### Abstract

This paper indicates that an appropriate design of metric leads to significant improvements in the adaptive projected subgradient method (APSM), which unifies a wide range of projection-based algorithms [including normalized least mean square (NLMS) and affine projection algorithm (APA)]. The key is to incorporate a priori (or a posteriori) information on characteristics of an estimandum, a system to be estimated, into the metric design. We propose a family of efficient adaptive filtering algorithms based on a parallel use of quadratic-metric projection, which assigns every point to the nearest point in a closed convex set in a quadratic-metric sense. We present two versions: (1) constant-metric and (2) variable-metric, i.e., the metric function employed is (1) constant and (2) variable among iterations. As a constant-metric version, adaptive parallel quadratic-metric projection (APQP) and adaptive parallel min-max quadratic-metric projection (APMQP) algorithms are naturally derived by APSM, being endowed with desirable properties such as convergence to a point optimal in asymptotic sense. As a variable-metric version, adaptive parallel variable-metric projection (APVP) algorithm is derived by a generalized APSM, enjoying an extended monotone property at each iteration. By employing a simple quadratic-metric, the computational complexity of the proposed algorithms is kept linear with respect to the filter length. Numerical examples demonstrate the remarkable advantages of the proposed algorithms in an application to acoustic echo cancellation.

Original language | English |
---|---|

Article number | 4244541 |

Pages (from-to) | 1665-1680 |

Number of pages | 16 |

Journal | IEEE Transactions on Audio, Speech and Language Processing |

Volume | 15 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2007 Jul |

Externally published | Yes |

### Fingerprint

### Keywords

- Acoustic echo cancellation
- Adaptive filtering
- Adaptive projected subgradient method (APSM)
- Quadratic- metric

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Acoustics and Ultrasonics

### Cite this

*IEEE Transactions on Audio, Speech and Language Processing*,

*15*(5), 1665-1680. [4244541]. https://doi.org/10.1109/TASL.2007.896655

**Adaptive parallel quadratic-metric projection algorithms.** / Yukawa, Masahiro; Slavakis, Konstantinos; Yamada, Isao.

Research output: Contribution to journal › Article

*IEEE Transactions on Audio, Speech and Language Processing*, vol. 15, no. 5, 4244541, pp. 1665-1680. https://doi.org/10.1109/TASL.2007.896655

}

TY - JOUR

T1 - Adaptive parallel quadratic-metric projection algorithms

AU - Yukawa, Masahiro

AU - Slavakis, Konstantinos

AU - Yamada, Isao

PY - 2007/7

Y1 - 2007/7

N2 - This paper indicates that an appropriate design of metric leads to significant improvements in the adaptive projected subgradient method (APSM), which unifies a wide range of projection-based algorithms [including normalized least mean square (NLMS) and affine projection algorithm (APA)]. The key is to incorporate a priori (or a posteriori) information on characteristics of an estimandum, a system to be estimated, into the metric design. We propose a family of efficient adaptive filtering algorithms based on a parallel use of quadratic-metric projection, which assigns every point to the nearest point in a closed convex set in a quadratic-metric sense. We present two versions: (1) constant-metric and (2) variable-metric, i.e., the metric function employed is (1) constant and (2) variable among iterations. As a constant-metric version, adaptive parallel quadratic-metric projection (APQP) and adaptive parallel min-max quadratic-metric projection (APMQP) algorithms are naturally derived by APSM, being endowed with desirable properties such as convergence to a point optimal in asymptotic sense. As a variable-metric version, adaptive parallel variable-metric projection (APVP) algorithm is derived by a generalized APSM, enjoying an extended monotone property at each iteration. By employing a simple quadratic-metric, the computational complexity of the proposed algorithms is kept linear with respect to the filter length. Numerical examples demonstrate the remarkable advantages of the proposed algorithms in an application to acoustic echo cancellation.

AB - This paper indicates that an appropriate design of metric leads to significant improvements in the adaptive projected subgradient method (APSM), which unifies a wide range of projection-based algorithms [including normalized least mean square (NLMS) and affine projection algorithm (APA)]. The key is to incorporate a priori (or a posteriori) information on characteristics of an estimandum, a system to be estimated, into the metric design. We propose a family of efficient adaptive filtering algorithms based on a parallel use of quadratic-metric projection, which assigns every point to the nearest point in a closed convex set in a quadratic-metric sense. We present two versions: (1) constant-metric and (2) variable-metric, i.e., the metric function employed is (1) constant and (2) variable among iterations. As a constant-metric version, adaptive parallel quadratic-metric projection (APQP) and adaptive parallel min-max quadratic-metric projection (APMQP) algorithms are naturally derived by APSM, being endowed with desirable properties such as convergence to a point optimal in asymptotic sense. As a variable-metric version, adaptive parallel variable-metric projection (APVP) algorithm is derived by a generalized APSM, enjoying an extended monotone property at each iteration. By employing a simple quadratic-metric, the computational complexity of the proposed algorithms is kept linear with respect to the filter length. Numerical examples demonstrate the remarkable advantages of the proposed algorithms in an application to acoustic echo cancellation.

KW - Acoustic echo cancellation

KW - Adaptive filtering

KW - Adaptive projected subgradient method (APSM)

KW - Quadratic- metric

UR - http://www.scopus.com/inward/record.url?scp=45749144203&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45749144203&partnerID=8YFLogxK

U2 - 10.1109/TASL.2007.896655

DO - 10.1109/TASL.2007.896655

M3 - Article

AN - SCOPUS:45749144203

VL - 15

SP - 1665

EP - 1680

JO - IEEE Transactions on Speech and Audio Processing

JF - IEEE Transactions on Speech and Audio Processing

SN - 1558-7916

IS - 5

M1 - 4244541

ER -