### Abstract

Suppose a vector of observations y = Hx + n stems from independent inputs x and n, both of which are Gaussian Mixture (GM) distributed, and that H is a fixed and known matrix. This work focuses on the design of a precoding matrix, F, such that the model modifies to z = HFx + n. The goal is to design F such that the mean square error (MSE) when estimating x from z is smaller than when estimating x from y. We do this under the restriction E[(Fx)^{T}Fx] ≤ P_{T}, that is, the precoder cannot exceed an average power constraint. Although the minimum mean square error (MMSE) estimator, for any fixed F, has a closed form, the MMSE does not under these settings. This complicates the design of F. We investigate the effect of two different precoders, when used in conjunction with the MMSE estimator. The first is the linear MMSE (LMMSE) precoder. This precoder will be mismatched to the MMSE estimator, unless x and n are purely Gaussian variates. We find that it may provide MMSE gains in some setting, but be harmful in others. Because the LMMSE precoder is particularly simple to obtain, it should nevertheless be considered. The second precoder we investigate, is derived as the solution to a stochastic optimization problem, where the objective is to minimize the MMSE. As such, this precoder is matched to the MMSE estimator. It is derived using the KieferWolfowitz algorithm, which moves iteratively from an initially chosen F_{0} to a local minimizer F*. Simulations indicate that the resulting precoder has promising performance.

Original language | English |
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Title of host publication | 2012 International Symposium on Information Theory and Its Applications, ISITA 2012 |

Publisher | IEEE Institute of Electrical and Electronic Engineers |

Pages | 81-85 |

ISBN (Electronic) | 978-4-88552-267-3 |

ISBN (Print) | 978-1-4673-2521-9 |

Publication status | Published - 1 Dec 2012 |

MoE publication type | A4 Article in a conference publication |

Event | 2012 International Symposium on Information Theory and Its Applications, ISITA 2012 - Honolulu, HI, United States Duration: 28 Oct 2012 → 31 Oct 2012 |

### Conference

Conference | 2012 International Symposium on Information Theory and Its Applications, ISITA 2012 |
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Country | United States |

City | Honolulu, HI |

Period | 28/10/12 → 31/10/12 |

### Fingerprint

### Cite this

*2012 International Symposium on Information Theory and Its Applications, ISITA 2012*(pp. 81-85). [6401056] IEEE Institute of Electrical and Electronic Engineers .

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*2012 International Symposium on Information Theory and Its Applications, ISITA 2012.*, 6401056, IEEE Institute of Electrical and Electronic Engineers , pp. 81-85, 2012 International Symposium on Information Theory and Its Applications, ISITA 2012, Honolulu, HI, United States, 28/10/12.

**Mean square error reduction by precoding of mixed Gaussian input.** / Flåm, John T.; Vehkaperä, Mikko; Zachariah, Dave; Tsakonas, Efthymios.

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceedings › Scientific › peer-review

TY - GEN

T1 - Mean square error reduction by precoding of mixed Gaussian input

AU - Flåm, John T.

AU - Vehkaperä, Mikko

AU - Zachariah, Dave

AU - Tsakonas, Efthymios

PY - 2012/12/1

Y1 - 2012/12/1

N2 - Suppose a vector of observations y = Hx + n stems from independent inputs x and n, both of which are Gaussian Mixture (GM) distributed, and that H is a fixed and known matrix. This work focuses on the design of a precoding matrix, F, such that the model modifies to z = HFx + n. The goal is to design F such that the mean square error (MSE) when estimating x from z is smaller than when estimating x from y. We do this under the restriction E[(Fx)TFx] ≤ PT, that is, the precoder cannot exceed an average power constraint. Although the minimum mean square error (MMSE) estimator, for any fixed F, has a closed form, the MMSE does not under these settings. This complicates the design of F. We investigate the effect of two different precoders, when used in conjunction with the MMSE estimator. The first is the linear MMSE (LMMSE) precoder. This precoder will be mismatched to the MMSE estimator, unless x and n are purely Gaussian variates. We find that it may provide MMSE gains in some setting, but be harmful in others. Because the LMMSE precoder is particularly simple to obtain, it should nevertheless be considered. The second precoder we investigate, is derived as the solution to a stochastic optimization problem, where the objective is to minimize the MMSE. As such, this precoder is matched to the MMSE estimator. It is derived using the KieferWolfowitz algorithm, which moves iteratively from an initially chosen F0 to a local minimizer F*. Simulations indicate that the resulting precoder has promising performance.

AB - Suppose a vector of observations y = Hx + n stems from independent inputs x and n, both of which are Gaussian Mixture (GM) distributed, and that H is a fixed and known matrix. This work focuses on the design of a precoding matrix, F, such that the model modifies to z = HFx + n. The goal is to design F such that the mean square error (MSE) when estimating x from z is smaller than when estimating x from y. We do this under the restriction E[(Fx)TFx] ≤ PT, that is, the precoder cannot exceed an average power constraint. Although the minimum mean square error (MMSE) estimator, for any fixed F, has a closed form, the MMSE does not under these settings. This complicates the design of F. We investigate the effect of two different precoders, when used in conjunction with the MMSE estimator. The first is the linear MMSE (LMMSE) precoder. This precoder will be mismatched to the MMSE estimator, unless x and n are purely Gaussian variates. We find that it may provide MMSE gains in some setting, but be harmful in others. Because the LMMSE precoder is particularly simple to obtain, it should nevertheless be considered. The second precoder we investigate, is derived as the solution to a stochastic optimization problem, where the objective is to minimize the MMSE. As such, this precoder is matched to the MMSE estimator. It is derived using the KieferWolfowitz algorithm, which moves iteratively from an initially chosen F0 to a local minimizer F*. Simulations indicate that the resulting precoder has promising performance.

UR - http://www.scopus.com/inward/record.url?scp=84873547730&partnerID=8YFLogxK

M3 - Conference article in proceedings

AN - SCOPUS:84873547730

SN - 978-1-4673-2521-9

SP - 81

EP - 85

BT - 2012 International Symposium on Information Theory and Its Applications, ISITA 2012

PB - IEEE Institute of Electrical and Electronic Engineers

ER -