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.

prob theory & stochastic1

Exam zone H ( z) = b 1 ¡ az ¡ 1 ; j zj &gt; j aj ; j aj &lt; 1; a 2 R 1 : R yy [k ]= ¾ b Azero-mean, WSS,variance ¾ 2 whiteGaussianrandomsig...