Solution 4                          

(a)   We know that the overall impulse response of cascaded systems is the convolution of the impulse responses of the individual systems.

Let the overall impulse response of the cascaded system be .

Therefore using the above mentioned property, we have

Since the convolution is associative in nature therefore we can say that

So first convolving with itself:

Let .

Since h₂[k] is non-zero for k = 0 and 1 only, therefore we can write that

Therefore:

Therefore, we have

Now h[n] is nonzero from 0 to 6 and is nonzero from 0 to 2.

Also, .

Therefore will be nonzero from 0 to 4.

( If a signal is non-zero in the interval (a , b )and it is convoluted with another signal which is non zero in the interval (c,d ) then the convoluted signal is non zero in the interval (a+c,b+d).)

Let

Now,

Therefore, we have

          (from the Figure (b) we have h[0] = 1)

.

.

.

Therefore,

(b)   We have

To get the response, we have to convolve it with h[n] which we obtained in the part (a)

Proceeding as in part (a) above, i.e. putting different values of n and correspondingly calculating y[n],finally we get the answer

y[0] =1 , y[1] = 4 , y[2] = 5, y[3] = 1 , y[4] = -3 , y[5] = -4 , y[6] = -3 , y[7] = -1.

Therefore,