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54 f(t)= -e eq if t <0 if t>0 (a < 0) HEL

54 f(t)= -e eq if t <0 if t>0 (a < 0) HEL

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the fourier transform pair for the given function (Solve 54)

 

One example is given for reference . Must follow the same rule :

Example 6.10: Fourier transform pair Find the Fourier transform pair for Solution By Eq. (6.30), we have Ch.6 Fourier Analysis [0 if t<0 0 - - if :>0 f(t)=< 1 f(a)=e-are-l dt = 1 f(0) = -√2! * √√2.T √√2r The inverse Fourier transform is determined via Eq. (6.31), as f(t)= 1 f(t)= √√2.r 1 1 = a +i 1 (a-iw)(cosat +i sin cot) a² +0² _jax dw= do- 1 X (a > 0) 0 This is the complex Fourier integral representation of f(t). The real form of the Fourier integral is easily derived from here as follows. Multiply and divide the integrand by the conjugate of a +ia, and use Euler's formula, to obtain 1 2л 18 1 a+ia which is the real form of the Fourier integral for f(t). I 1 a+io 1a cosat +ia sin at-iacos ax + @sin axt 2r 21-0 a² +0² doo el do Since sin ot and acos x are odd functions of , and cos at and csin or are even functions of this reduces to f(t)= a cosax+cosin ox a² + a² 199 do
Transcribed Image Text:Example 6.10: Fourier transform pair Find the Fourier transform pair for Solution By Eq. (6.30), we have Ch.6 Fourier Analysis [0 if t<0 0 - - if :>0 f(t)=< 1 f(a)=e-are-l dt = 1 f(0) = -√2! * √√2.T √√2r The inverse Fourier transform is determined via Eq. (6.31), as f(t)= 1 f(t)= √√2.r 1 1 = a +i 1 (a-iw)(cosat +i sin cot) a² +0² _jax dw= do- 1 X (a > 0) 0 This is the complex Fourier integral representation of f(t). The real form of the Fourier integral is easily derived from here as follows. Multiply and divide the integrand by the conjugate of a +ia, and use Euler's formula, to obtain 1 2л 18 1 a+ia which is the real form of the Fourier integral for f(t). I 1 a+io 1a cosat +ia sin at-iacos ax + @sin axt 2r 21-0 a² +0² doo el do Since sin ot and acos x are odd functions of , and cos at and csin or are even functions of this reduces to f(t)= a cosax+cosin ox a² + a² 199 do
of u 54 f(t)=- -at if t <0 if t>0 eq (a < 0)
Transcribed Image Text:of u 54 f(t)=- -at if t <0 if t>0 eq (a < 0)
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