Biomedical Imaging Center

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Presentation transcript:

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

BB Schedule for S18 Tue Topic Fri 1/16 Introduction 1/19 MatLab I (Basics) 1/23 System 1/26 Convolution 1/30 Fourier Series 2/02 Fourier Transform 2/06 Signal Processing 2/09 Discrete FT & FFT 2/13 MatLab II (Homework) 2/16 Network 2/20 No Class 2/23 Exam I 2/27 Quality & Performance 3/02 X-ray & Radiography 3/06 CT Reconstruction 3/09 CT Scanner 3/20 MatLab III (CT) 3/23 Nuclear Physics 3/27 PET & SPECT 3/30 MRI I 4/03 Exam II 4/06 MRI II 4/10 MRI III 4/13 Ultrasound I 4/17 Ultrasound II 4/20 Optical Imaging 4/24 Machine Learning 4/27 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

Sampling Theorem

What If P=2W

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

Good Case: True versus Sampled

Bad Case: True versus Sampled

Big Picture

Signal Sampling Continuous Signal Shah Function (Impulse Train) Sampled Function

Spectral Duplication Sampled Function There will be no overlap if Sampling Frequency There will be no overlap if

Nyquist Theorem If Aliasing When can we recover F(u) from FS(u)? Only if (Nyquist Frequency) We can use Then and Sampling frequency must be greater than

Why Non-unique?

Digitization Not Finished Yet

Discretizing Spectrum

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

Big Picture

Key Variables

Direct Fourier Transform of

Continuous FT of Sampled f(t)

Sampling in the Fourier Domain

Discrete Fourier Transform

Use of Integer Indices

Inverse Discrete Fourier Transform

Why 1/N?

Perspective 1: Discretization

Perspective 2: Harmonics

Orthonormal Basis

Discrete FT in Different Notations Vector of N Elements Only Needs N Basis Functions N Harmonic Orthogonal Basis Functions Are Enough Frequencies Differ by Constant Increment Forward & Inverse Transforms Are Symmetric

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 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!

FFT & IFFT

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

Circular Convolution

Zero Padding

Zero Padding Illustrated

Further Reading

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

Without Zero Padding xdft = fft(x); xdft = xdft(1:length(x)/2+1); 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

Zero Padding xdft = fft(x,2000); xdft = xdft(1:length(xdft)/2+1); 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

Note: Impossible into Possible

Convergence Issue

Increasingly Smaller, Not Enough

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

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

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

Art_X

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