THE FOURIER TRANSFORM - Signals, Systems, and Transforms - page 241

Harry Nyquist (born February 7, 1889, in Nilsby, Sweden; died April 4, 1976). was
educated in America in electrical engineering and physics. Nyquist worked at Bell
Laboratories from 1934 until 1954. His 1928 paper “Certain Topics in Telegraph
Transmission Theory” established the principles of sampling continuous signals to
convert them to digital signals. The Nyquist sampling theorem showed that the sam-
pling rate must be at least twice the highest frequency component present in the sam-
ple in order to reconstruct the original signal. Nyquist's paper is cited in the first
paragraph of Shannon's classic essay “The Mathematical Theory of Communication”
(1948). In 1932, Nyquist discovered how to determine when negative feedback ampli-
fiers are stable. His criterion, generally called the Nyquist stability theorem, is of great
practical importance. In addition to his theoretical work, he was a prolific inventor and
is credited with 138 patents relating to telecommunications.

Reference: “Nyquist, Harry.” Encyclopedia Britannica, 2007. Encyclopedia Online.
January 5, 2007: http://www.britannica.com/eb/article-9125110

T M /2 (T M = 1/f M ) seconds apart. Hence, according to Shannon's sampling theo-

rem, we must take at least two samples per cycle of the highest frequency compo-
nent in
Figure 5.24 illustrates the requirement that the sampling frequency must be
properly chosen. In this figure, the continuous sine wave represents a signal
The black dots on stems represent the sampled values we obtain
by sampling the signal with a sampling period, therefore,
(rad/s). Note that The dashed sine wave represents
the signal that would be recovered from the sample data. It is seen that, because the
signal was sampled with a sampling frequency less than the Nyquist frequency
the signal recovered from the sample data is not the original signal. In-
stead, a signal is recovered from the sample data, even though
that signal was not part of the original signal, f(t).

f(t).

f(t) = sin(0.9pt).

T S = 2.5 seconds;

v S = 2p/2.5 = 0.8p v S 6 2v M .

(v S 6 2v M ),

g(t) = sin(0.3pt)

Claude Elwood Shannon (born April 30, 1916, in Petoskey, Michigan; died
February 24, 2001) was mathematician and electrical engineer who laid the theoretical
foundations for digital circuits and information theory. Shannon was affiliated with
Bell Telephone Laboratories from 1941 until 1972. He became a member of the facul-
ty at Massachusetts Institute of Technology in 1958. In 1948, Shannon published “A
Mathematical Theory of Communication,” which built on the foundations of other re-
searchers at Bell Labs, such as Harry Nyquist and R.V.L. Hartley, but went far beyond
the earlier works. It established the basic results of information theory so completely
that his framework and terminology are still used.

Reference: “Shannon, Claude.” Encyclopedia Britannica, 2007. Encyclopedia
Britannica Online, January 5, 2007: http://www.britannica.com/eb/article-9125111 .