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I found a claim in a paper (BSSA, Vol 81, No. 6: "A Waveform Correlation Method for Identifying Quarry Explosions", By D.B. Harris) concerning finding filter coefficients. The statement is given without proof. I cannot locate a reference or theorem for the following, and have not been able thus far to justify this claim quantitatively.

Suppose:

\begin{equation}

\mathbf{v}(t) = \displaystyle \sum_{k=1}^{N} \int_{-T}^{T} \! a_{k}(t-s) \mathbf{u}_{k}(s) \, ds,

\end{equation}

Then maximizing the correlation coefficient over filter coefficients a:

\begin{equation}

\rho(a) = \max_{a} \frac{\left\langle {\mathbf{u}(t), \mathbf{v}(t) } \right\rangle} { \sqrt{\left\langle {\mathbf{u}(t), \mathbf{u}(t) } \right\rangle \, \left\langle {\mathbf{v}(t), \mathbf{v}(t) } \right\rangle}}

\end{equation}

Is equivalent to:

\begin{equation}

\min_{a} \int_{-T}^{T} \! \parallel \mathbf{u}(t) - \mathbf{v}(t) \parallel^{2} \, dt,

\end{equation}

Where the inner product is defined by:

\begin{equation}

\left\langle {\mathbf{u}(t), \mathbf{v}(t) } \right\rangle = \int_{-T}^{T} \! {\mathbf{u}(t)}^{T} \mathbf{v}(t) \, dt,

\end{equation}

Qualitatively, this makes sense, of course. I initially attempted to prove this in the frequency domain by making use of the convolution theorem, to reduce the problem into one that looks similar to a Rayleigh quotient. This effort did not yield the correct equations.

Suppose:

\begin{equation}

\mathbf{v}(t) = \displaystyle \sum_{k=1}^{N} \int_{-T}^{T} \! a_{k}(t-s) \mathbf{u}_{k}(s) \, ds,

\end{equation}

Then maximizing the correlation coefficient over filter coefficients a:

\begin{equation}

\rho(a) = \max_{a} \frac{\left\langle {\mathbf{u}(t), \mathbf{v}(t) } \right\rangle} { \sqrt{\left\langle {\mathbf{u}(t), \mathbf{u}(t) } \right\rangle \, \left\langle {\mathbf{v}(t), \mathbf{v}(t) } \right\rangle}}

\end{equation}

Is equivalent to:

\begin{equation}

\min_{a} \int_{-T}^{T} \! \parallel \mathbf{u}(t) - \mathbf{v}(t) \parallel^{2} \, dt,

\end{equation}

Where the inner product is defined by:

\begin{equation}

\left\langle {\mathbf{u}(t), \mathbf{v}(t) } \right\rangle = \int_{-T}^{T} \! {\mathbf{u}(t)}^{T} \mathbf{v}(t) \, dt,

\end{equation}

Qualitatively, this makes sense, of course. I initially attempted to prove this in the frequency domain by making use of the convolution theorem, to reduce the problem into one that looks similar to a Rayleigh quotient. This effort did not yield the correct equations.

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