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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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?

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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))

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 6 / 37 Industrial Example -- Equipment

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 7 / 37 Industrial Problem -- Result

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 9 / 37 Planned Solution Visual Model

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 10 / 37 What algorithm could be used Time domain filtering tap FIR N = size of the data ( 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

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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 13 / 37 FFT -- divide and conquer

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 14 / 37 Bit reverse addressing INPUT OUTPUT

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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; }

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 21 / 37 Signal Characteristics -- Time/Frequency MAGNITUDE

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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.

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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 = , a1 = , a2 =

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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 29 / 37 Naturally windowed -- frequency

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 30 / 37 Naturally windowed -- time

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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!!!!!!

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 32 / 37 Synchronously Sample -- Time/Frequency SAMPLED AT “ZEROS” IN WINDOW’S SPECTRUM

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 33 / 37 Synchronously Sample -- Frequency

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 34 / 37 Synchronously Sample -- Time

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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 “How RISCy is DSP”

6/3/2015 ENCM Custom DSP -- not necessarily speed Copyright 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?