Discrete Time System is an algorithm, which operates on a discrete time signal called as input signal according to some well-defined rules/operation. Impulse Response of a system is the reaction to any discrete time system in response to some external changes. Impulse Response is generally denoted as h(t) or h[n]. The output y[n] of any discrete LTI system is depended on the input (i.e. x(n)) and system’s response to unit impulse (i.e. h[n]). Illustrated in Figure 1.
We can determine the systems output y[n], if we know system’s impulse response, h[n], and the input, x[n]. To find the impulse response of the system we provide a Unit impulse to the input x[n]. Illustrated in Figure 2.
To Find a Impulse Response, we first solve the difference equation of the system and then use Z transform.
Now multiply both numerator and denominator by , we get
By partial fractions,
on simplifying we get A= 0.4 and B= 1,
Taking inverse Z transform,
[Obtained by using the property ]
We now put n= 0,1,2,3,….. in equation 1 to get values of h(n) and compute them in the following table:
Plotting these values shall give us the Impulse Response of given Discrete Time System. Figure 3 illustrates the plot(at the end).
syms n z %Take the equation of H H=1/(1-0.18*z^(-1)+0.16*(z^(-2))); disp('Impulse response h(n) is,'); % Compute Inverse Z-transform h=iztrans(H) % Simplify the fractions simplify(h); N=15; % Numerator Coefficients b=[0 0 1] % Denominator Coefficients a=[1 -0.8 0.16] %Generate Impulse Response [H,n]=impz(b,a,N) % Plot the response stem(n,H) xlabel('n') ylabel('h(n)') title('Impulse Response of Discrete Time System')
Impulse response h(n) is,
2*(2/5)^n + (2/5)^n*(n – 1)
0 0 1
1.0000 -0.8000 0.1600