# Wiener Deconvolution: Get Clean Signal with Jumps & Bursts

In summary, the system lowpasses the signal, and a deconvolution puts more emphasis on the high frequencies. The result is dampened by weighting the filter with the expected signal to noise ratio.
I have a system that acts as a low pass. I know the transient response quite well, and I am trying to do a Wiener deconvolution on the measurement data. I expect a clean signal with a few jumps and bursts. Basically piecewise continuous, with the occasional peak. For the Wiener deconvolution I need the spectral power density of the expected signal, and the spectral power density of the noise.
The noise looks white falling a bit at the high frequencies. Is there a somewhat generic term to use for the spectral power density of a generic signal? Maybe 1/abs(f+epsilon). I could simulate a bit, but maybe someone here has used deconvolution and knows what kind of power densities tend to work well, and how sensitive the outcome is to this.
Also: How well do these deconvolutions work, and what are the biggest problems?

Wait. Are you saying you have an unknown signal x that you are trying to estimate from a filtered noisy version y = x*h + n? And you know the impulse response h (or frequency response H)?

If so, you don't need Wiener's formulism, which attempts to estimate h and its inverse. Just apply h inverse to y (or H^(-1) to Y).

Well, it's not quite as easy. The system lowpasses, and doing a deconvolution puts more emphasis on the high frequencies. But as signals tend to be dominantly in the low frequencies, most of the stuff that gets amplified is noise. So basically the H^-1 applied to N term is too large.
The Wiener deconvolution is damping this effect by weighting the filter with the expected signal to noise ratio. I did a try with a 1/(1 + f^2) power spectrum for the signal but it does not look too good, I think I need more high frequencies.
I am not sure, but it seems to me that by Wiener's formalism you mean blind deconvolution, where the impulse response is iteratively guessed.

No, I was referring to Wiener deconvolution, aka Wiener filter.

It sounds like you have a firm idea of the spectrum of your signal. You say
1) it is predominantly low frequencies, below your filter cutoff
2) a single pole rolling off at 1 Hz removes too much high frequency

BTW, you have answered your question about how sensitive the results are to choice of signal PSD. It matters significantly.

Try a response that corresponds to your understanding of the signal spectrum. Or measure the filtered spectrum directly--if you are right, you'll see signal predominantly at low frequencies, then noise, then the filter rolls everything off. Fit a spectrum to it and use that in the Wiener deconvolution.

## 1. What is Wiener deconvolution?

Wiener deconvolution is a mathematical technique used to improve the quality of a signal by removing noise and other disturbances. It works by estimating the original signal from a noisy or distorted version using statistical and spectral information.

## 2. How does Wiener deconvolution work?

Wiener deconvolution works by using a mathematical model to estimate the original signal from a noisy or distorted version. This model takes into account the spectral information of the signal and the noise, as well as the statistical properties of both. The resulting estimate is a "clean" version of the original signal, with reduced noise and improved quality.

## 3. When should Wiener deconvolution be used?

Wiener deconvolution is most useful when the original signal is known or can be estimated accurately, but has been distorted or corrupted by noise or other disturbances. It is commonly used in signal processing, image restoration, and other fields where the quality of a signal needs to be improved.

## 4. What are the advantages of using Wiener deconvolution?

The main advantage of using Wiener deconvolution is that it can significantly improve the quality of a signal without requiring additional data. It is also a relatively simple and widely applicable technique, making it a popular choice for many scientists and engineers.

## 5. Are there any limitations to using Wiener deconvolution?

Wiener deconvolution is not a perfect solution and has some limitations. It works best when the original signal is known or can be estimated accurately and when the noise or disturbances follow a specific statistical model. It may not be effective in cases where the noise is unpredictable or has a complex structure. Additionally, Wiener deconvolution can be computationally expensive for large datasets.

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