DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Figure 16.2 Cosine
waves at exactly 0.5f S
cannot be accurately
represented in a sample
sequence if the phase or
amplitude of the sampled
waveform is arbitrary.
cos 0.5 ωt
Nyquist essentially applied to
signal processing a mathematical
discovery made in 1915 by E.T.
Whittaker. Later contributions
were made by Shannon (in the
U.S.) and Kotelnikov (in Russia).
+0°
+45°
+90°
0 1 2 3 4 5 ts Sampling at exactly 0.5f S
You might assume that a signal whose frequency is
exactly half the sampling rate can be accurately represented
by an alternating sequence of sample values,
say, zero and one. In Figure 16.2 above, the series of
samples in the top row is unambiguous (provided it is
known that the amplitude of the waveform is unity).
But the samples of the middle row could be generated
from any of the three indicated waveforms, and the
phase-shifted waveform in the bottom row has samples
that are indistinguishable from a constant waveform
having a value of 0.5. The inability to accurately analyze
a signal at exactly half the sampling frequency leads to
the strict “less-than” condition in the Sampling
Theorem, which I will now describe.
Harry Nyquist, at Bell Labs, concluded in about 1928
that to guarantee sampling of a signal without the
introduction of aliases, all of the signal’s frequency
components must be contained strictly within half the
sampling rate (now known as the Nyquist frequency). If
a signal meets this condition, it is said to satisfy the
CHAPTER 16 FILTERING AND SAMPLING 143