Page 39 - Introduction to Information Optics

1. Entropy Information and Optics

- 3 Vm - Vm 0 ^ m 3 Vm

Fig. 1.6. Periodic representation of the Fourier spectrum.

In words, the sampling frequency has to be at least equal or twice the highest
frequency limit of f(t).
By taking the Fourier transform of /(t), we have

f<*
F(v) = f(t)e\p(-i2nvt)dt.

Since f(t) is assumed band limited, F(v) may be repeated over the frequency
domain as shown in Fig. 1.6. Thus, the corresponding Fourier expansion can
be written as

(innv\
, v
V
V rn )
(1.86)
where

/*V m / \
K n= F(v)exp(-i —Uv.
J-v m V W
F(v) is the Fourier transform of f(t), which can be written as

f(t) = F(v)exp(i2nvt)dv. (1.87)
J - v m
In particular, at sampling points t = —n/2v m, we have

—- = F(v)exp -/ (1.88)
m/ .; — v