ADAPTIVE FILTERS I

1. ADAPTIVE FILTERS I The Adaptive Linear Combiners CHAPTER # 02 Instructor :Dr. Aamer Iqbal Bhatti Lecture 2

2. Optimal Filters Word optimal means doing a job in the best possible way. Before beginning search for such an optimal solution , the job must be defined. A mathematical scale must be established for what best means , and the possible alternatives must be spelled out. Unless there is agreement an these qualifiers a claim that a system is optimal is really meaningless. Lecture 2

3. Statement for the Optimal Problem A mathematical statement of optimal problem consists of Description of system constraints and possible alternatives A description of task to be performed A statement of the criterion for judging optimal performance Lecture 2

4. Linear Optimal Filtering: Statement of the Problem Output of the system is used to provide an estimate of the desired response Filter input and output represents single realization of respective stochastic processes i.e. the process to be filtered and the desired process after filtering of the input process. Lecture 2

5. Linear Optimal Filtering: Statement of the Problem Estimation is accomplished by an reducing an error with the statistical characteristics of its own. Mathematical scale for the performance of the filtering system is to minimize the estimation error produced by the filter approximations Lecture 2

6. Linear Optimal Filtering: Statement of the Problem Constraints which have been placed on the system are The system is linear , which makes the problem mathematically easy to tract The filter operates in discrete time Lecture 2

7. Choice of the Filter Parameters Choice for the finite or infinite impulse response of the filter depends on the practical considerations Selection of the statistical criterion for optimization of filter is influenced by mathematical tractability Lecture 2

8. Adaptive Linear Combiner Adaptive Linear Combiner appears in most of adaptive systems It is the single most important element in learning systems It is a non-recursive digital filter and its performance is quite simple Linear adaptive consists of a multiple summer and their associated weights Lecture 2

9. Adaptive Linear Combiner These weights are varied in order to change the response of the system to achieve some desired response Procedure for adapting weights is called “Weight Adjustment” Adaptive linear combiner is a linear system once its weights are adjusted During weight adjustment weights are function of the input vector Lecture 2

10. Adaptive Linear Combiner Lecture 2

11. Input Signal and Weight Vectors There are two types of inputs Simultaneous inputs from different sources Sequential Samples or single input Input vector for both of them is are given by Lecture 2

12. Input Signal and Weight Vectors First of these is for the spatial domain and second one is for temporal domain Response for spatial and temporal inputs are given by Lecture 2

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42. Cost Function for Performance of the Optimal Filter Possibilities for selection of index for performance for filter design are Mean square value of the estimation error Expectation of the absolute value for estimation error Expectation of third or higher power of the absolute value of the estimation error Lecture 2

43. Cost Function for Performance of the Optimal Filter Choice for the mean square error results in a second order dependence of the cost function on the unknown coefficients in impulse response of the filter Cost function has a distinct minimum that uniquely defines the optimum statistical design of the filter Lecture 2

44. Cost Function for Performance of the Optimal Filter Optimum Filter design Problem is summarized as Design a linear discrete time filter whose output provides an estimate of the desired response , given a set of input samples, such that the minimum mean square value of the estimation error is minimized Lecture 2

45. Formulation for the Optimum Open Loop Filter Lecture 2

46. Formulation for the Optimum Open Loop Filter • Input for the single input system is given by • For a single input system the adaptive processor may be implemented using a linear combiner which is an FIR filter with varying weights • Response for the non-recursive filter is given by the convolution sum formula Lecture 2

47. Formulation for the Optimum Open Loop Filter Output of the filter is supposed to converge to the desired process in some statistical sense This convergence criterion forms the COP (criterion for performance ) for the optimal filter. We adopt the convergence in the mean square sense for the optimal filter characteristics Lecture 2

48. COP Function • Estimation error for the optima ; filter is given by • It is assumed that the process represented by the time sequence ek , xk and dk are Stationary and Ergodic Lecture 2

49. COP Function • Mean square error is • Let Lecture 2

50. COP Function Vector P is the vector for the cross correlation matrix of input and output processes Above Matrix is known as Correlation Matrix of the input process. Let Lecture 2