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Fourier

Fourier. Fourier Analysis. Fourier Analysis. When we analyse a function using Fourier methods, the function is decomposed into its frequency components. This analysis is used in signal processing, filtering. Fourier Analysis.

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Fourier

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  1. Fourier

  2. Fourier Analysis

  3. Fourier Analysis • When we analyse a function using Fourier methods, the function is decomposed into its frequency components. • This analysis is used in signal processing, filtering.

  4. Fourier Analysis • Consider the two waveforms above, where the first waveform is made up of only one pure wave. • The second waveform (which could be from a steel pan or speech) is made up of one fundamental (pure) wave, with many other waves of higher frequency.

  5. Intensity Intensity frequency frequency Fourier Analysis • If we consider the Fourier of the waveforms we find the following:

  6. Fourier Analysis • Where is called the fundamental frequency. • are the harmonics.

  7. Fourier Series

  8. Fourier Series • The Fourier series is used to represent other functions. • This is achieved by a series of sines and cosines within the given interval, which can be used as an approximation to the function.

  9. Fourier Series • The general formula for the fourier series: • Where the coefficients are given by,

  10. Fourier Series • Writing a Fourier series means finding the the coefficients an, bn. • Consider the example. • Write the function f(x)=x as a Fourier series on the interval –Π,Π.

  11. Fourier Series • The Fourier series is used for approximating or analysing periodic functions.

  12. Fourier Transform

  13. Fourier Transform • While a Fourier is useful for periodic functions, the Fourier transform or integral is used for non-periodic functions.

  14. Fourier Transform • While a Fourier is useful for periodic functions, the Fourier transform or integral is used for non-periodic functions.

  15. Fourier Transform • The Fourier transform of a function is, • For compactness we use the complex exponent function. • The Fourier transform transforms from the t domain to the f domain.

  16. Fourier Transform • By convention time t is used as the functions variable and frequency as the transform variable. • However these can be reversed or variable such as position and momentum used. eg. position to frequency.

  17. Fourier Transform • A plot of the square of the modulus of the Fourier transform ( vs ) is called the power spectrum. • It gives the amount the frequency contributes to the waveform.

  18. Fourier Transform • The inverse Fourier is,

  19. Fourier Transform • Example: Fourier transform of .

  20. Fourier Transform • Example: Fourier transform of .

  21. Fourier Transform • Example: Fourier transform of .

  22. Fourier Transform • Example: Fourier transform of .

  23. Fourier Transform • However the delta function is,

  24. Discrete Fourier Transform

  25. Discrete Fourier Transform • If or is known analytically, the integral can be evaluated numerically or numerically using one of the previous techniques. If a table of values is know interpolation can be used to evaluate the function.

  26. Discrete Fourier Transform • However we consider the case for directly Fourier transforming functions that are only known at sampled points. • By sampling a function at N times, we determine N values for the Fourier transform of the function ( N independent values).

  27. Discrete Fourier Transform • If the samples are truly independent, the DFT produces a function with is periodic between the sampling period.

  28. Discrete Fourier Transform • If the samples are truly independent, the DFT produces a function with is periodic between the sampling period. • If the function we are analysing is actually periodic, the first N points should all be in one period to guarantee independence.

  29. Discrete Fourier Transform • The time interval T is the largest time over which we are sampling our function. • NB: for a periodic function it is the period of the function.

  30. Discrete Fourier Transform • The time interval T is the largest time over which we are sampling our function. • NB: for a periodic function it is the period of the function. • T therefore determines the lowest frequency.

  31. Discrete Fourier Transform • Assume that the function we wish to transform is measured or sampled at a discrete number of points N+1 times (N time intervals).

  32. Discrete Fourier Transform • Assume that the function we wish to transform is measured or sampled at a discrete number of points N+1 times (N time intervals). • Let be the time interval between samples.

  33. Discrete Fourier Transform

  34. Discrete Fourier Transform • Because we are sampling at discrete times we have a discrete set of frequencies.

  35. Discrete Fourier Transform • NB: = dt.

  36. Discrete Fourier Transform • NB: = dt. • Assuming that the samples are evenly spaced, .

  37. Discrete Fourier Transform • NB: = dt. • Assuming that the samples are evenly spaced, . • The inverse of the time interval gives the sampling frequency.

  38. Discrete Fourier Transform • The maximum sampling frequency is called the Nyquist frequency (when n = N/2).

  39. Discrete Fourier Transform • The maximum sampling frequency is called the Nyquist frequency (when n = N/2). • The Discrete Fourier Transform has the form,

  40. Discrete Fourier Transform • The Discrete Inverse Fourier Transform has the form,

  41. Discrete Fourier Transform • Example: Suppose we sample N=8 values. Then n = -4,-3,-2,-1,0,1,2,3,4 • We have 9 point, 1 more than we are allowed. Thus there is some redundant data.

  42. Discrete Fourier Transform • This is because the endpoints (-4 and 4) give the same information, the Nyquist frequency.

  43. Discrete Fourier Transform No new info

  44. Discrete Fourier Transform • The DFT can be applied to any complex values series. The computation time is proportional to the square of the number of points.

  45. Fast Fourier Transform

  46. Fast Fourier Transform • A much faster algorithm was developed by Cooley and Tukey called the Fast Fourier Transform (FFT).

  47. Fast Fourier Transform • A much faster algorithm was developed by Cooley and Tukey called the Fast Fourier Transform (FFT). • The most popular implementation is the radix-2 FFT. It’s computational time is proportional to .

  48. Fast Fourier Transform • The FFT algorithm is based on a divide and conquer approach. The initial problem is divided into smaller and smaller problem. These computations are recombined to give the final result.

  49. Fast Fourier Transform • The FFT algorithm is based on a divide and conquer approach. The initial problem is divided into smaller and smaller problem. These computations are recombined to give the final result. • The simplest implementation continually halves the dimensions of the DFT until it becomes unity.

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