Solution 5

(a) Let and be two inputs with corresponding outputs and respectively. Now, suppose the new input to this system is . Then, the corresponding output is given by, if and 0 otherwise. From this it is clear that and hence the system is linear.

(b) The system A is not time invariant. We can see this from the following example. Let , for n = 0, 1 and otherwise. Also assume that N=2. Thus, the output corresponding to x [ n ] will be if n = 0, 2 and otherwise. Now, let if n = 1, 2 and 0 otherwise. The corresponding output is if n = 2, 4 and y 1[ n ] = 0 otherwise. Thus, and hence this shows that the system is not time invariant.

(c) Refer to Fig. 1 and Fig. 2

Figure 1: Spectrum given in the problem.

Figure 2: Spectrum of the interpolated signal.

(d) Refer to Fig. 3.

Figure 3: Spectrum of the upsampled signal.