An Introduction to adaptive filtering & it’s applications By Asst.Prof.Dr.ThamerM.Jamel Department of Electrical Engineering University of Technology Baghdad – Iraq
Introduction Linear filters : the filter output is a linear function of the filter input Design methods: • The classical approach frequency-selective filters such as low pass / band pass / notch filters etc • Optimal filter design Mostly based on minimizing the mean-square value of the error signal
Wiener filter • work of Wiener in 1942 and Kolmogorov in 1939 • it is based on a priori statistical information • when such a priori information is not available, which is usually the case, it is not possible to design a Wiener filter in the first place.
Adaptive filter • the signal and/or noise characteristics are often nonstationaryand the statistical parameters vary with time • An adaptive filter has an adaptation algorithm, that is meant to monitor the environment and vary the filter transfer function accordingly • based in the actual signals received, attempts to find the optimum filter design
Adaptive filter • The basic operation now involves two processes : 1. a filtering process, which produces an output signal in response to a given input signal. 2. an adaptation process, which aims to adjust the filter parameters (filter transfer function) to the (possibly time-varying) environment Often, the (average) square value of the error signal is used as the optimization criterion
Adaptive filter • Because of complexity of the optimizing algorithms most adaptive filters are digital filters that perform digital signal processing • When processing analog signals, the adaptive filter is then preceded by A/D and D/A convertors.
Adaptive filter • The generalization to adaptive IIR filters leads to stability problems • It’s common to use a FIR digital filter with adjustable coefficients
Applications of Adaptive Filters: Identification • Used to provide a linear model of an unknown plant • Applications: • System identification
Applications of Adaptive Filters: Inverse Modeling • Used to provide an inverse model of an unknown plant • Applications: • Equalization (communications channels)
Applications of Adaptive Filters: Prediction • Used to provide a prediction of the present value of a random signal • Applications: • Linear predictive coding
Applications of Adaptive Filters: Interference Cancellation • Used to cancel unknown interference from a primary signal • Applications: • Echo / Noise cancellation hands-free carphone, aircraft headphones etc
LMS Algorithm • Most popular adaptation algorithm is LMS Define cost function as mean-squared error • Based on the method of steepest descent Move towards the minimum on the error surface to get to minimum gradient of the error surface estimated at every iteration
LMS Adaptive Algorithm • Introduced by Widrow & Hoff in 1959 • Simple, no matrices calculation involved in the adaptation • 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 input signals: • 1.) Filtering process, producingoutput signal. • 2.)Desired signal (Training sequence) • Adaptive process: recursive adjustment of filter tap weights
LMS Algorithm Steps • Filter output • Estimation error • Tap-weight adaptation
Stability of LMS • The LMS algorithm is convergent in the mean square if and only if the step-size parameter satisfy • Here max is the largest eigenvalue 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
Now lets find the solution by the steepest descend method STEEPEST DESCENT EXAMPLE • Given the following function we need to obtain the vector that would give us the absolute minimum. • It is obvious that • give us the minimum. • (This figure is quadratic error function (quadratic bowl) )
So our iterative equation is: STEEPEST DESCENT EXAMPLE • We start by assuming (C1 = 5, C2 = 7) • We select the constant . If it is too big, we miss the minimum. If it is too small, it would take us a lot of time to het the minimum. I would select = 0.1. • The gradient vector is:
Initial guess Minimum STEEPEST DESCENT EXAMPLE As we can see, the vector [c1,c2] converges to the value which would yield the function minimum and the speed of this convergence depends on .
Desired Combination of taps LMS – CONVERGENCE GRAPH Example for the Unknown Channel of 2nd order: Desired Combination of taps This graph illustrates the LMS algorithm. First we start from guessing the TAP weights. Then we start going in opposite the gradient vector, to calculate the next taps, and so on, until we get the MMSE, meaning the MSE is 0 or a very close value to it.(In practice we can not get exactly error of 0 because the noise is a random process, we could only decrease the error below a desired minimum)
Linear Combiner Interference SMART ANTENNAS Adaptive Array Antenna • Adaptive Arrays
Applications are many • Digital Communications (OFDM , MIMO , CDMA, and RFID) • Channel Equalisation • Adaptive noise cancellation • Adaptive echo cancellation • System identification • Smart antenna systems • Blind system equalisation • And many, many others
Introduction Wireless communication is the most interesting field of communication these days, because it supports mobility (mobile users). However, many applications of wireless comm. now require high-speed communications (high-data-rates).
What is the ISI Inter-symbol-interference, takes place when a given transmitted symbol is distorted by other transmitted symbols. Cause of ISI ISI is imposed due to band-limiting effect of practical channel, or also due to the multi-path effects (delay spread).
Definition of the Equalizer: the equalizer is a digital filter that provides an approximate inverse of channel frequency response. Need of equalization: is to mitigate the effects of ISI to decrease the probability of error that occurs without suppression of ISI, but this reduction of ISI effects has to be balanced with prevention of noise power enhancement.
Types of Equalization techniques Linear Equalization techniques which are simple to implement, but greatly enhance noise power because they work by inverting channel frequency response. Non-Linear Equalization techniques which are more complex to implement, but have much less noise enhancement than linear equalizers.
Equalization Techniques Fig.3 Classification of equalizers