Chapter 8 DFT : The Discrete Fourier Trasform
Roots of Uity Defiitio: A th root of uity is a complex umber x such that x The th roots of uity are: ω, ω,, ω - where ω e π /. Proof: (ω ) (e π / ) (e π ) (-). ω i 5 W 8 ω 3 ω ω 4-8 ω ω 5 ω 7 W 8 ω 6 -i I DSP, we ofte tae the pricipal root of uity as W e π /
Properties of -th Root of Uity There are exactly -th roots of uity: W, W, W,, W - W is called the pricipal -th root of uity The iverse of W : W - W - Proof: W W - W W d d W Proof: : W d π / d d π / d ( e ) ( e ) W K W / W - W e π / π / Proof: W / W (/)* / (by property 3) W - If > ad is eve, the (W ) W /,,,, r ( W ) r a iteger r ( W ) ( W ) Proof : ( ) W W W e 3
Circular Shift of a Sequece Cosider legth- sequeces defied for Values of such sequeces are ot defied for < ad If x is such a sequece, the for ay arbitrary iteger o, the shifted sequece x x o is o loger defied for the rage We thus eed to defie aother type of a shift that will always eep the shifted sequece i the rage 4
The desired shift, called the circular shift, is defied usig a modulo operatio: x x(( )) c o For > (right circular shift), the above equatio implies o x c x, o for x o +, for < o o periodic shift tae oe x x x x extesio period c 5
Illustratio of the cocept of a circular shift x x(( )) 6 x(( 4)) 6 x(( + 5)) x(( + )) 6 6 6
As ca be see from the previous figure, a right circular shift by o is equivalet to a left circular shift by sample periods o A circular shift by a iteger umber o greater tha is equivalet to a circular shift by (( )) o 7
Circular Time Shift Poits shifted out to the right do t disappear they come i from the left g 3 4 5-pt sequece delay by g((-)) 5 3 4 Lie a barrel shifter : origi poiter 8
Example: Circular Shift 9
Circular Time Reversal Time reversal is tricy i modulo- idexig - ot reversig the sequece: x 5-pt sequece made periodic -7-6 -5-4 -3 - - 3 4 5 6 7 8 9 Zero poit stays fixed; remaider flips Time-reversed periodic sequece x (( )) -7-6 -5-4 -3 - - 3 4 5 6 7 8 9 periodic reversal tae oe x x x x extesio period r
Discrete-Time Fourier Trasform (DTFT) The DTFT is useful for the theoretical aalysis of sigals ad systems. But, the defiitio of DTFT is The umerical computatio of DTFT has several problems: Problem : The summatio over is ifiite Problem : The idepedet variable ω is cotiuous DTFT ad z-trasform are ot umerically computable trasforms.
Approaches Problem requiremet : If the sequece had oly a fiite umber of o-zero terms! This might be a good assumptio sice i may cases, we cocer oly a fiite duratio of the sigal The sigal itself might be of a fiite duratio Oly a segmet is of iterest at a time Sigal is periodic ad thus has oly fiite umber of uique values Problem Requiremet : If we eeded to compute the spectra oly at a fiite umber of frequecies! This ca be achieved by samplig the DTFT i the frequecy domai or the z-trasform o the uit circle. The remaiig issue is that what will happe for the spectra at other frequecies. Aliasig?
DTFS for a periodic sequece satisfy both requiremets Aalysis equatio Sythesis equatio But, DTFS is defied for oly periodic sigals!!! Way to get there: X x W,,,, x X W,,,, Termiology DFS DTFS Aalyze periodic sequeces o the basis that a periodic sequece ca always be represeted by a liear combiatio of harmoically related complex expoetials Discrete Fourier Series (DFS). Exted the DFS to fiite-duratio sequeces Discrete Fourier Trasform (DFT), the solutio to the two problems!
Discrete Fourier Trasform (DFT) For fiite duratio sequeces, a alterative Fourier represetatio is DFT (Discrete Fourier Trasform) The summatio over is fiite DFT itself is a sequece, rather tha a fuctio of a cotiuous variable Therefore, DFT is computable ad importat for the implemetatio of DSP systems DFT correspods to the samples of the Fourier trasform DFT is very importat i DSP systems sice there is FFT (Fast Fourier Trasform), a extremely efficiet ad fast way of computig DFT.
8. Discrete Fourier Series (DFS) A discrete-time sigal is periodic if Due to the periodicity of the complex expoetials, we oly eed expoetials for discrete time Fourier series ~ x ~ x + Fudametal period: the smallest positive iteger Fudametal frequecy: ω π / The DFS ca represet a periodic discrete sigal by a liear combiatio of harmoically related complex expoetials. The Fourier series represetatio of cotiuous-time periodic sigals require ifiite umber of complex expoetials. But for discrete-time periodic sigals, we have ( π / )( + ) ( π / ) ( π ) ( π / ) e e e e m m
DFS aalysis ad sythesis pair is expressed as follows: ( π ) x X e X e / / < > ( π ) The Fourier series coefficiets ca be obtaied via X x e ( π / ) For coveiece we may use the otatio Aalysis equatio X x W Sythesis equatio x X W W e ( π / )
Example 8. : DFS of a periodic impulse trai x δ r r r else Sice the period of the sigal is / / / ( π ) ( π ) ( π ) δ X x e e e We ca represet the sigal with the DFS coefficiets as x δ r e ( π / ) r
Example 8. : Here let the Fourier series coefficiets be the periodic impulse trai Y, ad give by this equatio: Y δ r r Substitutig Y ito DFS equatio gives + y W W δ Comparig this result with the results for Example 8., we see that Yx ad yx.
Example 8.3 : DFS of a periodic rectagular pulse trai The DFS coefficiets ( π ) 4 ( ) e X π e e ( π /) e / 5 / 4 / ( π ) si ( π / ) si ( π /)
8. Properties of DFS Liearity (all sigals have the same period) DFS x X DFS x X DFS + + ax bx ax bx Shift of a Sequece DFS x X DFS π m/ m DFS x m e X π / e x X m m Duality DFS X DFS x X x
Periodic Covolutio Tae two periodic sequeces Let s form the product DFS DFS x X x X X X X 3 The periodic sequece with give DFS ca be writte as a periodic covolutio x x m x m 3 m Periodic covolutio is commutative x x m x m 3 m
Proof Substitute periodic covolutio ito the DFS equatio X 3 x 3 W x m x m W m Iterchage summatios The ier sum is the DFS of shifted sequece Substitutig X 3 x m x mw m m x mw W X X 3 x m x mw x mw X X X m m m
Example 8.4
Properties of DFS
8.3 Fourier Trasform of Periodic Sigals Periodic sequeces are ot absolute or square summable Hece they do t have a Fourier Trasform per se We ca represet them as sums of complex expoetials: DFS We ca combie DFS ad Fourier trasform (Geeralized) Fourier trasform of periodic sequeces Periodic impulse trai with values proportioal to DFS coefficiets X ω π π ( e ) X δ ω This is periodic with π sice DFS is periodic The iverse trasform ca be writte as πε π ε ω ω π π ω ( ) X e e dω X δ ω e dω π ε π ε πε π ω X δ ω e dω X e ε π
Example 8.5 Cosider the periodic impulse trai p δ r r The DFS was calculated previously to be Therefore the Fourier trasform is P for all P ω π π ( e ) δ ω
Relatio betwee Fiite-legth ad Periodic Sigals Cosider fiite legth sigal x spaig from to - Covolve with periodic impulse trai x x p x δ r x r r The Fourier trasform of the periodic sequece (called periodic extesio) is X ω ω ω ω π π ( e ) X ( e ) P ( e ) X ( e ) δ ω π ω π π X ( e ) X e δ ω This implies that π X ω X e X ( e ) π ω DFS coefficiets of a periodic expasio sigal ca be thought as equally spaced samples of the Fourier trasform of oe period ( the origial fiite duratio sequece) r
Example 8.6 Cosider the followig sequece 4 x else The Fourier trasform ( ) X e e The DFS coefficiets ( ) ( ω ) ω ω ω si 5 / si / ( 4 π /) si ( π / ) e si ( π /) X
8.4 Samplig the Fourier Trasform Cosider a aperiodic sequece with its Fourier trasform ( ) DTFT x X e ω Assume that a sequece is obtaied by samplig the DTFT ( ω ) ω ( π / ) Sice the DTFT is periodic, resultig sequece is also periodic We ca also write it i terms of the z-trasform ( / ) X π X ( z) ( π / ) X ( e ) z e The samplig poits are show i figure X could be the DFS of a sequece Write the correspodig sequece ( / ) ( ) X X e X e π x X e ( π / )
The oly assumptio made o the sequece is that DTFT exist ω ( ) X e ωm x m e ( / ) X X e π x m Combie equatio to get x x me e m ( π / ) ( π / ) m ( ) xm e xmp m m m Term i the parethesis is ( π / ) ( m) m ( π / ) ( ) δ p m e m r r So we get the periodic extesio of x δ x x r x r r r X e ( π / )
I this case, the Fourier series coefficiets for a periodic sequece are samples of the Fourier trasform of oe period
I this case, still the Fourier series coefficiets for x are samples of the Fourier trasform of x. But, oe period of x is o loger idetical to x. This is ust samplig i the frequecy domai as compared i the time domai discussed before.
Samples of the DTFT of a aperiodic sequece ca be thought of as DFS coefficiets of a periodic sequece obtaied through summig periodic replicas of origial sequece. periodic expasio If the origial sequece is of fiite legth ad we tae sufficiet umber of samples of its DTFT the origial sequece ca be recovered by x x else It is ot ecessary to ow the DTFT at all frequecies to recover the discrete-time sequece i time domai Discrete Fourier Trasform represetig a fiite legth sequece by samples of DTFT
Time-domai aliasig The relatioship betwee x ad oe period of x i the udersampled case is cosidered a form of time domai aliasig. Time domai aliasig ca be avoided oly if x has fiite legth, ust as frequecy domai aliasig ca be avoided oly for sigals beig badlimited. If x has fiite legth ad we tae a sufficiet umber of equally spaced samples of its Fourier trasform (specifically, a umber greater tha or equal to the legth of ), the the Fourier trasform is recoverable from these samples, equivaletly is recoverable from.
Time-Domai Aliasig vs. Frequecy-Domai Aliasig To avoid frequecy-domai aliasig Sigal is badlimited Samplig rate i time-domai is high eough To avoid time-domai aliasig Sequece is fiite Samplig iterval (π/) i frequecy-domai is small eough 36
8.5 The Discrete Fourier Trasform Cosider a fiite legth sequece x of legth x outside of For give legth- sequece associate a periodic sequece (a periodic extesio) x xr r The DFS coefficiets of the periodic sequece are samples of the DTFT of x Sice x is of legth there is o overlap betwee terms of x-r ad we ca write the periodic sequece as x x ( mod ) x ( ( ) ) To maitai duality betwee time ad frequecy We choose oe period of X as the Fourier trasform of x X X X X ( mod ) X ( ( ) ) else
The DFS pair ( π / ) x e X The equatios ivolve oly oe period so we ca write X ( π / ) xe else ( π / ) X e x else
The DFT pair ca also be writte as DFT x X The Discrete Fourier Trasform X ( π / ) xe else ( π / ) X e x else
Discrete Fourier Trasform (DFT) The Discrete Fourier Trasform (DFT) of a fiite legth sequece DFT X { x } x e for {,..., } π The Iverse Discrete Fourier Trasform (IDFT) is defied by IDFT x { X } X e for {,..., } π A DFT ad IDFT pair is show as DFT x X
4 Proof of Iverse Discrete Fourier Trasform The iverse discrete Fourier trasform (IDFT) is give by To verify the above expressio we multiply both sides of the above equatio by ad sum the result from to W, W X x W W X W x W X ) ( W X ) (
Maig use of the idetity ( ) W,, for r, otherwise r a iteger we observe that the RHS of the last equatio is RHS X δ l r but, for,,,-, there is oly oe o-zero term at l, ad RHS X Hece x W X 4
π / X xe, DFT: aalysis equatio time domai x X frequecy domai IDFT: sythesis equatio π / x X e, 43
Periodic Extesio of DFT Sequece We ca show that the defiitios of DFT ad IDFT iduces periodicity π / X xe, X + xe x e π ( + ) / π / π π / x e X e 44
Similarly π / x X e, x + X e Xe π ( + ) / π / π π / X e x e 45
DTFS ad DFT X is the Fourier series coefficiet (or spectral compoet) at the frequecy Thus except the scale factor, we have the followig relatioship periodic extesio x ˆ x tae oe period ω DTFS periodic extesio X X tae oe period Differeces betwee the DFT ad the DTFS the DFT assumes sigal periodicity while the DTFS requires periodicity the DFT scales the sythesis equatio by / whereas the DTFS scales the aalysis equatio. DFT IDFT 46
Example 8.7 The DFT of a rectagular pulse x is of legth 5 We ca cosider x of ay legth greater tha 5 Let s pic 5 Calculate the DFS of the periodic form of x 4 ( π /5) e X π e ( /5) π e 5, ± 5, ±,... else
If we cosider x of legth We get a differet set of DFT coefficiets Still samples of the DTFT but i differet places
8.6 Properties of DFT X X ad X X
Liearity x ( ) x ( ) max(, ) Duratio x ( ) DFT X( ) x( ) DFT X ( ) Duratio DFT ax ( ) + bx( ) ax( ) + bx ( ) 5
Review: Circular Shift of a Sequece x ( ) ~ x ( ) x(( m)) otherwise x() ~ x ( ) ~ x ( ) ~ x ( m ) 5
DFT of Circular Shifted Sequece x ( ) ~ x ( ) x(( m)) otherwise x( ) DFT X ( ) x m e X W X DFT ( / ) m m (( )), π ( ) ( ) 5
Duality x( ) DFT X ( ) X DFT ( ) x(( )), 53
Example: Duality Choose Rex () ReX() ReX() Imx () ImX() ImX() X () x(()) 54
otatios otatio : Circular reversal y() x(( )) x(( )) x(( ) ) y() x() y ( ) x(( )) x ( ) Properties of DFT x(( )) X X * * * x X X ( ) 55
Review: Eve ad Odd Compoets Ay real-valued sigal ca be expressed as the sum of a odd sigal ad a eve sigal. xt ( ) x( t) + x( t) x x + x e o e o where xe( t) { xt ( ) + x( t)} xe { x + x } aevefuctio ad xo( t) { xt ( ) x( t)} xo { x x } a odd fuctio 56
Eve ad Odd Compoets of a Fiite Sequece For a real fiite duratio sequece xe ( ) x ( ) + x(( )) x ( ) + x ( ) xo ( ) x ( ) x(( )) x ( ) x ( ) ( ) ( ) ( ) ( ) 57
Review: Eve ad Odd Compoets If x(t) is a complex valued fuctio: x(t)x R (t) + x I (t) x(t) is eve iff both the real ad imagiary parts of x(t) are eve. Similarly x(t) is odd iff both the real ad imagiary compoets of x(t) are odd. We ca also defie cougate symmetry, or Hermitia symmetry: real part of x(t) is eve but the imagiary part of x(t) is odd x R (t) x R (-t) ad x I (t) -x I (-t) i.e. x * (t) x * (-t) 58
Symmetry Properties Real-Imagiary Decompositio x () x() + x() R e o e o x ( ) + x ( ) + x ( ) + x ( ) X () X() + X() I R R I I R e o e o X ( ) + X ( ) + X ( ) + X ( ) Symmetry Properties (Proais p.45) I R R I I e o e o x( ) x ( ) + x ( ) + x ( ) + x ( ) R R I I e o e o X( ) X ( ) + X ( ) + X ( ) + X ( ) R R I I 59
e o DFTRe{ x( )} DFT x ( ) + x ( ) e o X + X R e o DFT Im{ x ( )} DFT x ( ) + x ( ) R o e X + X R I I I R I Hermitia Symmetry Hermitia ati-symmetry 6
Symmetry Relatios for DFT of Real Sigal The Fourier trasform of real sigals exhibits special symmetry which is importat to us i may cases. Basically, The trasform of a real sigal x is therefore cougate Symmetric (Hermitia symmetric) Real part Symmetric (eve) Imagiary part Atisymmetric (sew-symmetric, odd) Magitude Symmetric (eve) Phase Atisymmetric (odd) 6
Example x() ~ x( ), X ( ) X () 6
63 Liear Covolutio (Review) ) ( )* ( ) ( ) ( ) ( 3 x x m x m x x m ) ( ) ( ω e X x FT ) ( ) ( ω e X x FT ) ( ) ( ) ( ) ( )* ( ) ( 3 3 ω ω ω e X e X e X x x x FT
Circular Covolutio (Cyclic Covolutio) Circular covolutio of two fiite legth sequeces x x m x m 3 m m m ( ) ( ) x m x m ( ) ( ) x m x m, I liear covolutio, oe sequece is multiplied by a time reversed ad liearly shifted versio of the other. For covolutio here, the secod sequece is circularly time reversed ad circularly shifted. So it is called a -poit circular covolutio
The cyclic covolutio is ot the same as the liear covolutio of liear system theory. It is a byproduct of the periodicity of DFS/DFT. Whe the DFT X is used, the periodic iterpretatio of the sigal x is implicit: if the for ay iteger m: Thus, ust as the DFT X is implicitly period-, the iverse DFT is also implicitly period- the periodic extesio of x. h x y * h x y { } { },..., for IDFT e X X x π + ) ( m m x e e X e X π π π
Circular Covolutio Theorem x ( ) DFT X ( ) x ) DFT X ( ) both of legth ( x ( ) x ( m) x(( m)) x ( ) x ( ) 3 m x ( ) x ( ) x ( ) X ( ) X ( ) X ( ) DFT 3 3 66
Proof: Circular Covolutio Theorem Suppose we have two fiite duratio sequeces x ad x of legth. Suppose we multiply the two DFTs together: X 3 X X What happes whe I tae the IDFT of X 3? x ( m) X W 3 3 W XX m m l m x W x l W W l ( + l m) x x l W l 67
( + l m) W iff + l m p p Thus, W ( : a iteger) ( + l m), if l m + P (( m)), otherwise ad we have x3( m) x( ) x(( m)), the circular covolutio of x ad x 68
Circular Covolutio Example 69
Example: Circular Covolutio Theorem L6 x ) x ( X ( ) ( ) X ( ) otherwise L otherwise W L L x ( ) x ( ) x 3 ( ) x ( ) x ( ) x ( ) X 3 3 ( ) X ( ) X ( ) otherwise L 7
7 3 ( ) ( ) ( ) x x x + otherwise L W W W X X L L L L ) ( ) ( ) ( + otherwise L W W X X X L L L ) ( ) ( ) ( ) ( 3 otherwise L x x ) ( ) ( L6, L ) ( x L ) ( x ) ( x 3
Why is cyclic covolutio ot true liear covolutio? Because a wraparoud effect occurs at the eds : The procedure of each pair are summed aroud the circle. y x * h I a while, it will be see that ca be computed usig y x h.
DFT W X x δ 3 X W X X X )) (( )) (( 3 x x x x x x
Example 8. Circular covolutio of two rectagular pulses L6 L x x else DFT of each sequece π X X e else Multiplicatio of DFTs X3 X X else Ad the iverse DFT x 3 else
We ca augmet zeros to each sequece L The DFT of each sequece e X X e Multiplicatio of DFTs π L π X π L e e 3 π
Example : Cosider the two legth-4 sequeces repeated below for coveiece: g h 3 3 We will tae DFT approach 77
78 The 4-poit DFT G of g is give by Therefore, G, G + G + + 3, 4 + + G 4 / e g g G π + 4 6 4 4 3 / / e g e g π π + + 3 3 + + e e, / / π π
79 Liewise, Hece,, 6 + + + H, + H, H + H + + 3 4 / e h h H π + 4 6 4 4 3 / / e h e h π π + + 3 3 + + + e e e, / / π π π
G g 4 G g W 4 G g G3 g3 + H h 6 H h W 4 H h H3 h3 + W 4 is the 4-poit DFT matrix 8
8 If deotes the 4-poit DFT of the Thus 3 H G Y C, Y C y C 4 3 3 3 H G H G H G H G Y Y Y Y C C C C
8 The 4-poit IDFT of yields Y C 4 W* 4 3 3 C C C C C C C C y Y y Y y Y y Y 5 6 7 6 4 4
4 pt sequeces: g{ } h{ } m (( )) g m h m 3 3 h(( )) 4 3 g 4 h{4 7 5 4} h(( -)) 4 h(( )) 4 h(( 3)) 4 3 3 3 3 chec: g * h { 6 5 4 } 83
84 Example : ow let us exteded the two legth-4 sequeces to legth 7 by appedig each with three zero-valued samples, i.e. 6 4 3 g g e,, 6 4 3 h h e,,
We ext determie the 7-poit circular covolutio of ad : 6 h e From the above y g mh (( m)), 6 m e e 7 g e y g h + g h 6 + g h 5 + g 3 h 4 e e e e e e e e + g 4 h 3 + g 5 h + g 6 h e e e e e e g h 85
86 Cotiuig the process we arrive at, ) ( ) ( 6 + + h g h g y h g h g h g y + +, ) ( ) ( ) ( 5 + + 3 3 3 h g h g h g h g y + + +, ) ( ) ( ) ( ) ( 5 + + + 3 3 4 h g h g h g y + +, ) ( ) ( ) ( 4 + +, ) ( ) ( 3 3 5 + + h g h g y 3 3 6 ) ( h g y
As ca be see from the above that y is precisely the sequece obtaied by a liear covolutio of g ad h y L y L 87
DFT ad DFS Periodic Extesio: Give a fiite-legth sequece { x ;,..., } { } { } defie ~ the periodic sequece x ; < < + by x x x mod ~ x ~ does ot have a z-trasform or a coverget Fourier sum (why?). But it does have a DFS represetatio. The DFS that is the true frequecy represetatio of discrete-time periodic sigals. The DFT is ust oe period of the DFS. The legth- DFT X of the legth- sigal x cotais all the iformatio about x. It is coveiet to wor with. Wheever the DFT is used, actually the DFS ~ X X is beig used computatios ivolvig X are affected by the true periodicity of ~ the X coefficiets.
DFT ad z-trasform The DFT samples the z-trasform at evely spaced samples of the uit circle over oe revolutio: X / {,..., } X ( z) π for z e
DFT ad DTFT The DFT samples o period of the discrete-time Fourier trasform (DTFT) at evely spaced frequecies X / {,..., } ω X ( e ) ω π for
DTFT from DFT by Iterpolatio It ca readily be show that X( e ω ) ω π si X e ω π si ( ωπ/ )( ) / This is the dual of iterpolatio of origial time fuctio from sampled values by a ideal low-pass filterig IDTFT DTFT x samplig DFT IDFT X ( ω) X iterpolatio for a fiite-duratio sigal with o aliasig 9
Relatios to the DTFT Aliasig x y x + m X m DTFT IDTFT IDFT DFT -poit samplig X( ω) Y K X 9
Example : A periodic sequece is costructed from the sequece Discrete-Time Fourier Trasform (DTFT) Discrete Fourier Trasform (DFT) By comparig DTFT ad DFT, we fid, ~, < + < a r x x a u a x r ω ω ω ω ae e a e x e X ) ( ae e x X / ) / ( ~ ~ π π { },..., for ) ( / e X X π ω ω
Matrix Represetatio The DFT samples defied by X x W, ca be expressed i matrix form as X x W X W x x X W W x X where x ad x x : x X X X X : 94
ad is the DFT matrix give by W W e π / W ( ) W W W 4 ( ) W W W W ( ) ( ) ( ) W W W W i, i, : W ( i)( ) row ad colum idexes 95
Liewise, the IDFT relatio give by x X W, ca be expressed i matrix form as x W X where W is the IDFT matrix W ( ) ( ) ( ) W W W ( ) W W W 4 ( ) W W W W W W* 96
Example: Matrix Represetatio W W W W W 4 4 4 4 3 W4 W4 W4 W4 4 4 6 W4 W4 W4 W4 3 6 9 W4 W4 W4 W4 W W W W W 4 4 4 4 3 W4 W4 W4 W4 4 4 6 4 W4 W4 W4 W4 3 6 9 W4 W4 W4 W4 97
Example : The DFT matrices of dimesio, 3, 4 are as follows: Suppose Where we observe that the real part of X is eve-symmetric, ad the imagiary part is odd-symmetric the DFT of the real sigal. W + + 3 3 3 3 3 W W 4, 3 T x X Wx + 3 5 5 3 4 X X X X W x X
8.7 Liier covolutio usig the DFT Efficiet algorithms are available for computig the DFT of fiiteduratio sequece, therefore it is computatioally efficiet to implemet a covolutio of two sequeces by the followig procedure: Compute the poit discrete Fourier trasforms X ad X for the two sequeces give. Compute the product X 3 X X Compute the iverse DFT of X 3 The multiplicatio of discrete Fourier trasforms correspods to a circular covolutio. To obtai a liear covolutio, we must esure that circular covolutio has the effect of liear covolutio.
+ x x m x m 3 m
With aliasig Without aliasig
Implemetig LTI systems usig the DFT Let us cosider a L poit iput sequece x ad a P poit impulse respose h. The liear covolutio has fiite-duratio with legth L+P-. Cosequetly for liear covolutio ad circular covolutio to be idetical, the circular covolutio must have the legth of at least L+P- poits. i.e. both x ad h must be augmeted with sequece amplitude with zero amplitude. This process is ofte referred to as zero-paddig.
Liear covolutio is a ey operatio i may sigal processig applicatios Sice a DFT ca be efficietly implemeted usig FFT algorithms, it is of iterest to develop methods for the implemetatio of liear covolutio usig the DFT Let g ad h be two fiite-legth sequeces of legth ad M, respectively Pic L + M Defie two legth-l sequeces g, g e, L h, h e, M M L 6
Liear Covolutio of Two Fiite-Legth Sequeces The y g h y g h L C The correspodig implemetatio scheme is illustrated below g e g Zero-paddig ( + M ) with Legth- ( M ) zeros poit DFT ( + M ) y L h Zero-paddig h e ( + M ) poit IDFT with Legth-M ( ) zeros poit DFT Legth- ( + M ) The size of DFT to be used ca be ay iteger L +M- 7
Circular Covolutio as Liear Covolutio with Time Aliasig P L L x () x () x () P L L+P 8
Termiology: Fast Covolutio 9
Filterig Log Sequeces Sometimes we wat to filter a sequece that is very log could save up all the samples, the either» do a really log time-domai covolutio, or» use really big DFTs to do it i the frequecy domai but big DFTs may become impractical; besides we get log latecy: we have to wait a log time to get ay output Sometimes we wat to filter a sequece of idefiite legth ad the eve the methods above do t wor
Example: Filterig Log Sequeces Fiite-legth impulse respose h ad idefiite-legth sigal x to be filtered Implemetig Liear Time-Ivariat Systems Usig the DFT Theoretically, we store the etire samples ad the implemet the covolutio procedure usig a DFT for a large umber poits which is geerally impractical to compute. o filter samples ca be computed util all the iput samples have bee collected. Geerally, we would lie to avoid such a large delay i processig. The solutio of both problems is to use bloc covoltuio.
Bloc Covolutio Techiques All of samples will be segmeted ito sectio of appropriate legth (L). Each sectio ca the be covolved with the fiite-legth impulse respose ad the filtered sectios fitted together i a appropriate way. Overlap-Add Method Overlap-Save Method
Usig Liearity to Filter a Log Sigal 3
4
Bloc Covolutio - Overlap Add Method 5
Segmetig the Iput i OLA 6
Puttig the Output Pieces Together 7
Filterig the Segmets 8
Bloc Covolutio - Overlap Save Method 9
Liear Covolutio Example Agai
Aliased Covolutio
OLS Method - Segmetig the Iput 3
OLS Method - Extractig the Output 4
Compariso Overlap-add method is the procedure of decompositio of x ito ooverlappig sectios of legth L ad the result of covolvig each sectio with h which are overlapped ad added to costruct the output. 5
Overlap-save method is the procedure of decompositio of x ito overlappig sectios of legth L ad the result of covolvig each sectio with h which the portios of each filtered sectio to be discarded i formig the liear covolutio. 6