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# Mathematical Methods and Techniques in Signal Processing

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### Overview

• 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.
• Sampling theorems (a peek into Shannon and compressive sampling), 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 including convergence issues.
• Wavelets: Characterization of wavelets, wavelet transform, multi-resolution analysis.

INTENDED AUDIENCE : Post graduates and senior UGs with a strong background in basic DSP.PRE-REQUISITES : UG in Digital Signal Processing, familiarity with probability and linear algebraINDUSTRY SUPPORT : Any company using DSP techniques in their work, such as, TI, Analog Devices, Broadcom and many more.Rajeev Motwani and Prabhakar Raghavan,Randomized Algorithms

### COURSE LAYOUT

Week 1:Review of vector spaces, inner product spaces, orthogonal projections, state variable representation
Week 2: Review of probability and random processes
Week 3:Signal geometry and applications
Week 4:Sampling theorems multirate signal processing decimation and expansion (time and frequency domain effects)
Week 5:Sampling rate conversion and efficient architectures, design of high decimation and interpolation filters, Multistage designs.
Week 6:Introduction to 2 channel QMF filter bank, M-channel filter banks, overcoming aliasing, amplitude and phase distortions.
Week 7:Subband coding and Filter Designs: Applications to Signal Compression
Week 8:Introduction to multiresolution analysis and wavelets, wavelet properties
Week 9:Wavelet decomposition and reconstruction, applications to denoising
Week 10:Derivation of the KL Transform, properties and applications.
Week 11:Topics on matrix calculus and constrained optimization relevant to KL Transform derivations.
Week 12:Fourier expansion, properties, various notions of convergence and applications.

### Taught by

Shayan Garani Srinivasa

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