ECEN 444 - Fall 2022 semester

Homework 3:

Due date: Tuesday, August 23, 1:00pm.

Problem 1:

An LTI system has impulse response h[n] = 5(-1/2)"u[n]. Use the Fourier transform to

find the output of this system when the input is x[n] = (1/3)"u[n].

Problem 2:

(a) Determine the Fourier transform of the sequence

7[m] = {1;

1, 0≤ n ≤ M,

0, otherwise.

(b) Consider the sequence

-{[-(~7)].

M

0≤n≤ M,

otherwise.

Sketch w[n] and express W(el), the Fourier transform of w[n], in terms of R(e), the

Fourier transform of r[n]. (Hint: First express w[n] in terms of r[n] and the complex

exponentials e/(2/M) and e-/(2n/M).)

(c) Sketch the magnitude of R(e") and W (ei) for the case when M = 4.

Problem 3:

A linear time-invariant system is described by the input-output relation

y[n] = x[n] + 2x[n 1] + x[n-2].

(a) Determine h[n], the impulse response of the system.

(b) Is this a stable system?

(c) Determine H(ei"), the frequency response of the system. Use trigonometric identities

to obtain a simple expression for H(eja).

w[n] = 2

(d) Plot the magnitude and phase of the frequency response.

(e) Now consider a new system whose frequency response is H₁(e) = H(ei(+*)). De-

termine h₁ [n], the impulse response of the new system./n→ problem I

htn] = (4) un]

aunf

- (ung the

2

" acr

* Fth Cr

(

) (

| En] = 3(

@rn]

+

→ problem 2

ehs) (س) - سال) Y

( - ( د ) () ۷

( ): (مال) ۷

(س) R

R ()

R ( ) =

[:

S

1-2

no

1+

| -

√6

36

)+( )

)

)"]+2 (1) En

سالام =

`-

سالے

s

-11

le

(

"36

سالم - ا

•≤n≤ M

otherwise

(1+M) سال م

x[n] = ({} 4[n]

( )En] +2( )cn

R (2) sin(M! w)

sin

ليا

(سال) H

y ley

xcejuj

output/nW[n] = {{[1-105 (2)

W[n]=ren] ( { [l-cos (21)

20

w[e³") =R /ews +(1+/= the me

n=40

©

→ problem 3

Ⓒh[n] = ?

OENEM

otherwise

W( ² ) = R (~³W) ( = S(w) + { S(w +257) + 1 / 8 (w_2T ))

IR Cet) |

h with

-Ju

H(e) = 1 + 2e²

14 won] skelch

te

H(e) = 2ěvku (= eto +1 +1 ÷ 2)

(e") 2√ ° (1 ( 2 ² ²

سال

4

y[n] = x[n] + 2x[n − 1] + x[n-2].

y[A] =x[n] h[n]

Y[n] = x[n] (5[n]+28{n-1}+5[n-2])

h[n] = S[n] + 28 [n-13+5 [n-2]

yes stable system as h[n] absolutely sumable Ef

-j2w

رسالت + + + ) + 1 ) 2 = (H

H(e) = 2³ (1 + cos (w))

ارسالی با

^/4/2

finite-length./nH (e )

ना

phase:

ⒸH₁(el) = H(el(w+x))

h, [n] =

T

h₁ [n] = = π SHielus

could Ju

2πT

h₁ [n] =

1

2πT

h₁ [n] = ?

2πT SA Cel(WIT))/W

з неше

(en

ew dw

dw

swn

dw

↑Ince431

h₁ [n] =

1" h[n]

h₁ [n] = S[n] -2 5 In-1] + 5 In -2]

magatule

| He") = 2(1+ (os(w))