Signals and Systems - Electrical Engineering

430 CHAPTER 7: Sampling Theory

FIGURE 7.4

Anti-aliasing filtering

of non-band-limited

signal.

−Ωc

−Ωc

Ωc

Ωc

Ω

1

X(Ω) Xa(Ω)

H(s)

H(jΩ)

Ω

Ω

about 5 KHz (this range of frequencies provides understandable speech in phone conversations). Thus,

when sampling speech an anti-aliasing filter with a cut-off frequency of 5 KHz is chosen and the sampling

rate is then set to 10,000 samples/sec. Likewise, it is also known that an acceptable range of frequencies

from 0 to 22 KHz provides music with good fidelity, so that when sampling music signals the anti-aliasing

filter cut-off frequency is set to 22 KHz and the sampling rate to 44 K samples/sec or higher to provide

good-quality music.

Origins of the Sampling Theory—Part 1

The sampling theory has been attributed to many engineers and mathematicians. It seems as if mathematicians and

researchers in communications engineering came across these results from different perspectives. In the engineering com-

munity, the sampling theory has been attributed traditionally to Harry Nyquist and Claude Shannon, although other famous

researchers such as V. A. Kotelnikov, E. T. Whittaker, and D. Gabor came out with similar results. Nyquist’s work did not deal

directly with sampling and reconstruction of sampled signals but it contributed to advances by Shannon in those areas.

Harry Nyquist was born in Sweden in 1889 and died in 1976 in the United States. He attended the University of North

Dakota at Grand Forks and received his Ph.D. from Yale University in 1917. He worked for the American Telephone and

Telegraph (AT&T) Company and the Bell Telephone Laboratories, Inc. He received 138 patents and published 12 technical

articles. Nyquist’s contributions range from the fields of thermal noise, stability of feedback amplifiers, telegraphy, and

television, to other important communications problems. His theoretical work on determining the bandwidth requirements

for transmitting information provided the foundations for Claude Shannon’s work on sampling theory [33].

As Hans D. Luke [44] concludes in his paper “The Origins of the Sampling Theorem,” regarding the attribution of the

sampling theorem to many authors:

This history also reveals a process which is often apparent in theoretical problem in technology or physics: first

the practicians put forward a rule of thumb, then theoreticians develop the general solution, and finally someone

discovers that the mathematicians have long since solved the mathematical problem which it contains, but in

“splendid isolation.”

nExample 7.3

Consider the two sinusoids

x 1 (t)=cos( 0 t) −∞≤t≤∞

x 2 (t)=cos(( 0 +s)t) −∞≤t≤∞