# Electronics and Communication Engineering - Exam Questions Papers - Discussion

*h*[

*n*] and rational system function H(

*z*). Suppose it is known that H(

*z*) contains a pole at

*z*= 1/2 and a zero some where on the unit circle. The precise number and locations of all of the other poles is unknown.

The following statement which is false is __________ .

Statement (*d*) is false because a finite-duration sequence must have an ROC that includes the entire *z*-plane, except possible *z* = 0 and/or *z* = ∞

This is not consistent with having a pole at *z* = 1/2.

Statement (b) is true corresponds to the value of the *z*-transform of *h*[*n*] at *z* = 2.

Thus, its convergence is equivalent to the point *z* = 2 being in the ROC.

Since the system is stable and causal, all of the poles of *H*(*z*) are inside the unit circle, and the ROC includes all the points outside the unit circle, including *z* = 2.

Statement (*c*) is true. Since the system is causal, *h*[*n*] = 0 for *n* < 0.

Consequently, *h*[*n*] * *h*[*n*] = 0 for *n* < 0; i.e., system with *h*[*n*] * *h*[*n*] as its impulse response is causal.

The same is then true for *g*[*n*] = *n*[*h*[*n*]].

Furthermore, by the convolution property the system function corresponding to the impulse response *h*[*n*] * *h*[*n*] is H^{2}(*z*), and by the differentiation property the system function corresponding to *g*[*n*] is

From above equation we can conclude that the poles of *G*(*z*) are at the same locations as those of *H*(*z*), with the possible exception of the origin.

Therefore, since H(*z*) has all its poles inside the unit circle, so must *G*(*z*).

It follows that *g*[*n*] is the impulse response of a causal and stable system.

Statement (*a*) is true because there is a zero on the unit circle.