Review Of One Dimensional Fourier Analysis


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Review Of One Dimensional Fourier Analysis

  1. 1. ECE 665 Fourier Optics Spring, 2004 Thomas B. Fowler, Sc.D. Senior Principal Engineer Mitretek Systems
  2. 2. Course goal <ul><li>To provide an understanding of optical systems for processing temporal signals as well as images </li></ul><ul><li>Course is based on use of Fourier analysis in two dimensions to understand the behavior of optical systems </li></ul>
  3. 3. Nature of light and theories about it <ul><li>Fourier optics falls under wave optics </li></ul><ul><li>Provides a description of propagation of light waves based on two principles </li></ul><ul><ul><li>Harmonic (Fourier) analysis </li></ul></ul><ul><ul><li>Linearity of systems </li></ul></ul>Quantum optics Electromagnetic optics Wave optics Ray optics
  4. 4. Course organization <ul><li>13 weeks </li></ul><ul><li>Main text: Introduction to Fourier Optics , Joseph Goodman, McGraw-Hill, 1996 </li></ul><ul><li>Other material to be downloaded from Internet </li></ul><ul><li>Student evaluation </li></ul><ul><ul><li>Homework 20% </li></ul></ul><ul><ul><li>Midterm exam 40% </li></ul></ul><ul><ul><li>Final exam 40% </li></ul></ul>
  5. 5. Topics <ul><li>Week 1: Review of one-dimensional Fourier analysis </li></ul><ul><li>Week 2: Two-dimensional Fourier analysis </li></ul><ul><li>Weeks 3-4: Scalar diffraction theory </li></ul><ul><li>Weeks 5-6: Fresnel and Fraunhofer diffraction </li></ul><ul><li>Week 7: Transfer functions and wave-optics analysis of coherent optical systems </li></ul><ul><li>Weeks 8-9: Frequency analysis of optical imaging systems </li></ul><ul><li>Week 10: Wavefront modulation </li></ul><ul><li>Week 11: Analog optical information processing </li></ul><ul><li>Weeks 12-13: Holography </li></ul>
  6. 6. Week 1: Review of One-Dimensional Fourier Analysis <ul><li>Descriptions: time domain and frequency domain </li></ul><ul><li>Principle of Fourier analysis </li></ul><ul><ul><li>Periodic: series </li></ul></ul><ul><ul><ul><li>Sin, cosine, exponential forms </li></ul></ul></ul><ul><ul><li>Non-periodic: Fourier integral </li></ul></ul><ul><ul><li>Random </li></ul></ul><ul><li>Convolution </li></ul><ul><li>Discrete Fourier transform and Fast Fourier Transform </li></ul><ul><li>A deeper look: Fourier transforms and functional analysis </li></ul>
  7. 7. Basic idea: what you learned in undergraduate courses <ul><li>A periodic function f ( t ) can be expressed as a sum of sines and cosines </li></ul><ul><ul><li>Sum may be finite or infinite, depending on f ( t ) </li></ul></ul><ul><ul><li>Object is usually to determine </li></ul></ul><ul><ul><ul><li>Frequencies of sine, cosine functions </li></ul></ul></ul><ul><ul><ul><li>Amplitudes of sine, cosine functions </li></ul></ul></ul><ul><ul><ul><li>Error in approximating with finite number of functions </li></ul></ul></ul><ul><ul><li>Function f ( t ) must satisfy Dirichlet conditions </li></ul></ul><ul><li>Result is that periodic function in time domain, e.g., square wave, can be completely characterized by information in frequency domain, i.e., by frequencies and amplitudes of sine, cosine functions </li></ul>
  8. 8. Historical reason for use of Fourier series to approximate functions <ul><li>Breaks periodic function f(t) into component frequencies </li></ul><ul><li>Response of linear systems to most periodic waves can be analyzed by finding the response to each ‘harmonic’ and superimposing the results) </li></ul>
  9. 9. Basic idea: what you learned in undergraduate courses (continued) <ul><li>Periodic means that f ( t ) = f( t + T ) for all t </li></ul><ul><ul><li>T is the period </li></ul></ul><ul><ul><li>Period related to frequency by T = 1/ f 0 = 2  /  0 </li></ul></ul><ul><ul><li> 0 is called the fundamental frequency </li></ul></ul><ul><li>So we have </li></ul><ul><li>n  0 = 2n  /T is nth harmonic of fundamental frequency </li></ul>
  10. 10. How to calculate Fourier coefficients <ul><li>Calculation of Fourier coefficients hinges on orthogonality of sine, cosine functions </li></ul><ul><li>Also, </li></ul>
  11. 11. How to calculate Fourier coefficients (continued) <ul><li>And we also need </li></ul>
  12. 12. How to calculate Fourier coefficients (continued) <ul><li>Step 1. integrate both sides: </li></ul><ul><li>Therefore </li></ul>
  13. 13. How to calculate Fourier coefficients (continued) <ul><li>Step 2. For each n , multiply original equation by cos n  0 t and integrate from 0 to T: </li></ul>Therefore 0 0 0
  14. 14. How to calculate Fourier coefficients (continued) <ul><li>Step 3. Calculate b n terms similarly, by multiplying original equation by sin n  0 t and integrating from 0 to T </li></ul><ul><ul><li>Get similar result </li></ul></ul><ul><li>Some rules simplify calculations </li></ul><ul><ul><li>For even functions f ( t ) = f (- t ), such as cos t, b n terms = 0 </li></ul></ul><ul><ul><li>For odd functions f ( t ) = - f (- t ), such as sin t, a n terms = 0 </li></ul></ul>
  15. 15. Calculation of Fourier coefficients: examples <ul><li>Square wave (in class) </li></ul>1 -1 T T/2
  16. 16. Calculation of Fourier coefficients: examples (continued) <ul><li>Result </li></ul>Source: Gibbs phenomenon: ringing near discontinuity
  17. 17. Calculation of Fourier coefficients: examples (continued) <ul><li>Triangular wave (in class) </li></ul>T T/2 +V -V
  18. 18. Calculation of Fourier coefficients: examples (continued) <ul><li>Triangle wave result </li></ul><ul><ul><li>Note that value of terms falls off as inverse square </li></ul></ul>
  19. 19. Other simplifying assumptions: half-wave symmetry <ul><li>Function has half-wave symmetry if second half is negative of first half: </li></ul>
  20. 20. Other simplifying assumptions: half-wave symmetry <ul><li>Can be shown </li></ul>
  21. 21. Conditions for convergence <ul><li>Conditions for convergence of Fourier series to original function f ( t ) discovered (and named for) Dirichelet </li></ul><ul><ul><li>Finite number of discontinuities </li></ul></ul><ul><ul><li>Finite number of extrema </li></ul></ul><ul><ul><li>Be absolutely convergent: </li></ul></ul><ul><li>Example of periodic function excluded </li></ul>
  22. 22. Parseval's theorem <ul><li>If some function f ( t ) is represented by its Fourier expansion on an interval [- l , l ], then </li></ul><ul><li>Useful in calculating power associated with waveform </li></ul>
  23. 23. Effect of truncating infinite series <ul><li>Truncation error function  n ( t ) given by </li></ul><ul><ul><li>This is difference between original function and truncated series s n ( t ), truncated after n terms </li></ul></ul><ul><li>Error criterion usually taken as mean square error of this function over one period </li></ul><ul><li>Least squares property of Fourier series states that no other series with same number n of terms will have smaller value of E n </li></ul>
  24. 24. Effect of truncating infinite series (continued) <ul><li>Problem is that there is no effective way to determine value of n to satisfy any desired E </li></ul><ul><li>Only practical approach is to keep adding terms until E n < E </li></ul><ul><li>One helpful bit of information concerns fall-off rate of terms </li></ul><ul><ul><li>Let k = number of derivatives of f ( t ) required to produce a discontinuity </li></ul></ul><ul><ul><li>Then </li></ul></ul><ul><ul><ul><li>where M depends on f ( t ) but not n </li></ul></ul></ul>
  25. 25. Some DERIVE scripts <ul><li>To generate square wave of amplitude A, period T: </li></ul><ul><ul><li>squarewave(A,T,x) := A*sign(sin(2*pi*x/T)) </li></ul></ul><ul><li>For Fourier series of function f with n terms, limits c, d: </li></ul><ul><ul><li>Fourier(f,x,c,d,n) </li></ul></ul><ul><ul><li>Example: Fourier(squarewave(2,2,x),x,0,2,5) generates first 5 terms (actually 3 because 2 are zero) </li></ul></ul><ul><li>To generate triangle wave of amplitude A, period T: </li></ul><ul><ul><li>int(squarewave(A,T,x),x) </li></ul></ul><ul><ul><li>Then Fourier transform can be done of this </li></ul></ul>
  26. 26. Exponential form of Fourier Series <ul><li>Previous form </li></ul><ul><li>Recall that </li></ul>
  27. 27. Exponential form of Fourier Series (continued) <ul><li>Substituting yields </li></ul><ul><li>Collecting like exponential terms and using fact that 1/j = -j: </li></ul>
  28. 28. Exponential form of Fourier Series (continued) <ul><li>Introducing new coefficients </li></ul><ul><li>We can rewrite Fourier series as </li></ul><ul><li>Or more compactly by changing the index </li></ul>
  29. 29. Exponential form of Fourier Series (continued) <ul><li>The coefficients can easily be evaluated </li></ul>
  30. 30. Exponential form of Fourier Series (continued) <ul><li>Sometimes coefficients written in real and complex terms as </li></ul><ul><ul><li>where </li></ul></ul>
  31. 31. Exponential form of Fourier Series: example <ul><li>Take sawtooth function, f(t) = (A/T)t per period </li></ul><ul><li>Then </li></ul><ul><li>Hint: if using Derive, define  = 2  /T, set domain of n as integer </li></ul>
  32. 32. Fourier analysis for nonperiodic functions <ul><li>Basic idea: extend previous method by letting T become infinite </li></ul><ul><li>Example: recurring pulse </li></ul>t v 0 a/2 -a/2 T
  33. 33. Fourier analysis for nonperiodic functions (continued) <ul><li>Start with previous formula: </li></ul><ul><li>This can be readily evaluated as </li></ul>
  34. 34. Fourier analysis for nonperiodic functions (continued) <ul><li>Using fact that T = 2  /  0 , may be written </li></ul><ul><li>We are interested in what happens as period T gets larger, with pulse width a fixed </li></ul><ul><ul><li>For graphs, a = 1, V 0 = 1 </li></ul></ul>
  35. 35. Effect of increasing period T a/T a/T a/T
  36. 36. Transition to Fourier integral <ul><li>We can define f ( jn  0 ) in the following manner </li></ul><ul><ul><li>Since difference in frequency of terms  =  0 in the expansion. Hence </li></ul></ul>
  37. 37. Transition to Fourier integral (continued) <ul><li>Since </li></ul><ul><li>It follows that </li></ul><ul><li>As we pass to the limit,  -> d  , n  ->  so we have </li></ul>
  38. 38. Transition to Fourier integral (continued) <ul><li>This is subject to convergence condition </li></ul><ul><li>Now observe that since </li></ul><ul><li>We have </li></ul>
  39. 39. Transition to Fourier integral (continued) <ul><li>In the limit as T ->  </li></ul><ul><li>Since f (t) = 0 for t < -a /2 and t > a/2 </li></ul><ul><li>Thus we have the Fourier transform pair for nonperiodic functions </li></ul>
  40. 40. Example: pulse <ul><li>For pulse of area 1, height a , width 1/a, we have </li></ul><ul><li>Note that this will have zeros at  = 2an  n=0, + 1, + 2 </li></ul><ul><li>Considering only positive frequencies, and that “most” of the energy is in the first lobe, out to 2a  , we see that product of bandwidth 2a  and pulse width 1/a = 2  </li></ul>
  41. 41. Example of pulse width=1 width=0.2 1/2 -1/2 1/10 -1/10 1 5
  42. 42. Pulse: limiting cases <ul><li>Let a ->  , then f (t) -> spike of infinite height and width 1/a (delta function) -> 0 </li></ul><ul><ul><li>Transform -> line F(j  )=1 </li></ul></ul><ul><ul><li>Thus transform of delta function contains all frequencies </li></ul></ul><ul><li>Let a -> 0, then f(t) -> infinitely long pulse </li></ul><ul><ul><li>Transform -> spike of height 1, width 0 </li></ul></ul><ul><li>Now let height remain at 1, width be 1/a </li></ul><ul><ul><li>Then transform is </li></ul></ul>
  43. 43. Pulse: limiting cases (continued) <ul><li>Now, we are interested in limit as a -> 0 for  -> 0 and  > 0 </li></ul><ul><ul><li>First, consider case of small  : </li></ul></ul><ul><ul><li>So when a -> 0, 1/a ->  </li></ul></ul><ul><ul><li>As w moves slightly away from 0, it drops to zero quickly because of w/2a term in denominator (numerator < 1 at all times) </li></ul></ul><ul><li>So we get delta function,  (0) </li></ul>
  44. 44. Fourier transform of pulse width 0.1
  45. 45. Properties of delta function <ul><li>Definition </li></ul><ul><li>Area for any  > 0 </li></ul><ul><li>Sifting property </li></ul><ul><ul><li>since </li></ul></ul>
  46. 46. Some common Fourier transform pairs Source:
  47. 47. Some Fourier transform pairs (graphical illustration) function function transform transform Source: Physical Optics Notebook: Tutorials in Fourier Optics , Reynolds, et. al., SPIE/AIP
  48. 48. Fourier transform: Gaussian pulses
  49. 49. Properties of Fourier transforms <ul><li>Simplification: </li></ul><ul><li>Negative t : </li></ul><ul><li>Scaling </li></ul><ul><ul><li>Time: </li></ul></ul><ul><ul><li>Magnitude: </li></ul></ul>
  50. 50. Properties of Fourier transforms (continued) <ul><li>Shifting: </li></ul><ul><li>Time convolution: </li></ul><ul><li>Frequency convolution: </li></ul>
  51. 51. Convolution and transforms <ul><li>A principal application of any transform theory comes from its application to linear systems </li></ul><ul><ul><li>If system is linear, then its response to a sum of inputs is equal to the sum of its responses to the individual inputs </li></ul></ul><ul><ul><li>This was original justification for Fourier's work </li></ul></ul><ul><li>Because a delta function contains all frequencies in its spectrum, if you “hit” something with a delta function, and measure its response, you know how it will respond to any individual frequency </li></ul><ul><ul><li>The response of something (e.g., a circuit) to a delta function is called its “impulse response” </li></ul></ul><ul><ul><ul><li>Called “point spread function” in optics </li></ul></ul></ul><ul><ul><li>Often denoted h(t) </li></ul></ul>
  52. 52. Convolution and transforms (continued) <ul><li>The Fourier transform of the impulse response can be calculated, usually designated H(j  ) </li></ul><ul><li>Therefore if one knows the frequency content of an incoming “signal” u (t), one can calculate the response of the system </li></ul><ul><ul><li>The response to each individual frequency component of incoming signal can be calculated individually as product of impulse response and that component </li></ul></ul><ul><ul><li>Total response is obtained by summing all of individual responses </li></ul></ul><ul><li>That is, response Y(j  ) = H(j  )U(j  ) </li></ul><ul><ul><li>Where U(j  ) is sum of Fourier transforms of individual components of u (t) </li></ul></ul>
  53. 53. Convolution and transforms (continued) <ul><li>May be visualized as </li></ul>H (j  ) U (j  ) Y (j  )= H (j  ) U (j  ) System Input Response
  54. 54. Convolution and transforms (continued) <ul><li>Example </li></ul><ul><ul><li>Signal is square wave, u(t)=sgn(sin(x)) </li></ul></ul><ul><ul><li>This has Fourier transform </li></ul></ul><ul><ul><li>So response Y (j  ) is </li></ul></ul>
  55. 55. Convolution and transforms (continued) <ul><li>If incoming signal described by Fourier integral instead, same result holds </li></ul><ul><li>To get time (or space) domain answer, we need to take inverse Fourier transform of Y(j  ) </li></ul>
  56. 56. Convolution and transforms (continued) <ul><li>Can also be calculated in time (or space), i.e., non-transformed domain </li></ul><ul><li>Derivation </li></ul><ul><li>Now, we introduce new variables v and  , related to t and z by </li></ul>
  57. 57. Convolution and transforms (continued) <ul><li>Computing Jacobean to transform variables </li></ul><ul><ul><li>Implies that differential areas same for both systems of variables </li></ul></ul><ul><li>Thus since t = v-z = v -  we have </li></ul><ul><li>Where we have calculated the limits as follows </li></ul>
  58. 58. Convolution and transforms (continued) <ul><li>We may assume without loss of generality that u(z) = 0 for z<0 </li></ul><ul><ul><li>Otherwise we can shift variables to make it so </li></ul></ul><ul><ul><ul><li>Must assume that u(z) has some starting point </li></ul></ul></ul><ul><ul><li>Therefore the lower limit of integration in the inner integral is 0 </li></ul></ul><ul><li>We may also assume without loss of generality that h(t) = 0 for t<0 </li></ul><ul><ul><li>Therefore h(v-  ) = 0 for  > v </li></ul></ul>
  59. 59. Convolution and transforms (continued) <ul><li>Since the outer integral defines a Fourier transform, its inverse is just y (t), so we have </li></ul><ul><li>This is usually written with t as the inner variable, </li></ul><ul><li>This is called the convolution of h and u , usually written y(t) = h * u </li></ul><ul><li>Can readily be calculated on a computer </li></ul>
  60. 60. Convolution: old way (graphically)
  61. 61. Convolution: old way (continued) Source: P. S. Rha, SFSU, ENGR449_PDFs/EE449_L5_Conv.PDF
  62. 62. Convolution and transforms (new way) <ul><li>Use computer algebra programs </li></ul><ul><li>Some Derive scripts </li></ul><ul><ul><li>Step function: u(t):=if(t<0,0,1) </li></ul></ul><ul><ul><li>Pulse of width d, amplitude a: f1(t):=if(t>=0 and t<=d,a,0) </li></ul></ul><ul><ul><li>Triangle of width d, amplitude a: triangle(t):=if(t>=0 and t<=d/2,2at/d,(if(t>d/2 and t<d,2a-2at/d,0)0) </li></ul></ul><ul><ul><li>Convolution: convolution(t):=int(f1(t-  )*f2(  ),  ,0,t) </li></ul></ul><ul><li>Example </li></ul><ul><ul><li>f1 is pulse of width 1, amplitude 1 </li></ul></ul><ul><ul><li>f2 is pulse of width 2, amplitude 3 </li></ul></ul>
  63. 63. Convolution functions
  64. 64. Convolution: useful web sites <ul><li> </li></ul><ul><li> </li></ul><ul><li> </li></ul><ul><li> </li></ul>
  65. 65. Fourier and Laplace transforms <ul><li>Fourier transform does not preserve initial condition information </li></ul><ul><ul><li>Therefore most useful when “steady state” conditions exist </li></ul></ul><ul><ul><ul><li>This is typically the case for optical systems </li></ul></ul></ul><ul><ul><ul><li>But often not true for electrical networks </li></ul></ul></ul><ul><li>Comparison of definitions </li></ul>Laplace Fourier
  66. 66. Fourier and Laplace transforms (continued) <ul><li>Differences </li></ul><ul><ul><li>In Fourier transform, j  replaces s </li></ul></ul><ul><ul><li>Limits of integration are different, one-sided vs. two-sided </li></ul></ul><ul><ul><li>Contours of integration in inverse transform different </li></ul></ul><ul><ul><ul><li>Fourier along imaginary axis </li></ul></ul></ul><ul><ul><ul><li>Laplace along imaginary axis displaced by  1 </li></ul></ul></ul><ul><li>Conversion between Fourier and Laplace transforms </li></ul><ul><ul><li>Laplace transform of f ( t ) = Fourier transform of f ( t )e -  t </li></ul></ul><ul><ul><li>Symbolically, </li></ul></ul>
  67. 67. Fourier transforms of random sources (noise) <ul><li>Noise has frequency characteristics </li></ul><ul><ul><li>Generally continuous distribution of frequencies </li></ul></ul><ul><ul><li>Since transform of individual frequencies gives spikes, this allows us to separate signal from noise via Fourier methods </li></ul></ul><ul><li>Common types of noise </li></ul><ul><ul><li>White noise: equal power per Hz (power doubles per octave) </li></ul></ul><ul><ul><li>Pink noise: equal power per octave </li></ul></ul><ul><ul><li>Other “colors” of noise described at </li></ul></ul><ul><ul><li>Fourier transform distinguishes these </li></ul></ul>
  68. 68. Fourier transforms of random sources (noise) (continued) <ul><li>Frequency domain thus allows us to obtain information about signal purity that is difficult to obtain in time (or space) domain </li></ul><ul><ul><li>Noise </li></ul></ul><ul><ul><li>Distortion </li></ul></ul>
  69. 69. Fourier transforms of random sources (noise) (continued) Source:
  70. 70. Discrete and Fast Fourier Transforms <ul><li>Most Fourier work today carried out by computer (numerical) analysis </li></ul><ul><li>Discrete Fourier transform (DFT) is first step in numerical analysis </li></ul><ul><ul><li>Simply sample target function f ( t ) at appropriate times </li></ul></ul><ul><ul><li>Replace integral by summation </li></ul></ul><ul><ul><ul><li>Here t n = nT , where T =sampling interval, N = number of samples, and frequency sampling interval  = 2  /NT,  k = k  </li></ul></ul></ul>
  71. 71. Discrete and Fast Fourier Transforms (continued) <ul><li>Sampling frequency f s = 1/ T </li></ul><ul><li>Frequency resolution  f = 1/ NT = f s / N </li></ul><ul><li>For accurate results, sampling theorem tells us that sample frequency f s > 2 x f max , the highest frequency in the signal </li></ul><ul><ul><li>Implies that highest frequency captured f max < 1/2 T = f s /2 </li></ul></ul><ul><ul><ul><li>Otherwise aliasing will occur </li></ul></ul></ul><ul><li>To improve resolution, note that you can't double sampling frequency, as that also doubles N (for same piece of waveform) </li></ul><ul><ul><li>The only way to increase N without affecting f s is to increase acquisition time </li></ul></ul>
  72. 72. Discrete and Fast Fourier Transforms (continued) <ul><li>Note that DFT calculation requires N separate summations, one for each  k </li></ul><ul><li>Since each summation requires N terms, number of calculations goes up as N 2 </li></ul><ul><ul><li>Therefore doubling frequency resolution requires quadrupling number of calculations </li></ul></ul><ul><li>Method also assumes function f ( t ) is periodic outside time range (nT) considered </li></ul><ul><li>Also note that raw DFT calculation gives array of complex numbers which must be processed to give usual magnitude and phase information </li></ul><ul><ul><li>When only power information required, squaring eliminates complex terms </li></ul></ul>
  73. 73. Inverse discrete Fourier transform <ul><li>Calculated in straightforward manner as </li></ul><ul><li>This gives, of course, the original sampled values of the function back </li></ul><ul><ul><li>Other values can be determined by appropriate filtering </li></ul></ul>
  74. 74. Uses of DFT <ul><li>DFT usage may be visualized as </li></ul>DFT Spectrum Magnitude Phase Power Spectrum Power Spectral Density
  75. 75. Power measurements and DFT <ul><li>Power spectrum </li></ul><ul><ul><li>Gives energy (power) content of signal at a particular frequency </li></ul></ul><ul><ul><li>No phase information </li></ul></ul><ul><ul><li>Squared magnitude of DFT spectrum </li></ul></ul>
  76. 76. Power spectral density <ul><li>Derived from power spectrum </li></ul><ul><li>Generally normalized in some fashion to show relative power in different ranges </li></ul><ul><li>Measures energy content in specific band </li></ul>
  77. 77. Fast Fourier Transform (FFT) <ul><li>Developed by Cooley and Tukey in 1965 to speed up DFT calculations </li></ul><ul><li>Increases speed from O ( N 2 ) to O ( N log N ), but there are requirements </li></ul><ul><li>Useful reference: </li></ul>
  78. 78. Fast Fourier Transform (FFT) (continued) <ul><li>Requirements for FFT </li></ul><ul><ul><li>Sampled data must contain integer number of cycles of base (lowest frequency) waveform </li></ul></ul><ul><ul><ul><li>Otherwise discontinuities will exist, giving rise to “spectral leakage”, which shows up as noise </li></ul></ul></ul><ul><ul><li>Signal must be band limited and sampling must be at high enough rate </li></ul></ul><ul><ul><ul><li>Otherwise “aliasing” occurs, in which higher frequencies than those capturable by sampling rate appear as lower frequencies in FFT </li></ul></ul></ul><ul><ul><li>Signal must have stable (non-changing) frequency content </li></ul></ul><ul><ul><li>Number of sample points must be power of 2 </li></ul></ul>
  79. 79. Spectral leakage No discontinuities Discontinuities present Source: National Instruments
  80. 80. Fast Fourier Transform (FFT) (continued) <ul><li>We will not discuss exactly how the method works </li></ul><ul><li>Lots of software packages are available </li></ul><ul><ul><li>See this site for many of them </li></ul></ul><ul><ul><li>Contained in Mathcad package </li></ul></ul><ul><ul><li>Also available in many textbooks </li></ul></ul><ul><ul><li>Many modern instruments such as digital oscilloscopes have FFT built-in </li></ul></ul><ul><li>Averaging is frequently used to improve result </li></ul><ul><ul><li>Averages over several FFT runs with different data sets representing same waveform </li></ul></ul><ul><ul><ul><li>Sometimes with slightly staggered start times </li></ul></ul></ul>
  81. 81. FFT (continued) <ul><li>Also inverse FFT exists for going in opposite direction </li></ul><ul><li>Short Mathcad demo </li></ul><ul><li>Note that output of FFT is two-dimensional array of length ½ number of sample points + 1 </li></ul><ul><ul><li>The points in this array are the complex values F( j  k ) </li></ul></ul><ul><ul><li>But the  k values themselves do not appear </li></ul></ul><ul><ul><ul><li>Must be calculated by user </li></ul></ul></ul><ul><ul><ul><li>They are  k = k x frequency resolution = k x 2  /NT, k = 0...N/2 </li></ul></ul></ul>
  82. 82. FFT examples showing different resolution f ( x )=sin (  x /5), analysis done in MATHCAD 32 sample points, T =1 sec, f s =1 resolution 1/32 Hz 64 sample points, T= 1 sec, f s = 1 resolution 1/64 Hz
  83. 83. Fourier analysis: a deeper view <ul><li>Fourier series only one possible way to analyze functions </li></ul><ul><li>Best understood in terms of functional analysis </li></ul><ul><li>Let X be a space composed of real-valued functions on some interval [a,b] </li></ul><ul><ul><li>Technically, the set of Lebesgue-integral functions </li></ul></ul><ul><ul><li>Infinite-dimensional space </li></ul></ul><ul><li>Define an inner product (“dot product” in Euclidean space) as follows: </li></ul>
  84. 84. Fourier analysis: a deeper view (continued) <ul><li>This induces a norm on the space </li></ul><ul><li>Can be shown that this space is complete </li></ul><ul><ul><li>Complete normed space with norm defined by inner product is known as a Hilbert space </li></ul></ul><ul><li>An orthogonal sequence ( u k ) is a sequence of elements u k of X such that </li></ul>
  85. 85. Fourier analysis: a deeper view (continued) <ul><li>This series can be converted into an orthonormal sequence ( e k ) by dividing each element u k by its norm ||u k || </li></ul><ul><li>Consider an arbitrary element x  X , and calculate </li></ul><ul><li>Now formulate the sum </li></ul><ul><li>Then clearly if || x - x n ||  0 as n  the sum converges to x </li></ul>
  86. 86. Fourier analysis: a deeper view (continued) <ul><li>We have the following theorem: If ( e k ) is an orthonormal sequence in Hilbert space X , then </li></ul><ul><ul><li>(a) The series converges (in the norm on X ) if and only if the following series converges: </li></ul></ul><ul><ul><li>(b) If the series converges, then the coefficients  k are the Fourier coefficients so that x can be written </li></ul></ul>
  87. 87. Fourier analysis: a deeper view (continued) <ul><li>(c) For any x  X , the foregoing series converges </li></ul><ul><li>Lemma: Any x in X can have at most countably many (may be countably infinite) nonzero Fourier coefficients with respect to an orthonormal set ( e k ) </li></ul><ul><li>Note that we are not quite where we want to be yet, as we have not shown that every x  X has a sequence which converges to it </li></ul><ul><ul><li>For this we require another notion, that of totality </li></ul></ul>
  88. 88. Fourier analysis: a deeper view (continued) <ul><li>Note also that as of this point we have said nothing about the nature of the functions e k </li></ul><ul><ul><li>Any set which meets the orthogonality condition is OK, since it can be normalized </li></ul></ul><ul><ul><li>Note that (sin nt ), (cos nt ) meet condition, can be combined into new set containing all elements by suitable renumbering </li></ul></ul><ul><ul><li>Lots of other functions would work as well, such as triangle waves, Bessel functions </li></ul></ul>
  89. 89. Fourier analysis: a deeper view (continued) <ul><li>Most interesting orthonormal sets are those which consists of “sufficiently many” elements so that every element in the space can be approximated by Fourier coefficients </li></ul><ul><ul><li>Trivial in finite-dimensional spaces: just use orthonormal basis </li></ul></ul><ul><ul><li>More complicated in infinite dimensional spaces </li></ul></ul><ul><li>Define a total orthonormal set in X as a subset M  X whose span is dense in X </li></ul><ul><ul><li>Functions analogously to orthonormal basis in finite spaces </li></ul></ul><ul><ul><li>But Fourier expansion doesn't have to equal every element, just get arbitrarily close to it in sense of norm </li></ul></ul>
  90. 90. Fourier analysis: a deeper view (continued) <ul><li>Can be shown that all total orthonormal sets in a given Hilbert space have same cardinality </li></ul><ul><ul><li>Called Hilbert dimension or orthogonal dimension of the space </li></ul></ul><ul><ul><li>Trivial in finite dimensional spaces </li></ul></ul><ul><li>Necessary and sufficient condition for totality of an orthonormal set M is that there does not exist a non-zero x  X such that x is orthogonal to every element of M </li></ul>
  91. 91. Fourier analysis: a deeper view (continued) <ul><li>Parseval relation can be expressed as </li></ul><ul><li>Another theorem states that an orthonormal set M is total in X if and only if the Parseval relation holds for all x </li></ul><ul><ul><li>True for {(sin nt)/  , (cos nt)/  terms </li></ul></ul><ul><ul><li>Therefore these terms form total orthonormal set </li></ul></ul><ul><li>Key results </li></ul><ul><ul><li>Fourier expansion works because {(sin nt)/  , (cos nt)/  }terms from orthonormal basis for space of functions </li></ul></ul><ul><ul><li>Any other orthonormal set of functions can also serve as basis of Fourier analysis </li></ul></ul>
  92. 92. Fourier analysis: a deeper view (continued) <ul><li>Effect of truncating Fourier expansion </li></ul><ul><ul><li>Finite set ( e 1 ... e m ) no longer total </li></ul></ul><ul><ul><li>But it can be shown that the projection theorem applies </li></ul></ul>Space spanned by ( e 1 ... e m ) Function f ( x ) to be approximated Approximation error Approximation f m ( x )
  93. 93. Fourier analysis: a deeper view (continued) <ul><li>Projection theorem states that optimal representation of f ( x ) in lower-order space obtained when error || f – f m || is orthogonal to f m </li></ul><ul><li>This is guaranteed by orthonormal elements e i and the construction of the Fourier coefficients </li></ul><ul><li>Therefore truncated Fourier representation is optimal representation in terms of ( e 1 ... e m ) </li></ul><ul><li>References: </li></ul><ul><ul><li>Erwin Kreyszig, Introductory Functional Analysis with Applications </li></ul></ul><ul><ul><li>Eberhard Zeidler, Nonlinear Functional Analysis and its Applications , Vol. I, Fixed-Point Theorems </li></ul></ul>