Digital Signal Processing Reference
In-Depth Information
We can also use the sifting property to write this input signal as
q
x(t) =
3
v(t)d(t - t)dt = v(t).
-
q
So we can now write
q
y(t) =
3
x(t)h(t - t)dt,
(3.13)
-
q
which is the result that we are seeking.
■
This result is fundamental to the study of LTI systems, and its importance can-
not be overemphasized. The system response to any input is expressed as an
integral involving only the input function and the system response to an impulse
function From this result, we see the importance of impulse functions to the
investigation of LTI systems.
The result in (3.13) is called the
convolution integral
. We denote this integral
with an asterisk, as in the following notation:
x(t)
h(t).
q
y(t) =
L
x(t)h(t - t) dt = x(t)*h(t).
(3.14)
-
q
Next we derive an important property of the convolution integral by making a
change of variables in (3.13); let
s = (t - t).
Then
t = (t - s)
and
dt =-ds.
Equation (3.13) becomes
q
-
q
y(t) =
L
x(t)h(t - t) dt =
L
x(t - s)h(s)[-ds]
-
q
q
q
=
L
x(t - s)h(s) ds.
-
q
Next we replace
s
with
t
in the last integral, and thus the convolution can also be
expressed as
q
q
y(t) =
L
x(t)h(t - t) dt =
L
x(t - t)h(t) dt.
(3.15)
-
q
-
q
The convolution integral is symmetrical with respect to the input signal
x(t)
and the
impulse response
h(t),
and we have the property
y(t) = x(t)*h(t) = h(t)*x(t).
(3.16)
We derive an additional property of the convolution integral by considering the
convolution integral for a unit impulse input; that is, for
x(t) = d(t),
y(t) = d(t)*h(t).
