1、1,Ch.5 Signal-Space Analysis,5.1Introduction 5.2Geometric Representation of Signals 5.3Conversion of the Continuous AWGN Channel into a Vector Channel 5.4Likelihood Functions 5.5Coherent Detection of Signals in Noise: Maximum Likelihood Decoding 5.6Correlation Receiver 5.7Probability of Error 5.8Sum

2、mary and Discussion,2,5.1 Introduction,Fig. 5.1 Block diagram of a generic digital communication system.,Minimizing pe Optimum receiver in the minimum probability of error sense,3,5.2 Geometric Representationof Signals,Geometric representation To represent any set of M energy signals as linear combi

3、nations of N orthonormal basis functions, where N M. Gram-Schmidt orthogonalization procedure How to choose the N orthonormal basis functions for M energy signals,4,Basic Representations,Orthonormal,5,Illustration of Concepts,Figure 5.4 Illustrating the geometric representation of signals for the ca

4、se when N 2 and M 3.,6,Orthonormal Basis Functions,Fig. 5.3 Geometric representation of signals.,7,Orthonormal Basis Functions (Contd),The signal vector si si(t) is completely determined by si , length (or “absolute value”, “norm”) of si,8,Orthonormal Basis Functions (Contd),Euclidean distance betwe

5、en si and sk,9,Example: Schwarz Inequality,For real-valued signals:,For complex-valued signals:,For either case, the equality holds if and only if s2(t) = cs1(t), where c is any constant.,10,Example: Schwarz Inequality (Contd),11,Gram-Schmidt Orthogonalization Procedure,12,Example: 2B1Q Code,Fig. 5.

6、5 Signal-space representation of the 2B1Q code.,M = 4 and N = 1,13,5.3 Conversion of the Continuous AWGN Channel into a Vector Channel,In this section, we show that: In an AWGN channel, only the projections of the noise onto the basis functions of the signal set affect the sufficient statistics of t

7、he signal detection; the remainder of the noise is irrelevant.,14,Signal Analysis with AWGN Channel,Fig. 5.2 The AWGN Channel.,Noise element affecting signal detection,Irrelevant noise element,15,Statistical Characterization,X(t), W(t) : Random Processes x(t), w(t) : Sample Functions Xj, Wj : Random

8、 Variables xj, wj : Sample Values,Since Xj are Gaussian random variables, they are statistically independent.,16,Statistical Characterization (Contd),Observation vector,Memoryless channel,17,Theorem of Irrelevance,Insofar as signal detection in AWGN is concerned, only the projections of the noise on

9、to the basis functions of the signal set affects the sufficient statistics of the detection problem; the remainder of the noise is irrelevant.,18,5.4 Likelihood Functions,Likelihood function:,Log-likelihood function:,AWGN channel:,Given the observation vector x, which message symbol mi is transmitte

10、d,?,19,5.5 Coherent Detection of Signals in Noise: Maximum Likelihood Decoding,The signal constellation The signal detection problem The optimal decision rules,20,The Signal Constellation,Set of N orthonormal basis functions,A Euclidean space of dimension N,The set of message points in this space co

11、rresponding to the set of transmitted signals is called a signal constellation.,The observation vector x is represented by a received signal vector in the same Euclidean space.,21,The Signal Constellation (Contd),Fig. 5.7 Illustrating the effect of noise perturbation, depicted in (a), on the locatio

12、n of the received signal point, depicted in (b).,22,Given the observation vector x, perform a mapping from x to an estimate of the transmitted symbol, mi, in a way that would minimize the probability of error in the decision-making process.,The Signal Detection Problem,Probability of error when make

13、 the decision:,23,The Optimum Decision Rules,The maximum a posteriori probability (MAP) rule:,The maximum likelihood rule:,24,The Optimum Decision Rules (Contd),Graphical interpretation of the maximum likelihood decision rule Dividing the observation space Z into M-decision regions Z1, Z2, , ZM The

14、rule is: Observation vector x lies in region Zi if l(mk) is maximum for k = i,25,The Optimum Decision Rules (Contd),With AWGN channel:,Maximizing:,Equals minimizing:,Equals maximizing:,Fig. 5.8 An illustration of the decision rule with AGWN channel.,26,5.6 Correlation Receiver,Fig. 5.9 The optimal r

15、eceiver using correlators.,Detector/Demodulator,Signal transmission decoder,27,Correlation Receiver (Contd),Therefore, the correlator equals sampling at time T after a matched filter.,28,Correlation Receiver (Contd),Fig. 5.10 The optimal receiver using matched filters.,Detector/Demodulator,Signal tr

16、ansmission decoder,29,5.7 Probability of Error,Average probability of symbol error Invariance of the probability of error to rotation and translation Minimum energy signals Union bound on the probability of error Bit versus symbol error probabilities,30,Average Probability of Symbol Error,31,Invaria

17、nce of the Probability of Error to Rotation and Translation,Rotation:,Translation:,Distance invariance:,Error probability invariance,Maximum likelihood detection AWGN channel,32,Rotational Invariance,Figure 5.11 A pair of signal constellations for illustrating the principle of rotational invariance.

18、,33,Minimum Energy Signals,Energy of a signal constellation:,Translating the signal constellation by a vector amount a:,where,34,Minimum Energy Signals (Contd),Figure 5.12 A pair of signal constellations for illustrating the principle of translational invariance.,35,Union Bound on the Probability of

19、 Error,For AWGN channel, the symbol error probability is:,where,The above formulations are impractical to calculate,Using bounds,Union bound: one of the bounds, made by simplifying the region of integration in the top formulation,36,Union Bound (Contd),Fig. 5.13 Illustrating the union bound. (a) Constellation of four message points. (b) Three constellations with a common message point and one other message point retained from the original constellation.,The probability of x is closer t