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Optimal choice of test signals for linear channel estimation using second order statistics

Abstract

A time-varying acoustic channel may be estimated by an appropriate inference using the output from a periodic test signal. In this paper it is shown how to do this in a way that takes full account of the past history of the background noise and the past history of the channel. An explicit formula is obtained for the optimal linear estimator that may be used for rapid channel estimation for a given test signal when we know the autocovariance or power spectrum of the interfering noise and the autocovariance of the echo channel variation.

Given this closed formula for the optimal estimator of the channel impulse response, an efficient method for determining the optimal test signal, subject to a constraint on the test signal power, given the history of the channel and the noise, is developed. We show that if the second order statistics of the channel or the noise are known, then the optimal test signal is not white. The method includes an explicit formula for the optimal test signal given a fixed estimator. A model of channel variation which is realistic while having less complexity than a full second-order statistical model, and therefore is more amenable to robust estimation, is used in the experiments which illustrate the performance of the optimal test signals and the channel estimation method. Matrix calculus identities required for the derivation of this expression for the optimal estimator are stated and proved in the Appendixes 1 and 2.

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Correspondence to Stephen Braithwaite.

Appendices

Appendix 1: Some identities of traces and their derivatives with proofs

5.1 Identity 1.1

Let A be an arbitrary real matrix. Let x be an arbitrary real vector, and \(X = \operatorname{Circ}(x)\). Then

$$\frac{\partial}{\partial x} \operatorname{tr}(XA) = \operatorname{traces}(A) $$

Proof

 □

5.2 Identity 1.2

Let A be an arbitrary real matrix. Let x be an arbitrary real vector, and \(X = \operatorname{Circ}(x)\). Then

$$\frac{\partial}{\partial x} \operatorname{tr}\bigl(X^T A\bigr) = \operatorname{traces} \bigl(A^T\bigr) $$

Proof

 □

5.3 Identity 1.3

Let A and B be arbitrary real matrices. Let x be an arbitrary real vector, and \(X = \operatorname{Circ}(x)\). Then

$$\frac{\partial}{\partial x} \operatorname{tr}\bigl(X^T A X B\bigr) = \operatorname{traces} \bigl(B^T X^T A^T\bigr) + \operatorname{traces}\bigl(B X^T A\bigr) $$

Proof

$$\operatorname{tr}\bigl(X^T A X B\bigr) = \operatorname{tr}\bigl(X^T (A X B)\bigr) = \operatorname{tr}\bigl(X \bigl(B X^T A\bigr) \bigr) $$

Therefore, applying Identity 1.1 and Identity 1.2 using the product rule:-

$$\frac{\partial}{\partial x} \operatorname{tr}\bigl(X^T A X B\bigr) = \operatorname{traces} \bigl(B^T X^T A^T\bigr) + \operatorname{traces}\bigl(B X^T A\bigr) $$

 □

5.4 Identity 1.4

Let A and B be a arbitrary real matrices. Let x be an arbitrary real vector, and \(X = \operatorname{Circ}(x)\). Then

$$\operatorname{traces}\bigl(A X^T B\bigr) = G x $$

where the elements of the matrix G are given by

$$G_{s,k} = \sum_{i=0}^{n-1} \sum _{j=0}^{n-1} A_{i,j} B_{(j+k)\% n,(i+s)\%n} $$

Proof

We can define \(\operatorname{traces}()\) by

$$\operatorname{traces}(P)_{s} = \sum_{i=0}^{n-1} P_{i,(i+s)\%n} $$

for each index s. Therefore:-

 □

5.5 Identity 1.5

Let A and B be a arbitrary real matrices and d be an arbitrary real vector. Then

$$\operatorname{diag}\bigl(A \operatorname{Diag}(d) B\bigr) = \bigl(A \diamond B^T \bigr)d $$

Proof

Let \(D = \operatorname{Diag}(d)\)

 □

Appendix 2: Some further identities

6.1 Identity 2.1

Given a vector of random variables x and symmetric matrix A, then

$$E\bigl( x^{T} A x \bigr) = \operatorname{tr}(A C_x) + m^T A m $$

Where C x is the covariance matrix of x and m is the mean of x (Petersen and Pederson 2008; Identity 306).

6.2 Identity 2.2

Given arbitrary square matrices A, B and X, then (Petersen and Pederson 2008; Identity 93)

$$\frac{\partial}{\partial X} \operatorname{tr}(A X B) = A^T B^T. $$

6.3 Identity 2.3

Given arbitrary square matrices A and X, then (Petersen and Pederson 2008; Identity 101)

$$\frac{\partial}{\partial X} \operatorname{tr}\bigl(X A X^T\bigr) = X A^T + X A. $$

6.4 Identity 2.4

Given a square matrix Y that is a function of a scalar x, then (Petersen and Pederson 2008; Identity 53)

$$\frac{\partial}{\partial x} \bigl(Y^{-1}\bigr) = -Y^{-1} \frac{\partial Y}{\partial x}Y^{-1}. $$

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Braithwaite, S., Addie, R.G. Optimal choice of test signals for linear channel estimation using second order statistics. Optim Eng 15, 93–118 (2014). https://doi.org/10.1007/s11081-012-9193-3

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Keywords

  • Acoustic
  • Echo
  • Autocovariance
  • Channel estimation
  • Matlab