Probability Models (779819), страница 4

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For a wide-sense stationary process thecross-covariance function of Equation (3.57) becomesc xy ( m1 , m2 ) = c xy ( m1 − m2 ) = rxy ( m1 − m2 ) − µ x µ y(3.58)Example 3.8 Time-delay estimation. Consider two signals y1(m) andy2(m), each composed of an information bearing signal x(m) and an additivenoise, given byy1 (m)= x(m)+ n1 (m)(3.59)y2 (m)= A x(m − D) + n2 (m)(3.60)where A is an amplitude factor and D is a time delay variable. The crosscorrelation of the signals y1(m) and y2(m) yieldsrxy(m)DCorrelation lag mFigure 3.7 The peak of the cross-correlation of two delayed signals can be used toestimate the time delay D.64Probability Modelsry1 y 2 ( k ) = E [ y1 ( m ) y 2 ( m + k )]= E {[x ( m )+ n1 ( m ) ][ Ax ( m − D + k )+ n2 ( m + k ) ] }(3.61)= Arxx ( k − D ) + rxn 2 ( k ) + Arxn1 ( k − D ) + rn1 n 2 ( k )Assuming that the signal and noise are uncorrelated, we haver y1 y2 (k ) = Arxx (k − D). As shown in Figure 3.7, the cross-correlationfunction has its maximum at the lag D.3.4.7 Cross-Power Spectral Density and CoherenceThe cross-power spectral density of two random processes X(m) and Y(m) isdefined as the Fourier transform of their cross-correlation function:PXY ( f ) = E [ X ( f )Y * ( f )]=∞∑ rxy ( m ) e − j 2πfm(3.62)m =−∞Like the cross-correlation the cross-power spectral density of two processesis a measure of the similarity, or coherence, of their power spectra.

Thecoherence, or spectral coherence, of two random processes is a normalisedform of the cross-power spectral density, defined asC XY ( f ) =PXY ( f )PXX ( f ) PYY ( f )(3.63)The coherence function is used in applications such as time-delay estimationand signal-to-noise ratio measurements.3.4.8 Ergodic Processes and Time-Averaged StatisticsIn many signal processing problems, there is only a single realisation of arandom process from which its statistical parameters, such as the mean, thecorrelation and the power spectrum can be estimated. In such cases, timeaveraged statistics, obtained from averages along the time dimension of asingle realisation of the process, are used instead of the “true” ensembleaverages obtained across the space of different realisations of the process.65Expected Values of a Random ProcessThis section considers ergodic random processes for which time-averagescan be used instead of ensemble averages.

A stationary stochastic process issaid to be ergodic if it exhibits the same statistical characteristics along thetime dimension of a single realisation as across the space (or ensemble) ofdifferent realisations of the process. Over a very long time, a singlerealisation of an ergodic process takes on all the values, the characteristicsand the configurations exhibited across the entire space of the process. Foran ergodic process {x(m,s)}, we havestatistical averages[ x ( m, s )] = statistical averages[ x (m, s )]along time m(3.64)across space swhere the statistical averages[.] function refers to any statistical operationsuch as the mean, the variance, the power spectrum, etc.3.4.9 Mean-Ergodic ProcessesThe time-averaged estimate of the mean of a signal x(m) obtained from Nsamples is given by1 N −1µˆ X = ∑ x(m)(3.65)N m =0A stationary process is said to be mean-ergodic if the time-averaged value ofan infinitely long realisation of the process is the same as the ensemblemean taken across the space of the process.

Therefore, for a mean-ergodicprocess, we havelim E [ µˆ X ] = µ XN →∞lim var [ µˆ X ] = 0N →∞(3.66)(3.67)where µX is the “true” ensemble average of the process. Condition (3.67) isalso referred to as mean-ergodicity in the mean square error (or minimumvariance of error) sense. The time-averaged estimate of the mean of a signal,obtained from a random realisation of the process, is itself a randomvariable, with is own mean, variance and probability density function. If thenumber of observation samples N is relatively large then, from the centrallimit theorem the probability density function of the estimate µˆ X isGaussian. The expectation of µˆ X is given by66Probability Models1NE [ µˆ x ] = E N −1 1∑ x ( m)  = Nm =0N −1∑ E [ x(m)] =m =01NN −1∑ µx = µx(3.68)m =0From Equation (3.68), the time-averaged estimate of the mean is unbiased.The variance of µˆ X is given byVar[ µˆ x ] = E[ µˆ x2 ] − E 2[ µˆ x ](3.69)= E[ µˆ x2 ] − µ x2Now the term E [µˆ 2x ] in Equation (3.69) may be expressed as 1 NE[ µˆ x2 ] = E 1=NN −1 1∑ x(m)   Nm=0N −1k =0∑ x(k ) N −1|m|∑ 1− N  rxx (m)m = − ( N −1)(3.70)Substitution of Equation (3.70) in Equation (3.69) yieldsVar[ µˆ x2 ] =1N1=NN −11− | m |  r (m) − µ 2 xxxN m= − ( N −1) ∑N −1|m|∑ 1− N  c xx (m)m = − ( N −1)(3.71)Therefore the condition for a process to be mean-ergodic, in the meansquare error sense, is1 N −1  | m | (3.72)lim∑ 1− N  c xx (m) = 0N →∞ N m= − ( N −1) 3.4.10 Correlation-Ergodic ProcessesThe time-averaged estimate of the autocorrelation of a random process,estimated from N samples of a realisation of the process, is given by67Expected Values of a Random Processrˆxx (m) =1 N −1∑ x ( k ) x( k + m )N k=0(3.73)A process is correlation-ergodic, in the mean square error sense, iflim E [rˆxx (m)] = rxx (m)(3.74)lim Var[rˆ xx (m)] = 0(3.75)N →∞N →∞where rxx(m) is the ensemble-averaged autocorrelation.

Taking theexpectation of rˆxx (m) shows that it is an unbiased estimate, sinceN −1 1x ( k ) x ( k + m)  =∑ N k =0 NE[rˆxx (m)] = E  1N −1∑ E [ x(k ) x(k + m)] = rxx (m)k =0(3.76)The variance of rˆxx (m) is given by22Var[rˆxx (m)] = E [rˆxx(m)]− rxx( m)(3.77)2The term E [rˆxx(m)] in Equation (3.77) may be expressed asE[rˆxx2 (m)] = 12N==1N21NN −1 N −1∑ ∑ E [ x(k ) x(k + m) x( j ) x( j + m)]k =0 j =0N −1 N −1∑ ∑ E [ z (k , m) z ( j, m)](3.78)k =0 j =0N −11 − | k |  r ( k , m ) zzN k = − N +1 ∑where z(i,m)=x(i)x(i+m). Therefore the condition for correlation ergodicityin the mean square error sense is given by1lim N →∞  NN −1 1 − | k |  r ( k , m ) − r 2 ( m)  = 0 zzxxN k =− N +1 ∑(3.79)68Probability Models3.5 Some Useful Classes of Random ProcessesIn this section, we consider some important classes of random processesextensively used in signal processing applications for the modelling ofsignals and noise.3.5.1 Gaussian (Normal) ProcessThe Gaussian process, also called the normal process, is perhaps the mostwidely applied of all probability models.

Some advantages of Gaussianprobability models are the following:(a) Gaussian pdfs can model the distribution of many processesincluding some important classes of signals and noise.(b) Non-Gaussian processes can be approximated by a weightedcombination (i.e. a mixture) of a number of Gaussian pdfs ofappropriate means and variances.(c) Optimal estimation methods based on Gaussian models often resultin linear and mathematically tractable solutions.(d) The sum of many independent random processes has a Gaussiandistribution. This is known as the central limit theorem.A scalar Gaussian random variable is described by the following probabilitydensity function:f X ( x) = (x − µ x )2 1exp  −2π σ x2σ x2 (3.80)where µ x and σ 2x are the mean and the variance of the random variable x.The Gaussian process of Equation (3.80) is also denoted by N (x, µ x , σ 2x ).The maximum of a Gaussian pdf occurs at the mean µ x , and is given byf X ( µ x )=12π σ x(3.81)69Some Useful Classes of Random Processesf(x)F(x)12π σ x1.00.8410.6072π σx0.5µx-σxµxµx+σxxµx-σxµxµx+σxxFigure 3.8 Gaussian probability density and cumulative density functions.From Equation (3.80), the Gaussian pdf of x decreases exponentially withthe increasing distance of x from the mean value µ x .

The distributionfunction F(x) is given by (χ − µ x )2exp∫  − 2σ 22π σ x −∞x1FX ( x ) =x dχ(3.82)Figure 3.8 shows the pdf and the cdf of a Gaussian model.3.5.2 Multivariate Gaussian ProcessMultivariate densities model vector-valued processes. Consider a P-variateGaussian vector process {x=[x(m0), x(m1), . . ., x(mP–1)]T} with mean vectorµx, and covariance matrix Σxx.

The multivariate Gaussian pdf of x is givenbyf X ( x) =1(2π )P/2Σ xx1/ 21−1exp − ( x − µ x ) T Σ xx( x − µ x ) 2where the mean vector µx is defined as(3.83)70Probability Models E [ x(m0 )]  E[ x(m2 )] µx = E[ x(m P −1 )](3.84)and the covariance matrix Σ xx is given byc xx (m0 , m1 ) c xx (m0 , m0 )c (m , m )c xx (m1 , m1 )Σ xx =  xx 1 0 c xx (m P −1 , m0 ) c xx (m P −1 , m1 ) c xx (m0 , m P −1 )  c xx (m1 , m P −1 )  c xx (m P −1 , m P −1 ) (3.85)The Gaussian process of Equation (3.83) is also denoted by N (x, µx, Σxx). Ifthe elements of a vector process are uncorrelated then the covariance matrixis a diagonal matrix with zeros in the off-diagonal elements.

In this case themultivariate pdf may be described as the product of the pdfs of theindividual elements of the vector:()P −1f X x = [ x(m0 ) , , x(m P –1 )]T = ∏i =0 [ x(mi ) − µ xi ] 2 1exp −22π σ xi2σ xi(3.86)Example 3.9 Conditional multivariate Gaussian probability densityfunction. Consider two vector realisations x(m) and y(m+k) from twovector-valued correlated stationary Gaussian processes N (x, µ x , Σ xx ) andN (y, µ y , Σ yy ). The joint probability density function of x(m) and y(m+k) isa multivariate Gaussian density N ([x(m),y(m+k)], µ(x,y), Σ(x,y)), with meanvector and covariance matrix given byµ x µ ( x, y) =  µ y Σ xxΣ ( x, y) = Σ yx(3.87)Σ xy Σ yy (3.88)71Some Useful Classes of Random ProcessesThe conditional density of x(m) given y(m+k) is given from Bayes’ rule asf X Y ( x ( m) y ( m + k ) ) =f X,Y ( x (m), y (m + k ) )(3.89)fY ( y (m + k ) )It can be shown that the conditional density is also a multivariate Gaussianwith its mean vector and covariance matrix given byµ ( x y ) = E [x ( m) y ( m + k ) ](3.90)−1( y − µ y)= µ x + Σ xy Σ yyΣ( x y) =−1Σ xx − Σ xy Σ yyΣ yx(3.91)3.5.3 Mixture Gaussian ProcessProbability density functions of many processes, such as speech, are nonGaussian.

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