Question

For the system shown in Fig. 2, X(e"), the Fourier transform of the input signal x[n], is shown in Fig. 3 5.

Answer 1

Given:

Here in the problem the given signal x[n] in first interpolated by 'L' followed by decimation by 'M' to obtain x_d[n], with overall cut-off frequency (let w_c) :

From the scaling property of DTFT:

If,

then, , for L> 1 i.e Interpolation

And, if x[n] is decimated by M, then,

Thus,overall,

, where, k= 0,1,2,3......(eqn.A)

Now let us take the problem for different values of L and M given in the question.

a) M =5 and L=3

Since, M > L, the cut-off frequency will be:

for the signal x_d[n]:

Thus, from eqn A, the first pulse of x_d [n] lies between (-3w/5, 3w/5) for K=0,

This implies,  ,

Thus,

.

b) M=3 and L=5.

Since, L > M, the cut-off frequency will be:

for the signal x_d[n]:

Thus, from eqn A, the first pulse of x_d [n] lies between (-5w/3, 5w/3) for K=0,

This implies, ,

Thus,

.