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82
Signals and Spectra Chap. 2
Input waveform
x(t)
Linear system
Output waveform
y(t)
h(t)
H(f)
Some descriptions
for the input
X(f)
R x ()
“Voltage” spectrum
Autocorrelation function
Power spectral density
Figure 2–14 Linear system.
Some descriptions
for the output
Y(f)
R y ()
2–6 REVIEW OF LINEAR SYSTEMS
Linear Time-Invariant Systems
An electronic filter or system is linear when superposition holds—that is, when
y1t2 = [a 1 x 1 1t2 + a 2 x 2 1t2] = a 1 [x 1 1t2] + a 2 [x 2 1t2] (2–130)
where y(t) is the output and x(t) = a1 x 1 (t) + a 2 x 2 (t) is the input, as shown in Fig. 2–14. [ # ]
denotes the linear (differential equation) system operator acting on [ # ] . The system is said to
be time invariant if, for any delayed input x(t - t 0 ), the output is delayed by just the same
amount y(t - t 0 ). That is, the shape of the response is the same no matter when the input is applied
to the system.
A detailed discussion of the theory and practice of linear systems is beyond the scope of
this book. That would require a book in itself [Irwin, 1995, 2011]. However, some basic ideas
that are especially relevant to communication problems will be reviewed here.
Impulse Response
The linear time-invariant system without delay blocks is described by a linear ordinary differential
equation with constant coefficients and may be characterized by its impulse response
h(t). The impulse response is the solution to the differential equation when the forcing function
is a Dirac delta function. That is, y(t) = h(t) when x(t) = d(t) . In physical networks, the
impulse response has to be causal. That is, h(t) = 0 for t 6 0. † This impulse response may
be used to obtain the system output when the input is not an impulse. In that case, a general
waveform at the input may be approximated by ‡
† The Paley–Wiener criterion gives the frequency domain equivalence for the causality condition of the time
domain. It is that H( f) must satisfy the condition
‡ ∆t corresponds to dx of Eq. (2–47).
q
L-q
| ln |H(f)| |
1 + f 2 df 6q