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Unformatted text preview: MODULE 2 REVIEW OF CONTINUOUS SIGNALS AND SYSTEMS • Fourier Integral and Properties • Gibb's Phenomena • Uncertainty Principle • Fourier Transform Examples Page 2.1 INDEX Review of Continuous Signals and Systems Fourier Integral Inversion Theorem Proof of Fourier Inversion Formula Properties of the Fourier Integral Gibbs' Phenomena Ideal LowPass Filter Explaining Gibbs' Phenomena The Uncertainty Principle Statement of Uncertainty Principle Comments on Uncertainty Principle Fourier Transform Examples Impulse Function and Generalized Functions MAIN INDEX Page 2.2 REVIEW OF CONTINUOUS SIGNALS AND SYSTEMS index Reference : Papoulis, The Fourier Integral and its Applications. Notation • In this module we shall denote a continuoustime signal by (for example) x ( t ), and its Fourier transform by X ( Ω ). • No ambiguity with discretetime signals x ( n ) since only continuous signals are treated in this module. • Later, relationships between continuous and discrete (sampled) signals will be explored (Sampling Theorem, filter design). • Then specific notation will be used to discriminate continuous and discrete signals, e.g., x a ( t ) (continuous) vs. x d ( n ) (discrete) • However, we shall always use t = continuous time (occasionally r,s, τ ) n = discrete time (occasionally k,l,m ) Ω = continuous frequency ϖ = discrete frequency Page 2.3 Fourier Integral index • Given a continuoustime signal x ( t ), denote the Fourier integral or Fourier transform X ( Ω ) = ∫ R x ( t ) ej Ω t dt provided the integral exists for every Ω . • The function X ( Ω ) is generally complex: X ( Ω ) = Re { X ( Ω ) } + j Im { X ( Ω ) } = X R ( Ω ) + jX I ( Ω ) =  X ( Ω )  exp { j ∠ X ( Ω ) } where  X ( Ω )  = { [ X R ( Ω )] 2 + [ X I ( Ω )] 2 } 1/2 = { X ( Ω ) X *( Ω ) } 1/2 = magnitude spectrum of X ( Ω ) ∠ X ( Ω ) = tan1 { X I ( Ω )/ X R ( Ω ) } = phase spectrum of X ( Ω ) Page 2.4 Inversion Theorem index • The signal x ( t ) can be uniquely expressed in terms of its Fourier transform X ( Ω ) (if it exists): x ( t ) = π 2 1 ∫ R X ( Ω ) e j Ω t d Ω • If the Fourier integral exists, then x ( t ) and X ( Ω ) form a Fourier transform pair , denoted by x ( t ) ℑ ↔ X ( Ω ) and x ( t ) is the inverse Fourier transform of X ( Ω ). Existence of Fourier Integral • A sufficient (but not necessary ) condition for the existence of X ( Ω ) is that x ( t ) be absolutely integrable : ∫ R  x ( t )  dt < . ∞ • There are many functions that do not satisfy this condition but still have Fourier transforms (not Fourier integrals) most of these involve singularity functions (impulses) treated as distributions ....
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 Spring '04
 AlanC.Bovik
 Digital Signal Processing, Signal Processing, dt

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