Page 38 - Introduction to Information Optics

1.4. Band-limited Analysis 23

1.4.1. DEGREES OF FREEDOM

Let f(t) be a band-limited signal the spectrum of which extends from zero
frequency to a definite limit of v m. Assume that f(t) extends over a time interval
of T, where v mT » 1. Strictly speaking, a band-limited signal cannot be time
limited or vice versa. Now a question arises: How many sampling points (i.e.,
degrees of freedom) are required to describe the function /(r), over T,
uniquely? To answer this fundamental question we present an example.
First, we let f(t) repeat itself at every time interval of T; that is,

f(t) = f(t + T).

Thus, the function f(t) over the period T, can be expanded in a Fourier series:
M
fit) - X C»QXp(i2nnv 0t), (1.81)
n- -M
where v 0 = 1/T, and M — v mT.
From this Fourier expansion, we see that f(t) contains a finite number of
terms:

N = 2M + 1 =2v m T+ 1,
which includes the zero-frequency Fourier coefficient C 0. If the duration T is
sufficiently large, we see that the number of degree of freedom reduces to

N*2v mT. (1.82)

In other words, it requires a total of N equidistant sampling points of f(t\ over
T, to describe the function
T 1
'• = * = *r-

where t s is known as the Nyquist sampling interval and the corresponding
sampling frequency is

/, = 7 = 2v M, (1.84)
s
which is known as the Nyquist sampling rate or sampling frequency. Thus, we
see that

,/>2v m. (1.85)