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))
Fig: 1
Fig: 2
Fig: 3
Fig: 4