Least mean squares filter

Least mean squares (LMS) algorithms are a class of adaptive filter used to mimic a desired filter by finding the filter coefficients that relate to producing the least mean squares of the error signal (difference between the desired and the actual signal). It is a stochastic gradient descent method in that the filter is only adapted based on the error at the current time. It was invented in 1960 by Stanford University professor Bernard Widrow and his first Ph.D. student, Ted Hoff.

Problem formulation

Relationship to the least mean squares filter

The realization of the causal Wiener filter looks a lot like the solution to the least squares estimate, except in the signal processing domain. The least squares solution, for input matrix \scriptstyle\mathbf{X} and output vector \scriptstyle\mathbf{y} is

   \boldsymbol{\hat\beta} = (\mathbf{X} ^\mathbf{T}\mathbf{X})^{-1}\mathbf{X}^{\mathbf{T}}\boldsymbol y .

The FIR Wiener filter is related to the least mean squares filter, but minimizing its error criterion does not rely on cross-correlations or auto-correlations. Its solution converges to the Wiener filter solution. Most linear adaptive filtering problems can be formulated using the block diagram above. That is, an unknown system $\mathbf{h}(n)$ is to be identified and the adaptive filter attempts to adapt the filter $\hat{\mathbf{h}}(n)$ to make it as close as possible to $\mathbf{h}(n)$, while using only observable signals x(n), d(n) and e(n); but y(n), v(n) and h(n) are not directly observable. Its solution is closely related to the Wiener filter.

definition of symbols

$\mathbf{x}(n) = \left[x(n), x(n-1), \dots, x(n-p+1)\right]^T$

$\mathbf{h}(n) = \left[h_0(n), h_1(n), \dots, h_{p-1}(n)\right]^T,\quad \mathbf{h}(n) \in \mathbb{C}^p$

$y(n) = \mathbf{h}^H(n) \cdot \mathbf{x}(n)$

d(n) = y(n) + ν(n)

$e(n) = d(n) - \hat{y}(n) = d(n) - \hat{\mathbf{h}}^H(n) \cdot \mathbf{x}(n)$

Idea

The idea behind LMS filters is to use steepest descent to find filter weights $\mathbf{h}(n)$ which minimize a cost function. We start by defining the cost function as

$C(n) = E\left\{|e(n)|^{2}\right\}$

where e(n) is the error at the current sample 'n' and E{.} denotes the expected value.

This cost function (C(n)) is the mean square error, and it is minimized by the LMS. This is where the LMS gets its name. Applying steepest descent means to take the partial derivatives with respect to the individual entries of the filter coefficient (weight) vector

$\nabla_{\hat{\mathbf{h}}^H} C(n) = \nabla_{\hat{\mathbf{h}}^H} E\left\{e(n) \, e^{*}(n)\right\}=2E\left\{\nabla_{\hat{\mathbf{h}}^H} ( e(n)) \, e^{*}(n) \right\}$

where $\nabla$ is the gradient operator.

$\nabla_{\hat{\mathbf{h}}^H} (e(n))= \nabla_{\hat{\mathbf{h}}^H} \left(d(n) - \hat{\mathbf{h}}^H \cdot \mathbf{x}(n)\right)=-\mathbf{x}(n)$

$\nabla C(n) = -2E\left\{\mathbf{x}(n) \, e^{*}(n)\right\}$

Now, $\nabla C(n)$ is a vector which points towards the steepest ascent of the cost function. To find the minimum of the cost function we need to take a step in the opposite direction of $\nabla C(n)$. To express that in mathematical terms

$\hat{\mathbf{h}}(n+1)=\hat{\mathbf{h}}(n)-\frac{\mu}{2} \nabla C(n)=\hat{\mathbf{h}}(n)+\mu \, E\left\{\mathbf{x}(n) \, e^{*}(n)\right\}$

where $\frac{\mu}{2}$ is the step size(adaptation constant). That means we have found a sequential update algorithm which minimizes the cost function. Unfortunately, this algorithm is not realizable until we know $E\left\{\mathbf{x}(n) \, e^{*}(n)\right\}$.

Generally, the expectation above is not computed. Instead, to run the LMS in an online (updating after each new sample is received) environment, we use an instantaneous estimate of that expectation. See below.

Simplifications

For most systems the expectation function ${E}\left\{\mathbf{x}(n) \, e^{*}(n)\right\}$ must be approximated. This can be done with the following unbiased estimator

$\hat{E}\left\{\mathbf{x}(n) \, e^{*}(n)\right\}=\frac{1}{N}\sum_{i=0}^{N-1}\mathbf{x}(n-i) \, e^{*}(n-i)$

where N indicates the number of samples we use for that estimate. The simplest case is N = 1

$\hat{E}\left\{\mathbf{x}(n) \, e^{*}(n)\right\}=\mathbf{x}(n) \, e^{*}(n)$

For that simple case the update algorithm follows as

$\hat{\mathbf{h}}(n+1)=\hat{\mathbf{h}}(n)+\mu \mathbf{x}(n) \, e^{*}(n)$

Indeed this constitutes the update algorithm for the LMS filter.

LMS algorithm summary

The LMS algorithm for a pth order algorithm can be summarized as

Parameters:	p = filter order
	μ = step size
Initialisation:	$\hat{\mathbf{h}}(0)=0$
Computation:	For n = 0,1,2,...
	$\mathbf{x}(n) = \left[x(n), x(n-1), \dots, x(n-p+1)\right]^T$
	$e(n) = d(n)-\hat{\mathbf{h}}^{H}(n)\mathbf{x}(n)$
	$\hat{\mathbf{h}}(n+1) = \hat{\mathbf{h}}(n)+\mu\,e^{*}(n)\mathbf{x}(n)$

where $\hat{\mathbf{h}}^{H}(n)$ denotes the Hermitian transpose of $\hat{\mathbf{h}}(n)$.

Convergence and stability in the mean

Assume that the true filter $\mathbf{h}(n)=\mathbf{h}$ is constant, and that the input signal x(n) is wide-sense stationary. Then $E\{\hat{\mathbf{h}}(n)\}$ converges to $\mathbf{h}$ as $n\rightarrow \infty$ if and only if

$0<\mu<\frac{2}{\lambda_{\mathrm{max}}},$

where λ_max is the greatest eigenvalue of the autocorrelation matrix $R=E\{\mathbf{x}(n)\mathbf{x}^H(n)\}$. If this condition is not fulfilled, the algorithm becomes unstable and $\hat{\mathbf{h}}(n)$ diverges.

Maximum convergence speed is achieved when

$\mu=\frac{2}{\lambda_{\mathrm{max}}+\lambda_{\mathrm{min}}},$

where λ_min is the smallest eigenvalue of R. Given that μ is less than or equal to this optimum, the convergence speed is determined by μλ_min, with a larger value yielding faster convergence. This means that faster convergence can be achieved when λ_max is close to λ_min, that is, the maximum achievable convergence speed depends on the eigenvalue spread of R.

A white noise signal has autocorrelation matrix R = σ²I, where σ² is the variance of the signal. In this case all eigenvalues are equal, and the eigenvalue spread is the minimum over all possible matrices. The common interpretation of this result is therefore that the LMS converges quickly for white input signals, and slowly for colored input signals, such as processes with low-pass or high-pass characteristics.

It is important to note that the above upperbound on μ only enforces stability in the mean, but the coefficients of $\hat{\mathbf{h}}(n)$ can still grow infinitely large, i.e. divergence of the coefficients is still possible. A more practical bound is

$0<\mu<\frac{2}{tr\left[R\right]},$

where $tr\left[R\right]$ denotes the trace of R. This bound guarantees that the coefficients of $\hat{\mathbf{h}}(n)$ do not diverge (in practice, the value of μ should not be chosen close to this upper bound, since it is somewhat optimistic due to approximations and assumptions made in the derivation of the bound).

Normalised least mean squares filter (NLMS)

The main drawback of the "pure" LMS algorithm is that it is sensitive to the scaling of its input x(n). This makes it very hard (if not impossible) to choose a learning rate μ that guarantees stability of the algorithm (Haykin 2002). The Normalised least mean squares filter (NLMS) is a variant of the LMS algorithm that solves this problem by normalising with the power of the input. The NLMS algorithm can be summarised as:

Parameters:	p = filter order
	μ = step size
Initialization:	$\hat{\mathbf{h}}(0)=0$
Computation:	For n = 0,1,2,...
	$\mathbf{x}(n) = \left[x(n), x(n-1), \dots, x(n-p+1)\right]^T$
	$e(n) = d(n)-\hat{\mathbf{h}}^{H}(n)\mathbf{x}(n)$
	$\hat{\mathbf{h}}(n+1) = \hat{\mathbf{h}}(n)+\frac{\mu\,e^{*}(n)\mathbf{x}(n)}{\mathbf{x}^H(n)\mathbf{x}(n)}$

Optimal learning rate

It can be shown that if there is no interference (v(n) = 0), then the optimal learning rate for the NLMS algorithm is

μ_opt = 1

and is independent of the input x(n) and the real (unknown) impulse response $\mathbf{h}(n)$. In the general case with interference ($v(n) \ne 0$), the optimal learning rate is

$\mu_{opt}=\frac{E\left[\left|y(n)-\hat{y}(n)\right|^2\right]}{E\left[|e(n)|^2\right]}$

The results above assume that the signals v(n) and x(n) are uncorrelated to each other, which is generally the case in practice.

Proof

Let the filter misalignment be defined as $\Lambda(n) = \left| \mathbf{h}(n) - \hat{\mathbf{h}}(n) \right|^2$, we can derive the expected misalignment for the next sample as:

$E\left[ \Lambda(n+1) \right] = E\left[ \left| \hat{\mathbf{h}}(n) + \frac{\mu\,e^{*}(n)\mathbf{x}(n)}{\mathbf{x}^H(n)\mathbf{x}(n)} - \mathbf{h}(n) \right|^2 \right]$

$E\left[ \Lambda(n+1) \right] = E\left[ \left| \hat{\mathbf{h}}(n) + \frac{\mu\, \left( v^*(n)+y^*(n)-\hat{y}^*(n) \right) \mathbf{x}(n)}{\mathbf{x}^H(n)\mathbf{x}(n)} - \mathbf{h}(n) \right|^2 \right]$

Let $\mathbf{\delta}=\hat{\mathbf{h}}(n)-\mathbf{h}(n)$ and $r(n) = \hat{y}(n)-y(n)$

$E\left[ \Lambda(n+1) \right] = E\left[ \left| \mathbf{\delta}(n) - \frac{\mu\, \left( v(n)+r(n) \right) \mathbf{x}(n)}{\mathbf{x}^H(n)\mathbf{x}(n)} \right|^2 \right]$

$E\left[ \Lambda(n+1) \right] = E\left[ \left( \mathbf{\delta}(n) - \frac{\mu\, \left( v(n)+r(n) \right) \mathbf{x}(n)}{\mathbf{x}^H(n)\mathbf{x}(n)} \right)^H \left( \mathbf{\delta}(n) - \frac{\mu\, \left( v(n)+r(n) \right) \mathbf{x}(n)}{\mathbf{x}^H(n)\mathbf{x}(n)} \right) \right]$

Assuming independence, we have:

$E\left[ \Lambda(n+1) \right] = \Lambda(n) + E\left[ \left( \frac{\mu\, \left( v(n)-r(n) \right) \mathbf{x}(n)}{\mathbf{x}^H(n)\mathbf{x}(n)} \right)^H \left( \frac{\mu\, \left( v(n)-r(n) \right) \mathbf{x}(n)}{\mathbf{x}^H(n)\mathbf{x}(n)} \right) \right] - 2 E\left[\frac{\mu|r(n)|^2}{\mathbf{x}^H(n)\mathbf{x}(n)}\right]$

$E\left[ \Lambda(n+1) \right] = \Lambda(n) + \frac{\mu^2 E\left[|e(n)|^2\right]}{\mathbf{x}^H(n)\mathbf{x}(n)} - \frac{2 \mu E\left[|r(n)|^2\right]}{\mathbf{x}^H(n)\mathbf{x}(n)}$

The optimal learning rate is found at $\frac{dE\left[ \Lambda(n+1) \right]}{d\mu} = 0$, which leads to:

$2 \mu E\left[|e(n)|^2\right] - 2 E\left[|r(n)|^2\right] = 0$

$\mu = \frac{E\left[|r(n)|^2\right]}{E\left[|e(n)|^2\right]}$

See also

• Recursive least squares
• For statistical techniques relevant to LMS filter see Least squares.
• Similarities between Wiener and LMS
• Multidelay block frequency domain adaptive filter
• Kernel adaptive filter

References

• Monson H. Hayes: Statistical Digital Signal Processing and Modeling, Wiley, 1996, ISBN 0-471-59431-8
• Simon Haykin: Adaptive Filter Theory, Prentice Hall, 2002, ISBN 0-13-048434-2
• Simon S. Haykin, Bernard Widrow (Editor): Least-Mean-Square Adaptive Filters, Wiley, 2003, ISBN 0-471-21570-8
• Bernard Widrow, Samuel D. Stearns: Adaptive Signal Processing, Prentice Hall, 1985, ISBN 0-13-004029-0
• Weifeng Liu, Jose Principe and Simon Haykin: Kernel Adaptive Filtering: A Comprehensive Introduction, John Wiley, 2010, ISBN 0470447532
• Paulo S.R. Diniz: Adaptive Filtering: Algorithms and Practical Implementation, Kluwer Academic Publishers, 1997, ISBN 0-7923-9912-9

External links

• LMS Algorithm in Adaptive Antenna Arrays www.antenna-theory.com
• LMS Noise cancellation demo www.advsolned.com