Course10810810977 lectures
Mathematical Methods and Techniques in Signal Processing
Prof. Shayan Srinivasa Garani · IISc Bangalore
About
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 algebra
Lectures
Week 1 Introduction to Signal Processing, State Space Representation and Vector Spaces - I
- 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
Week 2 Vector Spaces - II
- 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
- Problem on sum of subspaces
Week 3 Vector Spaces - III and Signal Geometry
- Linear independence of orthogonal vectors
- Hilbert space and linear transformation
- Gram Schmidt orthonormalization
- Linear approximation of signal space
- Gram Schmidt orthogonalization of signals
- Problem on orthogonal complement
- Problem on signal geometry (4-QAM)
Week 4 Probability and Random Processes
- 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
- Problem on mean and variance
- Problem on MAP Detection
Week 5 Sampling Theorem and Multirate Systems - I
- Fourier transform of dirac comb sequence
- Sampling theorem
- Basics of multirate systems
- Frequency representation of expanders and decimators
- Decimation and interpolation filters
Week 6 Multirate Systems - II
- Fractional sampling rate alterations
- Digital filter banks
- DFT as filter bank
- Noble Identities
- Polyphase representation
- Efficient architectures for interpolation and decimation filters
- Problems on simplifying multirate systems using noble identities
- Problem on designing synthesis bank filters
Week 7 Multirate Systems - III
- Efficient architecture for fractional decimator
- Multistage filter design
- Two-channel filter banks
- Amplitude and phase distortion in signals
Week 8 Multirate Systems - IV
- 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
Week 9 Wavelets - I
- Introduction to wavelets
- Multiresolution analysis and properties
- The Haar wavelet
- Structure of subspaces in MRA
- Haar decomposition - 1
- Haar decomposition - 2
Week 10 Wavelets - II and Continuity of Functions
- 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
Week 11 Fourier Series - I
- 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
Week 12 Fourier Series - II and KL Transform
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IISc Bangalore
- Department
- Electrical Engineering
- Teacher
- Prof. Shayan Srinivasa Garani
- Level
- Graduate
- Language
- English
- Subjects
- Engineering
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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