READING ASSIGNMENT 3
Tue 02/10 Lecture
Topic:
discretetime sinusoids; sampling of continuoustime sinusoids
Textbook References:
sections 1.5, 1.6
Key Points:
•
The discrete time parameter
n
counts samples. The (angular) frequency parameter
ω
is an
angle increment (radians/sample). Physical time (seconds) is nowhere involved.
•
Frequencies
ω
and
ω
+ 2
π
are equivalent (i.e., produce the same signal) for real or complex
sinusoids in discrete time.
•
Frequencies
ω
and 2
π

ω
can be used alternatively to describe a real sinusoid in discrete
time:
cos(
ωn
+
φ
) = cos(

ωn

φ
) = cos((2
π

ω
)
n

φ
)
•
The eﬀective range of frequencies for a real sinusoid in discrete time is 0 (lowest) to
π
(highest).
•
A discretetime sinusoid is periodic if and only if
ω
is of the form
ω
=
k
N
·
2
π
for integers
k
and
N
. The fundamental period is the smallest value of
N
for which the above
holds.
•
Sampling a continuoustime sinusoid at a rate of
f
s
= 1
/T
s
(samples/second) produces a
discretetime sinusoid.
•
If the continuoustime sinusoid has angular frequency Ω = 2
πf
= 2
π/T
, the resulting discrete
time sinusoid has angular frequency
ω
= Ω
T
s
= 2
π
·
f
f
s
= 2
π
·
T
s
T
•
At high sampling rates, discretetime samples capture the variation of the continuoustime
signal in great detail.
•
Two diﬀerent sampling rates
f
s
= 1
/T
s
and
f
0
s
= 1
/T
0
s
will produce samples having the same
eﬀective frequency provided the sum
T
s
+
T
0
s
or the diﬀerence
T
s

T
0
s
is an integer multiple
of
T
= 1
/f
.
Theory and Examples:
1
. A discretetime signal is a sequence of values (samples)
x
[
n
], where
n
ranges over all integers.
A discretetime sinusoid has the general form
x
[
n
] =
A
cos(
ωn
+
φ
)
or, in its complex version,
z
[
n
] =
Ae
j
(
ωn
+
φ
)
Question:
How is
x
[
n
] related to
z
[
n
]?
1
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 Spring '08
 staff
 Frequency, Signal Processing, Cos, Ω, Discretetime sinusoid

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