LEAST MEAN-SQUARE (LMS) ADAPTIVE FILTERING

Steepest Descent The update rule for SD is where or SD is a deterministic algorithm, in the sense that p and R are assumed to be exactly known. In practice we can only estimate these functions.

Basic Idea The simplest estimate of the expectations is  To remove the expectation terms and replace them with the instantaneous values, i.e. Then, the gradient becomes Eventually, the new update rule is No expectations, Instantaneous samples!

Basic Idea However the term in the brackets is the error, i.e. then is the gradient of instead of as in SD.

Basic Idea Filter weights are updated using instantaneous values

Update Equation for Method of Steepest Descent Update Equation for Least Mean-Square

LMS Algorithm Since the expectations are omitted, the estimates will have a high variance. Therefore, the recursive computation of each tap weight in the LMS algorithm suffers from a gradient noise. In contrast to SD which is a deterministic algorithm, LMS is a member of the family of stochastic gradient descent algorithms. LMS has higher MSE (J(∞)) compared to SD (J min ) (Wiener Soln.) as n→∞  i.e., J(n) →J(∞) as n→∞  Difference is called the excess mean-square error J ex (∞)  The ratio J ex (∞)/ J min is called the misadjustment.  Hopefully, J(∞) is a finite value, then LMS is said to be stable in the mean square sense.  LMS will perform a random motion around the Wiener solution.

LMS Algorithm Involves a feedback connection. Although LMS might seem very difficult to work due the randomness, the feedback acts as a low-pass filter or performs averaging so that the randomness can be filtered-out. The time-constant of averaging is inversely proportional to μ. Actually, if  is chosen small enough, the adaptive process is made to progress slowly and the effects of the gradient noise on the tap weights are largely filtered-out. Computational complexity of LMS is very low → very attractive  Only 2M+1 complex multiplications and 2M complex additions per iteration.

LMS Algorithm

Canonical Model LMS algorithm for complex signals/with complex coef.s can be represented in terms of four separate LMS algorithms for real signals with cross-coupling between them. Write the input/desired signal/tap gains/output/error in the complex notation

Canonical Model Then the relations bw. these expressions are

Canonical Model

Analysis of the LMS Algorithm Although the filter is a linear combiner, the algorithm is highly non- linear and violates superposition and homogenity Assume the initial condition, then Analysis will continue using the weight-error vector input output

Analysis of the LMS Algorithm We have Let Then the update eqn. can be written as Analyse convergence in an average sense  Algorithm run many times→study their ensemble average behavior

Analysis of the LMS Algorithm Using It can be shown that Small step size assumption Here we use expectation, however, actually it is the ensemble average!.

Small Step Size Analysis Assumption I: step size  is small → LMS filter acts like a low-pass filter with very low cut-off frequency. Assumption II: Desired response is described by a linear multiple regression model that is matched exactly by the optimum Wiener filter where e o (n) is the irreducible estimation error and Assumption III: The input and the desired response are jointly Gaussian.

Small Step Size Analysis Applying the similarity transformation resulting from the eigendecom. on i.e. Then, we have where We do not have this term in Wiener filtering!. Components of v(n) are uncorrelated!

Small Step Size Analysis Components of v(n) are uncorrelated:  first order difference equation Solution: Iterating from n=0 natural component of v(n) forced component of v(n) stochastic force

Learning Curves Two kinds of learning curves  Mean-square error (MSE) learning curve  Mean-square deviation (MSD) learning curve Ensemble averaging → results of many (→∞) realizations are averaged. What is the relation bw. MSE and MSD? for  small

Learning Curves under the assumptions of slide Excess MSE  LMS performs worse than SD, there is always an excess MSE for  small ← use

Learning Curves Mean-square deviation D is lower-upper bounded by the excess MSE. They have similar response: decaying as n grows or

Convergence For  small Hence, for convergence The ensemble-average learning curve of an LMS filter does not exhibit oscillations, rather, it decays exponentially to the const. value or J ex (n)

Misadjustment Misadjustment, define  For small , from prev. slide or equivalently but then

Average Time Constant From SD we know that but then

Observations Misadjustment is  directly proportional to the filter length M, for a fixed  mse,av  inversely proportional to the time constant  mse,av slower convergence results in lower misadjustment.  Directly proportional to the step size  smaller step size results in lower misadjustment.  Time constant is inversely proportional to the step size   smaller step size results in slower convergence  Large  requires the inclusion of  k (n) (k≥1) into the analysis Difficult to analyse, small step analysis is no longer valid, learning curve becomes more noisy

LMS vs. SD Main goal is to minimise the Mean Square Error (MSE) Optimum solution found by Wiener-Hopf equations. Requires auto/cross-correlations. Achieves the minimum value of MSE, J min. LMS and SD are iterative algorithms designed to find w o.  SD has direct access to auto/cross-correlations ( exact measurements ) can approach the Wiener solution w o, can go down to J min.  LMS uses instantenous estimates instead (noisy measurements) fluctuates around w o in a Brownian-motion manner, at most J(∞).

LMS vs. SD Learning curves  SD has a well-defined curve composed of decaying exponentials  For LMS, curve is composed of noisy- decaying exponentials

Statistical Wave Theory As filter length increases, M→∞  Propagation of electromagnetic disturbances along a transmission line towards infinity is similar to signals on n infinitely long LMS filter. Finite length LMS filter (transmission line)  Corrections have to be made at the edges to tackle reflections,  As length increases reflection region decreases compared to the total filter. Imposes a limit on the step size to avoid instability as M→∞ If the upper bound is exceeded, instability is observed. S max : maximum component of the PSD S(ω) of the tap inputs u(n).