Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 67

Frequency Domain

X ð f Þ¼ X ð f Þ P ð f Þ
¼ X ð f Þ f s X

1

d ð f kf s Þ

kn ¼1

¼ f s X

1

X ð f kf s Þ d ð t nT s Þ

n ¼1

where we have been used Tables to find that:

X

d ð t nT s Þ F f s X

1 1

d ð f kf s Þ; ð 2 : 1 Þ

n ¼1 k ¼1

X ð f Þ d ð f kf s Þ¼ X ð f kf s Þ; ð 2 : 2 Þ

x ð t Þ d ð t nT s Þ¼ x ð nT s Þ d ð t nT s Þ: ð 2 : 3 Þ

The sampling process in the frequency domain is illustrated in Fig. ( 2.3 ). From
the above equations and Fig. ( 2.3 ) one can deduce that X s (f) is essentially a col-
lection of repeated versions of X(f) scaled by f s . These repeated versions are often
referred to as ''images''. Hence, when a signal x(t) is sampled, its spectrum X(f) is:

1. scaled by f s and
2. repeated every f s .

Cintinuous−time Spectrum

X ( f )

1

f , Hz

−B 0 B

Spectrum of Sampled Signal

X s ( f )

f s X(f+1.f s ) f s X ( f )

f s X(f−1.f s )

f s

f , Hz

−B 0 B f s −B f s +B

−f s /2 f s /2 2f s

f s

Normalised−frequency Spectrum
of Sampled Signal

X s (v)

f s X(v + 1) f s X( v ) f s X(v − 1) f s X(v − 2)

f s

v

−b 0 b

1− b 1+ b

Note: b = B / f s

−0.5 0.5 1 2

Fig. 2.3 The discrete-time Fourier transform (DTFT) of a sampled signal