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

68 ANALYSIS AND TRANSMISSION OF SIGNALS
For example, the signal ( sin at)/ t, violates condition (3.14 ), but does have a Fourier transform.
Any signal that can be generated in practice satisfies the Dirichlet conditions and therefore has
a Fourier transform. Thus, the physical existence of a signal is a sufficient condition for the
existence of its transform.
Linearity of the Fourier Transform (Superposition Theorem)
The Fourier transform is linear; that is, if
then for all constants a1 and a2, we have
and
(3.15)
The proof is simple and follows directly from Eq. (3.9a). This theorem simply states that
linear combinations of signals in the time domain correspond to linear combinations of their
Fourier transforms in the frequency domain. This result can be extended to any finite number
of terms as
I:akgk (t) ¢:::=}
k
I:ak Gkif)
k
for any constants {ad and signals {gk (t)}.
Physical Appreciation of the Fourier Transform
To understand any aspect of the Fourier transform, we should remember that Fourier representation
is a way of expressing a signal in terms of everlasting sinusoids, or exponentials.
The Fourier spectrum of a signal indicates the relative amplitudes and phases of the sinusoids
that are required to synthesize that signal. A periodic signal's Fourier spectrum has finite
amplitudes and exists at discrete frequencies (f and its multiples). Such a spectrum is easy
to visualize, but the spectrum of an aperiodic signal is not easy to visualize because it has a
continuous spectrum that exists at every frequency. The continuous spectrum concept can be
appreciated by considering an analogous, more tangible phenomenon. One familiar example
of a continuous distribution is the loading of a beam. Consider a beam loaded with weights
Di, D2, D3, . .. , D n units at the uniformly spaced points xi, x2, . .. , X n , as shown in Fig. 3.5a.
The total load WT on the beam is given by the sum of these loads at each of the n points:
n
Consider now the case of a continuously loaded beam, as shown in Fig. 3.5b. In this case,
although there appears to be a load at every point, the load at any one point is zero. This does
Figure 3.5
Analogy for
Fourier
transform.
(a)
(b)