VE451Su2010C7 - VE451 Lecture Notes Dianguang Ma Summer...

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Unformatted text preview: VE451 Lecture Notes Dianguang Ma Summer 2010 Chapter 7 The Discrete Fourier Transform: Its Properties and Application DTFT Computation • The frequency-domain representation of a discrete-time signal is given by the discrete-time Fourier transform (DTFT). However, the DTFT is not computationally convenient. DTFT Computation 1 We consider a length- signal ( ), 0,1, , 1. Its DTFT can be written in the simplified notation ( ) ( ) This expression may be computed at any desired value of in the Nyquist interval L j n n L x n n L X x n e ϖ ϖ ϖ-- = =- = ∑ L . π ϖ π- ≤ < It is customary in the context of developing computational algorithms to take advantage of the periodicity of ( ) and map the conventional symmetric Nyquist interval onto the right-sided one 0 X ϖ π ϖ π- ≤ < 2 . We will refer to the latter the DFT Nyquist interval. ϖ π ≤ < The positive-frequency subinterval remains unchanged, but the negative-frequency one gets mapped onto the second half of the DFT Nyquist interval. Often, we must compute the DTFT over a frequency range, . Suppose we want to compute the DTFT at frequencies that are equally spaced over this interval, that is, , 0,1 a b b a k a a bin N k k k N ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ≤ <- = + = + ∆ = bin 1 , , 1 where is the bin width, that is, the spacing between frequencies . ( ) ( ) , 0,1, , 1 k k L j n k n N X x n e k N ϖ ϖ ϖ ϖ-- =- ∆ = =- ∑ K K Discrete Fourier Transform (DFT) The N-point DFT of a length- signal is defined to be the DTFT evaluated at equally spaced frequencies over the DFT Nyquist interval 2 . The "DFT frequencies" are defined as follows 2 , 0,1 k L N k k N ϖ π π ϖ ≤ < = = , , 1 N- K 1 2 The bin width is also known as the computaional frequency resolution of the DFT. Thus, the N-point DFT will be ( ) ( ) , 0,1, , 1 k bin L j n k n N X x n e k N ϖ π ϖ ϖ-- = ∆ = = =- ∑ K In practice, the two lenfth and can be specified independently of each other: is the number of time-domain samples of a continuous-time signal ( ) and can even be infinite; is the number of f L N L x t N requency- domain samples of the DTFT ( ) (the computation of the DFT can be though of as the frequency-domain sampling of the ( )). X X ϖ ϖ Frequency-Domain Sampling: the DFT Let us consider an aperiodic discrete-time signal ( ) with DTFT ( ) ( ) We sample ( ) periodically. Since ( ) is periodic with period 2 , only samples in the fundamental frequency range j n n x n X x n e X X ϖ ϖ ϖ ϖ π ∞- =-∞ = ∑ are necessary. 1 2 / 1 2 1 2 / 2 / 1 2 / 2 ( ) ( ) ( ) ( ) ( ) j kn N k n N N N j kn N j kn N n n N lN N j kn N l n lN X X k x n e N x n e x n e x n e π π π π π ϖ-- =----- = = ∞ +-- =-∞ = = = + + ÷ + + = ∑ ∑ ∑ ∑ ∑ L L 1 2 ( )/ 1 1 2 / 2 / 1 2 / 1 2 / ( ) ( ) ( ) ( ) Let ( ) ( ), we have 2 ( ) N j k m lN N l m N N j km N j kn N l m l n N j kn N n l p l N j kn N p n x m lN e x m lN e x n lN e x n lN e x n x n lN X k x n e N π π π π π π ∞--- =-∞ = ∞...
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VE451Su2010C7 - VE451 Lecture Notes Dianguang Ma Summer...

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