### Abstract

Given (n + 1) consecutive autocorrelations of a stationary discrete-time stochastic process, how this finite sequence is extended so that the power spectral density associated with the resulting infinite sequence of correlations is nonnegative everywhere is discussed. It is well known that when the Hermitian Toeplitz matrix generated from the given autocorrelations is positive definite, the problem has an infinite number of solutions and the particular solution that maximizes the entropy functional results in a stable all-pole model of order n. Since maximization of the entropy functional is equivalent to maximization of the minimum mean-square error associated with one-step predictors, the problem of obtaining admissible extensions that maximize the minimum mean-square error associated with k-step (k ≤ n) predictors, that are compatible with the given autocorrelations, is studied. It is shown that the resulting spectrum corresponds to that of a stable autoregressive moving average (ARMA) (n, k-1) process.

Original language | English (US) |
---|---|

Pages (from-to) | 142-158 |

Number of pages | 17 |

Journal | IEEE Transactions on Signal Processing |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1992 |

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### ASJC Scopus subject areas

- Signal Processing
- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Signal Processing*,

*40*(1), 142-158. https://doi.org/10.1109/78.157189

**A new spectrum extension method that maximizes the multistep minimum prediction error--Generalization of the maximum entropy concept.** / Pillai, Unnikrishna; Shim, Theodore I.; Benteftifa, M. Hafed.

Research output: Contribution to journal › Article

*IEEE Transactions on Signal Processing*, vol. 40, no. 1, pp. 142-158. https://doi.org/10.1109/78.157189

}

TY - JOUR

T1 - A new spectrum extension method that maximizes the multistep minimum prediction error--Generalization of the maximum entropy concept

AU - Pillai, Unnikrishna

AU - Shim, Theodore I.

AU - Benteftifa, M. Hafed

PY - 1992/1

Y1 - 1992/1

N2 - Given (n + 1) consecutive autocorrelations of a stationary discrete-time stochastic process, how this finite sequence is extended so that the power spectral density associated with the resulting infinite sequence of correlations is nonnegative everywhere is discussed. It is well known that when the Hermitian Toeplitz matrix generated from the given autocorrelations is positive definite, the problem has an infinite number of solutions and the particular solution that maximizes the entropy functional results in a stable all-pole model of order n. Since maximization of the entropy functional is equivalent to maximization of the minimum mean-square error associated with one-step predictors, the problem of obtaining admissible extensions that maximize the minimum mean-square error associated with k-step (k ≤ n) predictors, that are compatible with the given autocorrelations, is studied. It is shown that the resulting spectrum corresponds to that of a stable autoregressive moving average (ARMA) (n, k-1) process.

AB - Given (n + 1) consecutive autocorrelations of a stationary discrete-time stochastic process, how this finite sequence is extended so that the power spectral density associated with the resulting infinite sequence of correlations is nonnegative everywhere is discussed. It is well known that when the Hermitian Toeplitz matrix generated from the given autocorrelations is positive definite, the problem has an infinite number of solutions and the particular solution that maximizes the entropy functional results in a stable all-pole model of order n. Since maximization of the entropy functional is equivalent to maximization of the minimum mean-square error associated with one-step predictors, the problem of obtaining admissible extensions that maximize the minimum mean-square error associated with k-step (k ≤ n) predictors, that are compatible with the given autocorrelations, is studied. It is shown that the resulting spectrum corresponds to that of a stable autoregressive moving average (ARMA) (n, k-1) process.

UR - http://www.scopus.com/inward/record.url?scp=0026705940&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026705940&partnerID=8YFLogxK

U2 - 10.1109/78.157189

DO - 10.1109/78.157189

M3 - Article

AN - SCOPUS:0026705940

VL - 40

SP - 142

EP - 158

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 1

ER -