082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

758 Chapter 24 The Fourier transform
WeshallnowseehowFouriertechniquescanstillbeusefulbyintroducingtheFourier
transformwhichisusedextensivelyincommunicationsengineeringandsignalprocessing.Forexample,itcanbeusedtoanalysetheprocessesof
modulation,whichinvolves
superimposinganaudiosignalontoacarriersignal,anddemodulation,whichinvolves
removing the carrier signaltoleave the audio signal.
24.2 THEFOURIERTRANSFORM--DEFINITIONS
f(t)
1
0
a
Figure24.1
Anon-periodic
function.
b
t
Under certain conditions it can be shown that a non-periodic function, f (t), can be
expressed notasthe sum ofsineand cosinewaves but asanintegral. Inparticular,
where
f(t)=
Provided
∫ ∞
0
A(ω) = 1 π
A(ω)cosωt +B(ω)sinωtdω (24.1)
∫ ∞
−∞
f(t)cosωtdt and B(ω) = 1 π
∫ ∞
(1) f(t)andf ′ (t) are piecewise continuous inevery finiteinterval, and
(2) ∫ ∞
|f (t)|dt exists
−∞
−∞
f (t)sinωtdt (24.2)
then the above Fourier integral representation of f (t) holds. At a point of discontinuity
of f (t) the integral representation converges to the average value of the right- and lefthand
limits. As with Fourier series, an equivalent complex representation exists which
is, infact, more commonly used:
Fourier integral representationof f (t):
where
f(t)= 1
2π
F(ω) =
∫ ∞
∫ ∞
−∞
−∞
F(ω)e jωt dω (24.3)
f (t)e −jωt dt (24.4)
Thereisnouniversalconventionconcerningthedefinitionoftheseintegralsandanumber
of variants are still correct. For instance, some authors write the factor in the
1
2π
1
second integral rather than the first, while others place a factor √ in both, giving
2π
some symmetry to the equations. There is also variation in the location of the factors
e −jωt ande jωt .Weshallusedefinitions(24.3)and(24.4)throughoutbutitisimportantto
be aware of possible differences when consulting other texts.
Equations (24.3) and (24.4) form what is called a Fourier transform pair. The
Fourier transform of f (t) is F(ω) which is sometimes written F{f (t)}. Similarly
f (t) in Equation (24.3) is the inverse Fourier transform of F(ω), usually denoted
F −1 {F(ω)}.