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

772 Chapter 24 The Fourier transform
f(t) = cos at
F(v)
p
t
–a a
v
Figure24.11
The spectrum ofcosat.
We see that the spectrum of cosat consists of single lines at ω = ±a corresponding to
a singlefrequency component (Figure 24.11).
Example24.13 Find F{sinat}.
Solution Subtractingthepreviousexpressionsfor F{e jta }and F{e −jta }andusingEuler’srelations
wefind
F{e jta }− F{e −jta } =2π(δ(ω −a)−δ(ω +a))
that is,
{ e jta −e −jta }
F = π (δ(ω −a)−δ(ω +a))
2j j
so that
F{sinat} = π (δ(ω −a)−δ(ω +a))
j
24.7 THERELATIONSHIPBETWEENTHEFOURIER
TRANSFORMANDTHELAPLACETRANSFORM
Wehavealreadynoted(Section24.2)thesimilaritybetweentheLaplacetransformand
the Fourier transform.Let usnow look atthis a littlemoreclosely. We have
F{f(t)} =
∫ ∞
−∞
f (t)e −jωt dt and L{f (t)} =
∫ ∞
0
f (t)e −st dt
InthedefinitionoftheLaplacetransform,theparametersiscomplexandwemaywrite
s=σ+jω,sothat
L{f(t)} =
∫ ∞
0
f(t)e −σt e −jωt dt
Thus an additional factor, e −σt , appears in the integrand of the Laplace transform. For
σ > 0thisrepresentsanexponentiallydecayingfactor,thepresenceofwhichmeansthat
theintegralexistsforawidervarietyoffunctionsthanthecorrespondingFourierintegral.
Example24.14 Find,ifpossible,
(a) the Laplace transform
(b) the Fourier transform
of f (t) =u(t)e 3t . Comment uponthe result.