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

252 SAMPLING AND ANALOG-TO-DIGITAL CONVERSION
Figure 6. 1
Sampled signal
and its Fourier
spectra.
g(t)
G (f)
t -
(a)
(bl
co-
J -
/5T, (I)
1 1 1 1 1 IJJ I
(c)
' g(t)
(, r-
w-
(d)
(e)
0
B fs J-
sampled signal g(t) and the original analog signal g(t) is
g(t) = g(t)8T s (t) = L g (nT s )8 (t - nT s ) (6. 1)
Because the impulse train 8T s (t) is a periodic signal of period T.,, it can be expressed as
an exponential Fourier series, already found in Example 3.11 as
00
1 "
81 (t) = T L.., e1nw , t
s n=-oo
2n
Ws = - = 2nfs
T s
(6.2)
Therefore,
g(t) = g (t)8T s (t)
1
00
=
Ts
L g(t)J n2 nf5 t
n=-oo
(6.3)
To find G(f), the Fourier transform of g(t), we take the Fourier transform of the summation
in Eq. (6.3). Based on the frequency-shifting property, the transform of the nth term is shifted