Discretetime Sinusoids
A discretetime sinusoid is generated when a continuoustime sinusoid is sampled. For example, given frequency
ω
(which we will assume to be positive without loss of generality) the continuoustime (complex) sinusoid is defined
as
u
(
t
) =
e
jωt
,
t
∈
(
∞
,
∞
)
.
If this signal is sampled with a sample period
t
s
, the following discretetime sinusoid is generated
u
p
:=
u
(
pt
s
) =
e
jωpt
s
,
p
∈
Z
(
Z
=set of integers
)
.
(69)
We introduce the notation
ω
s
=
2
π
t
s
to denote the
sampling frequency
or
sampling rate
in rad/s. If
ω > ω
s
/
2
,
though, we will show that it is possible to generate the same sequence
{
u
p
}
by sampling a continuoustime sinusoid
of
lower
frequency in the interval
[0
, ω
s
/
2]
. For example, rewrite (69) as
u
p
=
e
j
2
π
ω
ωs
p
,
p
∈
Z
,
and suppse
1
2
< ω/ω
s
≤
1
, i.e.
ω
∈
(
1
2
ω
s
, ω
s
]
. Define a new frequency, denoted
¯
ω
, as follows:
¯
ω
:=
ω
s

ω
→
¯
ω
ω
s
= 1

ω
ω
s
.
Note that
¯
ω
∈
[0
, ω
s
/
2)
so it is a lower frequency. Substituting this definition into the expression for
u
p
yields,
u
p
=
e
j
2
π
(
1

¯
ω
ωs
)
p
=
e
j
2
πp
e

j
2
π
¯
ω
ωs
p
=
e

j
2
π
¯
ω
ωs
p
p
∈
Z
.
In other words,
{
u
p
}
can be generated by sampling a sinusoid with frequency
¯
ω
.
As another example, suppose
ω
∈
[
ω
s
,
3
2
ω
s
]
and define
¯
ω
:=
ω

ω
s
which implies
¯
ω
∈
[0
,
1
2
ω
s
]
. Note that
¯
ω
ω
s
=
ω
ω
s

1
,
so substituting this relation into (69) yields
u
p
=
e
j
2
π
(
1+
¯
ω
ωs
)
p
=
e
j
2
πp
e
j
2
π
¯
ω
ωs
p
=
e
j
2
π
¯
ω
ωs
p
p
∈
Z
,
Do not distribute these notes
171
Prof. R.T. M’Closkey, UCLA