Digital Signal Processing Reference
In-Depth Information
x
(
t
)
T
/2
0
T
/2
T
t
Figure 4.1
Periodic function.
We define a function to be
periodic,
with the period
T
, if the relationship
is satisfied for all
t
. For example, the function
x(t)
x(t) = x(t + T)
cos
vt
is periodic
(v = 2pf = 2p/T),
because
cos
v(t + T) = cos
(vt + vT) = cos
(vt + 2p) = cos
vt.
Another example is shown in Figure 4.1, where the function is constructed of connected
straight lines. Periodic functions have the following properties (see Section 2.2):
1.
Periodic functions are assumed to exist for all time; in the equation
we do not limit the range of
t
.
x(t) = x(t + T),
2.
A function that is periodic with period
T
is also periodic with period
nT
,
where
n
is any integer. Hence for a periodic function,
x(t) = x(t + T) = x(t + nT),
(4.2)
with
n
any integer.
3.
We define
the fundamental period
T
0
as the minimum value of the period
T 7 0
that satisfies
x(t) = x(t + T).
The
fundamental frequency
is defined
as
v
0
= 2pf
0
= 2p/T
0
.
For the units of
T
0
in seconds, the units of
v
0
are
radians per second (rad/s) and of
f
0
are hertz (Hz).
The second property is seen for the function of Figure 4.1 and also for
cos
vt,
since
cos
v(t + nT) = cos(vt + nvT) = cos(vt + n2p) = cos vt.
We usually choose the period
T
to be the fundamental period
T
0
;
however, any
value
nT
0
,
with
n
an integer, satisfies the definition of periodicity.
Consider again the periodic signal of Figure 4.1. Suppose that this function is the
input to a stable LTI system and we wish to find the steady-state response. It is quite
difficult to find the exact steady-state response for this input signal, since the math-
ematical description of the signal is a sum of ramp functions. However, we may be
able, with
adequate accuracy,
to approximate this signal by a sinusoid, as indicated
