MATH 350: Introduction to Computational Mathematics

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1 MATH 350: Introduction to Computational Mathematics Chapter VIII: The Fast Fourier Transform Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2008

2 Outline 1 The (Continuous) Fourier Transform 2 The Discrete Fourier Transform 3 The FFT in MATLAB 4 Examples

3 Outline The (Continuous) Fourier Transform 1 The (Continuous) Fourier Transform 2 The Discrete Fourier Transform 3 The FFT in MATLAB 4 Examples

4 Introduction The (Continuous) Fourier Transform From your differential equations class you know the Laplace transform F(s) = L{f (t)} = 0 e st f (t)dt and perhaps remember some of its nice properties,

5 Introduction The (Continuous) Fourier Transform From your differential equations class you know the Laplace transform F(s) = L{f (t)} = 0 e st f (t)dt and perhaps remember some of its nice properties, such as translation e at F(s a),

6 Introduction The (Continuous) Fourier Transform From your differential equations class you know the Laplace transform F(s) = L{f (t)} = 0 e st f (t)dt and perhaps remember some of its nice properties, such as translation e at F(s a), derivatives t n f (t) ( 1) n d n ds n F(s),

7 Introduction The (Continuous) Fourier Transform From your differential equations class you know the Laplace transform F(s) = L{f (t)} = 0 e st f (t)dt and perhaps remember some of its nice properties, such as translation e at F(s a), derivatives convolution (f g)(t) = t n f (t) ( 1) n d n t 0 ds n F(s), f (τ)g(t τ)dτ F(s)G(s). These properties allow us to convert systems of linear ODEs to systems of linear algebraic equations.

8 The (Continuous) Fourier Transform Another very important integral transform is given by the Fourier transform.

9 The (Continuous) Fourier Transform Another very important integral transform is given by the Fourier transform. The Fourier transform takes a function in the "time-domain" and transforms it into another function its Fourier transform in the "frequency domain".

10 The (Continuous) Fourier Transform Another very important integral transform is given by the Fourier transform. The Fourier transform takes a function in the "time-domain" and transforms it into another function its Fourier transform in the "frequency domain". This terminology is due to the fact that the Fourier transform plays a fundamental role in signal processing.

11 The (Continuous) Fourier Transform Definition The Fourier transform ŷ of a square-integrable function y is defined as: ŷ(ω) = e iωt y(t)dt, < ω <. (1)

12 The (Continuous) Fourier Transform Definition The Fourier transform ŷ of a square-integrable function y is defined as: ŷ(ω) = e iωt y(t)dt, < ω <. (1) On the other hand, the inverse Fourier transform lets us reconstruct y from its Fourier transform ŷ: y(t) = 1 2π e iωt ŷ(ω)dω, < t <. (2)

13 The (Continuous) Fourier Transform Definition The Fourier transform ŷ of a square-integrable function y is defined as: ŷ(ω) = e iωt y(t)dt, < ω <. (1) On the other hand, the inverse Fourier transform lets us reconstruct y from its Fourier transform ŷ: Remark y(t) = 1 2π e iωt ŷ(ω)dω, < t <. (2) There are other (equivalent) ways to define the Fourier transform: ŷ(ω) = 1 2π e iωt y(t)dt and y(t) = 1 2π eiωt ŷ(ω)dω ŷ(f ) = e 2πift y(t)dt and y(f ) = e2πift ŷ(f )df

14 The (Continuous) Fourier Transform The Fourier transform also satisfies many nice properties that make it particularly useful in the context of signal processing.

15 The (Continuous) Fourier Transform The Fourier transform also satisfies many nice properties that make it particularly useful in the context of signal processing. For example, convolution (x y)(t) = x(τ)y(t τ)dτ ˆx(ω)ŷ(ω),

16 The (Continuous) Fourier Transform The Fourier transform also satisfies many nice properties that make it particularly useful in the context of signal processing. For example, convolution (x y)(t) = x(τ)y(t τ)dτ ˆx(ω)ŷ(ω), modulation e iω 0t y(t) ŷ(ω ω 0 ),

17 The (Continuous) Fourier Transform The Fourier transform also satisfies many nice properties that make it particularly useful in the context of signal processing. For example, convolution (x y)(t) = x(τ)y(t τ)dτ ˆx(ω)ŷ(ω), modulation e iω 0t y(t) ŷ(ω ω 0 ), scaling y(at) 1 ( ω ). a ŷ a

18 The (Continuous) Fourier Transform Example Consider the function y(t) = and compute its Fourier transform. { 1, if 1 2 t 1 2 0, otherwise,

19 The (Continuous) Fourier Transform Example Consider the function y(t) = and compute its Fourier transform. Solution { 1, if 1 2 t 1 2 0, otherwise, The definition of the Fourier transform (1) and the definition of y imply ŷ(ω) = e iωt y(t)dt

20 The (Continuous) Fourier Transform Example Consider the function y(t) = and compute its Fourier transform. Solution { 1, if 1 2 t 1 2 0, otherwise, The definition of the Fourier transform (1) and the definition of y imply ŷ(ω) = = e iωt y(t)dt e iωt dt

21 The (Continuous) Fourier Transform Solution (cont.) Now, Euler s formula and the fact that cosine is even and sine is odd give ŷ(ω) = [cos(ωt) i sin(ωt)] dt

22 The (Continuous) Fourier Transform Solution (cont.) Now, Euler s formula and the fact that cosine is even and sine is odd give ŷ(ω) = = [cos(ωt) i sin(ωt)] dt cos(ωt)dt

23 The (Continuous) Fourier Transform Solution (cont.) Now, Euler s formula and the fact that cosine is even and sine is odd give ŷ(ω) = = = 2 sin(ωt) ω [cos(ωt) i sin(ωt)] dt cos(ωt)dt 1 2 0

24 The (Continuous) Fourier Transform Solution (cont.) Now, Euler s formula and the fact that cosine is even and sine is odd give ŷ(ω) = = = 2 sin(ωt) ω [cos(ωt) i sin(ωt)] dt cos(ωt)dt = sin ω 2 ω 2.

25 The (Continuous) Fourier Transform Solution (cont.) Now, Euler s formula and the fact that cosine is even and sine is odd give ŷ(ω) = = = 2 sin(ωt) ω [cos(ωt) i sin(ωt)] dt cos(ωt)dt = sin ω 2 ω 2 These functions play an important role in many applications (e.g., signal processing). The function y is known as a square pulse or characteristic function of the interval [ 1 2, 1 2 ]..

26 The (Continuous) Fourier Transform Solution (cont.) Now, Euler s formula and the fact that cosine is even and sine is odd give ŷ(ω) = = = 2 sin(ωt) ω [cos(ωt) i sin(ωt)] dt cos(ωt)dt = sin ω 2 ω 2 These functions play an important role in many applications (e.g., signal processing). The function y is known as a square pulse or characteristic function of the interval [ 1 2, 1 2 ]. Its Fourier transform ŷ is known as the sinc function..

27 The (Continuous) Fourier Transform

28 The (Continuous) Fourier Transform Other information about the continuous Fourier transform and its connection to Fourier series is provided in Chapter 8.9 of [NCM].

29 Outline The Discrete Fourier Transform 1 The (Continuous) Fourier Transform 2 The Discrete Fourier Transform 3 The FFT in MATLAB 4 Examples

30 The Discrete Fourier Transform In many applications the data is given in discrete form (for example, digital audio for MP3, CD or DVD players).

31 The Discrete Fourier Transform In many applications the data is given in discrete form (for example, digital audio for MP3, CD or DVD players). Therefore, we now look at the Fourier transform of the discrete and periodic data y = [y 0,..., y n 1 ] T with y j = y( 2jπ n ), j = 0,..., n 1. Here time is specified by the n-vector t = [0, 2π n, π n,..., 2(n 1)π n ] T with entries t j = 2jπ n.

32 The Discrete Fourier Transform In many applications the data is given in discrete form (for example, digital audio for MP3, CD or DVD players). Therefore, we now look at the Fourier transform of the discrete and periodic data y = [y 0,..., y n 1 ] T with y j = y( 2jπ n ), j = 0,..., n 1. Here time is specified by the n-vector t = [0, 2π n, π n,..., 2(n 1)π n ] T with entries t j = 2jπ n. We can think of the data y j being sampled from the continuous function y at the time instances t j.

33 The Discrete Fourier Transform Definition The discrete Fourier transform (DFT) is given by n 1 n 1 ŷ k = e ikt j y j = ωn jk y j, k = 0,..., n 1, (3) j=0 j=0 where the numbers ω n = e 2πi n are so-called n-th roots of unity which lie equally spaced on the unit circle.

34 The Discrete Fourier Transform Definition The discrete Fourier transform (DFT) is given by n 1 n 1 ŷ k = e ikt j y j = ωn jk y j, k = 0,..., n 1, (3) j=0 j=0 where the numbers ω n = e 2πi n are so-called n-th roots of unity which lie equally spaced on the unit circle. Remark The MATLAB program fftmatrix.m) from [NCM] not only plots the roots of unity, but I creates a graph connecting the elements of each column of the Fourier matrix F of order n. If n is prime, we get a complete graph, otherwise a subgraph. The sparsity of the graph serves as an indicator for the speed of the FFT.

35 The Discrete Fourier Transform Definition The formula for the inverse discrete Fourier transform (inverse DFT) is given by y j = 1 n 1 n 1 e ikt j ŷ n k = ω jkŷ n k, j = 1,..., n (4) k=0 k=0 with ω n = e 2πi n, the complex conjugate of ω n (also n-th roots of unity).

36 The Discrete Fourier Transform Definition The formula for the inverse discrete Fourier transform (inverse DFT) is given by y j = 1 n 1 n 1 e ikt j ŷ n k = ω jkŷ n k, j = 1,..., n (4) k=0 k=0 with ω n = e 2πi n, the complex conjugate of ω n (also n-th roots of unity). Remark Since the data are periodic the Fourier domain will be discrete as well (since only waves e ikt with integer wavenumber k have period 2π).

37 The Discrete Fourier Transform Since n 1 ŷ k = ωn jk y j j=0 we can express the DFT ŷ in matrix-vector form ŷ = Fy, where the Fourier matrix F has entries F kj = ωn jk.

38 The Discrete Fourier Transform Since n 1 ŷ k = ωn jk y j j=0 we can express the DFT ŷ in matrix-vector form ŷ = Fy, where the Fourier matrix F has entries F kj = ω jk n. The definition of the inverse DFT implies y = 1 n F H ŷ with the Hermitian conjugate F H of F, or where I is the n n identity matrix. F H F = ni,

39 Outline The FFT in MATLAB 1 The (Continuous) Fourier Transform 2 The Discrete Fourier Transform 3 The FFT in MATLAB 4 Examples

40 The FFT in MATLAB Since the Fourier matrix has entries F kj = ω jk n MATLAB via we can generate F in omega = exp(-2*pi*i/n); j = 0:n-1; k = j ; F = omega.^(k*j)

41 The FFT in MATLAB Since the Fourier matrix has entries F kj = ω jk n MATLAB via we can generate F in omega = exp(-2*pi*i/n); j = 0:n-1; k = j ; F = omega.^(k*j) Alternatively, we can just use F = fft(eye(n))

42 Outline Examples 1 The (Continuous) Fourier Transform 2 The Discrete Fourier Transform 3 The FFT in MATLAB 4 Examples

43 Examples touchtone.m

44 Examples fftgui.m displays plots of the elements of the four vectors real(y), imag(y), real(fft(y)), imag(fft(y)).

45 Examples fftgui.m displays plots of the elements of the four vectors real(y), imag(y), real(fft(y)), imag(fft(y)). Example Setting only y 0 0 should give a constant (real) Fourier transform since n 1 ŷ k = e ikt j y j j=0

46 Examples fftgui.m displays plots of the elements of the four vectors Example real(y), imag(y), real(fft(y)), imag(fft(y)). Setting only y 0 0 should give a constant (real) Fourier transform since n 1 ŷ k = e ikt j onlyy 0 0 y j = e 0 y 0. j=0

47 Examples Example Setting only y 1 0 should give ŷ k = n 1 e ikt j y j j=0

48 Examples Example Setting only y 1 0 should give ŷ k = n 1 e ikt j y j j=0 onlyy 0 0 = e 2πik/n y 1

49 Examples Example Setting only y 1 0 should give ŷ k = n 1 e ikt j y j j=0 onlyy 0 0 = e 2πik/n y 1 Euler = [cos(2kπ/n) i sin(2kπ/n)] y 0.

50 Examples SpectralDiffDemo.m dftfilter.m

References I Appendix References W. L. Briggs and V. E. Henson. The DFT: An Owners Manual for the Discrete Fourier Transform. SIAM, Philadelphia, 1995. C. Moler.

51 References I Appendix References W. L. Briggs and V. E. Henson. The DFT: An Owners Manual for the Discrete Fourier Transform. SIAM, Philadelphia, C. Moler. Numerical Computing with MATLAB. SIAM, Philadelphia, Also M. Frigo and S. G. Johnson. FFTW: The fastest Fourier transform in the West (C library),

If s is a character that labels one of the buttons on the key pad, the corresponding row index k and column index j can be found with

If s is a character that labels one of the buttons on the key pad, the corresponding row index k and column index j can be found with Chapter 8 Fourier Analysis We all use Fourier analysis every day without even knowing it. Cell phones, disc drives, DVDs and JPEGs all involve fast finite Fourier transforms. This chapter discusses both