通信原理英文版课件：Ch5 Signal-Space Analysis

1、Ch.5 Signal-Space Analysis5.1Introduction5.2Geometric Representation of Signals5.3Conversion of the Continuous AWGN Channel into a Vector Channel5.4Likelihood Functions5.5Coherent Detection of Signals in Noise: Maximum Likelihood Decoding5.6Correlation Receiver5.7Probability of Error5.8Summary and D

2、iscussion15.1 IntroductionFig. 5.1 Block diagram of a generic digital communication system.Minimizing pe Optimum receiver in the minimum probability of error sense25.2 Geometric Representationof SignalsGeometric representationTo represent any set of M energy signals as linear combinations of N ortho

3、normal basis functions, where N M.Gram-Schmidt orthogonalization procedureHow to choose the N orthonormal basis functions for M energy signals3Illustration of ConceptsFigure 5.4 Illustrating the geometric representation of signals for the case when N 2 and M 3.4Orthonormal Basis FunctionsFig. 5.3 Ge

4、ometric representation of signals.5Orthonormal Basis Functions (contd)The signal vector sisi(t) is completely determined by si , length (or “absolute value”, “norm”) of si6Orthonormal Basis Functions (contd)Euclidean distance between si and skAngle between si and sk7Example: Schwarz InequalityFor re

5、al-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.8Gram-Schmidt Orthogonalization Procedure9Example: 2B1Q CodeFig. 5.5 Signal-space representation of the 2B1Q code.M = 4 and N = 1105.3 Conversion of the Continuous

6、AWGN Channel into a Vector ChannelIn 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 the signal detection; the remainder of the noise is irrelevant.11Signal Analysis with AWGN ChannelFig.

7、5.2 The AWGN Channel.Noise element affecting signal detectionIrrelevant noise element12Statistical CharacterizationX(t), W(t) : Random Processesx(t), w(t) : Sample FunctionsXj, Wj : Random Variablesxj, wj : Sample ValuesSince Xj are Gaussian random variables, they are statistically independent.13Sta

8、tistical Characterization (contd)Observation vectorMemoryless channel14Theorem of IrrelevanceInsofar as signal detection in AWGN is concerned, only the projections of the noise onto the basis functions of the signal set affects the sufficient statistics of the detection problem; the remainder of the

9、 noise is irrelevant.The AWGN channelThe vector channel155.4 Likelihood FunctionsLikelihood function:Log-likelihood function:AWGN channel:Given the observation vector x, which message symbol mi is transmitted?165.5 Coherent Detection of Signals in Noise: Maximum Likelihood DecodingThe signal constel

10、lationThe signal detection problemThe optimal decision rules17The Signal ConstellationSet of N orthonormal basis functionsA Euclidean space of dimension N The set of message points in this space corresponding to the set of transmitted signals is called a signal constellation.The observation vector x

11、 is represented by a received signal vector in the same Euclidean space.18The Signal Constellation (contd)Fig. 5.7 Illustrating the effect of noise perturbation, depicted in (a), on the location of the received signal point, depicted in (b).19Given the observation vector x, perform a mapping from x

12、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 ProblemProbability of error when make the decision:20The Optimum Decision RulesThe maximum a posteriori probability (MAP) rule:The maximum likelihoo

13、d rule:21The Optimum Decision Rules (contd)Graphical interpretation of the maximum likelihood decision ruleDividing the observation space Z into M-decision regions Z1, Z2, , ZMThe rule is:Observation vector x lies in region Zi if l(mk) is maximum for k = i22The Optimum Decision Rules (contd)With AWG

14、N channel:Maximizing:Equals minimizing:Equals maximizing:Fig. 5.8 An illustration of the decision rule with AGWN channel.235.6 Correlation ReceiverFig. 5.9 The optimal receiver using correlators.Detector/DemodulatorSignal transmission decoder24Correlation Receiver (Contd)Therefore, the correlator eq

15、uals sampling at time T after a matched filter.25Correlation Receiver (Contd)Fig. 5.10 The optimal receiver using matched filters.Detector/DemodulatorSignal transmission decoder265.7 Probability of ErrorAverage probability of symbol errorInvariance of the probability of error to rotation and transla

16、tionMinimum energy signalsUnion bound on the probability of errorBit versus symbol error probabilities27Average Probability of Symbol ErrorZi: Region in the observation space corresponding to decision mi.28Invariance of the Probability of Error to Rotation and TranslationRotation:Translation:Distanc

17、e invariance:Error probability invarianceMaximum likelihood detectionAWGN channel29Minimum Energy SignalsEnergy of a signal constellation:Translating the signal constellation by a vector amount a:where30Union Bound on the Probability of ErrorFor AWGN channel, the symbol error probability is:whereThe

18、 above formulations are impractical to calculateUsing boundsUnion bound: one of the bounds, made by simplifying the region of integration in the top formulation31Union 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 to sk than si, when si is sent is:andPairwise error probability32Union Bound (contd)The signal constellation i