Fourier Transform - ver 3

Fourier Transform - ver 3 - Fourier Transform Linear...

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Fourier Transform Linear Systems and Signals Dr. J. K. Aggarwal The University of Texas at Austin
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Fourier Transform ± The Fourier Transform of a function x ( t ) is given as ± The Inverse Fourier Transform is given as ± The relationship is one-to-one and unique ± A sufficient condition for the existence of FT is that () {} ( ) jt Fx t X e x t d t ω −∞ == 1 1 2 F Xx e X d t ωω π −∞ xd t t −∞ < ∞
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Example 1 ± 0 t () at ke u t xt ( ) 0 0 01 jt at j t aj t X xte d t ke e dt e k aj k k ω −∞ −− −+ = = = =− = + a>0
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Example 2 ± 2 τ 2 t () x t A - A ( ) 0 2 0 2 0 2 0 2 22 11 2 1c o s jt jj XA e d t A e d t ee AA A j ωω ω τωτ ωτ −− =− + + ⎛⎞ +− ⎜⎟ ⎝⎠ 2 ∫∫
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Example 3 ± () 02 22 1 11 1 21 2 X ω ωω −∞≤ ≤− ≤≤ =− ≤∞ 112 1 2 1 1 2 2212 1 2 jt xt X e d x t ed ee e jt jt jt ωωω π −∞ −− = =+ + + ∫∫
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Example 3 ...contd () 22 21 2 2 14 4 2sin sin sin 2 4sin2 jt j t jt jt j t jt xt e e e e e e jt jt jt jt j t jt jt jt tt π −− =− + +− + 1 2sin 2 sin t t t
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Example 4 ± Consider a pulse 2 τ 2 t () x t m V 2 2 2 2 sin 2 2 jt m m m X Ve d t e V j V ω = = =
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Example 4 ...contd ± 2 τ 2 t () xt m V T 0 Recall the Fourier Series for the periodic function sin sin , sin m m nn m V CC T V As T n XV ωω ττ ωτ 00 0 2 2 =⇒ = Τ 2 2 →∞ 2 ∴= 2
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Example 5 ± 0 t t Ae ε A () , t xt A e => 0 0 0 2 2 jj tt XA e e d t A e e d t AA A εω ω −− −∞ 2 =+ −+ = + ∫∫ ( ) ( ) , 2 Now as x t A for all t ωπ δ →0, ∴→ t −∞ A
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Example 5 ...contd Consider It is zero, except at
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Fourier Transform - ver 3 - Fourier Transform Linear...

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