Sampling - DSP Using MATLAB and Wavelets - page 160

5.1 Sampling

Converting an analog signal to a digital one is a necessary step for a computer to
analyze a signal: modern computers are digital machines, and can store only digital
values.
In the continuous function x(t), we replace t, the continuous variable, with nT s ,
a discrete value. We use n to index the discrete array, and T s is the sampling period,
the amount of time between two samples. The sampling time is the inverse of the
sampling frequency f s , that is, T s =

1

f s .
Before we can talk about sampling, let us call the frequency range of the sig-
nal that we are interested in the bandwidth. Sometimes, we simply use the highest
frequency as the bandwidth, implying that we are interested in all frequencies be-
tween 0 Hz and the maximum. Other times, it may be limited, e.g., the visible light
spectrum.
The term critical sampling applies to the case when the sampling rate is exactly
twice the bandwidth, B. This means that we will record the signal just fast enough
to properly reconstruct it later, i.e., f s = 2B. We get this value based on Nyquist's
criterion, f s 2B, and choosing the lowest possible sampling frequency.
Oversampling is taking samples more frequently than needed (or more than 2
times the bandwidth). This results in more samples than are needed. You can use
oversampling, but you may not want to due to hardware limitations, like not being
able to process it fast enough, or not being able to store it in memory. Oversampling
may be something you would want to do if you are working with cheap hardware,
such as an inexpensive digital-to-analog converter, which appears to be the case
with CD players [6]. An inexpensive digital-to-analog converter may use a simple
step function to recreate the signal. Therefore, the more samples it has to work
with, the smaller the time between the samples, and the better it approximates
the original. As a counter example, reading (and storing) the temperature every
microsecond would produce a massive volume of data with no gain for predicting
tomorrow's weather.
Undersampling occurs when we do not take samples often enough, less than twice
the bandwidth. You would not want to use this, since you cannot reconstruct the
signal. Also, any analysis done on an undersampled signal will likely be erroneous
(as we say in computer programming, \garbage in, garbage out").
x[n] = x(nT s ) describes the sampling process. T s is the sampling time, while
n is an index. This means that the continuous signal x is converted into a digital
representation. x[n] is not exactly the same as x(t). It cannot be, since x(t) is
continuous. At best, x[n] is a good approximation of x(t).
The period of a signal is the length of time before it repeats itself. For a sinusoid,