I need to identify a matched filter, and so have gone through the mathematics to do this as listed here https://en.wikipedia.org/wiki/Matched_filter which derives the optimal filter $h$ to apply to a time series $x$, where that time series is assumed to comprise a desirable signal $s$ and some noise $v$. This suggests the optimal matched filter is: \begin{equation} h = \frac{1}{\sqrt{s^{\mathrm{H}}\mathbf{\mathrm{R}}_v^{-1}s}} \mathbf{\mathrm{R}}_v^{-1}s \end{equation} and this should be applied to produce the filtered series $y$ thus: \begin{equation} y\left(n\right) = \sum_{k=-\infty}^{\infty} h\left(n-k\right)x\left(k\right) \end{equation} however, I don't know $s$. How, therefore, do I use this method (if at all)?
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3$\begingroup$ Matched filters are based on the assumption that the signal is known (else there is nothing to match to in designing the matched filter). $\endgroup$– Dilip SarwateCommented Jul 6, 2020 at 12:35
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$\begingroup$ Depending on the problem, if $s$ has some known structure but the parameters are unknown, you can think about first estimating the signal parameters then you can generate an "estimated/approximate" matched filter $\endgroup$– EngineerCommented Jul 6, 2020 at 14:14
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$\begingroup$ @Engineer Yes, I have a time series ($x$) for which I have a decomposition: it has three structural components (plus noise) and I have an autocovariance matrix for each component. The series ($x$) is measured river flows and comprises a sinusoidal (seasonal) cycle $s$, a river flow contribution from groundwater (known as "baseflow", $b$ and a contribution to river flow from storm runoff events, $r$, and some random fluctuations $n$. So, $x = s+b+r+n$. I want to filter $x$ such that I am left with $s+b$. I know the autocovariance functions for $s+b$ - does this help? $\endgroup$– hydrologistCommented Jul 6, 2020 at 14:27
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$\begingroup$ Oh I see, there is multiple things going on. So you know that $s$ is periodic, $b$ must be constant (?), $r$ is periodic (?), and $n$ is not periodic. Is this right? $\endgroup$– EngineerCommented Jul 7, 2020 at 14:43
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$\begingroup$ @Engineer Yes. I have just written an updated post, having done some extra reading - its here... dsp.stackexchange.com/questions/68900/… $\endgroup$– hydrologistCommented Jul 7, 2020 at 14:46
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3 Answers
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As stated by others, the signal $s[n]$ has to be known to calculate the corresponding matched filter.
An alternative is to use the power of deep learning to emulate a matched filter response as shown in this paper.
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Dilip Sarwate's comment, as usual, is correct. hydrologist, are you making this problem overly complicated? If you know the finite-duration samples of the signal $s[n]$ you want to detect then the finite-duration impulse response of your matched filter should be the "reversed-in-time" conjugate of $s[n]$.
answered Jul 6, 2020 at 19:53
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$\begingroup$ The desired signal is unknown, but it’s covariance is known. There is a longer comment on the question where the OP gives that detail $\endgroup$– EngineerCommented Jul 6, 2020 at 22:40
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When the signal is unknown but has known statistics (PDF, covraiance etc), you match with the estimation of the signal based on the received data, this is called the "estimator correlator" in literature. Refer to Stephen M Kay Detection theory for a more detailed explanation. The answer will be too long if I explain here.
