chapter 5ele 774 - adaptive signal processing 1 least mean-square adaptive filtering

Chapter 5 ELE 774 - Adaptive Signal Processing 1

Least Mean-SquareAdaptive Filtering

ELE 774 - Adaptive Signal Processing 2Chapter 5

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

Noexpectations,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 forMethod of Steepest Descent

Update Equation forLeast 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 (Jmin) (Wiener Soln.) as n→∞

i.e., J(n) →J(∞) as n→∞ Difference is called the excess mean-square error Jex(∞)

The ratio Jex(∞)/ Jmin 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.

unbiased


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

and its autocorrelation

inputoutput

Here we use expectation,however, actually it isthe ensemble average!.


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 sizeassumption

Here we use expectation,however, actually it isthe ensemble average!.


Small Step Size Analysis

Assumption I: step size is small (how small?) → LMS filter act 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 eo(n) is the irreducible estimation error and

Assumption III: The input and the desired response are jointly Gaussian.



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!

HW: Prove these relations.



Components of v(n) are uncorrelated:

first order difference equation (Brownian motion, thermodynamics)

Solution: Iterating from n=0

natural componentof v(n)

forced componentof 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 17. 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

Jex(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, Jmin.

LMS and SD are iterative algorithms designed to find wo. SD has direct access to auto/cross-correlations (exact measurements)

can approach the Wiener solution wo, can go down to Jmin.

LMS uses instantenous estimates instead (noisy measurements)

fluctuates around wo 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.

Smax: maximum componentof the PSD S(ω) of the tap inputs u(n).


H∞ Optimality of LMS

A single realisation of LMS is not optimum in the MSE sense Ensemble average is. The previous derivation is heuristic

(replacing auto/cross correlations with their instantenous estimates.)

In what sense is LMS optimum? It can be shown that LMS minimises

Maximum energy gain of the filter under the constraint

Minimising the maximum of something → minimax Optimisation of an H∞ criterion.


H∞ Optimality of LMS

Provided that the step size parameter satisfies the limits on the prev. slide, then

no matter how different the initial weight vector is from the unknown parameter vector wo of the multiple regression model, and

irrespective of the value of the additive disturbance (n), the error energy produced at the output of the LMS filter will never

exceed a certain level.


Limits on the Step Size

chapter 5ele 774 - adaptive signal processing 1 least mean-square adaptive filtering

Documents

lms algorithm slide

adaptive process

unbiased slide

canonical model slide

canonical model lms

meansquare slide

complex notation slide

separate lms algorithms