This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: VE451 Lecture Notes Dianguang Ma Summer 2010 Chapter 7 The Discrete Fourier Transform: Its Properties and Application DTFT Computation • The frequencydomain representation of a discretetime signal is given by the discretetime 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 rightsided one 0 X ϖ π ϖ π ≤ < 2 . We will refer to the latter the DFT Nyquist interval. ϖ π ≤ < The positivefrequency subinterval remains unchanged, but the negativefrequency 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 Npoint 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 Npoint 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 timedomain samples of a continuoustime 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 frequencydomain sampling of the ( )). X X ϖ ϖ FrequencyDomain Sampling: the DFT Let us consider an aperiodic discretetime 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 ∞ =∞ = ∞...
View
Full Document
 Summer '10
 DianguangMa
 Digital Signal Processing, Frequency, DFT, DFTs

Click to edit the document details