Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +,
Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these idex mappigs we ca write as [ [ x X [ [ X X + + + + ) )( ( [ x + [ x
Cooley-Tuey FFT Algorithms Sice,, ad, we have X [ + + x[ where ad 3
4 Cooley-Tuey FFT Algorithms The effect of the idex mappig is to map the -D sequece x[ ito a -D sequece that ca be represeted as a -D array with specifyig the rows ad specifyig the colums of the array Ier paretheses of the last equatio is see to be the set of -poit DFTs of the - colums: + G[, x[,
5 Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms ote: The colum DFTs ca be doe i place ext, these row DFTs are multiplied i place by the twiddle factors yieldig Fially, the outer sum is the set of -poit DFTs of the colums of the array:,, [, [ ~ G G +,, [ ~ [ G X
6 Cooley-Tuey FFT Algorithms The row DFTs, X [ +, ca agai be computed i place The iput x[ is etered ito a array accordig to the idex map: +, Liewise, the output DFT samples X[ eed to extracted from the array accordig to the idex map: +,
Cooley-Tuey FFT Algorithms Example- Let 8. Choose ad 4 The X[ + + x[4 8 3 for ad 3 4 7
Cooley-Tuey FFT Algorithms -D array represetatio of the iput is The colum DFTs are -poit DFTs give by G[ 3 x[ x[ x[ x[3 x[4 x[5 x[6 x[7, [ + ( ) [4 +, x x 3 8 These DFTs require o multiplicatios
Cooley-Tuey FFT Algorithms -D array of row trasforms is G[, G[, G[, G[, G[, G[, 3 G[,3 G[,3 After multiplyig by the twiddle factors array becomes 3 ~ ~ ~ ~ G[, G[, G[, G[,3 ~ ~ ~ ~ G[, G[, G[, G[,3 8 9
Cooley-Tuey FFT Algorithms ote: ~ [, G 8 G[, Fially, the 4-poit DFTs of the rows are computed: 3 ~ [ + [, X, G 4 3 Output -D array is give by 3 X[ X[ X[4 X[6 X[ X[3 X[5 X[7
Cooley-Tuey FFT Algorithms The process illustrated is precisely the first stage of the DIF FFT algorithm By choosig 4 ad, we get the first stage of the DIT FFT algorithm Alterate idex mappigs are give by +, +,
Prime Factor Algorithms Twiddle factors ca be elimiated by defiig the idex mappigs as < < A C To elimiate the twiddle factors we eed to express + >, B + >, D ( A + B )( C+ D )
Prime Factor Algorithms ow ( A + B )( C+ D ) AC AD BC BD It follows from above that if AC, BD, AD BC the ( A + B )( C+ D ) 3
Prime Factor Algorithms Oe set of coefficiets that elimiates the twiddle factors is give by A, B, C, D Here deotes the multiplicative iverse of reduced modulo If α the α or, i other words α β + where β is ay iteger 4
Prime Factor Algorithms For example, if 4 ad 3, the 3 4 3 sice 3 3 4 Liewise, if γ, the γ δ + where δ is ay iteger ow, AC ( ) ( δ + ) δ + Similarly, BD ( ) β + ) β + ( 5
Prime Factor Algorithms ext, AD ( ) α Liewise, BC ( ) γ Hece, X X[ C + D [ x[ A + B 6
Prime Factor Algorithms Thus, X [ C + D 7 where ad x[ A + B G[,, G[, x[ A + B
8 Prime Factor Algorithms Prime Factor Algorithms Example - Let. Choose ad The, A 3, B 4, C ad The idex mappigs are 4 3 9 3 3 4 4 4 4 3 D + 3, 4 3 + 3, 4 9
Prime Factor Algorithms -D array represetatio of iput is 3 x[ x[3 x[6 x[9 x[4 x[7 x[ x[ x[8 x[ x[ x[5 4-poit trasforms of the colums lead to 9 3 G[, G[, G[, G[3, G[, G[, G[, G[3, G[, G[, G[, G[3,
Prime Factor Algorithms Fial DFT array is 3 X[ X[9 X[6 X[3 X[4 X[ X[ X[7 X[8 X[5 X[ X[ 4-poit DFTs require o multiplicatios, whereas the 3-poit DFTs require 4 complex multiplicatios Thus, the algorithm requires 6 complex multiplicatios
Chirp z-trasform Algorithm Let x[ be a legth- sequece with a Fourier trasform e cosider evaluatio of M samples of that are equally spaced i agle o the uit circle at frequecies ω ωo + ω, M where the startig frequecy ω o ad the frequecy icremet ω ca be chose arbitrarily
Chirp z-trasform Algorithm Figure below illustrates the problem Im z plae (M ) o Re uit circle
3 Chirp Chirp z-trasform Algorithm -Trasform Algorithm The problem is thus to evaluate or, with defied as to evaluate, [ ) ( M e x e X j j ω ω ω j e j j e x e X o [ ) ( ω ω
4 Chirp Chirp z-trasform Algorithm -Trasform Algorithm Usig the idetity we ca write Lettig we arrive at / ) ( / / [ ) ( j j e x e X o ω ω ) ( [ + / [ [ j e x g o ω, [ ) ( / ) ( / j g e X ω M
Chirp z-trasform Algorithm Iterchagig ad we get jω / ( ) / X ( e ) g[, Thus, X ( e j ω ) correspods to the covolutio of the sequece g[ with the sequece / followed by multiplicatio by the sequece / as idicated below x[ g[ e jω / / o / X e j ω ( ) 5
Chirp z-trasform Algorithm / The sequece ca be thought of as a complex expoetial sequece with liearly icreasig frequecy Such sigals, i radar systems, are called chirp sigals Hece, the ame chirp trasform 6
Chirp z-trasform Algorithm For the evaluatio of X ( e jω ) / g[ ( ) the output of the system depicted earlier eed to be computed over a fiite iterval Sice g[ is a legth- sequece, oly a fiite portio of the ifiite legth sequece / is used i obtaiig the covolutio sum over the iterval M /, 7
Chirp z-trasform Algorithm Typical sigals g[ / g[ * - / 8 M
Chirp z-trasform Algorithm The portio of the sequece used i obtaiig the covolutio sum is from the iterval + M / 9 Let h[ as show below, /, ( M + ) otherwise h[ ( M )
Chirp z-trasform Algorithm It ca be see that g[ O / * g[ O* h[, M Hece, the computatio of the frequecy samples X ( e j ω ) ca be carried out usig a FIR filter as idicated below x[ e g[ h[ jω / / o y[ 3 where jω y[ X ( e ), M
Chirp z-trasform Algorithm Advatages - () M is ot required as i FFT algorithms () either or M do ot have to be composite umbers (3) Parameters ωo ad ω are arbitrary (4) Covolutio with h[ ca be implemeted usig FFT techiques 3
Chirp z-trasform Algorithm g[ / g[ * - / M 3