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Unformatted text preview: Ve216 Lecture Notes Dianguang Ma Spring 2009 Chapter 3 (Part IV) Fourier Representations of Signals and LTI Systems 3.8 Periodicity Properties of Fourier Rrepresentations • In general, representations that are continuous in one domain are nonperiodic in the other domain. Conversely, representations that are discrete in one domain are periodic in the other domain. TimeDomain Property FrequencyDomain Property continuous nonperiodic discrete periodic periodic discrete nonperiodic continuous 3.8 Periodicity Properties of Fourier Rrepresentations • Continuous or discretetime periodic signals have a series representation in which the signal is represented as a weighted sum of complex sinusoids having the same period as the signal. A discrete set of frequencies is involved in the series; hence, the frequencydomain representation involves a discrete set of weights or coefficients. ∑ ∑ ∫ ∑ < Ω < Ω ∞∞ = = ↔ = = ↔ = N n jk N n jk T t jk k t jk e n x N k X e k X n x dt e t x T k X e k X t x ] [ 1 ] [ ] [ ] [ ) ( 1 ] [ ] [ ) ( ϖ ϖ 3.8 Periodicity Properties of Fourier Rrepresentations • For nonperiodic signals, both continuous and discretetime Fourier transform representations involve weighted integrals of complex sinusoids over a continuum of frequencies. Accordingly, the frequencydomain representation is a continuous function of frequency. ∑ ∫ ∫ ∫ ∞∞ = Ω Ω ∞ ∞ Ω Ω ∞ ∞ ∞ ∞ = ↔ Ω = = ↔ = n n j j n j j t j t j e n x e X d e e X n x dt e t x j X d e j X t x ] [ ) ( ) ( 2 1 ] [ ) ( ) ( ) ( 2 1 ) ( π ϖ ϖ ϖ π ϖ ϖ 3.8 Periodicity Properties of Fourier Rrepresentations • The Fourier representations of discretetime signals are periodic functions of frequency. This is because the discretetime complex sinusoids used to represent the discretetime signals are 2πperiodic functions of frequency. That is, discretetime sinusoids whose frequencies differ by integer multiples of 2π are identical. ∑ ∫ ∑ ∑ ∞∞ = Ω Ω ∞ ∞ Ω Ω < Ω < Ω = ↔ Ω = = ↔ = n n j j n j j N n jk N n jk e n x e X d e e X n x e n x N k X e k X n x ] [ ) ( ) ( 2 1 ] [ ] [ 1 ] [ ] [ ] [ π 3.8 Periodicity Properties of Fourier Rrepresentations • The Fourier representations of continuoustime signals involve superpositions of continuoustime sinusoids. continuoustime sinusoids with distinct frequencies are always distinct; thus, the frequency domain representations of continuoustime signals are nonperiodic ....
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 Spring '09
 DianguangMa
 jω, Symmetry Properties, Integration Property

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