MATH 267
Due: March 28, 2012, in the class
ASSIGNMENT # 9
You have FIVE problems to hand-in. Hand in written solutions for grading at the BEGINNING of the
lecture on the due date. Illegible, disorganized or partial solutions will receive no credit.
*Staple your HW. You will get F IV E marks OFF if you do not staple your HW! Note that
the instructor will NOT provide stapler.
Note:
In the following, we do not distinguish between “length= N discrete-time signals” and “N -periodic discretetime signals”.
1. [Discrte Fourier transform for periodic signals]
(a) Find the discrete Fourier transform of the following periodic signals with period N .
i. x[n] = cos(2πn). N = 4
ii. y[n] = cos(πn/3) + sin(πn/2). N = 12
(Hint: Express sin and cos using complex exponentials and try to use ‘orthogonality’ to compute the
summation. )
(b) Suppose x[n] is a periodic discrete-time signal with period = N . Let x
b[k] be its discrete Fourier
PN −1
transform. Assume x[0] = N and k=0 |b
x[k]|2 = N . Find x[n] and x
b[k] for all 0 ≤ n, k ≤ N − 1.
(Hint: Use Parseval’s relation. This is similar to one of the class examples.)
(c) Let a[n] be a periodic signal with period N = 16 with


1 0 ≤ n ≤ 8,
a[n] = 0 9 ≤ n ≤ 12,


1 13 ≤ n ≤ 15.
Compute the discrete Fourier transform b
a[k]. (This is similar to one of the class examples.)
2. [NOT TO HAND IN] (This problem is similar to Example 1 in the online notes “Discrete-Time Fourier
Series and Transforms ”.)
Consider the “discrete square wave” function x[n] with one period given by
1, −N1 ≤ n ≤ N1
x[n] =
0, otherwise
for some positive integer N1 < N , where N is the fundamental period of x[n].
(a) Show that
(
x
b[k] =
2N1 +1
,
N
1 sin[2πk(N1 +1/2)/N ]
,
N
sin(πkN )
k = 0, ±N, ±2N, . . .
otherwise
(b) Use part (a) and the time shift property to compute the discrete Fourier series coefficients of the
function y[n] with one period given by:

0≤n≤8
 1,
0, 9 ≤ n ≤ 12
y[n] =

1, 13 ≤ n ≤ 15
3. [Periodic convolution]
Consider the folloing signals with period N = 4:
a = [1, 0, 1, −1],
(e.g. a[0] = 1, a[3] = −1, b[2] = 1 + i, etc. )
1
b = [2, i, 1 + i, 3]
(a) Calculate the periodic convolution a ∗ b by directly calculating the convolution sum.
(b) Calculate the Fourier coefficients b
a[k] and bb[k]. Use this to compute the Fourier coefficients ad
∗ b[k]
for a ∗ b by using the convolution property of the Fourier transform.
(c) Find a signal x[n] of period N = 4, such that (a ∗ x)[n] = b[n].
(Hint: you may want to use the convolution property of the Fourier transform/inversion. Remember
how we handle the circuit problem. This is similar.)
4. [Convolution of non-periodic signals] (This problem is related to the topic and examples we covered in the
class on Wed. March 21.)
Recall for integers n ∈ Z,
(
1 if n ≥ 0 ,
u[n] =
0 otherwise.
(
δ[n] =
1 if n = 0,
0 otherwise.
(
δn0 [n] =
1 if n = n0 ,
0 otherwise.
Recall the class example (u ∗ u)[n] = (n + 1)u[n].
100 times
}|
{
z
(a) Find (δ2 ∗ δ2 ∗ · · · ∗ δ2 )[n].
(b) Let f [n] = u[n − 2]. g[n] = u[n + 3].
i. Find (f ∗ u)[n].
ii. Find (f ∗ g)[n].
(c) Let
(
1
h[n] =
0
|n| ≤ 3 ,
otherwise.
Find (h ∗ u)[n]
i. first, by computing the convolution sum directly;
ii. second, by using the algebraic properties of the convolution and using (u ∗ u)[n] = (n + 1)u[n].
5. [Discrete-time Fourier transform for non-periodic signals] (We will cover this topic on Monday, March 26.)
(a) x[n] = δ2 [n] + δ−2 [n]
n
(b) y[n] = 15 u(n − 1)
|n+1|
(c) z[n] = 15
6. [Inverse discrete-time Fourier transform for non-periodic signals] (We will cover this topic on Monday,
March 26.)
Recall the discrete-time Fourier transforms of δn0 [n] and an u[n] (for |a| < 1) are e−iωn0 and
respectively.
1
1−ae−iω ,
Use these to find discrete-time signals x[n], y[n], z[n], whose Fourier transforms are given below. (Here,
each answer should be a signal defined on the set of integers: n ∈ Z.)
(a) x
b(ω) = cos2 ω + cos ω sin ω. (Hint: Can we express this as combination of complex exponentials?)
(b)
ei2ω
yb(ω) = 1 +
1 + 31 e−iω
(Hint; you may want to use time-shift property: see the table in page 12 in the online note ”
Discrete-time Fourier series and Fourier Transforms”.)
(c)
zb(ω) =
1
(1 +
1 −iω
)(1
2e
+ 13 e−iω )
.
(Hint: use partial fractions.)
*Staple your HW. You will get F IV E marks OFF if you do not staple your HW! Note that
the instructor will NOT provide stapler.
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