Question

Page ( 4 > of 4 5 - 2000 + /26/2020 4. Apply the analytic method of convolution to determine and sketch the convolution output y[n] = x[n]*h[n] for x[n] = (1/4)" u[n+2] h[n] = 9u[n-3] Sketch the first four non-zero terms of y[n].

Answer 1

Given

x[n=(4) un + 2]

h[n] = 9. [ – 3]

The convolution sum is given as

x[n] = x[n] * h[n]

»[n] = { "[k]x[n – k] k -00

Or

>[n] = ¿ *[k]H[n– k] k -00

We will use

>[n] = ¿ *[k]H[n– k] k -00

«[k] = () * r[k +2]

h[n – k] = 9.u[n - k-3]

So

y[n] u[k + 2]9.u[n- k – 3]

y[n] = u[k + 2]. u[n -k - 3]

We know that

k +20 u[k + 2] = lok +2 <0

u[k + 2] = k 2-2 k<-2

Using this, the summation will become

Ενη] => Σα)' αψη - k-3

Now

> -4-u 0z&- * -u 02 i}= [ɛ – 4 – u]n-

بیا IV [n - k - 3] = { بیا A

u[n- k – 3] = {1, ks n-3

So the summation becomes

„(3) 3 6 = [u]c

Expanding the sum, we get

|_$+-, 0+,()+,()+_ 4)+ ) 6 = 1496

Taking $1585233196743_blob.png$ outside

[__ --+, 0+0+0+,0+1). 6)6 – ga

comm+,@+,)+@)+, @) + 1] ºr x6 = {436

|_G) * ,+6+6+,(9)+1]**2 = fu36

The sum inside the bracket is a finite GP with a common ratio of $1585233207344_blob.png$

The sum of a finite GP is given as

S=1+r+p2 +p3 + .......... trin

1- 7+1 S= 1 - r

So

1-(7)" y[n] = 144 1-1

_.@) – 1)** **1 = [ujc

„¢) – 1|261 = [4]c

Since
x[n] = (4)" u[n +2]
and
h[n] = 9. [ – 3]
, the sequence x[n] starts at n = -2 and the sequence h[n] starts at n = 3. So the output will start at n = 1. So

Y[n] = 192ſ1 - (-)") forn 1

So

*1 = 192|1 - 0) 1-10

081 = [0) – 1261 – feje

681 = [0) – 126= f87€

s2161 = (0)– 1]201=(436

So the plot of first 4 non zero elements will be

y[n] 191.25 189 180 144 1 2 3 4 n-