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Optimal Filter Coefficients: Correlation versus Least Squares

  1. Aug 17, 2011 #1
    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. Turning the community was my last resort.
     
  2. jcsd
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