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95-1 Under-Graduate ProjectMid-Term Paper Reading Presentation

Adviser: Prof. An-Yeu WuMentor: 詹承洲

第二組溫仁揚 溫昌懌

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A New Approach to Pipeline FFT Processor

Shousheng He & Mats TorkelsonDepartment of Applied Electronics, Lund University, S-22100 Lund, SWEDE

N

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Outline What is FFT How to solve FFT by hardware Approaches to Pipeline FFT A closer look to Radix- Simulation by C language Future Work Conclusion Reference

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What is FFT

Discrete Fourier Transform is defined as:

with complexity Fast Fourier Transform (FFT) is an

efficient algorithm which reduced the complexity into

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What is FFT(cont’d) The Cooly-Tukey algorithm

by symmetry,

Hence, we re-express the size N transform into 2 size N/2 transforms

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How to Solve FFT by Hardwares

Divide & Conquer

Butterfly

Twiddle Factor

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W(n) –twiddle factor

Radix-22DIFFFT :

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Approach to Pipeline FFT R2MDC

R2SDF

R4MDC

R4SDF

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Approach to Pipeline FFT(cont’d)

R4SDC

Radix-

Still many other approaches, ex:

Radix- , Mixed Radix, Chirp-Z… etc.

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We focus on this structure today

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A Closer Look to Radix-22x(0)+x(8)[x(0)+x(8)]+[x(4)+(12)]

W(n)

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A Closer Look to Radix- (cont’d)

22A 256-point model of Radix-

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The size of register is 2n

W2(n)=e{-j*(n15*(k11+2*k12) )} W3(n)= e{-j*4*n3*(k21+2*k22)}

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First clks, push inputs into registerfile.

n2

A Closer Look to Radix- (cont’d)

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n2 12 n

Then start to do thebutterfly.

From +1 clk to next clks, push inputsinto register file 12 n

n2

Then start to do thebutterfly

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A Closer Look to Radix-22

input

BF2i

Register

BF2ii

Register

W(n)

output

counter

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Simulation by C language (cont’d)

Main function (A 64 points model)int main(void)

{ ……//declare variables, assign inputsfor(clk=0;clk<=2*N;clk++){ BF2i ( 5,clk>>5 ); //first BF #5 REGIST(5); //first regs BF2ii( 4,clk>>5 ,(clk>>4)%2 ); //second BF #4 REGIST(4); //second regs Zr[0][4] = W2 * Zr[0][4] ; //twiddle factor

} //end of for

//print outputs and make comparison to theoretical value

} //end of main

Do it 3 times with different parameters

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Simulation by C language (cont’d)

void BF2i(char t,bool s)// 4 inputs which are Xr(n), Xi(n), Xr(n+N/2),

Xi(n+N/2) ,clk signal in the fiqure{ if (s){ Zr[0][t]= Zr[0][t+1]+regr[t][pow(2,t)-1]; Zi[0][t]= Zi[0][t+1]+regi[t][pow(2,t)-1]; Zr[1][t]= regr[t][pow(2,t)-1]-Zr[0][t+1]; Zi[1][t]= regi[t][pow(2,t)-1]-Zi[0][t+1]; }// the vatue of Zs when s==1 else{ Zr[0][t]=regr[t][pow(2,t)-1]; Zi[0][t]=regi[t][pow(2,t)-1]; Zr[1][t]=Zr[0][t+1]; Zi[1][t]=Zi[0][t+1]; }// the value of Zs when s==0 }// end of BF2i

regr[t][2^t-1]regi[t][2^t-1]

Zr[0][t+1]Zi[0][t+1]

Zr[0][t]Zi[0][t]

Zr[1][t]Zi[1][t]

BF2i

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Simulation by C language (cont’d)

S=0 :

S=1 :

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Simulation by C language (cont’d)

BF2iivoid BF2ii(char t, bool s, bool g){if(g){ if(s){ Zr[0][t]=regr[t][pow(2,t)-1]+Zr[0][t+1]; Zi[0][t]=regi[t][pow(2,t)-1]+Zi[0][t+1]; Zr[1][t]=regr[t][pow(2,t)-1]-Zr[0][t+1]; Zi[1][t]=regi[t][pow(2,t)-1]-Zi[0][t+1]; } else {

…}

else{ … }}

Zr[0][t+1]Zi[0][t+1]

Zr[1][t]Zi[1][t]

regr[t][2^t-1]regi[t][2^t-1]

Zr[0][t]Zi[0][t]

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Simulation by C language (cont’d)

g=0:

g=1, s=0,1:

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Simulation by C language (cont’d)

Register

12 n

void REGIST(char num){ for (char l=pow(2,num-1); l>=1;l--) { // from MSB to LSB regr[num][l]=regr[num][l-1]; regi[num][l]=regi[num][l-1]; } // end of for loop regr[num][0]=Zr[1][num]; regi[num][0]=Zi[1][num];}//end of function

-1 … 1 012 n

Zr[1][num]Zi[1][num]

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Simulation by C language (cont’d)

Simulation results:

16 points

completed 64 points with error

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Future Work

Develop more points Perform it in HDL language, ex: Verilo

g Final Presentation

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Conclusion

Under how to solve FFT by hardware

Develop W(n) to 64 points Construct Radix22 ‘s model C simulation with some faults

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Reference Shousheng He & Torkelson, M.,”A new approach to pipeline

FFT processor,” Proceedings of IPPS’96, 15-19 April 1996, pp766-770.

Shousheng He & Torkelson, M.,”Designing Pipeline FFT Processor for OFDM (de)Modulation,” Signals, Systems, and Electronics’98 29 Sep-2 Oct 1998,pp 257-262.

施信毓 ,”95-1 Under-Graduate Project, Final Project: single-path Delay Feedback FFT”,04 Oct 2006.

http://www.clecom.co.uk/science/autosignal/help/FFT_Algorithms.htm

http://en.wikipedia.org/ http://zh.wikipedia.org/