Digital Signal Processing Reference
In-Depth Information
The convolution property states that if
f
1
(t)
Î
f
"
F
1
(v)
f
2
Î
f
"
F
2
(v),
and
then convolution of the time-domain waveforms has the effect of multiplying their
frequency-domain counterparts. Thus,
f
1
(t)*f
2
(t)
Î
f
"
F
1
(v)F
2
(v),
(5.16)
where
q
q
f
1
(t)*f
2
(t) =
L
f
1
(t)f
2
(t - t)dt =
L
f
1
(t - t)f
2
(t)dt.
-
q
-
q
Also, by applying the duality property to (5.16), it is shown that multiplication of
time-domain waveforms has the effect of convolving their frequency-domain repre-
sentations. This is sometimes called the
multiplication property
,
1
2p
F
1
(v)*F
2
(v),
f
1
(t)f
2
(t)
Î
f
"
(5.17)
where
q
q
F
1
(v)*F
2
(v) =
L
F
1
(l)F
2
(v - l)dl =
L
F
1
(v - l)F
2
(l)dl.
-
q
-
q
Engineers make frequent use of the convolution property in analyzing the interac-
tion of signals and systems.
The time-convolution property of the Fourier transform
EXAMPLE 5.8
Chapter 3 discusses the response of linear time-invariant systems to input signals. A block
diagram of a linear system is shown in Figure 5.9(a). If the output of the system in re-
sponse to an impulse function at the input is described as then is called the
impulse response
of the system. The output of the system in response to any input signal
can then be determined by convolution of the impulse response,
h(t),
h(t)
h(t),
and the input signal,
x(t):
q
y(t) = x(t)*h(t) =
L
x(t)h(t - t)dt.
-
q
Using the convolution property of the Fourier transform, we can find the frequency spectrum
of the output signal from
Y(v) = X(v)H(v),
