3.5.3 The Gradient of the Cost Function To evaluate gk , first the portion of cost function that depends on sk must be formed: N J sk-sjkl112] E Ljk PISkSjkl exp[+ Lki 1 exp lSi-Skill2 -2 Pi Isi-Skill 1 N0 gkI and gk2 the components of gk are found by taking the derivative of above expression with respect to the components Lk Pk 1 exp[- DSk-Sk II ks-skD N. L p 1 exp [- ISi-Skil2] LkiPi i Ski 11 No 1 1 jsk-S I112 N 1 1-L +- 11 DHs2-sk~iI{ No (3.73) where aXjk) + (ykYk ( ul = (Xk-Xjk) (1 - ayj)k )Xk) )aXk aXk aXkl aYki l Lik PkS I S1 exp [- Sk-SJk12] V2LkiPI 1 exp N Si-Ski 2 V2Lk P ls-ski 1 N. 1 11 ISk-Sjk 1-L N, I 1 1 Isi-ski ll No (3.76) where (aXjk) + (Yk-Yjk) ayk of sk N j.1 N -E U2 i-i (3.74) (3.75) and N j.1 N 1-1 (1 - ayJk) ayk (3.77) V1 = (xk-_xjk) v2=(xxk)-) a +a(y-Y) (- (3.78) The gradient of the cost function depends not only on the nearest points on the boundary but also on their derivatives. Let us find the derivative of the components of sji with respect to the components of s,. Differentiating eq. (3.71) and (3.72) with respect to xi and yj we get N ISSk12 N w-s 12 Lik W exp - I ]pk=LJk wlexp k ]Pk k-1 N0 k-1 N, (3.79) where Wl = (X-Xk) a+ (Yyk)ay (3.80) IS-Sk12 N Is -Sk 2 E Lik w2 exp [ ]Pk= Lk w2 exp[- ]Pk k-1 N k-1 N. (3.81) where w2= (x-xk) i+ (Y-Yk) X ay1 (3.82) ax N ax -1) EN (Lik-Ljk) (Y-yk) exp PkI-sU ax1 k-1 N N ay- zl ss +(x-xN) (Lik-Ljk) ( -2 (y-yk)---)exp[- ISSk k-1 ax1 N. N. N E (LIk-Ljk) (x-xk) exp - is-skI2 Nx ax1 N. kI (y-yi) E (Lik-Lk) ( ax -2 (x-xk) -l S-exp[I- N k-i 1 0 0o IV. No (3.83) 1 where zl = (x-xk) (-Lx) axi + (Y-yk) ( ay ax1 -yE k-i (Lik-Ljk) (Y-Yk) exp[- iss Pk ak-1 No N ays-ski X-Xi) E (Lk a -2 ( o) ) exp [ NS = ( ay -1) (LikLik) (x-xk)exp [] N k-iS 12___(Y N xz Is-skll: i) (Lik-Ljk) (l. -2 (x-xk) -L' ) exp [ ' k-1 ay, 1 IV. 12]Pk 2 -]Pk (3.85) where Z2 = (X-Xk) (- x) + (Y-Yk) ay1 (er) ay1 (3.86) It can be shown that (refer to Appendix B) D 2 ] Pk 2 ] Pk (3.84) axi =0 IIxj,=0' axi ayX (3.87) and aIYJ1 =0 "~ 0 axi aYJ (3.88) Equations (3.75) and (3.78) imply that v2=u2=0. Contrary to the equations for finding sji, the equations for finding its derivatives are linear. 3.5.4 The Iterative Algorithm To find the signal constellation we will start from an initial guess. An iterative rule (either gradient search or conjugate gradient search algorithm) is selected. An improved signal set is found using the iterative rule. Depending on the type of constraint (peak or average power constraint) the constellation is modified accordingly (eq.(15) or eq.(30)). The iteration is continued until a minima is reached. The above procedure is valid under the assumption of large channel SNR. When channel SNR is not large, obtaining an analytical expression for the gradient is not feasible because of the complexity of the shape of the decision regions. An obvious solution is an exhaustive search which is 66 computationally very expensive and long. However, one can use a modified form of the above procedure to save on the computational expenses. In the modified procedure, the average cost is computed based on the received data and the gradient based update rule is replaced with the directional set (Powell's) method in multidimensions [20) which does not require the gradient. 3.5.5 Application to DPCM System In a DPCM system some errors are more costly than others, therefore we could benefit from utilizing a Bayes receiver. In Chapter Five an optimum method of selecting the cost matrix will be presented. 3.6 Results In this section we present some numerical results, elaborate on the design procedure and discuss the performance of the methods described in this chapter. The probability of error, in general, is not a convex function of the signal set, therefore the algorithm can converge at local as well as global minima. The multi start technique can be applied to the global optimization problem [16]. In this technique one selects an optimization technique and runs it from a number of different starting points. The 67 set of all terminating points hopefully includes the global minimum point. our implementation of the technique is somewhat different. To come up with a set of start points, First a search procedure is implemented that finds signal sets having small error rates initiating from random start points. About thirty start points are selected. Then we run the gradient search techniques from these starting points. To run the algorithms we start with a small step size and monitor the changes in probability of error as a function of iteration number. if P, changes very slowly we can proceed to increase the step size. On the other hand, we may need to decrease the step size if P, starts to oscillate around a minima or goes unstable. The best constellations for a number of sources with three, four, five, seven and eight signals were found. Figure 3.6 shows the best constellations for three, five and seven signals with the probability distributions given in Table I. We compared the speed of convergence of the conjugate gradient method to that of the conventional gradient search method for a given value of step size. The normalized conjugate gradient method converges to a solution much faster than the normalized gradient method. A set of initial start points and the optimum constellation for seven equally likely signals are shown are in Figure 3.7. The conjugate gradient method is faster about an order of magnitude. To evaluate the gain in using a non-symmetric signal TABLE 3. 1. VALUES OF SIGNAL-TO-NOISE RATIO AND THE PROBABILITY DISTRIBUTIONS OF NON-SYMMETRIC SOURCE. TABLE 3.2. SOURCE STATISTICS FOR THE FOUR SIGNAL SOURCE. source statisticsI .35, .35, .15, .15 Source Number SNR(dB) Probability Distribution 1 12.2 .8,.1 1.1 2 9 .9,.025,.025,.025,.025 3 8 _.96,.01,.01,.O01,.O1 4 15.5 .674, .142, .142, .018, .018, .003, .003 -1 -8.5 8 8.5 I 1.5 (a) x -I -8.5 8 8.5 1 1.5 Figure 3.6. Best signal constellations for non-symmetric sources. Probability distributions are given in Table 3.1. (a) Source 1.(b) Source 2. (c) Source 3. (d) source 4. x x 8.5 - -8.5F -1.5 -8.5I- -1.51 -1.5 (C) -i b, (d) 8.5 1 1.5 Figure 3.6--continued -6.5- -1.5 - 18-1 I. . tS. IV -9.5 8 9.5 1 1.5 a 288 488 688 number of iterations B8B 1800 Figure 3.7. (a) Seven signal constellation for an equally likely source. O's represent the start points and x's represent the final constellation. (b) The solid and dashed curves show the error rate as a function of number of iterations for gradient search and conjugate gradient method respectively. CI 0 K 0 x 0 Ox a 0 xx a 72 design relative to a equally likely signal selection, a first order Gauss-Markov source with a correlation coefficient of 0.9 was synthesized. The output was encoded using a first order DPCM encoder. A four signal constellation was designed with the source statistics shown in Table 3.2. The performance of the non-symmetric constellation was compared with that of the well known equally likely constellation (for four equally likely signals, the best constellation is formed by the vertices of a square (16)). Figure 3.9 demonstrates a comparison between the systems in terms of output signal-tonoise ratio (output SNR). For large values of channel signalto-noise ratio (channel SNR) the two design procedures result in an identical performance. As the channel SNR decreases the curves representing the performance of non-symmetric and equally likely signal design separate and the difference between the two systems gets larger. The non-symmetric signal design is 3 dB (in terms of output SNR) superior to equally likely signal design for the noisiest channel considered (a channel SNR of 4 dB). It is seen that significant improvement in performance can be obtained for noisy channels by utilizing non-symmetric signal design. The amount of improvement is a function of source statistics. A comparison between the equally likely signals and unequal signal probabilities shows that if the signal statistics are not very different, the shape of constellation is not appreciably different from the equally likely case. But 73 if the signal probabilities are very different from the equally likely case the shape of the constellation could be very different from the equally likely constellation. Figure 3. 6 (b) and Figure 3. 6 (c) show constellations f or two dif f erent five signal sources. The constellations for the extremely nonsymmetric source is completely different from the other one. Generally speaking the geometry of. the constellations depends upon the power constraint. For example, the optimum signal constellations for five equally likely signals subject to average power and peak power constraints are displayed in Figure 3. 8 (a) and Figure 3. 8 (b) . Clearly, the choice of power constrain affects the geometry. Simulations show that for large values of signal-to-noise ratio the average cost signal selection does not result in any improvement relative to the minimum error signal design. For other values of signal-to-noise ratio the decision regions form unusual shapes which vary with the signal-to-noise ratio. Figure 3.10 shows the decision regions for a four signal constellation with the statistics listed in Table 3.2. Because of the complexity of the shape, obtaining an analytical expression for the gradient was not feasible and as a result the modified procedure in section 3.4.10 was utilized. To compare the minimum error with the minimum cost signal selection, a' first order Gauss-Markov source with the correlation coefficient of 0.9 was generated. The output of the source was encoded with a first order DPCM system. An 74 eight-level MSQE (mean-squared quantization error) optimized quantizer was used and the output (reconstruction) signal-tonoise ratio of the two systems were evaluated over a wide range of channel SNR. The results graphed in Figure 3.11 indicate that for large values of channel signal-to-noise ratio the two systems are almost identical. However, for smaller values of signalto-noise ratio the minimum cost design is superior to the MAP system. In this case, the improvement in the output signal-tonoise ratio is around 0.5 dB for the intermediate values of channel signal-to-noise ratio and 1 dB for low values of channel SNR. (b) Figure 3.8. The best signal constellations for five equally likely signals subject to average power (a), and peak power (b) constraints. .4. .2- x .4 x x .2 .4 - 10 /' 8 41 3 4 5 a 7 a 9 10 11 12 Figure 3.9. Performance results for non-symmnetric signal selection (the solid curve) and the equally likely signal design (the dashed curve) over a wide range of channel signalto-noise ratio. ï¿½sloquAs ;ouTqsTp Aq paXau 9aP suoTbaa UoTsT ap Dq; puP s,* Aq pa-uasaadaa 9- sTuBs ;9LU, "EP TT=UNS laUUpqo (q) pup sp s'c=uNs IaUUptqP (P) "ubTSsp TlU6Ts -4soo uinUITUTUI 10_ SUOT6ba UOTST09P alq 0 " a DnbTd (q) TO 0 TOgL oeIeeooeo@oeobIIe H. *666,46 666 .oo.iogI. XXI IXXXXXXIXXXIXIXIXX: XIXEXIXXXXH*HXHXX IXXXXXEXIXMXIMXXXXXXXXXEXXXXXXIXXIXX IXI IXIx!XIXEX~XIXXIXEXVXEXXXXIXI X EXXXXIXVXXXIXXXXIXXXXXXXXXXXKXIXXX 4XX X~XXXXIX! KlU! MIN XE XXXIXEXMMXIX XIXXIXXXXIXIXIXXXIKXEXXXXXXXXMXX ExXXXXXXIXI~XIXXEEXIXI XXXlXI I.*. .:: t++ 5.1.OOOOVit4l . 4+ 94+.X I KIMI 1 11 I X .I .XV .I x . N4 '' I" MXXX II XEI X IXXI XIX II X X ix X ENIX xx XXv I "NxX IX IX I XX I XIX I XXXI"IVI WXE "xxxxEXvXX IIXIII . IIIIxIX.II.I 0400000000000oooox:-+D404+ M J-0ï¿½0,:ï¿½:::::::=:::::t 4iaiaai . ++t. + Iit s X4IIIXIIXXXXiXIXXE XXIXXIIIIXXiI~ XXIXIXlxXXXIXIXIXXvXIXXXxIXvXIXXIXIXX YIXXIIIXXXXXXIXXI XXXXIIKKXXXXX xxxx xxxxx (~cxxix xxxixxXxxxvxxxlxix XXXXX XXXxlxxXXIIIIXXI XXXIX!X! XIxXX XXXXIEIIIIXXXIXIxxxxxxxXXIXIIXXXX xx.Xvx~x. x x Ymmi xxxxxw wxwi~xi Xx.Xl XIXII XXIXXIIXIIXIXXXIIXIXxvXIIYXX -xxixlxixxxxlxlxx~xlxxxxlx X I~XXXI x 1 R S XxEIXXXIXXVXIXXIIXIXIXIXXIXI XXXXIXXX .xxxxxvvxxvlvxAvlxxxx 5 8 7 8 9 10 11 12 13 14 choan.l SNR Figure 3.11. Comparison between minimum cost signal selection (the solid curve) and minimum error signal design(the dashed curve) for a first order Gauss-Markov source with correlation coefficient of 0.9. CHAPTER FOUR ALPHABET SIZE SELECTION FOR VIDEO SIGNAL CODING 4.1 Introduction Previously the methods of signal selection for nonsymmetric source were studied. Here we are going to utilize those methods in the deign of an image transmission system. An image transmission system typically consists of three parts, a source coder, a channel coder and a modulator. Source coding is the first step in the transfer of information from a source. The purpose of source coding is to remove as much redundancy as possible from the source. DPCM of video signals with as few as seven quantization levels has been shown to produce pictures virtually indistinguishable from the original under standard viewing conditions [21]. Since the output of DPCM system is a nonsymmetric source, an entropy coder must be used to take advantage of that redundancy. The well known Shannon-Fano method and the Huffman procedure are examples of entropy coding techniques [22]. In channel coding the goal is to correct errors introduced in the channel by inserting redundancy in the data. Channel coding is accomplished by either decreasing the 80 information rate or increasing the bandwidth. Block and convolutional codes are the two major classes of error correction codes. Modulation is the process of mapping a base band signal into a band pass signal. To transmit the channel encoded signals one must design a signal constellation that minimizes error rate under a power constraint. This issue has been addressed in Chapter Three. Even though efficient techniques for source coding, channel coding and signal design exist, it is not known how the choice of alphabet size affects the performance of a communication system. In other words, given a nonsymmetric memoryless source with a known probability distribution, it is not clear what alphabet size results in smallest error rate subject to equal average power, bandwidth and the information rate. It is the purpose of this chapter to explore the relationship between signal constellation size and a system performance measure (the error rate) for video signals under different bandwidth constraints. 4.2 Preliminaries 4.2.1 Description of the Communication Systems The block diagrams of the system under consideration are shown in Figures 4.1-4.2. Huffman optimal source coding is used to encode the source signals. 81 The BCH codes are chosen for channel coding (If extra bandwidth is available) because codes close to any rate can be found. A q-ary BCH code is denoted by (n,kq,tq) where kq represents the number of information symbols, n represents the bolck size and tq represents the the error correction capability of the code. The receivers are coherent in-phase/quadrature detectors (except for BPSK where only the in-phase branch is needed) and perfect carrier and symbol synchronization is assumed. The code words for the nonbinary systems are not equally likely. A minimum error procedure is the optimum way of decoding at the receiver but a maximum likelihood procedure can be used as a sub-optimal procedure. The Berlekamp-Massey procedure will be used for decoding of the received data [23][25]. 4.2.2 Communication Channel The communication channel is modeled as a q-ary independent error channel and the source alphabet and channel output alphabet are the same. If a symbol si is transmitted there is a probability of pi that s, is received (i1j) and a probability of Pki that sk is received (ixk), and so on and so forth. Therefore, the probability that si is received correctly is (4.1) q P i=I-- Pn n=1 noi Notice that pji depends on the modulation scheme. Here a two dimensional signal constellation is utilized. The channel matrix is (4.2) P= [pjj] , S2 ï¿½ s3 ï¿½ PNN Figure 4.1. The Model of the communication channel. where Pji is given by Pji Nï¿½ 1 exp[- iSiI2] 'n jjsi-sill 4No It is also obvious that Pji= Pij 1 i~q l~j~q. (4.3) (4.4) 4.3 Analysis For a fair comparison of systems employing different alphabet sizes, average power, bandwidth and information rate are held equal for all systems. In practice the bandwidth constraint could be either very strict or can be somewhat relaxed. Therefore two cases are considered: I. When the bandwidth constraint is strict and can not be relaxed. The binary system against which all other systems will be compared uses source coding, BPSK modulation and no channel coding due to strict bandwidth limitations. For the same information rate higher alphabet systems send longer pulses and require less bandwidth. Therefore these systems can use the extra bandwidth for error-correction codes (Figure 4.1). Here nonbinary BCH codes are used for error correction. For a nonbinary system of alphabet size q a q-ary (n,kq,) code must be found to satisfy the equal information rate: n hb (4.5) kq hq where hb and hq are the average message length for binary and q-ary systems respectively. (b) Figure 4.2. The transmitters for systems employing (a) binary (b) nonbinary alphabets. In variable length coding (such as Huffman coding) a symbol decoding error propagates in the block. Therefore, it is appropriate to use word error probability as the figure of merit. For the binary system P,, is given by Pw= 1- (l P.) 1(4.6) where p, is the crossover probability. For a BPSK system p, is known to be equal to po=Q (v -) (4.7) where e is the signal-to-noise ratio. II. When the bandwidth constraint is not very strict and can be somewhat relaxed. Let us allow some bandwidth expansion for the binary system (Figure 4.2). As a result the nonbinary 85 systems employing higher order alphabets will enjoy even greater bandwidth expansion. Ei Sure Huf Emn - oig BP encoding Figure 4.3. Transmitter for the binary system when the bandwidth can be expanded. Let the bandwidth expansion for the binary system be percent. To satisfy the equal information rate constraint, binary BCH (n,k,t) code and q-ary BCH (nq,kq,t,) code must be found such that: =(4.8) kq h q The binary system makes no error unless t+l or more of the n total bits in a word are in error. The probability of t+l or more symbols error is given by n where p, is the crossover probability. Usually P, can be approximated by (t1 n-t1 Pe P "<1. (4.1 0) Let the word error probability for the system using the q-ary alphabet be Pwq and let n =M (4.11) nlq where n and n q are the block lengths for the binary and the qary systems respectively and m is an integer. Then the word error rate for the q-ary system can be expressed as Pq111U Pwq M. (4.12) 4.4 Implementation Issues As it was mentioned DPCM of video signals with as few as seven quantization levels has been shown to produce pictures virtually indistinguishable from the original. The quantizer uses seven quantization levels : a zero difference and three graded sizes, each of positive and negative differences. In order to make results meaningful, we used pictures of different contents. Low, medium and high detail pictures were used in the simulations. Table 4.1 gives the probabilities associated with each quantization level for three different scenes namely a low detail scene MICHAEL, a medium detail 87 NARROWS scene and a high detail scene BANKSIAS [21]. It is observed that: P(Y0) > P(Y1) =P(Y-1) > P(Y2) =P(Y-2) > P(Y3) =P(Y-3) (4.13) where p(yi) denotes the probability of ith difference level. The primitive irreducible polynomials over the nonbinary fields are given in Table 4.2. The minimal polynomials and the generator polynomials of the nonbinary codes are given in Tables 4.4, 4.5, 4.6 and 4.7. Notice that GF(4) the ground field for GF(42) in itself is an extended version of GF(2). The elements of GF(22) are 0,I,A and B where B=A2. The arithmetic tables for GF(4) are provided in Table 4.3. Alphabets of size 6 are not used in the simulations because 6 is neither a prime or power of a prime number and does not lead to BCH code implementation. TABLE 4.1. DIFFERENCE SYMBOL PROBABILITIES. TABLE 4.2. PRIMITIVE IRREDUCIBLE POLYNOMIALS OVER NONBINARY FIELDS. Field Polynomial GF(2) x2 +x +1 GF(3) x3 +2x +1 GF(4) x2 +x +A GF(5) x2 +x +2 GF(7) x2 +x +3 Picture p (y.) P (Y+I) P (Y+2) P (Y+3) P (Y-I) P (Y-2) P (Y-3) Low .674 .142 .018 .003 Detail Medium .584 .172 .032 .004 detail High .5 .166 .064 .02 Detail TABLE 4.3. ADDITION AND MULTIPLICATION TABLES FOR GF(4). + 0 1 A B 0 0 1 A B 1 1 0 B A A A B 0 1 B B A 1 0 X 0 1 A B 0 0 0 0 0 1 0 1 A B A 0 A B 1 B 0 B 1 A TABLE 4.4. (A) MINIMAL POLYNOMIALS FOR ELEMENTS OF GF(27). THE POWERS OF a ARE SHOWN ON THE LEFT COLUMN AND THE COEFFICIENTS OF MINIMAL POLYNOMIALS ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS . (B) GENERATOR POLYNOMIALS FOR 3ARY CODES. COEFFICIENTS OF G(X) ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS. THE POWERS OF a ARE GIVEN IN THE MIDDLE COLUMN. GF(3') Minimal Polynomial 1,3,9 1 2 01 2,6,18 2 1 1 1 4,10,12 2 01 1 13 1 1 14,16,22 2 2 0 1 5,15,19 1 1 2 1 17,23,25 1 0 2 1 (a) 3-ary code Powers of a g(x) (26,19,2) 13,14,15,16 120111201 (26,14,3) 14, 15, 16, 17, 18, 19 ,1122002000021 (b) TABLE 4.5. (A) MINIMAL POLYNOMIALS FOR ELEMENTS OF GF(16). THE POWERS OF a ARE SHOWN ON THE LEFT COLUMN AND THE COEFFICIENTS OF MINIMAL POLYNOMIALS ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS . (B) GENERATOR POLYNOMIALS FOR 4-ARY CODES COEFFICIENTS OF G(X) ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS. THE POWERS OF ax ARE GIVEN IN THE MIDDLE COLUMN. GF(4') Minimal Polynomial 0 1 1 1,4 All 2,8 Bl 11 3,12 1 B1 5 Al1 6,9 1 A1 7,13 A A 1 10 B I 11,14 BB I 4-ary Code Powers of a g(x) (15,12,1) 0,1 A B 0 1 (15,9,2) 1,2,3,4 1 A A 1 1 B 1 (15,7,3) 0,1,2,3,4,5 A 0 B 1 B B 1 1 TABLE 4.6. (A) MINIMAL POLYNOMIALS FOR ELEMENTS OF GF(25). THE POWERS OF a ARE SHOWN ON THE LEFT COLUMN AND THE COEFFICIENTS OF MINIMAL POLYNOMIALS ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS . (B) GENERATOR POLYNOMIALS FOR 5-ARY CODES. COEFFICIENTS OF G(X) ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS. THE POWERS OF a ARE GIVEN IN THE MIDDLE COLUMN. GF(52) Minimal Polynomial 0 4 1 1,5 2 1 1 2,10 4 3 1 3,15 3 0 1 4,20 1 4 1 6 3 1 7,11 3 2 1 8,16 1 1 1 9,21 2 0 1 12 1 1 13,17 2 4 1 14,22 4 2 1 18 2 1 19,23 3 3 1 (a) 5-ary code Powers of a g(x) (24,15,3) 0,1,2,3,4,5 1322210121 (24,12,4) 0,1,2,3,4,5,6,7 4102441132021 TABLE 4.7. (A) MINIMAL POLYNOMIALS FOR ELEMENTS OF GF(49). THE POWERS OF a ARE SHOWN ON THE LEFT COLUMN AND THE COEFFICIENTS OF MINIMAL POLYNOMIALS ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS . (B) GENERATOR POLYNOMIALS FOR 7-ARY CODES. COEFFICIENTS OF G(X) ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS. THE POWERS OF a ARE GIVEN IN THE MIDDLE COLUMN. GF(7') Minimal Polynomial 1,7 3 11 2,14 2 5 1 3 66 1 4 4 01 5 53 1 6 14 1 8 4 1 9,15 63 1 10 41 1 11 54 1 12 1 01 13 3 21 1116 15 1 (a) 7-ary code Powers of a g(x) (48,31,5) 1 through 10 235134333561123361 (48,27,6) 1 through 12 3202052111645042656531 (48,24,8) 1 through 16 3655056652662113433534431 |

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set of all terminating points hopefully includes the global minimum point. Our implementation of the technique is somewhat different. To come up with a set of start points, First a search procedure is implemented that finds signal sets having small error rates initiating from random start points. About thirty start points are selected. Then we run the gradient search techniques from these starting points. To run the algorithms we start with a small step size and monitor the changes in probability of error as a function of iteration number, if Pe changes very slowly we can proceed to increase the step size. On the other hand, we may need to decrease the step size if Pe starts to oscillate around a minima or goes unstable. The best constellations for a number of sources with three, four, five, seven and eight signals were found. Figure 3.6 shows the best constellations for three, five and seven signals with the probability distributions given in Table I. We compared the speed of convergence of the conjugate gradient method to that of the conventional gradient search method for a given value of step size. The normalized conjugate gradient method converges to a solution much faster than the normalized gradient method. A set of initial start points and the optimum constellation for seven equally likely signals are shown are in Figure 3.7. The conjugate gradient method is faster about an order of magnitude. To evaluate the gain in using a non-symmetric signal 49 ln(pk) (pj-skj)2> In(Pi)-JLÂ£ (Pj-s^.)2 i*J. oi-X OJ7-1 (3.37) Assume point H is the intersection of decision regions. For point H the inequality turns into an equality and the sums on either side will be the distances between H and signals sÂ¡ and Sj, therefore AH2-BH2=N0 ln( ) P2 Similarly we obtain (3.38) AH2~CH2=N0 in( ) . P3 (3.39) In addition to the two above MAP constraints there are four geometric constraints (eq.(3.1) through eq.(3.4)), and an additional constraint to minimize the average energy. The origin of the constellation is shifted to a location that minimizes the average power. Then the average power function evaluated at that point is set to 1/N. p1a+p2b+p3c=Pave (3.40) where 30 Fig. 2.2. Lenna. Fig. 2.3. Aerial map. 89 TABLE 4.3. ADDITION AND MULTIPLICATION TABLES FOR GF(4) + 0 1 A B 0 0 1 A B 1 1 0 B A A A B 0 1 B B A 1 0 X 0 1 A B 0 0 0 0 0 1 0 1 A B A 0 A B 1 B 0 B 1 A 65 and dx. dx 1=0 dx. By, 1=0, (3.87) p-o p-o. dx, dy, (3.88) Equations (3.75) and (3.78) imply that v2=u2=0. Contrary to the equations for finding SjÂ¡, the equations for finding its derivatives are linear. 3.5.4 The Iterative Algorithm To find the signal constellation we will start from an initial guess. An iterative rule (either gradient search or conjugate gradient search algorithm) is selected. An improved signal set is found using the iterative rule. Depending on the type of constraint (peak or average power constraint) the constellation is modified accordingly (eq.(15) or eq.(30)). The iteration is continued until a minima is reached. The above procedure is valid under the assumption of large channel SNR. When channel SNR is not large, obtaining an analytical expression for the gradient is not feasible because of the complexity of the shape of the decision regions. An obvious solution is an exhaustive search which is 115 (a) (b) Figure 4.12. comparison of systems employing non-binary alphabets against the binary system when the bandwidth can somewhat be expanded. The solid curve represents the binary system, (a) ternary (26,14,3) code and binary (15,11,1) code are used and a block is 390 bits long; (b) 4-ary (15,7,3) and binary (15,11,1) are used and the block length is 225 bits long. A high detail picture is used. 92 TABLE 4.6. (A) MINIMAL POLYNOMIALS FOR ELEMENTS OF GF(25) THE POWERS OF a ARE SHOWN ON THE LEFT COLUMN AND THE COEFFICIENTS OF MINIMAL POLYNOMIALS ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS .(B) GENERATOR POLYNOMIALS FOR 5-ARY CODES. COEFFICIENTS OF G(X) ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS. THE POWERS OF a ARE GIVEN IN THE MIDDLE COLUMN. GF(52) Minimal Polynomial 0 4 1 1/5 2 11 2,10 4 3 1 3,15 3 0 1 to o 14 1 6 3 1 7,11 3 2 1 8,16 111 9,21 2 0 1 12 1 1 13,17 2 4 1 14,22 4 2 1 18 2 1 19,23 3 3 1 (a) 5-ary code Powers of a g(x) (24,15,3) 0,1,2,3,4,5 1322210121 (24,12,4) 0,1,2,3,4,5,6,7 4102441132021 (b) 136 O 0.02 0.0-4- O.OS O.OS 0.1 (a) Figure 5.7. The picture SNR vs. channel error rate, the dotted curves represent the corrupted picture, (a) The solid curve represents the MMAP receiver, (b) the solid curve represents the MSNR receiver and the dashed curve represents the Sayood and Borkenhagen method [12]. 47 1 are related by: l=AHs[3. (3.30) The point G(x,y), can be found as follows: pave=Pi I U--|)2+y2] +p21 (^+-|)2+y2] +p3 [x2+(y--^)2] (3.31) 3Pave Bx =0 (Pi-Pa) 1 (3.32) dP*ve By =0 v-EihH y 2 (3.33) Substituting x and y into (30) and solving for 1 we get 1=JFPÂ¡ (3.34) where k is given by (4/3) Pi i (Pi"P2-l) 2+3p32] +P2 [ (p1-p2+l)2+3p32] +p3 [ (P!-p2)2+3 (p3-l)2] (3.35) 141 (c) Figure 5.11continued 80 information rate or increasing the bandwidth. Block and convolutional codes are the two major classes of error correction codes. Modulation is the process of mapping a base band signal into a band pass signal. To transmit the channel encoded signals one must design a signal constellation that minimizes error rate under a power constraint. This issue has been addressed in Chapter Three. Even though efficient techniques for source coding, channel coding and signal design exist, it is not known how the choice of alphabet size affects the performance of a communication system. In other words, given a nonsymmetric memory less source with a known probability distribution, it is not clear what alphabet size results in smallest error rate subject to equal average power, bandwidth and the information rate. It is the purpose of this chapter to explore the relationship between signal constellation size and a system performance measure (the error rate) for video signals under different bandwidth constraints. 4.2 Preliminaries 4.2.1 Description of the Communication Systems The block diagrams of the system under consideration are shown in Figures 4.1-4.2. Huffman optimal source coding is used to encode the source signals. 32 TABLE 2.1. A EIGHT LEVEL MSQE QUANTIZER IS USED. THE FIRST AND THE SECOND COLUMNS SHOW THE ACTUAL SOURCE STATISTICS FOR THE PICTURES. THE THIRD COLUMN SHOWS THE STATISTICS USING A LAPLACIAN DISTRIBUTION FOR SOURCE. Lenna Aerial Map Model Statistics .3324 .2800 .2549 .3574 .2795 .2549 .0946 .1416 .1510 .1060 . 1390 . 1510 .0337 .0578 .0744 .0331 .0573 . 074 .0225 .0220 .0197 .0197 .0224 .0197 TABLE 2.2. STATISTICS FOR LENNA PICTURE. A SEVEN LEVEL QUANTIZER FOUND BY PSYCHOVISUAL EXPERIMENTS IS USED. Laplace Eq. (13) Actual Statistics .5541 .5967 .6578 . 1965 .1558 .1544 .1965 . 1558 . 1418 .0259 .0270 .0193 .0259 .0270 .0200 .0005 .0019 .0026 .0005 .0019 .0038 13 1.6.2 Convolutional Codes A convolutional encoder consists of some shift registers and modulo-2 summers. For the general case, k bits at a time are entered into the shift register, and the code rate is k/n. The state of the encoder is dependent upon the contents of the shift registers. Convolutional codes can be described by a code tree. It is seen that the tree contains redundant information which can be eliminated by merging, at any level, all nodes corresponding to the same encoder state. The redrawing of the tree with merging paths has been called a trellis by Forney. The problem of decoding a convolutional code can be thought of as attempting to find a path through the trellis or the tree by making use of some decoding rule. The Viterbi algorithm [5] which is shown to be a maximum likelihood decoder for convolutional codes, involves computing a metric between the received signal and the trellis path entering each state. In the event that two paths terminating on a given state are redundant, the one having the largest metric is stored (the surviving path). This selection of survivor is performed on for all paths entering each of the other states. The decoder continues in this way to advance deeper into the trellis, making decisions by eliminating the least likely paths. The complexity of Viterbi algorithm is an exponential 146 method that makes use of (M,L) algorithm was also presented. Then, a modified MAP method for Markov sources is described. The MMAP is in fact a special case of tree decoding where no search is performed. A maximal signal-to-noise (MSNR) receiver for DPCM system is developed that minimizes the distortion power due to channel errors and the appropriate cost matrix for this receiver is computed. These methods were applied to picture transmission over noisy channels and were compared to a recent method. The SNR graphs and the subjective examination of the enhanced pictures demonstrate that the procedures are quite effective and are superior to the method given in [12]. The Markov modelling the transmitted pictures was found to improve the SNR of the received pictures. Further more, it was observed that the gain increases with the increase in the order of Markov model. For instance, in the case of MSNR receiver, the third order modelling of sources was about 2 dB superior to utilizing a first order model. The MSNR receiver was found to be the more effective than the MMAP receiver for a given order of Markov source by about 1 dB. However, the performance of the suboptimum tree decoder is very close to MSNR receiver. Considering the ease of implementation and the superiority of results, MSNR is certainly the method of choice. CHAPTER THREE SIGNAL SELECTION FOR NONSYMMETRIC SOURCES 3.1 Introduction In many applications one has a bandlimited channel and has to achieve the least error rate for a given signal-to- noise ratio. Design of high speed modems is one example where the designer is faced with the problem of selecting an efficient set of signals with in-phase and quadrature components. The objective in signal design is to find the optimum signal constellation in presence of additive white Gaussian noise under a power constraint. Two dimensional modulation formats such as MPSK and QAM have been studied before [15]. These formats confine the signal points to a certain geometry and are not optimum in the sense of minimum error rate. There has been a few attempts to solve the signal design problem under peak or average power constraint without constraining the signal points to a special geometry such as a circle or a certain lattice. Foschini et al [16] presented an iterative approach for signal selection. A gradient search procedure is given that incorporates a radial contraction technique to meet the average signal power constraint. Kernighan and Lin [17] came up with a heuristic procedure for 35 124 Q-ary alphabet {aj} for j=l,2,...,Q. The average cost (or loss) in decoding to aj is given by N di=ELiJt^(rilvi=aJt) P k-1 where Ljk is the cost in deciding that aj was transmitted where actually ak was transmitted. For each received symbol, the Bayes receiver computes dw d2,...,dQ and decides a, was transmitted if d, is the smallest of all dj. In other words, the Bayes receiver chooses =a2 if d, When the received sequence is a Markov sequence of order M the present symbol depends on M previous symbols. Thus expression (5.11) needs to be modified to reflect the redundancy in the sequence N df=E hk P^iWi =ak) p(vd=ak\*=Â£,_! *=i (5.12) 5.5 A Maximal Siqnal-to-Noise Ratio (MSNR) Decoder In this section a maximal signal-to-noise ratio (MSNR) receiver for DPCM (differential pulse code modulation) of pictures is developed. To do so the modified Bayes classifier for Markov sources found earlier will be utilized and a cost CHAPTER TWO DPCM VIDEO SIGNAL: A NONSYMMETRIC INFORMATION SOURCE 2.1 Introduction The purpose of this chapter is to demonstrate that DPCM (differential pulse code modulation) of pictures results in a nonsymmetric information source. To do so, some introductory material is presented first. Since a quantizer is an important component of a DPCM system, it will be examined in some detail. Quantizers will be introduced, different criteria for the design will be mentioned and the procedure for finding an optimal quantizer (in MSQE sense) will be explained step by step. A DPCM encoder will be analyzed and a model for the resulting source statistics will be given. Eventually the model will be compared with actual picture statistics and results will be compared. 2.2 Basics of Quantizers Quantization is the process of rounding sample values to a finite number of discrete values. This process is not an information preserving process and the reconstructed signal is 21 149 average power were compared. Two realistic situations are considered, when one is operating under tight bandwidth constraint and when the bandwidth constraint is somewhat loose. The systems employing various alphabet sizes were analyzed and computer simulations were performed using pictures with different amount of details. In the bandlimited case the ternary alphabet was seen to be the superior system. In the other case, ternary system was somewhat superior to the binary system for relatively small bandwidth expansion. For larger bandwidth expansions, the binary system outperformed all the others. 6.1.3 Decoding of Correlated Sequences It is known that the source coding process does not remove all the redundancy from the signal. An optimum tree decoding method for decoding of the correlated sequence. A practical solution to the problem using the (M,L) algorithm is discussed. A modified MAP (MMAP) receiver for Markov sequences and a maximal signal-to-noise ratio (MSNR) receiver for DPCM signals were also derived. The methods were applied to the picture transmission problem. Considerable amount of improvement (from both qualitative and quantitative point of view) was obtained. The maximum SNR receiver was found to be superior to the other methods. 95 chosen to prescribe the consecutive roots, and therefore for the primitive codes where we choose a primitive element of GF(qm) the block length is n=qm-l. 4.5.1 Encoding Nonbinarv BCH Codes A BCH code or any other cyclic code can be encoded by using the generator polynomial g(x) in the manner indicated by basic definition of a cyclic code, namely, v(x)=u(x) g(x) (4.15) That is we associate a polynomial u(x) over GF(q) of degree k-1 with the set of k information symbols to be transmitted and multiply by degree r polynomial g(x) forming the degree n-1 code polynomial v(x). However, this results in a nonsystematic code structure, and therefore it is preferred instead to form codewords using v(x) = [xru(x) mod g(x) ] + xru (x) (4.16) It is seen that this encoding operation places the k information symbols in the k highest order terms of code polynomial, while the parity check symbols are confined to the r lowest order terms. Encoding can be implemented with a division circuit using linear feedback shift register. An encoding circuit using a 29 2.6 Results In this section we will compute the statistics of the quantizers for the model developed earlier and will compare them to actual source statistics for real world pictures. Two types of quantizer were used, a MSQE quantizer and a quantizer that was found by pschovisual experiments [14]. The material used were two eight-bit pictures; a low detail picture LENNA and a high detail picture AERIAL MAP (Fig.2.2 and Fig. 2.3). Both pictures consist of 512x512 pixels. An optimized eight-level quantizer for the Laplacian distribution was chosen. Model parameters can be estimated by o2= o0=< (Â§(n)-Â§(n))2> . Table 2.1 contains theoretical and actual source statistics. Theoretical values are seen to be reasonably close to actual source statistics. Then a seven-level quantizer [14] that is shown to work well with different pictures was selected. The histograms for the two pictures were prepared. The Laplacian distribution and the distribution given in eq (2.13) were compared with the histograms (Figures 2.4 2.5). They both seem to be a fair approximation to the histograms. Table 2.2 gives the statistics given by equation eq. (2.13), Laplacian pdf and the picture. The model based statistics are close to the actual source statistics. 139 Figure 5.9continued (c) Eventually the improvement process was judged based on the subjective quality of the improved pictures. The encoded version of LENNA picture is given in Figures 5.10. The corrupted pictures, the enhanced version of them using Sayood and Borkenhagen method [11], the MMAP and MSNR receivers for channel error rates of p=.01 and p=.l are shown (Figures 5.10- 5.11). An examination of the pictures shows that the three enhancement methods are effective. Further more, it is observed that MMAP and MSNR are certainly more effective in improving than the method given in [12]. Among our two receivers MSNR seems to work better. The MSNR does a good job in removing the horizontal streaks which are very unpleasant to a viewer. The subjective examination of pictures seems to agree well with conclusions derived from the SNR graphs. 85 systems employing higher order alphabets will enjoy even greater bandwidth expansion. Figure 4.3. Transmitter for the binary system when the bandwidth can be expanded. Let the bandwidth expansion for the binary system be /3 percent. To satisfy the equal information rate constraint, binary BCH (n,k,t) code and q-ary BCH (nq,kq,tq) code must be found such that: -|=P+1 (4.8) The binary system makes no error unless t+1 or more of the n total bits in a word are in error. The probability of t+1 or more symbols error is given by E (|pe(l-Pe)(n-) i-t+i \ (4.9) where pe is the crossover probability. Usually Pw can be 98 5.correct the indicated errors. Figure 4.5. The linear feedback shift register for generating a sequence of syndrome values in Berlekamp-Massey algorithm. The Peterson's direct method for solving coefficients of error location polynomial requires a large number of multiplications and divisions and it is not appropriate for computer use either. In Berlekamp-Massey algorithm one will find the coefficients of error location polynomial by finding the tap gains in a linear feedback shift register [24]. For a given sequence of syndrome values, there are a determinable number of polynomials of various lengths that will generate the syndromes. This corresponds to the fact that there are in general a number of error patterns that can account for a given set of syndrome values. However, the lowest-weight error pattern corresponds to the given syndrome. Once the coefficients of the error location polynomial has been determined, the roots of the polynomial must be found. Chien has developed a recursive algorithm for finding 152 j v/2o(a+v/2ou) e"2 du= J 2o2 u e a a v/2o v/Jo du + J v/2 a a e a v/2o du. (A.5) With a simple variable change the first integral can be computed J u e~u* du= J --iev dv= -e 2oi _1 2 a a* 2o2 (A.6) and by definition the second integral is: / /O du=&erf(--2-, ~) 2 /2 Thus (A.7) / (x-a)2 2o2 x e _ a- dx= o2 e 22 + V2* a o erf(-, ) 2 /2o (A .8) 110 (a) (b) Figure 4.9. Systems with non-binary alphabets are compared against the binary system. Word error rate is plotted versus average power when the bandwidth is fixed. A word is considered to be one block long. The solid curve represents binary system and the dashed curve represents (a) a ternary system using (26,19,2) code; (b) a 4-ary system using (15,9,2) code; (c) a 5-ary system using (24,12,4) code for medium detail picture and (d) a 7-ary system using (48,24,8) code. The results are given for high detail picture. ACKNOWLEDGEMENTS I wish to express my gratitude to my advisor, Dr. Scott L. Miller, for his encouragement, support and friendship. I am also grateful to Dr. Couch for his guidance and insight. In addition, I would like to thank Dr. Childers and Dr. Najafi for their time and interest in serving in my committee. Special thanks also go to Dr. Sigmon who has offered his valuable assistance throughout the course of this study. iii 129 (a) (b) Figure 5.2. The average autocorrelation function for the picture (top curve) and the encoded picture (the bottom curve). (a) horizontal direction, (b) vertical direction. 62 of sv y-i N I sk~si -S>2*iPi, !s ,exp[- 2-1 *ki I' llsj-s*J2 ] [ +A] Â¡Si-S^ll2 P< where (3.73) and i=(*k-Xjt) (1^) +(y*"yJ*> ("'^) ' (3.74) a^i-^ju) + (3.75) =-n tP*-irb-iexPt-!Â£Â£5r^l It -- +--1 S* [s^-s^ll2 i70 J-l N ~E V2 LJci Pi ||Si_Sjci|| eXp[~' N0 ISi-Sjtil'.j [ 1 l-l N iSiSitili2 N (3.76) l= where 100 TABLE 4.8. THE BCH CODES, THE CODE RATES AND THE QUANTITY hb/hq FOR (A) MEDIUM DETAIL IMAGE; (B) HIGH DETAIL IMAGE AND (C) LOW DETAIL IMAGE. Code Code rate hb/ha (26,19,2) 1.36 1.38 (15,9,2) 1.66 1.66 (24,15,3) 1.6 1.71 (48,27,6) 1.77 1.78 (a) Code Code rate hb/ha (26,19,2) 1.36 1.49 (15,9,2) 1.66 1.83 (24,12,4) 2 1.94 (48,24,8) 2 2.14 (b) Code Code rate hb/ha (26,19,2) 1.36 1.3 (15,9,2) 1.66 1.51 (24,15,3) 1.6 1.54 (48,31,5) 1.54 1.58 (c) 86 approximated by Pw~ (t+1) ! n I . C*1 (n-t-1)! pe Let the word error probability for the system using the q-ary alphabet be Pwq and let =m (4.11) "g where n and nq are the block lengths for the binary and the q- ary systems respectively and m is an integer. Then the word error rate for the q-ary system can be expressed as Pq=l-(1-Pvq)a. (4.12) 4.4 Implementation Issues As it was mentioned DPCM of video signals with as few as seven quantization levels has been shown to produce pictures virtually indistinguishable from the original. The quantizer uses seven quantization levels : a zero difference and three graded sizes, each of positive and negative differences. In order to make results meaningful, we used pictures of different contents. Low, medium and high detail pictures were used in the simulations. Table 4.1 gives the probabilities associated with each quantization level for three different scenes namely a low detail scene MICHAEL, a medium detail 101 TABLE 4.9. PICTURE PROBABILITIES BEFORE AND AFTER CHANNEL ENCODING. (A) 3-ARY CODE (26,19,2) FOR THE LOW DETAIL IMAGE; (B) 4-ARY CODE (15,12,1) FOR THE LOW DETAIL IMAGE; (C) 5-ARY CODE (24,15,3) FOR THE HIGH DETAIL IMAGE AND (D) 7-ARY CODE (48,24,8) FOR THE HIGH DETAIL IMAGE. before encoding after encoding .6903 .5943 . 1747 .2173 .1350 .1884 (a) before encoding after encoding .6641 .5814 .1535 .1725 . 1392 . 1617 .0432 .0844 (b) before encoding after encoding .5108 .3942 .1684 . 1802 . 1684 .1800 .0944 .1343 .0580 . 1113 (c) 97 the observed syndrome values. Therefore, decoding is viewed as a problem of solving a set of simultaneous syndrome equations, one where the set of unknowns include error values as well as error locations. Let v(x), e(x) and r(x) represent the transmitted word, error pattern and the received word respectively where the coefficients are elements in GF(q), we have r (x) = v(x)+e(x) (4.17) The error polynomial e(x) has nonzero terms only in those positions where errors have occurred, so that if there are t errors in the received word, we can write the syndrome values as c sk=J2 YiXi' ^o'V1 mo+d-2 (4.18) -1 where XÂ¡ is the error location for the ith error and YÂ¡ is its value. Syndrome decoding of a nonbinary code proceeds as follows: 1. calculate syndrome values. 2.Determine the error location polynomial from syndrome values. 3. solve for the roots of this equation (error locations). 4.calculate error values. 119 5.2 Optimum Decoding of Markov Sources Here it will be shown that the optimal decoding of Markov sequences in the sense of minimizing the error rate requires tree searching. A practical solution to the search problem is presented and a path metric for each branch is derived. The path with maximum cumulative metric using the (M,L) heuristic search routine is show to be a suboptimum solution to the problem. Let v=(v,, v2 . ., vN) and r= (r,,r2, . ., rN) denote the transmitted and the received sequences respectively where symbols vÂ¡ and r/s are from the same Q-ary alphabet. The error probability of a decoder is given by P(E) =Â£P(Â£|Â£)P(x) (5.1) x where P(r) is independent of the decoding rule. To minimize P(E) one must minimize P(E\z) =P(V*v\z) (5.2) Minimizing is equivalent to maximizing By Bayes rule (5.3) Since the channel is memoryless 114 (a) b UB) (b) Figure 4.11. A comparison of systems employing non-binary alphabets against the binary system when the bandwidth can somewhat be expanded. The solid curve represents the binary system, (a) ternary (26,14,3) code and binary (15,11,1) code are used and a block is 390 bits long; (b) 4-ary (15,7,3) and binary (31,26,1) are used for a low detail picture and block length is 465 bits long. The results are given for a high detail picture. 74 eight-level MSQE (mean-squared quantization error) optimized quantizer was used and the output (reconstruction) signal-to- noise ratio of the two systems were evaluated over a wide range of channel SNR. The results graphed in Figure 3.11 indicate that for large values of channel signal-to-noise ratio the two systems are almost identical. However, for smaller values of signal- to-noise ratio the minimum cost design is superior to the MAP system. In this case, the improvement in the output signal-to- noise ratio is around 0.5 dB for the intermediate values of channel signal-to-noise ratio and 1 dB for low values of channel SNR. 106 (a) (b) Figure 4.6. Systems with nonbinary alphabets are compared against the binary system. Word error rate is plotted versus average power when the bandwidth is fixed. A word is considered to be one block long. The solid curve represents binary system and the dashed curve represents (a) a ternary system using (26,19,2) code for medium detail picture; (b) a 4-ary system using (15,9,2) code for low detail image; (c) a 5-ary system using (24,15,3) code for medium detail picture and (d) a 7-ary system using (48,27,6) code for medium detail picture. 143 (b) Figure 5.12--continued (c) 14 function of the code's constraint length. For large values of constraint length (K>>10) one might consider other decoding algorithms. The complexity of sequential decoders is relatively independent of constraint length, so codes with much larger constraint length can be used. Also this technique is more suitable than Viterbi algorithm for low bit error rates. Sequential decoding [5] was first introduced by Wozencraft but the most widely used algorithm to date is due to Fano. It is an efficient method for finding the most probable code word, given the received sequence, without searching the entire tree. The explored path is probably only local; that is, the procedure is sub-optimum. The search is performed in a sequential manner, always operating on a single path, but the decoder can back up and change previous decisions. Each time the decoder moves forward, a tentative decision is made. If an incorrect decision is made, subsequent extensions of the path will be wrong. The decoder will eventually be able to recognize the situation. When this happens, a substantial amount of computation is needed to recover the correct path. Backtracking and trying alternate paths continue until it finally decodes successfully. Convolutional codes using either Viterbi or sequential decoding have the ability to utilize whatever soft-decision information might be available to the decoder. It is not surprising that they have been used widely even though their 15 theory is not as mathematically as profound as that of the block codes. Most good convolutional codes have been found by computer search rather than algebraic construction. 1.7 Transmission Errors in a DPCM System Differential PCM systems are affected differently by bit errors than PCM systems because the DPCM decoder loop causes an error propagation, while a PCM error does not propagate in time. Subjectively, DPCM is more error-robust than PCM in speech coding, but less robust than PCM for image coding. Assume a channel error changes channel input u(n) to a wrong value v(n). Due to linearity of the decoder filter, the correct computation of output is superposed by an error output caused by input c(n)=u(n)-v(n) to the decoder loop. Since the decoder is an all-pole filter, there will be an infinite sequence of error samples at the output, with decaying amplitudes. In the case of first-order prediction, the effect, on a future value, at time m, is described by [3] C(m)=C(n) a (m~n); mzn (1.28) Transmission errors therefore propagate in the reconstructed DPCM waveform. This kind of error smearing is perceptually desirable in speech coding where a PCM error spike of large magnitude is more annoying than a low amplitude error smeared over a long duration. 93 TABLE 4.7. (A) MINIMAL POLYNOMIALS FOR ELEMENTS OF GF(49) THE POWERS OF a ARE SHOWN ON THE LEFT COLUMN AND THE COEFFICIENTS OF MINIMAL POLYNOMIALS ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS .(B) GENERATOR POLYNOMIALS FOR 7-ARY CODES. COEFFICIENTS OF G(X) ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS. THE POWERS OF a ARE GIVEN IN THE MIDDLE COLUMN. GF(72) Minimal Polynomial 1,7 3 11 2,14 2 5 1 3 6 6 1 4 4 0 1 5 5 3 1 6 14 1 8 4 1 9,15 6 3 1 10 4 11 11 5 4 1 12 10 1 13 3 2 1 16 5 1 (a) 7-ary code Powers of a g(x) (48,31,5) 1 through 10 235134333561123361 (48,27,6) 1 through 12 3202052111645042656531 (48,24,8) 1 through 16 3655056652662113433534431 (b) 57 a circle with radius 1 N PnX 3.4.5 Gradient Search Algorithm for the Peak Power Constraint Even though the average power constraint is used more often, in certain applications such as space communication the peak power constraint is much more realistic. Given a peak power constraint the transmitted signal points must be placed inside the circle such that the error rate is minimum. We would like to modify the iterative procedure developed earlier to accommodate the peak power constraint. All we need to do is to further modify the modified constellation at each iteration to insure that peak signal power is bounded. Let the peak power be P=l/N. The signal set is modified using the iterative rule (gradient search or conjugate gradient search), let M= max {Js (k+1) l2} (3.62) 1 siiN We will modify the signal set again in the following manner 37 modifications to the method given in [16]. Then a gradient search method for a peak power constraint is developed. Eventually a gradient search method that finds a signal constellation for an average cost function subject to a peak or average power constraint is presented. In the end, a few examples are given and conclusions are drawn. 3.2 Maximum Likelihood Signal Design for Three Signals Here a three signal constellation is designed for a 3 symbol source for transmission on a white Gaussian noise channel. The signal constellation is depicted in Figure 3.1. From geometrical considerations 1+a2+a3+P1+P2 + P3=27i, (3.1) AH cosP 1=BH cosax, (3.2) AH cosa2=CH cosP2, (3.3) BH cosP3=CH cosa3. (3.4) Let r(t) = (r,,r2) be the received waveform and p(t) be a particular value of r(t). Suppose p(t) = (p,,p2) is received in the symbol interval, pr(rj,r2|mÂ¡) is the conditional joint probability density of random variables defining r(t) and s,, s2 s3 are the signal vectors denoted by A, B and C on the 54 [vF^]" vFk [VF^.J c vF*., (3.52) in which ak is the step size and hk is the quantity in this algorithm that combines the information from current and previous steps to define a new direction. Similar to gradient search algorithm, the power may alter with k. Therefore the signal vector needs to be normalized at each step of the algorithm. Let us summarize the procedure for iterative methods: 1. Set k=0 and select the starting points. 2. Determine the search direction by calculating hk for the conjugate gradient method and VF for the gradient method. 3. Find the improved signals coordinates and normalize them. If the improvement is smaller than a tolerance level stop, otherwise set k=k+l and go to step 2. 3.4.3 Analytical Expressions for the Gradient Vector Let us find the gradient vector for the ML receiver first. VF the gradient of Pe is a vector of 2N components in signal space, in which each signal occupies two dimensions. The gradient as a vector of N two dimensional vector components is represented by 69 l.S i 0.5 8 -8.5 -1 -1.5 1.5 1 8.5 8 -0.5 -1 -1.5 -1.5 -1 -8.5 0 0.5 1 1.5 X X .5 -1 -8.5 0 8.5 1 1. (a) (b) Figure 3.6. Best signal constellations for non-symmetric sources. Probability distributions are given in Table 3.1 (a) Source l.(b) Source 2. (c) Source 3. (d) source 4. 16 In picture coding, on the other hand, error propagation is perceptually very undesirable, taking the form of very visible streaks or blotches with one and two dimensional predictors. 1.8 Optimum Prediction for Noisy Channels One of the early approaches to system optimization under noisy conditions was presented by Chang and Donaldson [6]. Because of the importance of the result and the relevance to this dissertation a summary of the methods is given here. Let rÂ¡ denote the received signal, fÂ¡ denote the impulse response of the DPCM decoder. The output of the decoder is therefore given by Â£i = ri*.fi=(si+ni) *fi =s1*fi +n1 =x +qi+ni*fi (1.29) Let us define x'Â¡ as follows var x/i=ici -xx =g_Â£ +ni fi (Xi)=E[Oi2}+2^fkEiQiNi.k} k 1 k (1.30) 46 We experimented with a variety of signal-to-noise ratios and probability sets. Figure 3.3 shows the optimum a, for three probability sets over a wide signal-to-noise ratio range. In all cases optimum a, approaches 60 degrees as signal to noise ratio goes up. Even when the optimum angle is somewhat different from 60 degrees from the performance point of view the two system are almost indistinguishable. Figure 3.3. a, (deg.) versus average power (dB). (a) the top curves, source statistics {.9,.05,.05}, (b) The middle curve, source statistics {.8,.l,.l} and (c) the bottom curve, source statistics {.6,.2,.2}. The presented results suggest that the three signals should be placed on the vertices of an equilateral triangle (Figure 3.4). The origin of the signal constellation can be shifted to a new location that minimizes the average energy without affecting the probability of error. Let us define 1 to be 1= AB = BC = AC in triangle ABC. Since ABC is an equilateral triangle, AH and 75 0.4 0.2 -0.2 - -0.4 -0.5 0.5 (a) 0.4 0.2 - 0 - -0.2 - -0.4 - -0.5 0 0.5 (b) Figure 3.8. The best signal constellations for five equally likely signals subject to average power (a) and peak power (b) constraints. 61 0 6 By & =0, =0, =0. (3.70) Eliminating X we will come up with the following set of equations Â£/r r w x v Â£ (Lik~Ljk) (x~xk) exp (- Jc-l ivo (3.71) N I c I where sÂ¡=(xÂ¡,yÂ¡) and s=(x,y)=sji. 3.5.3 The Gradient of the Cost Function To evaluate gk first the portion of cost function that depends on sk must be formed: 4^ r 1 r 1 > Ljy Py-T. ?exp [- \\sk-sjki N J-l N X-^ X r IISv-SjhII2, +EL*i Pi T-- ii exp [ ---] J-X N gk* and gk2 the components of gk are found by taking the derivative of above expression with respect to the components 70 X X X X -Z -1 0 1 2 (C) X X X X X X X -1.5 -1 -0.5 0 0.5 1 1.5 (d) Figure 3.6continued 135 Figure 5.6. The picture SNR vs. channel error rate. The dashed curve represents the MMAP receiver utilizing a first order Markov model and the solid curve represents the Sayood and Borkenhagen method [12]. Both MMAP and MSNR receivers improve the SNR of the received picture a great deal. For instance the MAP receiver improves the picture SNR by about 10 dB for a bit error rate of p=.l (Figure 5.7(a)). The maximal SNR receiver, however, results in somewhat higher value of SNR compared to the modified MAP receiver for a given order of Markov source (Figure 5.7(b)). Figure 5.8 shows that the MSNR is about 1 dB superior to the MMAP receiver. It is also observed that the MSNR receiver is at least 2 dB superior to method given in [12] (Figure 5.7(b)). 150 6.2 Directions of Future Research Here we addressed the problem of signal design for nonsymmetric source without considering the channel effects. One could reformulate the problem to incorporate the different effects due to having less than a perfect channel. One could also investigate the effects of imperfect carrier frequency and phase estimates. On the other hand, the computational complexity of the minimum cost signal design problem is tremendous. It will be useful to reduce its computational cost or come up with the simplified versions of it. To take advantage of the redundancy in signal, we used the conditional statistics of the signal. However, it may be possible to utilize the redundancy in other forms. Moreover, the use of redundancy and asymmetry in other methods of image coding is yet to be explored. 130 Figure 5.3. The DPCM prediction error. take advantage of the redundancy in the encoded signal by modeling the encoded signal as a Markov source. 5.7 Picture Transmission Over Noisy Channels 5.7.1 Implementation Issues The picture was encoded to three bit DPCM samples and was sent over the channel. The communication channel was assumed to be a binary symmetric channel. We experimented with noisy channels where the error rate ranged from p=0.01 to p=0.1. It has been known for sometime that the effect of the transmission error can be reduced by using a somewhat smaller prediction coefficient. Chang and Donaldson [6] found the optimum prediction coefficient that minimizes the effect of the channel errors. A prediction coefficient of .91 was used CHAPTER ONE REVIEW OF MODULATION TECHNIQUES, SOURCE AND CHANNEL CODING IN DIGITAL COMMUNICATIONS 1.1 Introduction The goal of this chapter is to present the background necessary to follow the work presented in this dissertation, introduce the research objectives and give a brief description of the chapters. Since two-dimensional modulation has been utilized in this work, these formats are reviewed, their spectral efficiency are calculated and the upper bound on their spectral efficiency is given. Picture transmission is one of the applications presented here. To familiarize the reader with the field, a number of source coding techniques have been described. Error correction over noisy channels is an important topic and has been utilized in this thesis. The gain from channel coding has been explained and different methods of error correction have been discussed. The effects of channel errors in a DPCM system and optimum prediction in noisy channels was presented, because the enhancement of picture quality in DPCM systems has been addressed in this dissertation. 1 Ill (ffl) (C) (d) Figure 4.9--continued I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ^j4n) ^Kermit Sigmon Associate Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1993 Winfred M. Phillips Dean, College of Engineering Dean, Graduate School word error me 113 51 3.4 Minimum Error Signal Selection 3.4.1 Preliminaries Let us denote the signals in the constellation by s,,s2, ...,sN where sÂ¡=(xÂ¡,yÂ¡) The average power is given by n (3.41) n-1 Where {p, ,p2 ,...,pN} is the probability distribution of the signal set and ||sn|| is the magnitude of sn. Following the convention of (16] we choose to set P,ve =1/N and define the signal-to-noise ratio to be SNR=10 log10 (3.42) Na where N0 is the one sided power spectral density of white noise. By definition the error rate (probability of symbol error) is 'A iv (3.43) ie=E Pn Pr (error |sn) The union bound gives the upper bound for the conditional probability of error. For a maximum likelihood receiver we have: 56 I SkSl sk-si Isjt-SjH and t,=is-sf\\2+N0ln(l-Â£) , Pi t2 = lsJt-sil2"-woln(;fi) 3.4.4 Starting Points The average signal power is equal to 1/N or Ep.kiH- n-1 iv Thus, individual signals must satisfy Solving the inequality for sn we get \ pnN n ^ pnN (3.56) (3.57) (3.58) (3.59) (3.60) (3.61) which states that starting points must be selected from inside 28 O'Neal [13] has shown experimentally that the pdf of e(n) can be approximated with a Laplacian distribution p(x) =^-exp(- ^1*1) (2.11) v/2 o To find the statistics of the levels an optimum quantizer must be placed in the DPCM system. If we utilize a MSQE quantizer, the statistics associated with each level will be Po . P P, = f -i- exp () dx= exp (-/2y) dy 0 0^2 a (2.12) = -i (exp(-y2a) -exp(-./2P) , where a and /? are tabulated in [11]. We have also verified through simulations that the choice of a gamma distribution for e(n) and uniform pdf for Q results in a satisfactory approximation to the density of e(n) In other words P(*> = / | 8no |x| (2.13) A numerical method must be employed to find the statistics associated with each level. 91 TABLE 4.5. (A) MINIMAL POLYNOMIALS FOR ELEMENTS OF GF(16). THE POWERS OF a ARE SHOWN ON THE LEFT COLUMN AND THE COEFFICIENTS OF MINIMAL POLYNOMIALS ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS .(B) GENERATOR POLYNOMIALS FOR 4-ARY CODES COEFFICIENTS OF G(X) ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS. THE POWERS OF a ARE GIVEN IN THE MIDDLE COLUMN. GF(42) Minimal Polynomial 0 1 1 1,4 All 2,8 B 1 1 3,12 1 B 1 5 A 1 6,9 1 A 1 7,13 A A 1 10 B 1 11,14 B B 1 (a) 4-ary Code Powers of a g(x) (15,12,1) o H A B 0 1 (15,9,2) 1,2,3,4 1 A A 1 1 B 1 (15,7,3) 0,1,2,3,4,5 A0B1BB11 (b) 33 Figure 2.2. The histogram for LENNA picture is compared with (a) Laplacian Distribution (b) Distribution given in eq. (2.13). Histogram is shown with a broken line. 10 / a given total number of bits for Gaussian variables, the optimum assignment is done by making the average quantization error of each coefficient the same. This requires that the bits be assigned to the coefficients in proportion to the logarithm of their variance. 1.6 Channel Coding Channel coding is a method of inserting structured redundancy into the source data so that transmission errors can be identified and corrected. Block coding and convolutional coding are two important subcategories of channel coding techniques. 1.6.1 Block Codes With block coding the source data is first segmented into blocks of k bits; each block can represent any of M=2k distinct messages. The encoder transforms each message into a larger block of n digits. This set of 2k coded messages is called a code block. The (n-k) digits, which the encoder adds to each message block, are called redundant digits; they carry no new information. The ratio of data bits to total bits within a block, k/n, is called code rate. The code itself is referred to as an (n,k) code. To demonstrate the performance improvement possible with 96 shift register with r=n-k stages is shown in Figure 4.4. Each box in the circuit is a q-ary storage device. The multiplications and additions are done in GF(q), the symbol field of the code. The operation of the circuit is as follows: 1. Shift the k information symbols into the encoder and simultaneously into the channel. As soon as the k information symbols have entered the shift register, the r=n-k symbols in the register are the check symbols. 2. Disable the feedback circuit. 3. Shift the contents of the register out and into the channel. Figure 4.4. The encoder for q-ary BCH codes. 4.5.2 Decoding BCH Codes Given the set of syndrome values for the received word, the decoding task is to find the most likely error pattern, within the error correction limit of the code, that produces 38 constellation. A maximum likelihood receiver sets the message estimate to mk if for i=l,2,3, pr{r=p1, r2=p2\ mk) > pz{r1=pi, x2=p2\ i*k. (3.5) Assuming the noise components are Gaussian and statistically independent, Pr(Pi-'Pal jn)=lFexp{--^ [(p1-siJ)2+(p2-si2)2]}. (3,6) The decision rule then becomes t(Pi-sw)2+(p2-s^)2] > -~[(p1-s)2+(p2-si)2] i*k. JV0 iVo (3.7) Notice that the sum of squared terms on either side of the c Figure 3.1. A three signal constellation inequality are the square of the Euclidean distances between the received signal and signals sÂ¡ and sk. Thus the decision rule can be rewritten as: 133 5.7.2 Results DPCM decoders with first, second and third order Markov source models were implemented and compared based upon the SNR of the reconstructed signal and subjective impression. Considerable improvements in the SNR of the received signal and subjective impression is obtained. Figure 5.5(a) and 5.5(b) give a comparison of the various orders of Markov source. It is seen that the improvements increase with the increase in the order of Markov source. The second order modelling of the source is clearly superior to first order modelling. However, third order modelling of the source is only slightly better than that obtained with the second order modelling. The fourth and higher order modelling of the source were not implemented because of two reasons. Almost all of the redundancy is in the adjacent symbols and the overhead cost becomes tremendous (the overhead associated with the fourth dimensional statistics is about one hundred fifty percent). The MMAP was compared to the Viterbi decoding method given in [12]. The MMAP with a first order Markov source is comparable in performance to Sayood and Borkenhagen's receiver [12] (Figure 5.6), but it has the advantage of running faster and requiring no memory. 127 e= (^jc-yj)2 pk\i Pi- k-0 1-0 (5.19) The SNR can be maximized by minimizing en2 One can think of en2 as an average cost function where Lki=(yk-yi)2- (5.20) Therefore SNR is maximized by minimizing the cost function and setting the cost according to (5.20). 5.6 Redundancy in the Encoded Signals It has been shown that assuming an AR model for the signals the output of the encoder contains some redundancy [12]. Here we will examine the encoded signal and its autocorrelation function to get a feel for the extent of residual redundancy. Standard 512x512 pixel eight-bit pictures such as LENNA were encoded using a first order DPCM encoder. The average normalized autocorrelation in horizontal and vertical direction were found and plotted (Figure 5.2). This shows that certain amount of correlation among the adjacent encoded symbols exist. The DPCM prediction error signal is shown in 23 2.3 Approaches to Quantizer Design The quantizers can be designed based on a mean squared error criterion. This results in overspecification of the low detailed areas of picture and consequently a small amount of granular noise but relatively poor reproduction of edges. It has been realized for sometime that for a better picture quality, quantizers should be designed on the basis of psychovisual criteria [7]. One method of designing psychovisual quantizers is to minimize a weighted mean squared quantization error, where the weights are derived from subjective experiments [8]. Such optimization would be similar to mean squared error criteria, where the density function is replaced by a weighing function. 2.4 MSOE Quantizer Design An optimal quantizer is defined to be a quantizer with the smallest mean squared quantization error. It is desired to find the quantizer that minimizes the mean-squared quantization error for a given probability density function and number of levels. The mean-squared quantization error is given by 43 For a given signal to noise ratio the probability of word error depends only on a, and a2 because a,, a2 and a3 add up to n. To find the values of a, a2 a3 that minimize the probability of the word error, we need to differentiate Pw with respect to a, and a2 and set them equal to zero: U-0 (u(l + tan2^) e -u'tan'a -u(l+tan2(ax+a2)) e -uJtanJ(a1+o2)\ (3.23) 158 [36] Melsa, J. L. and Cohn, D., Decision and Estimation Theory. Me Graw-Hill Book Company, New York 1978. [37] Modestino, J. W., and Daut, D. G., "Combined Source- Channel Coding of Images," IEEE Trans. Commun.. Vol. COM- 27, pp. 1644-1659, 1979. 125 matrix (or its elements Ljk) will be derived. Let the alphabet (Sj / 3-2 ' represent output levels (ylfy2f ,y in a DPCM system. Suppose the receiver makes an error and aj is chosen where ak was actually sent. The reconstruction error amounts to yj-yk. The error affects the following samples. The contribution of error to following symbols will be a(yj-yk), a2(Yj-yk) a3(yj-yk) and so on and so forth, where a is the prediction coefficient in the DPCM system [3]. The total distortion power due to a channel error is therefore given by N= (yj-yk)2+a2 (yj-yk)2+a4 (yj-yk)2 +. . (5.13) = Wj~yk)2 [l+a2+a4+. .] =(yj-yk)2( 1 d. If we set Lj k=(yk-y^2 (5.i4) the modified Bayes receiver will minimize the average cost which is distortion power due to channel errors in this case. Thus, the maximal SNR receiver calculates d^dj, ..., dQ (the average loss for each symbol) and chooses the symbol with the least average loss where average loss is 142 (d) Figure 5.11continued (a) Figure 5.12. The effectiveness of the picture enhancement process in a very noisy situation (p=.l) is demonstrated, (a) Corrupted picture, (b) enhanced picture using method in [12], and (c) enhanced picture using MSNR receiver. 40 region IÂ¡ (Figure 3.2), pw=1-'EPiffPr(Pi'P2Â¡ mi)dpdp2, i-l J J (3.15) Figure 3.2. One decision region. By substituting the expressions for probability density function eq. (3.6) into eq. (3.15) we get ^//exp[~ (Pl^J>2~^idpdp2 (3-16> 1 1 PlP2 or equivalently 2 1.2 Two-Dimensional Modulation Formats M-ary phase-shift keying (MPSK) and quadrature amplitude modulation (QAM) are among the most popular two dimensional formats. In MPSK the transmitted signal is given by s(t) =Re{g( t) eJ"ct} (1.1) where g(t) =A e:70(c), (1.2) In other words, in MPSK while the amplitude is maintained constant the phase of signal can take on one of the M values in a symbol interval. The MPSK for M=4 is called quadrature phase-shift keying (QPSK). In MPSK signal points are confined to the circumference of a circle. But in QAM the transmitted signal is s(t) =Re{g( t) eJct (1.3) where gr( t) =x( t) +jy( t) , (1.4) or TABLE OF CONTENTS ACKNOWLEDGEMENTS iii ABSTRACT . vi CHAPTER ONE REVIEW OF MODULATION TECHNIQUES, SOURCE AND CHANNEL CODING 1 1.1 Introduction 1 1.2 Two-Dimensional Modulation Formats . . .. 2 1.3 Source Coding 5 1.4 DPCM 5 1.5 Transform Coding 8 1.6 Channel Coding 10 1.7 Transmission Errors in a DPCM system .... 15 1.8 Optimum Prediction for Noisy Channels .... 16 1.9 Research Objectives 18 1.10 Description of Chapters 20 CHAPTER TWO DPCM VIDEO SIGNAL: A NONSYMMETRIC INFORMATION SOURCE 21 2.1 Introduction 21 2.2 Basics of Quantizers 21 2.3 Approaches to Quantizer Design 23 2.4 MSQE Quantizer Design 23 2.5 Analysis of DPCM Encoder 26 2.6 Results 29 2.7 Discussion 31 CHAPTER THREE SIGNAL DESIGN FOR NONSYMMETRIC SOURCES 35 3.1 Introduction 35 3.2 Maximum Likelihood Signal Design for Three Signals 37 3.3 A Numerical Approach Based on Lagrange Multipliers Method 48 3.4 Minimum Error Signal Selection 51 3.5 Minimum Average Cost Signal Selection .... 58 3.6 Results 66 iv 140 Figure 5.10. The DPCM encoded version of LENNA in the absence of channel errors. Figure 5.11. The effectiveness of the picture enhancement process in a relatively noisy situation (p=.01) is demonstrated, (a) Corrupted picture, (b) enhanced picture using method in [12], (c) enhanced picture using MMAP receiver and (d) enhanced picture using MSNR receiver. 53 (3.47) where Si (k) s2{k) (3.48) sN(k) denotes the signal vector at the kth step of the algorithm, ak is the step size and VF is the gradient of Pc. Since the signal power may change with k, the signal vector is normalized at each step of the algorithm v/i7(p1jsi (k+1) 12+p2\s2 (k+1) J2+. . +pN\s(k+l) |2) (3.49) To speed up the convergence, instead of a conventional gradient search algorithm, the Fletcher and Reeves conjugate gradient method [18] can be utilized. In this method the information about the second derivative is used indirectly. The algorithm is described by (3.50) where hfr vFfr+tt j^Â£_i (3.51) and 48 Figure 3.4. The three signals form a equilateral triangle. 3.3 A Numerical Approach Based on Lagrange Multipliers Method In this approach the problem of signal selection is viewed as a constrained optimization problem. The constraints are incorporated into the optimization problem by the use of Lagrange multipliers method. Let us outline the design of a MAP receiver with three signals with unequal probabilities (Figure 3.1). A MAP receiver decision rule is m=mk if, for i=l,2,3, Pi (ri=Pi' r2=P21 mk)pk> Pr(ri=Pi,r2=p2| mi)pi i*k. (3.36) Upon substituting (6) into (43), we get 71 e.s - -1.5 -1 -8.5 8 8.5 1.5 (a) (b) Figure 3.7. (a) Seven signal constellation for an equally likely source. 0's represent the start points and x's represent the final constellation, (b) The solid and dashed curves show the error rate as a function of number of iterations for gradient search and conjugate gradient method respectively. To my parents for their love, patience and support. 109 o (dB> (a) e (dB) (b) Figure 4.8. Systems with non-binary alphabets are compared for the fixed bandwidth case. The first system is represented with a solid curve and the second system is represented with a dashed curve (a) 7-ary and ternary systems and (b) 7-ary and 4-ary systems respectively. 87 NARROWS scene and a high detail scene BANKSIAS [21]. It is observed that: p(y0) > p(y1)*p(y-1)> P(y2) *P(y.2) > p(y3)p(y.3) (4*13) where p(yÂ¡) denotes the probability of ith difference level. The primitive irreducible polynomials over the nonbinary fields are given in Table 4.2. The minimal polynomials and the generator polynomials of the nonbinary codes are given in Tables 4.4, 4.5, 4.6 and 4.7. Notice that GF(4) the ground field for GF(42) in itself is an extended version of GF(2) . The elements of GF(22) are 0,1, A and B where B=A2. The arithmetic tables for GF(4) are provided in Table 4.3. Alphabets of size 6 are not used in the simulations because 6 is neither a prime or power of a prime number and does not lead to BCH code implementation. 27 P5=Pe* Pq- (2-9) And the probability of quantum level i is determined by Pi= f p5(x) dx. (2.10) e (n) x(n) p (b) Figure 2.1. Block diagram of a DPCM system, (a) encoder and decoder. 90 TABLE 4.4. (A) MINIMAL POLYNOMIALS FOR ELEMENTS OF GF(27). THE POWERS OF a ARE SHOWN ON THE LEFT COLUMN AND THE COEFFICIENTS OF MINIMAL POLYNOMIALS ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS .(B) GENERATOR POLYNOMIALS FOR 3- ARY CODES. COEFFICIENTS OF G(X) ARE SHOWN IN ASCENDING ORDER ON THE RIGHT COLUMNS. THE POWERS OF a ARE GIVEN IN THE MIDDLE COLUMN. GF(33) Minimal Polynomial 1/3,9 12 0 1 2,6,18 2 111 4,10,12 2 0 11 13 1 1 14,16,22 2 2 0 1 5,15,19 112 1 17,23,25 10 2 1 (a) 3-ary code Powers of a g(x) (26,19,2) 13,14,15,16 20111201 (26,14,3) 14,15,16,17,18,19 1122002000021 (b) 137 Figure 5.8. The picture SNR vs. channel error rate. The solid curve represents the MSNR receiver and the dotted curve represents the MMAP receive. The minimum error decoding procedure with a limited search is expected to outperform the MMAP receiver. Table 5.2 compares the performance of a MMAP receiver with a minimum error receiver that utilizes a (M=750, L=7) search. For smaller error rate the improvements are around 0.5 dB but for very noisy channels the search does not seem to be beneficial. TABLE 5.2. THE PICTURE SNR SUBJECT TO VARIOUS CHANNEL ERROR RATES FOR MMAP AND MIN ERROR RECEIVER. A (M=750,L=7) SEARCH WAS PERFORMED FOR THE MINIMUM ERROR RECEIVER. p .01 o to .03 .05 t-' o .1 MMAP 22.23 20.22 19.44 18.41 17.52 16.69 Min. error 22.82 20.98 20.1 18.7 17.68 16.72 107 (c) Figure 4.6--continued. (d) 144 5.8 Side Information The transmitter must compute and transmit higher dimensional statistics of the picture together with the picture. The amount of side information depends upon system parameters. Consider a DPCM system with the following parameters: Number of pixels =NxN n= length of DPCM words q= number of quantizer levels m= digits of accuracy in statistics p= order of Markov model. The increase in bit rate is equal to R=gp log2 (10,n-l) (5.22) The percentage of overhead is therefore given by % overhead= gplog2 (10m-l) 3.32 m qp NxNxn m2 n (5.23) The effect of noisy statistics on the performance of the system was studied. It turns out that the system performance deteriorates rapidly as the channel noise increases. Therefore, one must protect the conditional statistics against channel noise. To reach the performance that is achievable by utilizing noise-free statistics, the following coding strategy 78 chonntl SNR Figure 3.11. Comparison between minimum cost signal selection (the solid curve) and minimum error signal design(the dashed curve) for a first order Gauss-Markov source with correlation coefficient of 0.9. Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Emami, Shahriar TITLE: Efficient communications for nonsymmetric information sources with application to picture transmission (record number: 1933237) PUBLICATION DATE: 1993 i, Sho^hfi AY" } as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation -oftrdigital rehve-eopy: fgltKZffion allows the University of Florida to generate image- and text-based versions as appflOpiieteJaffid to provide and enhance access using search software. This grant of permissions prohibits use-of the digitized versions fbr commercial use or profit. Signature 1 rtnV & Printed or Typed Name of Copyright'Holder/Incensee Personal information blurred Date of Signature Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Preservation Department University of Florida Libraries P.O. Box 117007 Gainesville, FL 32611-7007 5/28/2008 3 s(t)=x(t) cos (o>ct) -y( t) sin(oct) . (1.5) The waveforms x(t) and y(t) are x( t) =]Txi h(t-iT) (1.6) i and y(t) =52y h(t-iT), (1.7) i where T is the symbol interval in seconds and h(t) is the pulse shape. Let us find the spectral efficiency of MPSK and QAM with rectangular pulses. The null-to-null transmission bandwidth of MPSK and QAM is (1.8) where M=2 is the number of points in the signal constellation. The spectral efficiency is therefore given by R 1 bi ts/sec ^ Bt 2 Hz (1.9) When operating over a bandlimited channel and the overall pulse shape satisfies the raised cosine rolloff filter characteristics, the bandwidth of the modulating signal is 145 was found empirically. Each element of statistics is coded into a log2(10m-l) bit natural binary code. The first (log2(10m-l))/2 bits are coded using a (5,1) repeat code and the rest are left uncoded. Thus, the percentage of side information is determined by % overhead= A0. (5.24) N2 n The amount of overhead for a 512x512 pixel picture, a seven level quantizer and four decimal digits of accuracy is given in Table 5.3. Even with a third order Markov model the amount of side information does not exceed twenty one percent of the total picture information. TABLE 5.3. PERCENTAGE OVERHEAD AS A FUNCTION OF ORDER OF MARKOV SOURCE MODEL USED IN THE SYSTEM. order of Markov source percentage overhead first 1 % second 3 % third 21 % 5.9 Summary An optimum method for decoding correlated sequences with the aid of Markov source modeling was derived and it was shown to require tree searching. A suboptimum implementation of the 112 (a) Figure 4.10. Systems with non-binary alphabets are compared against the binary system. Word error rate is plotted versus average power when the bandwidth is fixed. A word is considered to be one block long. The solid curve represents binary system and the dashed curve represents (a) a ternary system using (26,19,2) code; (b) a 4-ary system using (15,9,2) code; (c) a 5-ary system using (24,15,3) code and (d) a 7-ary system using (48,31,5) code. A low detail picture is used. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Scott L. Miller, Chair Associate Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of' scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Â¡A) ^AiusL 1 Leon Couch Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. <&>i J Donald G. Childers Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosoj 'Fazil Najafi v Associate Professor of Civil Engineering 83 4.3 Analysis For a fair comparison of systems employing different alphabet sizes, average power, bandwidth and information rate are held equal for all systems. In practice the bandwidth constraint could be either very strict or can be somewhat relaxed. Therefore two cases are considered: I. When the bandwidth constraint is strict and can not be relaxed. The binary system against which all other systems will be compared uses source coding, BPSK modulation and no channel coding due to strict bandwidth limitations. For the same information rate higher alphabet systems send longer pulses and require less bandwidth. Therefore these systems can use the extra bandwidth for error-correction codes (Figure 4.1) . Here nonbinary BCH codes are used for error correction. For a nonbinary system of alphabet size q a q-ary (n,kq,tq) code must be found to satisfy the equal information rate: n Ab (4.5) where hb and hq are the average message length for binary and q-ary systems respectively. 60 with Sj. p (sj | Sj) can be approximated with a Q function P(S |s.)=C>(lÂ£LJ!zi!) (3.67) substituting back into the average cost expression we get C=A 2 N n 'E'Ehi p _.1 exp[-BSi SjiÂ¡2] (3-68) Ji 1 ISv-S,J 3.5.2 The Nearest Points on the Boundaries The first step in computing the average cost function is to find the set of nearest point to each and every signal point in the constellation on the boundaries. There are N signal points in the constellation and for each one there are (N-l) of nearest points on (N-l) boundaries, therefore a total of N(N-l) of nearest points must be found. The problem of finding SjÂ¡ which is the nearest point to sÂ¡ on the boundary between sÂ¡ and Sj can be formulated like a Lagrange multiplier problem. We would like to minimize ||s-sÂ¡|| subject to a constraint (eq. (3.65)). Let us form the auxiliary function and differentiate it with respect to x,y and X 156 [12] Sayood, K., and Borkenhagen, J. C., "Use of Residual Redundancy In The Design of Joint Source/Channel Coders," IEEE Trans.Commun.. Vol. 39, No. 6, pp.838- 846, 1991. [13] O'Neal, J. B., "Predictive Quantizing Systems (Differential Pulse code Modulation) for the Transmission of Television Signals," The Bell Systems Technical Journal. Vol. XLV, pp.689-721, 1966. [14] Limb, J. 0., Vision Oriented Coding of Visual Signals, Ph.D. dissertation. University of Western Australia, 1966. [15] Arthurs, E., and Dym, H., "On Optimum Detection of Digital Signals in Presence of White Gaussian Noise-A geometric interpretation and a study of three basic data transmission systems," IRE Trans, on Communications Systems. Vol. CS-10, No.4, pp.336-372, 1962. [16] Foschini, G.J., Gitlin, R.D., and Weinstein, S.B., "Optimization of Two -Dimensional Signal Constellation in presence of Gaussian Noise," IEEE Trans. Commun.. Vol. COM-22, pp.28-38, 1974. [17] Kernighan, B.W., and Lin, S., "Heuristic Solution of a Signal Design Optimization Problem," The Bell Systems Technical Journal. Vol. 52, pp. 1145-1159, 1973. [18] Fletcher, R., and Reeves, C.M., "Function Minimization by Conjugate Gradients," Computer Journal. Vol. 7, No. 2, pp.149-154, 1964. [19] Couch, II, L.W.. Digital and Analog Communications Systems. Macmillan Publishing Co.,Inc., New York, 1990. [20] Press, W. H., Teukolsky, S. A., and Vetterling, W. T., Numerical Recipes. Cambridge University Press, Cambridge, UK, 1986. [21] Hullett, J. L., "Impairment by Transmission errors of Differentially Quantized Television Signals," Proceedings of the Symposium on Picture Bandwidth Compression. Cambridge, MA, 1969. [22] Thomas, J. B., An Introduction to Statistical Communication Theory. John Wiley & Sons, Inc. New York, 1969 . [23] Berlekamp, E. R., Algebraic Coding Theory. McGraw-Hill Book Company,Inc., New York, 1968. 105 assumptions of equal information rate, bandwidth and average power. Pictures of different contents were used and two cases were considered, when the bandwidth is strictly fixed and when the bandwidth can be somewhat expanded. The results suggest that the nonbinary alphabets are more appropriate for picture tranmission over noisy channels when the bandwidth constraint is tight. This has to do with the fact that under equal information rate the nonbinary systems utilize error correction codes and the binary system does not. Among the nonbinary systems, the system with ternary alphabet performed best. On can explain this trend in the following manner. The ternary system has the least code rate among the nonbinary sytems. A little bit of coding is useful; but it will not be beneficial any longer, if you overdo it. In case two, for a modest bandwidth expansion the ternary system is superior to the binary system above a threshold. However, for larger bandwidth expansion factors the binary system seems to be more efficient than the systems utilizing nonbinary alphabets for image transmission. 19 two dimensions for nonsymmetric sources. It is desired to find an algorithmic solution to the minimum error signal constellation for average and peak power constraints. In addition the general case where the cost function is not necessarily the error rate is discussed. Even though efficient techniques for source coding, channel coding and signal design exists, it is not known how the choice of alphabet size affects a communication system. We would like to compare communication systems with various alphabet sizes for the transmission of video signals on the basis of equal information rate, bandwidth and average power. Two realistic situations will be considered, when one is operating under tight bandwidth constraint and when the constraint is somewhat loose. System performance can be improved using standard error correction techniques at the cost of increasing the bandwidth or reducing the information rate. However, we would like to use inherent asymmetry and redundancy in the transmitted picture to improve the reception. We will model the data as a Markov source and derive the optimum method for decoding the data. We will also find a receiver that instead of minimizing the error rate maximizes the SNR (signal-to-noise) ratio. xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008221000001datestamp 2009-01-28setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Efficient communications for nonsymmetric information sources with application to picture transmission dc:creator Emami, Shahriardc:publisher Shahriar Emamidc:date 1993dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082210&v=0000130784206 (oclc)001933237 (alephbibnum)dc:source University of Florida 128 Figure 5.3. This also indicates that some correlation is present in the encoded picture. Although significant amount of redundancy has been removed from the picture some structure is still present in the encoded signal. The decoding methods given in this chapter 59 3.5.1 Average Cost Function The Bayes receiver which minimizes the average cost is given by ^=EEp(sil si)Pihi' (3.64) j"i j-i where LjÂ¡ is cost involved when the receiver picks Sj when sÂ¡ was actually sent. The boundaries of the decision regions are no longer straight lines; instead they are two dimensional curves. For instance, the boundary between sÂ¡ and s is given by Â£L* ^(^|sjc)PrE% (rlsjcJp*, (3.65) k-l k-1 where conditional probability of error is P(rjsk) = exp(--S ) (3.66) nN0 N0 Channel noise could cause the received signal to move to an adjacent decision region and result in an error. For large values of signal-to-noise ratio; however, errors are almost entirely the result of displacement of a received signal to an adjacent decision region from the nearest point on a boundary to the transmitted signal. Let sÂ¡ be the transmitted signal and SjÂ¡ be the point with shortest distance on the boundary 9 international standard for transform coding systems. In addition to that it minimizes the blocking artifact that, results when the boundaries between the subimages become visible. 1.5.2 Size of Subpictures Computer simulations on real pictures show that the mean square error produced by transform coding improves with the size of subpicture. However, the improvement is not significant as subpicture is increased beyond blocksize of 16x16. Subjective quality of pictures, however, does not appear to improve with the size of block beyond 4x4 pixels [2]. 1.5.3 Bit Allocation One method of choosing the coefficients for transmission, is to evaluate the coefficient variances on a set of average picture, and then discard all the coefficients whose variance is lower than a certain value. Such a scheme is called zonal filtering [3]. Having decided which coefficients to transmit, we must then design a quantizer for each of them. This could be done by dividing a given total number of bits among all the coefficients. In order to minimize the mean square error for CHAPTER FOUR ALPHABET SIZE SELECTION FOR VIDEO SIGNAL CODING 79 4.1 Introduction 79 4.2 Preliminaries 80 4.3 Analysis 83 4.4 Implementation issues 86 4.5 Nonbinary BCH Codes 94 4.6 Results 99 4.7 Summary 104 CHAPTER FIVE EFFICIENT DECODING OF CORRELATED SEQUENCES 116 5.1 Introduction 116 5.2 Optimum Decoding of Markov Sequences . 119 5.3 A Modified MAP (MMAP) Receiver ...... 122 5.4 A Minimum Cost Decoder 12 3 5.5 A Maximum Signal-To-Noise Ratio (MSNR) Receiver 124 5.6 Redundancy in the Encoded Signals 127 5.7 Picture Transmission over noisy Channels 130 5.8 Side Information 144 5.9 Summary 145 CHAPTER SIX CONCLUSIONS AND SUMMARY 147 6.1 Summary of the Work 147 6.2 Directions of Future Research 150 APPENDIX A. EVALUATION OF AN INTEGRAL 151 APPENDIX B. EVALUATION OF THE DERIVATIVES ..... 153 REFERENCES 155 BIOGRAPHICAL SKETCH 159 v 24 x2 Â£-1 *H eq2 = f (x-l1)2p{x) dx+Y, f (x-li)2p(x) dx 1~Z Xi ** f 2 1 ^ +f (x-lL)2p(x)dx. *L Our purpose is to choose quantum levels 1Â¡ and interval boundaries xÂ¡ so that eq.(2.1) is minimized. The above expression can be differentiated to obtain a set of necessary conditions that must hold for the optimum quantizer. By applying Leibniz's rule-we get (2.2) 92, = -2 f (x~li)2p(x) dx=0, i=l, 2 . L, de dx 2- = t (x-li.1)2-(xi-li)2)p(x) dx=0, i=2, L. (2.3) where x,=-o and xL+1=>. Equation (2.3) is equivalent to i=2,3 L, (2.4) which says that interval boundaries should fall midway between the adjacent quantum levels. Alternatively, li=2xi-li.i, i=2,3 . L. (2.5) CHAPTER SIX CONCLUSIONS AND DISCUSSIONS 6.1 Summary of the Work Some important issues related to nonsymmetric information sources such as signal design, alphabet size selection, and decoding of information were investigated. New methods for signal design including a procedure that minimizes a cost function subject to different power criteria were developed. The role of alphabet size in a communication system for a asymmetric source was studied. It was found that the performance is sensitive to size of the alphabet. Subject to a fixed bandwidth, information rate and average power constraint, the ternary system was found to superior to all the others. Optimum tree decoding of correlated sequences was addressed and modified MAP (MMAP) receiver and a maximal SNR (MSNR) receiver were derived for decoding of correlated sources. Experimental results showed the effectiveness of the method. Further more, they were found to be superior to a recently reported technique. 147 138 To get a feel for the improvement process, a horizontal line of the video signal, the corrupted version of it and the enhanced version of the line using MMAP receiver are shown in Figure 5.9. The enhanced signal looks almost like the reference signal indicating that the decoding process is quite effective. (b) Figure 5.9. (a) A horizontal line of video signal, (b) corrupted version of the video line and (c) the enhanced version of the line. 42 utana. Pv=1^ J exp [-(u- -^-)2] J exp[-e2] u-0 N o --utanPj de du ucanttj f exp [- (u- ^-)2] f exp [-e2] de du 71 O N ..Lp, utana, f exp [-{u- }2] f exp[-e!] V-o N w de du. Now by definition *2 erf(x1,x2) = (exp (-t2) dt. Therefore, the following is the equation for Pw: (3.20) (3.21) Pw=l--^- f exp [- (u- -^-)2]erf(-u tanpi#u tana2) du 2v^uJ>0 N 2/rt ji /2LZJ2 exp [- (u-, ^)2] erf(-u tanp,, u tana.) du \ jW_ i-n f exp [-(u- 2\/u-L0 N Atf2 )2]eri(-u tanP2,u tana,) du. (3.22) BIOGRAPHICAL SKETCH Shahriar Emami received his BSEE with honors from the University of Florida in August 1986. He then entered Graduate School at the University of Florida and received his MS in December 1987. Meanwhile he worked as a graduate teaching assistant conducting electrical engineering laboratories and grading for graduate and undergraduate courses. Afterwards, Shahriar entered the Ph.D. program and focused primarily on digital communications. He has been working as a research assistant for Dr. Scott L. Miller at the Telecommunication Research Lab and is expected to receive his Ph.D. in August 1993. 159 45 Ju e~iu~k)2e~uil du= e k2l (i+D k* e ^y/2Â¥ k 2(1+1) 2 (1+J)3 erf ( v/Tl+IT ) (3.27) If we substitute for integrals from eq. (3.27) into eq. (3.23) and eq. (3.24) we will end up with the following set of nonlinear equations: (Pi~p3) e AH2 N 71 AH2 2\| Nt (Pi+P2) e ah2 Icosa.lerfi- |cosal|,>) 1 11 N No -(P2+P3) e Na AH2, |cos (a1+a2)|erf (-. (cos (a1+a2)|/~) N No / (3.28) -sin2a2 AH2 (p1-p2)e N + -| (Pi+p3) e N '|cosa2|erf(- ^^|cosa2|,~) ^ \ V N AH2, -Ajf. sin2 AH2 N ^ (cos (a1+a2)|/) =0 (3.29) Given a signal-to-noise ratio and a probability set the optimum angles can be found using eq. (3.28) and eq. (3.29). 22 only as good as the quantized samples allow. In other words, there remains some error, the quantization error between the original and the reconstructed waveform which is related to the parameters of the quantizer. Let the analog signal be modeled as a random waveform and let p(x) be the probability density function of the signal. The process of quantization subdivides the range of the values of x into a number of discrete intervals. If a particular sample value of the analog signal falls anywhere in a given interval, it is assigned a single discrete value corresponding to that interval. The intervals fall between boundaries denoted by x,,x2, . .,xL+,, where there are L intervals. The quantized values are denoted by 1,,12,...,1L and are called quantum levels or representative levels. The width of an interval is xi+1 -xÂ¡ and is called interval's step size. If all the steps are equal and, in addition, the quantum level separations are all the same, the quantizer is said to be uniform; otherwise it is a nonuniform quantizer. It is possible to design a quantizer for a given probability density function and a given number of levels. The optimal quantizer is non-uniform unless the signal has a uniform pdf. If a uniform quantizer is used instead the mean squared quantization error will be larger than that of the optimal nonuniform quantizer. 131 according to formula given in [6]. The MMAP receiver and the MSNR receiver with first, second and third order Markov models were utilized. For the first order Markov modelling, the decoder uses the previous symbol along the same line. Figure 5.4 shows the neighboring symbols used in second and third order modelling of the source. The codeword assignment is given in Table 5.1. Other codeword assignments such as natural and gray level coding assignment were also tried for purpose of comparison, but the given assignment consistently resulted in the largest SNR for the reconstructed signal. The conditional probability of error is given by p(ri|vi)=p(l-p) where a is the Hamming distance between rÂ¡ and vÂ¡ and n is the number of bits in a DPCM word. We used a seven-level quantizer given in [14] that was found experimentally and is seen to give satisfactory results for variety of pictures. We define signal-to-noise ratio (SNR) as the ratio between the power in the original picture to the power in the difference between the original and the received picture. 120 P(z\y) ^PUJvj) (5.4) For an Mth order Markov sequence P(lf) =JIP(vi\vi-i'vi-2 v-m^ (5.5) i Substituting (5.4) and (5.5) into (5.3) we get p(y|x) = nPir^mvJvw Til) / Vi-M> Taking the logarithm of both sides gives (5.6) logP(:z|z) =52 log (PirJvj) P(vi|vi.1, . ., vi.M)) -log P(n) . i (5.7) One might be tempted to use Viterbi algorithm for decoding; However, Viterbi decoding is not applicable to this problem because it makes a decision at each node without considering the effect of the next symbol. The next symbol could have a drastic effect on the cumulative metric. Therefore, the use of Viterbi decoding will not yield the optimum path. Since the decision metric for each sample depends not only on the present sample but also the previous ones, a code tree must be formed and delayed decision decoding must be employed. The code tree consists of a number of nodes and branches. From each node there are Q branches which extend the 77 2f xxxxxxixxxxixxkkikxxihxxkixxxxxxxkxxixx XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX . XXXXXXXXXXXXXXXXXXKXXXKXXXXXXXKKXXXXXXX 1.5 -XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XXXXXXXXXXXKXXKXXXXXXXXXHKXXXXXXXXXXXXK xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ..xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 1-MKXXXXXXXXXXXXXKXXXXXXXXXXXXXXXXXXXXXX xxxxxxxxxxxxxxxxxxxaxxxxxxxxxxxxxxxxxxx .XXXXXXXXXXXXXXXXXXf xxxxxxxxxxkx+44 444+ XXXXX44644444+0*000 444444440004M 0 00 94- -1.5 MM xxxxxxxxxxxxxxxxxxx 44444XXXXXXXXXXXXXX 4+4H+44444xxxk*xxx 0.5 4444444444444444444 HIM 444444444444t44444 utXM 0004000090600000000 oooooooooooooootoo OOOOOOODOOOOOOOOOOO -I -0.5 0 i (a) 2 1.5 1 a xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XXXIXXXXXXIXXXXXXXXXX1XXXXXXXXKXXXXXXIH xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxixixxxxxxx XX1XXXXXXXXXXXXXXXXXXXXXXXXXX1XXXXXXXXX -1.5 xxxxxxxxxxxixxxixxxxxxixxxixxXxxxixxxxx xxxxxxxxxxxxxxxxxxx 5 44444444444444XXXXX .samwmmtm .mm nuta*"*" 400 xxxxxxxxxxxxxxxxxxx XXXX44I000900990940- -0.5 0.5 (b) Figure 3.10. The decision regions for minimum cost signal design, (a) channel SNR=3.5 dB and (b) channel SNR=11 dB. The signals are represented by *'s and the decision regions are marked by distinct symbols. 26 2.5 Analysis of DPCM Encoder Consider an information source with alphabet {a1; a2, . , aQ}. The information source is said to be nonsymmetric if the source symbols are not equally likely. It will be shown that the output of a DPCM encoder can be viewed as a nonsymmetric information source. Assume that in a DPCM encoder a predictor of order M is used. Let us model the quantizer as. a additive noise source e(n)=e(n)+q(n) (2.7) where e(n) and Â§(n) are the input and the output of the quantizer respectively and q(n) is the quantization noise. It has been shown that [12] e(n) =e(n) bx q(n-i) (2*8) -i where e(n) is the difference signal in a DPCM without the quantizer and bÂ¡ are the prediction coefficients. The distribution of e(n) is therefore given by the M convolution of the pdf's of e(n) and Â£2=]^ g(n-i) ; -1 121 node to the next time sample. The path with the largest cumulative metric of X) log (PUi | v) PivJ, . ., v^H)) i will be the optimum path. Associated with each branch of the tree is a branch metric. The branch metrics of the tree are given by Tirjvj) =log(P(ri|vi) Piv^v^, . ., v^)) (5.8) Once the tree is appropriately populated, the next task is to select a sequence that minimizes the additive metric. An exhaustive search over all candidate sequences will provide the optimum solution. However, the enormous computational burden of the exhaustive search suggests the use of a non- exhaustive search procedure that concentrates on the most promising portions of the tree. There are a number of tree searching algorithms. We choose to use the (M,L) algorithm [34]. This algorithm being a breadth-first procedure examines all the branches extending from the current M nodes. At this point only the best M of these branches in the sense of largest cumulative path metric are saved. An example is presented in Figure 5.1 where the branching factor of the tree is three. At each depth only three branches are saved and the rest are discarded. 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Here a two dimensional signal constellation is utilized. The channel matrix is P= tPij] (4.2) N Figure 4.1. The Model of the communication channel, where PjÂ¡ is given by Pa = exp [--] n iSj-Sjl 4 Nn (4.3) It is also obvious that Pji = Pij Izizq (4.4) 50 a= (x-AH sinP3) 2+y2 b= (x+BH sinal) 2+y2 c= (x-CH sin (Pi+a2+P2) ) 2+ (y-AH cosP 1+CH cos (Pi+a2+P2) ) 2 x=p1AH sin$1-p2BH sina 1+p3C/ sin (Px+oc2+P2) y=p2AH cosP1-p3 CH cos (P1+a2+P2) . Now the problem is to minimize a function of nine variables under seven equality constraints. The method of Lagrange multipliers can be applied to this problem. We need to define a new cost function F of sixteen variables and take the partial derivatives with respect to the sixteen variables and set them equal to zero. cost function=F(alt a2, a3, px, P2, P3, AH, BH, CH, X2, X3, X4, X5, X6, A If we eliminate X/s among the sixteen equations we end up with a nonlinear set of equations of the other nine variables which must be solved numerically. This method becomes quite complicated as the number of the signals in the constellation grows. It was mainly used to verify the solutions obtained by the gradient based method when N was relatively small. 103 consideration perforin. The expressions plotted are the word error probability. When the bandwidth constraint is strict, the ternary alphabet is seen to be superior to the binary system (Figure 4.6(a)). It is also seen that the systems employing the alphabets of size four, five and seven are more efficient than the binary system above a certain coding threshold (Figures 4.6(b)-4.6(d)). Moreover, the coding threshold seems to be larger for larger alphabets sizes. The nonbinary systems are expected to outperform the binary system since they utilize error correction codes and the binary system does not. Nonbinary systems were also compared to each other. The results indicate that the ternary system is superior to the other ones Figure 4.8(a) indicates that ternary system outperforms the 7-ary system but the improvements decrease slowly as the signal-to-noise ratio increases. 7-ary system outperforms systems employing alphabets of size four and five above reasonable thresholds. The comparison between the systems with 7-ary and 4-ary alphabets is shown in Figure 4.8(b). For the other case when the bandwidth constraint can be somewhat relaxed the situation is different. There is always a threshold above which the nonbinary is superior to the binary system. For a twenty four percent bandwidth expansion the ternary system is still superior to the binary system above a threshold but the improvements over the binary system 44 J^=-Â£k JT e"(u_/^r>a(u(i+tan2a2)e-wa) 2 u-0 du _ -(u- /aW2)2 ^fe V^0 (-u(l+tan2(a1+a2)e-uW(l+a2)) du U-0 T J u-0 (-u(l+tan2 (a1+a2)) e-uW(-1+2>+u(l+tan2a2) e-uW2 da) =0. (3.24) Inspection shows that above equations consist of only one type of integral which can be expressed in a computable form, fu e (u-k)2e-u2l g ((l*i) u2*k2-2uk) du (3.25) By producing a perfect square in the exponent J g-((l+J)u2+k2-2uk) du= e k2l (1*1) / -(1*1)(u- (1+i) du (3.26) and using the results obtained in Appendix A we get: 123 However, if some redundancy is present in the encoded signal a more reasonable criteria can be formulated. Assuming that the data has a memory of M and bÂ¡.,,, bM., are the previous decisions, the modified MAP rule will be, set =ak if P(-rK=a*> P(vi=ak\vi-i=bi-i' vi-M=bi-n) > p(ri|vi=a.j)p(vi=aj|vi_1=.bi_lf . ., v^b^) (5.10) for j=l, 2, . ,Q and j?k. The symbol-by-symbol MMAP method is in fact a special case of tree decoding procedure for Markov sequences developed earlier where no search is performed and the past symbol decisions are used as an aid for the current symbol decision. 5.4 A Minimum Cost Decoder In a video transmission system some errors might be more costly than others. Therefore instead of minimizing the usual error rate a different criteria must be formed and minimized. A Bayes receiver minimizes the average cost, however, to implement Bayes receiver one must know the source statistics and must be willing to specify the cost associated with each decision [36]. Let the transmitted and received symbols be from the same CHAPTER FOUR ALPHABET SIZE SELECTION FOR VIDEO SIGNAL CODING 4.1 Introduction Previously the methods of signal selection for nonsymmetric source were studied. Here we are going to utilize those methods in the deign of an image transmission system. An image transmission system typically consists of three parts, a source coder, a channel coder and a modulator. Source coding is the first step in the transfer of information from a source. The purpose of source coding is to remove as much redundancy as possible from the source. DPCM of video signals with as few as seven quantization levels has been shown to produce pictures virtually indistinguishable from the original under standard viewing conditions [21]. Since the output of DPCM system is a nonsymmetric source, an entropy coder must be used to take advantage of that redundancy. The well known Shannon-Fano method and the Huffman procedure are examples of entropy coding techniques [22]. In channel coding the goal is to correct errors introduced in the channel by inserting redundancy in the data. Channel coding is accomplished by either decreasing the 79 102 Table 4.9continued before encoding after encoding .5 .3214 . 166 . 1543 . 166 . 1543 . 064 .1034 . .064 . 1033 .02 .0817 CM O .0816 (d) 55 >1 92 VF= ?N where 9k=(9k'9k) (3.53) gk is obtained by taking the derivative of Pe with respect to sk (Pjc+P) Sk~Si rexp [--|s*-Sil i*k 4Wn ] [ -i-1 l^-sj2 2N0 (3.54) Similarly one can find the kth component of VF for a MAP receiver i*k P*-Zs*-*iexP[" t . ta t2 . ] [-4+ T-5-] &N0\sk-sA2 2Njsk-Si i r + 4JV0|sJt-si|a t| 2i\y|sJt-sJ2 +Pi-VSiexPt' (3.55) where 148 6.1.1 Signal Selection for Nonsvmmetric Sources Gradient based signal selection methods for nonsymmetric sources that minimize the symbol error rate subject to a peak or an average power constraint were developed. The gradient and conjugate gradient versions of the algorithms were compared and the conjugate gradient version was found to be much faster. The generalized version of the algorithm that finds a signal set that minimizes a cost function (not necessarily the symbol error rate) under a power constraint was also addressed. It was seen that the nonsymmetric signal design gains significant improvements over the egually likely signal selection when the channel is noisy. The minimum cost signal design also demonstrated performance improvement relative to minimum error signal selection over noisy channels. 6.1.2 Alphabet Size Selection for Nonsvmmetric Sources Even though efficient techniques for source coding, channel coding and signal design exists, it is not known how the choice of alphabet size affects a communication system. To investigate the role of alphabet size in a communication system for nonsymmetric information sources, communication systems with various alphabet sizes for transmission of video signals on the basis of equal information rate, bandwidth and 20 1.10 Description of Chapters A review of background material relevant to this dissertation is given in Chapter One. Two dimensional modulation techniques, source coding techniques for images, standard error correcting techniques and the effect of channel errors on predictive system are among the topics addressed in this chapter. In Chapter Two a DPCM system will be analyzed and a model for the statistics of the source will be derived. It will be shown theoretically and empirically that DPCM of video signals produces nonsymmetric sources. The issues of signal design are addressed in Chapter Three. Algorithmic solutions to signal design for nonsymmetric information sources under average and peak power constraints for minimizing the error rate and average cost are presented. The study on the role of alphabet size for nonsymmetric sources in a communication system is given in Chapter Four. In Chapter Five various methods for decoding Markov sequences are presented. The application to the transmission of video signals over noisy channels and a comparison of the method is also given. Chapter Six contains a summary of presented approaches, conclusions and comments regarding the future research directions. 157 [24] Massey, J. L.,"Shift Register Synthesis and BCH Decoding," IEEE Transactions on Information Theory. Vol. IT-15, pp. 122-127, 1969. [25] Chien, R. T., "Cyclic Decoding Procedures for the Bose- Chaudhuri-Hocquenghem codes," IEEE Transactions on Information Theory. Vol. IT-10, pp. 357-363, 1964. [26] Huffman, D.A., "A method for the Construction of Minimum Redundancy Codes," Proceedings of the I.R.E.. Vol. 40, pp. 1098-1101, 1952. [27] Lippmann, R., "Influence of Channel Errors on DPCM Picture Coding," ACTA Electron.. Vol. 19, No. 4, pp. 289- 294, 1976. [28] Stammnitz, P., "Error Protection of 34 Mbits/s DPCM Encoded TV Signals with Multiple Error Correcting BCH Codes," Proc. 1980 Zurich Sem. Digital Commun.. 1980. [29] Lippman, R.,"A Technique for Channel Error Correction in Differential PCM Picture Transmission," Proc. Int Conf. Commun.. Seattle, WA, 1973. [30] Fenwick, D. M., Steele, R., and Vasanji, N., "Partial Correction of DPCM Video Signals Using a Walsh Corrector," Radio Electron Eng.. Vol. 48, pp. 271-276, 1978. [31] Ngan, K. N., and Steel, R., "Enhancement of PCM and DPCM Images Corrupted By Transmission Errors," IEEE Trans. Commun.. Vol. COM-30, pp. 257-264, 1982. [32] Steele, R., Jayant, N. S. and Schmidt, C. E, "Statistical Block Protection Code for DPCM-Coded Speech," Bell System Tech. J.. Vol. 58, No. 7, pp. 1647- 1657, 1979. [33] Moore, C. C. and Gibson, J. D., "Self-Orthogonal Convolutional coding for DPCM-AQB Speech Encoder," IEEE Trans. Commun.. Vol. COM-32, pp. 980-982, 1984. [34] Anderson, J. B. and Bodie, J. B., "Tree Encoding of Speech," IEEE Trans. Inform. Theory. Vol. IT-21, pp. 379-387, 1975. [35] Modestino, J. W., Bhaskaran, V. and Anderson, J. B., "Tree Encoding of Images in the Presence of Channel Errors," IEEE Trans. Inform. Theory, Vol. IT-27, pp. 677-697, 1981. 81 The BCH codes are chosen for channel coding (If extra bandwidth is available) because codes close to any rate can be found. A q-ary BCH code is denoted by (n,kq,tq) where kq represents the number of information symbols, n represents the bolck size and tq represents the the error correction capability of the code. The receivers are coherent in-phase/quadrature detectors (except for BPSK where only the i'n-phase branch is needed) and perfect carrier and symbol synchronization is assumed. The code words for the nonbinary systems are not equally likely. A minimum error procedure is the optimum way of decoding at the receiver but a maximum likelihood procedure can be used as a sub-optimal procedure. The Berlekamp-Massey procedure will be used for decoding of the received data [23]- [25]. 4.2.2 Communication Channel The communication channel is modeled as a q-ary independent error channel and the source alphabet and channel output alphabet are the same. If a symbol sÂ¡ is transmitted there is a probability of pjÂ¡ that Sj is received (ip^j) and a probability of pki that sk is received (i?*k), and so on and so forth. Therefore, the probability that sÂ¡ is received correctly is APPENDIX A EVALUATION OF AN INTEGRAL We would like to evaluate the following integral: (x-a)2 fx e 22 dx. o Let us make the following change of variable: (A.l) u= x-a (A.2) y/2 0 Therefore x and dx can be expressed in terms of u and du: x=y/2a u+a (A*3) dx=/2a du. Upon substituting values for x and dx into the integral we get (x-a)2 Jx e 202 dx= J y/2o(a+\/2ou) e"2 du. (A.4) 0 a v/Jo The integral can be broken into the following two integrals: 151 58 Si (k+1) sf~M Sz (k+1) Sk*i~ yfWM s(k+l) y/TTR to meet the peak power constraint. (3.63) 3.5 Minimum Average Cost Signal Selection The communication channels are almost never error free. The probability of error (error rate) is a measure of system \ performance. Various error types are usually weighted equally but in some communication system certain errors are more costly than the others. The Bayes receiver allows us to rank the different error types [19]. To utilize the Bayes receiver, one must know the source statistics and a reasonable estimate of a cost matrix must be obtained. The Bayes receiver requires more apriori knowledge about the communication system than others but it results in a superior performance if there is no mismatch between the design and operating conditions. The goal is to find a signal constellation that minimizes the average cost subject to a peak or average power constraint. First a workable expression for the average cost will be obtained. 104 are not as much as the fixed bandwidth case (Figure 4.5(a)). For larger bandwidth expansion factors the coding threshold becomes large. For very large bandwidth expansion factors the binary system is superior to ternary systems. The binary system turns out be superior to larger alphabet systems (4-ary, 5-ary and 7-ary) with a little bandwidth expansion below a threshold. Figure 4.7(b) demonstrates that the binary system is superior to system with 4-ary alphabet below a certain threshold. Notice that the threshold for these systems are much larger than the threshold value for ternary system. Further more, the threshold value increases as either the bandwidth expansion factor or the alphabet size go up. Up until here we have only examined the graphs for the medium detail picture. The results for low and high detail picture are given in Figures 4.9-4.12. A comparison between the graphs obtained for different pictures indicate that there is a slight variability among the graphs which has to do with the amount of detail in each picture. However, the relationship between the system performance and the alphabet size does not vary from picture to picture. 4.7 summary Communication systems employing alphabets of different sizes are compared for transmission of video signals under the 126 N dj=]C (yk-yj)2 p(rAvi=ak') p(vi=ak\vi-i=b (5.15) The given proof is only applicable to first order DPCM systems. Let us give a more general proof. It has been shown that the signal-to-noise ratio of a DPCM system is given by [37] where a,2 is the variance of the signal, ae2 is the variance of the input to the quantizer and (5.17) in which eq represents the mean-squared quantization noise, em is mutual error term and ec represents noise due to channel noise. The only term in the SNR expression that depends on the choice of receiver is ec ec is given by (5.18) where D(z1(z2) is the discrete system transfer function and 154 c=Â£ <*-**> and Â£=Â£ (irtit) (y-yj,)exp[- Sjci 3p^. *-1 ivo (B. 5) Since the right hand side of the equations are zero, the solution to the set of equations (B.l) is dxji dxJ =0 dxit -^=, dYj (B 6) and OX, dy dVj *=0 (B.7) 88 TABLE 4.1. DIFFERENCE SYMBOL PROBABILITIES. Picture P(Yo) p (y+i)= p(y.i) p(y+2) = p(y.2) p(y+3) = p(y.3) Low Detail . 674 . 142 .018 .003 Medium detail .584 .172 . 032 .004 High Detail .5 . 166 .064 . 02 TABLE 4.2. PRIMITIVE IRREDUCIBLE POLYNOMIALS OVER NONBINARY FIELDS. Field Polynomial GF (2) x2 +x +1 GF (3) x3 +2x +1 GF (4) x2 +x +A GF (5) x2 +x +2 GF (7) x2 +x +3 7 prediction. The expansion of this equation gives the following condition for optimum aj, N Â£ a. R^ik-j) =Rxx(k) ; k=l,2,...,N (1.19) j-i or '^(0) 1^(1) ***(2) RjocIN- 1)' (a \ ai RjociO) R^d) R^iN-2) a2 = ^(2) rJW-1) R^iN-2) R^N- 3) Rjcx'i 0) , aN, ***(*>, (1.20) in matrix notation, Rxxa=r*x (1-21) where rxx={Rxxd)): N (1.22) The equations are called normal equations, Yule-Walker prediction equations or Wiener-Hopf equations. The mean squared error decreases significantly by using up to three elements in predictive coding. However, if the coefficients are not exactly matched to the statistics of a picture, the decrease in mean squared error is not significant by using three previous elements as compared to one [1]. 8 1.5 Transform Coding In transform coding a picture is divided into subpictures and then each of these subpictures are transformed into a set of independent coefficients. The coefficients are then quantized and coded for the transmission. An inverse transformation is applied to recover intensities of picture element. Much of the compression is a result of dropping coefficients from transmission that are small and coarsely quantizing the others as required by the picture quality. It is desirable to have a transform which compacts most of the image energy in as few coefficients as possible. Another consideration is the ease of implementation. 1.5.1 Optimum Transform Optimum transform (K-L transform) is explicitly known, but computationally it is very demanding. This undesirable feature has prevented any hardware implementation of the optimum transform. It is mainly studied in simulations to obtain bounds. The most practical transform coding techniques are based on the DCT (discrete cosine transform), which provides a good compromise between information packing ability and computational complexity. In fact, the properties of DCT have proved of such practical value that it has become the 68 TABLE 3.1. VALUES OF SIGNAL-TO-NOISE RATIO AND THE PROBABILITY DISTRIBUTIONS OF NON-SYMMETRIC SOURCE. Source Number SNR(dB) Probability Distribution 1 12.2 .8,.1 1 2 9 .9, .025,.025,.025,.025 3 8 .96,.01,.01,.01,.01 4 15.5 .674,.142,.142,.018,.018,.003,.003 TABLE 3.2. SOURCE STATISTICS FOR THE FOUR SIGNAL SOURCE. source statistics .35,.35,.15,.15 94 4.5 Nonbinarv BCH Codes The nonbinary BCH codes are a special class of cyclic codes that can be constructed for any symbol alphabet defined on a finite field, say GF(q), which can be a prime field or some extension of a prime field. As a generalization of binary case a t-error correcting BCH code on GF(q) is a cyclic code , and all the codewords have roots that include 2t consecutive powers of some element j3 contained in GF(qm) an extension field of GF(q). It will be convenient to distinguish between the two fields by calling the GF(q) the symbol field and GF(qm) the locator field. As with the binary codes, BCH codes on GF(q) can be primitive or nonprimitive, depending on whether a primitive or nonprimitive element of GF(qm) is used to specify the connective roots of codewords. A t-error correcting code may have either odd or even minimum design distance, given by d=2t+l or d=2t+2, respectively. The generator polynomial of a BCH code on GF(q) is defined as the least common multiple of minimal polynomial of amo, amo+1,..., amo+d-2 (the sequence of powers can begin with any arbitrary power a) g (x) =LCM [ The block length of the code is the order of the element 76 chanrxl SNR Figure 3.9. Performance results for non-symmetric signal selection (the solid curve) and the equally likely signal design (the dashed curve) over a wide range of channel signal- to-noise ratio. APPENDIX B EVALUATION OF THE DERIVATIVES Differentiating equations (5.38) and (5.39) with respect to Xj and yj we get the following system of equations A- A- C C dx ji dx4 +B- dx. dy +B- dx a dxj +D dx dy dYji dxj dVli- 0 dyj=0 pi-0 dxj dyJA-0 dy* (B.l) where N =Â£ (Lik~Ljk) (y-y*) exp [- jt-i lls-sj2 N -Ix-Xj Â£ Uik~Ljk) 2 (y-yk) (x-xk) ls-st jt-i N exp[-- Nn }pk (B. 2) , x-v , 2 (X~Xk) 2 r Â¡S-SjuJ2 , -(y-yi>Â£ (1_)exP[- k-1 N B=(x-Xi) Y; (Lik-Ljk) (1- 2 (y-yk)2, r Ils-Sjtl2 k=l N ) exp [- IP* "Â£ (Lik~Lj*) U-x*)exp(- s-s, k-1 ]pk (B. 3) N + (y-yi)Â£ ^ik-Ljk) k-l 2(x-xk) (y-y.),)exp[_,.-^",]p^ lls-sJI2 153 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFICIENT COMMUNICATIONS FOR NONSYMMETRIC INFORMATION SOURCES WITH APPLICATION TO PICTURE TRANSMISSION By Shahriar Emami August 1993 Chairperson: Dr. Scott L. Miller Major Department: Electrical Engineering This dissertation is concerned with issues related to nonsymmetric information sources. Signal design, alphabet size selection and decoding of information from these sources are among the topics covered in this dissertation. Although the techniques presented here are applicable to any nonsymmetric source, the emphasis is placed on video sources. Initially a model for the statistics of DPCM (Differential Pulse Code Modulation) of video signals is derived and it is shown that DPCM of video signals results in a nonsymmetric source. The problem of signal selection for nonsymmetric sources in two dimensions is considered. Iterative methods for finding the minimum error signal (and minimum cost) constellation subject to an average (or a peak) power constraint are presented. vi 17 Let us assume the channel noise is uncorrelated and the difference signal samples are statistically independent, then variXj =E[Oi2] +2E[QiNi] +E[Ni2]'Â£fk2 (1.31) k The second term is called the mutual error and can be shown to be approximately zero if the quantizer is near optimum. The error power reduces to varx'i) =E[QÂ¡] +ElNi2}'Â£fk2. (1.32) k The sum can be evaluated using an identity ijt2=l+a2+a4+a6 + k 1 1-a2 =Jb. (1.33) The expression for the reconstruction error variance then becomes var(Xf_) =var(Qi) +b var(N) (1.34) We now define the following quantities to relate the quantizer and noise variances to the differential signal variance, var(Q) =kg var(E-), var (N) =Kn var(EÂ£) . (1.35) The DPCM prediction gain is also given by CHAPTER FIVE EFFICIENT DECODING OF CORRELATED SEQUENCES 5.1 Introduction Differential pulse code modulation (DPCM) of digitized pictures results in a stream of binary words which represent the difference between the current pixel and the previous ones along the same scan line. Communication channels are almost never error free. A channel error results in a streak whose termination may extend to the end of the current scan line because in a DPCM system the channel errors propagate. Visually this effect is very disturbing. As a result one must deal with the errors introduced in the channel in the design of a DPCM system. Standard error correction techniques have been used to combat the channel errors in DPCM systems. The use of error correction codes such as BCH codes requires addition of redundancy to the data stream. The redundancy turned out to be twenty five percent for a bit error rate of 0.01 for a reasonable picture quality [27],[28]. It is possible to utilize the natural redundancies that are present in the video signals to enhance the received signals. Lippmann [29] introduced a method for locating 116 36 solving signal design problem under a peak power constraint. Previous investigations on signal design have focused on signal selection for equally likely signals [15]-[17]. There are some applications where the information source is non- symmetric. A practical instance in which such a model proves rewarding is in the transmission of video signals. It was demonstrated in Chapter Two that DPCM of digitized video signals results in a nonsymmetric source. In this case signals should be mapped into a two dimensional signal constellation in an optimum manner. In other words the goal is to determine the signal constellation that minimizes the probability of error (or a given cost function) in presence of additive white Gaussian noise under an average power(or a peak power) constraint, given N signals with unequal probabilities. To illustrate the difficulty of direct approach we will design a ML receiver for a three signal constellation. We will also describe a numerical method that uses the Lagrange multipliers method for optimization. These two methods are appropriate for smaller signal sets. Then a number of iterative algorithms are developed. First a normalized conjugate gradient search algorithm and a gradient search algorithm will be presented that can be applied to signal sets of any size and with any probability distribution. The methods presented here are applicable to the design of both MAP (maximum a posteriori) and ML (maximum likelihood) receivers. These methods are generalizations and 122 Because of two interesting features the (M,L) is favored in practice. The outputs are released in a synchronous manner and a fixed number of branches are visited at each depth. The algorithm in one form or another has been used in speech and image coding previously [34],[35]. Figure 5.1. A tree with branching factor of three is shown. At each depth only three branches are saved. 5.3 A Modified MAP (MMAP) Receiver A MAP receiver chooses <Â£=<3* if p(ri\vi=ak) p(>p(ri|vi=aJ)p(v^a^) (5.9) for j=l,2,...,Q and j?k. 64 (-^_1)E iLik-Ljk> (y-y*) exp [ - OJCi k-l Z, |g-g*l )pk + (x-xi)'^2 (Lik~Ljk) (~g~~~2 (y~y*) it1) exP t~ B *-l oxi -iVo "o = (-^-)E {Lik~Ljk) (x-xk)exp[--S ^kÂ¡ ]pk oxi k-l -w~ ]pj, Jr-1 axi "o |g-g*|2 i ]p* (3.83) where (3.84) 3V ^ | o_o |2 (-0)E ^Lik~hk) (y*-yjt> exp [ ----- ] p + (x-xi) E (-2 (y-y*) ^) exp [ Bs^Sjc!1 ] Pk = (-^~1)E (Lik~Ljk) (x-xk)exp[-^-Â£A.]pk (y-Vi) E (< 3r "2 (*~*Jc> > exp [- ,S*k>I ] p* jc-i yj "o (3.85) where z2-U-xt)({|)+(y-yÂ¡[)(^l) It can be shown that (refer to Appendix B) (3.86) 99 the roots of a polynomial [25]. The next step is to find the error values. Berlekamp [23] has derived a formula for the magnitude of errors if the location of errors are known. Eventually to correct the errors we will subtract the error polynomial from the received polynomial. 4.6 Results To encode the pictures, a DPCM encoder was utilized to generate a stream of symbols. The Huffman optimal procedure [26] was applied to the encoded pictures to produce ternary, 4-ary, 5-ary and 7-ary streams of symbols and the average message length per symbol were calculated. Then ternary, 4- ary, 5-ary and 7-ary BCH codes were used to channel encode the image data. The code rates were selected to be as close as possible to the quantity hb/hq. The q-ary BCH codes and their rates for all the pictures are given in Table 4.8. The symbol probabilities after channel encoding were calculated. Table 4.9 shows the symbol probabilities before and after encoding. It turns out that the probabilities still follow the structure given by (4.13) but the output source is not quite as asymmetric as the input source. After encoding, the larger probabilities get slightly smaller and the smaller probabilities get slightly bigger. Figures 4.4-4.6 illustrate how the systems under 12 coding is not apparent yet. Since the code corrects all single errors within a block of 15 bits, the message error rate for the coded system will be Pc.m=E(^5) Pc,/(l-Pc.jb)15-Jc=1.94xl0-6 (1.27) It is seen by comparing the error rates that the message error rate has improved by a factor of 58 through the use of a block code. Most of the research on block codes has been concentrated on a subclass of linear codes known as cyclic codes. A cyclic code word, after any number of cyclic shifts, has the property of remaining a valid code word from the original set of code words. Cyclic codes are attractive because they can be easily implemented with feedback shift registers. The decoding methods are simple and efficient. Examples of cyclic and related codes are BCH, Reed- Solomon, Hamming, Reed-Muller, Golay, quadratic residue and Goppa codes. The classes form overlapping sets so that a particular code may be a BCH code and also a quadratic residue code. Recent applications of these codes to digital communication include a (31,15) Reed-Solomon code for joint tactical information distribution system(JTIDS) and a (127,112) BCH code for INTELSAT V system [4]. 72 design relative to a equally likely signal selection, a first order Gauss-Markov source with a correlation coefficient of 0.9 was synthesized. The output was encoded using a first order DPCM encoder. A four signal constellation was designed with the source statistics shown in Table 3.2. The performance of the non-symmetric constellation was compared with that of the well known equally likely constellation (for four equally likely signals, the best constellation is formed by the vertices of a square [16]). Figure 3.9 demonstrates a comparison between the systems in terms of output signal-to- noise ratio (output SNR). For large values of channel signal- to-noise ratio (channel SNR) the two design procedures result in an identical performance. As the channel SNR decreases the curves representing the performance of non-symmetric and equally likely signal design separate and the difference between the two systems gets larger. The non-symmetric signal design is 3 dB (in terms of output SNR) superior to equally likely signal design for the noisiest channel considered (a channel SNR of 4 dB) It is seen that significant improvement in performance can be obtained for noisy channels by utilizing non-symmetric signal design. The amount of improvement is a function of source statistics. A comparison between the equally likely signals and unequal signal probabilities shows that if the signal statistics are not very different, the shape of constellation is not appreciably different from the equally likely case. But 84 (a) (b) Figure 4.2. The transmitters for systems employing (a) binary (b) nonbinary alphabets. In variable length coding (such as Huffman coding) a symbol decoding error propagates in the block. Therefore, it is appropriate to use word error probability as the figure of merit. For the binary system Pw is given by Pw=l-(l-pe)n (4.6) where pc is the crossover probability. For a BPSK system pc is known to be equal to pe=OW2e) (4.7) where e is the signal-to-noise ratio. II. When the bandwidth constraint is not very strict and can be somewhat relaxed. Let us allow some bandwidth expansion for the binary system (Figure 4.2). As a result the nonbinary 4 B_ (i+r)D (1.10) where D=R/1 and r is the rolloff factor. Since BT =2B, the transmission bandwidth of QAM is d (1+r)R nT j (i.ii) and the spectral efficiency with raised cosine filtering is given by log^AT bj ts/sec (1.12) 1 (1+r) Hz The spectral efficiency increases with the number of the points in the constellation. However, one can not increase the spectral efficiency by increasing the number of points in the signal constellation too much, because as you place more signals in the constellation the error rate increases. For reliable communications, the information rate must be kept below the channel capacity. Therefore, the spectral efficiency is upper bounded by ti N where S/N is the signal-to-noise power ratio. Two dimensional formats are well suited for high speed data transmission because of their efficient use of bandwidth. However, they require coherent detection that implies the need for synchronization circuits. 18 VJ2Â£Ht>.1*a.-2ap1, (1.36) p variEj 1 where p, is equal to R(0) Putting all this together yields tra r / Y t \ var(X,i)=(k+bkn)' 1 (1-37) gp The second term above is the dominant term because the effect of channel noise is much more destructive to the reconstruction of the image than the effect due to quantization noise, var(X'j) =K vax(Xi) 1 2a Pi a (1.38) 1-a2 To minimize the variance of the reconstruction error we will set the derivative of this expression with respect to a to zero. The optimum value of a turns out to be a= ^-Pi2 Pi (1.39) 1.9 Research Objectives This dissertation is concerned with issues related to nonsymmetric information sources. To motivate the work on nonsymmetric sources, it is shown that DPCM of digitized video signals results in a nonsymmetric information source. One of the main goals is to address the problem of signal design in 41 P =1- Pi tzM I Pi- p1tana] J exp [-- Pa PitanPi (Pi-Aii) Jp| Nn ) dp2dp1 P2 nN0 / Pl-0 PjCanaj J exp[- PiPitanPj (Pi-Aii)2 ] dp2dpx P3 nN0 M / Pl-0 Pjtanaj J exp[- P2--PitanPj (Pi-AH) j^]dp2dPl. lvo (3.17) By making the following changes of variables e = -7= dp 2=desfiro W (3.18) u=-^; dp1=duv/^ W> (3.19) the probability of error in terms of the new variables becomes 25 Equation (2.2) is readily solved for 1Â¡ *1.1 j x p (x) dx , =1,2,...,!. (2.6) *ii f p(x)dx *i The solution of the equations for the general nonuniform quantizer is difficult. However, a procedure to obtain a solution by computer iteration has been introduced by Llyod and Max [9],[10]. For a specified probability density function and a fixed value of L, 1, is first selected arbitrarily, with x,=-oo, we solve eq. (2.6) for x2. Next x2 and 1, are used in eq. (2.5) to obtain 12. The process is repeated to obtain x3 from eq. (2.6) and 13 from eq. (2.5). Continued iteration finally stops when 1L is obtained from eq. (2.5). If 1, has been correctly guessed, then 1L will satisfy the equation with xL+, =oo. if it does not, 1, is corrected to a new value and the process is repeated until 1L satisfies eq. (2.6). This procedure satisfies conditions eq. (2.5) and eq. (2.6) which are necessary for optimality. Max [10] used the above procedure to find the quantum levels for a zero-mean Gaussian message for quantizers up to 36 levels. Paez and Glisson [11] used the procedure to find optimum levels for signals having either a gamma density or a Laplace density for L=2,4,8,16, and 32. 34 (b) Figure 2.3. The histogram for AERIAL MAP picture is compared with (a) Laplacian Distribution (b) Distribution given in equation (13). Histogram is shown with a broken line. EFFICIENT COMMUNICATIONS FOR NONSYMMETRIC INFORMATION SOURCES WITH APPLICATION TO PICTURE TRANSMISSION By SHAHRIAR EMAMI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1993 63 (3.78) The gradient of the cost function depends not only on the nearest points on the boundary but also on their derivatives. Let us find the derivative of the components of SjÂ¡ with respect to the components of sÂ¡. Differentiating eq. (3.71) and (3.72) with respect to xÂ¡ and yÂ¡ we get (3.79) where (3.80) (3.81) where w2= (x-x (3.82) 11 channel coding, let us pick a (15,11) single error correcting code. Assume a BPSK modulation, a signal-to-noise ratio of 43,776 ( S/N0=43,776) and a data rate of R=4800 b/s. Let Pub and Pum represent the bit and message error rate for the uncoded system and Pcb Pcm represent the bit and message error rate for the coded system, respectively, without coding (1.23) pu,b=0\ -^)=1.02xl0-5 (1.24) and PUia=l-(l-PUtb) ll=l. 12x10-4. (1.25) With coding i?c=4800x (15/11) =6545 b/S (1.26) ) =1.36xlO-4 The bit error rate for the coded system is inferior to that of the uncoded system and the performance improvement due to 6 can be adjusted to give a smaller quantization error than would be possible when quantizing the input directly. 1.4.1 Optimum Prediction We are interested in linear prediction of the form N Â£(n) ='Â£, aj x(n-j) (1-14) 3-1 which is the weighted sum of N previous samples. The weights aj are linear prediction coefficients. The filter is optimized by finding the weights that minimize prediction error in a mean squared sense oe2=E[(x(n)-St(n))2) (1-15) Since mean squared error is a function of aj and do 2 -=5-^=0; =1,2 N (1.16) da is a necessary condition for minimum MSE (mean-squared error). Evaluating the derivative gives E[-2ix(n) -Â£(n))8j?(ri) ] (1.17) a Equating this to zero yields, E[{x(n) -x(n) )x(n-i) ] =0; i=l, 2, . N. (1.18) This is called the orthogonality principles which states that minimum error must be orthogonal to all data used in the 108 (a) (b) Figure 4.7. A comparison of systems employing non-binary alphabets against the binary system when the bandwidth can somewhat be expanded. The solid curve represents the binary system, (a) ternary (26,14,3) code and binary (15,11,1) code are used and a block is 390 bits long; (b) 4-ary (15,7,3) and binary (31,26,1) are used and the block length is 465 bits long. The reults are given for a medium detail picture. 117 erroneous pixels. In this method a pixel is compared with pixels on the previous and following lines. If the differences are above a threshold an erroneous pixel is found. The codeword assignment is such that errors will be either small or large. Because of high correlation between adjacent lines, the pixel in error is replaced by the corresponding pixel in the previous line. Walsh-Hadamard transformation have been used by Fenwick, Steel and Vansanji [30]. In this method DPCM blocks are transformed and sent over the channel. DPCM blocks are transformed in a manner that when the receiver recovers the transform coefficients, the presence of an error is detected and its magnitude is known. Correlation techniques are used to find the erroneous sequence. Periodic PCM updates have been used by some researchers before. A good example is the work by Ngan and Steele [31]. They used several PCM updates per line. If the difference between the PCM update word and DPCM output is greater than a threshold, that potion of the line is in error. The pixel value in error is replaced by a combination of the adjacent pixels. The percentage increase in bit rate due to side information was about 12 percent. Many of the systems presented take advantage of signal statistics. Steel, Jayant and Schmidt [32] introduced a statistical block protection procedure. In this work the statistical features of an input block such as the average and I 134 (a) (b) Figure 5.5. The variation of picture SNR as a function of channel error rate. The receiver utilizing first (the dotted curve), second (the dashed curve) and third (the solid curve) order Markov modelling of the source, (a) MMAP receiver, (b) MSNR receiver. References [1] Habibi, A., "Compression of Nth-Order DPCM Encoder with Linear Transformation and Block Quantization Techniques," IEEE Trans, on Commum. Technol. Vol. Com- 19, pp. 948-956, 1971. [2] Wintz, P. A.,"Transform Picture Coding," Proc. IEEE. Vol. 60, No. 7, pp. 809-820, 1972. [3] Jayant, N. S., and Noll, P., Digital Coding of Waveforms. Englewood Cliffs, NJ: Prentice-Hall, 1984. [4] Bhargava, V. K., Haccoun, D., Mattyas, R., and Nuspl, P., Digital Communications by Satellite Modulation. Multiple Access and Coding. Wiley, New York, 1981. [5] Lin, S., and Costello, D. J., Error Control Coding. Englewood Cliffs, NJ: Prentice-Hall, 1983. [6] Chang, K. Y., and Donaldson, R. W., Analysis, "Optimization and Sensitivity Study of Differential PCM Operating on Noisy Communication Channels," IEEE Trans. Commun.. Vol. COM-20, pp. 338-350, 1972. [7] Netravali, A. N., and Limb, J. O., "Picture Coding: A Review," Proceedings of the IEEE. Vol. 68, No. 3, pp.366- 406, 1980. [8] Limb, J. 0., "Source Receiver Encoding of television Signals," Proc. IEEE. Vol. 55, pp. 364-379, 1967. [9] Lloyd, S.P., "Least Squares Quantization of PCM," IEEE Trans. Information Theory. Vol. IT-28, No. 2, pp. 129- 136, 1982. [10] Max, J., "Quantization for Minimum Distortion," IRE Trans. Information Theory. Vol. IT-6, No. 1, pp. 7-12, 1960. [11] Paez, M. D., and Glisson, T. H., "Minimum Mean-Squared- Error Quantization in Speech PCM and DPCM Systems," IEEE Trans. Commun.. Vol. COM-20, No. 2, pp. 225-230, 1972. 155 5 1.3 Source Coding The purpose of source coding is to remove as much redundancy as possible from the message. Efficient coding of messages provides the opportunity for significantly decreasing the transmission costs. Two main approaches to picture coding predictive coding and transform coding will be addressed here. 1.4 DPCM There is considerable correlation between adjacent samples of speech or image data, and indeed the correlation is significant even between samples that are several sampling intervals apart. The meaning of this high correlation is that, in an average sense, the signal does not change rapidly from sample to sample so that the difference between adjacent samples should have a lower variance than the variance of the signal itself. The predicted value is the output of the predictor system, whose input is a quantized version of the input signal. The difference signal may also be called the prediction error signal, since it is the amount by which the predictor fails to exactly predict the input. Since the variance of error signal is smaller than the variance of signal, a quantizer with a given number of levels 132 (a) O
Figure 5.4. The solid circles indicate the neighboring symbolsused in computing conditional statistics, (a) second order Markov model and (b) third order Markov model. TABLE 5.1. CODEWORD ASSIGNMENT FOR VARIOUS QUANTIZER LEVELS. Level Probability codeword 1 . 6602 000 2 .1761 110 3 .1205 101 4 .0274 011 5 .0087 111 6 .0054 100 7 .0013 010 (3.44) For large values of signal-to-noise ratio the conditional probability of error will be egual to the upper bound. By using approximation for Q function when the argument is large and plugging back in (3.43) we get Pe=N N n-i i*n exp [- lSi-SnH 4 N 3 (3.45) Similarly it can be shown that the symbol error rate for a MAP receiver is given by Nr N Pe~\\l^Vn Z ' n-i l* X> Â£ tei-sj lsrsar+N0lnljÂ¡i) exp [ (ls snll2 +Naln (~))2 Px 4W0Dsi-sn|2 (3.46) 3.4.2 Gradient Search Algorithms for an Average Power Constraint In search of the minimum, the gradient of the probability of error is obtained analytically and an iterative gradient search algorithm which modifies the constellation at each iteration is used to find the optimum constellation. In the gradient search algorithm, the iterative rule is given by 39 d(p,si)> d(p,sk) i*k. (3.8) where d(x,y) is the Euclidean distance between x and y. Point H is the intersection of the decision regions and it is therefore on the boundaries of the decision regions. For the boundary points the above inequality changes to an equality, d(H, A) =d(H, B) d(H, A) =d(H, C) (3.9) or AH = BH = CH, (3.10) and this results in ai=Pi' 2 P2' a3=P3. (3.13) Define Pw to be the probability of word error. The probability of symbol error is one minus the probability that a symbol is correct. The probability of a correct symbol is obtained by averaging all of the conditional probabilities of a correct symbol, we have PW=1~Y^P(C I mjpt. (3.14) i-i But P(C|mÂ¡) is the result of point p falling in the decision (3.11) (3.12) 31 2.7 Discussion It was shown that the pdf of input to the quantizer can be fairly approximated by equation eq. (2.13) (which is the convolution of a gamma and a uniform pdf) or simply a Laplacian distribution. The theoretical statistics derived from the models agreed well with the actual source statistics. It is seen that DPCM of video signals does in fact produce a nonsymmetric source. 66 computationally very expensive and long. However, one can use a modified form of the above procedure to save on the computational expenses. In the modified procedure, the average cost is computed based on the received data and the gradient based update rule is replaced with the directional set (Powell's) method in multidimensions [20] which does not require the gradient. 3.5.5 Application to DPCM System In a DPCM system some errors are more costly than others, therefore we could benefit from utilizing a Bayes receiver. In Chapter Five an optimum method of selecting the cost matrix will be presented. 36 Results In this section we present some numerical results, elaborate on the design procedure and discuss the performance of the methods described in this chapter. The probability of error, in general, is not a convex function of the signal set, therefore the algorithm can converge at local as well as global minima. The multi start technique can be applied to the global optimization problem [16]. In this technique one selects an optimization technique and runs it from a number of different starting points. The Even though efficient techniques for source coding, channel coding and signal design exists, it is not known how the choice of alphabet size affects a communication system. Image transmission systems with various alphabet sizes are compared on the basis of equal information rate, bandwidth and average power. The systems employing various alphabet sizes are analyzed and computer simulations are performed using pictures with different amount of details. An optimum procedure for decoding Markov sequences is developed and the path metric is derived. A heuristic tree searching algorithm is employed to obtain a suboptimum solution. Two other techniques for decoding Markov sequences, a symbol-by-symbol modified MAP (MMAP) receiver using higher order statistics and a maximum signal-to-noise ratio (MSNR) receiver, are also given. The decoding procedures were applied to image communication over noisy channels. In summary, the major contributions of this dissertation were the development of signal selection methods for nonsymmetric sources, derivation of procedures for decoding of correlated sources and application of these procedures to the picture communication in noisy situations. vii 73 if the signal probabilities are very different from the equally likely case the shape of the constellation could be very different from the equally likely constellation. Figure 3.6(b) and Figure 3.6(c) show constellations for two different five signal sources. The constellations for the extremely non- symmetric source is completely different from the other one. Generally speaking the geometry of the constellations depends upon the power constraint. For example, the optimum signal constellations for five equally likely signals subject to average power and peak power constraints are displayed in Figure 3.8(a) and Figure 3.8(b). Clearly, the choice of power constrain affects the geometry. Simulations show that for large values of signal-to-noise ratio the average cost signal selection does not result in any improvement relative to the minimum error signal design. For other values of signal-to-noise ratio the decision regions form unusual shapes which vary with the signal-to-noise ratio. Figure 3.10 shows the decision regions for a four signal constellation with the statistics listed in Table 3.2. Because of the complexity of the shape, obtaining an analytical expression for the gradient was not feasible and as a result the modified procedure in section 3.4.10 was utilized. To compare the minimum error with the minimum cost signal selection, a first order Gauss-Markov source with the correlation coefficient of 0.9 was generated. The output of the source was encoded with a first order DPCM system. An 118 the maximum magnitude of the differences is transmitted. At the receiver end the maximum value is used to detect the most seriously affected pixel, and the average statistic is used to correct these samples approximately. Moore and Gibson [33] have used convolutional code for DPCM of speech. In their DPCM system some of the bits used for source coding are used for error correction using self- orthogonal convolutional codes. Sayood and Borkenhagen [12] have presented a technique for error correction of DPCM systems. No explicit channel coding is performed. The output of the source coder is argued to contain redundancy since certain codeword combinations are more likely than the others. The decoding is formulated similar to the Viterbi algorithm and a path metric that depends on signal statistics is given. Substantial performance gain in image coding is reported. A number of methods for decoding correlated sources have been proposed in this chapter. A minimum error procedure for the decoding of correlated sequences will be derived which requires tree decoding. A modified MAP receiver (MMAP) and a maximal signal-to-noise ratio (MSNR) will also be presented. The application of the proposed methods to picture transmission over noisy channels will discussed in section 6. The results will compared to the performance of a recently proposed technique. |