lecture07: Theory: FFTs, Random Processes, Parseval, Rayleigh and Plancheral
This lecture covers some of the theory that we have
neglected so far. In particular: Fast Fourier Transforms, some
theory related to random processes and the spectral representation,
and the Parseval, Rayleigh and Plancheral theorems. We further
consider the theoretical basis for the Fourier transform by
considering how and why we could generalize it.
Topics covered
- Multiplication
- Fast Fourier Transform
- The Fast Fourier Transform
- Fast Fourier Transform
- Radix-2
- Radix-2 as factorization
- Picture of factorization
- Radix-2 bit reversal
- Other algorithms
- Matlab commands for FFT
- Matlab example
- Symmetry
- Matlab example 2
- Random processes
- Densities and distributions
- Moments of the process
- Covariance
- Stationarity
- Marginal distribution
- Gaussian processes
- Useful fact
- Spectral density
- Spectral density properties
- Why the relationship?
- Wiener-Kintchine theorem
- White noise
- Example: white noise
- Spectral density
- Filtered noise
- Examples
- Example 1: filtered white noise
- Example 2: filtered white noise
- Example 3: filtered white noise
- Color of noise
- Parseval's theorem
- Rayleigh's theorem
- Rayleigh's theorem proof
- Parseval-Rayleigh theorem
- Parseval-Rayleigh theorem proof
%
- Circular convolution
%
- Examples (i)
%
- Examples (ii)
%
- Examples (iii)
%
- Examples (iv)
- Plancherel
- Parseval-Rayleigh-Plancherel
- Linear filter characterization
- Revue of linear algebra
- Vector spaces and function spaces
- Examples
- Normed spaces
- Examples
- Examples (cont.)
- Norms
- Inner products
- Examples
- Orthogonality
- Basis
- Generalized Fourier transform
- Generalization of P-R
- Alternate transforms
- Examples of integral transforms
- Transforms
- Eigenvalues and Eigenvectors
- Diagonalization
- FT as diagonalization
%
- Fourier Transform
- What if?