0122229现代数字通信与编码理论 November 21,2011 XDU,Winter 2011 Lecture Notes Chapter 5 Coded-Modulation for Band-Limited AWGN Channels We now introduce the bandwidth-efficient coded-modulation techniques for ideal AWGN channels. The idea of combined coding and modulation design was first suggested by J.L Massey in 1974,and then realized with stunning results by Ungerboeck and Imai.The common core is to optimize the code in Euclidean space. On band-limited channels,nonbinary signal alphabets such as M-PAM must be used.The M-ary signaling and the potential coding gain in the bandwidth-limited regime have been discussed in Chapterfrom the information-theoretic point of view 5.1编码调制的基本原理 Traditionally,coding and modulation have been considered as two separate parts of a simple modul ator,at th rec the received wa eform is first de odulated,and the error correction code is decoded.In this scenario,the modulator and demodulator are usually devised to convert a waveform channel into a discrete channel,and the error correction encoder/decoder are designed,based on maximizing the minimum Hamming distance,to correct the errors that occurred in the discrete channel.Higher improvement in performance is ved by l a the 最近，随者数 速率的日益提高 要求通信系统具有较高的频谱利用率。为了在提 高系统功率效率的同时，不宽展系统所占用的带宽，人们提出了编码调制技术。Wh coded modulation schemes,significant coding gains(so the BER performance improvement) can be achieved without increasing bandwidth(or sacrificing bandwidth efficiency) 编码调制遵循下面两个基本原理： 通过扩展信号星月 (即增加调制信号集中的信号个数)而不是通过增加系统的带宽 来提供编码所要求的信号元余。 Example 5.1:Consider the situation where a stream of data is to be transmitted with throughput of 2 bits/s/Hz over an AWGN channel.One possible solution is to use an uncoded syste em.As a coded solutior mploy a rate-2/3 convolutional code with QPSK,i.e.,2 bits/s/Hz;moreover,both schemes require the same bandwidth. 5-1
5-1 0122229 现代数字通信与编码理论 November 21, 2011 XDU, Winter 2011 Lecture Notes Chapter 5 Coded-Modulation for Band-Limited AWGN Channels We now introduce the bandwidth-efficient coded-modulation techniques for ideal AWGN channels. The idea of combined coding and modulation design was first suggested by J. L. Massey in 1974, and then realized with stunning results by Ungerboeck and Imai. The common core is to optimize the code in Euclidean space. On band-limited channels, nonbinary signal alphabets such as M-PAM must be used. The M-ary signaling and the potential coding gain in the bandwidth-limited regime have been discussed in Chapter 2 from the information-theoretic point of view. 5.1 编码调制的基本原理 Traditionally, coding and modulation have been considered as two separate parts of a digital communication system. At the transmitter, an error-correcting encoder is followed by a simple modulator; at the receiver, the received waveform is first demodulated, and then the error correction code is decoded. In this scenario, the modulator and demodulator are usually devised to convert a waveform channel into a discrete channel, and the error correction encoder/decoder are designed, based on maximizing the minimum Hamming distance, to correct the errors that occurred in the discrete channel. Higher improvement in performance is achieved by lowering the code rate at a cost of bandwidth expansion. 最近，随着数据速率的日益提高，要求通信系统具有较高的频谱利用率。为了在提 高系统功率效率的同时，不宽展系统所占用的带宽，人们提出了编码调制技术。With coded modulation schemes, significant coding gains (so the BER performance improvement) can be achieved without increasing bandwidth (or sacrificing bandwidth efficiency). 编码调制遵循下面两个基本原理： 通过扩展信号星座（即增加调制信号集中的信号个数）而不是通过增加系统的带宽 来提供编码所要求的信号冗余。 Example 5.1：Consider the situation where a stream of data is to be transmitted with throughput of 2 bits/s/Hz over an AWGN channel. One possible solution is to use an uncoded QPSK system. As a coded solution, we may employ a rate-2/3 convolutional code with an 8-PSK signal set. Note that this coded 8-PSK scheme yields the same throughput as uncoded QPSK, i.e., 2 bits/s/Hz; moreover, both schemes require the same bandwidth
QPSK 调制器 FEC 8-PSK 编码器 调制器 Figure5.1.1 从第二章中的调制信号星座的容量分析可知，信号星座点个数增加一倍所提供的冗 余己足够实现在不增加系统带宽的条件下，逼近容量限的性能。 再进 一步扩展星座，所 得到的性能增益将很少。因此，在通常的编码调制系统中采用的是码率为kk+1)的信道 码。 ■将编码与调制作为一个整体进行联合优化设计。 Although the expansion of a signal set (e.g.,from QPSK to 8-PSK)provides the redundancy required for coding.it shrinks the distance betweer the ignal points if the average signal energy is kept constant.This reduc ction in distance should be compensated by coding advantage if the coded scheme is to provide a benefit. 如果是按照传统方法，简单地在一个纠错编码器后级联一个M元调制器，而纠错编码器 是基于汉明距离准则进行设计，则所得到的结果往往会令人失望。 ,The use of hard-decision demodulation prior to the decoding in a coded scheme causesalosofSNRToavoidsuchahard-decsionlos,itism aryt女 soft-outpu detector decoder directly on the soft-output samples of the channel.The decision rule of the optimum decoder depend on the Euclidean distance. Example5.2:对于上例中的) For the coded 8-PSK scheme above.if we choose the rate-2/3 convolutional code of Fig.5.1.2(a)which is designed based on maximizing free Hamming distance,and the mapping of 3 output bits of the convolutional encoder to the 8-PSK signal points is done as shown in Fig.5.1.2(b).then we can find that the minimum Euclidean distance between a pair of paths forming an error event(which is sometimes called free Euclidean distance)is d=0∑d,0 =d(xo)+d(xo.x)+d(x2x) =△+0+=1.172E
5-2 Figure 5.1.1 从第二章中的调制信号星座的容量分析可知，信号星座点个数增加一倍所提供的冗 余已足够实现在不增加系统带宽的条件下，逼近容量限的性能。再进一步扩展星座，所 得到的性能增益将很少。因此，在通常的编码调制系统中采用的是码率为k/(k+1)的信道 码。 将编码与调制作为一个整体进行联合优化设计。 因为 Although the expansion of a signal set (e.g., from QPSK to 8-PSK) provides the redundancy required for coding, it shrinks the distance between the signal points if the average signal energy is kept constant. This reduction in distance should be compensated by coding advantage if the coded scheme is to provide a benefit. 如果是按照传统方法，简单地在一个纠错编码器后级联一个M元调制器，而纠错编码器 是基于汉明距离准则进行设计，则所得到的结果往往会令人失望。 另外，The use of hard-decision demodulation prior to the decoding in a coded scheme causes a loss of SNR. To avoid such a hard-decision loss, it is necessary to employ soft-output detector. TCM integrates demodulation and decoding in a single step and decoder operates directly on the soft-output samples of the channel. The decision rule of the optimum decoder depend on the Euclidean distance. Example 5.2：（对于上例中的） For the coded 8-PSK scheme above, if we choose the rate-2/3 convolutional code of Fig.5.1.2(a) which is designed based on maximizing free Hamming distance, and the mapping of 3 output bits of the convolutional encoder to the 8-PSK signal points is done as shown in Fig.5.1.2(b), then we can find that the minimum Euclidean distance between a pair of paths forming an error event (which is sometimes called free Euclidean distance) is 2 2 { }{ } min ( , ) i i free E i i x x i d d xx ≠ ′ = ∑ ′ 222 07 00 21 2 2 0 0 (,) (,) (,) 0 1.172 EEE s d xx d xx d xx E =++ =Δ + +Δ =
c2) 8-PSK c=[ccc] (a)Encoder structure 0 E 101 110 8-PSK (b)Signal mapping rule and the trellis diagram of the coded scheme Fig.5.1.2 A rate-2/3 convolutional coded 8-PSK scheme To compare the coded and uncoded schemes it is common to use the coding gain parameter,which is defined as the difference in SNR for an objective target bit error rate between a coded system and an uncoded system. coding gainSNR-SNR At high SNR,this gain is termed the asymptotic coding gain (ACG)and is expressed as d1E,) y=10log (diE. dB For the coded scheme above.-23 dB.This result shows the 2 performance degradation of the coded scheme (optimized based on the free Hamming distance)compared to the uncoded one Massey pointed out that it was necessary to integrate the design of encoder and modulator. and to treat the code and modulation scheme as an entirety,as shown in Fig.5.1.Thus, 整体方案就应该基于maximizing the minimum Euclidean distance between coded signal 5-3
5-3 T T 8-PSK Signal (1) Set i c (2) i c (3) i c (1) i a (2) i a xi (3) (2) (1) [ ] iii c = ccc (a) Encoder structure 8-PSK Δ0 Es Δ1 Δ2 Δ3 (000) (001) (010) (011) (100) (101) (110) (111) 4 0 0 6 2 6 2 5 1 0 4 3 7 1 5 7 3 1 0 2 S0 S1 S2 S3 (b) Signal mapping rule and the trellis diagram of the coded scheme Fig. 5.1.2 A rate-2/3 convolutional coded 8-PSK scheme To compare the coded and uncoded schemes it is common to use the coding gain parameter, which is defined as the difference in SNR for an objective target bit error rate between a coded system and an uncoded system. uncoded coded coding gain | | SNR SNR − At high SNR, this gain is termed the asymptotic coding gain (ACG) and is expressed as ( ) ( ) 2 10 2 ,min / 10log / free s coded E s uncoded d E d E γ = dB For the coded scheme above, 10 1.172 10log 2.3 dB 2 γ = =− . This result shows the performance degradation of the coded scheme (optimized based on the free Hamming distance) compared to the uncoded one. Massey pointed out that it was necessary to integrate the design of encoder and modulator, and to treat the code and modulation scheme as an entirety, as shown in Fig. 5.1. Thus, 系统 整体方案就应该基于maximizing the minimum Euclidean distance between coded signal
hat is best viewed in -space context.所以编码调制也称为信号空间第码。 In TCM schemes,the code and an expanded signal set are jointly designed as a physical unit.The design criterion is to maximize the free Euclidean distance between coded signal sequences rather than Hamming distance.The resulting code can provide a significant coding gain and the loss from the expansion of the signal set can be overcome. Example 5.3: 我们从编码调制的角度，考虑图5.1.1中的编码器与调制器的联合设计。 As an alternative coded scheme we may use the 8-PSk TCM scheme shown in Fig 5 1 3 which was introduced by Ungerboeck.We will see in Section 5.5 that this TCM scheme can provide an asymptotic coding gain of=3(dB). a c) 8-PSK T e=Icoccm] Fig.5.1.3 The 4-state TCM encoder for 8-PSK The performances of various TCM schemes are shown in Fig.5.1.4.It is seen from Fig. 5.1.4 that the improvement of coding is evident.Note that the coding schemes shown in Figure5 achieves the codingins withour requring more bandwidth than the unc uncoded 54
5-4 sequences rather than Hamming distance来设计. More recently, it has been recognized that the design of coded modulation schemes for the AWGN channel is a problem that is best viewed in the geometric signal-space context. 所以编码调制也称为信号空间编码。 In TCM schemes, the code and an expanded signal set are jointly designed as a physical unit. The design criterion is to maximize the free Euclidean distance between coded signal sequences rather than Hamming distance. The resulting code can provide a significant coding gain and the loss from the expansion of the signal set can be overcome. Example 5.3: 我们从编码调制的角度，考虑图 5.1.1 中的编码器与调制器的联合设计。 As an alternative coded scheme, we may use the 8-PSK TCM scheme shown in Fig. 5.1.3, which was introduced by Ungerboeck. We will see in Section 5.5 that this TCM scheme can provide an asymptotic coding gain of γ = 3 (dB). T T 8-PSK Signal (1) Set i c (2) i c (3) i c (1) i a (2) i a xi (3) (2) (1) [ ] iii c = ccc Fig. 5.1.3 The 4-state TCM encoder for 8-PSK The performances of various TCM schemes are shown in Fig. 5.1.4. It is seen from Fig. 5.1.4 that the improvement of coding is evident. Note that the coding schemes shown in Figure 5.1.4 achieves the coding gains without requiring more bandwidth than the uncoded QPSK system
10 10 TCM QPSK te 10 TCM Zero error range SNR[dB] Figure 5.1.4:Bit error probability of Quadrature Phase-Shift Keying (QPSK)and selected 8-PSK coded modulation (TCM).rellis-tubo coded (TTCM).and boc turbo coded(BTC)systems as a 5.1.1两种基本实现方法 Similar to the case of binary codes,we introduce interdependences between consecutive signal points in order to increas e the distance b etween the clo stpa nces of sign points.A perspective from signal-space coding may provide more insight into codec modulation schemes.In order to obtain large coding gain,the codes should be designed in a subspace of signal space with high dimensionality,where a larger minimum distance in relation to signal power can be obtained.The dimensionality 2BTo can be increased for fixed bandwidth B by in g the time interval To,making it multiple symbol intervals. For moderate coding gain at moderate complexity, Iwo ba ways to generate modulation (or signal-space)codes in conjunction with passband QAM modulation are as follows: ■直接来自于几何考虑：A sequence of N/2two-dimensional transmitted symbols can be considered as a single point in an N-dimensional constellation.Each element of the constellation alphabet(called a codeword)isa vector inN(or).Aset of K input bits are used to select one of2 codewords in the multidimensional constellation.A typical example of the multidimensional constellation is the lattice code.它类似于二进制编码 5.5
5-5 Figure 5.1.4: Bit error probability of Quadrature Phase-Shift Keying (QPSK) and selected 8-PSK trellis-coded modulation (TCM), trellis-turbo coded (TTCM), and block turbo coded (BTC) systems as a function of the normalized signal-to-noise ratio. 5.1.1 两种基本实现方法 Similar to the case of binary codes, we introduce interdependences between consecutive signal points in order to increase the distance between the closest pair of sequences of signal points. A perspective from signal-space coding may provide more insight into coded modulation schemes. In order to obtain large coding gain, the codes should be designed in a subspace of signal space with high dimensionality, where a larger minimum distance in relation to signal power can be obtained. The dimensionality 2BT0 can be increased for fixed bandwidth B by increasing the time interval T0, making it multiple symbol intervals. For moderate coding gain at moderate complexity, Two basic ways to generate modulation (or signal-space) codes in conjunction with passband QAM modulation are as follows: 直接来自于几何考虑：A sequence of N/2 two-dimensional transmitted symbols can be considered as a single point in an N-dimensional constellation. Each element of the constellation alphabet (called a codeword) is a vector in RN (or CN/2). A set of K input bits are used to select one of 2K codewords in the multidimensional constellation. A typical example of the multidimensional constellation is the lattice code. 它类似于二进制编码