ECE130B
January 20, 2010
Home Work 3
Due date
: January 27, 2010 by 5p.m. at the home work box.
Reading assignment
. Chapter 3 of the text book.
Exercise 1.
Suppose you sample the continuous signal
e
jω
0
t
at the rate of
f
s
samples per
second (with the first sample at
t
=0
).
i. Find all values of
ω
0
for which the resulting discrete signal will be periodic.
ii. Find all values of
ω
0
for which the resulting discrete signal will have the fixed period
N
.
Exercise 2.
Suppose
e
j
2
π
N
kn
is a discrete time signal with fixed values of
N
and
k
(so
n
is the
discrete time variable) that was obtained by sampling a continuous signal of the form
e
jω
0
t
at
the rate
f
s
samples per second. Find all possible values of
ω
0
in terms of
N
,
k
and
f
s
.
Discrete time Fourier series
. Let
x
[
n
]
be a discrete periodic signal of period
N
. That is
x
[
n
+
N
] =
x
[
n
]
for all values of
n
. The signal
x
[
n
]
can be written as the sum of discrete
harmionic periodic oscillators of period
N
via the formula
x
[
n
]=
summationdisplay
k
=0
N

1
a
k
e
j
2
π
N
kn
,
(1)
where the
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 Winter '08
 Staff
 Fourier Series, Signal Processing, 1 j, Fourier series coefficients

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