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

40 SIGNALS AND SIGNAL SPACE
First of all, it is clear that the set of exponentials iJ nw ot (n = 0, ±1, ±2, ...) is orthogonal
over any interval of duration T o = 21r / w o , that is,
m -1- n
m=n
(2.58)
Moreover, this set is a complete set. 1·2 From Eqs. (2.53) and (2.56), it follows that
a signal g(t) can be expressed over an interval of duration T o second(s) as an
exponential Fourier series
where [see Eq. (2.53)]
g (t) = L D n J nw ot
n=-oo
00
= L D n
J n2 nfo t
n=-oo
D n = _!__ { g(t)e-j n2 nfo t dt
(2.60)
To }T o
(2.59)
The exponential Fourier series in Eq. (2.59) consists of components of the form iJ n Znfot with
n varying from -oo to oo. It is periodic with period T o .
Example 2.3 Find the exponential Fourier series for the signal in Fig. 2.13b.
Figure 2.13
A periodic
signal.
In this case, To = Jr, 21rf o = 21r /T o = 2, and
<p(t) = L D n
ei2nt
n=-oo