Fast Fourier Transform (FFT)
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1 IC Desig Lab. Fast Fourier Trasfor FFT, - e e π π G Twiddle Factor : e π Radi- DIT FFT:
2 IC Desig Lab. π cos si π Syetry Property Periodicity Property u u u * Cougate Syetric Property a a b b a b b a b a a b c a b b c d d a b b c a b c d d
3 IC Desig Lab. Fast Fourier Trasfor FFT, G G, ' ' ' ' G G G G G G G G '' '' '' '' G G G ' ' ' ' G G G G '' '' '' '' G G
4 IC Desig Lab. Sigal Flow Graph of -poit Radi- DIT FFT G log cople ultiplicatios & log cople additios. Require oly storage for storig cople ubers.
5 IC Desig Lab. Radi- DIT FFT: Fast Fourier Trasfor FFT The splittig ito sus over eve ad odd tie idees is called deciatio i tie DIT. Observatios. I place coputatio a & b are A & B, a & b are ot eeded ad A & B ca be stored i their registers - i place coputatio Require oly storage for storig cople ubers. Data Shufflig otice order of iput data st deciatio -,,,,,, 5, 7 d Deciatio -,,,,, 5,, 7 Shufflig of data has well defied order Figure as a biary uber, order of deciated data obtaied by reversig the biary represetatio of the ide, is stored i bit reversed order Usig i place coputatio ad accessig data i bit reversed order eable DFT coputatio i atural order Other fors possible but ay require ore eory
6 IC Desig Lab. Fast Fourier Trasfor FFT Divide & Coquer Approach Deciatio-I-Frequecy: e π
7 IC Desig Lab. Fast Fourier Trasfor FFT Divide & Coquer Approach: G } { G
8 IC Desig Lab. Sigal Flow Graph of -poit Radi- DIF FFT
9 IC Desig Lab. Fast Fourier Trasfor FFT Divide & Coquer Approach: ardware Copleity log cople additios log cople ultiplicatios Observatios atural order & i bit reversed order Coputatio is i place Reverse coputatio is possible atural & reversed order is possible
10 IC Desig Lab. Fast Fourier Trasfor FFT Radi- ad Split-Radi Approaches: { }, { } { }
11 IC Desig Lab. Radi- FFT Algorith { } { } { } { }
12 IC Desig Lab. Radi- FFT Algorith { } { } { } { }
13 IC Desig Lab. Sigal Flow Graph of -poit Radi- FFT By applyig,,, ad recursively
14 IC Desig Lab. Split-Radi FFT Algorith { } { }
15 IC Desig Lab. Sigal Flow Graph of -poit Split-Radi FFT By applyig,, ad recursively Radi- Butterfly
16 IC Desig Lab. Sigal Flow Graph of -poit Split-Radi FFT By applyig,, ad recursively
17 IC Desig Lab. Equivalet Butterfly Structures betwee Split-Radi ad Radi- FFT s
18 IC Desig Lab. Sigal Flow Graph of -poit Radi- FFT
19 IC Desig Lab. Sigal Flow Graph of -poit Split-Radi FFT
20 IC Desig Lab. Iverse Fast Fourier Trasfor IFFT, FFT : - e i π * * * * * * * *, : IFFT FFT e e - e i i i π π π * FFT * IFFT ipleetatio by FFT
21 IC Desig Lab. Sigal Flow Graph of the Radi- FFT
22 IC Desig Lab
23 IC Desig Lab
24 IC Desig Lab Ver..
25 IC Desig Lab Ver Radi- FFT
26 IC Desig Lab Ver Radi- FFT
27 IC Desig Lab. PE PE PE PE PE PE PE PE R SDF Pipelie FFT Architecture for 5 i i PE PE PE PE PE PE PE PE Radi- Pipelie FFT Architecture for 5
28 IC Desig Lab.
29 IC Desig Lab. Variable-Legth FFT Processor
30 IC Desig Lab. Variable-Legth FFT Processor
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