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If a pure sine wave is put through a rectangular window (e.g. read into a spectrum analyser), it becomes distorted. This distortion can be observed by studying the frequency response.

No longer is the spectrum just one fundamental frequency, but a number of harmonic frequencies which are generally thought of as multiples of the fundamental i.e.

If f1 is the fundamental then the second harmonic, f2 = 2 * f1

Third harmonic, f3 = 3 * f1 etc.

However on closer inspection the harmonics are not exact multiples of the fundamental.

Is there a method for selecting the frequencies of the harmonics ? I thought it was just a matter of looking at the spectrum and reading off the frequencies relating to the maximum amplitude ?

But having used MATLAB / SIMULINK this is not the case. I started with a sine wave with a frequency of 498 Hz and read it into the spectrum analyser in SIMULINK with the following results :

F1 = 564.138
F2 =1118.55 = 1.98*F1
F3 =1672.96 = 2.96*F1
F4 =2227.37 = 3.95*F1
F5 =2762.33 = 4.9*F1
F6 =3307.02 = 5.86*F1

Although the harmonics are not exact multiples of the fundamental, they are very close to it, but the strange thing is that the amplitudes at these frequencies are not the maximums apart from F1 and F4.

If the frequencies at which the maximums occur were used then the following result would have been achieved :

F1 =564.138

F2 =1216 = 2.16*F1
F3 =1731 = 3.07*F1
F4 =2227 = 3.95*F1
F5 =2733 = 4.84*F1
F6 =3229 = 5.72*F1

The frequencies here are less of a multiple than the first table above, but at least they occur at the maximum amplitudes, something that most people would understand recognise.

The frequencies in the first table are closer to multiples of the fundamental, but only 2 of them occur (F1 and F4) at maximum amplitudes. Not sure how the other 4 frequencies / amplitudes are determined. Is it purely down to the individual person to decide on the frequency ? Surely not as these values are used to calculate other important performance parameters such as total harmonic distortion.

Maybe the differences are so small they are considered negligible ?


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However on closer inspection the harmonics are not exact multiples of the fundamental.
Only on your spectrum analyzer...


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In the REAL world, rather than the computer game world of simulators, harmonics are an exact integer multiple of the fundamental frequency.



The thing is I have derived equations for the DFT and DTFT and used EXCEL to do the number crunching to compare with the simulator results - it turns out the EXCEL calcs are almost an exact copy of SIMULINK.
EXCEL F1 = 560 Hz, Simulink F1 = 564 Hz.

I have also come across literature on the internet which state than when a sinewave is distorted by the "windowing" effect of a spectrum analyser the resulting harmonics do not occur at exact multiples of the fundamental - see attachment.


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