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
中的分组码。 A vector of K input W/2 complex bits symbols P/S To QAM Constellation modulator Fig5.1.5 Another way is to extend the dimensionality of the transmitted signal by basing it on a achine (M)Theedd the PM mpnhrent increase in the number of points in the FSM k+r selecto Information Subset label bit sequence (e Coded signal m-k Signal point sequence selector Mapper function x=f(c) Fig.5.1.6 5.1.2 Overview ofCoded Modulation Techniques The existing coded modulation schemes for band-limited AWGN channels can be broadly classified into four categories: 1)Lattice codes 2)Trellis/TCM codes(trellis-coded modulation) TCM was proposed by Ungerboeck is usuallyused as the underlying FSM.The term trellis-coded modulation originates from the fact that these coded sequences consist of modulated symbols rather than binary digits.In other words,in TCM schemes,the trellis branches are labeled with redundant nonbinary modulated symbols rather than with binary coded symbols tcm codes can be decoded by naximu m-likelihood de sing Viterbi alg Trellis codes are to lattices as binary convolutionl codes are to block codes 3)Turbo-TCM 4)Multilevel codes(also known as BCM) Multilevel codes was proposed by H.Imai in 1977.The underlying strategy is to protect 5-6
5-6 中的分组码。 N-dimensional Constellation P/S K input bits A vector of N/2 complex symbols To QAM modulator Fig. 5.1.5 Another way is to extend the dimensionality of the transmitted signal by basing it on a finite-state machine (FSM). The extra bits produced by the FSM implies an inherent increase in the number of points in the constellation. FSM Constellation selector Signal point selector k k+r m-k Mapper function x=f(c) Coded signal sequence {xi } m bits per symbol Information bit sequence Subset label sequence {ci } Fig. 5.1.6 5.1.2 Overview of Coded Modulation Techniques The existing coded modulation schemes for band-limited AWGN channels can be broadly classified into four categories: 1) Lattice codes 2) Trellis/TCM codes (trellis-coded modulation) TCM was proposed by Ungerboeck in 1982, in which a convolutional code is usually used as the underlying FSM. The term trellis-coded modulation originates from the fact that these coded sequences consist of modulated symbols rather than binary digits. In other words, in TCM schemes, the trellis branches are labeled with redundant nonbinary modulated symbols rather than with binary coded symbols. TCM codes can be decoded by a maximum-likelihood decoder using Viterbi algorithm. Trellis codes are to lattices as binary convolutional codes are to block codes. 3) Turbo-TCM 4) Multilevel codes (also known as BCM) Multilevel codes was proposed by H. Imai in 1977. The underlying strategy is to protect
each label bit of the signal point by an individual binary code,so mutiple encoders (at different levels)are employed.At the receiver,the received sequence of signal points are sually decoded by a multis age decoder In multilevel coding(MLC)schemes,any code,e.g,block codes,convolutional codes,or concantenated codes.can be used as component codes.Since in early MLC schemes the FEC codes used were usually block codes,the MLC scheme is also referred to as block coded modulation (BCM) 5)Bit-interleaved coded modulation(BICM)(with iterative decoding) BICM was first proposed by Zehavi in199 for coding for fading channels,in which the output stream of a binary encoder is bit-interleaved and then mapped to an M-ary constellation.Its basic idea is to increase the code diversity.(For Rayleigh fading channels, the code performance depends strongly its minimum Hamming distance rather than the minimum Euclidean distance.)The information-theoretic aspects of BICM have been analyzed by Caire Recently,it has been recognized that the BICM based on turbo-like codes and iterative decoding provides an effective realization method for Gallager's coding theorem(proposed in 1968 for discrete memoryless channels).With this scheme,very good performance can be achieved on both awgn and fading channels Fig.5.1.7 depicts the performances of some of typical coded modulation schemes 7 M-QAM bound V33 64-QAM <4M4.37 32.0AM ·e0(4,3 + 16QAM 404,36) BPSK 43为 TCW32.7刀Mc OPSK P=10 40 12 14 16 20 E/N (dB) spectral efficiencies achieved by 5.2 Coding Gain and Shaping Gain In coded modulations,we can use the following approaches for improving signal 5.7
5-7 each label bit of the signal point by an individual binary code, so mutiple encoders (at different levels) are employed. At the receiver, the received sequence of signal points are usually decoded by a multistage decoder. In multilevel coding (MLC) schemes, any code, e.g., block codes, convolutional codes, or concantenated codes, can be used as component codes. Since in early MLC schemes the FEC codes used were usually block codes, the MLC scheme is also referred to as block coded modulation (BCM). 5) Bit-interleaved coded modulation (BICM) (with iterative decoding) BICM was first proposed by Zehavi in 1992 for coding for fading channels, in which the output stream of a binary encoder is bit-interleaved and then mapped to an M-ary constellation. Its basic idea is to increase the code diversity. (For Rayleigh fading channels, the code performance depends strongly its minimum Hamming distance rather than the minimum Euclidean distance.) The information-theoretic aspects of BICM have been analyzed by Caire. Recently, it has been recognized that the BICM based on turbo-like codes and iterative decoding provides an effective realization method for Gallager’s coding theorem (proposed in 1968 for discrete memoryless channels). With this scheme, very good performance can be achieved on both AWGN and fading channels. Fig. 5.1.7 depicts the performances of some of typical coded modulation schemes. Figure 5.1.7: Theoretical limits on spectral and power efficiency for different signal constellations and spectral efficiencies achieved by various coded and uncoded transmission methods. 5.2 Coding Gain and Shaping Gain In coded modulations, we can use the following approaches for improving signal
constellation designs ■The first idea he agonal (Alternatively,we could keep the variance constant,in which case the hexagonal constellation would have a larger minimum distance than the squared constellation.)This decrease in power for the same minimum distance or increase in minimum distance for the same power through changing the relative spacing of the points is called codinggain The 2nd approach is to change the shape or outline of the constellation without changing the relative positioning of points.The circular constellation will have a lower variance than the squared constellation.On the same grounds.a circular constellation will have the lowest variance of any shaping region for a square grid of points.The resulting reduction in power is called shaping gain Significantly,shaping onstellation changes the marginal density of the data symbol This isillustrated in Fig.5.9 Coding and shaping gain can be combined.e.g.by changing the points in the circularly shaped constellation to a hexagonal grid while retaining the circular shaping. Usually,channel coding deals with the internal arrangement of the points,whereas The egions idea is to employ multidimensional signal constellation.A data symbol drawn from an N-dim constellation is transmitted once every N2 signaling interval.When we design a 2D constellation,and choose the N/2 successive symbols to be an arbitrary sequence of 2D symbols drawn from that constellation,the resulting N-dim constellation is the N2-fold Cartesian product of 2D constellations.From Chapter2,we know that the nance of this N-dim tellation is the san as the und erlying2D。 ation llation that is not constrained to have this Cartesian product structure.When N>2,it is called a multidimensional signal constellation. Greater shaping and coding gains can be achieved with a multidimensional signal onstellation than with a 2D constellation.How eve multidim ensional constellations from a complexity that increases exponentially with dimensionality Square Hexagonal Circular Fig 5.8 Three 2D constellations with the same minimum distance and 256 points 5-8
5-8 constellation designs. The first idea is to change the relative spacing of points in the constellation. The hexagonal constellation leads to the reduced variance with the same minimum distance. (Alternatively, we could keep the variance constant, in which case the hexagonal constellation would have a larger minimum distance than the squared constellation.) This decrease in power for the same minimum distance or increase in minimum distance for the same power through changing the relative spacing of the points is called coding gain. The 2nd approach is to change the shape or outline of the constellation without changing the relative positioning of points. The circular constellation will have a lower variance than the squared constellation. On the same grounds, a circular constellation will have the lowest variance of any shaping region for a square grid of points. The resulting reduction in power is called shaping gain. Significantly, shaping the constellation changes the marginal density of the data symbol. This is illustrated in Fig. 5.9. Coding and shaping gain can be combined, e.g., by changing the points in the circularly shaped constellation to a hexagonal grid while retaining the circular shaping. Usually, channel coding deals with the internal arrangement of the points, whereas shaping treats regions. The 3rd idea is to employ multidimensional signal constellation. A data symbol drawn from an N-dim constellation is transmitted once every N/2 signaling interval. When we design a 2D constellation, and choose the N/2 successive symbols to be an arbitrary sequence of 2D symbols drawn from that constellation, the resulting N-dim constellation is the N/2-fold Cartesian product of 2D constellations. From Chapter 2, we know that the performance of this N-dim constellation is the same as the underlying 2D constellation. An alternative is to design an N-dim signal constellation that is not constrained to have this Cartesian product structure. When N>2, it is called a multidimensional signal constellation. Greater shaping and coding gains can be achieved with a multidimensional signal constellation than with a 2D constellation. However, multidimensional constellations suffer from a complexity that increases exponentially with dimensionality. Square Hexagonal Circular Fig.5.8 Three 2D constellations with the same minimum distance and 256 points
Square Circular Fig.59The one-dimensional probability distribution for an unshaped and shaped 2D constellation 5.2.1 Ultimate Shaping Gain Here,we just provide a preliminary discussion on the maximum shaping gain,we will discuss this in detail in the next section.Without loss of generality,the derivations are done for one-dimensional constellations ystem is ass med to use auniform distribution of signal points.The shaped ystemshou(disrete) both distributions have to have the same entropy. When considering constellations with a large number of signal points,it is convenient to approximate the distribution by a continuous probability density function(pdf).Hence we ompare a continuous uniform pdf with a Gaussian one. Let Ebe the verage energy of the reference system.Then the differential entropy of its transmitted symbolsx is given by MX)=g:2c)=g:2E) If x-N(0,E)(Gaussian distributed with average energy E).its entropy is equal to h(X)=log2(2πeE) Since the above entropies should be equal,we have E2-形=1.53dB 6 The quantity is called the ultimate shaping gain,which is achieved for a continuous Gassian pdf. 5.9
5-9 Square Circular Fig. 5.9 The one-dimensional probability distribution for an unshaped and shaped 2D constellation. 5.2.1 Ultimate Shaping Gain Here, we just provide a preliminary discussion on the maximum shaping gain; we will discuss this in detail in the next section. Without loss of generality, the derivations are done for one-dimensional constellations. The baseline system is assumed to use a uniform distribution of signal points. The shaped system should exhibit a (discrete) Gaussian distribution. In order to transmit at the same rate, both distributions have to have the same entropy. When considering constellations with a large number of signal points, it is convenient to approximate the distribution by a continuous probability density function (pdf). Hence we compare a continuous uniform pdf with a Gaussian one. Let Eu be the average energy of the reference system. Then the differential entropy of its transmitted symbols x is given by ( ) ( ) 2 2 2 1 1 ( ) log 12 log 12 2 2 x u hX E = = σ If ( ) 0, g x ∼ N E (Gaussian distributed with average energy Eg), its entropy is equal to 2 ( ) 1 ( ) log 2 2 g h X eE = π Since the above entropies should be equal, we have , 1.53 dB 6 u s g E e E π γ ∞ ≡=≈ The quantity s, γ ∞ is called the ultimate shaping gain, which is achieved for a continuous Gassian pdf
5.3 Lattice Constellations It is from Shannon's capacity theorem that an optimal block code for a bandwidth-limited AWGN channel consists of a dense packing of code points within a sphere in a high-dimensional Euclidean space.Most of the densest known packings are lattices. An n-dimensional (n-D)lattice A is an infinite discrete set of points(vectors,n-tuples)in the real Euclidean n-space R"that has the group property. Example 5.3.1:The set of all integers,=,is a one-dimensional lattice,since is a discrete subgroup of R.The set 22 of all integer-valued two-tuples(m,n)with n Z is a 2-dimensional lattice.More generally,the set Z"of all integer-valued n-tuples is an n-D lattice. The lattice RZ.whereR1 1-1A is obtained by rotating Z2 by /4 and scaling by √5.Clearly,R2z2=2z Definition I:Let g1sjsm be a set of linearly independent vectors in R"(so that msn).The set of points ==2,e2 (5.1) is called an m-dimensional lattice,and gsmis called a basis of the lattice. That is,A是基向量的整数线性组合。The matrix withg,as rows g G Lg. is called a generator matrix for the lattice.在后续讨论中，we will deal with full-rank lattices. i.e.,m=n.So a general n-D lattice that spans R"may be expressed as A=x=aGaeZ" (5.1) 例如，the lattice2 has the generator-0] 10 A coset of a lattice A,denoted by A+x,is a set of all points obtained by adding a fixed point x to all lattice points aA.Geometrically,the coset A+x is a translate of A by x.If 文10
5-10 5.3 Lattice Constellations It is known from Shannon’s capacity theorem that an optimal block code for a bandwidth-limited AWGN channel consists of a dense packing of code points within a sphere in a high-dimensional Euclidean space. Most of the densest known packings are lattices. An n-dimensional (n-D) lattice Λ is an infinite discrete set of points (vectors, n-tuples) in the real Euclidean n-space Rn that has the group property. Example 5.3.1: The set of all integers, {0, 1, 2,.} Z = ± ± , is a one-dimensional lattice, since Z is a discrete subgroup of R. The set Z2 of all integer-valued two-tuples (n1, n2) with i n ∈Z is a 2-dimensional lattice. More generally, the set Zn of all integer-valued n-tuples is an n-D lattice. The lattice RZ2 , where 1 1 1 1 R ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ − , is obtained by rotating Z2 by π/4 and scaling by 2 . Clearly, R2 Z2 = 2Z2 . Definition 1: Let { ,1 } j g ≤ ≤j m be a set of linearly independent vectors in Rn (so that m n ≤ ). The set of points 1 m jj j j a a = ⎧ ⎫ Λ= = ∈ ⎨ ⎬ ⎩ ⎭ x g ∑ Z (5.1) is called an m-dimensional lattice, and { ,1 } j g ≤ j m≤ is called a basis of the lattice. That is, Λ是基向量的整数线性组合。The matrix with gj as rows 1 2 m G ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ g g g # is called a generator matrix for the lattice. 在后续讨论中，we will deal with full-rank lattices, i.e., m=n. So a general n-D lattice that spans Rn may be expressed as { }n Λ= = ∈ xaa G Z (5.1) 例如，the lattice Z2 has the generator 1 0 0 1 G ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . A coset of a lattice Λ, denoted by Λ+x, is a set of all points obtained by adding a fixed point x to all lattice points a∈Λ. Geometrically, the coset Λ+x is a translate of Λ by x. If