Computational Methods CMSC/AMSC/MAPL 460. Fourier transform

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1 Computational Methods CMSC/AMSC/MAPL 460 Fourier transform Ramani Duraiswami, Dept. of Computer Science Several slides from Prof. Healy s course at UMD

2 Fourier Methods Fourier analysis ( harmonic analysis ) a key field of math Has many applications and has enabled many technologies. Basic idea: Use Fourier representation to represent functions Has fast algorithms to manipulate them (the fast Fourier Transform) Requires a complete course (Signal Processing in EE or Math 464 Introduction to Fourier Analysis

3 Basic idea Function spaces can have many different types of bases We have already met monomials and other polynomial basis functions Fourier introduced another set of basis functions : the Fourier series These basis functions are particularly good for describing things that repeat with time

4 Fourier s Representation F(t)= A 0 /2 + A Cos( t) + A 2 Cos(2 t) + A 3 Cos(3 t) + + B Sin( t) + B 2 Sin(2 t) + B 3 Sin(3 t) + For coefficients that go to 0 fast enough these sums will converge at each value of t. This defines a new function, which must be a periodic function. (Period 2 π ) Fourier s claim: ANY periodic function f(t) can be written this way

5 Music

6 Mr. Fourier s Representation Sin( t) + Represent f(t) = t for t < π and 2π periodic

7 Mr. Fourier s Representation Sin( t)

8 Mr. Fourier s Representation Sin( t) - /2 Sin( 2 t)

9 Mr. Fourier s Representation Sin( t) - /2 Sin( 2 t)

10 Mr. Fourier s Representation Sin( t) - /2 Sin( 2 t) +/3 Sin( 3 t)

11 Mr. Fourier s Representation Sin( t) - /2 Sin( 2 t) + /3 Sin( 3 t)

12 Mr. Fourier s Representation 20 th degree Fourier expansion

13 How do you get the Coefficients for a given f? A 0 /2 + A Cos( t) + A 2 Cos(2 t) + A 3 Cos(3 t) + + B Sin( t) + B 2 Sin(2 t) + B 3 Sin(3 t) + Fourier s claim: ANY periodic function f(t) can be written this way (SYNTHESIS) The coefficients are uniquely determined by f: (ANALYSIS) A k = / 2 π 0 2 π f (t) cos( k t) d t B k = / 2 π 0 2 π f (t) sin ( k t) d t

14 Fourier Analysis: match data with sinusoids 0.75 s(t) Time t S[k] = s (t) cos( k t) d t Frequency k

15 Complex Notation For f, periodic with period p Fourier transform f(t) -> F[k] F[k] = /p 0 p f (t) e -2 π i k t/p d t = /p 0 p f (t) cos(2 π k t/p) dt i / p 0 p f (t) sin(2 π k t/p) dt Inverse Fourier transform F[k] -> f(t) f(t) = Σ k in Z F[k] e 2 π i k t/p

16 Sampling Fourier representations work just fine with sampled data Simple connection to Fourier of the continuous function it came form Familiar example: Digital Audio

17 Measuring and Discretizing Input field Physical Field (continuum) PHYSICAL LAYER

18 Sample Physical Field (continuum) PHYSICAL LAYER

19 Quantize Physical Field (continuum) PHYSICAL discretized waveform LAYER

20 Code and output Physical Field (continuum) Digital Representation PHYSICAL LAYER 3, 8, 0, 9, 3,, 2

21 Sampling Often must work with a discrete set of measurements of a continuous function

22 Sampling R Takes a function defined on and creates a function defined on Z S h f(t) φ[n] = f(n h) h

23 Sampling In this case, it is a periodic function on Z, (Assuming p/h = N) N φ[n] = f(n h) φ[n+n] = φ[n] h

24 DFT N- Σ p f (t) e -2 π i k t/p φ[n] e d t 0 n =0-2 π i k nh /p f(t) φ[n] = f(n h) h

25 DFT and its inverse for periodic discrete data Inverse is like Fourier series, but with only p terms Φ[k] = N- Σ n =0 φ[n] e -2 π i k n h /p p = N h N- = Σ n=0 φ[n] e -2 π i k n / N This is automatically periodic in k with period N

26 DFT: Discrete time periodic version of Fourier time domain /Ν N Σ γ[k] e k =0-2 π i k m/n = Γ [m] frequency domain γ[k], on P N i.e. on Z, Period N γ[k]= Σ k =0 N Γ [k] e 2 π i m k/n Γ [m] on P N i.e. on Z, Period N

27 Two PERIODIC time versions of Fourier p f (t) e -2 π i k t/p d t /p f(t), period p S p/n Σ k in Z /Ν F[k] e 2 π i k t/p N- Σ k =0 γ[k] e P N -2 π i k m/n F[k], on Z 0.2 γ[k], on P N i.e. on Z, Period N time domain Σ N- Γ [k] e 2 π i m k/n Γ [m] on P N i.e. on Z, Period N k =0 frequency domain

28 Discrete time Numerical Fourier Analysis DFT is really just a matrix multiplication! Γ [m] = /Ν Γ[0] Γ[] Γ[2]... Γ[N-] = Freq. index N- Σ k =0 e -2 π i k m/n γ[k] time index γ[0] γ[] γ[2]... γ[n-] Γ = F N γ

29 Numerical Harmonic Analysis FFT: Symmetry Properties permits Divide and Conquer Sparse Factorization F mn = ( F m I n ) T mn n ( I m F n ) L mn m F n Naive FFT

30 Structured matrices Fast algorithms have been found for many dense matrices Typically the matrices have some structure Definition: A dense matrix of order N N is called structured if its entries depend on only O(N) parameters. Most famous example the fast Fourier transform

Computational Methods CMSC/AMSC/MAPL 460

Computational Methods CMSC/AMSC/MAPL 460 Fourier transform Balaji Vasan Srinivasan Dept of Computer Science Several slides from Prof Healy s course at UMD Last time: Fourier analysis F(t) = A 0 /2 + A