Sürekli faz modülasyonunun çok düzeyli kodlanması
Multilevel coding of continuous phase modulation
- Tez No: 100736
- Danışmanlar: PROF.DR. ÜMİT AYGÖLÜ
- Tez Türü: Doktora
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Kod çözme, Kodlama, Sürekli faz modülasyonu, Decoding, Coding, Continuous phase modulation
- Yıl: 1999
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Sürekli faz modülasyonu, sabit zarf, iyi band ve güç verimliliği özellikleri ne deniyle, gezgin radyo ve uydu iletişimi gibi band ve güç sınırlı ortamlar için oldukça uygun bir modülasyon türüdür. Üstelik dış kodlama işlemi yardımıyla güç verimliliği daha da artınlabilmektedir. Çok düzeyli kodlama/ çok aşamalı kodçozme, modülasyonlu işaretin birden fazla kodlayıcı tarafından belirlendiği ve alıcı kısımda da çok aşamalı olarak çözüldüğü bir kanal kodlama tekniğidir. Bu teknik, kodlama kazancı yüksek (hata olasılığı düşük) ve alıcı donanımı basit sistem tasarımlarına olanak tanımaktadır. Bu çalışmada, literatürde bulunan yöntemlere yeni bir alternatif olarak sü rekli faz modülasyonunun çok düzeyli olarak kodlanması ve çok aşamalı olarak çözülmesi önerilmektedir. Bu amaçla, sürekli faz modülasyonunun en çok bilinen biçimi olan, frekans darbe yanıtının dikdörtgen olarak seçildiği CPFSK modü lasyonu için, çok düzeyli kodlama/çok aşamalı kodçozme tekniğine dayalı olarak AWGN ve sönümlemeli kanallar için yeni sistemler tasarlanmıştır. Sistemler, J ve P ortak bölenleri olmayan iki tamsayı olmak üzere, modülasyon indisi J/P olan M bilgi genlik düzeyli CPFSK modülasyonu için M > 2P koşulu altında elde edilmiştir. Tasarlanan sistemlerin hata başarım incelemeleri, analitik bit hata olasılığı üst sınırları ve bilgisayar benzetimleri yardımıyla yapılmıştır. Bu amaçla, çok düzeyli sistemlerin AWGN ve sönümlemeli kanallardaki bit hata olasılığı üst sınır larının hesaplanabilmesine olanak tanıyan yeni bir yöntem önerilmiştir. Böylece, üst sınırlardan, bilgisayar benzetiminden ve ayrıca kodçözücülerin donanım kar maşıklığının bir ölçütü olan“kodçozme karmaşıklığı”tanımından yararlanarak, yeni sistemlerin, literatürde daha önce var olan kodlara kıyasla hata başarımı ve kodçozme karmaşıklığı üstünlükleri sağladıkları gösterilmiştir. Ayrıca, literatür de daha önce tek düzeyli durumda belirtilenlerin aksine, kodlama ile sürekli faz modülasyonunun kolayca birleştirilebileceği ve değişik bit/simge oranlarında kod tasarımının kolayca gerçekleştirilebileceği gösterilmiştir. xıv
Özet (Çeviri)
MULTILEVEL CODING OF CONTINUOUS PHASE MODULATION SUMMARY Because of their attractive bandwidth efficiency properties, continuous phase modulation (CPM) schemes have gained considerable interest in modern digi tal telecommunications. The phase continuity of CPM signals improves spectral properties, and introduces memory which induces error control capability and therefore power efficiency. In a channel with a nonlinear transmitter power am plifier, constant envelope signals must be used. Even though some modulation techniques without memory (e.g. PSK) have constant envelope, the use of the transmitter filter destroys constant envelope property. Since CPM has excellent spectral properties without the need for filtering of the transmitted signal, it is well suited choice for such cases. The power efficiency of CPM can be improved by combining coding with CPM. In recent years, mainly three different optimiza tion techniques for coded CPM have been proposed for additive white gaussian noise (AWGN) channels. In the first technique, the external encoder and CPM modulator are optimized for the predetermined CPM parameters, the rate and the constraint length of the encoder, by looking for the code achieving the maximum free Euclidean dis tance (d/B) which is the key parameter in error performance evaluation at high signal-to-noise ratio for AWGN channels. In the second method, called matched encoding, the combined coding and CPM schemes maximize djE for a given to tal number of trellis states. In the third technique, CPM is decomposed into a time invariant continuous phase encoder (CPE) and a time invariant memoryless modulator (MM). This decomposition allows the direct combination of memory parts of the CPM modulator and the external convolutional encoder. By using this method, the overall encoder can be designed to obtain maximum df£. All of the optimization methods cited above are partly based on trellis coded modula tion (TCM) which is a well known power/bandwidth efficient coded modulation technique. Another power/bandwidth efficient coded modulation technique is the multi level coding/multistage decoding which leads to large coding gain and low decod ing complexity at the expense of an increase in the decoding delay. The multilevel coding scheme employs, at each signalling interval, one or more output bits of each of several binary error-control encoders to construct the signal to be trans mitted. The Hamming distances of component binary codes are linearly related to the Euclidean distances between partitioned signal subsets. An important advantage of the multilevel coding is the possibility of suboptimum multistage decoding of each code with decoded information transferred from one stage to xvthe next. This permits the decoding complexity to be reduced at each stage and therefore for the overall system. In coding theory, although the most frequently assumed channel model is AWGN channel, fading channel is also another important channel model for many communication systems such as mobile satellite, HF radio and meteor burst links. Due to the strictly band-limiting condition caused by the growing demand of the users, good spectral properties have further importance besides power efficiency properties for this kind of channels. Therefore, CPM is very suitable for fading channels. As mentioned, the design criterion for the TCM technique on the AWGN channel is the maximization of d/jg. However, the codes optimized for the AWGN channel are not generally optimum for fading channels for which the effective code length (ECL) and the squared product distance (d%) are the most significant criteria instead of djE. ECL is defined as the number of nonzero pairwise distances between the symbols along the shortest erroneous path and those along the correct path and d% is defined as the product of the nonzero squared pairwise distances between the symbols along the path pair with length ECL. In the literature, there is not any research on coding of CPM for fading channels except for a few papers. So, this is an open area for coding theory. In this thesis, multilevel coding of continuous phase frequency shift keying (CPFSK) which is the rectangular frequency pulse shaped case of CPM is pro posed in order to obtain systems having high coding gain and low decoding com plexity for AWGN and/or fading channels. The proposed systems are obtained under the constraint M > IP for M-ary CPFSK with modulation index h = J/P, where J and P are relatively prime positive integers. It is also shown that com bining multilevel coding with CPFSK provides the design simplicity. For fading channels, it is assumed that the fading effect on the phase of the received signal is compensated for by some technique, e.g. phase-locked loop. So, the fading only effects the amplitude of the received signal. It is also considered the case of independent fading of adjacent symbols (ideal fading) under ideal interleaver/deinterleaver conditions and coherent detection with ideal channel state information (CSI) where the fading amplitude is determined at the receiver. The transmitted M-ary CPFSK signal having“tilted phase”over the interval nT < t < (n + 1)T can be written as «(0 = ^coa[2xfxt + Khd^r^ + 9n] (1) where E is the symbol energy, T is the symbol duration, fc is the carrier fre quency and f1 = fc - (M - l)h/2T is the modified carrier frequency. an E {±1,±3,...,±(M- 1)} is the nth information symbol and fin G {0, 1,...,M- 1} is the modified nth information symbol which is defined as Şn - (an + (M - 1))/2. 0n denotes the memory term of the modulation and satisfies 0n+l = (0n + ^h/3n)mod2-n. (2) Then, the phase trellis corresponding to this equation is time invariant and can be shown as in Fig. 1 where M > IP. Note that, for h = J/P, 0n 6 {0, (2Trh)mo<i2iT, (47r/i)m0(f27r)...} with P different phase values for all n. M branches xvioriginate from each state and there are M/P parallel branches between the states. s(t) can be one of the MP different signals which are represented as {s0, âı,..., sjv/p-i} in a modulation interval. {5.-si'---sM-i} ?^ branches o Mbr. i / 0 °n (KJ (* - p >r> °n.) ^ 1st slate 0 (2TCh>n>od2TT <K X> (2TCh)mod27T 2nd State Pth state Fig. 1. M-ary CPFSK phase trellis (h = J/P, M > 2P). M-ary CPFSK signal set So can be divided into Q-level nonoverlapping subsets to form a partition chain So/ Si/... /Sq,where 5,- denotes the ith level partitioned set (i = 0,1,..., Q), with minimum subset distances A0 < Ax <... < Aq-i, respectively. Here, A,_i denotes the minimum Euclidean distance between two subsets produced by the partition 5,_i/5,- (i = 1,2,...,Q). If the partitioning is performed as dividing the (sub)sets by two at each level, excluding the last one, in that case Q equals to log2(2M/P). At the last level, each subsets of Sq-i have P2 signals and these signals and the phases can be represented by a P-state trellis diagram without parallel transitions. Each of these subsets can be considered as trellis of the last level's combined (inherent) continuous phase encoder (CPEq) and the memoryless mapper structure (MMq). The coding rate of CPEq is \ogiP /2\ogiP. Each of the M/P subsets of Sq-x are equivalent to a P-ary CPFSK signal set with modulation index h = J/P in terms of internal Euclidean distances and phase states. For example, for h = 1/2, each of the subsets of Sq-i is equivalent to 2-ary CPFSK, h=l/2 (fast frequency shift keying (FFSK) or minimum shift keying (MSK)) signal set. Such a partitioning leads to design simplicity for coded M=4 or higher level CPFSK modulation schemes. Note that the phase continuity of the overall M-ary CPFSK system is satisfied at the last level. xvnk, bits 5 R1 = p _ k2bits=5 2“ * n k2/n2 !* ÜL -Q-1 kQ-1 bİtS ”0-1 Wb-i = - ov yQ- kQ bits Coded P-aryCPFSK,h = J/P (CQ-CPEQ) Vq Vq.1 rtog2 MP at c“5. Q. o in I/) >. o E 2 s(t.J3> Fig. 2. Multilevel coded M-ary CPFSK system (Q = log2(2M/P)). The proposed multilevel coded system structure having Q coding levels is shown in Fig. 2. To each partitioning level 5,_i/5,-,i = 1,2,...,Q a binary convolutional code C, is associated with free Hamming distance djjji and rate Ri = ki/rn. In order to achieve high coding gain and low decoding complexity, convolutional codes which maximize the performance criteria for a given code rate and having the suitable constraint length for a given Hamming distance are used. To reduce further the decoding complexity punctured convolutional codes are also used. At the last level, P-ary CPFSK, h = JjP signal set can be coded by a convolutional encoder (Cq) with rate Rq = kç/nç. The combination of Cq and CPEq (Cq + CPEq) is represented as ”coded P-ary CPFSK, h = J/P“ in the figure. For the last level, previously given P-ary CPFSK, h = J/P trellis codes can be used. In Fig. 2, MM represents memoryless mapping rule of the overall system. If /i; denotes the number of the ith encoder output bits per channel symbol, it can be easily shown that /*. ' 1, » = 1,2,...Q-1 log2P, i - Q (3) for the proposed system. Rs denoting the multilevel system rate in bits/symbol is also given by xvinQ R. = Y,Ril*i. (4) «=ı For AWGN channels, the squared free Euclidean distance (djE) of the overall system is given by d2fF< = min [rf2/r.,] = 1TŞ_l[*l-ld/HUdJBQ] (5) where dfEi is the squared free Euclidean distance of the ith level trellis code. dfEQ is obtained from coded P-ary CPFSK, h = J/P trellis. The normalized squared free Euclidean distance (<Pjen) xa defined as <Pjen - d2jE/2Eb where E\> is the energy per information bit. The equation E = RaEb can be easily derived. For fading channels, the effective code length (ECL) of the overall system is given by ECL = min [ECLA 1<»'<Q = x$t§Jdi»»ECL0\ (6) where ECL{ is the effective code length of the zth level trellis code. ECLq is obtained from coded P-ary CPFSK, h = J/P trellis. For i = 1,2,...,Q - 1, squared product distance (d£,-) is equal to (A;_1)d'*n. For the last level, d^q is obtained from coded P-ary CPFSK, h = J/P trellis. The squared product distance of the overall system is equal to the squared product distance of the coding level which gives ECL. At the receiver, the multistage decoding technique is used. In the multistage decoded structure, there are Q decoders and the decoder of the ith level, decodes ith level code (i = 1, 2,..., Q). At the last decoding level, the decoder uses trellis of the combined Cq + CPEq structure. Each decoder uses output information from the previous level decoders as a side information. If the Viterbi Algoritm is used at each decoder, the decoding complexity per received signal can be defined as the number of comparisons per received signal. It can be shown that for a binary convolutional code C,- with rate JR,- = fcf/n,-, the decoding complexity is equal to £t = 2”i(2*<-l)/*,/n,- (7) where 2“' denotes the number of states of the ith level trellis code. When d is a punctured convolutional code with rate fcj/n,-, the decoding complexity is calculated from Z,,- = 2^,771,-. (8) The sum of L,'s for i = 1, 2,..., Q gives the total decoding complexity, L. xixTable 1. Some examples of two-level systems for AWGN channels. (4-ary CPFSK, /i=l/2) As an example, 5 new systems are represented in Table 1. These systems are designed for 4-ary CPFSK, h=l/2 modulation on AWGN channels. In Table 1, Vi denotes the number of memory elements of the ith level trellis code. First level codes (Cx) are classical convolutional codes. Second level codes are FFSK trellis codes taken from the Morales- Moreno and Pasupathy's paper. For example, Sys. - 1 achieves djEN of 4.71 at a decoding complexity of 1.86 and bits/symbol rate of 11/14. Vx for Sys. - 2 is twice that of Sys. - 1. Therefore, Sys. - 2 outperforms Sys. - 1 in error performance point of view, even though <PjEN values are the same. In the literature, for 4-ary CPFSK, h=l/2, the best trellis codes are designed by N.Naraghi-Pour using matched encoding method. These codes are taken as reference codes in this work. For example, Sys. - 1 achieves 10/o(7(4. 71/2.67) = 2.46 dB of asymptotic coding gain (ACG) with respect to the reference code with i?s=2/3, ^^^=2.67 and L=2. It has also bit rate and decoding complexity advantages. Sys. - 2 has the same advantages except for the decoding complexity. Sys. - 3 and Sys. - 4 have 1.76 dB ACG with respect to another reference code having i?s=2/3, <PjEN=Ç> and L=S at a larger bits/symbol rate. Similarly, Sys. - 5 has ACG and decoding complexity advantages over the corresponding reference codes. In this thesis, 60 new systems with various bits/symbol rates are proposed for 4-ary CPFSK, h=l/2 on AWGN channels. The new systems have larger ACGs and lower decoding complexities with respect to the best reference codes. 20 new systems are proposed for 8-ary CPFSK, h=l/2 and 16-ary CPFSK, h=l/2. Similarly, 20 systems are proposed for 8-ary CPFSK, h=l/4 and 16-ary CPFSK, h=l/4 and 5 new sytems are proposed for 16-ary CPFSK, /i=l/8. The new systems have ACG and decoding complexity advantages over their counterparts given in the literature. The ACG is a fair performance measure only at high signal to noise ratio values. A more general performance measure for the trellis codes are the upper bounds on the error probabilities derived by the transfer function approach. The xxperformance in terms of bit error probability has been evaluated by Zehavi-Wolf approach which can be applied to a broad class of trellis codes with some symme try properties, namely, ”uniform error properties“ (UEP). For a multilevel system, the bit error probability was firstly analized by Kofman, Zehavi and Shamai for 8-PSK modulation. In this thesis, a new error probability upper bound between the partitioned subsets is proposed. By using this approach, for AWGN channels, the bit error probability of the decoder at the ith level (i = 1,2,..., Q) is upper bounded by **hQ[\ J exP\ ae>\t t 2EN0 i rK AENo ' dl (9) 7=1 where Q(x) = (l/V2~ir)fTexp(-t2/2yit and Ti(W^\l) is the transfer function of the ith level code. W^ is the bit error probability upper bound between the ith level subsets which can be calculated from w{<) = W^exp(~AlJ*No) (10) where A^ denotes the total number of signals with distance A,_i between each of the partitioned subset elements. Note that, the ith level decoder is related to a total number of MP2_,+1 signals. For the last level (i = Q), Ti(W{i),I) can be obtained from the error state diagram of the Cq -f CPEq trellis. However, for the last level, W^ is equal to exp(-E/4;No). I is an abstract parameter whose power corresponds to the number of information bits which differ in the binary information sequence, associated with the all-zero transition and the particular state transition under consideration. yV0 is the one-sided white noise spectral density. Similar to that generally done in the literature, it is assumed that the side information carried by the preceeding decoders is always correct. The overall bit error probability upper bound is given by ft-^ipr (11) By using the above method, the bit error probability upper bound curves of some of the new systems are obtained and it is confirmed that the new systems outperform the reference codes especially at high signal to noise ratio values. In order to verify the analytical results, in the presence of AWGN, computer simulations are also done for some systems. At the receiver side, the decision depth for the Viterbi Algorithm is chosen to be equal to 10 signaling intervals. It is seen from the simulation curves that, for example, at a bit error probability of Pj, = 10-5, the coding gain of Sys. - 2 given in Table 1, over the corresponding reference code is equal to 1.25 dB. Furthermore, by using 50*56 block interleaver at the first coding level, this gain is improved to 1.4 dB. Sys - 5 achieves approx imately 0.7 dB coding gain over the corresponding reference codes. When 50*60 interleaver is used at the first level, this gain becomes about 1 dB. Similar coding gains are obtained for other simulated systems. xxiTable 2. Some examples of two-level systems for fading channels. (4-ary CPFSK, /i=l/2) For fading channels, 5 new systems are shown as an example, in Table 2. These systems are designed for 4-ary CPFSK, /i=l/2 and the same as in Table 1. Actually, there is an important advantage since these systems can be used on fading and AWGN channels with the best error performance. As seen from this Table, Sys. - 1 achieves ECL=4, d% = (ME4 at a decoding complexity of 1.86 and bits/symbol rate of 11/14. No coding scheme has been reported in the literature, for M-ary CPFSK on Rayleigh and Rician fading channels. Therefore, the new systems are compared with the reference codes designed by Naraghi- Pour for AWGN channels. Sys. - 1 and Sys. - 2 have larger ECL and d% values with respect to the reference code with R”=2/3, ECL=3, d% = 16E3, L-1 and Sys. - 1 has decoding complexity advantage. Sys. - 3 and Sys. - 4 have ECL=8 and d^ = 4096ü?8 values which provides very good error performance for fading channels. 60 new systems are designed for 4-ary CPFSK, h=l/2 and 25 new systems are designed for other considered modulations. These systems have ECL, d% and L value advantages with respect to the corresponding reference codes. The ECL and d% are fair performance measures only at high signal to noise ratio values. Therefore, the bit error probability upper bounds of the new systems are obtained by using the transfer function approach. In this thesis, a new error probability upper bound between the partitioned subsets is proposed. For fading channels, the bit error probability of the decoder at the ith level (i = 1, 2,..., Q) is upper bounded by Pbi< 1 dTijWM,!) kt dl (12) /=i where W(') is the bit error probability upper bound between the partitioned subsets for fading channels which can be given as xxnV? A? where 7 = E/ANq. In this equation, if represents the ratio of the power in direct and specular component to that in the diffuse component. When there is no direct and specular component (#=0), the Rayleigh fading which corresponds to a severly fading, occurs. K=10 is Rician fading case and it is the typical value of the mobile satellite channel. For the last level (i = Q), Ü7U) _ 1 + K e-Kfr/(i+KHi) 1 + K + ti (14) where £ is the squared Euclidean distance between the signal elements normalized to E. The overall bit error probability upper bound is given by Equation (11). By using this technique, bit error probability upper bound curves of the new systems are obtained for K=0 (Rayleigh) and K=10 (Rician) cases. From this curves, it is shown that the proposed systems have error performance advantage with respect to the reference codes. For fading channels, in order to verify the analytical results, computer sim ulations are also performed for some systems. The simulation results show that the new systems outperform significantly the reference codes. For example, when K=0y at a probability of Pj, = 10~4, Sys. - 1 and Sys - 2 achieve 2.5 dB and 4 dB coding gains, respectively over the reference code. When K=10, these gains are 0.7 dB and 1.4 dB. Sys. - 4 has approximately 6 dB of coding gain over the reference code at Pj, = 10-4 for Rayleigh channels. For Rician channels, this gain is approximately 1.8 dB. Similarly, Sys. - 5 achieves coding gains up to 4.5 dB and 1.5 dB for Rayleigh and Rician channels, respectively. The other new systems have similar coding gains with respect to the reference codes. At the last part, performances of the new systems designed for ideal fading channels are analized in the presence of correlative fading. In such case, inter- leaver/deinterleaver pairs have to be used at all levels of the system. At the last level, interleaver is replaced between external convolutional encoder (Cq) and continuous phase encoder (CPEq) of the P-ary CPFSK, h = J/ P. There fore, at the last decoding level, combined trellis structure can not be used and a separation of P-ary CPFSK, h = J/P demodulator and trellis decoder Cq is needed. This separation degrades the error performance of the system. However, the degradation is only at the last level, so, it does not effect the overall system's performance very much. Furthermore, if Cq is designed so that to be matched to CPEq, this effect becomes lowered. The analytical bit error probability upper bounds of some new systems are obtained and it is shown that, as expected, there is a moderate degradation in the performance. Although the new systems have bits/symbol rate, error performance and de coding complexity advantages, their decoding delays are large due to the multi stage decoding technique and (in some case) the interleaver/deinterleaver pairs. Therefore, the proposed systems are not very suitable for digital speech applica tions. However, the new systems are well suited to mobile satellite applications, digital audio, video broadcasting and data packet transmission. xxmFor future researches, some recommendations are as follows: The multilevel systems for M-ary CPFSK, h = J/P which satisfy M < P condition can be inves tigated. Partial response CPM and multi-h CPM can be coded by using multilevel coding technique. For fading channels, phase noise effects should be investigated. Noncoherent detection and no channel state information cases can also be con sidered. In order to further improve error performance, iterative decoders should be used at each decoding level. Furthermore, turbo encoder/decoder pairs can be used at each level. The last techniques further increase decoding delay. Therefore, the techniques which minimize the decoding delay can be developed. xxiv
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