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

3 .1 Aperiodic Signal Representation by Fourier Integral 67
where w = 2nf. Expressing a + jw in the polar form as J a
2 + w
2
e l tan -I< l, we obtain
G(f) =
1
Ja 2 + (2rrf) 2
e - j ta n-1 ( 2 ;1 )
(3. l 3ba)
Therefore,
IG(f)I =
1
Ja 2 + (2rrf) 2
and
Figure 3.4
e-at u(t) and its
Fourier spectra.
g(t)
0
(a)
t
(b)
The amplitude spectrum IG(f)I and the phase spectrum 0 g (f) are shown in Fig. 3.4b.
Observe that IG(f)I is an even function off , and 0 g (f) is an odd function off , as expected.
Existence of the Fourier Transform
In Example 3. 1 we observed that when a < 0, the Fourier integral for e-atu(t) does not
converge. Hence, the Fourier transform for e-at u(t) does not exist if a < 0 (growing exponentially).
Clearly, not all signals are Fourier transformable. The existence of the Fourier transform
is assured for any g(t) satisfying the Dirichlet conditions, the first of which is*
1_: lg(t) I dt < oo
(3.14)
To show this, recall that je- 1 2 11".fr I = 1. Hence, from Eq. (3.9a) we obtain
IG(f)I :s. 1_: lg(t) I dt
This shows that the existence of the Fourier transform is assured if condition (3 .14) is satisfied.
Otherwise, there is no guarantee. We have seen in Example 3. I that for an exponentially growing
signal (which violates this condition) the Fourier transform does not exist. Although this
condition is sufficient, it is not necessary for the existence of the Fourier transform of a signal.
* The remaining Dirichlet conditions are as follows: In any finite interval, g(t) may have only a finite number of
maxima and minima and a finite number of finite discontinuities. When these conditions are satisfied, the Fourier
integral on the right-hand side of Eq. (3.9b) converges to g(t) at all points where g(t) is continuous and converges to
the average of the right-hand and left-hand limits of g(t) at points where g(t) is discontinuous.