Optimal Filter Coefficients: Correlation versus Least Squares

[Squatchmichae]

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.

 

[fresh_42]

I can only see a connection via the polarization identity: $$4 \langle \mathbf{u},\mathbf{v} \rangle = ||\mathbf{u}+\mathbf{v}||^2- ||\mathbf{u}-\mathbf{v}||^2 = 4 \cdot ||\mathbf{u}||\cdot ||\mathbf{v}||\cos \sphericalangle ( \mathbf{u},\mathbf{v})$$ which says that $\rho(a)$ is maximal if the angle is close to $90°$ or if $||\mathbf{u}+\mathbf{v}||^2$ is maximal and $||\mathbf{u}-\mathbf{v}||^2$ minimal, and vice versa. Both conditions depend also on the length of the sum.