Sampling Theory and Analog-to-Digital Conversion - Signal Processing of Random Physiological Signals - page 40

P

P ( t )

T

FIGURE 6.4 : The realistic Delta function model. Note the uniform magnitude and finite pulse
width

is known as a “delta function.” The model is attractive in analytical expressions, but the
delta function (model) is really impossible to generate in the real world with real circuit
components. The Realist model is considered as modulation of the original signal by
a uniform train of pulses, with a pulse finite width of “ P ” and interval of the sample
period “ T ,” as shown in Fig. 6.4.
Each pulse represents the instant of time that the value of an analog signal is
multiplied by the delta function. It may be easier to think of the process as grab-
bing a value from the analog signal at each pulse such that the resulting collec-
tion of values is no longer continuous in time but only exists at discrete points in
time.
Sampling Theorem : The sampling theory is generally associated with the works of Nyquist
and Shannon. The Sampling Theorem for band-limited signals of finite energy can be
interpreted in two ways:

1. Every signal of finite energy and Bandwidth W Hz may be completely recovered

from the knowledge of its samples taken at the rate of 2 W per second (called
the Nyquist rate). The recovery is stable in the sense that a small error in read-
ing sample values produces only a corresponding small error in the recovered
signal.

2. Every square-summable sequence of numbers may be transmitted at the rate of

2 W per second over an ideal channel of bandwidth W Hz. By being represented
as the samples of a constructed band limited signal of finite length.