# Why the power line interference harmonics appear gaussian?

I have a recorded signal from a data acquisition system.

I am getting this noise in my signal: The picture shows the FFT of the signal. I checked the frequencies of the noise and they are 50 Hz apart which are the harmonics of 50Hz. The signal is not filtered, it is a raw signal without processing. I do not understand why or what could cause the harmonics to appear as shown below, I would've expected the harmonics, at least, the odd ones to decrease as the frequency increases but they are not.

Is this noise normal? What could cause it to appear as so? Could it be pure hardware problem in the DAQ?

• What are: the sampling frequency; the scale on the vertical axis; the height of the dominant spectral line (it doesn't seem to end at '7'); the known characteristics of the measured signal? Why isn't the frequency scale logarithmic?
– Chu
Sep 15, 2020 at 7:33
• Sampling frequency is 44kHz, Vertical axis is linear (uV), height of the fundamental reaches 59 uV. Measured signal is physiological brain recorded signal. I just preferred linear, no specific reason it is not log scale. Sep 15, 2020 at 10:05

1st: Harmonics are folded back at sampling frequency from higher orders. if sampling and signal frequency do not have a least common multiple, they will not superpose. if your sampling freq. is like 10kHz and the signal is 50Hz they will.

The Theory is, the Fourier transform is from minus infinity to plus infinity (at least real signals are). But when digitized we quantise both amplitude (the ADC steps) and time (sampling period). Also we implicit multiply the signal with the dirac impulse (don't be hard on me, I haven't looked up the formulas for several years :-)). The fourier we do on the quantised signal is from 0 to some time X, which causes the fourier transform to produce mirrored results at half of the sampling frequency. And all these mirrored spectra are "folded" into the lower ones. So this effect is, because we use the fourier transform on quantised signals, while it is intended for not quantised signals. Note: the FFT is just a faster way to calculate Fourier on quantised signals - it doesn't introduce any further error.

So chances are there, you are seeing superposed harmonics.

This is why you need a analog low pass filter in the ADC frontend to suppress this effect. You can't resolve that in the digital domain.

2nd: if you just take the input signal from the ADC as it is, you implicit have applied a rectangular window. https://en.wikipedia.org/wiki/Window_function In the spectrum, the rectangular window appears as sin(x)/x function which will be seen as an envelope of your spectrum. some other common window functions: A nice List, I scanned more than 20 years ago for my diploma: 3nd: parts of your system will amplify different frequencies different. maybe there are some poles near some of the multiples of 50Hz. hard to tell whithout knowing what and where you are measuring.

So Your spectrum is a superposition of the real spectrum plus side effects, and you need to know these side effects and select measures which will reduce em. unfortunately each measure to reduce side effects will introduce another error on you spectrum, so you only can trade one error for another.

• I am sampling at 44kHz and will look more into superposed harmonics to see if it is the source. And I did use the default windowing of the FFT function so I will try changing window type or size to see if that helps. Thank you for the answer! Sep 15, 2020 at 6:33
• Following back, I have tested several different windows to see if the effect changes but the results remained the same. My sampling frequency is much higher than 50Hz so I am excluding that, so I am left with a possible hardware problem. Thank you for your reply and time, I have learned new things. Sep 16, 2020 at 14:11
• well the windows change amplitude or frequency resolution by e.g. reducing the effect of one frequency band blleding energy into another. It also helps if two sample arrays have a little time gap and such there is a step between the last and first sample. But as you can see there are always more or less variations of sin(x)/x... some more compressed, some wider,... but essentially similar. So I am not surprised you find the general effect to remain. Thats OK. Sep 16, 2020 at 15:01