## A Multistage Representation of the Wiener Filter Based on Orthogonal Projections (1998)

Venue: | IEEE Transactions on Information Theory |

Citations: | 41 - 3 self |

### BibTeX

@ARTICLE{Goldstein98amultistage,

author = {J. Scott Goldstein and Senior Member and Irving S. Reed and Louis L. Scharf},

title = {A Multistage Representation of the Wiener Filter Based on Orthogonal Projections},

journal = {IEEE Transactions on Information Theory},

year = {1998},

volume = {44},

pages = {2943--2959}

}

### Years of Citing Articles

### OpenURL

### Abstract

The Wiener filter is analyzed for stationary complex Gaussian signals from an information-theoretic point of view. A dual-port analysis of the Wiener filter leads to a decomposition based on orthogonal projections and results in a new multistage method for implementing the Wiener filter using a nested chain of scalar Wiener filters. This new representation of the Wiener filter provides the capability to perform an information-theoretic analysis of previous, basis-dependent, reduced-rank Wiener filters. This analysis demonstrates that the recently introduced cross-spectral metric is optimal in the sense that it maximizes mutual information between the observed and desired processes. A new reduced-rank Wiener filter is developed based on this new structure which evolves a basis using successive projections of the desired signal onto orthogonal, lower dimensional subspaces. The performance is evaluated using a comparative computer analysis model and it is demonstrated that the low-complexity multistage reduced-rank Wiener filter is capable of outperforming the more complex eigendecomposition-based methods.

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Citation Context ...the observed vector random process that is used to estimate the scalar random process . Because of the assumed Gaussianity, the self-information or entropy of the signal process is given by (see [12]�=-=��[15]) -=-and the entropy of the vector input process is (2) (3) (4) (5) (6) (7) (8) (9) (10) where denotes the determinant operator. Next define an augmented vector by (11) Then, using (1)–(3) and (11), the ... |

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Citation Context ...s defined in (12). Note that the solution in (61) and (62) can be achieved by (59) using the method described in Appendix A or by a slight modification of the Householder tridiagonalization technique =-=[21]��-=-�[24]. That is to say, that the decomposition presented here as a component of this new multistage Wiener filter represents a generalization of the unitary Householder tridiagonalization method for ar... |

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Citation Context ...t problem. In other words, the decompositions considered were based on Gram–Schmidt, Householder, Jacobi, or principal-components analyses of the observed-data covariance matrix (for example, see [1=-=]–[5]-=- and the references contained therein). Treating the Wiener filter as a dual-port problem, however, seems more logical since the true problem at hand is not determining the best representation of the ... |

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Citation Context ...esired signal and its estimate, is the classical Wiener filter for complex stationary processes. The resulting error is The minimum mean-square error (MMSE) is where the squared canonical correlation =-=[8]��-=-�[11] is As will be seen in Section III-D, the squared canonical correlation provides a measure of the information present in the observed vector random process that is used to estimate the scalar ran... |

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Citation Context ...ector in the direction of , given by (17) and is an operator which spans the nullspace of ; i.e., is the blocking matrix which annihilates those signal components in the direction of the vector [18], =-=[19]-=- such that . Two fast algorithms for obtaining such a unitary matrix are described in [20] which use either the singularvalue decomposition or the QR decomposition. For a which is nonsingular, but not... |

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Citation Context ...65) Then the error-synthesis filterbank can be expressed in the form (66) The error-synthesis filterbank is now seen to provide planar rotations, similar to the Jacobi or Givens transforms [5], [21], =-=[25]-=-. Note that the matrices which form the multistage decomposition may be chosen to be unitary or nonunitary as a design choice and that the matrices which form the error-synthesis filterbank are unimod... |

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Citation Context ...unit vector in the direction of , given by (17) and is an operator which spans the nullspace of ; i.e., is the blocking matrix which annihilates those signal components in the direction of the vector =-=[18]-=-, [19] such that . Two fast algorithms for obtaining such a unitary matrix are described in [20] which use either the singularvalue decomposition or the QR decomposition. For a which is nonsingular, b... |

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Citation Context ...ed signal and its estimate, is the classical Wiener filter for complex stationary processes. The resulting error is The minimum mean-square error (MMSE) is where the squared canonical correlation [8]�=-=��[11]-=- is As will be seen in Section III-D, the squared canonical correlation provides a measure of the information present in the observed vector random process that is used to estimate the scalar random p... |

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Citation Context ...; i.e., is the blocking matrix which annihilates those signal components in the direction of the vector [18], [19] such that . Two fast algorithms for obtaining such a unitary matrix are described in =-=[20]-=- which use either the singularvalue decomposition or the QR decomposition. For a which is nonsingular, but not unitary, a new, very efficient, implementation of the blocking matrix is presented in App... |

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Citation Context ...ovariance matrix developed originally by Hotelling and Eckart [29], [30]. A new method to achieve a reduction in the number of degrees of freedom, or filter order, in a Wiener filter is introduced in =-=[31]-=- and [32]. This method utilizes a measure, termed the cross-spectral metric, to determine the smallests2952 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 7, NOVEMBER 1998 number of degrees of ... |

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Citation Context ... , an implementation of the blocking matrix is now found which demonstrates a low computational complexity. The MMSE is preserved when the nonunitary matrix is any arbitrary, nonsingular matrix [19], =-=[33]-=-. To establish this fact, recall that the MMSE for the filter in Fig. 1 is given by (7) (A.1) If the matrix in Fig. 2 is an arbitrary, nonsingular matrix, then the covariance matrix is given by and it... |

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(Show Context)
Citation Context ...ined in (12). Note that the solution in (61) and (62) can be achieved by (59) using the method described in Appendix A or by a slight modification of the Householder tridiagonalization technique [21]�=-=��[24]-=-. That is to say, that the decomposition presented here as a component of this new multistage Wiener filter represents a generalization of the unitary Householder tridiagonalization method for arbitra... |

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Citation Context ...xtension of a linear estimator. A. Previous Approaches to Rank Reduction The first approaches to the rank-reduction problem were motivated by the array processing application and were somewhat ad hoc =-=[27]-=-, [28]. More statistical approaches to this problem were introduced next which were based on the principal-components analysis of the covariance matrix developed originally by Hotelling and Eckart [29... |

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Citation Context ...ergy in each orthogonal coordinate. An interesting decomposition which at first appears similar in form to that presented in this paper is developed for constrained adaptive Wiener filters in [6] and =-=[7]-=-. This decomposition also treats the constrained Wiener filter as a single-port problem and does not use the constraint (in place of the desired signal, as detailed in Appendix B) in basis determinati... |

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Citation Context ...method is also termed reduced-rank or reduced-dimension Wiener filtering. Another rank-reduction technique utilizes rank-shaping and shrinkage methods. The data-adaptive shrinkage method presented in =-=[26]-=- results in a nonlinear filter that uses modedependent, nonlinear companders to estimate something akin to the Wiener filter gain. This technique, which is not discussed further in this paper, represe... |

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(Show Context)
Citation Context ...ation energy in each orthogonal coordinate. An interesting decomposition which at first appears similar in form to that presented in this paper is developed for constrained adaptive Wiener filters in =-=[6]-=- and [7]. This decomposition also treats the constrained Wiener filter as a single-port problem and does not use the constraint (in place of the desired signal, as detailed in Appendix B) in basis det... |

2 |
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(Show Context)
Citation Context ...on of a linear estimator. A. Previous Approaches to Rank Reduction The first approaches to the rank-reduction problem were motivated by the array processing application and were somewhat ad hoc [27], =-=[28]-=-. More statistical approaches to this problem were introduced next which were based on the principal-components analysis of the covariance matrix developed originally by Hotelling and Eckart [29], [30... |

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(Show Context)
Citation Context ...t the resulting Wiener filter is unconstrained, as is further explored in the example given in Section IV-C and Appendix B. It is further noted that other constraints also may be decomposed similarly =-=[17]-=-. Thus the constrained Wiener filter can be represented as an unconstrained Wiener filter with a prefiltering operation determined by the constraint. It is seen next that the unconstrained Wiener filt... |