Computation of DFT. point DFT. W has a specific structure and because

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1 DSP 5 Spig 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 VLSI implemetatio DFT as a Liea Tasfomatio Mati epesetatio of DFT Defiitio of DFT: whee Let ad 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 5 Spig 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 Fo each we eed comple multiplicatios ad - comple additios. eal multiplicatios ad - eal additios.

3 DSP 5 Spig Computatio of DFT 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 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: odd eve e e / / / / -poit DFT G H -poit DFT CTU EE

4 DSP 5 Spig 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 CTU EE

5 DSP 5 Spig Computatio of DFT At each stage: ~ comple multis ad adds Total: ~ log comple multis ad adds --> log umbe of poits Diect Computatio: Comple Multis FFT: Comple Multis Speed Impovemet Facto Buttefly: Basic uit i FFT Two multiplicatios: CTU EE 5

6 DSP 5 Spig Computatio of DFT Oe multiplicatio: 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 CTU EE 6

7 DSP 5 Spig Computatio of DFT 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. 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 5 Spig Computatio of DFT CTU EE 8 Radi- Decimatio-i-fequecy Algoithms Dividig the output sequece ito smalle pieces. If is eve. Similaly if is odd.

9 DSP 5 Spig Computatio of DFT CTU EE 9 Let h g 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.

10 DSP 5 Spig 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 5 Spig Computatio of DFT Computatio of =5-poit DFT by meas of -poit ad 5-poit DFTs. CTU EE

12 DSP 5 Spig Computatio of DFT CTU EE Etesio: If poit DFT - multiplicatios fo umbe of Let 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 5 Spig 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. The Goetzel Algoithm e e If we defie = fo < ad ad u u y y

14 DSP 5 Spig Computatio of DFT H z z If is comple we eed eal multiplicatios ad eal additios to compute each y. To compute y we eed to compute y y y-. e eed eal multiplicatios ad eal additios to compute. Remas: less efficiet tha the diect method. Avoid the computatio o stoage of the coefficiets. To educe the umbe of multiplicatios H z z z z cos / z z z If is comple we oly eed eal multiplicatios ad eal additios to implemet the poles of the system. The comple multiplicatio by eeds ot be pefomed at evey iteatio. To compute we eed eal multiplicatios ad eal additios fo the poles ad eal multiplicatios ad eal additios fo the zeo. Remas: Avoid the computatio o stoage of the coefficiets Oly eed to compute ad save ad cos /.. CTU EE