Mathematical methods and techniques in signal processing [E9 252 (3:0) – Spring 2021]
Shayan S. Garani
Digital signal processing at the undergrad level.
- Review of basic signals, systems and signal space: Review of 1-D signals and systems, review of random signals, multi-dimensional signals, review of vector spaces, inner product spaces, orthogonal projections and related concepts.
- Basics of multi-rate signal processing: sampling, decimation and interpolation, sampling rate conversion (integer and rational sampling rates), oversampled processing (A/D and D/A conversion), and introduction to filter banks.
- Signal representation: Transform theory and methods (FT and variations, KLT), other transform methods.
- Wavelets: Characterization of wavelets, wavelet transform, multi-resolution analysis.
- Statistical signal modeling: The least squares method, Pade’s approximation, Prony’s method, Shanks’ method, iterative pre-filtering, all-pole modeling and linear prediction, autocorrelation and covariance methods, FIR least squares inverse filter design, applications and examples.
- Inverse problems (signal reconstruction): underdetermined least squares, pseudo-inverse (SVD), min-norm solutions, regularized methods, reconstruction from projections, iterative methods such as projection onto convex sets, expectation-maximization and simulated annealing.
- Moon & Stirling, Mathematical Methods and Algorithms for Signal Processing, Prentice Hall, 2000.
- Monson Hayes, Statistical Digital Signal Processing and Modeling, John Wiley and Sons, 1996.
- A. Boggess & F. J. Narcowich, A First Course in Wavelets with Fourier Analysis, Prentice Hall, 2001.
- G. Strang, Introduction to Linear Algebra, 2016.
- H. Stark & J. W. Woods, Probability and Random Processes with Applications to Signal Processing, 2014.
- P. P. Vaidyanathan, Multirate systems and filter banks, 1993.
|Topics Covered||Lecture Notes|
|Week 1||Introduction to signal processing; Basics of signals and systems; Linear time-invariant systems; Modes in a linear system; Introduction to state space representation; State space representation; Non-uniqueness of state space representation; Introduction to vector space||MMTSP_Week_1|
|Week 2||Linear independence and spanning set; Unique representation theorem; Basis and cardinality of basis; Norms and inner product spaces; Inner products and induced norm; Cauchy Schwartz inequality; Orthonormality||MMTSP_Week_2|
|Week 3||Linear independence of orthogonal vectors; Hilbert space and linear transformation; Gram Schmidt orthonormalization; Linear approximation of signal space; Gram Schmidt orthogonalization of signals||MMTSP_Week_3|
|Week 4||Basics of probability and random variables; Mean and variance of a random variable; Introduction to random process; Statistical specification of random processes; Stationarity of random processes||MMTSP_Week_4|
|Week 5||Fourier transform of dirac comb sequence; Sampling theorem
Basics of multirate systems; Frequency representation of expanders and decimators; Decimation and interpolation filters
|Week 6||Fractional sampling rate alterations; Digital filter banks; DFT as filter bank; Noble Identities; Polyphase representation; Efficient architectures for interpolation and decimation filters||MMTSP_Week_6|
|Week 7||Efficient architecture for fractional decimator; Multistage filter design; Two-channel filter banks; Amplitude and phase distortion in signals||MMTSP_Week_7|
|Week 8||Polyphase representation of 2-channel filter banks, signal flow graphs and perfect reconstruction; M-channel filter banks; Polyphase representation of M-channel filter bank; Perfect reconstruction of signals; Nyquist and half band filters; Special filter banks for perfect reconstruction||MMTSP_Week_8|
|Week 9||Introduction to wavelets; Multiresolution analysis and properties; The Haar wavelet; Structure of subspaces in MRA; Haar decomposition – 1; Haar decomposition – 2||MMTSP_Week_9|
|Week 10||Wavelet Reconstruction; Haar wavelet and link to filter banks; Demo on wavelet decomposition; Problem on circular convolution; Time frequency localization; Basic analysis: Pointwise and uniform continuity of functions||MMTSP_Week_10|
|Week 11||Basic Analysis : Convergence of sequence of functions; Fourier series and notions of convergence; Convergence of Fourier series at a point of continuity; Convergence of Fourier series for piecewise differentiable periodic functions; Uniform convergence of Fourier series of piecewise smooth periodic function||MMTSP_Week_11|
|Week 12||Convergence in norm of Fourier series; Convergence of Fourier series for all square integrable periodic functions; Matrix Calculus; KL transform; Applications of KL transform; Demo on KL Transform||MMTSP_Week_12|