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

266 SAMPLING AND ANALOG-TO-DIGITAL CONVERSION
In the frequency domain, we have
00
G1 (f) = H(f) L Qn G(f - nfs)
n=-oo
00
= sine (n/T p
) L Qn G(f - nfs)
n=-(X)
(6.23)
Because
g(t) = L g1 (kT,)8(t - kT 5 )
we can apply the sampling theorem to show that
k
1
V
G(f) = - L G1 (J + mfs)
Ts m
1
= T L sine 2
s m
[ (2nf + m2nfs)T · ]
P L Qn G(f + mfs - nfs)
n
= :E(: L Qn sinc [(nf+(n+C)nj) Tp]) cu + cfs ) (6.24)
C s n
The last equality came from the change of the summation index C = m - n.
We can define frequency responses
1
Fc (f) = T s L Qn sine [(nf + (n + C)n:f,)T p
]
n
This definition allows us to conveniently write
G(J) = L Fe (J)G1 (f + Cf,)
t
(6.25)
For the low-pass signal G(J) with bandwidth B Hz, applying an ideal low-pass (interpolation)
filter will generate a distorted signal
Fo(f)G(f)
(6.26a)
in which
(6.26b)
It can be seen from Eqs. (6.25) and (6.26) that the practically sampled signal already contains
a known distortion Fo(f).