Rensselaer Polytechnic Institute
Department of Electrical, Computer, and Systems Engineering
ECSE 4530: Digital Signal Processing, Fall 2014
Homework #3: due Thursday, Sep. 25th , at the beginning of class.
Show all work for full credit!
The problems marked (**) are only for those taking the 6000 level of the course.
1. (20 points) Compute the Discrete-Time Fourier Transform (DTFT) of the following signals using
the DTFT formula (that is, without using property tables).
(a) x[n] = u[n] − u[n − 6]
(b) x[n] = 2n u[−n]
(c) x[n] = αn sin(ω0 n)u[n], where |α| < 1 and ω0 is real
(d) x[n] = [−2 − 1 0 1 2]
2. (10 points) Let x[n] = [−1 2 −3 2 − 1]. Determine each of the following quantities without
explicitly computing X (ω).
(a) X (0)
X (ω)
Rπ
(c) −π X (ω) d ω
(b)
6
(d) X (π)
Rπ
(e) −π |X (ω)|2 d ω
3. (10 points) An FIR filter is described by the difference equation
y[n] = x[n] + x[n − 10]
(a) Compute and sketch the magnitude and phase response of the filter.
(b) Determine the response of the signal to the input
³π ´
³π
π´
x[n] = cos
n + 3 sin n +
10
3
10
Hint: Part (b) isn’t complicated if you understand the concept of frequency response.
4. (10 points**) Consider the signal x[n] = [1 0 − 1 2 3], which has the DTFT given by X (ω) =
X R (ω) + j X I (ω). Using symmetry properties, determine the signal y[n] that has the DTFT
Y (ω) = X I (ω) + e j 2ω X R (ω)
That is, do not evaluate X (ω) explicitly at any point!
5. (10 points**) Define the “center of gravity” c of a discrete-time signal as
∞
X
c=
nx[n]
n=−∞
∞
X
x[n]
n=−∞
(a) Express c in terms of X (ω).
(b) Determine c for the signal whose DTFT (inside [−π, π]) is X (ω) = (1 − ω2 )(u(ω + 1) − u(ω − 1)).