Ve216LectureNotesChapter3Part4

Ve216LectureNotesChapter3Part4 - Ve216 Lecture Notes...

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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.-------------------------------------------------------------------------------------- Time-Domain Property Frequency-Domain Property----------------------------------------------------------------------------------------------- continuous nonperiodic discrete periodic periodic discrete nonperiodic continuous-------------------------------------------------------------------------------------- 3.8 Periodicity Properties of Fourier Rrepresentations • Continuous- or discrete-time 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 frequency-domain 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 discrete-time Fourier transform representations involve weighted integrals of complex sinusoids over a continuum of frequencies. Accordingly, the frequency-domain 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 discrete-time signals are periodic functions of frequency. This is because the discrete-time complex sinusoids used to represent the discrete-time signals are 2π-periodic functions of frequency. That is, discrete-time 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 continuous-time signals involve superpositions of continuous-time sinusoids. continuous-time sinusoids with distinct frequencies are always distinct; thus, the frequency- domain representations of continuous-time signals are nonperiodic ....
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Ve216LectureNotesChapter3Part4 - Ve216 Lecture Notes...

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