Mitra《数字信号处理-基于计算机的方法》翻译 武汉大学 孙洪ch3 randomsignal.pdf

现代数字信号处理1 武汉大学 电子信息学院研究生课程 2002 孙洪 1 现代数字信号处理现代数字信号处理 孙洪 Modern Digital Signal Modern Digital Signal ProcessingProcessing Chap 3 Fundamentals of Random Signal Chap 3 Fundamentals of Random Signal Analysis and ProcessingAnalysis and Processing 武汉大学 电子信息学院研究生课程 2002 孙洪 2 IntroductionIntroduction lRandom signal The random variabletakes value equal to with a probability The stochastic processwith probabilities and power spectral density the model of nx x nxp n nx xpx nx lnxnxEFTnxPSD 2121 xxpxx x nx 武汉大学 电子信息学院研究生课程 2002 孙洪 3 3 1 Random Variables and 3 1 Random Variables and Probability ModelsProbability Models lRandom variables The random variabletakes value equal to Discrete valued random variable if takes a discrete set of values x x xx x k x 武汉大学 电子信息学院研究生课程 2002 孙洪 4 Probabilistic ModelsProbabilistic Models lCumulative distribution function cdf lProbability density function pdf lProbability mass function pmf 3 1 random variable xxxFx Pr dx xdF xf x x Ni xxxXxX Pr 21 L 武汉大学 电子信息学院研究生课程 2002 孙洪 5 The Joint Probability Mass The Joint Probability Mass Function Function pmfpmf lThe joint pmf lThe marginal pmf lBayes rule 3 3 random signal jYjiYXiXijXYjiYX yPyxPxPxyPyxP M j jYjiYX M j jiYXiX yPyxPyxPxP 11 M i iXijXY iXijXY iXijXY jY jiYX xPxyP xPxyP xPxyP yP yxP 1 1 武汉大学 电子信息学院研究生课程 2002 孙洪 6 Statistical AveragesStatistical Averages lThe expected or mean value lMoments The mth order moments Central moments 3 1 random variable continuousxdxxxf discretexpx xE x k kk x dxxfxxEr x mmm x dxxfxxE x m x m x m x 现代数字信号处理2 武汉大学 电子信息学院研究生课程 2002 孙洪 7 lNote lThe variance lStandard deviation lThe relation between moments and central moments 3 1 random variable 22 xExE 2 2 2 var xxx xEx 2 xx 2 2 2 xxx 武汉大学 电子信息学院研究生课程 2002 孙洪 8 lCharacteristic functions FT lMoment generating function Laplace T 3 1 random variable dxexfeE xj x xj x dxexfeEs jsx x jsx x LL LL 2 2 2 2 1 2 1 m x m xx m jsx x r m s r s s m sxsx sxEeEs 武汉大学 电子信息学院研究生课程 2002 孙洪 9 Properties of the Characteristic Properties of the Characteristic FunctionFunction lPossess all the properties associated with the Fourier and Laplace transform lSpecially The characteristic function always exists since It is maximum at the origin since 3 1 random variable 1 dxxfdxxf xx 1 0 xx 0 xfx 武汉大学 电子信息学院研究生课程 2002 孙洪 10 Some Useful Random VariablesSome Useful Random Variables lUniformly distributed variable lNormal Gaussian random variable lCauchy random variable 3 1 random variable elsewhere bxa ab xfx 0 1 12 2 2 2 abba xx 2 2 2 1 exp 2 1 x x x x x xf 22 2 1 exp xxx j 22 1 x xfx j x exp 武汉大学 电子信息学院研究生课程 2002 孙洪 11 3 2 Random Vectors3 2 Random Vectors lDefinition A collection of random variables lA random M vector lDistribution and density function T M xxxX 21 L XXXF xxxxxxF X MMMX Pr Pr 111 LL 1 XFXf XxxX M L 1 dXXfX 武汉大学 电子信息学院研究生课程 2002 孙洪 12 Statistical descriptionStatistical description lMean vector lAutocorrelation and autocovariance matrices 2ndorder moments 3 2 random vectors MM X xE xE XE MM 11 H XXX H XXX RXXE MMM M H X rr rr XXER L MOM L 1 111 现代数字信号处理3 武汉大学 电子信息学院研究生课程 2002 孙洪 13 Statistical description 2 Statistical description 2 lCorrelation and covariance matrices lUncorrelated lOrthogonal 3 2 random vectors H YXXY H YXXY RYXE LMM L H XY yxEyxE yxEyxE XYER L MOM L 1 111 H YXXYXY R 0 0 XY R 武汉大学 电子信息学院研究生课程 2002 孙洪 14 Transformation of a Random VectorTransformation of a Random Vector Monotonic transformationMonotonic transformation 3 2 random vectors XfX X 1 1 Ygf Y Yg Yf XY XgY YgX 1 武汉大学 电子信息学院研究生课程 2002 孙洪 15 Linear TransformationsLinear Transformations ManyMany toto One Mapping One Mapping 3 2 random vectors A YAf Yf X Y det 1 AJ M M M M x y x y x y x y detdet 1 11 1 L MOM L XfX X J Ygf YfAXXgY X Y 1 2 1 det A YAf Yf X Y For real valued For Complex valued 武汉大学 电子信息学院研究生课程 2002 孙洪 16 Sum of Independent Random Sum of Independent Random VariablesVariables lMean lVariance lProbability density function 3 2 random vectors 21 xxy yfyfyf xxy 21 M k kkx cy 1 M k xky k c 1 M k xky k c 1 2 2 2 武汉大学 电子信息学院研究生课程 2002 孙洪 17 Central limit theoremCentral limit theorem lLet be a collection of random variables such that a are mutually independent and b have the same distribution and c the mean and variance of each random variable exist and are finite that is and for all Then the distribution of the normalized sum approaches that of a normal random variable with zero mean and unit standard deviation as 3 2 random vectors M y M k XkM xy 1 M xxx 11 L M kk x 1 kx 2 kx 2 1MkL M 武汉大学 电子信息学院研究生课程 2002 孙洪 18 3 3 Discrete3 3 Discrete time Stochastic time Stochastic Processes random signal Processes random signal lis a random variable time is fixed and value is variable lis a sample sequence value is fixed and time is variable lis a number both time and value are fixed lis a stochastic process both time is and value are variable k nx i nx ik nx k nx n n n n 现代数字信号处理4 武汉大学 电子信息学院研究生课程 2002 孙洪 19 Probability FunctionsProbability Functions lThe kth order distribution function lThe kth order pdf 3 3 random signal kkkkx xnxxnxnnxxF Pr 1111 LLL 1 1 11 11 k nxnx nnxxF nnxxf k kkx k kkx L LL LL 武汉大学 电子信息学院研究生课程 2002 孙洪 20 SecondSecond Order Statistical Order Statistical DescriptionDescription lThe mean value lThe variance lThe autocorrelation sequence lThe autocovariance sequence lThe cross correlation sequence lThe cross covariance sequence 3 3 random signal nxEn x 2 2 nnxEn xx nxnxEnnrxx 21 2121221121 nnnnrnnxnnxEnn xxxxxxxx nynxEnnrxy 21 2121221121 nnnnrnnynnxEnn yxxyyxxy 2nd order 武汉大学 电子信息学院研究生课程 2002 孙洪 21 StationarityStationarity lStationary of order N lStrict sense stationary SSS For all orders N 1 2 lWide sense stationary WSS 3 3 random signal x nxE 2 2121 xxx xxx lrl lnxnxEnxlnxElrnnrnnr 1111kNkkxNNx nnxxfnnxxf LLLL 2 var x nx 武汉大学 电子信息学院研究生课程 2002 孙洪 22 Properties of autocorrelation Properties of autocorrelation sequencessequences lFor WSS process for all lA conjugate symmetric lNonnegative definite 3 3 random signal 00 2 2 xxx r 0 11 M k M m mxk amkra lrr xx 0l lrlr xx 武汉大学 电子信息学院研究生课程 2002 孙洪 23 Ergodicity Ergodicity and Time Averagesand Time Averages lThe time average lErgodic random processes In the mean In the correlation Joint ergidicity 3 3 random signal N Nn N N12 1 lim lnxnxElnxnx nxEnx lnynxElnynx 武汉大学 电子信息学院研究生课程 2002 孙洪 24 Sample RealizationsSample Realizations 01002003004005006007008009001000 3 2 1 0 1 2 3 01002003004005006007008009001000 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1 Gaussian Uniform 3 3 random signal 现代数字信号处理5 武汉大学 电子信息学院研究生课程 2002 孙洪 25 FrequencyFrequency Domain Description of Domain Description of Stationary ProcessesStationary Processes lPower Spectral Density PSD lProperties of the autoPSD 3 3 random signal deRlrelreR lj xx l lj x j x 2 1 j x j x eReR 00 2 12 nxErdeR x j x 0 j x eR 武汉大学 电子信息学院研究生课程 2002 孙洪 26 FrequencyFrequency Domain Description of Domain Description of Stationary Processes 2 Stationary Processes 2 lA second order White Noise Process lCross power Spectral Density lThe normalized cross spectrum 3 3 random signal llnwnwElrnwE wwx 2 2 ww WN 2 w j we R deRlrelreR lj xyxy l lj xy j xy 2 1 j yx j xy eReR j y j x j xyj xy eReR eR eC 武汉大学 电子信息学院研究生课程 2002 孙洪 27 3 4 Linear Systems with Stationary 3 4 Linear Systems with Stationary Random InputsRandom Inputs lTime Domain Analysis Output mean value Input output cross correlation Output autocorrelation 0HkhknxEkh x k x k y lrlhlrlrlhlr xxyxxxxy lrlrlrlhlr xxhxyyy lhlhlrh 武汉大学 电子信息学院研究生课程 2002 孙洪 28 An equivalent LTI system for An equivalent LTI system for autocorrelation filtrationautocorrelation filtration lrx lry lrxy lrh lh lh 3 4 processing of random signal 武汉大学 电子信息学院研究生课程 2002 孙洪 29 Output probability density Output probability density functionfunction lA Gaussian process a gaussian process lIf x n is IID the probability