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The following image is an example of a fluorescence signal trace from the photomultiplier tube detector (PMT). Each spike represents one fluorescently labeled cell.

enter image description here

This signal model is definitely not additive white Gaussian, although the noise term is Gaussian. The signals are low variance but the point is that when the signal amplitude increases, the noise variance rapidly decreases. In the given example, above the intensity value of $0.6$ all three Gaussian signals are almost noise free. Around $0.4-0.5$ intensity of the noise is reduced but can be seen if zoomed. Below $0.4$ the noise is significantly apparent.

How can this signal-noise relation be properly modeled? For example for additive noise we have $$x[n]=s[n]+w[n]$$ where $x$ is the received signal $s$ is the error free signal and $w$ is the Gaussian noise.

My aim is to simulate such signals in Matlab using a theoretical model. However, I was not able to see and/or find any model via google search or on some related books.

The physics behind this is due to detection of laser signal via photodiodes or PMTs. I was thinking that the noise source was probably the quantum noise but I am not sure because for the quantum noise, the noise variance does not decrease when the total number of photons $N$ increases. Instead it increases as $\sqrt N$.

What I tried was dividing the noise values by $k+s[n]^2$ before adding to the signal, where $k$ is some constant. I did this over additive Gaussian noise model. The results were okay. But I am not able to justify this theoretically. It is just my intuitive understanding.

Sources:

https://www.cut.ac.cy/faculties/fet/mem/research/research-units-and-laboratories/biolisys/research/imaging/cytometry/?languageId=1

https://en.wikipedia.org/wiki/Laser-induced_fluorescence

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  • $\begingroup$ PMTs have several sources of noise, as do the amplifiers that process their signals. "almost noise-free" and "noise is significantly apparent" may not be the best way to judge. You have so few bins in one event that the rise and fall dominate the image. Your amplifier probably has a shaping circuit with a transfer function already optimized for the PMT's signal in terms of peak-height or total area, depending on how your ADC is gated and using the signal. Because PMTs have been the go-to device for so much science over the last half-century, there are tons of books and tutorials out there. $\endgroup$ – uhoh Oct 10 '18 at 16:17
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    $\begingroup$ Are you able to repeat the fluorescence measurement in a controlled experiment so that you can then (with enough samples) estimate the noise of the signal from the experimental results? This would be the S+N result, and with the laser signal off as shown in the flat areas of your plot the estimation of the noise is straightforward. There are additional computations worth doing to confirm stationarity for purposes of such an estimation. As uhoh correctly indicated, the Signal estimate will be the signal after the electronics interface. $\endgroup$ – Dan Boschen Oct 12 '18 at 11:35
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    $\begingroup$ Further, when you estimate SNR like this, it is important to consider the noise density for the noise (so W/Hz). An FFT of the result with noise alone can be helpful to indicate how the noise is shaped versus frequency. $\endgroup$ – Dan Boschen Oct 12 '18 at 11:38

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