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Rick314

WaveForms FM Modulation Index

Question

In WaveForms 3.9.1, what does the FM Modulation Index setting actually set?

The screenshot explains why this is unclear.  On the Electonics Explorer board a single jumper wire connects AWG1 to the Scope 1 AC input.  Frequency Modulation (FM) Modulation Index is defined as (peak frequency deviation) / (modulation frequency) or delta_f / f_mod.  A value of 2.405 is the first Bessel null, meaning on a spectrum analyzer the carrier level goes to zero as shown in the image.  This is a common test scenario.  So given that f_mod = 987.6 Hz, delta_f = 987.6 * 2.405 = 2.375 kHz.  I found the carrier null by small changes to the Amplitude/Index FM setting in WaveForms, and 20.3% nulls the carrier lower than either 20.2% or 20.4%.  What logic makes a setting of 20.3% result in 2.375 kHz FM frequency deviation?  I thought it might be "peak frequency deviation = 20.3% of the carrier frequency" but that would be 0.203*12.34 kHz = 2.505 kHz so that doesn't work.  Both Factory and my own calibration settings produce the same result.

FM_1st_Bessel_Null.png

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Hi @Rick314

In your test the modulation frequency too high for the frequency transformation and produces odd result.
The FFT is performed on the entire (windowed) capture, approximating the entire waveform length with sine waves of different frequencies.
Ideally the input waveform should be stable for the transformation period.

The FM produces output of = Carrier x (100% +/- index), like 1kHz carrier with 20% outputs between 800Hz to 1.2kHz

 

Here the top frequency is 20kHz, input period is 200ms:

FM 1kHz:

image.thumb.png.af7169a06aaf3579517745d7e1e5ba92.png

 

FM 10Hz:

image.thumb.png.066a45e8a1bee54c0108a526d1783816.png

 

Here you can see 500Hz modulation in FFT and Spectrogram which performs sliding FFT on smaller period resulting 800Hz - 1.2kHz

image.thumb.png.131fcc48dd46e2d7849e2edac018ad3a.png

image.png.ae8e99017392b5f0f474a67ee0650081.png

 

With quick measurement you can also see 800Hz 1.2kHz

image.thumb.png.ce2db97d9dac3bac4f29a98cb4091f62.png

 

image.thumb.png.d91d2e8da9075a5a4bd95313428526da.png

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Thank you for your response @attila but it led to more questions.

My understanding is that in my test case, the AWG is sweeping 12.34 kHz +/- 20.3% or +/- 2.505 kHz.  There is no problem with the AWG and it is behaving as its settings indicate.

> your test the modulation frequency too high for the frequency transformation and produces odd result.

So the issues relate to the Spectrum Analyzer and its FFT.

> The FFT is performed on the entire (windowed) capture, approximating the entire waveform length with sine waves of different frequencies.

I think of the FFT as the 40 Ms/second sampler starting, sampling, stopping, processing, and posting one measurement.  What do you mean by "entire (windowed) capture"?  It would be best if you can point me to somewhere in the Reference Manual for such details.  I am missing why you inserted "(windowed)".

> Ideally the input waveform should be stable for the transformation period.

What do you mean by "stable"?  The input is an FM signal, 12.34 kHz carrier, 987.6 Hz modulation frequency, 2.505 kHz deviation.  It is a "stable" signal as I understand the word.

What do you mean by "the transformation period"?  The sample period (or what I think of as the "window") of the 40 Ms/second sampler, or the time when the FFT (the "transformation") is happening?

A key question: Given an AWG FM signal of 12.34 kHz carrier frequency, 987.6 Hz modulation frequency, and 19.25% "Index" (+/- 2.375 kHz deviation, or a Modulation Index of 2.375/987.6 = 2.405), how can I configure the Spectrum Analyzer to accurately indicate the first Bessel null, where the carrier level is at a minimum?

> Here the top frequency is 20kHz, input period is 200ms:

I understand what you are showing, but not "input period".  Do you mean "sample period"?  And how do you know it is 200ms?  Again, pointing to the Reference Manual explanation would help.

Thanks again for the time you are putting into this.

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@attila -- It appears the default FFT settings used in my test case introduce errors in the displayed spectrum analyzer result.  I did not understand your reply, and think I do understand FM, spectrum analysis, and FFTs.  If possible, please answer my questions.  I want to understand the source of the error in the spectrum analyzer display produced by the Electronic Explorer board and Waveforms software.

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Hi @Rick314

The FFT/DFT tries to identify the sine wave frequencies that make up the captured signal.

Having 1kHz FM on a 100ms capture period it means you have 100 cycles of modulation.
After each cycle the carrier starts with a different phase, so the FFT can't correctly match sine wave on the capture.
If you look at the frequency measurement which is performed progressively you see ~9kHz/~11kHz, but the FFT shows wrongly much wider spectrum.

image.thumb.png.742f732dd1d4be9738876f89859735fd.png

 

Here the FM frequency is 10Hz which is identical to the capture period of 100ms.
With this the FFT can match more correctly sine waves to the capture and the 9/11kHz boundaries are is identified correctly.

Before the FFT process a Window is applied on the capture to reduce the calculations errors due to non full period components, by smoothing the ends of the captures.
Here to illustrate the process, I added as M1 a "FlatTop" window and M2 applied on the data.
Because of the window the spectrum is not symmetrical, only about half of the capture reaches the FFT.
http://download.ni.com/evaluation/pxi/Understanding FFTs and Windowing.pdf

image.thumb.png.ab2077978465fed118df2d8f40dddce8.png

 

Setting the FM to 20Hz (50ms) we will have about one period of modulation in the windowed domain. 

image.thumb.png.79f16493284ff895e429212411acad2e.png

 

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