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EE 102 spring 2001-2002 Handout #23 Lecture 11 The Fourier transform definition examples the Fourier transform of a unit step the Fourier transform of a periodic signal properties the inverse Fourier transform 11–1
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The Fourier transform we’ll be interested in signals defined for all t the Fourier transform of a signal f is the function F ( ω ) = −∞ f ( t ) e jωt dt F is a function of a real variable ω ; the function value F ( ω ) is (in general) a complex number F ( ω ) = −∞ f ( t ) cos ωt dt j −∞ f ( t ) sin ωt dt • | F ( ω ) | is called the amplitude spectrum of f ; F ( ω ) is the phase spectrum of f notation: F = F ( f ) means F is the Fourier transform of f ; as for Laplace transforms we usually use uppercase letters for the transforms ( e.g. , x ( t ) and X ( ω ) , h ( t ) and H ( ω ) , etc.) The Fourier transform 11–2
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Fourier transform and Laplace transform Laplace transform of f F ( s ) = 0 f ( t ) e st dt Fourier transform of f G ( ω ) = −∞ f ( t ) e jωt dt very similar definitions, with two differences: Laplace transform integral is over 0 t < ; Fourier transform integral is over −∞ < t < Laplace transform: s can be any complex number in the region of convergence (ROC); Fourier transform: lies on the imaginary axis The Fourier transform 11–3
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therefore, if f ( t ) = 0 for t < 0 , if the imaginary axis lies in the ROC of L ( f ) , then G ( ω ) = F ( ) , i.e. , the Fourier transform is the Laplace transform evaluated on the imaginary axis if the imaginary axis is not in the ROC of L ( f ) , then the Fourier transform doesn’t exist, but the Laplace transform does (at least, for all s in the ROC) if f ( t ) = 0 for t < 0 , then the Fourier and Laplace transforms can be very different The Fourier transform 11–4
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