Discrt Fourir Trasform uo Vascoclos UCSD
Liar Shift Ivariat (LSI) systms o of th most importat cocpts i liar systms thory is that of a LSI systm Dfiitio: a systm T that maps [ ito y[ is LSI if ad oly if it is liar it is shift ivariat { } { } { } [ [ [ [ [ [ by ay bt at b a T + + + { } [ [ m m y m m T
D covolutio th opratio y [ [ h [ is th D covolutio of ad h w will dot it by y [ [ h [ this is of grat practical importac: for a LSI systm th rspos to ay iput ca b obtaid by th covolutio with this impuls rspos th IR fully charactrizs th systm it is all that I d to masur 3
4 Fourir Trasforms for LSI systms it is quivalt to wor i th spatial or frqucy domai th Discrt-Spac Fourir Trasform is th D tsio of th Discrt-Tim Fourir Trasform ot that this is a cotiuous fuctio of frqucy icovit to valuat umrically i DSP hardwar w d a discrt vrsio this is th D Discrt Fourir Trasform (D-DFT) ) ( ) ( [ [ ) ( ω ω ω ω ω ω ω ω ω ω d d
D-DFT th D-DFT is obtaid by samplig th DSFT at rgular frqucy itrvals [ ( ω ω) ω ω this turs out to ma th D-DFT hardr to wor with tha th DSFT bcaus w ar samplig i frqucy w hav aliasig i spac this mas that v though th squc [ is fiit w ar ffctivly worig with a priodic squc th DFT thrfor ihrits all th proprtis of th frqucy rprstatios of priodic squcs it is bttr udrstood by first cosidrig th D Discrt Fourir Sris (D-DFS) 5
D-DFS it is th atural rprstatio for a priodic squc a squc [ is priodic of priod if ot that [ [ + [ + ( r r ) [ r mas o ss for a priodic sigal th sum will b ifiit for ay pair r r ithr th D DSFT or th Z-trasform will wor hr r 6
7 D-DFS th D-DFS solvs this problm ot that [ is also priodic outsid li th DSFT proprtis of th D-DFS ar idtical to thos of th D-DFS with th straightforward tsio of sparability ( ) [ [ [
Priodic covolutio li th Fourir trasform th ivrs trasform of multiplicatio is covolutio DFS [ [ ( ) Y ( ) howvr w hav to b carful about how w dfi covolutio sic th squcs hav o d th stadard dfiitio y [ [ h [ mas o ss.g. if ad h ar both positiv squcs this will always b ifiit 8
9 Priodic covolutio to dal with this w itroduc th ida of priodic covolutio istad of th rgular dfiitio which from ow o w rfr to as liar covolutio priodic covolutio oly cosidrs o priod of our squcs th oly diffrc is i th summatio limits [ [ * h y [ [ h h o
Priodic covolutio ot that th squc which rsults from th covolutio is also priodic it is importat to rmmbr th followig w wor with a sigl priod (th fudamtal priod) to ma thigs maagabl but rmmbr that w hav priodic squcs it is li if w wr pig through a widow if w shift or flip th squc w d to rmmbr that th squc dos ot simply mov out of th widow but th t priod wals i!!! ot that this ca ma th fudamtal priod chag cosidrably shift by ()
Discrt Fourir Trasform th DFT is dfid as (hr (ω ω ) is th DSFT) which ca b writt as [ [ [ [ othrwis othrwis ) ( [ ω ω ω ω
Discrt Fourir Trasform comparig this with th DFS [ [ [ [ othrwis othrwis [ [ [ [
3 Discrt Fourir Trasform w s that isid th bos th two trasforms ar actly th sam if w dfi th idicator fuctio of th bo w ca writ [ othrwis R [ [ [ R [ [ [ R
Discrt Fourir Trasform ot from [ [ R [ [ [ R [ that worig i th DFT domai is quivalt to worig i th DFS domai tractig th fudamtal priod at th d w ca summariz this as priodiciz i this way I ca wor with th DFT without havig to worry about aliasig trucat DFS [ [ [ [ trucat priodiciz 4
5 Discrt Fourir Trasform this tric ca b usd to driv all th DFT proprtis.g. what is th ivrs trasform of a phas shift? lt s follow th stps ) priodiciz: this causs th sam phas shift i th DFS [ [ [ [ trucat priodiciz DFS priodiciz trucat [ [ m m Y [ [ m m Y
Discrt Fourir Trasform priodiciz trucat DFS [ [ [ [ trucat priodiciz ) comput th ivrs DFS: it follows from th proprtis of th DFS (pag 4 o Lim) that w gt a shift i spac y [ [ m m 3) trucat: th ivrs DFT is qual to o priod of th shiftd priodic tsio of th squc y [ [ m m R [ i summary th w squc is obtaid by maig th origial priodic shiftig ad taig th fudamtal priod 6
Eampl priodiciz shift by () trucat ot that what lavs o o d trs o th othr 7
Eampl for this raso it is calld a circular shift circular shift by () ot that this is way mor complicatd tha i D to gt it right w rally hav to thi i trms of th priodic tsio of th squc it shows up i most proprtis of th DFT.g. what is th ivrs DFT of th product of two DFTs? 8
Discrt Fourir Trasform priodiciz DFS [ [ [ [ trucat w us our tric agai ) priodiciz: [ [ H[ Y [ [ H[ Y trucat priodiciz ) comput th ivrs DFS: this is ust th priodic covolutio y [ [ o y[ 9
Discrt Fourir Trasform priodiciz trucat DFS [ [ [ [ trucat priodiciz 3) trucat: th ivrs DFT is qual to o priod of th shiftd priodic tsio of th squc y i summary th w squc is obtaid by maig th origial squcs priodic computig th priod covolutio ad taig th fudamtal priod this is th circular covolutio of [ ad h[ [ ( [ o h[ ) R [ [ h[ ( [ o h[ ) R [
Discrt Fourir Trasform w thrfor hav th proprty that th product of two DFTs is th DFT of th circular covolutio of th two squcs ot that circular covolutio o priod of priodic covolutio hc thr is rally ot much that is w priodiciz th squcs ad apply what w lard for th covolutio of DFSs.g. [ h [ (3) (4) () ()
Circular covolutio stp ): prss squcs i trms of ( ) [ h[ ( [ o h[ ) R [ (3) (4) () () h [ [ w t procd actly as for priodic covolutio
Circular covolutio stp ): ivrt h( ) o h [ h[ (3) (4) () () () () () () (3) (4) (4) (3) h [ h [ g [ h [ 3
Circular covolutio stp 3): shift g( ) by ( ) o h [ h[ () () this sds whatvr is at () to ( ) () () (4) (3) (4) (3) g [ h [ g [ h [ 4
Circular covolutio.g. for ( ) () () () (4) (3) [ h[ but hr w rcall that w ar worig with priodic squcs us priodicity to fill valus missig i th flippd squc o h [ h[ 5
Circular covolutio stp 4): w ca fially poit-wis multiply th two sigals ad sum o h [ h[.g. for ( ) () (3) (4) () () 3+4 [ h[ y [ 6
Circular covolutio fially w tract th fudamtal priod [ h[ ( [ o h[ ) R [ ot that th squc vr grows byod our origial widow this is fudamtally diffrt from liar covolutio it is th raso why w d to do circular shifts ot that bcaus of this it is vry diffrt to ) covolv two sigals ) ta th DFTs multiply ad ta ivrs DFT lt s s what happs o MATLAB 7
Circular covolutio >> [ 3; 3 3 ; 5 5; h [ 3 4; 5 ; 3 ; z cov(h) z 3 7 5 4 3 7 6 9 44 53 7 8 39 5 4 7 8 5 >> H fft(h); fft(); Y.*H; y ifft(y) y 49 6 6 54 46 63 69 56 44 8
Discrt Fourir Trasform why do w car about th DFT? ) w d a discrt rprstatio of th frqucy spctrum if w ar to implmt algorithms o computrs th DSFT caot b usd for this bcaus it is cotiuous ) thr ar vry fast algorithms to comput th DFT i D DSP you may hav mtiod th Fast Fourir Trasform (FFT) it is a fast algorithm to comput th DFT if th squc has poits istad of O( ) complity it has O( log) this has mad a trmdous historical diffrc FFT spdup o or two gratios of DSP hardwar Q: is thr a two dimsioal FFT? 9
3 Fast Fourir Trasform to aswr this w loo at th prssio of th DFT ot that this ca b computd with giv f[ is th D DFT of [ i.. th D-DFT of row of th squc w hav s somthig li this wh w studid sparability [ [ [ [ ) ( f 44 4 3 44 4
Fast Fourir Trasform th ida is to crat a itrmdiat squc f[ whos rows ar th DFTs of th rows of D-DFT [ t w raliz that f [ [ f [ 3
Fast Fourir Trasform is ust th D DFT of colum of f[ D-DFT f [ [ this mas that th D-DFT ca b computd with a squc of D-DFTs ot that THIS DOES OT REQUIRE SEPARABILITY this proprty is valid for ay squc it has obvious implicatios o th computatioal complity of th DFT 3
Fast Fourir Trasform ot that th D-DFT rquirs D-DFTs o siz followd by D-DFTs of siz wh ths ar implmtd with th FFT total complity is O( O( O( log log log ) + i.. w hav th sam typ of prssio as i D i summary th D-FFT simply cosists of ) applyig th D-FFT to th rows of th squc ) O( ) applyig th D-FFT to th colums of this itrmdiat squc ) + O( log log ) ) 33
Proprtis of th DFT 34
Proprtis of th DFT 35
Proprtis of th DFT 36
Discrt Cosi Trasform du to its computatioal fficicy th DFT is vry popular howvr it has strog disadvatags for som applicatios it is compl it has poor rgy compactio rgy compactio is th ability to pac th rgy of th spatial squc ito as fw frqucy cofficits as possibl this is vry importat for imag comprssio w rprst th sigal i th frqucy domai if compactio is high w oly hav to trasmit a fw cofficits istad of th whol st of pils 37
38 Discrt Cosi Trasform a much bttr trasform from this poit of viw is th DCT it is dfid by with w will tal mor about it i th t class [ [ [ [ + + + + othrwis C w w othrwis C ) ( cos ) ( cos [ [ ) ( cos ) ( cos [ [ w w
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