NPTEL Syllabus

Adaptive Signal Processing - Web course

COURSE OUTLINE

Adaptive signal processing concerns with processing of signals where theprocessing parameters are adjusted continuously to suit time varying signalenvironmental conditions.

Central to adaptive signal processing is the concept of adaptive linear combiner,often called adaptive filter where the combiner (filter) coefficients are trainedcontinuously so that the filter can estimate and track an unknown, target signal.

The study of adaptive signal processing involves development of variousadaptation algorithms and assessing them in terms of convergence rate,computational complexity, robustness against noisy data, hardware complexity,numerical stability etc.

This course will develop two main classes of adaptive filter algorithms, namelythe LMS and the RLS algorithms. Towards this, it will develop all necessarymathematical tools, in particular, random variables, stochastic processes andcorrelation structure. The filtering problem is developed in the form of computingorthogonal projection on a signal subspace.

To make this treatment self-contained, a rigorous treatment to linear algebra andvector spaces will be presented. For the LMS algorithm, convergence analysisfor filter weights (in mean) as well stability analysis for the steady state meansquare error will be taken up. Various versions of the LMS transversal forms likethe normalized LMS, sign LMS, block LMS etc will be considered.

As a counterpart to the transversal filter, the lattice filter/predictor will bedeveloped next, where relation of the lattice filter with autoregressive time seriesmodeling and human speech production process will be highlighted. Adaptiveversion of the lattice like the so-called gradient adaptive lattice will also bedeveloped.

In the case of the RLS algorithm, the optimal filtering problem will be formulatedin terms of the pseudo-inverse of a data matrix. The RLS lattice filter will bedeveloped by using time and order updating of an orthogonal projectionoperator, for which exact time and order update relations of the latticecoefficients will be derived.

The course will present several examples of adaptive filter applications likechannel equalization, echo cancellation, noise cancellation, interferencesuppression etc.

COURSE DETAIL

ModuleNo.

Topic/s No.ofLectures

1 Introduction to Adaptive Filters.

1. Adaptive filter structures, issues and examples.

2. Applications of adaptive filters.

a. Channel equalization, active noise control.

b. Echo cancellation, beamforming.

3

NPTELhttp://nptel.iitm.ac.in

Electronics &Communication

Engineering

Pre-requisites:

1. First course in Signals andSystems as well as DSP is aprerequisite.

2. Familiarity with linear algebraand random process theory willbe helpful.

Coordinators:

Prof. Mrityunjoy ChakrabortyDepartment of Electronics & ElectricalCommunication EngineeringIITKharagpur

2 Discrete time stochastic processes.

1. Re-visiting probability and random variables.

2. Discrete time random processes.

3. Power spectral density - properties.

4. Autocorrelation and covariance structures ofdiscrete time random processes.

5. Eigen-analysis of autocorrelation matrices.

5

3 Wiener filter, search methods and the LMS algorithm.

1. Wiener FIR filter (real case).

2. Steepest descent search and the LMS algorithm.

3. Extension of optimal filtering to complex valuedinput.

4. The Complex LMS algorithm.

4

4 Convergence and Stability Analyses.

1. Convergence analysis of the LMS algorithm.

2. Learning curve and mean square error behavior.

3. Weight error correlation matrix.

4. Dynamics of the steady state mean square error(mse).

5. Misadjustment and stability of excess mse.

5

5 Variants of the LMS Algorithm.

1. The sign-LMS and the normalized LMS algorithm.

2. Block LMS.

3. Review of circular convolution.

4. Overlap and save method, circular correlation.

5. FFT based implementation of the block LMSAlgorithm.

5

6 Vector space framework for optimal filtering.

1. Axioms of a vector space, examples, subspace.

2. Linear independence, basis, dimension, direct sumof subspaces.

3. Linear transformation, examples.

4. Range space and null space, rank and nullity of alinear operator.

5. Inner product space, orthogonality, Gram-Schmidtorthogonalization.

6. Orthogonal projection, orthogonal decomposition ofsubspaces.

7. Vector space of random variables, optimal filtering

7

as an orthogonal projection computation problem.

7 The lattice filter and estimator.

1. Forward and backward linear prediction, signalsubspace decomposition using forward andbackward predictions.

2. Order updating the prediction errors and predictionerror variances, basic lattice section.

3. Reflection coefficients, properties, updatingpredictor coefficients.

4. Lattice filter as a joint process estimator.

5. AR modeling and lattice filters.

6. Gradient adaptive lattice.

6

8 RLS lattice filter.

1. Least square (LS) estimation, pseudo-inverse of adata matrix, optimality of LS estimation.

2. Vector space framework for LS estimation.

3. Time and order updating of an orthogonalprojection operator.

4. Order updating prediction errors and predictionerror power.

5. Time updating PARCOR coefficients.

5

Total 40

References:

1. "Adaptive Filter Theory" by S. Haykin, Prentice Hall, Englewood Cliffs, NJ,1991 (end Ed.).

2. "Adaptive Filters – Theory and Applications", by B. Farhang-Boroujeny,John Wiley and Sons, 1999.

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