# What is the advantage of Wiener filter for noise reduction of a 1D signal?

I have a 1D signal, acquired by an accelerator sensor that measure the vibration of a structure. What is the advantage of using a Wiener filter for noise reduction compared to the other (more classic) filtering methods?

I have a signal + noise $$u[n]= s[n] +w[n]$$ measurement and a noise only $$w[n]$$ measurement, taken at rest.

The Wiener filter considers statistical behaviours of the noise and the signal, and thus, (theoretically) achieves optimum separation of them for a class of signals and systems, which is not the case for more classical approaches.

Wiener filter frequency response is such that, at those frequencies where noise power is dominant (a.k.a. low SNR), the gain is reduced, and the output is suppressed; causing noise-reduction. Whereas for frequencies at which the signal power is dominant (a.k.a. high SNR), then the gain is closer to one, and output is closer to the input.

Thus, to determine the gain of the Wiener filter, at a particular frequency, you must know the Power Spectral Densities (PSDs) of the noise and the desired signal.

Typically, you won't know these in advance very well; but they can be estimated from available data. The result is a departure from the optimum (ideal) performance.

• Thanks. I guess the auto-correlation functions are equivalent to PSDs, correct? What if I can only estimate the auto-correlation function of the noise only? I can't know that of the signal, since it is my objective i.e the result of the denosing ... Feb 3 at 1:36
• Yes, the PSD is the Fourier transform of the ACF (they exist only for WSS random processes). Assuming the noise and the signal are independent (uncorrelated in practice), then knowing the noise spectrum, will/may also enable you to estimate signal spectrum. Note that you have the (noisy) signal at hand already. Feb 3 at 1:50
• I understand that if I have the signal + noise and the noise only, then the auto-corr of the signal only can be obtained by that of the signal+noise minus the auto-corr of the noise only, under the assumptions that they are uncorrelated. I would appreciate if you can write this by equations in your answer, so it will be clearer. To do this, I modified the question specifying what I have available and the names of the variables. Thanks again. Feb 3 at 2:43

A remarkably terse specification of the instrumentation (sensors, probes) and techniques used to measure "the vibration of a structure" leaves open the issues of applicability of the Wiener filter vs "the other (more classic) filtering methods" to processing of "the-vibration-of-a-structure" data.

Well established techniques are not readily amenable to classification on a list of unconditional advantages/disadvantages. Rather, one can compose the lists of features, itemizing the relevance of the solution for a set of application scenarios. The Wiener filter is adaptive, and this feature makes it well suited in changing environments. On the other hand, being an estimator, the Wiener filter guesses at the denoised signal waveform, but not without restrictions: it minimizes MSE when both processes, a signal of interest and the noise, are Gaussian; it implements the linear signal processing model. To implement a casual filter variety -- the classic Wiener filter -- the additional measurements are needed to trace the signal/noise statistics.

Depending on application, one may have to consider the other techniques, like nonlinear estimators, ARMA, recursive estimation solutions, or even the Kalman filter (predictor/corrector), although the latter is not adaptive in its basic implementation. But it is not a recommendation: the OP provides too little information to advise them on the preferential SP techniques and write down "the equations".

• Thank you very much for your answer. I have a measurement (signal+noise) u[n] taken with an accelerometer when the structure vibrates, and a noise only w[n], in which the structure does not vibrate, so I just pick up the background noise due to the instrumentation. I can estimate the auto-correlation functions then Ruu and Rnu, but then I have to estimate Rsu to apply the Wiener filter, which, under the assumption of uncorrelated s and n, can be derived as Rsu=Rwu-Rnu, I guess. Can you confirm this? Feb 3 at 15:58
• I don't know whether it is more convenient to use the Wiener filter in the FIR formulation or in the non-causal formulation. What do you suggest? If you like to add some equations to your answer, I would appreciate it, and promote it as a final answer. Feb 3 at 15:58
• Measurement signal processing and communications engineering have much more in common than what one can usually imagine. So thanks also for the link to the article. Feb 3 at 16:00
• ain't interested in a conditional appraisal of my answer. SE is not, or, rather, I believe, it should not be the marketplace, "upvotes in exchange for equations". But I like your other comments (except for said offer), and edited the answer. First I understood your question as an invitation to a free discourse on the filtering technique subject matter, but, if you want working formulas, you need to demonstrate your attempts on the solution. Your "signal + noise u[n]= s[n] +w[n] measurement" cannot be credited as an attempt. Feb 4 at 4:19