Processor Architecture Needed to handle FFT algoarithm

1. Processor Architecture Needed to handleFFT algoarithm M. Smith

2. Tackled already this term Three types of DSP algorithms Long loops, multiplication and addition intensive, regular (simple) memory accesses – e.g. 300 taps in FIR algorithms Short loops involving multiplications and additions – e.g. 3 stages in IIR algorithms DSP Introduction, M. Smith, ECE, University of Calgary, Canada

3. Comparing IIR and FIR filters Infinite Impulse Responsefilters – few operations to produce output frominput for each IIR stage 3 – 7 stages Finite Impulse Responsefilters – many operations to produce output frominput. Long FIFO buffer whichmay require as many operations As FIR calculation itself. Easy to optimize DSP Introduction, M. Smith, ECE, University of Calgary, Canada

4. Discrete Fourier Transform • FIR and IIR algorithms directly manipulate the data in “the time domain”. • FIR -- Process M data points using N point FIR filter – involves M * (N-1) additions M * N multiplications M * N * 2 + M memory accesses Algorithm takes a time of Order (M * N) • Very slow if manipulating large amount of data DSP Introduction, M. Smith, ECE, University of Calgary, Canada

5. Frequency domain analysis • Apply discrete Fourier transform (implemented via FFT) • Transform to frequency domain takes time Order (M log M) • Perform FIR in frequency domain takes time Order (M) • Transform back to time-domain takes time Order (M log M) • FFT (Order (M log M) is orders of magnitude faster that FIR (Order (M log N) DSP Introduction, M. Smith, ECE, University of Calgary, Canada

6. DSP Introduction, M. Smith, ECE, University of Calgary, Canada

7. DSP Introduction, M. Smith, ECE, University of Calgary, Canada

8. 4 point DFT to show concepts DSP Introduction, M. Smith, ECE, University of Calgary, Canada

9. Simplify using special complex exponential properties DSP Introduction, M. Smith, ECE, University of Calgary, Canada

10. Running FFT on data stored in array DSP Introduction, M. Smith, ECE, University of Calgary, Canada

11. 8 point FFT with log 8 (= 3) stages • 3 stages – with N / 2 butterflies / stageOrder (N log N) in time DSP Introduction, M. Smith, ECE, University of Calgary, Canada

12. Architectural characteristics needed to handle FFT efficiently DSP Introduction, M. Smith, ECE, University of Calgary, Canada

13. Add / subtract in one instruction • The following instruction is illegal as a single instruction • XFR4 = R2 + R3, XFR5 = R6 + R7;; • Note: comma and NOT semi-colon, means “one instruction” using 6 registers; Not enough data paths to get data into ALU (4 in -- 2 out) • XFR4 = R2 + R3; XFR5 = R6 + R7;; ILLEGAL • FFT Butterfly add is special instruction • XFR4 = R2 + R3, XFR5 = R2 – R3;; • Uses only “4 registers”, 2 in, 2 out DSP Introduction, M. Smith, ECE, University of Calgary, Canada

14. Memory accesses • Stage 1 • Fetch X data at location k and k + N /2 • Store X data at location k and k + N /2 • Stage 2 • Fetch X data at location k and k + N /4 • Store X data at location k and k + N /4 • Stage 3 -- Final stage • Fetch X data at location k and k + N /8 • Store X data at bit-reversed location k and k + N /4 DSP Introduction, M. Smith, ECE, University of Calgary, Canada

15. First issue – how do you store complex numbers? • One option • Use 16-bit values • Store real part in top 16-bits • Store imaginary part in bottom 16 bits • Access data on J-bus • Access complex sinusoids on J-bus • Access both components (R and I) in one cycle • TigerSHARC has the ability to do 16-bit complex additions and multiplications as specific instructions – INTEGER only • Can Use both X and Y compute blocks DSP Introduction, M. Smith, ECE, University of Calgary, Canada

16. Integer operations a pain – tend to overflow • Option 2 – floating point • Store Real component in location X and imaginary component in location Y • Use R1:0 = Q[J4 += 4];; • Store first imaginary number in X0 and Y0 • Store second imaginary number in X1 and Y1 • FR3 = R1 + R0;; – performs complex floating point addition in single cycle • L[J5] = R3;; stores complex answer back DSP Introduction, M. Smith, ECE, University of Calgary, Canada

17. Integer operations a pain – tend to overflow • Option 3 – floating point • Access Real component along J- bus from “data1” and Imaginary component along K-bus from “data 2” • Use XR3:0 = Q[J4 += 4]; YR3:0 = Q[K4 += 4]; ; • Store first imaginary number in X0 and Y0 • Store second, third and fourth imaginary number in XR1, YR1;; XR2, YR2;; XR3, YR3 • Which option is best? Depends? How handle bring in complex sinusoids DSP Introduction, M. Smith, ECE, University of Calgary, Canada

18. Bit reverse addressing DSP Introduction, M. Smith, ECE, University of Calgary, Canada

19. Bit reverse addressing – Check manual for “accurate details” before MII • Only possible with J0, J1, J2 and J3 registers (also K0, K1, K2, K3) • You must start the array on a N aligned boundary otherwise it does not work • J0 = address pointer • JB0 = base register – point to start of array • JL0 = length of array register • JM0 = special circular buffer modify register ???? • XR4 = BR [J0 += 1]; • Bit-reverse addressinbg only works on POST-MODIFY (permits next address to be calculated in parallel) DSP Introduction, M. Smith, ECE, University of Calgary, Canada

20. Issues handling “FFT Butterfly DSP Introduction, M. Smith, ECE, University of Calgary, Canada

21. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

22. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

23. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

24. Wrong again • This is using the “Radix 2” form of the algorithm – breaks down into 2-pt DFT • There is also a Radix 4 form of the algorithm – which is faster again FFT Introduction, M. Smith, ECE, University of Calgary, Canada

25. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

26. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

27. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

28. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

29. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

30. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

31. FFT Introduction, M. Smith, ECE, University of Calgary, Canada

32. Many special TigerSHARC features to handle FFT FFT Introduction, M. Smith, ECE, University of Calgary, Canada