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Sec. 2–6 Review of Linear Systems 83
(2–131)
which indicates that samples of the input are taken at ∆t-second intervals. Then, using the
time-invariant and superposition properties, we find that the output is approximately
This expression becomes the exact result as ∆t becomes zero. Letting n ¢t = l, we obtain
(2–132)
(2–133)
An integral of this type is called the convolution operation, as first described by Eq. (2–62) in
Sec. 2–2. That is, the output waveform for a time-invariant network can be obtained by convolving
the input waveform with the impulse response for the system. Consequently, the impulse
response can be used to characterize the response of the system in the time domain, as
illustrated in Fig. 2–14.
Transfer Function
q
x(t) = a x(n ¢t)[d(t - n ¢t] ¢t
q
y(t) = x(l)h(t - l) dl K x(t) * h(t)
L
-q
-q
q
y(t) = a x(n ¢t)[h(t - n ¢t)] ¢t
n=0
The spectrum of the output signal is obtained by taking the Fourier transform of both sides of
Eq. (2–133). Using the convolution theorem of Table 2–1, we get
or
Y(f) = X(f)H(f)
H(f) = Y(f)
X(f)
(2–134)
(2–135)
where H(f) = [h(t)] is said to be the transfer function or frequency response of the network.
That is, the impulse response and frequency response are a Fourier transform pair:
h(t) 4 H(f)
Of course, the transfer function H(f) is, in general, a complex quantity and can be written in
polar form as
H(f) = ƒH(f)ƒ e jlH(f)
where ƒH(f)ƒ is the amplitude (or magnitude) response and
u(f) = lH(f) = tan -1 c Im{H(f)}
Re{H(f)} d
(2–136)
(2–137)