# Wiener filters in canonical coordinates for transform coding, filtering, and quantizing

TL;DR: Canonical correlations are used to decompose the Wiener filter into a whitening transform coder, a canonical filter, and a coloring transform decoder, which produces new formulas for error covariance, spectral flatness, and entropy.

Abstract: Canonical correlations are used to decompose the Wiener filter into a whitening transform coder, a canonical filter, and a coloring transform decoder. The outputs of the whitening transform coder are called canonical coordinates; these are the coordinates that are reduced in rank and quantized in our finite-precision version of the Gauss-Markov theorem. Canonical correlations are, in fact, cosines of the canonical, angles between a source vector and a measurement vector. They produce new formulas for error covariance, spectral flatness, and entropy.

## Summary (2 min read)

### Introduction

- Publisher Item Identifier S 1053-587X(98)01999-0. measurement coordinates to produce a quantized Wiener filter or a quantized Gauss–Markov theorem.
- The abstract motivation for studying canonical correlations is that they provide a minimal description of the correlation between a source vector and a measurement vector.
- The vector is the coherence between and , or the cross correlation between the white random scalar and the white random vector : (7) This basic idea may be iterated to write as (8) (9) where is the squared coherence between the scalar and the vector .
- The source vector and the measurement vector are generated by Mother Nature.

### A. Standard Coordinates

- In standard coordinates, the Wiener filter and the error covariance matrix are (14) (15) We shall call Fig. 2(a) the Wiener filter in standard coordinates.the authors.the authors.
- The linear transformation (16) resolves the source vector and the measurement vector into orthogonal vectors and , with respective covariances and (17).

### B. Coherence Coordinates

- The coherence matrixmeasures the cross-correlation between thewhite vectors and : (20) Using coherence, the authors can refine the Wiener filter and its corresponding error covariance matrix as (21) (22).
- The corresponding Wiener filter, in coherence coordinates, is illustrated in Fig. 2(b).
- The first stage whitens bothand to produce the coherence coordinatesand , the second stage filters with the coherence filter to produce the estimator error and the estimator , and the third stage colors these to produce and .
- The refined linear transformation from to is (23).
- It also shows that the covariance matrix for the error in coherence coordinates is .

### C. Canonical Coordinates

- The authors achieve one more level of refinement by replacing the coherence matrix by its SVD: (27) (28) diag (29) The corresponding Wiener filter, in canonical coordinates, is illustrated in Fig. 2(c).
- The diagonal structure of this covariance matrix shows that the estimator error and the measurement are also uncorrelated, meaning that the estimatorand the error orthogonally decompose the canonical coordinate.
- This canonical correlation matrix is also the Wiener filter for estimating the canonical source coordinates from the canonical measurement coordinates.

### A. Linear Dependence

- This formula tells us that what matters is theintradependence within as measured by its direction cosines, theintradependence within as measured by its direction cosines, and theinterdependence betweenand as measured by the direction cosines between and .
- These latter direction cosines are measured in canonical coordinates, much as principal angles between subspaces are measured in something akin to canonical coordinates.

### B. Relative Filtering Errors

- The prior error covariance for the message vectoris , and the posterior error covariance for the error is .
- The volumes of the concentration ellipses associated with these covariances are proportional to and .
- The relative volumes depend only on the direction cosines : (40).

### C. Entropy and Rate

- The entropyof the random vector is (41) Normally, the authors write this entropy as the conditional entropy of given , plus the entropy of .
- Thus, the rate at which brings information about is determined by the direction cosines or squared canonical correlations between the source and the measurement.
- The authors observe that tr consists of three terms: the infinite-precision filtering error, the bias-squared introduced by rank reduction, and the variance introduced by quantizing.
- If the bit rate is specified, then the slicing level is adjusted to achieve it.
- The SVD representation for becomes a Fourier representation; therefore (55) where coherence spectrum; spectral mask diag ; and Fourier matrices.

### A. Error Variance, Spectral Flatness, and Entropy

- The Toeplitz matrix has the error variance on its diagonal.
- This formula shows the error spectrum to be the product of the source spectrum and an error spectrum, where the latter is determined by the squared coherence spectrum.
- The spectral flatness of the error spectrum is (62) which is the ratio of prediction error variance to prior variance.

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### Additional excerpts

...where the squared canonical correlation [8]–[11] is...

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...INTRODUCTION CANONICAL correlations were introduced by Hotelling [1], [2] and further developed by Anderson [3]....

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