(Z-transform) The system function of a discrete-time LTI system is given by H(z)

Question

(Z-transform) The system function of a discrete-time LTI system is given by H(z) = 1 + 2z^-1/1 + 5/2 z^-1 + z^-2, |z| > 2 (a) Convert the above system function into equation relating X(z) and Y(z) Assume the z-transform of x(t) is X(z) and the z-transform of y(t) is Y(z). (b) What is the difference equation describing the system? (Z-transform) (a) What is the requirement on the region of convergence of a system function H(z) for a discrete-time LTI system to be stable? (b) What is the requirement on the region of convergence of a system function H(z) for a discrete-time LTI system to be causal? (c) The system function of a discrete-time LTI system is given by H(z) = 1/(1 - 1/2 z^-1)(1 + 1/3 z^-1), |z| > 1/2 Sketch the pole zero plot of H(z) and shade the region of convergence. (d) Is the system in part (c) stable? Why? (e) Is the system in part (c) causal? Why?

Explanation / Answer

Q.6 a) We are given H(z) as :

H(z) = ( 1 + 2 z-1 ) / ( 1 + 5/2 z-1 +z-2 )

H(z) = Y(z) / X(z)

Therefore

Y(z) / X(z) = ( 1 + 2 z-1 ) / ( 1 + 5/2 z-1 +z-2 )

Y(z)( 1 + 5/2 z-1 +z-2 ) = X(z) ( 1 + 2 z-1 )

Y(z) + 5/2 z-1Y(z) +z-2Y(z) = X(z) + 2 z-1 X(z)    ....................1

This is the required relation between Y(z) & X(z)

Q.6 b)

From a we have following relation

Y(z) + 5/2 z-1Y(z) +z-2Y(z) = X(z) + 2 z-1 X(z)

Taking inverse Z transform of above equation and applying time shifting property inverse Z transform z-K X(z) =x(n-k)

we get,

y(n) + 5/2y(n-1) +y(n-2) = x(n)) + 2 x(n-1)              .......................2

This is the required diffrence equation describing the system.

Q.7 a)

Region of convergence of a system function H(z) for a discrete time LTI system to be stable is that, all the poles of H(z) must lie inside unit circle in Z plane.

This is the rquirement for stability.

Q.7 b)Region of convergence of a system function H(z) for a discrete time LTI system to be causal is that, H(z) does not negative values of n i.e. n<0 .

This is the rquirement for causality.

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