Generation of random sequences: While deterministic signals such as square pulse

Question

Generation of random sequences: While deterministic signals such as square pulses, sine waveforms, sinuses and cosines are used in specific applications, almost all other real-life signals from econometric series to radar returns, from genetic codes to multimedia signals in consumer electronics are information-bearing random signals. Time sequences: Generate a sequence of 1000 equally spaced samples of a Gauss Markov process using the recursive relation: xn-axn_1 + wnin = 1,2, , 1000. Here assume thatxo 0, and (wn) is a sequence of independent identically distributed Gaussian random variables. You can use the randn function in MATLAB to generate zero-mean, unit-variance random variables. Below is given a low-pass filter (a>0) excited by a white noise sequence, and the filter has a real pole at z- a. Plot the output waveform for the following values of a: a-0.5, a = 0.95, a 0.995. Comment on the effect of the resulting time sequence. Autocorrelation function: In order to estimate the autocorrelation function of this discrete process, one can apply the formula: Rx(m)= limien1XnXn -m N-oo N E(xnxn-m). However, since we have a finite number of samples an approximation would be Rx(m) _ ??-m xnxn-murn-0,1, , 64, Notice we shorten the data and consider only the overlapping samples in the window starting at n-1 and the window starting at n-m+l. The theoretical autocorrelation for such a signal model is also known N-m as: Rx(m1) = ?alml, where ? is the input process variance. Plot the autocorre!atio sequence in three separate graphs for the values of a- 0.5, a 0.95, a 0.995, but superpose in each graph the analytical and estimated Rr(m) and R (m) correlation functions.

Explanation / Answer

input of time sequences

clc;

close all;

clear all;

a = [0.5 0.95 0.995]

; w = randn(1000,1);

%

x(1) = 0; % this is the value of x0, since MATLAB

canon take zero index for i = 1:length(a)

for k = 2:1000;

x(k) = a(i) * x(k-1) + w(k);

end

figure;plot(x);title(['filter output with a = ',num2str(a(i))]);

end

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