Chapter 9 Computation of the Discrete. Fourier Transform

Transcription

1 Chapter 9 Coputatio of the Discrete Fourier Trasfor Itroductio Efficiet Coputatio of the Discrete Fourier Trasfor Goertzel Algorith Deciatio-I-Tie FFT Algoriths Deciatio-I-Frequecy FFT Algoriths Ipleetatio of the DFT Usig Covolutio Effects of Fiite Register Legth

2 . Itroductio Topics Efficiet coputatio ethod of the DFT. Direct Coputatio Each F requires ultiplicatios. F requires ultiplicatios. Fast Fourier Trasfor Efficiet coputatio algoriths for the DFT. I ters of ultiplicatios ad additios.

3 . Efficiet Coputatio of the Discrete Fourier Trasfor DFT j π / ( ) e 0 0 j π / ( ) e F ( ) 0 j π / 0 where e ;, 0,,,, Cople Fors ( ) 0 ( Re { } Re { } I{ }I{ }) + j ( R e { }I{ } + I{ }Re{ } Aalysis cople ultiplicatios (-) cople additios

4 . Efficiet Coputatio of the 4 Discrete Fourier Trasfor (c.) Potetial: Syetry ad Periodicity ( ) * Approach ( + ) ( + ) The decopositio of a -poit DFT ito successively saller DFTs. E. l+l ad Mp+q ( ) ; ( p, q ) ( l, ) p 0 ql M L 0 0 q q Mpl pl / L M / M ; ; ad L M lq q ( p, q ) ( l, ) M l 0 0 lp L ( Mp+ q )( L+ l ) pl L LM cople ult. LM cople ult. ML cople ult.

5 . Goertzel Algorith 5 Apply the periodicity to reduce the copleity. Defie Sice Real Multiplicatios for Poles Used to copute ay desired set of saples of the Fourier trasfor of a sequece

6 4. Deciatio-I-Tie FFT Algoriths 6 Decopositio ( / ) ( / ) r 0 r 0 r 0 ( ) r + r + r ( ) ( ) ( / ) ( / ) r + r + r 0 r 0 r ( r + ) Properties j ( π / ) j π /( / ) e e Results / ( / ) ( / ) r / r 0 r 0 ( ) r + r + G + H r /

7 4. Deciatio-I-Tie FFT Algoriths 7 Recursive Coputatio

8 4. Deciatio-I-Tie FFT Algoriths 8 Each stage of the FFT process has iputs ad outputs, so we eed eactly storage locatios atay oe poit i the calculatios. It is possible to re-use the sae storage locatios at each stage to reduce eory overhead. Ay algorith which uses the sae eory to store successive iteratios of a calculatio is called a i-place algorith. Coputatio ust be doe i a specific order

9 4. Deciatio-I-Tie FFT Algoriths 9 I-Place Coputatio Bit Reversed Alterative Fors

10 5. Deciatio-I-Frequecy FFT 0 Algoriths Decopositio ( ) 0 ( / ) ( r ) ( r ) ( r ) + ( / ) ( /) ( r ) + + ( / ) 0 Properties Results 0 0 / 0 r + ( / ) r + ( / ) r r r ( / ) ( r ) ( + + ( / )) 0 r / ( r ) Decopositio ( r + ) ( / ) + 0 Properties / ( r + ) Results 0 ( r + ) ( / ) ( r + ) / + ( / ) 0 ( / ) ( / )( r + ) ( r + ) 0 + ( / ) ( / ) + ( / ) 0 ( / ) ( r + ) ( r + ) + ( / )( r + ) r ( r + ) ( + ( / )) 0 ( / ) ( r + ) ( + ( / )) 0 ( + ) r

11 5. Deciatio-I-Frequecy FFT Algoriths Recall that the DFT is DIT FFT algorith is based o the decopositio of the DFT coputatios by forig sall subsequeces i tie doai ide : l or l+ Oe ca cosider dividig the output sequece, i frequecy doai, ito saller subsequeces: r or r+:

12 5. Deciatio-I-Frequecy FFT Algoriths Recursive Coputatio

13 5. Deciatio-I-Frequecy FFT Algoriths

14 5. Deciatio-I-Frequecy FFT 4 Algoriths to log: "64: 4,096 reduces to 84 "56: 65,56,048 "04:,048,576 0,40 "4096: 6,777,6 49,5 "684: 68,45,456 9,76

15 5. Deciatio-I-Frequecy FFT 5 Algoriths Alterative Fors

16 6. Ipleetatio of the DFT Usig 6 Covolutio

17 6. Ipleetatio of the DFT Usig 7 Covolutio Give two sequeces ad of legth ad respectively Direct ipleetatio requires cople ultiplicatios Cosider usig FFT to covolve two sequeces: Pic, a power of, such that + - Zero-pad ad to legth Copute -poit FFTs of zero-padded ad, oe obtais ad Multiply ad Apply the IFFT to obtai the covolutio su of ad Coputatio copleity: (/) log + + (/)log

18 6. Ipleetatio of the DFT Usig Covolutio Covolutio For GF(p) for reide E. GF(5), prie eleet GF(7), prie eleet. E. For ad >0 iograd Fourier Trasfor Regularity for the addressig. Loose relatio betwee Multiplicatio ad Add. ot siilar for differet. O() istead of O(Log) 0 ' ' 0 ' ' ' ' ' ' '

19 Chirp Algorith Iterchage the idices ad 6. Ipleetatio of the DFT Usig Covolutio 0,,... ;, ) ( 0 0 Δ + M e e j j ω ω ω ω ω ) ( + 0. M 0 / ) ( / ) ( j g e ω 0 / ) ( / 0 / ) ( / / 0 ) ( j j g e e ω ω

20 6. Ipleetatio of the DFT Usig Covolutio 0 Fiite Sequece ocausal For

21 6. Ipleetatio of the DFT Usig Covolutio Fiite Sequece with Causal for Delay the FIR sequece Output is delay

22 6. Ipleetatio of the DFT Usig Covolutio Further delay for DFT coputig he

23 6. Ipleetatio of the DFT Usig Covolutio

24 7. Effects of Fiite Register legth i DFT Direct Coputig Oe Multiplicatio Ter Quatizatio Effects } } Re{ I{ } I{ } { ( }) } I{ I{ } { } { ( e j e e + R + R R 0 0 / j e π ), si } {, cos } (I{, ) si } I{, cos } { ( 4 e j e Q ε π ε π ε π ε π + R R 4

25 6. Effects of Fiite Register legth i DFT Direct Coputig Quatizatio oise Assuptio Uifor distributio for Variace is Errors are ucorrelated each other ad errors are ucorrelated with iput ad output sigals. The error variace of oe ter The average error B B < < ε / B ( ) ( ) 4,,,,, ε ε ε ε ε { } B B 4, ε ξ { } { } B F 0, ε ξ ξ 5

26 6. Effects of Fiite Register legth i DFT Dyaic Rage Cotrol If the 0 < ; 0,,..., The iput should be scaled dow to avoid overflow. ' 6

27 Quatizatio Effects i Fied-Poit FFT Algoriths The output oise is Output Rage Cotrol 6. Effects of Fiite Register legth i DFT ( ) r q p p q p p + { } B F ξ { }, B B q σ ε ξ ), a( ), a( ), a( ), a( q p q p q p q p { } B B F ξ 7

28 7. Effects of Fiite Register legth i DFT Quatizatio Effects i Fied-Poit FFT Algoriths (scale at this stage by 0.5. { } B v B v B v B v v B v v v B F ) 0.5 ( (0.5) (0.5) (0.5) σ σ σ σ σ ξ 8

29 7. Effects of Fiite Register legth i 9 DFT DFT /S Ratio FFT /S Ratio

30 7. Effects of Fiite Register legth i DFT 0 Bloc Floatig Poit. Scale by a half wheever a overflow is detected.

31 7. Effects of Fiite Register legth i DFT Coefficiet Quatizatio Mea-square error to ea square error of output sigals ξ { } v B F 6

32 Rears Itroductio Efficiet Coputatio of the Discrete Fourier Trasfor Goertzel Algorith Deciatio-I-Tie FFT Algoriths Deciatio-I-Frequecy FFT Algoriths Ipleetatio of the DFT Usig Covolutio Effects of Fiite Register Legth