Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 397

in Figs. 9.3(c) and (d) for the following two cases:

ω s ≥ 4 πβ ;

case I

case II ω s < 4 πβ.

When the sampling rate ω s ≥ 4 πβ , no overlap exists between consecutive repli-
cas in X s ( ω ). However, as the sampling rate ω s is decreased such that ω s < 4 πβ ,
adjacent replicas overlap with each other. The overlapping of replicas is referred
to as aliasing, which distorts the spectrum of the original baseband signal x ( t )
such that x ( t ) cannot be reconstructed from its samples. To prevent aliasing, the
sampling rate ω s ≥ 4 πβ . This condition is referred to as the sampling theorem
and is stated in the following. †

Sampling theorem A baseband signal x ( t ) , band-limited to 2 πβ radians/s, can
be reconstructed accurately from its samples x (kT) if the sampling rate ω s ,in
radians/s, satisfies the following condition:

ω s ≥ 4 πβ. (9.6a)

Alternatively, the sampling theorem may be expressed in terms of the sampling
rate f s = ω s / 2 π in samples/s, or the sampling interval T s . To prevent aliasing,

sampling rate (samples/s) f s ≥ 2 β ; (9.6b)

or

sampling interval T s ≤ 1 / 2 β. (9.6c)

The minimum sampling rate f s (Hz) required for perfect reconstruction of the
original band-limited signal is referred to as the Nyquist rate .

The sampling theorem is applicable for baseband signals, where the sig-
nal contains low-frequency components within the range 0 − β Hz. In some
applications, such as communications, we come across bandpass signals that
also contain a band of frequencies, but the occupied frequency range lies
within the band β 2 − β 1 Hz with β 1 = 0. In these cases, although the max-
imum frequency of β 2 Hz implies the Nyquist sampling rate of 2 β 2 H zit
is possible to achieve perfect reconstruction with a lower sampling rate (see
Problem 9.8).

† The sampling theorem was known in various forms in the mathematics literature before its
application in signal processing, which started much later, in the 1950s. Several people
developed independently or contributed towards its development. Notable contributions,
however, were made by E. T. Whittaker (1873-1956), Harry Nyquist (1889-1976), Karl
Kupfm uller (1897-1977), V. A. Kotelnikov (1908-2005), Claude Shannon (1916-2001), and
I. Someya.