When comparing analog vs discrete time, we find that there
are many similarities. Often we only need to substitute the varible
t with n and integration with summation. Still there are some
important differences that we need to know.
As the complex exponential signal is truly central to signal processing
we will study that in more detail.
Analog
The complex exponential function is defined:
xt=ⅇⅈΩt
x
t
Ω
t
.
If Ω(rad/second) is increased the rate of oscillation will increase continuously.
The complex exponential function is also periodic for
any
value of Ω. In figure
Figure 1 we have plotted
ⅇⅈπt
t
and
ⅇⅈ3πt
3
t
(the real parts only). In
Figure 1 we see that
the red plot, corresponding to a higher value of Ω, has a higher rate
of oscillation.
Discrete time
The discrete time complex exponential function is defined:
xn=ⅇⅈωn
x
n
ω
n
.
If we increase ω (rad/sample) the rate of oscillation
will increase and decrease periodically.
The reason is:
ⅇⅈω+2πkn=ⅇⅈωnⅇⅈ2πkn=ⅇⅈωn
ω
2
k
n
ω
n
2
k
n
ω
n
, where
n,k∈ℤ
n,k
.
This implies that the complex exponential with digital
angular frequency ω is identical to
a complex exponential with
ω1=ω+2π
ω1
ω
2
, see
Figure 2
The rate of oscillation will increase until
ω=π
ω
, then it decreases and repeats after 2π. In
Figure 3
we see that as we increase the angular frequency towards π the rate of
oscillation increases. If you download the Matlab files included at the
end of this module you can adjust the parameters and see that the rate
of oscillation will decrease when exceeding π (but less than 2π).
consequence:
We need to consider discrete time exponentials at an (digital angular) frequency interval of 2π only.
Low (digital angular) frequencies will correspond to ω near even multiplies of π.
High (digital angular) frequencies will correspond to ω near odd multiplies of π.