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Understanding Discrete Fourier Transform (DFT) Fundamentals: Practical Guide

Learn about DFT properties, steps for calculating DFT, digital filtering, ideal filters, and common filters. Explore MATLAB examples and comparison of filters.

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Understanding Discrete Fourier Transform (DFT) Fundamentals: Practical Guide

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  1. Digital Signal Processing

  2. Discrete Fourier Transform Discrete Fourier Transform Inverse Discrete Fourier Transform

  3. Properties of DFT • DFT has the same number of datapoints as the signal • The signal is assumed to be periodic with a period of N • X[k] corresponds to the amplitude of the signal at frequency f=k/(NT) • The frequency resolution of the DFT is Df=1/(NT), i.e. the # of samples determines the frequency resolution

  4. Steps for Calculating DFT • Determine the resolution required for the DFT, establish a lower limit on the # of samples required, N. • Determine the sampling frequency to avoid aliasing • Accumulate N samples • Calculate DFT

  5. Matlab Example of FFT

  6. Digital Filtering a1*y(n) = b1*x(n) +b2*x(n-1) + ... + bnb+1x(n-nb) - a2*y(n-1) - ... – ana+1*y(n-na) A=[a1,a2,..., ana+1] Filter parameters B=[b1,b2,..., bnb+1] X=[x(n-nb),..., x(n-1), x(n)]: input signal Y=[y(n-na),..., y(n-1), y(n)]: filtered signal

  7. Ideal Filters • Low pass filter • High pass filter • Bandpass filter • Bandstop filter

  8. Butterworth filter: Common Filters • Chebyshev filter:

  9. Comparison of Common Filters

  10. MATLAB example of Filtering

  11. MATLAB Example of Undersampling

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