# What process could generate this kind of noise in the multitaper spectrum?

I am recording some data for scientific purposes (it is neurological data), and the data is corrupted by some very strange noise. At multiples of about 700 Hz (slightly differing on different days) there are box like peaks in the multitaper spectrum. The width of each box is proportional to its frequency. The highest peak is at 5 times the base frequency, and every other repetition of the box decreases in power, the rest has the same power. I have attached an idealized image of how the spectrum of the noise looks below (in reality the width of the boxes is much smaller).

I am wondering what could cause this kind of noise. I would be interested in both practical ideas as to what device may be causing this (btw the data is recorded in a hospital) as well as theoretical insights into how this may happen (modulation, filtering etc etc).

Thank you and best regards!

Edit: Adding a few more details about the real data.

The data is recorded with implanted electrodes (256 channels) which connect to a so called headstage. During the recording the headstage is connected to two recording devices (128 ch each). I should mention that some channels are significantly more affected by this noise, and in general the channels from the first device are affected much more strongly. These devices are connected with a BNC cable which somehow ensures synchronization of the internal clocks. Collaborators of mine have suggested that the noise is due to this synchronization, but it is also present without this cable, so i do not think that this is necessarily the case. Unfortunately I do not know much more about the specific inner workings of this recording set-up. I can currently not play around with the set-up but will be able to do so in the near future, so suggestions what to try then are also welcome.

This is a plot of the real multitaper spectrum, where the base frequency of the peaks was 668. The 1/f shape is what is expected in this kind of data. In the image it appears as if the noise is narrow spikes, but they are in fact boxes with a width proportional to their frequency, as described above. The sharp spike at 10000Hz is some (separate?) source of noise that I also do not know the origin of. I thought it best to focus on one problem at a time.

Edit 2: Additional images to show the box shape and width of boxes. These are from another day where the box shape is particularly pronounced, not all days are that boxy. Here the base frequency of the noise as around 699Hz.

• can you please add to your question the full signal processing chain for this plot, i.e. how the data gets from the electrode (?) to your PC and how you made this plot? Commented Feb 13, 2023 at 14:48
• still a bit confused by how you get the upper figure – I understand you have up to 128 channels of time signal, right? If that's the case, how do we arrive at the frequency-domain plot, exactly? What is it that you do? Commented Feb 13, 2023 at 15:45
• The upper figure is something I made "by hand" to illustrate what I mean with the boxes getting wider and their respective powers). It is not derived from the data Commented Feb 13, 2023 at 16:21
• ok, so I'm going to ignore it! (That upper figure is what I'd call a decoy, honestly.) Your noise is not rectangular in spectrum at all, I guess (otherwise, there'd have to be a very clean sinc-shape signal autocorrelation in time, and that would be hell of a hint on what to look for – namely an artifact in your digital signal processing that produces that). That's a bit the risk of making your own "simplified" illustration when you're still looking for an explanation for a phenomenon – you might add significant properties that aren't there in the actual signal. So: let's look at the signal! Commented Feb 13, 2023 at 16:28
• To me this looks like you have noise components every 666.66… Hz; the fact that you say the noisy regions get wider proportionally to frequency (Is that really the case, or is this just an "eyeballing good guess"?) would suggest that's the result of your noise actually being a narrowband interferer at the 666.66 Hz fundamental with some phase noise, which goes through a nonlinear system, leading to intermodulation, and because higher-order intermodulation products self-intermodulate the phase noise, the higher harmonics get "wider in spectrum". Commented Feb 13, 2023 at 16:32

## 1 Answer

I suspect that the noise may be a frequency-drifting periodic signal with a fundamental frequency of around 660 Hz.

A periodic function $$x$$ satisfies $$x(t + P) = x(t)$$ for a fundamental period $$P > 0$$. It follows that $$x(t + nP) = x(t)$$ for any integer $$n$$. $$P$$ is called a fundamental period to distinguish it from other $$nP > 0$$ that are also periods of the periodic function. The fundamental period is the shortest period of the periodic function.

The fundamental period is related to the fundamental frequency $$f_P$$ by $$f_P = 1/P$$. A periodic function can be written as a weighted sum of sinusoidal components that are harmonic frequencies of the fundamental frequency, see Fourier series. Harmonic frequencies have the form $$nf_P$$, where $$n$$ is an integer, meaning that they are multiples of the fundamental frequency. Strictly speaking the fundamental frequency is the first harmonic, but one may also say harmonic to refer to harmonic frequencies or sinusoidal components of frequency $$nf_P$$ with $$n > 1$$. A noise signal may naturally contain harmonics or harmonics may arise due to non-linearities in the signal path.

For fundamental frequency $$f_P$$ that drifts smoothly and slowly enough as function of time $$t$$, doing a local frequency analysis of the signal around time $$t_0$$ will look as if the signal had fixed frequency $$f_P = f_0$$. Over a longer interval $$t_0 \le t \le t_1$$ the fundamental frequency will sweep from $$f_0$$ to $$f_1$$ and each $$n$$th harmonic will sweep from $$nf_0$$ to $$nf_1$$. Frequency-analyzing the signal over the interval in a way that averages local power spectral estimates sampled uniformly over the time interval will for a linear sweep appear as boxes of widths $$|nf_1 - nf_0| = n|f_1 - f_0|$$, which is proportional to $$n$$. This would explain why the higher harmonics have wider boxes. Even if the drift is not linear, if the drift is smooth (handwavy, not referring to any mathematical definition), then at a small enough time interval it will appear arbitrarily close to linear.

To solve noise problems, I would always have a look at the waveform itself and at a spectrogram to understand the noise signal, and have a listen if the signal is audio-frequency, but you can also test if you have a case of drifting periodic-signal noise by analyzing the first and last halves of the signal separately. This should give half-width boxes.

• Wow Olli, what a great insight.
– Jdip
Commented Feb 14, 2023 at 6:25