Learn · Mathematical Methods and Techniques in Signal Processing
Lecture 76 of 77Week 12 Fourier Series - II and KL Transform
Applications of KL transform
Prof. Shayan Srinivasa Garani · IISc Bangalore
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All the lectures
- 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
- 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
- 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)
- 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
- Fourier transform of dirac comb sequence
- Sampling theorem
- Basics of multirate systems
- Frequency representation of expanders and decimators
- Decimation and interpolation filters
- 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
- Efficient architecture for fractional decimator
- Multistage filter design
- Two-channel filter banks
- Amplitude and phase distortion in signals
- 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
- Introduction to wavelets
- Multiresolution analysis and properties
- The Haar wavelet
- Structure of subspaces in MRA
- Haar decomposition - 1
- Haar decomposition - 2
- 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
- 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
- Convergence in norm of Fourier series
- Convergence of Fourier series for all square integrable periodic functions
- Problem on limits of integration of periodic functions
- Matrix Calculus
- KL transform
- Applications of KL transform
- Demo on KL Transform