18-290/396: Signals and Systems
CMU/ECE, fall 09
Instructor: Markus P¨
uschel
Assignment 10
Total: 100 points
Issue date: Oct 27
Due date: Nov 4 (beginning of class)
Homework policy: http://www.ece.cmu.edu/∼ece290/f09/homeworks.html
1. (5 pts) How many hours did you spend on this homework?
2. (12 pts) Let X(ejw ) denote the Fourier transform of the signal x[n] depicted in the below figure.
Perform the following calculations without explicitly evaluating X(ejw ):
(a) Evaluate X(ej0 ).
(b) Find arg(X(ejw )).
Rπ
(c) Evaluate −π X(ejw ) dw.
(d) Find X(ejπ ).
3
2
x[n]
1
0
−1
−2
−10
−8
−6
−4
−2
0
2
4
6
8
10
n
3. (30 pts) Compute the Discrete-Time Fourier Transform of the following signals and simplify it as much
as possible:
(a) x[n] = n 21n u[n].
(b) x[n] = u[n + 1] − u[n − 1]
(
n, 1 ≤ n ≤ 3
(c) x[n] =
0, else
(
|n − 3|, −3 < n < 3
(d) x[n] =
0,
else
1
πn
(e) x[n] = sin( πn
2 ) + 2 cos( 4 )
4. (21 pts) Compute the inverse Discrete-Time Fourier Transform of the following frequency responses.
Note that the responses are (of course) periodic with period 2π.
P∞
(a) X(ejw ) = k=−∞ (−1)k δ(w − πk
4 )
(
1, π/4 ≤ |w| ≤ 3π/4
(b) X(ejw ) =
0, else
(c) X(ejw ) = cos2 (2w) + sin3 (3w)
5. (16 pts) Consider the signal (x[n]) depicted in Fig. 5. Let the Fourier transform of the the signal be
written as
X(ejw ) = A(w) + jB(w).
Sketch the function of time (y[n]) corresponding to the following Fourier transforms:
(a) Y (ejw ) = 2A(w) − jB(w)
(b) Y (ejw ) = A(w)ejw + jB(w)
(c) Y (ejw ) = A(w/2) + jB(w)
(d) Y (ejw ) =
dA(w)
dw
+
dB(w)
dw
3
2
x[n]
1
0
−1
−2
−6
−4
−2
0
n
2
4
6
6. (8 pts) Assuming the DTFT pair x[n] ⇐⇒ X(ejw ), show the following:
(a) x[n + k] + x[n − k] ⇐⇒ 2X(ejw ) cos(kw)
(b) x[n + k] − x[n − k] ⇐⇒ 2jX(ejw ) sin(kw)
7. (8 pts) Assming the DTFT pair x[n] ⇐⇒ X(ejw ), determine whether each of the following statements
is true or false. Justify your answer.
(a) If X(ejw ) = X(ej(w−π/4) ), then x[n] = 0 for |n| > 0
(b) If X(ejw ) = X(ej4w ), then x[n] = 0 for |n| > 0
2

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