Digital Signal Processing Reference
In-Depth Information
We define the
power
of a signal as the usual ratio of energy over time,
taking the limit over the number of samples considered:
N
−
1
x
1
2
N
l
g
r
,
y
i
d
.
,
©
,
L
s
2
P
x
=
lim
N
[
n
]
(2.20)
→∞
−
N
Clearly, signals whose energy is finite, have zero total power (i.e. their en-
ergy dilutes to zero over an infinite time duration). Exponential sequences
which are not decaying (i.e. those for which
1 in (2.7)) possess infinite
power (which is consistent with the fact that they describe an unstable be-
havior). Note, however, that many signals whose energy is infinite
do
have
finite power and, in particular, periodic signals (such as sinusoids and com-
binations thereof ). Due to their periodic nature, however, the above limit
is undetermined; we therefore
define
their power to be simply the
average
energy over a period
. Assuming that the period is
N
samples, we have
|
a
|
>
N
−
1
x
]
1
N
2
P
x
=
[
n
(2.21)
n
=
0
2.2
Classes of Discrete-Time Signals
The examples of discrete-time signals in (2.1) and (2.2) are two-sided, infi-
nite sequences. Of course, in the practice of signal processing, it is impos-
sible to deal with infinite quantities of data: for a processing algorithm to
execute in a finite amount of time and to use a finite amount of storage, the
input must be of finite length; even for algorithms that operate on the fly,
i.e. algorithms that produce an output sample for each new input sample,
an implicit limitation on the input data size is imposed by the necessar-
ily limited life span of the processing device.
(4)
This limitation was all too
apparent in our attempts to plot infinite sequences as shown in Figure 2.1
or 2.2: what the diagrams show, in fact, is just a meaningful and representa-
tive portion of the signals; as for the rest, the analytical description remains
the only reference. When a discrete-time signal admits no closed-form rep-
resentation, as is basically always the case with real-world signals, its finite
time support arises naturally because of the finite time spent recording the
signal: every piece of music has a beginning and an end, and so did every
phone conversation. In the case of the sequence representing the Dow Jones
index, for instance, we basically cheated since the index did not even exist
for years before 1884, and its value tomorrow is certainly not known - so that
(4)
Or, in the extreme limit, of the supervising engineer . . .
