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Let $y(t) = (h * x)(t) + n(t)$ be some observed signal where $h(t)$ is some filter / impulse response, $x(t)$ is some input signal we are interested in, and $n(t)$ is noise.

In order to recover $x$ given $y$ and $h$, I can define a regularized inverse filter, (in Fourier domain)

$$G(\nu) = \frac{H^*(\nu)}{|H(\nu)|^2 + \mu}$$

where $H(\nu)$ is the Fourier transform of $h(t)$ and $G(\nu)$ is the Fourier transform of $g(t)$. Here, we expect that $(g * y)(t) \approx x(t)$ for reasonable choices of $\mu$.

So, essentially this is some "regularized" form of the inverse filter. I want to be able to cite this (and learn more). What is this called? What are some academic references that derive or refer to this type of regularized inverse filter? A book, perhaps a well-cited paper?

Is the a formally recognized term and literature reference for this?

I know this is similar to Wiener deconvolution, where $\mu$ is the inverse of S/N ratio. I am, instead, looking for a name or reference to the general form written above where $\mu$ is some free regularization parameter. I am certain I have seen this formulation several times in different contexts. I vaguely remember a reference to "Tikhonov regularization".

I've found a few papers and questions on dsp.SE that casually throw this equation around without naming it or citing it.

PS: I wish there was a regularization tag.

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    $\begingroup$ I added the regularization tag as you requested. $\endgroup$
    – Royi
    Apr 5, 2023 at 9:47

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The proper way to analyze this is actually, in my opinion, not in the frequency domain.

We usually work in the frequency domain since those formulas were derived there in the formulation due to:

  1. Work on stochastic models where the spectrum of the cross / auto correlation function is deterministic.
  2. Work on infinite signals so the convolution theorem work easily with no border issues.

In practice, what you have here is just the spectrum equivalent of my derivation at How Is the Formula for the Wiener Deconvolution Derived.

You just set a different prior for the data. So instead of having this model:

$$ \hat{\boldsymbol{x}} = \arg \min_{\boldsymbol{x}} \frac{1}{2 {\sigma}_{n}^{2}} \left\| \boldsymbol{h} \ast \boldsymbol{x} - \boldsymbol{y} \right\|_{2}^{2} + \frac{1}{ 2 {\sigma}_{x}^{2} } \left\| \boldsymbol{x} \right\|_{2}^{2} $$

Use the following:

$$ \hat{\boldsymbol{x}} = \arg \min_{\boldsymbol{x}} \frac{1}{2} \left\| \boldsymbol{h} \ast \boldsymbol{x} - \boldsymbol{y} \right\|_{2}^{2} + \frac{\mu}{2} \left\| \boldsymbol{x} \right\|_{2}^{2} $$

The solution to this model, in frequency domain, is just your formula.

In the case of Gaussian Prior, the MMSE and MAP are the same.
So if you want to cite something you may use what's used in Wikipedia - Maximum a Posteriori Estimation page.

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  • $\begingroup$ A nice explanation of how the filter is derived. But doesn't actually answer my question. I am looking for an academic reference or name for this type of filter for the purpose of correctly citing in a paper. $\endgroup$
    – XYZT
    Apr 6, 2023 at 21:57
  • $\begingroup$ This is a regular MAP Estimator (for the Gaussian case MAP is also the MMSE). It dates back 100 of years. No need for a special academic reference, just derive it yourself in 2 liners. $\endgroup$
    – Royi
    Apr 7, 2023 at 7:01

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