Computation of DFT. W has a specific structure and because

Transcription

1 DSP 7 Computatio of DFT CTU EE Computatio of DFT Efficiet algoithms fo computig DFT Fast Fouie Tasfom. a Compute oly a few poits out of all poits b Compute all poits hat ae the efficiecy citeia? umbe of multiplicatios umbe of additios Chip aea i VSI implemetatio DFT as a iea Tasfomatio Mati epesetatio of DFT Defiitio of DFT: K K whee et M M ad M M M M Thus poit IDFT - poit DFT - Because the mati tasfomatio has a specific stuctue ad because has paticula values fo some ad we ca educe the umbe of aithmetic opeatios fo computig this tasfom.

2 DSP 7 Computatio of DFT CTU EE Eample Oly additios ae eeded to compute this specific tasfom. This is a well-ow adi- FFT Thus the DFT of is 6 Fast Fouie Tasfom -- Highly efficiet algoithms fo computig DFT Geeal piciple: Divide-ad-coque Specific popeties of Comple cougate symmety: Symmety: Peiodicity: Paticula values of ad : e.g. adi- FFT o multiplicatios Diect computatio of DFT Re Im Im Re Im Im Re Re K Fo each we eed comple multiplicatios ad - comple additios. eal multiplicatios ad - eal additios. e will show how to use the popeties of to educe computatios. Radi- algoithms: Decimatio-i-time; Decimatio-i-fequecy Composite algoithms: Cooley-Tuey; Pime facto iogad algoithm Chip tasfom algoithm

3 DSP 7 Computatio of DFT Radi- Decimatio-i-time Algoithms -- Assume -poit DFT ad Idea: -poit DFT -poit DFT -poit DFT -poit DFT -poit DFT -poit DFT -poit DFT Sequece: Eve ide: Odd ide: eve odd K Q e π e π / / / / -poit DFT G H -poit DFT CTU EE

4 DSP 7 Computatio of DFT Compaiso: a Diect computatio of -poit DFT fequecy samples: ~ comple multiplicatios ad comple adds b Diect computatio of -poit DFT: ~ comple multiplicatios ad comple adds additioal comple multis ad comple adds ~ Total: comple multis ad adds c log -stage FFT Sice we ca futhe bea -poit DFT ito two so o. -poit DFT ad At each stage: ~ comple multis ad adds Total: ~ log comple multis ad adds --> log CTU EE

5 DSP 7 Computatio of DFT umbe of poits Diect Computatio: Comple Multis FFT: Comple Multis Speed Impovemet Facto Buttefly: Basic uit i FFT Two multiplicatios: Oe multiplicatio: CTU EE 5

6 DSP 7 Computatio of DFT I-place computatios Oly two egistes ae eeded fo computig a buttefly uit. m m p q m m p p m m q q Advatage: less stoage! I ode to etai the i-place computatio popety the iput data ae accessed i the bit-evesed ode. ote: The outputs ae i the omal ode same as the positio Positio Biay equivalet Bit evesed Sequece ide 6 Rema: Ide iput data is placed at positio 6. e may also place the iputs i the omal ode; the the outputs ae i the bit-evesed ode. CTU EE 6

7 DSP 7 Computatio of DFT If we ty to maitai the omal ode of both iputs ad outputs the i-place computatio stuctue is destoyed. CTU EE 7

8 DSP 7 Computatio of DFT CTU EE 8 Radi- Decimatio-i-fequecy Algoithms Dividig the output sequece ito smalle pieces. K If is eve. Q Similaly if is odd. et h g

9 DSP 7 Computatio of DFT e ca futhe bea ito eve ad odd goups Agai we ca educe the two-multiplicatio buttefly ito oe multiplicatio. Hece the computatioal compleity is bout log. The i-place computatio popety holds if the outputs ae i bit-evesed ode whe iputs ae i the omal ode. CTU EE 9

10 DSP 7 Computatio of DFT CTU EE FFT fo Composite -- Cooley-Tuey Algoithm: ide : Feq. Time ide : Rema: ad Goal: Decompose -poit DFT ito two stages: -poit DFT -poit DFT -poit facto twiddle -poit Pocedue Compute -poit DFT: ow tasfom G Multiply twiddle factos: ~ G G Compute -poit DFT: colum tasfom ~ G

11 DSP 7 Computatio of DFT Computatio of 5-poit DFT by meas of -poit ad 5-poit DFTs. CTU EE

12 DSP 7 Computatio of DFT CTU EE Etesio: If poit DFT - multiplicatios fo umbe of et colum tasfm :. twiddle factos :. ow tasfom :. I geeal i i i I fact the tem should be because eaagig the buttefly stuctue would mae half of the baches becomig. Special Case: Radi-: ad log multiplicatios because equies o multiplicatios. Radi-: ad log multiplicatios because equies o multiplicatios. This FFT has fewe stages tha Radi- > fewe multiplicatios

13 DSP 7 Computatio of DFT CTU EE Ivese FFT IDFT: DFT: Hece tae the cougate of : DFT Tae the cougate of the above equatio: FFT DFT Thus we ca use the FFT algoithm to compute the ivese DFT.