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

774 Chapter 24 The Fourier transform
Fourier transform is applicable to functions with domain −∞ < t < ∞. In some applications
where, for example, t represents not time but a spatial variable, it is often
necessary towork with negative values.
The inverse Fourier transformisgiven by
F −1 {F(ω)} = 1 ∫ ∞
F(ω)e jωt dω (24.10)
2π −∞
ThecorrespondinginverseLaplacetransformrequiresadvancedtechniquesinthetheory
of complex variables which are beyond the scope of this book. The existence of Equation
(24.10) is not quite as advantageous as it may seem because it is often difficult to
perform the required integration analytically.
24.8 CONVOLUTIONANDCORRELATION
Convolution is an important technique in signal and image processing. It provides a
means of calculating the response or output of a system to an arbitrary input signal if
the impulse response is known. The impulse response is the response of the system to
animpulsefunction.Convolvingtheinputsignalandtheimpulseresponseresultsinthe
response to the arbitrary input. Correlation is a second important technique. It can be
used to determine the time delay between a transmitted signal and a received signal as
might occur inradarorsonar detection equipment.
24.8.1 Convolutionandtheconvolutiontheorem
If f (t) andg(t) are two real piecewise continuous functions, their convolution, which
wedenoteby f ∗g, isdefinedasfollows:
Theconvolution of f (t)andg(t):
f∗g=
∫ ∞
−∞
f(λ)g(t − λ)dλ
f ∗gisitselfafunctionoft, andtoshowthisexplicitly wesometimes write (f ∗g)(t).
Note that convolution is an integral with respect to the dummy variable λ. In general,
as can be seen from the limits of integration, λ varies from −∞ to ∞, but in particular
caseswewillseethatthisintervalofintegrationcanbereduced.Intheexampleswhich
follow the precisemeaning ofthe terms f (λ)andg(t − λ) will become apparent.
Because convolution is commutative, that is f ∗g = g ∗ f, the convolution can be
defined equivalently as
∫ ∞
−∞
f(t − λ)g(λ)dλ
Incaseswhen f (t)andg(t)arezerofort < 0thisexpressionreducestothatdefinedfor
the Laplace transform inSection21.9.