PowerPoint Presentation

Adaptive filter

Ahmed Shamel10/10/2016

Adaptive filter Theory By Simon Haykin

Adaptive filter By B.Farhang boroujeny

Adaptive Linear & Nonlinear Filters By (Frank) XiangYang Gao

10/10/2016References:

10/10/2016 Overview

Before we start we must understand some concept : The term filter is a black box that takes an input signal ,processes it, and then returns an output signal that in some way modifies the input. For example, if the input signal is noisy, then one would want a filter that removes noise, but otherwise leaves the signal unchanged.

we may use a filter to perform three basic information-processing tasks:

Filtering, which means the extraction of information about a quantity of interest at time t by using data measured up to and including time t.

10/10/2016Smoothing, which differs from filtering in that information about the quantity of interest need not be available at time t , This means that in the case of smoothing there is a delay in producing the result of interest.

Prediction, which is the forecasting side of information processing. The aim here is to derive information about what the quantity of interest will be like at some time t + T in the future, for some z > 0, by using data measured up to and including time t.

10/10/2016Filter

linearNon linearWe may classify filters into linear and nonlinear.

A filter is said to be linear if the 1) filtered 2) smoothed 3) predicted quantity at the output of the device is a linear function of the observations applied w the filter input. Otherwise, the filter is nonlinear.

Fixed versus Adaptive Filter Design10/10/2016

W0 , W1 , W2 ,.. Wn-1Fixed

Determine the values of the coefficients of the digital filter that meet the desired specifications and the values are not changed once they are implementedThe coefficient values are not fixed. They are adjusted to optimize some measure of the filter performance using incoming input data and error. W0(n) , W1(n) , W2(N),.Wn-1(N)Adaptive

The figure shows a filter emphasizing the way it is used in typical problems.

The filter is used to reshape certain input signals in such a way that its output is a good estimate of the given desired signal.

10/10/2016Introduction To adaptive filterFilter (A.F)Input signalOutput signalDesired signalError signal+

Introduction To adaptive filter10/10/2016 An adaptive filter is a digital filter with self-adjusting characteristics.

It adapts automatically, to changes in its input signals.

A variety of Adaptive algorithms have been developed for the operation of adaptive filters, e.g., LMS , RLS, etc.

*LMS (least Mean Square) *RLS (Recursive Least Squares)

Contains 2 main component :1- Digital filter(with adjustable coefficients).2- Adaptive Algorithm.10/10/2016

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Noise Cancelling and power

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Applications of Adaptive Filters: 1) Identification10/11/2016Used to provide a linear model of an unknown plant.Applications: System identification

10/11/2016Applications of Adaptive Filters: 2) Inverse ModelingUsed to provide an inverse model of an unknown plantApplications : Channel Equalization

Used to provide a prediction of the present value of a random signal

Applications : Signal detection

10/12/2016Applications of Adaptive Filters: 3) Prediction

Subtracts Noise from Received signal adaptively to improve SNR10/12/2016Applications of Adaptive Filters: 4) Echo (Noise) cancellation

A good example to illustrate the principles of adaptive noise cancelling is the noise removal from the pilot's microphone in the airplane. Due to the high environmental noise produced by the airplane engines, the pilots voice in the microphone is distorted with a high amount of noise ,and can be very difficult to understand . In order to overcome the problem , an adaptive filter can be used.10/12/2016

Approaches to adaptive filter 10/13/2016Adaptive Filtering

Stochastic Gradient ApproachLeast Square Estimation(Least Mean Square Algorithms) (Recursive Least Square Algorithm)

LMSNLMSTVLMSVSSNLMSRLSFTRLSLinear

Non linear

Neural Networks

Stochastic Gradient10/12/2016Most commonly used type of Adaptive Filters

Define cost function as mean-squared error Difference between filter output and desired responseBased on the method of steepest descentMove towards the minimum on the error surface to get to minimumRequires the gradient of the error surface to be knownMost popular adaptation algorithm is LMSDerived from steepest descentDoesnt require gradient to be know: it is estimated at every iteration

Least-Mean-Square (LMS) Algorithm.10/12/2016

In the family of stochastic gradient algorithms Approximation of the steepest descent method Based on the MMSE criterion.(Minimum Mean square Error) Adaptive process containing two important signals:1.) Filtering process, producing output signal.2.) Desired signal (Training sequence)

Adaptive process: Recursive adjustment of filter tap weightsThe LMS Algorithm consists of two basic processes that is followed in the adaptive equalization processes:Training : It refers to adapting to the training sequence.Tracking: keeps track of the changing characteristics of the channel.

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LMS Algorithm Steps:10/12/2016Filter output

Estimation error

Tap-weight adaptation

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Transversal Filter

Stability of LMS:10/12/2016The LMS algorithm is convergent in the mean square if and only if the step-size parameter satisfy

Here max is the largest Eigen value of the correlation matrix of the input data.More practical test for stability is

Larger values for step size Increases adaptation rate (faster adaptation) Increases residual mean-squared error

10/12/2016LMS Disadvantage: Slow Convergence

Demands using of training sequence as reference ,thus decreasing the communication BW.LMS Advantage: Simplicity of implementation

Not neglecting the noise like Zero forcing equalizer

Stable and robust performance against different signal conditions

The design of a Wiener filter requires a priori information about the statistics of the data to be processed. The filter is optimum only when the statistical characteristics of the input data match the a priori information on which the design of the filter is based.

10/13/2016Wiener filter

Many adaptive algorithms can be viewed as approximation to the discrete Wiener filter.

Tries to minimize the mean of the square of the error (Least Mean Square)

Assuming an FIR filter structure with N coefficient (weights) the output signal is given by:10/13/2016

Squaring the error:10/13/2016(The N-length Cross-Correlation Vector)

(The N * N autocorrelation matrix).Mean :

The mean square10/13/2016

The wiener-Hopf solution10/13/2016

Issues with the wiener Hopf solution10/13/2016

The windrow-Hopf LMS algorithm10/13/2016Base on the the steepest descent algorithm

WhereU determines Stability and rate convergence.If u is too large, we observe too much fluctuation.If u is too small, rate of convergence too slow.

Least Square Estimation10/13/2016

Recursive Least Square (RLS) Algorithm10/13/2016

Recursive Least Square (RLS) Algorithm10/13/2016

Gama (typically between 0.98 and 1) is referred to as the forgetting factor.

The previous samples contribute less and less to the new weights:

when Y=1, we have infinite memory and this weighting scheme reduce to exract Least Squares solution.10/13/2016

Comparison against LMS10/13/2016RLS has rapid rate convergence, compared to LMS.

RLS is computationally more expensive than LMS.

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Thank YOU ^_^10/13/2016