# 2.1 First order convergence analysis of the lms algorithm

 Page 1 / 1

## Analysis of the lms algorithm

It is important to analyze the LMS algorithm to determine under what conditions it is stable, whether or not it convergesto the Wiener solution, to determine how quickly it converges, how much degredation is suffered due to the noisy gradient,etc. In particular, we need to know how to choose the parameter  .

## Mean of w

does ${W}^{k}$ , $k\to$ approach the Wiener solution? (since ${W}^{k}$ is always somewhat random in the approximate gradient-based LMS algorithm, we ask whether the expectedvalue of the filter coefficients converge to the Wiener solution)

$({W}^{k+1})=\overline{{W}^{k+1}}=({W}^{k}+2{}_{k}{X}^{k})=\overline{{W}^{k}}+2({d}_{k}{X}^{k})+2(-({W}^{k}^T{X}^{k}{X}^{k}))=\overline{{W}^{k}}+2P+-(2({W}^{k}^T{X}^{k}{X}^{k}))$

## Patently false assumption

${X}^{k}$ and ${X}^{k-i}$ , ${X}^{k}$ and ${d}^{k-i}$ , and ${d}_{k}$ and ${d}_{k-i}$ are statistically independent, $i\neq 0$ . This assumption is obviously false, since ${X}^{k-1}$ is the same as ${X}^{k}$ except for shifting down the vector elements one place and adding one new sample. We make this assumptionbecause otherwise it becomes extremely difficult to analyze the LMS algorithm. (First good analysis not makingthis assumption: Macchi and Eweda ) Many simulations and much practical experience has shown that the results one obtains withanalyses based on the patently false assumption above are quite accurate in most situations

With the independence assumption, ${W}^{k}$ (which depends only on previous ${X}^{k-i}$ , ${d}^{k-i}$ ) is statitically independent of ${X}^{k}$ , and we can simplify $({W}^{k}^T{X}^{k}{X}^{k})$

Now ${W}^{k}^T{X}^{k}{X}^{k}$ is a vector, and

$({W}^{k}^T{X}^{k}{X}^{k})=(\left(\begin{array}{c}\\ \sum_{i=0}^{M-1} {w}_{i}^{k}{x}_{k-i}{x}_{k-j}\\ \end{array}\right))=\left(\begin{array}{c}\\ \sum_{i=0}^{M-1} ({w}_{i}^{k}{x}_{k-i}{x}_{k-j})\\ \end{array}\right)=\left(\begin{array}{c}\\ \sum_{i=0}^{M-1} ({w}_{i}^{k})({x}_{k-i}{x}_{k-j})\\ \end{array}\right)=\left(\begin{array}{c}\\ \sum_{i=0}^{M-1} \overline{{w}_{i}^{k}}{r}_{\mathrm{xx}}(i-j)\\ \end{array}\right)=R\overline{{W}^{k}}$
where $R=({X}^{k}{X}^{k}^T)$ is the data correlation matrix.

Putting this back into our equation

$\langle {W}^{k+1}\rangle =\langle {W}^{k}\rangle +2P+-(2R\langle {W}^{k}\rangle )=I\langle {W}^{k}\rangle +2P$
Now if $\langle {W}^{k}\rangle$ converges to a vector of finite magnitude ("convergence in the mean"), what does it converge to?

If $\langle {W}^{k}\rangle$ converges, then as $k\to$ , $\langle {W}^{k+1}\rangle \approx \langle {W}^{k}\rangle$ , and $\langle {W}^{}\rangle =I\langle {W}^{}\rangle +2P$ $2R\langle {W}^{}\rangle =2P$ $R\langle {W}^{}\rangle =P$ or $\langle {W}_{\mathrm{opt}}\rangle =R^{(-1)}P$ the Wiener solution!

So the LMS algorithm, if it converges, gives filter coefficients which on average arethe Wiener coefficients! This is, of course, a desirable result.

## First-order stability

But does $\langle {W}^{k}\rangle$ converge, or under what conditions?

Let's rewrite the analysis in term of $\langle {V}^{k}\rangle$ , the "mean coefficient error vector" $\langle {V}^{k}\rangle =\langle {W}^{k}\rangle -{W}_{\mathrm{opt}}$ , where ${W}_{\mathrm{opt}}$ is the Wiener filter $\langle {W}^{k+1}\rangle =\langle {W}^{k}\rangle -2R\langle {W}^{k}\rangle +2P$ $\langle {W}^{k+1}\rangle -{W}_{\mathrm{opt}}=\langle {W}^{k}\rangle -{W}_{\mathrm{opt}}+-(2R\langle {W}^{k}\rangle )+2R{W}_{\mathrm{opt}}-2R{W}_{\mathrm{opt}}+2P$ $\langle {V}^{k+1}\rangle =\langle {V}^{k}\rangle -2R\langle {V}^{k}\rangle +-(2R{W}_{\mathrm{opt}})+2P$ Now ${W}_{\mathrm{opt}}=R^{(-1)}$ , so $\langle {V}^{k+1}\rangle =\langle {V}^{k}\rangle -2R\langle {V}^{k}\rangle +-(2RR^{(-1)}P)+2P=(I-2R)\langle {V}^{k}\rangle$ We wish to know under what conditions $\langle {V}^{k}\rangle \to \langle 0\rangle$ ?

## Linear algebra fact

Since $R$ is positive definite, real, and symmetric, all the eigenvalues arereal and positive. Also, we can write $R$ as $(Q^{(-1)}, Q)$ , where  is a diagonal matrix with diagonal entries ${}_{i}$ equal to the eigenvalues of $R$ , and $Q$ is a unitary matrix with rows equal to the eigenvectors corresponding to theeigenvalues of $R$ .

Using this fact, ${V}^{k+1}=(I-2(Q^{(-1)}, Q)){V}^{k}$ multiplying both sides through on the left by $Q$ : we get $Q\langle {V}^{k+1}\rangle =(Q-2Q)\langle {V}^{k}\rangle =(1-2)Q\langle {V}^{k}\rangle$ Let ${V}^{\text{'}}=QV$ : ${V}^{\text{'}k+1}=(1-2){V}^{\text{'}k}$ Note that ${V}^{\text{'}}$ is simply $V$ in a rotated coordinate set in $^{m}$ , so convergence of ${V}^{\text{'}}$ implies convergence of $V$ .

Since $1-2$ is diagonal, all elements of ${V}^{\text{'}}$ evolve independently of each other. Convergence (stability) bolis down to whether all $M$ of these scalar, first-order difference equations are stable, and thus $(0)$ . $\forall i, i$

1 2 M
V i ' k + 1 1 2 i V i ' k These equations converge to zero if $\left|1-2{}_{i}\right|< 1$ , or $\forall i\colon \left|{}_{i}\right|< 1$  and ${}_{i}$ are positive, so we require $\forall i\colon < \frac{1}{{}_{i}}$ so for convergence in the mean of the LMS adaptive filter, we require
$< \frac{1}{{}_{\mathrm{max}}}$
This is an elegant theoretical result, but in practice, we may not know ${}_{\mathrm{max}}$ , it may be time-varying, and we certainly won't want to compute it. However, another useful mathematicalfact comes to the rescue... $\mathrm{tr}(R)=\sum_{i=1}^{M} {r}_{\mathrm{ii}}=\sum_{i=1}^{M} {}_{i}\ge {}_{\mathrm{max}}$ Since the eigenvalues are all positive and real.

For a correlation matrix, $\forall i, i\in \{1, M\}\colon {r}_{\mathrm{ii}}=r(0)$ . So $\mathrm{tr}(R)=Mr(0)=M({x}_{k}{x}_{k})$ . We can easily estimate $r(0)$ with $O(1)$ computations/sample, so in practice we might require $< \frac{1}{M(r(0))}$ as a conservative bound, and perhaps adapt  accordingly with time.

## Rate of convergence

Each of the modes decays as $(1-2{}_{i})^{k}$

The initial rate of convergence is dominated by the fastest mode $1-2{}_{\mathrm{max}}$ . This is not surprising, since a dradient descent method goes "downhill" in the steepest direction
The final rate of convergence is dominated by the slowest mode $1-2{}_{\mathrm{min}}$ . For small ${}_{\mathrm{min}}$ , it can take a long time for LMS to converge.
Note that the convergence behavior depends on the data (via $R$ ). LMS converges relatively quickly for roughly equal eigenvalues. Unequaleigenvalues slow LMS down a lot.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!