MATH 267
Due: March 21, 2012, in the class
ASSIGNMENT # 8
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: Throughout the assignment, the function u(t) denotes the unit step function:
1, t ≥ 0
u(t) =
0, t < 0
1. [NOT TO HAND-IN] [Scaling property of the delta function]
(a) For α > 0, show that
Z
∞
δ(αt)dt =
−∞
1
α
(Hint: This is a one line proof. Use change of variables.)
(b) For α < 0, show that
Z
∞
δ(αt)dt = −
−∞
1
α
(Hint: This is a one line proof. Use change of variables.)
(c) From parts (a) and (b), we conclude that, for α 6= 0,
Z ∞
1
.
δ(αt)dt =
|a|
−∞
2. [Delta function and integral]
(a) Find the value of the integral
R∞
−∞
δ(t − π)eit/3 dt.
R∞
d
u(t)dt.
sin(t + π2 ) dt
i
h
R∞
(c) Find the value of the integral −∞ δ(t − 1) u(1 − 2t ) + u(t − 1) + 2 dt. (Hint: Here applies the class
R∞
example about −∞ δ(t)u(t)dt.)
ih
i
R∞ h
(d) Find the value of the integral −∞ u(t + 1) + rect(t) δ(1 − 2t) + 2δ(t + 1) dt. (Hint: The result of
(b) Find the value of the integral
−∞
Problem 1 is relevant here.)
(e) Find the value of the integral
R∞ h
3δ(t − 2) +
−∞
d
dt u(t
ih
i
+ 1) u(t + 1) − u( 2t − 1) dt.
3. [Delta function and Fourier transform]
(a) Let f (t) = δ(t − 1) + 3 + δ(1 − 2t) + ei2t . Find fb(ω).
(b) Let gb(ω) = δ(2ω − 1) + 1 + δ(2 − 2ω) + eiω . Find g(t).
(c) Let h(t) = u(t + 1) + u(2t + 1). Find the values b
h(π) and b
h(π/2). (Hint: Here, the class example
about u
b(ω) applies.)
4. [Discrete Fourier transform]
(In the following, we do not distinguish between “length= N discrete-time signals” and “N -periodic
discrete-time signals”.)
For the given discrete-time signal x, find x
b, i.e. its discrete Fourier transform (or in other words, discrete
Fourier series).
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(a) x = [0, 1, 0, 0].
(b) x = [1, 1, 1, 1].
(c) x = [1, −1, 1, −1].
(d) x = [1, 2, 3, 4].
(e) x = [1, 13 , 312 , · · · , 3110 , 3111 ].
5. [Inverse discrete Fourier transform] Given x
b in the following, find the original signal x by using the inverse
discrete Fourier transform.
(a) x
b = [0, 0, 3, 0].
(b) x
b = [1, 1, 1, 1].
(c) x
b = [1, 14 , 412 , · · · , 4110 , 4111 ]
6. [Discrete complex exponentials]
(a) Compute
10 h
2π
i
X
2π
2π
2π
2π
e−i 11 2n + ei 11 3n ei 11 2n + e−i 11 3n + ei 11 8n
n=0
(b) Compute
10 h
2π
i
X
2π
2π
2π
2π
e−i 10 2n + ei 10 3n ei 10 2n + e−i 10 3n + ei 10 7n
n=0
7. [NOT TO HAND-IN] Two N -periodic (or length= N ) signals x, y and their discrete Fourier transforms
x
b, yb are given. Prove:
np
f [n] = e2πi N x[n]
f [n] =
1
N
N
−1
X
x[m] y[n − m]
=⇒
fb[k] = x
b[k − p]
(frequency shifting)
=⇒
fb[k] = x
b[k] yb[k]
(convolution)
=⇒
fb[k] =
m=0
f [n] = x[n] y[n]
N
−1
X
x
b[m] yb[k − m]
(multiplication)
m=0
Note the typos are corrected above, in red color!
*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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