Forums

Frequency Modulation Spectra

Started by Dr. Darth December 19, 2008
The frequency spectrum of an FM signal with a sinusoidal modulating signal 
is a discrete infinite frequency series where the carrier and sidebands are 
given by the Bessel functions, which are themselves a function of the 
modulation index.

But when the FM signal is modulated by TWO sinusoidal  signals, is the 
spectrum represented by TWO infinite frequency series?  Or is the spectrum 
more complex than that, representing the sum of the modulating signals?



Dr. Darth wrote:

> The frequency spectrum of an FM signal with a sinusoidal modulating signal > is a discrete infinite frequency series where the carrier and sidebands are > given by the Bessel functions, which are themselves a function of the > modulation index. > > But when the FM signal is modulated by TWO sinusoidal signals, is the > spectrum represented by TWO infinite frequency series?
There will be also cross products of both components of modulation.
> Or is the spectrum > more complex than that, representing the sum of the modulating signals?
It is much more complicated. AFAIK there is no closed form solution for this case. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Thank you.  Question in more specific form:

An FM signal is modulated with a sinusoidal modulating signal, producing 
carrier and sideband amplitudes given by the Bessel functions.

Now, let's add a second sinusoidal modulating signal.  My question is what 
that does to the original sideband series.

Do the original sidebands remain discrete components at the same frequencies 
(spaced at the modulating frequency)?  And what has happened to the carrier 
component now, since both modulating signals in the general case will have 
Jo terms.



On Dec 19, 5:47&#2013266080;pm, "Dr. Darth" <intentionally_l...@blank.com> wrote:
> Thank you. &#2013266080;Question in more specific form: > > An FM signal is modulated with a sinusoidal modulating signal, producing > carrier and sideband amplitudes given by the Bessel functions. > > Now, let's add a second sinusoidal modulating signal. &#2013266080;My question is what > that does to the original sideband series. > > Do the original sidebands remain discrete components at the same frequencies > (spaced at the modulating frequency)? &#2013266080;And what has happened to the carrier > component now, since both modulating signals in the general case will have > Jo terms.
interesting question, FM is a non-linear process so you can't just "ADD" the results.. But you can visuallize what happens more easily if you think about one high freq modulation tone, like 20kHz and another very low like 20 Hz. All the sidebands of the 20 kHz will be slowly swept back and forth by 20 Hz. In a sense it is like FM from the second tone is applied to each and every sideband of the first tone. If you can, best thing to do is get a generator and spectrum anlayzer and play.. Mark
On Fri, 19 Dec 2008 14:47:22 -0800, Dr. Darth wrote:

> Thank you. Question in more specific form: > > An FM signal is modulated with a sinusoidal modulating signal, producing > carrier and sideband amplitudes given by the Bessel functions. > > Now, let's add a second sinusoidal modulating signal. My question is > what that does to the original sideband series. > > Do the original sidebands remain discrete components at the same > frequencies (spaced at the modulating frequency)? And what has happened > to the carrier component now, since both modulating signals in the > general case will have Jo terms.
Please reply to the correct message, and retain context next time. This is USENET, even if you are going through Google Groups. Because FM is not a linear process, when you add the second modulating signal it messes up those original sidebands. _If_ the two modulating sinusoids have an "easy" harmonic relationship (i.e. 500Hz and 750Hz), then I'm pretty sure you'll have discrete components separated at the fundamental of the new modulating wave (250Hz in this example). The rest is hard to predict. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Dr. Darth wrote:

> Thank you. Question in more specific form: > > An FM signal is modulated with a sinusoidal modulating signal, producing > carrier and sideband amplitudes given by the Bessel functions. > > Now, let's add a second sinusoidal modulating signal. My question is what > that does to the original sideband series.
Depending on the particular frequency relation and the modulation indexes, the spectrum can or can not have the original components.
> Do the original sidebands remain discrete components at the same frequencies > (spaced at the modulating frequency)? And what has happened to the carrier > component now, since both modulating signals in the general case will have > Jo terms.
It depends. The spectrum will have all sorts of components from the row NxFcarrier +/- MxF1 +/- KxF2, however the distribution of phases and amplitudes will be different from the cases of either single F1 or single F2. What is the point of your question? Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
On Dec 20, 11:04&#2013266080;am, "Dr. Darth" <intentionally_l...@blank.com> wrote:
> The frequency spectrum of an FM signal with a sinusoidal modulating signal > is a discrete infinite frequency series where the carrier and sidebands are > given by the Bessel functions, which are themselves a function of the > modulation index. > > But when the FM signal is modulated by TWO sinusoidal &#2013266080;signals, is the > spectrum represented by TWO infinite frequency series? &#2013266080;Or is the spectrum > more complex than that, representing the sum of the modulating signals?
Do you mean adding at baseband or adding a second FM carrier modulated signal?
I was referring to adding a baseband signal to the same modulator - in other 
words, two modulating sinusoids into one modulator.


"HardySpicer" <gyansorova@gmail.com> wrote in message 
news:ccb56bd0-5cfd-4961-b908-87b64dd32270@z6g2000pre.googlegroups.com...
On Dec 20, 11:04 am, "Dr. Darth" <intentionally_l...@blank.com> wrote:
> The frequency spectrum of an FM signal with a sinusoidal modulating signal > is a discrete infinite frequency series where the carrier and sidebands > are > given by the Bessel functions, which are themselves a function of the > modulation index. > > But when the FM signal is modulated by TWO sinusoidal signals, is the > spectrum represented by TWO infinite frequency series? Or is the spectrum > more complex than that, representing the sum of the modulating signals?
Do you mean adding at baseband or adding a second FM carrier modulated signal?
Dr. Darth wrote:
> I was referring to adding a baseband signal to the same modulator - in other > words, two modulating sinusoids into one modulator.
The math becomes intractable. The sidebands are not symmetric about the carrier. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Jerry Avins <jya@ieee.org> wrote in
news:iE83l.65356$JU5.59230@newsfe20.iad: 

> Dr. Darth wrote: >> I was referring to adding a baseband signal to the same modulator - >> in other words, two modulating sinusoids into one modulator. > > The math becomes intractable. The sidebands are not symmetric about > the carrier. > > Jerry
Really? I thought the spectra would be symmetric as long as the modulating waveform was real. I haven't looked at a spectrum analyser in years though. To the OP: the maths is intractable in the general case, but we might be able to make some simplifications if certain conditions apply. If the small angle approximation holds (i.e. the peak phase modulation is much less than perhaps 0.1 radian), then we can regard the entire process as linear and the spectra add just like in AM (or QAM). This is the same condition that allows us to use the approximation: sin x = x = tan x This is called NBFM (narrow band FM). C.F. WBFM (wide band FM), in which this assumption does not hold. Regards, Allan