Digital Signal Processing Reference
In-Depth Information
Examples
Example 2.1: Discrete-time in the Far West
The fact that the “fastest” digital frequency is 2
l
g
r
,
y
i
d
.
,
©
,
L
s
can be readily appreciated
in old western movies. In classic scenarios there is always a sequence show-
ing a stagecoach leaving town. We can see the spoked wagon wheels starting
to turn forward faster and faster, then stop and then starting to turn back-
wards. In fact, each frame in the movie is a snapshot of a spinning disk with
increasing angular velocity. The filming process therefore transforms the
wheel's movement into a sequence of discrete-time positions depicting a
circular motion with increasing frequency. When the speed of the wheel is
such that the time between frames covers a full revolution, the wheel ap-
pears to be stationary: this corresponds to the fact that the maximum digi-
tal frequency
π
0.
As the speed of the real wheel increases further, the wheel on film starts to
move backwards, which corresponds to a negative digital frequency. This is
because a displacement of 2
π
+
α
between successive frames is interpreted
by the brain as a negative displacement of
ω
=
2
π
is undistinguishable from the slowest frequency
ω
=
: our intuition always privileges
the most economical explanation of natural phenomena.
α
Example 2.2: Building periodic signals
Given a discrete-time signal
x
[
n
]
and an integer
N
>
0 we can always for-
mally write
∞
y
[
n
]=
x
[
n
−
kN
]
k
=
−∞
The signal
y
,ifitexists,isan
N
-periodic sequence. The periodic signal
y
[
n
]
is “manufactured” by superimposing infinite copies of the original sig-
nal
x
[
n
]
[
n
]
spaced
N
samples apart. We can distinguish three cases:
(a) If
x
is finite-support and
N
is bigger than the size of the support,
then the copies in the sum do not overlap; in the limit, if
N
is exactly
equal to the size of the support then
y
[
n
]
corresponds to the periodic
extension of
x
[
n
]
[
n
]
considered as a finite-length signal.
(b) If
x
is finite-support and
N
is smaller than the size of the support
then the copies in the sum do overlap; for each
n
,thevalueof
y
[
n
]
is
be the sum of at most a finite number of terms.
[
n
]
(c) If
x
[
n
]
has infinite support, then each value of
y
[
n
]
is be the sumof an
infinite number of terms. Existence of
y
[
n
]
depends on the properties
of
x
[
n
]
.
