Digital Signal Processing Reference
In-Depth Information
By definition, this output is equal to
h(t),
the impulse response:
y(t) = d(t)*h(t) = h(t).
(3.17)
This property is independent of the functional form of
h(t).
Hence, the convolution
of any function
g(t)
with the unit impulse function yields that function
g(t).
Because
of the time-invariance property, the general form of (3.17) is given by
y(t - t
0
) = d(t - t
0
)*h(t) = h(t - t
0
).
This general property may be stated in terms of a function
g(t)
as
d(t)*g(t) = g(t)
and
(3.18)
d(t - t
0
)*g(t) = g(t - t
0
)*d(t) = g(t - t
0
).
The second relationship is based on (3.16).
Do not confuse convolution with multiplication. From Table 2.3, the multipli-
cation property (sifting property) of the impulse function is given by
d(t - t
0
)g(t) = g(t
0
)d(t - t
0
)
and
g(t - t
0
)d(t) = g(-t
0
)d(t).
The convolution integral signifies that the impulse response of an LTI discrete
system, contains a
complete input-output description
of the system. If this
impulse response is known, the system response to any input can be found, using
(3.16).
The results thus far are now summarized:
h(t),
1.
A general signal
x(t)
can be expressed as a function of an impulse function:
q
[eq(3.8)]
x(t) =
L
x(t)d(t - t) dt.
-
q
2.
By definition, for a continuous-time LTI system,
d(t) : h(t).
(3.19)
The system response
y(t)
for a general input signal
x(t)
can be expressed as
q
[eq(3.15)]
y(t) =
L
x(t)h(t - t) dt
-
q
q
=
L
x(t - t)h(t) dt.
-
q
