Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 19

where Xðf Þ is assumed to be the original baseband spectrum while X s ðf Þ is its sampled signal spec-
trum, consisting of the original baseband spectrum Xðf Þ and its replicas Xðf nf s Þ . Since Equation
(2.2) is a well-known formula, the derivation is omitted here and can be found in well-known texts
(Ahmed and Nataranjan, 1983; Ambardar, 1999; Alkin, 1993; Oppenheim and Schafer, 1975; Proakis
and Manolakis, 1996).
Expanding Equation (2.2) leads to the sampled signal spectrum in Equation (2.3) :

1
T Xðf þ f s Þþ

1
T Xðf Þþ

1
T Xðf f s Þþ /

X s ðfÞ¼ / þ (2.3)

Equation (2.3) indicates that the sampled signal spectrum is the sum of the scaled original spectrum and
copies of its shifted versions, called replicas . Three possible sketches based on Equation (2.3) can be
obtained. Given the original signal spectrum Xðf Þ plotted in Figure 2.6 (a), the sampled signal
spectrum according to Equation (2.3) is plotted in Figure 2.6 ( b), where the replicas 1

T Xðf Þ , 1 T Xðf f s Þ ,

1

T Xðf þ f s Þ , . , have separations between them. Figure 2.6 (c) shows that the baseband spectrum and its
replicas, 1

T Xðf Þ , 1 T Xðf f s Þ , 1

T Xðf þ f s Þ , . , are just connected, and finally, in Figure 2.6 (d), the original

X ()

(a)

1.

Bf =

max

f

B

− B 0

Xf

s ()

Lowpass filter

1
T

(b)

f s
2

f

− B B

− f B
s

− f s − f B

s

f B
s

− f s f B

s

+

0

Xf

s ()

Folding frequency/Nyquist limit

1
T

(c)

f

−− f B

s

− f s f B

s

+

f s

− B 0 B

Xf

s ()

1
T

(d)

f

f B
s

+

−− f B

s

− f s − B − f B

s

0 f B

s

− B f s

FIGURE 2.6
Plots of the sampled signal spectrum.