Draft:Matched Moving Average

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this section is supposed to be inserted in | the article on "moving average" after the sub-section "Weighted moving average". /\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/

• Matched Moving Average

When periodic patterns (temporal or spatial ones) are sampled with small sampling rates (temporal case) or low sampling ratios (spatial case) and the task is to suppress the periodic component by moving window averaging of the sampled signal (e.g. for identification of superimposed information), the width of the averaging window has to be adapted to the periodicity of the periodic pattern. If we assume e.g. a sampling rate/ratio of 2.4, application of standard moving window averaging methods is providing wrong results since only integer window widths are available . A method to achieve suppression of periodic patterns in the case of non-integer (fractional) sampling ratios/rates above 2 is introduced in this paper of which this article summarizes the procedure.

We construct a series of weights for averaging one-dimensional (equally spaced) data sets with M elements, shown as a row vector, k, comprising M elements with the fractional elements $r$ ($r\in R$) located at the periphery and all other elements being 1:

$$\mathbf {k} =(k_{1},k_{2},...,k_{M-1},k_{M})=(r,1,...,1,r)$$

With the sampling ratio of the periodic modulation, R_s, the order of the next larger odd integer kernel, M, is: M = 2 · ceiling[R_s/2 + 0.5] - 1.

The fractional elements, r, are obtained as:

$r(f|M)={\frac {sin((M-2)\pi f)}{sin((M-2)\pi f)-sin(M\pi f)}}\$ with f being the inverse of the sampling ratio of the periodic modulation, R_s, $f={\frac {1}{R_{s}}}$

The kernel (i.e. its elements, weights) should be normalized by dividing them by the sum of the weights $$\mathbf {k_{n}} ={\frac {1}{2r+(M-2)}}(r,1,...,1,r)$$

In case of 2-dimensional data-sets (images) filtering can be carried out individually for both dimensions in a sequence, alternatively a 2-dimensional kernel, K, can be constructed from the row-vector k:

$\mathbf {K} \equiv \mathbf {k} \otimes \mathbf {k} \equiv \mathbf {k} ^{\mathsf {T}}\mathbf {k} ={\begin{pmatrix}k_{1}\\k_{2}\\...\\k_{M-1}\\k_{M}\end{pmatrix}}{\begin{pmatrix}k_{1}&k_{2}&...&k_{M-1}&k_{M}\end{pmatrix}}={\begin{pmatrix}k_{1}k_{1}&k_{1}k_{2}&...&k_{1}k_{M-1}&k_{M}k_{M}\\k_{2}k_{1}&k_{2}k_{2}&...&k_{2}k_{M-1}&k_{2}k_{M}\\\vdots &\vdots &...&\vdots &\vdots \\k_{M-1}k_{1}&k_{M-1}k_{2}&...&k_{M-1}k_{M-1}&k_{M-1}k_{M}\\k_{M}k_{1}&k_{M}k_{2}&...&k_{M}k_{M-1}&k_{M}k_{M}\\\end{pmatrix}}={\begin{pmatrix}r^{2}&r&...&r&r^{2}\\r&1&...&1&r\\\vdots &\vdots &...&\vdots &\vdots \\r&1&...&1&r\\r^{2}&r&...&r&r^{2}\\\end{pmatrix}}$

The normalized kernel matrix, $\mathbf {K_{n}}$ being:

$\mathbf {K_{n}} ={\frac {1}{(M-2)^{2}+4r(M-2)+4r^{2}}}{\begin{pmatrix}r^{2}&r&...&r&r^{2}\\r&1&...&1&r\\\vdots &\vdots &...&\vdots &\vdots \\r&1&...&1&r\\r^{2}&r&...&r&r^{2}\\\end{pmatrix}}$

The method is simple and can easily be implemented in spreadsheet software for filtering of one-dimensional data. In the case of 2-dimensional data sets (e.g. images) filtering can separated, i.e. it can be carried out sequentially for each of both dimensions.