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Digital and analog signal processing
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Module 6
The Discrete Fourier Transform and Fast Algorithms
1
From Gauss to Cooley-Tukey: A History of the FFT
Trace the historical development of the Fast Fourier Transform (FFT) from Gauss to the Cooley-Tukey algorithm.
2
Deriving the DFT from DTFT Sampling
Derive the Discrete Fourier Transform (DFT) by sampling the DTFT spectrum.
3
DFT, DTFT, and Z-Transform Relationship
Analyze the relationship between the DFT, DTFT, and z-transform, interpreting the DFT as samples of the z-transform on the unit circle.
4
Spectral Leakage and the Picket-Fence Effect
Analyze spectral leakage and the picket-fence effect in DFT-based spectral analysis.
5
Window Functions for Spectral Analysis
Apply various window functions (e.g., Hamming, Hanning) to reduce spectral leakage and analyze their trade-offs.
6
Circular Convolution: DFT and Distinction from Linear Convolution
Derive the concept of circular convolution from the DFT multiplication property and distinguish it from linear convolution.
7
Zero-Padding for DFT-based Linear Convolution
Apply zero-padding to compute linear convolution using the DFT/FFT.
8
DIT FFT: Algorithm and Complexity Analysis
Derive the decimation-in-time (DIT) FFT algorithm and analyze its computational complexity compared to the direct DFT.
Previous module
Discrete-Time Signals and Systems in the z-Domain
Next module
Digital Filter Design Techniques