Solution 3

(a) It is given that for the input , the output of the LTI system is of the form . From this fact we can calculate the transfer function H ( z ) to be

Let the above equation be equation (1).

Since the ROC of Y(z) is and that of X ( z ) is and since the ROC of H ( z ) must be such that its intersection with the ROC of X ( z ) is contained in the ROC of Y ( z ) we must have that the ROC of H ( z ) is .

It is also given that the output of this system to the input is y [ n ] = 0 for all n . Since the function is an eigen function for a discrete time LTI system, the output to this input is . From this we can infer that, .

Using this in (1) we can calculate the value of a from to be .

(b) The response y [ n ] to the input x [ n ] = 1, for all n is given by y [ n ] = H (1) for all n . Using the value of a obtained in the previous part we can find the value of H (1) to be -1. Hence, y [ n ] = -1.