(Re)Introduction to Integral Transforms: Fourier and Laplace

1. (Re)Introduction to Integral Transforms:Fourier and Laplace COMP417 Instructor: Philippe Giguère

2. Outline • Transforms in context of problem solving • Convolution • Dirac delta function d(t) • Fourier Transform • Sampling • Laplace Transform

3. Why use Transforms? • Transforms are not simply math curiosity sketched at the corner of a woodstove by ol’ Frenchmen. • Way to reframe a problem in a way that makes it easier to understand, analyze and solve.

4. General Scheme using Transforms Problem Solution of the equation Result Equation of the problem Transformation Inverse transformation = EASY Transformed equation Solution of the transformed equation = HARD

5. Which Transform to Use?

6. Typical Problem • Given an input signal x(t), what is the output signal y(t) after going through the system? • To solve it in the time domain(t) is cumbersome!

7. Integrating Differential Equation? • Let’s have a simple first order low-pass filter with resistor R and capacitor C: • The system is described by diff. eq.: • To find a solution, we can integrate. Ugh!

8. Convolution • Math operator (symbol *) that takes two input functions (x(t) and h(t)) and produces a third (y(t)) • Expresses the amount of overlap of one function x(t) as it is shifted over another function h(t). • Way of « blending » one function with another.

9. “Frame-by-frame” convolution • Can be visualized as flipping one function (x), sliding it, and doing the dot product. movie

10. More convolution examples Smoothing

11. (Moving Average of GOOG)

12. Convolution is heavily used in image processing (Gaussian Blur in 2D) = *

13. Dirac Delta function d(t)

14. Convolution with Dirac delta d(t) • Convolving a signal with Dirac delta d(t) simply yields the same signal. movie

15. Delay operator as convolutionwith d(t-t) • Convolving with a shifted delta d(t-t) shifts the original signal: delay. movie

16. Convolution Properties • Convolution is a linear operation and therefore has the typical linear properties: • Commutativity • Associativity • Distributivity • Scalar multiplication

17. Using Convolution to Solve • Again same first order low-pass filter: • The system is described by its impulse response: • Solution is convolution impulse resp. with x(t)

18. Use a Convolution to Solve • Convolution is expensive to compute. • Little intuition about output signal y(t).

19. Fourier Transform • Jean-Baptiste Fourier had crazy idea (1807): • Any periodic function can be rewritten as a weighted sum of sines and cosines of different frequencies. • Called Fourier Series

20. Square-Wave Deconstruction

21. Other examples

22. FT expands this idea • Take any signal (periodic and non-periodic) in time domain and decompose it in sines + cosines to have a representation in the frequency domain.

23. FT: Formal Definition • Convention: Upper-case to describe transformed variables: • Transform: F{ x(t) } = X(w) or X(f) (w=2pf) • Inverse: F-1{Y(w) orY(f)}= y(t)

24. FT gives complex numbers • You get complex numbers • Cosine coefficients are real • Sine coefficients are imaginary

25. Complex plane • Complex number can be represented: • Combination of real + imaginary value: x +iy • Amplitude + Phase A and j

26. Alternative representation of FT • Complex numbers can be represented also as amplitude + phase.

27. Example Fourier Transform Fast moving vs slow moving signals

28. Example Fourier Transform Time Domain t Frequency Domain w Real Real Real

29. Example Fourier Transform

30. Example Fourier Transform

31. Example Fourier Transform

32. Example Fourier Transform Note: FT is imaginary for sine

33. Example Fourier Transform Time Domain t Frequency Domain w Real Real « DC component »

34. FT of Delay d(t-t) • Amplitude + phase is easier to understand: (click movie) • Amplitude: • Gives you information about frequencies/tones in a signal. • Phase: • More about when it happens in time.

35. Important FT Properties • Addition • Scalar Multiplication • Convolution in time t • Convolution in frequency w

36. FT timefrequency duality

37. FT: Reframing the problem in Frequency Domain * Problem Solution of the equation Result x(t),h(t) Fourier Transform Inverse Fourier Transform = EASY X(w), H(w) X(w)H(w) x = HARD Completely sidesteps the convolution!

38. FT: Another Example What is the amplitude spectrum |Y(f)| of a voice signal (bandlimited to 5 kHz) when multiplied by a cosine f=15 kHz? (Note: this is Amplitude Modulation  AM radio)

39. FT: Solution (Look Ma! No Algebra!)

40. FT Gaussian Blur = * Space Frequency

41. Sampling Theorem • In order to be used within a digital system, a continuous signal must be converted into a stream of values. • Done by sampling the continuous signal at regular intervals. • But at which interval?

42. Sampling Theorem • Sampling can be thought of multiplying a signal by a d pulse train:

43. Aliasing • If sampling rate is too small compared with frequency of signal, aliasing WILL occur:

44. Fourier Analysis of Sampling • The FT of a pulse train with frequency fs is another pulse train with interval 1/fs:

45. Fourier Analysis of Sampling • Aliasing will happen if fs <2 fmax • Nyquist frequency = fs/2

46. A few sampling frequencies • Telephone systems: 8 kHz • CD music: 44.1 kHz • DVD-audio: 96 or 192 kHz • Aqua robot: 1 kHz • Digital Thermostat (HMTD84) : 0.2 Hz

47. Laplace Transform • Formal definition: • Compare this to FT: • Small differences: • Integral from 0 to ¥to for Laplace • f(t) for t<0 is not taken into account • -s instead of -iw

48. Common Laplace Transfom Name f(t) F(s) Impulse d 1 Step Ramp Exponential Sine Damped Sine

49. Laplace Transform Properties • Similar to Fourier transform: • Addition/Scaling • Convolution • Derivation

50. Transfer Function H(s) • Definition • H(s) = Y(s) / X(s) • Relates the output of a linear system (or component) to its input. • Describes how a linear system responds to an impulse. • All linear operations allowed • Scaling, addition, multiplication. H(s) X(s) Y(s)