Digital Signal Processing Reference
In-Depth Information
We have established that the output of a linear system can be expressed in the
frequency domain as the product of the system's transfer function and the Fourier
transform of the input signal
[eq(5.45)]
V
2
(v) = H(v)V
1
(v).
Consider now the result when the input is an impulse function In this
case, and, therefore, In other words, the transform of the
output is exactly equal to the transfer function when the input is an impulse.
Considering this, we call the inverse Fourier transform of the transfer function the
impulse response
, denoted by
v
1
(t) = d(t).
V
1
(v) = 1
V
2
(v) = H(v).
h(t).
Thus,
h(t)
Î
f
"
H(v).
Using the convolution property to find the inverse Fourier transform of (5.45)
yields
q
v
2
(t) = v
1
(t)*h(t) =
L
v
1
(t)h(t - t)dt.
(5.47)
-
q
Of course, this agrees with the convolution integral in (3.13).
If we know the impulse response or its Fourier transform, the transfer func-
tion of a linear system, we can find the output for any given input by evaluating
either (5.45) or (5.47).
Using the Fourier transform to find the response of a system to an input signal
EXAMPLE 5.17
Consider the system shown in Figure 5.29(a). The electrical network in the diagram re-
sponds to an impulse of voltage at the input, with an output of
, as shown in Figure 5.29(b). Our task is to determine the frequency
spectrum of the output of this system for a step function input of voltage. Thus,
as shown in Figure 5.29(c). Because the impulse response of this linear
network is known, the output response to any input can be determined by evaluating the
convolution integral:
x(t) = d(t),
h(t) = (1/RC)e
-t/RC
u(t)
x(t) = Vu(t),
q
y(t) = x(t)*h(t) =
L
x(t)h(t - t)dt.
-
q
This looks like an onerous task; however, we are saved by the convolution property of the
Fourier transform. From (5.16),
Y(v) = f
5
x(t)*h(t)
6
= X(v)H(v).
