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Integral Transform

Integral Transform. Dongsup Kim Department of Biosystems, KAIST Fall, 2004. Problem in Transform space. Relatively easy solution. Solution in Transform space. Inverse transform. Integral transform. Solution of original problem. Original problem. Difficult solution.

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Integral Transform

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  1. Integral Transform Dongsup Kim Department of Biosystems, KAIST Fall, 2004

  2. Problem in Transform space Relatively easy solution Solution in Transform space Inverse transform Integral transform Solution of original problem Original problem Difficult solution Integral Transforms • Integral transform: • K(,t): Kernel • Mapping a function f(t) in t-space into another function g() in -space • Fourier, Wavelet, Z-transform, Laplace, Hilbert, Radon, etc • An original problem is difficult to solve in the original coordinate. • Often, the transform of the problem can be solved more easily. • Then, the inverse transform returns the solution from the transform coordinates to the original system.

  3. Fourier Transform, I • A physical process can be described either in the time domain by a function of time t, h(t), or in the frequency domain as a function of frequency f, H(f) • h(t) and H(f) are two different representations of the same process. • One goes back and forth between these two representations by means of the Fourier transform, • Dirac delta function: • Using angular frequency =2f,

  4. 1 d(t) cos(w0t) w t +w0 -w0 0 0 w t Examples, I • FT of Dirac delta function: • FT of cos(0t)

  5. exp(iw0t) Im t 0 Sum Re t 0 Examples, II • FT of exp(2if0t)=exp(i0t) F {exp(iw0t)} w w0 0 FT w 0

  6. Properties, II • Correspondence between symmetries in the two domains: • Scaling and shifting

  7. w t w t t w Scaling Shortpulse Medium-lengthpulse Longpulse

  8. Properties, III • With two functions h(t) and g(t), and their FT H(f) and G(f), the convolution, g*h, is defined by • Convolution theorem: the FT of the convolution is the product of the individual FTs. • The correlation, Corr(g,h) • Correlation theorem (for two real functions, g and h): • Autocorrelation, Wiener-Khinchin theorem: • Parseval’s theorem:

  9. Properties, IV • FT of derivatives; • in general • application: partial differential equation; wave equation

  10. Sampling theorem, I • Suppose function h(t) is sampled at evenly spaced intervals in time; • 1/: Sampling rate • For any sampling interval , there is a special frequency fc, called Nyquist critical frequency, given by • ex: critical sampling of a sine wave of Nyquist critical frequency is two sample points per cycle. • A function f is “bandwidth limited” if its Fourier transform is 0 outside of a finite interval [-L, L] • Sampling Theorem: If a continuous function h(t), sampled at an interval , is bandwidth limited to frequency smaller than fc, i.e., H(f)=0 for all |f|>fc, then the function h(t) is completely determined by its samples hn.

  11. Sampling theorem, II • For bandwidth limited signals, such as music in concert hall, sampling theorem tells us that the entire information content of the signal can be recorded by sampling rate -1 equal to twice the maximum frequency pass by the amplifier. • For the function that is not bandwidth limited to less then the Nyquist critical frequency, frequency component that lies outside of the frequency range, -fc < f < fc is spuriously moved into that range (aliasing). • Demo applet http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/nyquist/nyquist_limit_java_plugin.html

  12. Discrete Fourier transform, I • Suppose we have N consecutive sampled values where  is the sampling interval, and assume N is even. • With N numbers of input, we can produce no more than N independent number of outputs. Therefore, we seek estimates only at the discrete values; • Then, discrete Fourier Transform (DFT) is • ; DFT maps N complex numbers (the hk’s) into N complex numbers (the Hn’s). • ;trapezoidal approximation of the integral

  13. Discrete Fourier transform, II • It’s periodic in n, with period N; H-n = HN-n, n=1,2,… • With this conversion, one lets the n in Hn vary from 0 to N-1. Then n and k vary exactly over the same range. • With this convention, • zero frequency  n=0 • positive frequencies, 0 < f < fc  1  n  N/2-1 • negative frequencies, –fc < f < 0  N/2+1  n  N-1 • the value n = N/2  both f = fc and f = -fc • The DFT has symmetry properties almost exactly the same as the continuous Fourier transform.

  14. Discrete Fourier transform, III • The discrete inverse Fourier transform is • Proof: • Parseval’s theorem: • Proof?

  15. Fast Fourier Transform (FFT), I • DFT appears to be an O(N2) process. • Danielson and Lanczos; DFT of length N can be rewritten as the sum of two DFT of length N/2. • We can do the same reduction of Hk0 to the transform of its N/4 even-numbered input data and N/4 odd-numbered data. • For N = 2R, we can continue applying the reduction until we subdivide the data into the transforms of length 1. • For every pattern of log2N number of 0’s and 1’s, there is one-point transformation that is just one of the input number hn

  16. Fast Fourier Transform (FFT), II • For N=8 • Since WN/2 = -1, Hk0 and Hk1 have period N/2, • Diagrammatically (butterfly), • There are N/2 butterflies for this stage of the FFT, and each butterfly requires one multiplication

  17. Fast Fourier Transform (FFT), III • So far, • The splitting of {Hk} into two half-size DFTs can be repeated on Hk0 and Hk1 themselves,

  18. Fast Fourier Transform (FFT), III • So far, • {Hk00} is the N/4-point DFT of {h0, h4,…, hN-4}, • {Hk01} is the N/4-point DFT of {h2, h6,…, hN-2}, • {Hk10} is the N/4-point DFT of {h1, h5,…, hN-3}, • {Hk11} is the N/4-point DFT of {h3, h7,…, hN-1}, • Note that there is a reversal of the last two digits in the binary expansions of the indices j in {hj}.

  19. Fast Fourier Transform (FFT), IV

  20. Fast Fourier Transform (FFT), V • If we continue with this process of halving the order of the DFTS, then after R=log2N stages, we reach where we are performing N one-point DFTs. • One-point DFT of the number hj is just the identity hj hj • Since the reversal of the order of the bits will continue, all bits in the binary expansion of j will be arranged in reverse order. • Therefore, to begin the FFT, one must first rearrange {hj} so it is listed in bit reverse order. • For each of the log2N stages, there are N/2 multiplications, hence there are (N/2)log2N multiplications needed for FFT. • Much less time than the (N-1)2 multiplications needed for a direct DFT calculation. • When N=1024, FFT=5120 multiplication, DFT=1,046,529  savings by a factor of almost 200.

  21. Fast Fourier Transform (FFT), VI

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