1.11 Applications of Fourier Series and Integrals 121
C. The Sampling Theorem
One of the most important results of information theory is the sampling the-
orem, which is based on a combination of the Fourier series and the Fourier
integral in their complex forms. What the electrical engineer calls a signal is
just a functionf(t)defined for allt. If the function is integrable, there is a
Fourier integral representation for it:
f(t)=∫∞
−∞C(ω)exp(iωt)dω,C(ω)=^1
2 π∫∞
−∞f(t)exp(−iωt)dt.Asignaliscalledband limitedif its Fourier transform is zero except in a
finite interval, that is, if
C(ω)= 0 , for|ω|>.Thenis called the cutoff frequency. Iffis band limited, we can write it in
the form
f(t)=∫
−C(ω)exp(iωt)dω (1)becauseC(ω)is zero outside the interval−<ω<.Wefocusourattention
on this interval by writingC(ω)as a Fourier series:
C(ω)=∑∞
−∞cnexp(
inπω
)
, −<ω<. (2)The (complex) coefficients are
cn=^1
2 ∫
−C(ω)exp(−inπω
)
dω.The point of the sampling theorem is to observe that the integral forcnac-
tually is a value off(t)at a particular time. In fact, from the integral Eq. (1),
we see that
cn=1
2 f(−nπ
)
.
Thus there is an easy way of finding the Fourier transform of a band-limited
function. We have
C(ω)=1
2
∑∞
−∞f(−nπ
)
exp(inπω