Random processes spectral characteristics

Jntuh ptsp unit 4

1. Random Process Spectral Characteristics Dr. Vijaykumar R. Urkude Professor , VMTW

2. Content Introduction Power Density Spectrum Average Power of the Random Process Properties of the Power Density Spectrum Cross Power Density Spectrum Average Cross Power Properties of Cross Power Density Spectrum Spectral Characteristic of System Response Problems

3. Introduction This unit explores the important concept of characterizing random processes in the frequency domain. These characteristics are called spectral characteristics. All the concepts in this unit can be easily learnt from the theory of Fourier transforms. Consider a random process X (t). The amplitude of the random process, when it varies randomly with time, does not satisfy Dirichlet’s conditions. Therefore it is not possible to apply the Fourier transform directly on the random process for a frequency domain analysis. Thus the autocorrelation function of a WSS random process is used to study spectral characteristics such as power density spectrum or power spectral density ( psd ).

4. Power Density Spectrum The power spectrum of a WSS random process X (t) is defined as the Fourier transform of the autocorrelation function R XX (τ) of X(t). It can be expressed as We can obtain the autocorrelation function from the power spectral density by taking the inverse Fourier transform i.e. Therefore, the power density spectrum S XX (ω) and the autocorrelation function R XX (τ) are Fourier transform pairs.

5. Power spectral density can also be defined as Where X T (ω) is a Fourier transform of X(t) in interval [-T,T]

6. Average Power of the Random Process The average power P XX of a WSS random process X(t) is defined as the time average of its second order moment or autocorrelation function at τ =0. We know that from the power density spectrum, At τ =0 Therefore average power of X(t) is

7. Properties of power density spectrum S XX ( ω ) ≥ 0 The power spectral density at zero frequency is equal to the area under the curve of the autocorrelation R XX ( τ ). i.e The power density spectrum of a real process X(t) is an even function i.e. S XX (-ω)= S XX (ω) S XX (ω) is always a real function

8. If S XX (ω) is a psd of the WSS random process X(t), then (or) The time average of the mean square value of a WSS random process equals the area under the curve of the power spectral density. If X(t) is a WSS random process with psd S XX (ω), then the psd of the derivative of X(t) is equal to ω 2 times the psd S XX (ω). that is The power density spectrum and the time average of the autocorrelation function form a Fourier transform pair (also known as Wiener- Khintchine relation)

9. Cross Power Density Spectrum Definition 1 Consider two real random processes X(t) and Y(t). which are jointly WSS random processes, then the cross power density spectrum is defined as the Fourier transform of the cross correlation function of X(t) and Y(t).and is expressed as by inverse Fourier transformation, we can obtain the cross correlation functions as Therefore the cross psd and cross correlation functions forms a Fourier transform pair.

10. Definition 2 If X T (ω) and Y T (ω) are Fourier transforms of X(t) and Y(t) respectively in interval [-T,T], Then the cross power density spectrum is defined as

11. Average Cross Power The average cross power P XY of the WSS random processes X(t) and Y(t) is defined as the cross correlation function at τ =0. That is Also

12. Properties of Cross Power Density Spectrum S XY ( ω ) = S YX (- ω ) = S YX * ( ω ) The real part of S XY (ω) and real part S YX (ω) are even functions of ω, i.e. Re [S XY (ω)] and Re [S YX (ω)] are even functions. The imaginary part of S XY (ω) and imaginary part S YX (ω) are odd functions of ω, i.e. Im [S XY (ω)] and Im [S YX (ω)] are odd functions. If X(t) and Y(t) are Orthogonal then S XY (ω) = 0 and S YX (ω) = 0 If X(t) and Y(t) are uncorrelated and have constant mean values, then S XY ( ω ) = FT{A[R XY ( t,t + τ )]} and S YX ( ω ) = FT{A[R YX ( t,t + τ )]}. Also if X(t) and Y(t) are jointly WSS random processes, then S XY ( ω ) = FT[R XY ( τ )] and S YX ( ω ) = FT[R YX ( τ )]

13. Problems The psd of X(t) is given by Find the autocorrelation function A random process has autocorrelation function Find psd and sketch plots Find the autocorrelation function and power spectral density of the random process X(t) = A cos ( ω t + θ ), where θ is a random variable over the ensemble and is uniformly distributed over the range (0,2 π )

14. Problems The autocorrelation function of a WSS random process is R XX ( τ ) = a exp(-( τ /b) 2 ). Find the power spectral density and normalized average power of the signal. Two independent stationary random processes X(t) and Y(t) have psds S XX ( ω ) and S YY ( ω ) as given below respectively with zero mean. Let another random process U(t) = X(t) + Y(t). Find ( i ) psd of U(t), (ii) S XY ( ω ) and (iii) S XU ( ω ). Determine the psd of a WSS random process whose autocorrelation function is R XX ( τ ) = Ke -K| τ | . A stationary random process X(t) has autocorrelation function R XX ( τ ) = 10 + 5 cos (2 τ ) + 10e -2| τ | . Find the dc, ac and average power of X(t).

15. Problems The spectral density of a WSS random process X(t) is given by Find the autocorrelation and average power of the process. The psd of a WSS random process is a) What are the frequencies in X(t) b) Find men, variance and average power of X(t).

16. Spectral Characteristics of System Response Consider that the random process X(t) is a WSS process with the auto correlation function RXX( τ ) applied through an LTI system. The output process Y(t) is also a WSS and the processes X(t) and Y(t) are jointly WSS. One can obtain the spectral characteristics of output response Y(t) by applying Fourier transform to a correlation function.

17. Power Density Spectrum of a Response Consider that a random process X(t) is applied on an LTI system having a transfer function H( ω ) If the power spectrum of the input process is S XX ( ω ), then the power spectrum of a output response is S YY ( ω ) = |H( ω )| 2 S XX ( ω ) The average power in the system response is Similarly the cross power spectral density function of input and output is S XY ( ω ) = S XX ( ω )H( ω ) S YX ( ω ) = S XX ( ω )H(- ω )

18. Problems A random process X(t)is applied to a network with impulse response h(t) = e - bt u (t), where b > 0 is constant. The cross correlation X(t) with the output Y(t) is known to have the form R XY ( τ ) = u( τ ) τ e -b τ . Find the autocorrelation of response of the network. A random process X(t) whose mean value is 2 and autocorrelation function is R XX ( τ ) = 4e -2| τ | is applied to a system whose transfer function is 1/2+j ω . Find the mean value, autocorrelation, power density spectrum and average power of the output signal Y(t).