122 Chapter 1 Fourier Series and Integrals
=^1
2 ∑∞
−∞f(nπ
)
exp(−inπω
)
, −<ω<.By utilizing Eq. (1) again, we can reconstructf(t):
f(t)=∫
−C(ω)exp(iωt)dω=^1
2
∑∞
−∞f(
nπ
)∫
−exp(
−inπω
)
exp(iωt)dω.Carrying out the integration and using the identity
sin(θ )=(eiθ−e−iθ)
2 i ,we find
f(t)=∑∞
−∞f(
nπ
)sin(t−nπ)
t−nπ. (3)This is the main result of the sampling theorem. It says that the band-limited
functionf(t)may be reconstructed from the samples offatt=0,±π/,....
It is difficult to determine what functions are actually band limited. However,
the process usually works quite well.
In practice, we must use a finite series to approximate the function
f(t)∼=∑N
−Nf(nπ
)sin(t−nπ)
t−nπ. (4)
Since the sampled values all come from the interval−Nπ/toNπ/,the
series cannot attempt to approximate the function outside that interval. An
animation on the CD shows the effects of choosingNand.
Example.
The function
f(t)= t(^2) + 2 t
( 1 +t^2 )^2
is not band limited but can be approximated satisfactorily from a finite por-
tion of the sum as in Eq. (4). Figure 13 shows results forN=100 (that
is, 201 terms) and=4 and 10. The target function is dashed. Notice the
improvement.