Understanding Signal Sampling and the Sampling Theorem

38655 bmed 2300 02 lecture 8 discrete ft ge wang n.w
1 / 47
Embed
Share

Explore the concepts of signal sampling, the Sampling Theorem, and spectral duplication in signal processing. Learn about Nyquist Theorem and the implications of aliasing in this comprehensive study.

  • Signal Processing
  • Sampling Theorem
  • Spectral Duplication
  • Nyquist Theorem
  • Aliasing

Uploaded on | 0 Views


Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.

E N D

Presentation Transcript


  1. 38655 BMED-2300-02 Lecture 8: Discrete FT Ge Wang, PhD Biomedical Imaging Center CBIS/BME, RPI wangg6@rpi.edu February 9, 2018

  2. BB Schedule for S18 Tue 1/16 1/23 1/30 2/06 2/13 2/20 2/27 3/06 3/20 3/27 4/03 4/10 4/17 4/24 Topic Introduction System Fourier Series Signal Processing MatLab II (Homework) No Class Quality & Performance CT Reconstruction MatLab III (CT) PET & SPECT Exam II MRI III Ultrasound II Machine Learning Fri 1/19 1/26 2/02 2/09 2/16 2/23 3/02 3/09 3/23 3/30 4/06 4/13 4/20 4/27 Topic MatLab I (Basics) Convolution Fourier Transform Discrete FT & FFT Network Exam I X-ray & Radiography CT Scanner Nuclear Physics MRI I MRI II Ultrasound I Optical Imaging Exam III Office Hour: Ge Tue & Fri 3-4 @ CBIS 3209 | wangg6@rpi.edu Kathleen Mon 4-5 & Thurs 4-5 @ JEC 7045 | chens18@rpi.edu

  3. Sampling Theorem

  4. What If P=2W

  5. Derivation of the Sampling Theorem https://dsp.stackexchange.com/questions/37480/formulating-a-function-on-matlab-for-the-shannon-interpolation-formula

  6. Good Case: True versus Sampled

  7. Bad Case: True versus Sampled

  8. Big Picture

  9. Signal Sampling ( ) x f Continuous Signal x Shah Function (Impulse Train) ( ) s x ( ) ( ) x = x nT s = n x T Sampled Function ( ) x ( ) ( ) f x s x ( ) f x ( ) = = f x nT s = n

  10. Spectral Duplication Sampled Function ( ) x ( ) ( ) f x s x ( ) f x ( ) = = f x nT s = n 1 T n T ( ) ( ) ( ) ( ) = = F u F u S u F u u S = n ( ) u ( ) u A FS F AT u u u u max max 1T Sampling Frequency 1 T There will be no overlap if u max 2

  11. Nyquist Theorem 1 T ( ) u FS u If max 2 Aliasing u u max 1 T When can we recover F(u) from FS(u)? 1 T u Only if (Nyquist Frequency) max 2 We can use 12 T u ( ) T = C u Otherwise 0 ( ) x ( ) u ( ) u ( ) ( ) C u = = IFT f F F F u Then and S 2u Sampling frequency must be greater than max

  12. Why Non-unique?

  13. Digitization Not Finished Yet

  14. Discretizing Spectrum

  15. From Continuous to Discrete g(t) f(t) G(t) F(t) Continuous Discrete

  16. Big Picture 1 P = = / N T TP 1 T = = = / M N P PT

  17. Key Variables 1 P = = / N T TP 1 T = = = / M N P PT 1 N 1 1 P T = = t u

  18. Direct Fourier Transform of

  19. Continuous FT of Sampled f(t) n n t u P ( ) f u 2 2 i i = n = n T ( ) f t c e c e n n = =

  20. Sampling in the Fourier Domain 1 N 1 1 P T = = t u

  21. Discrete Fourier Transform

  22. Use of Integer Indices e e e e e e e e e [ ] f m = [ ] f n

  23. Inverse Discrete Fourier Transform 1 e e e e e e e e e f m = [ ] f n [ ]

  24. Why 1/N?

  25. Perspective 1: Discretization

  26. Perspective 2: Harmonics

  27. Orthonormal Basis

  28. Discrete FT in Different Notations Vector of N Elements Only Needs N Basis Functions 1 N = 1 N ikn = 2 N H h e n k N Harmonic Orthogonal Basis Functions Are Enough 0 k 1 N = n Frequencies Differ by Constant Increment ikn = 2 N h H e k n 0 Forward & Inverse Transforms Are Symmetric

  29. FFT Fast Fourier Transform (FFT) is an efficient algorithm for performing a discrete Fourier transform FFT published by Cooley & Tukey in 1965 In 1969, the 2048 point analysis of a seismic trace took 13 hours. Using the FFT, the same task on the same machine took 2.4 seconds!

  30. FFT & IFFT

  31. Application 1: Discrete Convolution https://www.mathworks.com/matlabcentral/answers/38066-difference-between-conv-ifft-fft-when-doing-convolution

  32. Circular Convolution

  33. Zero Padding

  34. Zero Padding Illustrated

  35. Further Reading

  36. Application 2: Spectral Analysis Fs = 1e3; t = 0:0.001:1 0.001; x = cos(2*pi*100*t)+sin(2*pi*202.5*t); Plot(x(1:100)); https://www.mathworks.com/help/signal/ug/amplitude-estimation-and-zero-padding.html

  37. Without Zero Padding xdft = fft(x); xdft = xdft(1:length(x)/2+1); xdft = xdft/length(x); xdft(2:500) = 2*xdft(2:500); freq = 0:Fs/length(x):Fs/2; plot(freq,abs(xdft)) hold on plot(freq,ones(length(x)/2+1,1),'LineWidth',2) xlabel('Hz') ylabel('Amplitude') hold off

  38. Zero Padding xdft = fft(x,2000); xdft = xdft(1:length(xdft)/2+1); xdft = xdft/length(x); xdft(2:500) = 2*xdft(2:500); freq =0:Fs/(2*length(x)):Fs/2; plot(freq,abs(xdft)) hold on plot(freq,ones(2*length(x)/2+1,1),'LineWidth',2) xlabel('Hz') ylabel('Amplitude') hold off

  39. Note: Impossible into Possible

  40. Convergence Issue

  41. Increasingly Smaller, Not Enough

  42. Wheat & Chessboard Problem Exponential growth never can go on very long in a finite space with finite resources.

  43. https://see.stanford.edu/Course/EE261 https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf

  44. https://see.stanford.edu/Course/EE261 https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf

  45. Art_X

  46. ArtX HW: Do an Overview Poster DFT & FFT Signal Processing Fourier Series Fourier Transform Periodic Non-periodic Convolution Shift-invariant Linear System Due Next Fri Function/System

Related


More Related Content