Signal Processing and the Fourier Transform

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1 Signal Processing and the Fourier Transform a theme for an applied mathematics course Marcus Pendergrass Hampden-Sydney College MD/DC/VA Section Meeting April 18, 2009 University of Mary Washington

2 Applied Mathematics at H-SC one of two possible tracks to a math major many students do a double-major in a science (e.g. physics, chemistry, biology) and applied math prep for graduate school in a math-intensive discipline other applied math majors go on to graduate school in statistics, or to actuarial and related careers

3 Math Applied Mathematics a topics course Catalog description: Mathematical models and topics in advanced mathematics with application to the natural and social sciences. prerequisites are calculus III and linear algebra typically small class sizes

4 Math Applied Mathematics - Fall 09 theme : signal processing and the Fourier transform theme provides focus to a topics course signal processing provides a rich and intuitive context for teaching the Fourier transform Fourier time domain / frequency domain perspective provides an organizing framework for understanding classical signal processing meta-theme : applied courses are great places to introduce new mathematics

5 Outline of the talk A vignette The Fourier transform as an ideal sinusoid detector Projects class project : removing distortion caused by variable recording speed individual projects : modeling a nonlinear amplifier; data compression using Fourier transforms; fast multiplication with the FFT

6 Fourier Transform as a Sinusoid Detector Pedagogical motivation the Fourier transform decomposes a signal (function) into an ensemble of complex exponentials (sinusoids) x(t) = X(f) e 2πift df where Fourier transform of x X(f) =F[x](f) = amplitude in x of the complex sinusoid e 2πift at frequency f

7 Fourier Transform as a Sinusoid Detector Idea for the detector we re looking for a sinusoid at frequency f0 (the detection frequency) in the incoming signal x(t) correlate the incoming signal against a local replica of the sinusoid we re trying to detect do this for a specified amount of time T, the detection time high values of the correlation should indicate the presence of a sinusoid at the desired frequency low value of the correlation should indicate the absence of a sinusoid

8 A Sinusoid Detector A sinusoid detector - frequency matched case replica correlation } multiply integrate I incoming signal I = T/2 T/2 cos 2 (2πf 0 t) dt = nice healthy-sized positive number

9 A Sinusoid Detector A sinusoid detector - frequency mismatch case replica multiply integrate I incoming signal I = T/2 T/2 cos(2πf 1 t) cos(2πf 0 t) dt = sickly zero-ish kind of number

10 A Sinusoid Detector A sinusoid detector - phase mismatch case replica multiply integrate I incoming signal I = T/2 T/2 cos(2πf 0 t + π/2) cos(2πf 0 t) dt = 0 (flat dead!)

11 A Sinusoid Detector A sinusoid detector - improved in-phase replica multiply integrate I incoming signal multiply integrate Q quadrature replica I2 + Q 2 I 2 + Q A2 T 2 for all phase alignments I Q θ = phase alignment

12 A Sinusoid Detector A sinusoid detector - detector output versus detection frequency and phase I Q I2 + Q 2 The incoming signal is a sinusoid at 440 hertz. Horizontal axis is the detection frequency. Detection time was T = 0.1 seconds.

13 A Sinusoid Detector How does the detector react to other signals? Sawtooth wave with fundamental frequency 440 hz.

14 A Sinusoid Detector How does the detector react to other signals? A complex musical chord

15 From Sinusoid Detector to Fourier transform A sinusoid detector - output versus detection frequency and detection time The detection time T goes from 0.1 seconds to seconds in this animation

16 From Sinusoid Detector to Fourier transform x(t) = incoming signal I = T/2 T/2 x(t) cos(2πf 0 t) dt Q = T/2 T/2 x(t) sin(2πf 0 t) dt I + iq = T/2 T/2 x(t) e 2πif 0t dt x(t) e 2πif 0t dt = F[x](f 0 )=X(f 0 )

17 Fourier Transform as a Sinusoid Detector Next steps complex representation of the Dirac delta e 2πif 0t e 2πift dt = δ(f f 0 ) discussion of weak convergence (convergence in distribution) derivation of inverse transform properties

18 Class Project - The Wobble Problem Removing distortions caused by variable recording speed wow and flutter distortion ( Wobble ) can be corrected if some kind of reference is embedded in the distorted recording assume that the original (undistorted) signal contained a sinusoid that is relatively in the clear in frequency domain

19 Class Project - The Wobble Problem Goal : take the class through one cycle of a problem solving iteration formulate > model > analyze > develop algorithm > simulate and test algorithm > repeat Also want to build proficiency in using some of the tools of the trade Matlab CAS (Maple, Mathematica...) Also want to simulate the work process that an applied mathematician working in industry might experience

20 Class Project - The Wobble Problem Models needed signal library sinusoids, sawtooth, square wave, linear chirp, geometric chirp numerical approximation to the Fourier transform FFT basic filters low pass, bandpass (brick wall) record/playback model distortion model, Gaussian noise model

21 Class Project - The Wobble Problem Algorithm development basic analysis in class led to the dewobble algorithm each student implemented the algorithm in Matlab best features from each were merged into a version 1.0 reference implementation of the algorithm Algorithm testing scenarios noise test : how well does the algorithm work when the reference sinusoid is corrupted by additive Gaussian noise? interference test : how well does the algorithm work when a structured interferer corrupts the reference?

22 Class Project - Some Results Noise Test high wobble Recovery Error medium wobble low wobble SNR (db)

23 Class Project - Some Results Interference Test undistorted reference and interferer distorted reference and interferer recovered reference and interferer actual and recovered wobble

24 Class Project - Some Results Interference Test undistorted reference and interferer distorted reference and interferer recovered reference and interferer actual and recovered wobble

25 Individual Projects Robert Hembree, Mathematical Modeling of Nonlinear Amplifiers * Henry Skiba, Fast Multiplication of Large Integers Using the FFT Alex Smith, Audio Compression Using the Fourier Transform * * = presenting at this meeting

26 Conclusions An organizing theme provides unity and focus for an Applied Mathematics course Signal processing and Communications abound with great applied mathematics Introduce new mathematics in your Applied Math course! the application context will enhance student understanding Thank you!

Problem Set 8 #4 Solution

Problem Set 8 #4 Solution Solution to PS8 Extra credit #4 E. Sterl Phinney ACM95b/100b 1 Mar 004 4. (7 3 points extra credit) Bessel Functions and FM radios FM (Frequency Modulated) radio works by encoding