# Reducing noise from the same frequency band as signal

Sorry for the vague question (as I'm not even quite sure what I want to do is possible), just asking for some general direction to take my research.

For a brief description, my signal resembles exponentially distributed noise, bandlimited, with a triangle shaped spectrum around the baseband. I'm looking to remove additive white Gaussian noise within the same frequency band as the signal, ideally without affecting the signal bandwidth.

Below are some simulated examples of a section of my signal and noise:

and their bandwidth

So the question is whether I can, somehow, attempt to improve the SNR, given that they share the same frequency band, and my signal is not known beforehand (though it differs from the noise in distribution and spectral shape).

EDIT Averaging successive acquisitions is not an option, as what we want to measure is a local time delay of the blue trace. As such, just averaging would interfere with the measured delay.

EDIT 2 I see people commenting that the SNR is fairly high in this case, just want to clarify: This is the current use case and we are currently happy with the performance. In the case of our system, without going too much into irrelevant details, if we increase the spatial resolution the SNR will deteriorate (up to 10 or 20dB), so maybe the representation isn't accurate of what I was trying to convey. I will post new figures as soon as I am back in the office, or if I get an opportunity tomorrow of a noisier case.

• For a matched filter, wouldn't I need to know my uncorrupted signal beforehand? The case I presented was a simulation. In general, I only have noisy signals after the acquisition. Mar 14, 2018 at 16:17
• Very interesting question! If you could get independent samples from the noise source I think you could use an adaptive filter
– VMMF
Mar 15, 2018 at 1:28
• Both the Signal and Noise goes through the same filter?
– Royi
Mar 17, 2018 at 11:29
• Looking at your first plot, I cannot imagine that the noise would affect any quantification you'd do with the signal. Where does the signal come from, what do you want to do with it, and why is this small amount of noise problematic? Mar 17, 2018 at 14:15
• The measurement is one of local time delay and it is bounded by the Cramer Rao Lower Bound. The CRLB is a function of SNR, bandwidth and time window for correlation. Due to the way the system itself works and the way its sensitivity scales with these parameters as well, it ends up ultimately being limited by SNR and the correlation window. However, a larger correlation window is undesirable, as it affects the spatial resolution of our system. We want to reduce the size of the correlation window and attempt to maintain (or avoid losing too much) sensitivity, by processing, if possible. Mar 17, 2018 at 14:24

Once you resample to the common Nyquist rate (or nearly), you can consider it a mixture model. Iirc, the optimal detector for a Laplacian signal in Gaussian noise is the soft threshold function.

• Can you elaborate or link to some paper regarding some use of mixture models for denoising? I find some things for 2D denoising, but since I'm not familiar with the topic, I'm not sure if it applies. Mar 23, 2018 at 14:29

Amplifying Mark Borgerding's answer, even if you increase noise by 30dBs so that the average energies become similar, the peaks will still be relatively uncorrupted. This is because the noise energy is evenly distributed over time while the energy of the exponential signal is highly concentrated in the peaks. That means that (soft) thresholding at some number like 0.2 or 0.5 should still allow relatively accurate estimation on the events that exceed this level.

You can use a Kalman filter to track your signal, a dual Kalman filter to estimate your signal and your noise or an extended Kalman filter where your estimate your signal and use an autoregressive model whose coefficents are also tracked by the Kalman filter.

This works better if you have a good insight about the behaviour of the signal, by other words, if you can have a mathematical model for the signal. In any case using an AR process can help to reach a successful solution.

Here you can find some useful information:

http://ieeexplore.ieee.org/abstract/document/4404093/

A Wiener filter can reduce the in-band noise.