Signals and Systems - Electrical Engineering

438 CHAPTER 7: Sampling Theory

Origins of the Sampling Theory — Part 2

As mentioned in Chapter 0, the theoretical foundations of digital communications theory were given in the paper “A Math-

ematical Theory of Communication” by Claude E. Shannon in 1948 [51]. His results on sampling theory made possible the

new areas of digital communications and digital signal processing.

Shannon was born in 1916 in Petoskey, Michigan. He studied electrical engineering and mathematics at the University

of Michigan, pursued graduate studies in electrical engineering and mathematics at MIT, and then joined Bell Telephone

Laboratories. In 1956, he returned to MIT to teach.

Besides being a celebrated researcher, Shannon was an avid chess player. He developed a juggling machine, rocket-powered

frisbees, motorized Pogo sticks, a mind-reading machine, a mechanical mouse that could navigate a maze, and a device that

could solve the Rubik’s CubeTMpuzzle. At Bell Labs, he was remembered for riding the halls on a unicycle while juggling

three balls [23, 52].

7.3.1 Sampling of Modulated Signals

The given Nyquist sampling rate condition applies to low-pass or baseband signals. Sampling of

band-pass signals is used for simulation of communication systems and in the implementation of

modulation systems in software radio. For modulated signals it can be shown that the sampling rate

depends on the bandwidth of the message or modulating signal, not on the absolute frequencies

involved. This result provides a significant savings in the sampling, as it is independent of the car-

rier. A voice message transmitted via a satellite communication system with a carrier of 6 GHz, for

instance, would only need to be sampled at about a 10-KHz rate, rather than at 12 GHz as determined

by the Nyquist sampling rate condition when we consider the frequencies involved.

Consider a modulated signalx(t)=m(t)cos(ct)wherem(t)is the message and cos(ct)is the carrier

with carrier frequency

c>> max

wheremaxis the maximum frequency present in the message. The sampling ofx(t)with a sampling

periodTsgenerates in the frequency domain a superposition of the spectrum ofx(t)shifted in fre-

quency bysand multiplied by 1/Ts. Intuitively, to avoid aliasing the shifting in frequency should

be such that there is no overlapping of the shifted spectra, which would require that

c+max−s< c−max ⇒s> 2 max or Ts<

π

max

Thus, the sampling period depends on the bandwidthmaxof the messagem(t)rather than on the

maximum frequency present in the modulated signalx(t). A formal proof of this result requires the

quadrature representation of band-pass signals typically considered in communication theory [16].

If the messagem(t)of a modulated signalx(t)=m(t)cos(c)has a bandwidthBHz,x(t)can be reconstructed

from samples taken at a sampling rate

fs≥ 2 B

independent of the frequencycof the carriercos(c).