Question

Find the impulse response of the following system if 5. hi (n) 6(n) 35(n- 1) h2(n) 3"u(n) n h3(n) u(n) h4(n) nu(n) hs(n) (n)nu(n- 1)8(n - 2) h4 (n) h2 (n) h2(n) h3(n) h5 (n)

Answer 1

We know impulse response of a cascaded system is convolution of their individual responses and impulse response of the parallel system is addition or subtraction of their individual responses.

Hence the above system has the impulse response as

We know impulse response of a cascaded system is convolution of their individual responses and impulse response of the parallel system is addition or subtraction of their individual responses. Hence the above system has the impulse response as h [n] = h_1 [n] * h_2 [n] * h_3 [n] * [h_5 [n] - h_4 [n]] Now, we will determine convolution as h_1 [n] * h_2 [n] = (delta [n] - 3 delta [n - 1] * 3^n u [n] h_1 [n] * h_2 [n] = delta [n] * 3^n u [n] - 3 delta [n - 1] * 3^n u [n] h_1 [n] * h_2 [n] = 3^n u [n] - 3 times 3^n - 1 u [n - 1] h_1 [n] * h_2 [n] = 3^n (u [n] - 3^n u [n - 1] h_1 [n] * h_2 [n] = 3^n (u [n] - u [n - 1]) h_1 [n] * h_2 [n] = 3^n delta [n] Also now h_1 [n] * h_2 [n] *

Now, we will determine convolution as

  

Also now

Now

$h_{5}[n] -h_{4}[n] = -\delta[n]-n\delta[n]-\delta[n-2]$

Hence the overall impulse response will be

$h[n]=(3/2)^{n}u[n]*( -\delta[n]-n\delta[n]-\delta[n-2])$

$h[n]=-(3/2)^{n}u[n]-n(3/2)^{n}u[n]-(3/2)^{n-2}u[n-2]$

Note:The given figure and the typed question has some sign conflicts, here I used the data from the figure.