# Tensor-Based Adaptive Filtering Algorithms

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## Abstract

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## 1. Introduction

## 2. System Model

## 3. Tensor-Based LMS Algorithms

Algorithm 1: NLMS-T algorithm. |

Initialization: Set ${\widehat{\mathbf{h}}}_{i}\left(0\right),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$, based on (21)–(22) Set $0<{\alpha}_{i}\le 1,\phantom{\rule{4pt}{0ex}}{\delta}_{i}>0,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$ For n = 1,2,…, number of iterations: Compute ${\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$, based on (19) ${e}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right)=d\left(n\right)-{\widehat{\mathbf{h}}}_{i}^{T}(n-1){\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right)$, for any $i=1,2,\dots ,N$ ${\widehat{\mathbf{h}}}_{i}\left(n\right)={\widehat{\mathbf{h}}}_{i}(n-1)+\frac{{\alpha}_{i}{\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right){e}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right)}{{\delta}_{i}+{\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}^{T}\left(n\right){\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right)},\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$ $\widehat{\mathbf{g}}\left(n\right)={\widehat{\mathbf{h}}}_{N}\left(n\right)\otimes {\widehat{\mathbf{h}}}_{N-1}\left(n\right)\otimes \cdots \otimes {\widehat{\mathbf{h}}}_{1}\left(n\right)$ |

## 4. Tensor-Based RLS Algorithm

Algorithm 2: RLS-T algorithm. |

Initialization: Set ${\widehat{\mathbf{h}}}_{i}\left(0\right),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$, based on (21)–(22) ${\mathbf{R}}_{i}^{-1}\left(0\right)={\xi}_{i}^{-1}{\mathbf{I}}_{{L}_{i}},\phantom{\rule{4pt}{0ex}}{\xi}_{i}>0,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$ ${\lambda}_{i}=1-\frac{1}{K{L}_{i}},\phantom{\rule{4pt}{0ex}}K\ge 1,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$ For n = 1, 2,…, number of iterations: Compute ${\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$, based on (19) ${e}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right)=d\left(n\right)-{\widehat{\mathbf{h}}}_{i}^{T}(n-1){\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right)$, for any $i=1,2,\dots ,N$ ${\mathbf{k}}_{i}\left(n\right)=\frac{{\mathbf{R}}_{i}^{-1}(n-1){\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right)}{{\lambda}_{i}+{\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}^{T}\left(n\right){\mathbf{R}}_{i}^{-1}(n-1){\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right)},\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$ ${\widehat{\mathbf{h}}}_{i}\left(n\right)={\widehat{\mathbf{h}}}_{i}(n-1)+{\mathbf{k}}_{i}\left(n\right){e}_{{\widehat{\mathbf{h}}}_{i}}\left(n\right),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$ ${\mathbf{R}}_{i}^{-1}\left(n\right)=\frac{1}{{\lambda}_{i}}\left[{\mathbf{I}}_{{L}_{i}}-{\mathbf{k}}_{i}\left(n\right){\mathbf{x}}_{{\widehat{\mathbf{h}}}_{i}}^{T}\left(n\right)\right]{\mathbf{R}}_{i}^{-1}(n-1),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$ $\widehat{\mathbf{g}}\left(n\right)={\widehat{\mathbf{h}}}_{N}\left(n\right)\otimes {\widehat{\mathbf{h}}}_{N-1}\left(n\right)\otimes \cdots \otimes {\widehat{\mathbf{h}}}_{1}\left(n\right)$ |

## 5. Beyond the Identification of Rank-1 Tensors

## 6. Simulation Results

## 7. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Impulse responses used in simulations of the multiple-input/single-output (MISO) system identification scenario (for $N=4$): (

**a**) ${\mathbf{h}}_{1}$ (of length ${L}_{1}=16$) contains the first 16 coefficients of the first impulse response from G168 Recommendation [40], (

**b**) ${\mathbf{h}}_{2}$ (of length ${L}_{2}=8$) is a randomly generated impulse response, (

**c**) ${\mathbf{h}}_{3}$ (of length ${L}_{3}=4$) has the coefficients computed as ${h}_{3,{l}_{3}}=0.{9}^{{l}_{3}-1}$, with ${l}_{3}=1,2,\dots ,{L}_{3}$, (

**d**) ${\mathbf{h}}_{4}$ (of length ${L}_{4}=4$) has the coefficients computed as ${h}_{4,{l}_{4}}=0.{5}^{{l}_{4}-1}$, with ${l}_{4}=1,2,\dots ,{L}_{4}$, and (

**e**) $\mathbf{g}$ (of length $L={L}_{1}{L}_{2}{L}_{3}{L}_{4}=2048$) is the global impulse response, which results based on (10).

**Figure 2.**Impulse responses used in simulations of the single-input/single-output (SISO) system identification scenario, with $\underline{L}=1000$: (

**a**) network echo path from G168 Recommendation [40] and (

**b**) measured acoustic echo path available on-line at www.comm.pub.ro/plant (accessed on 5 March 2021).

**Figure 3.**Performance of the LMS-T and LMS algorithms (using different step-size parameters), for the identification of the global impulse response $\mathbf{g}$. The input signals are white Gaussian noises, $N=4$, and $L=2048$.

**Figure 4.**Performance of the LMS-T and LMS algorithms (using different step-size parameters), for the identification of the global impulse response $\mathbf{g}$. The input signals are AR(1) processes, $N=4$, and $L=2048$.

**Figure 5.**Performance of the LMS-T and LMS algorithms (using different step-size parameters), for the identification of the global impulse response $\mathbf{g}$. The input signals are AR(1) processes, $N=5$, and $L=4096$.

**Figure 6.**Performance of the NLMS-T and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response $\mathbf{g}$. The input signals are white Gaussian noises, $N=4$, and $L=2048$.

**Figure 7.**Performance of the NLMS-T and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response $\mathbf{g}$. The input signals are AR(1) processes, $N=4$, and $L=2048$.

**Figure 8.**Performance of the NLMS-T and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response $\mathbf{g}$. The input signals are AR(1) processes, $N=5$, and $L=4096$.

**Figure 9.**Performance of the NLMS-T algorithm using different normalized step-size parameters (with equal values of ${\alpha}_{i},\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$), for the identification of the global impulse response $\mathbf{g}$. The input signals are AR(1) processes, $N=4$, and $L=2048$.

**Figure 10.**Performance of the NLMS-T algorithm using different normalized step-size parameters (with different values of ${\alpha}_{i},\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,N$), for the identification of the global impulse response $\mathbf{g}$. The input signals are AR(1) processes, $N=4$, and $L=2048$.

**Figure 11.**Performance of the RLS-T algorithm (using different forgetting factors), for the identification of the global impulse response $\mathbf{g}$. The input signals are white Gaussian noises, $N=4$, and $L=2048$.

**Figure 12.**Performance of the RLS-T, NLMS-T, and RLS algorithms, for the identification of the global impulse response $\mathbf{g}$. The input signals are AR(1) processes, $N=4$, and $L=2048$.

**Figure 13.**Performance of the RLS-T, NLMS-T, and RLS algorithms, for the identification of the global impulse response $\mathbf{g}$. The input signals are AR(1) processes, $N=5$, and $L=4096$.

**Figure 14.**Performance of the RLS-T, NLMS-T, tensor LMS [7], and RLS algorithms, for the identification of the impulse response $\mathbf{h}=\mathbf{g}+\mathbf{f}$. The vector $\mathbf{f}$ is randomly generated (Gaussian distribution), with the variance $\zeta {\u2225\mathbf{g}\u2225}_{2}/L$, where $\zeta =0.001$. The input signals are AR(1) processes, $N=4$, and $L=2048$.

**Figure 15.**Performance of the RLS-T, NLMS-T, tensor LMS [7], and RLS algorithms, for the identification of the impulse response $\mathbf{h}=\mathbf{g}+\mathbf{f}$. The vector $\mathbf{f}$ is randomly generated (Gaussian distribution), with the variance $\zeta {\u2225\mathbf{g}\u2225}_{2}/L$, where $\zeta =0.001$. The input signals are AR(1) processes, $N=5$, and $L=4096$.

**Figure 16.**Performance of the RLS-T, NLMS-T, tensor LMS [7], and RLS algorithms, for the identification of the impulse response $\mathbf{h}=\mathbf{g}+\mathbf{f}$. The vector $\mathbf{f}$ is randomly generated (Gaussian distribution), with the variance $\zeta {\u2225\mathbf{g}\u2225}_{2}/L$, where $\zeta =0.005$. The input signals are AR(1) processes, $N=5$, and $L=4096$.

**Figure 17.**Performance of the RLS-T, NLMS-T, tensor LMS [7], and RLS algorithms, for the identification of the impulse response $\mathbf{h}=\mathbf{g}+\mathbf{f}$. The vector $\mathbf{f}$ is randomly generated (Gaussian distribution), with the variance $\zeta {\u2225\mathbf{g}\u2225}_{2}/L$, where $\zeta =0.01$. The input signals are AR(1) processes, $N=5$, and $L=4096$.

**Figure 18.**Performance of the RLS-T algorithm for the identification of the impulse response $\mathbf{h}=\mathbf{g}+\mathbf{f}$. The vector $\mathbf{f}$ is randomly generated (Gaussian distribution), with the variance $\zeta {\u2225\mathbf{g}\u2225}_{2}/L$, using different values of $\zeta $. The theoretical error (misalignment) is marked with a dashed line. The input signals are AR(1) processes, $N=5$, and $L=4096$.

**Figure 19.**Performance of the RLS-NKP and RLS algorithms for the identification of the network impulse response $\underline{\mathbf{h}}$ from Figure 2a, with the length $\underline{L}=1000$. The RLS-NKP algorithm uses ${\underline{L}}_{1}=40$, ${\underline{L}}_{2}=25$, ${\underline{\lambda}}_{1}=1-1/\left(10P{\underline{L}}_{1}\right)$, and ${\underline{\lambda}}_{2}=1-1/\left(10P{\underline{L}}_{2}\right)$. The forgetting factor of the RLS algorithm is $\lambda =1-1/\left(10\underline{L}\right)$. The input signal is an AR(1) process and $\mathrm{ENR}=20$ dB.

**Figure 20.**Performance of the RLS-NKP and RLS algorithms for the identification of the acoustic impulse response $\underline{\mathbf{h}}$ from Figure 2b, with the length $\underline{L}=1000$. The RLS-NKP algorithm uses ${\underline{L}}_{1}=40$, ${\underline{L}}_{2}=25$, ${\underline{\lambda}}_{1}=1-1/\left(10P{\underline{L}}_{1}\right)$, and ${\underline{\lambda}}_{2}=1-1/\left(10P{\underline{L}}_{2}\right)$. The forgetting factor of the RLS algorithm is $\lambda =1-1/\left(10\underline{L}\right)$. The input signal is an AR(1) process and echo-to-noise ratio $\left(\mathrm{ENR}\right)=20$ dB.

**Figure 21.**Performance of the RLS-NKP and RLS algorithms for the identification of the acoustic impulse response $\underline{\mathbf{h}}$ from Figure 2b, with the length $\underline{L}=1000$. The RLS-NKP algorithm uses ${\underline{L}}_{1}=40$, ${\underline{L}}_{2}=25$, ${\underline{\lambda}}_{1}=1-1/\left(10P{\underline{L}}_{1}\right)$, and ${\underline{\lambda}}_{2}=1-1/\left(10P{\underline{L}}_{2}\right)$. The forgetting factor of the RLS algorithm is $\lambda =1-1/\left(10\underline{L}\right)$. The input signal is a speech sequence and $\mathrm{ENR}=20$ dB.

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**MDPI and ACS Style**

Dogariu, L.-M.; Stanciu, C.-L.; Elisei-Iliescu, C.; Paleologu, C.; Benesty, J.; Ciochină, S.
Tensor-Based Adaptive Filtering Algorithms. *Symmetry* **2021**, *13*, 481.
https://doi.org/10.3390/sym13030481

**AMA Style**

Dogariu L-M, Stanciu C-L, Elisei-Iliescu C, Paleologu C, Benesty J, Ciochină S.
Tensor-Based Adaptive Filtering Algorithms. *Symmetry*. 2021; 13(3):481.
https://doi.org/10.3390/sym13030481

**Chicago/Turabian Style**

Dogariu, Laura-Maria, Cristian-Lucian Stanciu, Camelia Elisei-Iliescu, Constantin Paleologu, Jacob Benesty, and Silviu Ciochină.
2021. "Tensor-Based Adaptive Filtering Algorithms" *Symmetry* 13, no. 3: 481.
https://doi.org/10.3390/sym13030481