1. Processor Architecture Needed to handle FFT algoarithm M. Smith

2. FFT algorithms  There are more FFT algorithms performed per day than any other algorithm in the world  Therefore, of course, custom parts of the processors to handle this situations FFT Introduction, M. Smith, ECE, University of Calgary, Canada 2

3. 1 MRI session FFT Introduction, M. Smith, ECE, University of Calgary, Canada 3 http://www.core.org.cn/NR/rdonlyres /Nuclear-Engineering/22-56JFall- 2005/EA28B7B3-39E5-4999-B858- 5E3B248E5408/0/chp_mri.jpg

4. Magnetic resonance (MR) and DFT discrete Fourier transform  Place a body in a steady magnetic field – 1.5 Telsa, (3 or 7)  All the protons spin (precess) around magnetic field at a frequency of around10 MHz  If sent in an RF (90 degree) pulse at 10 MHz – will be absorbed by system.  Some energy then emitted by system as sinusoid at 10 MHZ  Do DFT – single pulse whose height proportional to number of protons in body (amount of hydrogen in body = amount of water in body)  Used to non-destructively measuring water content in wheat FFT Introduction, M. Smith, ECE, University of Calgary, Canada 4

5. Magnetic resonance imaging (MRI) and DFT discrete Fourier transform  Place four bodies in a steady magnetic field – 1.5 Telsa, (3 or 7) Apply 90 pulse Apply DFT on response One signal at 10 MHz  Place four bodies in a steady magnetic field of 1.5 Telsa Apply 90 pulse Apply a “field gradient” in X direction Apply DFT on response Four signals 10 +G.X1, 10 + G.X2, 10 + G.X3, 10 + G.X4 where Xi is the x position of object FFT Introduction, M. Smith, ECE, University of Calgary, Canada 5

6. Magnetic resonance imaging (MRI) and DFT discrete Fourier transform  Place four bodies in a steady magnetic field of 1.5 Telsa Apply 90 pulse Apply a “field gradient” Gx in X direction Apply DFT on response Four signals 10 +Gx.X1, 10 + Gx.X2, 10 + Gx.X3, 10 + Gx.X4 where Xi is the x position of object Apply 90 pulse Now add “a field gradient in both X and Y directions Apply DFT on response Four signals 10 +Gx.X1 + Gy.Y1, 10 + Gx.X2 + Gy.Y2, 10 + Gx.X3 + Gy.Y3, 10 + Gx.X4 + Gy.Y3 where Xi is the x position of object and Yi is the i position of object FFT Introduction, M. Smith, ECE, University of Calgary, Canada 6

7. 1 MRI session  Occurs for about 20 minutes  Echo planar imaging (EPI) Generates 19 2 D slices of the brain in about 60 seconds Each image is 256 pixels by 256 pixels x 19 Each image requires 256 * 256 * 19 DFTs / minute DFT is only ONE part of the algorithm  My research is using MRI for stroke diagnosis FFT Introduction, M. Smith, ECE, University of Calgary, Canada 7 http://www.core.org.cn/NR/rdonlyres /Nuclear-Engineering/22-56JFall- 2005/EA28B7B3-39E5-4999-B858- 5E3B248E5408/0/chp_mri.jpg http://www.magnet.fsu.edu/educatio n/tutorials/magnetacademy/mri/imag es/mri-scanner.jpg

8. 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 8

9. 9 Comparing IIR and FIR filters Infinite Impulse Response filters – few operations to produce output from input for each IIR stage 3 – 7 stages Finite Impulse Response filters – many operations to produce output from input. Long FIFO buffer which may require as many operations As FIR calculation itself. Easy to optimize

10. 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 10

11. 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 M) DSP Introduction, M. Smith, ECE, University of Calgary, Canada 11

12. DSP Introduction, M. Smith, ECE, University of Calgary, Canada 12

13. DSP Introduction, M. Smith, ECE, University of Calgary, Canada 13

14. 4 point DFT to show concepts DSP Introduction, M. Smith, ECE, University of Calgary, Canada 14

15. Simplify using special complex exponential properties DSP Introduction, M. Smith, ECE, University of Calgary, Canada 15

16. Running FFT on data stored in array DSP Introduction, M. Smith, ECE, University of Calgary, Canada 16

17. 8 point FFT with log 8 (= 3) stages  3 stages – with N / 2 butterflies / stage Order (N log N) in time DSP Introduction, M. Smith, ECE, University of Calgary, Canada 17

18. Architectural characteristics needed to handle FFT efficiently DSP Introduction, M. Smith, ECE, University of Calgary, Canada 18

19. Add / subtract in one instruction  The following instruction is illegal as a single instruction F4 = F2 + F3, F5 = F6 – F7; ILLEGAL Needs bits to describe 6 registers (6 * 4 bits)  FFT Butterfly add is special instruction F4 = F11 + F12, F5 = F11 – F12; Uses only “4 registers”, 2 in, 2 out (4 * 2 bits)  2 bits – how come? F4 = F12 + F11, F5 = F12 – F11; ILLEGAL Fx = F11-8 + F15-12, Fy = F11-8 - F15-12 DSP Introduction, M. Smith, ECE, University of Calgary, Canada 19

20. 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 20

21. 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 (NOT SHARC) Can use both X and Y compute blocks DSP Introduction, M. Smith, ECE, University of Calgary, Canada 21

22. Integer operations a pain – tend to overflow -- TigerSHARC syntax  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 22

23. Integer operations a pain – tend to overflow -- TigerSHARC syntax  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 23

24. Bit reverse addressing DSP Introduction, M. Smith, ECE, University of Calgary, Canada 24

25. Bit reverse addressing – Check manual for “accurate details” before MII  Only possible with I0, I1, I2? and I3? registers (also I8, I9, I10?, I11?)  You must start the array on a N aligned boundary otherwise it does not work I0 = address pointer B0 = base register – point to start of array L0 = length of array register M0 = special circular buffer modify register ???? F4 = BR [I0 += 1]; // Correct SHARC syntax??? Bit-reverse addressing only works on POST-MODIFY (permits next address to be calculated in parallel) DSP Introduction, M. Smith, ECE, University of Calgary, Canada 25

26. Issues handling “FFT Butterfly DSP Introduction, M. Smith, ECE, University of Calgary, Canada 26

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

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

29. FFT Introduction, M. Smith, ECE, University of Calgary, Canada 29 Only possible on TigerSHARC

30. 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 30

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

32. FFT Introduction, M. Smith, ECE, University of Calgary, Canada 32

33. FFT Introduction, M. Smith, ECE, University of Calgary, Canada 33 TIGERSHARC

34. DSP Co-processor on SHARC FFT Introduction, M. Smith, ECE, University of Calgary, Canada 34

35. FFT Introduction, M. Smith, ECE, University of Calgary, Canada 35

36. Will discuss FFT accelerator later  Was not available on previous SHARCS  Way to go with future processors Cheap processors + co-processors Cheap microcontroller + FPGA component FFT Introduction, M. Smith, ECE, University of Calgary, Canada 36