Representing polynominals with DFT (Discrete Fourier Transform) and FFT (Fast Fourier Transform) Arne Andersson

Transcription

1 Represetig polyomils with DFT Discrete Fourier Trsform d FFT Fst Fourier Trsform Are Adersso

2 Iformtiostekologi Polyomil A Emple: 4 Coefficiet represettio:,,,4 Poit-vlue represettio:,, ,,,7,, Istitutioe för iformtiostekologi

3 Iformtiostekologi Istitutioe för iformtiostekologi Evlutig polyomils Coefficiet represettio: Use Horer s rule. Time: Θ Poit-vlue represettio: The best wy is to covert ito coefficiet represettio iterpoltio A

4 Addig polyomils Iformtiostekologi Coefficiet represettio: Add the coefficiets. Time: Θ Poit-vlue represettio: Add the y- vlues provided tht the -coordites re the sme Time: Θ Istitutioe för iformtiostekologi

5 Multiplyig polyomils Iformtiostekologi Coefficiet represettio: Stdrd schoolbook multiplictio multiply ech coefficiet i oe polyomil with ll coefficiets i hte other: Time: Θ Poit-vlue represettio:. We eed eteded poit vlues represettio of ech polyomil, where the umber of poits equls the umber of poits i the ifl product i.e. The degree of the product. Multiply the y-vlues of ech poit Time: Θ Istitutioe för iformtiostekologi

6 Summry Iformtiostekologi Evlute Coefficiet Θ Add Θ Θ Multiply Θ Θ Poit vlue Wht if I eed to multiply d evlute polyomils? C we covert betwee represettios? Istitutioe för iformtiostekologi

7 Covertig betwee represettios versio Iformtiostekologi Coefficiet -> Poit Vlue evlutio: Just evlute i poits. Time: Θ Poit Vlue -> Coefficiet iterpoltio: Lgrge s formul. Time: Θ A j k k j k k j j Istitutioe för iformtiostekologi

8 Wht c we hope for Iformtiostekologi Miti polyomil i coefficiet represettio for fst evlutio Whe we muptiply, covert to poit-vlue repersettio, multiply, d covert bck I order to utilize this, we eed fster coversio betwee represettios th Θ Istitutioe för iformtiostekologi

9 Fst Fourier Trsform Iformtiostekologi The Fst Fourier Trsform FFT llows coversio o O log time. Istitutioe för iformtiostekologi

10 FFT: Bsic ide Iformtiostekologi Use poit represettio t crefully selected set of poits: Comple roots of uity. Comple th root of umber ω such tht ω uity There re comple th roots of Resides o the uit circle uity - i -i Istitutioe för iformtiostekologi

11 Iformtiostekologi Istitutioe för iformtiostekologi Rewrite the polyomil s lower-degree poyomils i [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]. degree re polyimls of d where odd epoets... eve epoets.... odd epoets... eve epoets / A A A A A

12 Iformtiostekologi..d use ice properties of comple roots of uity I order to evlute A t the th root of uity ω, ω, ω,..., ω we evlute the two / - degree polyomils t,,,..., ω ω ω ω But, by the ture of comple roots the Hlvig Lemm, there re oly / distict umbers! Hlvig Lemm : If is eve, the squres of the comple th roots re the /th roots of uity. Istitutioe för iformtiostekologi

13 The cost of evlutig polyomil i poits roots of uity: Iformtiostekologi. Recursively evlute two ployomils of degree /. Combie the results. T T T θ log / θ Istitutioe för iformtiostekologi

14 Iformtiostekologi Also, by usig properties of comple root of uity, we c iterpolte polyomil quickly T θ log Istitutioe för iformtiostekologi

15 Two polyomils, coefficet represettio θ Product, coefficet represettio Iformtiostekologi FFT log θ Two polyomils, poit-vlue represettio DFT θ FFT log θ Product, poit-vlue represettio DFT Istitutioe för iformtiostekologi