Exam zone
Filtering of Random Signals Consider a discrete-time LTI system with a system function of the form: H ( z) =
b ; j zj > j aj ; j aj < 1; a 2 R 1 : 1 ¡ az¡ 1
Azero-mean, WSS,variance ¾2 whiteGaussianrandomsignal x [n ]istheinput to the system H ( z) producing an output signal y[n ]. It can be shown that the ACF of the output process is: R yy [k ]=
¾2 b2 j k j a ; k 2 I: 1 ¡ a2
The PSD of the output y[n ] in this case can be shown to be: Pyy ( z) =
¾2 b2 ; a< (1 ¡ az¡ 1 )(1 ¡ az)
j zj <
1 : a
On the unit circle, this expression for the PSD reduces to: Pyy ( ej! ) =
¾2 b2 ;! 1 ¡ 2a cos(! ) + a2
2 [¡ ¼;¼]:
The average power of the output random signal is therefore: y Pave =
¾2 b2 : 1 ¡ a2
%********************************************************** % MATLAB MACRO FOR FILTERING OF RANDOM SIGNALS % Author: Balu Santhanam % Date : 10/12/2001 % EECE-541, Fall 2001 %********************************************************** format long % Input signal is a zero-mean, WSS, Gaussian random signal % with unit variance N = 2000; x = awn(zeros(1,N),[0,1],'AWGN'); % Filter signal through a LTI system b = 1; a = [1,-0.5]; y = filter(b,a,x); % Estimate output ACF and PSD R_yy = xcorr(y,'coeff'); [F,P_yy] = psd(y,512,1,128,1); P_act = abs(fft(b,512) ./ fft(a,512)).^2; %**********************************************************
INPUT PROCESS: = 0, 2 = 1, AWGN x
OUTPUT HISTOGRAM 500
x
4 450
3 400
2
350
1
300
0
250
200
−1
150 F R E Q U E N C Y O F O B S E R V A TI O N S
−2 S A M P L E R E A LI Z A TI O N O F y[ n]
100
−3
50
−4
0
500
1000 TIME SAMPLE
1500
2000
0 −4
(a)
−3
−2
INPUT PROCESS: = 0, 2 = 1, AWGN x
−1 RANGE
0 OF
1 OUTPUT
2
3
4
(b)
OUTPUT PSD: Nfft= 512, N win = 128
x
8
1 0.9
Pest Pact
6
[0]
0.8 yy
4
[k] / R
0.7 0.6
(e j ) (d B)
yy
2
yy
0.5
P
0.4 0.3
0
O U T P U T A C F: R
−2
0.2 −4 0.1 0 −10
−5
0 5 ACF LAG VARIABLE (k)
10
(c)
−6 0
0.1
0.2 NORMALIZED
0.3 0.4 FREQUENCY
0.5
(d)
INPUT−OUTPUT CROSS COVARIANCE 1
0.9
0.8
0.7
yx
[0]
0.6
R yx[k] / R
0.5
0.4
0.3
0.2
0.1
0 −10
−8
−6
−4
−2 0 2 LAG VARIABLE (k)
4
6
8
10
(e)
Figure 1: Filtering of white noise: (a) sample realization of y [n ], (b) histogram of the output process y [n ], (c) output ACF, (d) output PSD, (e) output-input cross-covariance.