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

110 ANALYSIS AND TRANSMISSION OF SIGNALS
Setting x = t + i in Eq. (3.72a) yields
1/f g
(i) = L: g(x)g(x - i) dx
In this equation, x is a dummy variable and could be replaced by t. Thus,
1/f g
(i) = L: g(t)g(t ±i)dt
(3.72b)
This shows that for a real g (t), the autocorrelation function is an even function of i, that is,
1/f g
(i) = 1/f g
(-i)
(3.72c)
There is, in fact, a very important relationship between the autocorrelation of a signal and
its ESD. Specifically, the autocorrelation function of a signal g(t) and its ESD W g
(J) form a
Fourier transform pair, that is,
(3.73a)
Thus,
W g (J) = F {i/fg (i)} = L: 1/f g
(i)e - f l :n:f r d i
(3.73b)
1/f g (i) = F - 1 { Wg (J)} = L: W g
(J)e-j 2 :n:f r df
(3.73c)
Note that the Fourier transform of Eq. (3.73a) is performed with respect to i in place of t.
We now prove that the ESD 1J.J g (j) = I G(j) 1 2 is the Fourier transform of the autocorrelation
function 1/f g
( i). Although the result is proved here for real signals, it is valid for complex signals
also. Note that the autocorrelation function is a function of i, not t. Hence, its Fourier transform
is J i/fg (i)e-1 2 :n:fr d-r. Thus,
.r[i/f g
(i)] = L: e- J
2 :n:fr [L: g(t)g(t + i) dt] di
= L: g(t) [L: g(i + t)e-j l :n:fr di] dt
The inner integral is the Fourier transform of g( i + t) , which is g( i) left-shifted by t. Hence, it
is given by G(j)J 2 :n:fi, in accordance with the time-shifting property in Eq. (3.32a). Therefore,
This completes the proof that
.r[i/f g
(i)] = G(j) L: g(t)ef l :n:ftdt = G(j)G(-f) = 1 Gif) l 2
(3.74)
A careful observation of the operation of correlation shows a close connection to convolution.
Indeed, the autocorrelation function 1/f g
( i) is the convolution of g ( i) with g ( -i)
because
g(i) * g(-i) = L: g(x)g[-(i - x)] dx = L: g(x)g (x - i) dx = 1/f g
(i)
Application of the time convolution property fEq. (3.44)] to this equation yields Eq. (3.74).