Signals and Systems - Electrical Engineering

424 CHAPTER 7: Sampling Theory

(a)

(b)

(c)

1

−Ωmax Ωmax

X(Ω)

Ω

No aliasing

Ωmax

Ωs≥ 2 Ωmax

Xs(Ω)

1/Ts

Ωs Ω

· · · · · ·

Aliasing

Ωs< 2 Ωmax

Xs(Ω)

1/Ts

Ωs − Ωmax

Ωs=Ωmax Ω

· · · · · ·

FIGURE 7.2

(a) Spectrum of band-limited signal, (b) spectrum of sampled signal when satisfying the Nyquist sampling rate

condition, and (c) spectrum of sampled signal with aliasing (superposition of spectra, shown in dashed lines,

gives a constant shown by continuous line).

possible to recover the original continuous-time signal from the sampled signal, and thus the sampled

signal does not share the same information with the original continuous-time signal. This phenomenon is

calledfrequency aliasingsince due to the overlapping of the spectra some frequency components of the

original continuous-time signal acquire a different frequency value or an “alias.”

n When the spectrum of x(t)does not have a finite support (i.e., the signal is not band limited) sampling

using any sampling period Tsgenerates a spectrum of the sampled signal consisting of overlapped shifted

spectra of x(t). Thus, when sampling non-band-limited signals frequency aliasing is always present. The

only way to sample a non-band-limited signal x(t)without aliasing—at the cost of losing information

provided by the high-frequency components of x(t)— is by obtaining an approximate signal xa(t)that

lacks the high-frequency components of x(t), thus permitting us to determine a maximum frequency for it.

This is accomplished byantialiasing filteringcommonly used in samplers.

A band-limited signalx(t)—that is, its low-pass spectrumX()is such that

|X()|= 0 for||> max (7.11)