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Topic: Why Do Partials Exist In FM signals? (Read 9342 times) previous topic - next topic
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Why Do Partials Exist In FM signals?

I would like to know why partials occur at all in FM signals.  Just in case anyone is unclear on FM I will give a brief overview:

FM or frequency modulation can be thought of simply as a single sine wave that has a constantly changing frequency.   We achieve this by having a 'carrier' wave at a constant frequency e.g 50Hz, and a 'modulator' frequency e.g 1Hz.  We use the modulator wave to modulate the frequency value of the carrier, so in this example the carrier frequency will rise up from 50Hz and then fall below 50Hz, and it will do this repeatedly at the same rate as the modulator frequency (which in this case is 1Hz, so it will happen once every second).  How far above and below the 50Hz frequency deviates is determined by a third value called the modulation index (which is the level of the modulating frequency). 

Here's what's confusing me...  Once we crank up the modulator frequency, all we are essentially doing is making the carrier (which is rising and falling in frequency once per second at the 1Hz modulating frequency) rise and fall in frequency at a faster rate.  Eventually when we make this rising and falling happen so fast, we can no longer perceive the variation in pitch, and 'partials' emerge around the carrier frequency which appear to be constant frequencies by themselves.  Now, I know that we can predict exactly where partials will occur (they occur at intervals of the modulating frequency around the carrier), and I know we can predict the amplitude of the partials given our input parameters (according to Bessel functions).  I also know that both the modulating frequency and index in turn affect the bandwidth, but what I cannot find out for the life of me is an explanation for why we get any partials at all.  With for e.g a carrier of 5000Hz and a modulator of 2000Hz, we will get partials at 7000Hz, 9000Hz, 11000Hz etc and vice versa in the negative direction at 3000Hz, 1000Hz etc, but HOW?  Why don't we instead end up with every frequency that is being swept through in the rise and fall manner?  What is it about the intervals of the modulating frequency that cause partials to be pronounced in those parts of the spectrum and not in others?  I think it at least stands to reason that it is a result of an evolution over time, but I'd like a precise understanding of what is happening over time to produce partials in an FM signal. 

Hope that made sense, and perhaps someone can help me out here.  I'm hoping to do my Honours project on this subject matter, but I absolutely have to understand this first.

Re: Why Do Partials Exist In FM signals?

Reply #1
What the heck are "partials"?  If you mean harmonics they already exist in the audio signal and are encoded in the RF signal using whatever form of modulation or demodulation that is chosen.  A decoded digital signal contains the harmonics that were recorded in the original signal too, assuming the process of modulation/demodulation was done properly.  If you aren't talking about harmonics what the heck are you on about?

Ed Seedhouse
VA7SDH

Re: Why Do Partials Exist In FM signals?

Reply #2
What the heck are "partials"?  If you mean harmonics they already exist in the audio signal and are encoded in the RF signal using whatever form of modulation or demodulation that is chosen.  A decoded digital signal contains the harmonics that were recorded in the original signal too, assuming the process of modulation/demodulation was done properly.  If you aren't talking about harmonics what the heck are you on about?

Here is the obligatory wikipedia reference for what partials are. https://en.wikipedia.org/wiki/Harmonic_series_(music)#Partial

 It doesn't strictly mean the same thing as harmonic but in this context, sure, that's basically what I'm talking about.  I think you have misunderstood that I am referring to FM audio synthesis and not FM radio transmission.  There are no harmonics encoded or present in our signals (carrier and modulating frequencies), as our signals are pure tones/sine waves and therefore single frequencies.   When I say partials, I'm talking about the frequencies that emerge around a carrier at intervals of the modulating frequency as sidebands.  


Re: Why Do Partials Exist In FM signals?

Reply #4
A partial according to that definition is just a harmonic. I think that wiki article is not very good however, and I am not sure that partial even means anything in this case.
If you want to understand frequency modulation, take a look here:
https://en.wikipedia.org/wiki/Frequency_modulation#Theory

The Wikipedia articleis not great, however my source of the word 'partial' in the context of FM synthesis comes directly from academic literature and university lectures.  I would not debate the validity of the term because it describes exactly what we observe by its definition; in the complex FM signal, we acknowledge the presence of specific sideband frequencies which are partial sinusoidal components of that signal.  The only other term to consider using here would be 'sidebands'.  A partial can be a harmonic, but it can also be an inharmonic frequency which is why we shouldn't use that term when describing the spectral sidebands in FM signals.  Your second link refers to partials as 'spectral components'.

By the way... when I gave a description of how FM works in my original post, did it not occur to you that I knew enough about FM to know how to find the Wikipedia page for it?  Not that I don't appreciate the help but suffice it to say I'm a little bit further ahead than that!  I'm just asking specifically /why/ partials occur in the spectral locations that they do.  There is a wealth of information available describing mathematical relationships between a complex FM signal, the frequencies that make that signal up, and the partial components that emerge as a result of the modulation BUT there is nothing  I can find that explains exactly /why/ partials or side bands emerge in the spectrum as a result of modulating frequency at high rates over short time.  That is precisely the information I'm looking for.



 


Re: Why Do Partials Exist In FM signals?

Reply #5
Yes I assumed from your terminology above that you were not familiar with the topic. If you've read that derivation above, then maybe you could clarify what is your question?

Re: Why Do Partials Exist In FM signals?

Reply #6
If the modulating frequency is to close to carrier frequency you have practically a ring modulator. So in your example in original post, modulating frequency is almost half the carrier, and you get partials at carrier plus / minus modulating frequency.
If age or weaknes doe prohibyte bloudletting you must use boxing

Re: Why Do Partials Exist In FM signals?

Reply #7
If the modulating frequency is to close to carrier frequency you have practically a ring modulator. So in your example in original post, modulating frequency is almost half the carrier, and you get partials at carrier plus / minus modulating frequency.

I only use a 1Hz modulating frequency as a way of imagining the frequency of the carrier sweeping up and down at a tangible rate; it's just to get an idea of what is happening over time when we have a much higher modulating frequency.  

Re: Why Do Partials Exist In FM signals?

Reply #8
Yes I assumed from your terminology above that you were not familiar with the topic. If you've read that derivation above, then maybe you could clarify what is your question?

 My question is simply put: why does sweeping back and forth through a range of frequencies at very fast rates(FM) create 'sidebands' at intervals of the modulating frequency, and not for e.g at some other arbitrary spectral locations?    e.g  if the carrier is 5000Hz and the modulator is 3000 Hz, we get the first pair of sidebands at 2000Hz and at 8000Hz.  Why do we specifically get something happening(a frequency) 2000 times a second and another thing happening 8000 times a second due to the fact we are sweeping up and down through the frequency ranges at either side of the carrier wave, at a rate of 3000 times per second?   What is happening in the signal that makes 2000Hz and 8000Hz audible?  Why are the second pair of sidebands lower in amplitude that the first pair?

 I'm just asking how this works in terms of waves interfering to produce the spectral result that we get as determined by the mathematics we already know.  I'm trying to visualize the process of a sideband being created essentially. 

Re: Why Do Partials Exist In FM signals?

Reply #9
You must realize that when you change the frequency of a sine wave you change its spectrum, right?  There is no "break point" at which "partials" become evident, it's simply the case that as you up the modulation derivative, they get wider and wider.
-----
J. D. (jj) Johnston

Re: Why Do Partials Exist In FM signals?

Reply #10
I would like to know why partials occur at all in FM signals. 

I believe that by partial, you are referring to what is otherwise called sideband(s).

Sidebands are a result of modulation, whether AM or FM.

Modulation is a form of nonlinear distortion, and since nonlinear distortion adds additional tones to the modulated signal, sidebands have to be added.

FM is very complex because heavier FM modulation adds more different sidebands. 

This article may help:      http://www.radio-electronics.com/info/rf-technology-design/fm-frequency-modulation/spectrum-bandwidth-sidebands.php


Re: Why Do Partials Exist In FM signals?

Reply #11
I would like to know why partials occur at all in FM signals. 

I believe that by partial, you are referring to what is otherwise called sideband(s).

Sidebands are a result of modulation, whether AM or FM.


If I could try to rephrase the question, I'm essentially trying to visualize this process in the waveform over time.  I'm having difficulty imaging how the sidebands form as a product of the modulation.  

Essentially I'm asking about how the modulation works in a physical sense to create the FM wave form bit by bit.  This is why my examples always used low modulating frequencies - because we can see exactly what the modulation is doing to the carrier, and that is moving the frequency up and down.  I can imagine that - it's a waveform where the crests and troughs get gradually closer together up until a maximum frequency, and then get wider apart until a minimum frequency.  What I can't quite imagine is how  within that range of frequencies, the 'sideband' frequencies become audible while the other frequencies we are sweeping through don't? 

I find that the links people share with me can break down the process mathematically, but the focus is never on visualizing the process of how modulation physically creates a sideband component by varying the carrier frequency. 

 

Re: Why Do Partials Exist In FM signals?

Reply #12
First study the relation of Bessel Functions to FM or Phase Modulation.  That will tell you what sidebands are generated for a given deviation of the carrier by a given audio frequency.

Also, realize that as you are changing the frequency of the carrier "on the fly", you are creating instantaneous distortion of the sine wave carrier, it is no longer a sine wave, hence it has distortion which by definition creates "sidebands".  In fact the sidebands of a FM modulated carrier go out to infinity, but with decreasing amplitude, and nulls at certain frequencies.  That's where the Bessel functions come in.  The narrower you filter the modulation products, as in the IF stages of a receiver, the more distortion will be present in the recovered audio when you demodulate.

Re: Why Do Partials Exist In FM signals?

Reply #13
What I can't quite imagine is how  within that range of frequencies, the 'sideband' frequencies become audible while the other frequencies we are sweeping through don't? 

I find that the links people share with me can break down the process mathematically, but the focus is never on visualizing the process of how modulation physically creates a sideband component by varying the carrier frequency. 
 

I think the problem is that you're assuming that the instantaneous frequency and the power spectrum are the same thing, when really they're distinct concepts.  For a periodic signal, the power spectrum is determined by the Fourier series coefficients and represents the signal expressed as a sum of pure sinusoids.  The instantaneous frequency however is just the derivative of phase with respect to time.  The two concepts are related, but if you just sample values of the instantaneous frequency over some time interval you do not get the Fourier series coefficients. 

Re: Why Do Partials Exist In FM signals?

Reply #14
Don't feel bad if you can't get a non-mathematical feel for the specifics of what is going on.  FM modulation is only the tip of the iceberg.  You might also try modern physics or even digital sampling on for size.

Re: Why Do Partials Exist In FM signals?

Reply #15
The only real answer to such questions lies in the mathematics of electromagnetic radiation (and wave motion in general) as developed by James Clerk Maxwell and later included in special and general relatively by Einstein.  If you can't understand the math at that level, welcome to the club, but there is no "intuitive" explanation.  It's just predicted by the math and confirmed by actual observation.

Of course the math itself was developed to explain the prior observations.  When Maxwell developed his equations that unified electricity and magnetism in a single force, out popped a form of radiation that moves at the speed of light, and we now know that as electromagnetic radiation, also known as "radio".
Ed Seedhouse
VA7SDH

Re: Why Do Partials Exist In FM signals?

Reply #16
I would say that EM is another good example of things that fall right out in the math but are extremely difficult to visualize.

Other than that this topic is specifically about modulation, not radio.

Re: Why Do Partials Exist In FM signals?

Reply #17
... Also, realize that as you are changing the frequency of the carrier "on the fly", you are creating instantaneous distortion of the sine wave carrier, it is no longer a sine wave, hence it has distortion which by definition creates "sidebands". ...

When I read echogecko's original question I realised that although I knew the math I also had the same question. The description above brought it together for me. Although the math is different, it describes AM as well. Thank you.
Regards,
   Don Hills
"People hear what they see." - Doris Day

Re: Why Do Partials Exist In FM signals?

Reply #18
Right - the partial products (where the name partials comes from) are the result of a nonlinearity when mixing the carrier and modulating signal together.  Without a nonlinearity (which can be in the modulator or caused by dynamic changes in the waveforms) all you would get would be the modulating signal and the carrier as distinct signals.  Obviously, this would be hard to do in a real world system since nonlinearities are in almost all analog systems to some extent.

Incidentally, a ring modulator is essentially a double-sideband modulator, or an AM modulator which suppresses the carrier.  A frequency shifter (so-called) is like a single-sideband system, wherein the output is a copy of the input but shifted by the modulation signal, which can be AC or DC.

Re: Why Do Partials Exist In FM signals?

Reply #19
An enlightening thing to try:

take a 2^20 length array in octave

use it as the frequency domain representation of a time signal

Treat the 2^19+1 value as fs/2, and set fs to 48000 (not necessary but makes the math easy.

put a line equal to 1 at the nearest line to 500 Hz in the postive frequency space.

put a line equal to .25 4 Hz lower, and .25 4 Hz higher. (same spacing in the FFT, please, around the center frequency.

conjugate the negative frequencies, and take the ifft. look at the waveform.

Now, chance one (either one) of the .25 values to -.25

tahe the ifft of that.

Look at that waveform.

One is AM, the other is narrowband FM.

Just for hoots scale them up and listen to them.

If you consider, they have precisely the same magnitude spectrum, too, yes?

There are several things to be learned from that.

Hmm. Let me write that out for you:

clc;
clear all
close all

l=2^20;

fs2=24000;
fs=2*fs2;

xt(1:l)=0;

ctr=round(l*500/fs)+1;
del=round(l*4/fs);

xt(ctr)=1;
xt(ctr-del)=.25;
xt(ctr+del)=.25;

xt(l:-1:(l/2+2))=conj(xt(2:(l/2)));

subplot(4,1,1)
plot(xt((ctr-3*del):(ctr+3*del)));
axis([ 1 6*del+1 -1 1])

x=ifft(xt);
subplot(4,1,2)
plot(x);

xt(ctr+del)=-.25;

xt(l:-1:(l/2+2))=conj(xt(2:(l/2)));
subplot(4,1,3)
plot(xt((ctr-3*del):(ctr+3*del)));
axis([ 1 6*del+1 -1 1])


y=ifft(xt);
subplot(4,1,4)
plot(y)

Hopefully that's self-explanatory?
-----
J. D. (jj) Johnston

Re: Why Do Partials Exist In FM signals?

Reply #20
I'm basically rephrasing what others have said before me, but it helps me to mentally visualise the reason for sidebands in a non-mathematical way by thinking thus...

Take a carrier wave and don't modulate it. You have no sidebands. Shift the carrier to a different fixed frequency. You still have no sidebands. Instantaneously step the carrier between these two frequencies at one of the two zero-crossing points. You still have no sidebands.

Waggle the carrier around in an analogue manner between these two frequencies. You have sidebands. Why? Because the carrier is no longer a pure sine wave but a distorted version of a sine wave. This carrier distortion forms the sidebands. Simples... unless I'm wrong.  ;)

Re: Why Do Partials Exist In FM signals?

Reply #21
... Take a carrier wave and don't modulate it. You have no sidebands. Shift the carrier to a different fixed frequency. You still have no sidebands. Instantaneously step the carrier between these two frequencies at one of the two zero-crossing points. You still have no sidebands. ...

You'll get sidebands at the moment of switching... :-)
Regards,
   Don Hills
"People hear what they see." - Doris Day