Recursive least squares (RLS) is anadaptive filter algorithm that recursively finds the coefficients that minimize aweighted linear least squarescost function relating to the input signals. This approach is in contrast to other algorithms such as theleast mean squares (LMS) that aim to reduce themean square error. In the derivation of the RLS, the input signals are considereddeterministic, while for the LMS and similar algorithms they are consideredstochastic. Compared to most of its competitors, the RLS exhibits extremely fast convergence. However, this benefit comes at the cost of high computational complexity.

RLS was discovered byGauss but lay unused or ignored until 1950 when Plackett rediscovered the original work of Gauss from 1821. In general, the RLS can be used to solve any problem that can be solved byadaptive filters. For example, suppose that a signal${\displaystyle d(n)}$ is transmitted over an echoey,noisy channel that causes it to be received as

${\displaystyle x(n)=\sum _{k=0}^q{b}_n(k)d(n-k)+v(n)}$

where${\displaystyle v(n)}$ representsadditive noise. The intent of the RLS filter is to recover the desired signal${\displaystyle d(n)}$ by use of a${\displaystyle p+1}$-tapFIR filter,${\displaystyle \mathbf{w}}$:

${\displaystyle d(n)\approx \sum _{k=0}^{p}w(k)x(n-k)={\mathbf{w}}^{\mathit{T}}{\mathbf{x}}_n}$

where${\displaystyle {\mathbf{x}}_n=[x(n)x(n-1)\dots x(n-p){]}^T}$ is thecolumn vector containing the${\displaystyle p+1}$ most recent samples of${\displaystyle x(n)}$. The estimate of the recovered desired signal is

${\displaystyle \hat{d}(n)=\sum _{k=0}^p{w}_n(k)x(n-k)={\mathbf{w}}_n^{\mathit{T}}{\mathbf{x}}_n}$

The goal is to estimate the parameters of the filter${\displaystyle \mathbf{w}}$, and at each time${\displaystyle n}$ we refer to the current estimate as${\displaystyle {\mathbf{w}}_n}$ and the adapted least-squares estimate by${\displaystyle {\mathbf{w}}_{n+1}}$.${\displaystyle {\mathbf{w}}_n}$ is also a column vector, as shown below, and thetranspose,${\displaystyle {\mathbf{w}}_n^{\mathit{T}}}$, is arow vector. Thematrix product${\displaystyle {\mathbf{w}}_n^{\mathit{T}}{\mathbf{x}}_n}$ (which is thedot product of${\displaystyle {\mathbf{w}}_n}$ and${\displaystyle {\mathbf{x}}_n}$) is${\displaystyle \hat{d}(n)}$, a scalar. The estimate is"good" if${\displaystyle \hat{d}(n)-d(n)}$ is small in magnitude in someleast squares sense.

As time evolves, it is desired to avoid completely redoing the least squares algorithm to find the new estimate for${\displaystyle {\mathbf{w}}_{n+1}}$, in terms of${\displaystyle {\mathbf{w}}_n}$.

The benefit of the RLS algorithm is that there is no need to invert matrices, thereby saving computational cost. Another advantage is that it provides intuition behind such results as theKalman filter.

The idea behind RLS filters is to minimize acost function${\displaystyle C}$ by appropriately selecting the filter coefficients${\displaystyle {\mathbf{w}}_n}$, updating the filter as new data arrives. The error signal${\displaystyle e(n)}$ and desired signal${\displaystyle d(n)}$ are defined in thenegative feedback diagram below:

The error implicitly depends on the filter coefficients through the estimate${\displaystyle \hat{d}(n)}$:

${\displaystyle e(n)=d(n)-\hat{d}(n)}$

The weighted least squares error function${\displaystyle C}$—the cost function we desire to minimize—being a function of${\displaystyle e(n)}$ is therefore also dependent on the filter coefficients:

${\displaystyle C({\mathbf{w}}_n)=\sum _{i=0}^n{\lambda }^{n-i}{e}^2(i)}$

where is the "forgetting factor" which gives exponentially less weight to older error samples.

The cost function is minimized by taking the partial derivatives for all entries${\displaystyle k}$ of the coefficient vector${\displaystyle {\mathbf{w}}_n}$ and setting the results to zero

${\displaystyle \frac{\mathrm{\partial }C({\mathbf{w}}_n)}{\mathrm{\partial }{w}_n(k)}=\sum _{i=0}^{n}2{\lambda }^{n-i}e(i)\cdot \frac{\mathrm{\partial }e(i)}{\mathrm{\partial }{w}_n(k)}=-\sum _{i=0}^{n}2{\lambda }^{n-i}e(i)x(i-k)=0k=0,1,\dots ,p}$

Next, replace${\displaystyle e(n)}$ with the definition of the error signal

${\displaystyle \sum _{i=0}^n{\lambda }^{n-i}\left[d(i)-\sum _{\ell =0}^p{w}_n(\ell )x(i-\ell )\right]x(i-k)=0k=0,1,\dots ,p}$

Rearranging the equation yields

${\displaystyle \sum _{\ell =0}^p{w}_n(\ell )\left[\sum _{i=0}^n{\lambda }^{n-i}x(i-\ell )x(i-k)\right]=\sum _{i=0}^n{\lambda }^{n-i}d(i)x(i-k)k=0,1,\dots ,p}$

This form can be expressed in terms of matrices

${\displaystyle {\mathbf{R}}_x(n){\mathbf{w}}_n={\mathbf{r}}_{dx}(n)}$

where${\displaystyle {\mathbf{R}}_x(n)}$ is the weightedsample covariance matrix for${\displaystyle x(n)}$, and${\displaystyle {\mathbf{r}}_{dx}(n)}$ is the equivalent estimate for thecross-covariance between${\displaystyle d(n)}$ and${\displaystyle x(n)}$. Based on this expression we find the coefficients which minimize the cost function as

${\displaystyle {\mathbf{w}}_n={\mathbf{R}}_x^{-1}(n){\mathbf{r}}_{dx}(n)}$

This is the main result of the discussion.

The smaller${\displaystyle \lambda }$ is, the smaller is the contribution of previous samples to thecovariance matrix. This makes the filtermore sensitive to recent samples, which means more fluctuations in the filter co-efficients. The${\displaystyle \lambda =1}$ case is referred to as thegrowing window RLS algorithm. In practice,${\displaystyle \lambda }$ is usually chosen between 0.98 and 1.^[1] By using type-IImaximum likelihood estimation the optimal${\displaystyle \lambda }$ can be estimated from a set of data.^[2]

The discussion resulted in a single equation to determine a coefficient vector which minimizes the cost function. In this section we want to derive a recursive solution of the form

${\displaystyle {\mathbf{w}}_n={\mathbf{w}}_{n-1}+\mathrm{\Delta }{\mathbf{w}}_{n-1}}$

where${\displaystyle \mathrm{\Delta }{\mathbf{w}}_{n-1}}$ is a correction factor at time${\displaystyle n-1}$. We start the derivation of the recursive algorithm by expressing the cross covariance${\displaystyle {\mathbf{r}}_{dx}(n)}$ in terms of${\displaystyle {\mathbf{r}}_{dx}(n-1)}$

${\displaystyle {\mathbf{r}}_{dx}(n)}$	${\displaystyle =\sum _{i=0}^n{\lambda }^{n-i}d(i)\mathbf{x}(i)}$
	${\displaystyle =\sum _{i=0}^{n-1}{\lambda }^{n-i}d(i)\mathbf{x}(i)+{\lambda }^{0}d(n)\mathbf{x}(n)}$
	${\displaystyle =\lambda {\mathbf{r}}_{dx}(n-1)+d(n)\mathbf{x}(n)}$

where${\displaystyle \mathbf{x}(i)}$ is the${\displaystyle p+1}$ dimensional data vector

${\displaystyle \mathbf{x}(i)=[x(i),x(i-1),\dots ,x(i-p){]}^T}$

Similarly we express${\displaystyle {\mathbf{R}}_x(n)}$ in terms of${\displaystyle {\mathbf{R}}_x(n-1)}$ by

${\displaystyle {\mathbf{R}}_x(n)}$	${\displaystyle =\sum _{i=0}^n{\lambda }^{n-i}\mathbf{x}(i){\mathbf{x}}^T(i)}$
	${\displaystyle =\lambda {\mathbf{R}}_x(n-1)+\mathbf{x}(n){\mathbf{x}}^T(n)}$

In order to generate the coefficient vector we are interested in the inverse of the deterministic auto-covariance matrix. For that task theWoodbury matrix identity comes in handy. With

${\displaystyle A}$	${\displaystyle =\lambda {\mathbf{R}}_x(n-1)}$ is${\displaystyle (p+1)}$-by-${\displaystyle (p+1)}$
${\displaystyle U}$	${\displaystyle =\mathbf{x}(n)}$ is${\displaystyle (p+1)}$-by-1 (column vector)
${\displaystyle V}$	${\displaystyle ={\mathbf{x}}^T(n)}$ is 1-by-${\displaystyle (p+1)}$ (row vector)
${\displaystyle C}$	${\displaystyle ={\mathbf{I}}_1}$ is the 1-by-1identity matrix

The Woodbury matrix identity follows

${\displaystyle {\mathbf{R}}_x^{-1}(n)}$	${\displaystyle =}$	${\displaystyle {\left[\lambda {\mathbf{R}}_x(n-1)+\mathbf{x}(n){\mathbf{x}}^T(n)\right]}^{-1}}$
	${\displaystyle =}$	${\displaystyle {\displaystyle \frac{1}{\lambda }}\left\{{\mathbf{R}}_x^{-1}(n-1)-{\displaystyle \frac{{\mathbf{R}}_x^{-1}(n-1)\mathbf{x}(n){\mathbf{x}}^T(n){\mathbf{R}}_x^{-1}(n-1)}{\lambda +{\mathbf{x}}^T(n){\mathbf{R}}_x^{-1}(n-1)\mathbf{x}(n)}}\right\}}$
		${\displaystyle }$

To come in line with the standard literature, we define

${\displaystyle \mathbf{P}(n)}$	${\displaystyle ={\mathbf{R}}_x^{-1}(n)}$
	${\displaystyle ={\lambda }^{-1}\mathbf{P}(n-1)-\mathbf{g}(n){\mathbf{x}}^T(n){\lambda }^{-1}\mathbf{P}(n-1)}$

where thegain vector${\displaystyle g(n)}$ is

${\displaystyle \mathbf{g}(n)}$	${\displaystyle ={\lambda }^{-1}\mathbf{P}(n-1)\mathbf{x}(n){\left\{1+{\mathbf{x}}^T(n){\lambda }^{-1}\mathbf{P}(n-1)\mathbf{x}(n)\right\}}^{-1}}$
	${\displaystyle =\mathbf{P}(n-1)\mathbf{x}(n){\left\{\lambda +{\mathbf{x}}^T(n)\mathbf{P}(n-1)\mathbf{x}(n)\right\}}^{-1}}$

Before we move on, it is necessary to bring${\displaystyle \mathbf{g}(n)}$ into another form

${\displaystyle \mathbf{g}(n)\left\{1+{\mathbf{x}}^T(n){\lambda }^{-1}\mathbf{P}(n-1)\mathbf{x}(n)\right\}}$	${\displaystyle ={\lambda }^{-1}\mathbf{P}(n-1)\mathbf{x}(n)}$
${\displaystyle \mathbf{g}(n)+\mathbf{g}(n){\mathbf{x}}^T(n){\lambda }^{-1}\mathbf{P}(n-1)\mathbf{x}(n)}$	${\displaystyle ={\lambda }^{-1}\mathbf{P}(n-1)\mathbf{x}(n)}$

Subtracting the second term on the left side yields

${\displaystyle \mathbf{g}(n)}$	${\displaystyle ={\lambda }^{-1}\mathbf{P}(n-1)\mathbf{x}(n)-\mathbf{g}(n){\mathbf{x}}^T(n){\lambda }^{-1}\mathbf{P}(n-1)\mathbf{x}(n)}$
	${\displaystyle ={\lambda }^{-1}\left[\mathbf{P}(n-1)-\mathbf{g}(n){\mathbf{x}}^T(n)\mathbf{P}(n-1)\right]\mathbf{x}(n)}$

With therecursive definition of${\displaystyle \mathbf{P}(n)}$ the desired form follows

${\displaystyle \mathbf{g}(n)=\mathbf{P}(n)\mathbf{x}(n)}$

Now we are ready to complete the recursion. As discussed

${\displaystyle {\mathbf{w}}_n}$	${\displaystyle =\mathbf{P}(n){\mathbf{r}}_{dx}(n)}$
	${\displaystyle =\lambda \mathbf{P}(n){\mathbf{r}}_{dx}(n-1)+d(n)\mathbf{P}(n)\mathbf{x}(n)}$

The second step follows from the recursive definition of${\displaystyle {\mathbf{r}}_{dx}(n)}$. Next we incorporate the recursive definition of${\displaystyle \mathbf{P}(n)}$ together with the alternate form of${\displaystyle \mathbf{g}(n)}$ and get

${\displaystyle {\mathbf{w}}_n}$	${\displaystyle =\lambda \left[{\lambda }^{-1}\mathbf{P}(n-1)-\mathbf{g}(n){\mathbf{x}}^T(n){\lambda }^{-1}\mathbf{P}(n-1)\right]{\mathbf{r}}_{dx}(n-1)+d(n)\mathbf{g}(n)}$
	${\displaystyle =\mathbf{P}(n-1){\mathbf{r}}_{dx}(n-1)-\mathbf{g}(n){\mathbf{x}}^T(n)\mathbf{P}(n-1){\mathbf{r}}_{dx}(n-1)+d(n)\mathbf{g}(n)}$
	${\displaystyle =\mathbf{P}(n-1){\mathbf{r}}_{dx}(n-1)+\mathbf{g}(n)\left[d(n)-{\mathbf{x}}^T(n)\mathbf{P}(n-1){\mathbf{r}}_{dx}(n-1)\right]}$

With${\displaystyle {\mathbf{w}}_{n-1}=\mathbf{P}(n-1){\mathbf{r}}_{dx}(n-1)}$ we arrive at the update equation

${\displaystyle {\mathbf{w}}_n}$	${\displaystyle ={\mathbf{w}}_{n-1}+\mathbf{g}(n)\left[d(n)-{\mathbf{x}}^T(n){\mathbf{w}}_{n-1}\right]}$
	${\displaystyle ={\mathbf{w}}_{n-1}+\mathbf{g}(n)\alpha (n)}$

where${\displaystyle \alpha (n)=d(n)-{\mathbf{x}}^T(n){\mathbf{w}}_{n-1}}$is thea priori error. Compare this with thea posteriori error; the error calculatedafter the filter is updated:

${\displaystyle e(n)=d(n)-{\mathbf{x}}^T(n){\mathbf{w}}_n}$

That means we found the correction factor

${\displaystyle \mathrm{\Delta }{\mathbf{w}}_{n-1}=\mathbf{g}(n)\alpha (n)}$

This intuitively satisfying result indicates that the correction factor is directly proportional to both the error and the gain vector, which controls how much sensitivity is desired, through the weighting factor,${\displaystyle \lambda }$.

The RLS algorithm for ap-th order RLS filter can be summarized as

Parameters:	${\displaystyle p=}$ filter order
	${\displaystyle \lambda =}$ forgetting factor
	${\displaystyle \delta =}$ value to initialize${\displaystyle \mathbf{P}(0)}$
Initialization:	${\displaystyle \mathbf{w}(0)=0}$,
	${\displaystyle x(k)=0,k=-p,\dots ,-1}$,
	${\displaystyle d(k)=0,k=-p,\dots ,-1}$
	${\displaystyle \mathbf{P}(0)=\delta I}$ where${\displaystyle I}$ is theidentity matrix of rank${\displaystyle p+1}$
Computation:	For${\displaystyle n=1,2,\dots }$
	${\displaystyle \mathbf{x}(n)=\left[\begin{array}{c}x(n)\\ x(n-1)\\ ⋮\\ x(n-p)\end{array}\right]}$
	${\displaystyle \alpha (n)=d(n)-{\mathbf{x}}^T(n)\mathbf{w}(n-1)}$
	${\displaystyle \mathbf{g}(n)=\mathbf{P}(n-1)\mathbf{x}(n){\left\{\lambda +{\mathbf{x}}^T(n)\mathbf{P}(n-1)\mathbf{x}(n)\right\}}^{-1}}$
	${\displaystyle \mathbf{P}(n)={\lambda }^{-1}\mathbf{P}(n-1)-\mathbf{g}(n){\mathbf{x}}^T(n){\lambda }^{-1}\mathbf{P}(n-1)}$
	${\displaystyle \mathbf{w}(n)=\mathbf{w}(n-1)+\alpha (n)\mathbf{g}(n)}$.

The recursion for${\displaystyle P}$ follows analgebraic Riccati equation and thus draws parallels to theKalman filter.^[3]

Thelattice recursive least squaresadaptive filter is related to the standard RLS except that it requires fewer arithmetic operations (orderN).^[4] It offers additional advantages over conventional LMS algorithms such as faster convergence rates, modular structure, and insensitivity to variations in eigenvalue spread of the input correlation matrix. The LRLS algorithm described is based ona posteriori errors and includes the normalized form. The derivation is similar to the standard RLS algorithm and is based on the definition of${\displaystyle d(k)}$. In the forward prediction case, we have${\displaystyle d(k)=x(k)}$ with the input signal${\displaystyle x(k-1)}$ as the most up to date sample. The backward prediction case is${\displaystyle d(k)=x(k-i-1)}$, where i is the index of the sample in the past we want to predict, and the input signal${\displaystyle x(k)}$ is the most recent sample.^[5]

${\displaystyle {\kappa }_f(k,i)}$ is the forward reflection coefficient

${\displaystyle {\kappa }_b(k,i)}$ is the backward reflection coefficient

${\displaystyle e_f(k,i)}$ represents the instantaneousa posteriori forward prediction error

${\displaystyle e_b(k,i)}$ represents the instantaneousa posteriori backward prediction error

${\displaystyle {\xi }_{b_{min}}^d(k,i)}$ is the minimum least-squares backward prediction error

${\displaystyle {\xi }_{f_{min}}^d(k,i)}$ is the minimum least-squares forward prediction error

${\displaystyle \gamma (k,i)}$ is a conversion factor betweena priori anda posteriori errors

${\displaystyle v_i(k)}$ are the feedforward multiplier coefficients.

${\displaystyle \epsilon }$ is a small positive constant that can be 0.01

The algorithm for a LRLS filter can be summarized as

Initialization:
	For$i=0,1,\dots ,N$
	${\displaystyle \delta (-1,i)={\delta }_D(-1,i)=0}$ (if$x(k)=0$ for)
	${\displaystyle {\xi }_{b_{min}}^d(-1,i)={\xi }_{f_{min}}^d(-1,i)=\epsilon }$
	${\displaystyle \gamma (-1,i)=1}$
	${\displaystyle e_b(-1,i)=0}$
	End
Computation:
	For$k\ge 0$
	${\displaystyle \gamma (k,0)=1}$
	${\displaystyle e_b(k,0)=e_f(k,0)=x(k)}$
	${\displaystyle {\xi }_{b_{min}}^d(k,0)={\xi }_{f_{min}}^d(k,0)=x^2(k)+\lambda {\xi }_{f_{min}}^d(k-1,0)}$
	${\displaystyle e(k,0)=d(k)}$
	For$i=0,1,\dots ,N$
	${\displaystyle \delta (k,i)=\lambda \delta (k-1,i)+\frac{e_b(k-1,i)e_f(k,i)}{\gamma (k-1,i)}}$
	${\displaystyle \gamma (k,i+1)=\gamma (k,i)-\frac{e_b^2(k,i)}{{\xi }_{b_{min}}^d(k,i)}}$
	${\displaystyle {\kappa }_b(k,i)=\frac{\delta (k,i)}{{\xi }_{f_{min}}^d(k,i)}}$
	${\displaystyle {\kappa }_f(k,i)=\frac{\delta (k,i)}{{\xi }_{b_{min}}^d(k-1,i)}}$
	${\displaystyle e_b(k,i+1)=e_b(k-1,i)-{\kappa }_b(k,i)e_f(k,i)}$
	${\displaystyle e_f(k,i+1)=e_f(k,i)-{\kappa }_f(k,i)e_b(k-1,i)}$
	${\displaystyle {\xi }_{b_{min}}^d(k,i+1)={\xi }_{b_{min}}^d(k-1,i)-\delta (k,i){\kappa }_b(k,i)}$
	${\displaystyle {\xi }_{f_{min}}^d(k,i+1)={\xi }_{f_{min}}^d(k,i)-\delta (k,i){\kappa }_f(k,i)}$
	Feedforward filtering
	${\displaystyle {\delta }_D(k,i)=\lambda {\delta }_D(k-1,i)+\frac{e(k,i)e_b(k,i)}{\gamma (k,i)}}$
	${\displaystyle v_i(k)=\frac{{\delta }_D(k,i)}{{\xi }_{b_{min}}^d(k,i)}}$
	${\displaystyle e(k,i+1)=e(k,i)-v_i(k)e_b(k,i)}$
	End
	End

The normalized form of the LRLS has fewer recursions and variables. It can be calculated by applying a normalization to the internal variables of the algorithm which will keep their magnitude bounded by one. This is generally not used in real-time applications because of the number of division and square-root operations which comes with a high computational load.

The algorithm for a NLRLS filter can be summarized as

Initialization:
	For$i=0,1,\dots ,N.$
	${\displaystyle \overline{\delta }(-1,i)=0}$ (if$x(k)=d(k)=0$ for)
	${\displaystyle {\overline{\delta }}_D(-1,i)=0}$
	${\displaystyle {\overline{e}}_b(-1,i)=0}$
	End
	${\displaystyle {\sigma }_x^2(-1)=\lambda {\sigma }_d^2(-1)=\epsilon }$
Computation:
	For$k\ge 0$
	${\displaystyle {\sigma }_x^2(k)=\lambda {\sigma }_x^2(k-1)+x^2(k)}$ (Input signal energy)
	${\displaystyle {\sigma }_d^2(k)=\lambda {\sigma }_d^2(k-1)+d^2(k)}$ (Reference signal energy)
	${\displaystyle {\overline{e}}_b(k,0)={\overline{e}}_f(k,0)=\frac{x(k)}{{\sigma }_x(k)}}$
	${\displaystyle \overline{e}(k,0)=\frac{d(k)}{{\sigma }_d(k)}}$
	For$i=0,1,\dots ,N$
	${\displaystyle \overline{\delta }(k,i)=\delta (k-1,i)\sqrt{(1-{\overline{e}}_b^2(k-1,i))(1-{\overline{e}}_f^2(k,i))}+{\overline{e}}_b(k-1,i){\overline{e}}_f(k,i)}$
	${\displaystyle {\overline{e}}_b(k,i+1)=\frac{{\overline{e}}_b(k-1,i)-\overline{\delta }(k,i){\overline{e}}_f(k,i)}{\sqrt{(1-{\overline{\delta }}^2(k,i))(1-{\overline{e}}_f^2(k,i))}}}$
	${\displaystyle {\overline{e}}_f(k,i+1)=\frac{{\overline{e}}_f(k,i)-\overline{\delta }(k,i){\overline{e}}_b(k-1,i)}{\sqrt{(1-{\overline{\delta }}^2(k,i))(1-{\overline{e}}_b^2(k-1,i))}}}$
	Feedforward filter
	${\displaystyle {\overline{\delta }}_D(k,i)={\overline{\delta }}_D(k-1,i)\sqrt{(1-{\overline{e}}_b^2(k,i))(1-{\overline{e}}^2(k,i))}+\overline{e}(k,i){\overline{e}}_b(k,i)}$
	${\displaystyle \overline{e}(k,i+1)=\frac{1}{\sqrt{(1-{\overline{e}}_b^2(k,i))(1-{\overline{\delta }}_D^2(k,i))}}[\overline{e}(k,i)-{\overline{\delta }}_D(k,i){\overline{e}}_b(k,i)]}$
	End
	End

• Adaptive filter
• Kernel adaptive filter
• Least mean squares filter
• Zero-forcing equalizer

• Hayes, Monson H. (1996). "9.4: Recursive Least Squares".Statistical Digital Signal Processing and Modeling. Wiley. p. 541.ISBN 0-471-59431-8.
• Simon Haykin,Adaptive Filter Theory, Prentice Hall, 2002,ISBN 0-13-048434-2
• M.H.A Davis, R.B. Vinter,Stochastic Modelling and Control, Springer, 1985,ISBN 0-412-16200-8
• Weifeng Liu, Jose Principe and Simon Haykin,Kernel Adaptive Filtering: A Comprehensive Introduction, John Wiley, 2010,ISBN 0-470-44753-2
• R.L.Plackett,Some Theorems in Least Squares, Biometrika, 1950, 37, 149–157,ISSN 0006-3444
• C.F.Gauss,Theoria combinationis observationum erroribus minimis obnoxiae, 1821, Werke, 4. Gottinge

• Digital signal processing
• Filter theory
• Statistical signal processing

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