Temelband İletişim Sistemlerinde Kafes Kodlama Tekniğine Dayanan Yeni Hat Kodlarının Tasarımı

Abstract

Bu tezde, sayısal haberleşme sistemlerinin hata ba şarımlarının yükseltilmesine yönelik olarak son yıllarda ortaya atılan kafes kodlamalı modülasyon yöntemi hat kod- lamalı temelband sayısal sistemlere uygulanmaktadır. R=n/n+l (n=l,2,3,) kodlama oranları için, kaynaktan üreti len her ikili teri dizisine, kodlayıcı kafes yapısına bağ lı olarak, 2 elemanlı hat kodu alfabesinden n+1 uzun luklu bir üçlü (+,0,-) hat kodu karşı düşürülmektedir. Kodlayıcı sonlu durumlu bir sistem olarak ele alınmakta, kod sözcükleri temelband iletişim gereksinimlerinin sağ lanması amacıyla tasarlanan bir durum geçiş modeline göre seçilerek durumlara atanmaktadır. Viterbi algoritmasıyla kod çözüleceği varsayılarak sistemin hata başarımı, kodla yıcının ürettiği üçlü simge dizileri arasındaki Öklid uzaklığı artırılarak düzeltilmekte, kafes kodlamasız duru ma göre 3-4.77 dB lik kodlama kazançları sağlanmaktadır. Kafes yapıda hat kodlarının analitik olarak ifade edilebi lecekleri gösterilmekte, serbest Öklid uzaklıklarının he saplanmasını sağlayan bir algoritma verilmektedir. Doğru sal olmayan R=l/2 ve R=3/4 oranlı hat kodlarının, hata ağırlıkları açısından simetrik bir yapıya sahip oldukları ortaya çıkarılmakta, buna dayanarak hata olasılıkları için analitik üst sınır ifadeleri verilmektedir. Kodlar, güç spektral yoğunlukları ve seyirme başarımları açısından in celenmektedir.

In recent years, "trellis coded modulation" (TCM) method for improving the error performance of band-limited digital communication systems has found a large application area in M-AM, PSK and QAM systems. The detection errors caused by the channel noise can be lowered by increasing the distance between channel signals. The trellis coded modulation method assumes the coding and modulation operations as an entity and uses the redundancy obtained by expanding the signal set in order to increase the minimum distance between coded channel signal sequences. At the receiver, the Viterbi decoder operates directly on the received channel signal sequences instead of the corresponding binary sequences. Then, the code design problem turns out to be searching the encoders with good Euclidean distance properties instead of the encoders with good Hamming distance properties. At the same information rate, important coding gains were obtained without bandwidth expansion. Signal set con taining twice the necessary channel signals is used according to a finite-state encoder, thus, for M-AM, PSK and QAM systems, coding gains of 3-4 dB are obtained by means of simple trellis structure. Coding gains up to 6 dB are possible using more complex trellis codes. In this thesis, the trellis coded modulation method is applied to the line coded baseband transmission systems. In baseband digital transmission systems, the signal to be transmitted must have zero direct-current (dc) component in order to avoid any dc power feeding over the line, must have small power spectral components a.t low frequencies to reduce low-frequency noise and ufficient timing content to provide the extraction of lock information. If the baseband digital signal is a1 si c. transmitted as a binary unipolar sequence, these requirements are not satisied. For this purpose, line requirements are not satısıea. tor tnxs purpose, line coding techniques are used. A line encoder transforms binary sequences feeded at rate R bit/sec, to its input, to R' symbol/sec rate L-level (L>2) sequences at its out put and provides a redundancy of R'log~ L-Rİog2 2 in information rate. This redundancy is used in order to fulfil the basic requirements cited above. Existing line encoders provide the desirable properties due to their finite-state structure without increasing the free Euclidean distance between sequences they produce. If the Viterbi decoding is assumed, the system error performance is lower-bounded relative to the free Euclidean distance of the encoder as follows: P(e) â N(ds)Q(ds/2a) 2 where a represents the variance of Gaussian noise process and K the average number of errors with distance dc, Q(.) is the Gaussian error function. s" The free Euclidean distance of a trellis encoder is the minimum Euclidean distance between sequence of pairs starting at a common state in the trellis and going over distinct states until ending at a common state, k. mi denoting i th symbol of the codeword km at the m th step, the free distance is given as -m d_ = min {Z ( £ d2(k., k*.))}1/2 s r i i /r t f i - t mı m:L {^m}^m} m 1=1 2 Here d (...) is the squared Euclidean distance between two channel symbol. In this work, the basic requirements for baseband digital signal transmission and the error performance improvement by increasing the number of channel signals are considered together. The design of the new line codes based on trellis coded modulation includes the following steps: and the restriction of the consecutive symbols w same polarity). The code alphabet must contain 2 words with zero polarity, 2n codewords with +1 codewords with -1 oolarities. and 2 2. The partitioning of the codeword alphabet to the subsets with maximum Euclidean distance, with respect to the proposed trellis encoder model. 3. The assignment of the codewords of each state- subset to the state transitions in order to maximize the minimum distance between codewords leading to the same state. On the other hand, the equally use of each codeword and the symmetric distance structure of the trellis encoder is important in order to reach the codes with optimum free distance. But, augmenting the state number of the encoder does not always leads to the codes with better distance properties. - vi - In the second chapter, a codeword assignment model for the branchs of the trellis encoder is proposed. According to this model, only codewords with disparities equal to -1,0 or 1 is used and the value of the running digital sum is restricted to values -1,0,1. For R=n/n+l, (n=l,2,3) coding rates, according to the trellis structure of the encoder, each n-length binary data sequence is encoded into a n+1 length ternary (+,0,-) line codeword chosen among the line code alphabet with 2n elements. For R=l/2 and R=3/4 rate line coding, codewords are chosen as ternary sequences produced at the output of the known paired-selected ternary (PST) encoder, in one and two steps, respectively. For R=2/3 rate, selected codewords satisfy the symmetry and distance properties best. In all cases, some codewords are used twice. This leads to partially overlapping subsets in set partitioning and to better codes than the distinct subsets case. Thus, for the same bandwidth, with some loss in data rate, the free Euclidean distance is increased by appropriate assignments of codeword subsets to the branchs of the trellis encoder. For R=l/2 rate line code, the maximum value of the free distance (d =8) is reached for four-state encoder and a coding gain of 3 dB is obtained compared to the paired- selected ternary "code. For R=2/3 rate, an eight-state encoder give the optimum solution. The reached value of the squared free Euclidean distance is equal to 6 and a coding gain of 4.77 dB is obtained compared to the uncoded case. In R=3/4 rate design, the maximum value of the free distance (d~=6) is realized with an eight-state encoder with parallel transitions. Parallel transitions mean that the last bits of binary 3-bit length sequences are not affected by the trellis coding. These last bits determine which of the two codewords in the corresponding subset will be chosen. The coding gain obtained with respect to the uncoded case is 4.77 dB and 3.52 dB with respect to the PST encoder. Further in the same chapter, it is shown that the designed line codes can be analytically expressed and a new algorithm for the calculation of free Euclidean distance between line coded sequences is developed. A com puter simulation is made for the baseband communication system with R=n/n+l rate TCM line encoders operating in an additive white Gaussian noise environment. When using the Viterbi decoding, the error-event probability is assumed as a performance criterion. It is assumed that an error event occurs if the Viterbi algorithm decides to a state which does not belong to the correct path. In terms of the error curves, it is shown that R=n/n+l (n=l,2,3) rate TCM line encoded systems have better error performance than the corresponding uncoded cases. For the convolutional codes, the free Hamming distance and the upper bounds on the error probabilities are determinated based on the state transition diagram of the encoder. Because of the linearity of these binary codes the all-zero sequence can be assumed as the correct code sequence. The generating function obtained from the state transition diagram provide the distance, step length - vii - and the number of bit errors of any incorrect sequence compared to the correct sequence. For the nonlinear trellis codes, these properties can not be obtained by the same simple approach. For a nonlinear encoder, the Euclidean distance between any incorrect sequence and the correct sequence is generally dependent upon the correct sequence. In other words, the all-zero sequence can not be assumed as the correct sequence without loss of generality. Then, the evaluation of. the upper bounds on error probabilities necessitates the analysis of very complex diagrams specially at high encoder state numbers. In Chapter 3, a method for computation of the upper bounds on error-event and bit-error probabilities of non linear trellis codes with the same complexity as the linear trellis codes is generalized for the new line encoders which have partially distinct and partially over lapping state-subsets. It is shown that R=l/2 and R=3/4 rate line encoders satisfy the error weight profile equivalency condition necessitated by this method. For these encoders, the modified generating functions are nese encoders, cne moaxriea generating xuncrions are btained from a signal-flow graph with state number equal _o the state number of their trellis diagram and the analytical expressions for the error upper bounds are O to Chapter 4 is concerned with the evaluation of power spectral densities for the new line codes proposed. For each R=n/n+l (n=l,2,3) rate, it is aimed at determine the code with minimum spectral components at low frequencies among codes having equal and maximum free distances. For R=l/2 rate, the encoder having minimal components at low frequencies is chosen among four encoders with equal and maximum free distance. For R=2/3 and R=3/4 rates, in spite of the fact that there are more than one encoder with maximum free distance, their power spectras are equal. The TCM line encoders are also compared with known line encoders. The bipolar encoder power spectra is the best at low frequencies. The new line encoders have smaller spectral components at these frequencies with respect to 4B3T, FOMOT, MS43 encoders. Only R=l/2 rate TCM encoder has smaller components than the PST encoder at frequencies around zero. In Chapter 5, the jitter performance of the base band transmission systems with line encoders based on trellis coded modulation method, is analysed. Usually, the analytical approach is not practical for the codes having complex statistical properties such as our case. There fore, in order to obtain the clock information, the base band transmission system including the TCM line encoders and a timing circuit is simulated by a computer program. The timing circuit is formed by a square-law device