1. 6/3/20151 This presentation will probably involve audience discussion, which will create action items. Use PowerPoint to keep track of these action items during your presentation In Slide Show, click on the right mouse button Select “Meeting Minder” Select the “Action Items” tab Type in action items as they come up Click OK to dismiss this box This will automatically create an Action Item slide at the end of your presentation with your points entered. Customizing DSP algorithms does not always mean speed A look at DFT / FFT issues M. R. Smith, Electrical and Computer Engineering, University of Calgary, Alberta, Canada smithmr@ucalgary.ca

2. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 2 / 37 Overview Introduction Industrial Example of DFT/FFT DFT -- FFT Theory Straight application Proper application “The KNOW-WHEN” application Future Talks The implications on DSP processor architecture How are actual DSP processors optimized for FFT operations?

3. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 3 / 37 References Work originally done for “Beta Monitors”, Calgary Talk first given to AMD FAE Meeting, Santa Clara Published in Microprocessors and Microsystems FFT - fRISCy Fourier Transforms Copy made available

4. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 4 / 37 Testing and using DSP Algorithms Typical testing pattern -- use something simple Simple test of algorithm correctness Time Signal = sum of sinusoids In test, expect, and get, sharp peaks in spectrum Algorithms used in my research DFT -- Discrete Fourier Transform FFT -- Fast Fourier Transform ARMA -- Autoregressive Moving Average Wavelet

5. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 5 / 37 Testing and using DSP Algorithms Typical testing pattern Simple test of algorithm correctness Time Signal = sum of sinusoids In test, expect, and get, sharp peaks in spectrum IN REAL LIFE -- this is not a valid test as following example shows and many people working in the field don’t get the best out of their algorithms because they don’t realize that. DFT -- Discrete Fourier Transform Implemented directly (Order(N x N) ) operations Implemented by FFT (Order(N x log 2 N))

6. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 6 / 37 Industrial Example -- Equipment

7. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 7 / 37 Industrial Problem -- Result

8. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 8 / 37 Planned Solution -- Theory Unwanted “noise” on a data set can be removed if the “noise” has particular frequency characteristics Improvement is obtained By transforming to the frequency domain, Cutting out (filtering) the unwanted “noise” and then, Inverse transforming to recover the original data form Actually faster to operate in Frequency domain than Time domain (You can show algorithms to be equivalent) Frequency domain -- more memory needed

9. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 9 / 37 Planned Solution Visual Model

10. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 10 / 37 What algorithm could be used Time domain filtering 40 -- 300 tap FIR N = size of the data (1000+ -- infinite) Complexity Order(N x Tap Length) 1024 * 300 = 300,000 operations Frequency domain filtering N-sized DFT Complexity Direct Order(2 * N * N) = 2,000,000 operations FFTOrder(2 * (N log N)) = 20,000 operations

11. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 11 / 37 Direct DFT and FFT Time savings -- Number of complex multiplications NDIRECTRadix 2%Change 4164400% 321024801300% 128163844482100% 10241048576512020488% Key issue -- How can you handle the memory accesses and operations associated with the complex multiplications of data and Fourier Coefficients? -- Data/Instruction Conflicts

12. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 12 / 37 Fast DFT algorithm implementation DFT -- Require Order(N ^ 2) FFT -- Divide and Conquer Principle N pt DFT can be decimated into 2 of N/2 pt DFT plus “some twiddling on N terms” N/2 pt DFT = 2 * N/4 DFT “plus twiddling” N/4 pt DFT = 2 * N / 8 etc Order(N x log N) PROVIDED you can handle bit reverse addressing efficiently

13. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 13 / 37 FFT -- divide and conquer

14. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 14 / 37 Bit reverse addressing INPUT OUTPUT000 100001010 110011101 011110111

15. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 15 / 37 Algorithm -- Different forms x, y == real/imaginary parts of the input wr, wi = precalculated cosine/sine values m = log2(N) where N is the number of points (power of 2) n2 = N for (k = 0; k < m; k++) {/* Outer loop */ n1 = n2; n2 = n2 / 2; ie = n / n1; ia = 1; for (j = 0; j < n2; j++) {/* Middle loop */ c = wr[ia]; s = wi[ia]; ia += ie; for (i = j; i < N; i += n1) {/* inner loop */ l = i + n2/* Butterfly offset */ xt = x[i] - x[l];/* Common */ yt = y[i] - y[l]; x[i] += x[l];/* Upper */ y[i] += y[l]; x[l] = c * xt + s * yt;/* Lower */ y[l] = c * yt - s * xt; }

16. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 16 / 37 What processors can be used? CISC Complex instruction set processor Basic and complex functions Control logic requires much real estate Many cycle instructions DSP Digital signal processing chip Specifically designed for DSP Specialized resources provided Dual cycle instructions (many now one) RISC Reduced instruction set processor Simple instructions done well Instructions complete in single cycle Intelligent compiler needed

17. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 17 / 37 Real life application of Theory Take 370 data points Pad to 512 with zeros to size of algorithm Use standard FFT algorithm Zero unwanted “noise” components Use standard inverse FFT Transform “Angle” measurement to “Volume” Area between hystersis loop is associated with compressor efficiency

18. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 18 / 37 Frequency domain -- filtering Distortions associated with “edge effects” mean that frequency domain signal is not clean. Last point and first point of data -- connected in discrete domain “Cut” will remove more than just “resonance” components

19. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 19 / 37 Time Domain Result Channel resonance -- old problem greatly reduced New distortions evident at edges of data

20. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 20 / 37 Real Life versus Theory Perfect data infinitely long perfectly sampled Actual data Nyquist must be met (sample fast enough to cover signal and noise characteristics) finite length of the data manipulated Can be analysed using Fourier Theory by treating as infinitely long signal multiplied by a square window

21. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 21 / 37 Signal Characteristics -- Time/Frequency MAGNITUDE

22. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 22 / 37 Windowing -- implied and deliberate Windowing the data in the “TIME” domain spreads the “SPECTRUM” MAIN LOBE -- width of main lobe determines resolutions, or how close two similar sized peaks can be placed but yet be separated SIDE LOBES -- height of side lobes determine how close a small peak can be placed to a large peak and be believed as not being a “false” peak (side lobe) Choose a window with the narrowest main lobe and smallest side lobe MRI, seismic, telecommunications all have similar problems This form of data distortion often missed by naive users KEY REFERENCE -- HARRIS -- Proc. IEEE 666, p51, 1978

23. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 23 / 37 Windowing occurs -- when? ALL DATA ANYBODY GATHERS is always windowed NO EXCEPTIONS -- finite length in either time or frequency domain DFT (and many other algorithms) treat data AS CYCLIC No problems if CYCLIC model results in continuous data across the cycles (Nth order continuity is needed) Discontinuities in data cause BIG problems in frequency domain -- in particular padding with zeros in order to use any DFT algorithm Some diseases in MRI are mimicked by truncation artifacts

24. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 24 / 37 How to fix Chose a better window Naturally window Take data in a way that the data goes more smoothly to zero at end Synchronously sample Very special case -- and possible for this data set Different DSP algorithm Not always stable -- MA, AR, ARMA, Burg, wavelet etc.

25. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 25 / 37 Windows W(m) = a0 + a1 cos (2 PI m / N) + a2 cos (4 PI m / N ) (0 <= m < N) BEWARE -N/2 <= m < N/2 -- flips sign of a1 Normal (Rect.)a0 = 1, a1 =0, a2 = 0 Simple a0 = 0.54, a1 = -0.46, a2 = 0; Blackman-Harris 3 term -- optimized a0 = 0.44959, a1 = -0.49364, a2 = 0.05677

26. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 26 / 37 Windowing -- 2 cycles Remember to “window” NOT cut out the channel resonance in Frequency Domain too!

27. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 27 / 37 Natural Window 1. Rearrange the way you sample so that data “naturally goes to same DC level” near ends 2. Remove DC offset then pad with zeros Resolution between peaks in the frequency domain is function of data length. This example uses 2.5 cycles of the original data sequence

28. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 28 / 37 Naturally window -- Match ends at “DC” Not always possible with “real data” Advantage -- no data distortion occurring when window gets applied. Actually does occur, but is hidden -- see later

29. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 29 / 37 Naturally windowed -- frequency

30. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 30 / 37 Naturally windowed -- time

31. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 31 / 37 Synchronously Sample the Data As an engineer, you have to be able to reach back into your “theory” and recognize when this sort of thing is possible and correct! Not a solution for most data sets There must be a “TRUE”, exact, cyclic property present in the original data set. Algorithm must be applied “exactly correctly” Windowing is still there! All the windowing distortions are still present -- BUT!!!!!!

32. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 32 / 37 Synchronously Sample -- Time/Frequency SAMPLED AT “ZEROS” IN WINDOW’S SPECTRUM

33. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 33 / 37 Synchronously Sample -- Frequency

34. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 34 / 37 Synchronously Sample -- Time

35. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 35 / 37 Synchronously Sample Not possible for most situations There is a “TRUE” cyclic property present in data Don’t Pad with zeros -- use 740 pt DFT This industrial example 370 points round the cycle Would a specialized FFT algorithm improve things? (2 x 2 x 5 x 7 x 7) Implemented directly using a 740 point DFT Customer satisfied with integer implementation on Z80

36. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 36 / 37 This sort of customization -- NOT NORMALLY POSSIBLE What are the characteristics of general DSP algorithms? What needs to be present on a processor to meet those requirements? Covered in earlier lecture See IEEE Micro Magazine, Dec. 1992 “How RISCy is DSP”

37. 6/3/2015 ENCM515 -- Custom DSP -- not necessarily speed Copyright smithmr@ucalgary.ca 37 / 37 Overview Introduction Industrial Example of DFT/FFT DFT -- FFT Theory Straight application Proper application “The KNOW-WHEN” application Future talks The implications on DSP processor architecture How are actual DSP processors optimized for FFT operations?