ece431notesCh2_2007_09_13

Ece431notesCh2_2007_ - CHAPTER2 TheDFT,Convolution,andWindowing InChapter1,-time, -.However,computers canonlyevaluate finite sums x n,itsDTFTis X f

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CHAPTER2 TheDFT,Convolution,andWindowing InChapter1,wesawthatsignalprocessingforcontinuous-time,bandlimitedwaveforms andsystemscanbeaccomplishedbydiscrete-timesignalprocessing.However,computers canonlyevaluate finite sums. Recallthatforaninfinitesequence x n ,itsDTFTis X ( f ) : = ∞ ∑ m =- ∞ x m e- j 2 π fm ≈ M ∑ m =- M x m e- j 2 π fm forlarge M .Thesumontherightcontains2 M + 1terms.Lateritwillbemoreefficientto usesumswithanevennumberofterms.Forthisreason,weusetheapproximation X ( f ) ≈ M- 1 ∑ m =- M x m e- j 2 π fm . Inordertowritethisasasumstartingfromzero,wecanmakethechangeofvariable n = m + M toget X ( f ) ≈ 2 M- 1 ∑ n = x n- M e- j 2 π f ( n- M ) . Althoughtheforegoingapproximationinvolvesonlyafinitesum,acomputercannotevalu- ateitforallvaluesof f inoneperiod,say | f |≤ 1 / 2.Instead,thecomputercanonlyevaluate thesumforfinitelymanyvaluesof f .Let N : = 2 M ,andlet f = k / N toget 1 X ( k / N ) ≈ N- 1 ∑ n = x n- M e- j 2 π k ( n- M ) / N = e j 2 π kM / N N- 1 ∑ n = x n- M e- j 2 π kn / N . (2.1) Regardingthislastsumasafunctionof k ,observethatithasperiod N ;i.e.,replacing k with k + N doesnotchangethevalueofthesum.Soweonlyneedtoevaluatethesumfor k = ,..., N- 1. 2.1.TheDiscreteFourierTransform(DFT) Givenafinitesequence y ,..., y N- 1 ,its discreteFouriertransform (DFT)is Y k : = N- 1 ∑ n = y n e- j 2 π kn / N . (2.2) 1 Since N = 2 M , e j 2 π kM / N = e j π k =(- 1 ) k . 12 2.1TheDiscreteFourierTransform(DFT) 13 Itiseasytoshowthat Y k isaperiodicfunctionof k withperiod N .Weshowbelowin Section2.1.5thatthesequence y n canberecoveredfromtheDFTsequence Y ,..., Y N- 1 by the inverseDFT (IDFT) y n = 1 N N- 1 ∑ k = Y k e j 2 π kn / N . (2.3) Ofcoursetheright-handsideisaperiodicfunctionof n withperiod N .Hence,although y n isonlydefinedfor n = ,..., N- 1,weoftenthinkofitasbeinganinfinite-duration periodicsignalwithperiod N . 2.1.1.SummingaPeriodicSequenceoveraPeriod Let z n haveperiod N .Thenforany m , m +( N- 1 ) ∑ n = m z n = N- 1 ∑ n = z n . Thisismosteasilyseenpictorially.Forexample,if z n hasperiod N = 5,weseefromthe diagram n :012 34567 89 z n : a a 1 a 2 a 3 a 4 a a 1 a 2 a 3 a 4 that z + ··· + z 4 and z 3 + ··· + z 7 arebothequalto a + ··· + a 4 . AsimpleapplicationoftheforegoingistotheIDFTformula(2.3)when N = 2 M + 1. Then y n = 1 N M ∑ k =- M Y k e j 2 π kn / N , wherewehaveusedthefactthatsince Y k and e j 2 π kn / N haveperiod N ,sodoestheirproduct. 2.1.2.ComputationoftheDFTinM ATLAB If y =[ y ,..., y N- 1 ] ,thentheDFTof y , Y =[ Y ,..., Y N- 1 ] ,canbecomputedinM ATLAB withthecommand Y=fft(y) .Here FFT standsfor fastFouriertransform .TheFFT isaspecialalgorithmthatcomputestheDFTveryquickly. SincetheDFTisperiodic, Y- 1 = Y- 1 + N = Y N- 1 Y- 2 = Y- 2 + N = Y N- 2 . . . Y- N / 2 = Y- N / 2 + N = Y N / 2 . So,toplot Y k for k =- N / 2to k = N / 2- 1,weneedtotake Y =[ Y ,..., Y N / 2- 1 , Y N / 2 ,..., Y N- 1 ] andconvertitto [ Y N / 2 ,..., Y N- 1 , Y , Y 1 ,..., Y N / 2- 1 ] . ThisisdonewiththeM...
View Full Document

This note was uploaded on 04/15/2008 for the course ECE 431 taught by Professor Gubner during the Spring '08 term at Wisconsin.

Page1 / 7

Ece431notesCh2_2007_ - CHAPTER2 TheDFT,Convolution,andWindowing InChapter1,-time, -.However,computers canonlyevaluate finite sums x n,itsDTFTis X f

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online