B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

3
ANALYSIS AND
TRANSMISSION OF
SIGNALS
E
lectrical engineers instinctively think of signals in terms of their frequency spectra
and think of systems in terms of their frequency responses. Even teenagers know about
audio signals having a bandwidth of 20 kHz and good-quality loud speakers responding
up to 20 kHz. This is basically thinking in the frequency domain. In the last chapter we
discussed spectral representation of periodic signals (Fourier series). In this chapter we extend
this spectral representation to aperiodic signals.
3.1 APERIODIC SIGNAL REPRESENTATION BY
FOURIER INTEGRAL
Applying a limiting process, we now show that an aperiodic signal can be expressed as a
continuous sum (integral) of everlasting exponentials. To represent an aperiodic signal g(t)
such as the one shown in Fig. 3.la by everlasting exponential signals, let us constrnct a
new periodic signal gT0 (t) formed by repeating the signal g(t) every To seconds, as shown in
Fig. 3.1 b. The period To is made long enough to avoid overlap between the repeating pulses. The
periodic signal gT 0 (t) can be represented by an exponential Fourier series. If we let To ➔ oo,
the pulses in the periodic signal repeat after an infinite interval, and therefore
lim gT0(t) = g(t)
To--'>OO
Thus, the Fourier series representing gTo(t) will also represent g(t) in the limit To ➔ oo.
The exponential Fourier series for gT 0 (t) is given by
in which
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gT o (t) = L D n J ncv ot
D n = -
1 !
To/2
To -To/2
n=-oo
gT o (t)e-; n cvot dt
(3.1)
(3.2a)