# Frequency Modulation Spectra

Started by December 19, 2008
```>
> > The math becomes intractable. The sidebands are not symmetric about
> > the carrier.
>
> > Jerry
>
> Really? &#2013266080;I thought the spectra would be symmetric as long as the
> modulating waveform was real. &#2013266080;I haven't looked at a spectrum analyser in
> years though.
>
Yes I was thinking about that too..   I thought the sidebands would
always be symmetrical as long as we were talking about pure FM with no
AM combined  and the modulating waveform was symmetrical around 0.
But what if the modulating wavform is f + 2f such that it's not
symmetrical around 0 and the + deviation is not equal to the -
deviation, then the sidebands might be asyymetrical..

I think I'm going to hook  up a generator to the spectrum analyzer and
see.

```
```On Dec 19, 4:29&#2013266080;pm, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
wrote:

> It is much more complicated. AFAIK there is no closed form solution for
> this case.

Very true.  The expression for the spectrum of the
general FM signal is messy because of the need
to take various special cases (e.g. narrowband FM
versus wideband FM) into account.  But some special
cases can be obtained in "closed form".  For example,
if the modulator signal is

x(t) = M_1 sin(2 pi f_1t) + M_2 sin(2 pi f_2 t)

where |M_1| + |M_2| < 1 (so that |x(t)| < 1), then
sin(2 pi f_c t - pi x(t)), a **phase-modulated** signal at
carrier frequency f_c  with a maximum phase deviation
of plus/minus pi can be expressed as an unmodulated
carrier signal (2/pi) sin(2pi f_c t) together with a sum
of signals

J_m(pi M_1)J_n(pi m_2)sin(2 pi [f_c - mf_1 - nf_2]t)

at frequencies f_c - mf_1 - nf_2 where m and n range from
-infinity to +infinity, and J_m, J_n denote ordinary Bessel
functions of orders m and n respectively.  Since frequency
modulation is a special case of phase modulation, this result
does give the spectrum of at least one FM signal.  Which
one is left as an exercise for the reader.

As noted by many respondents in this discussion, there are
tones separated by mf_1 from the carrier, by nf_2 from the
carrier (harmonics of the modulating tones), and also by
mf_1 + nf_2 (intermodulation products of the modulating tones).
Additional matters worth mentioning are that (i) every multiple of
the greatest common divisor of f_1 and f_2 can be expressed
as mf_1 + nf_2, so that if f_c = 1 MHz, f_1 = 100 Hz and
f_2 = 225 H_z, then there will be spectral lines every 25 Hz,
(note that -2f_1 + f_2 = 25 Hz) and (ii) the amplitude of a tone
at a particular frequency is the sum of a series of Bessel function
products in general.  This is because (in our example)
9f_1 - 4f_2 = 0 and so not only does -2f_1 + f_2 = 25 Hz,
but 7f_1 -3f_2 = 25 Hz as well, and 16f_1 - 7f_2 = 25 Hz too,
and so on.  Nitpickers are welcome to think of the case when f_1
f_2 are incommensurate.....

The result above generalizes to more than two tones.  More details
can be found in a paper available at
(http://www.ifp.uiuc.edu/~sarwate/pubs/zsdvs.ps)
(see especially Section 4 and Eqs (71) and (72).  Surfers and
the frequency spectrum of PWM signals, and the phase modulation
result is an incidental byproduct...)

Hope this helps...

--Dilip Sarwate

```
```dvsarwate@yahoo.com wrote:
> On Dec 19, 4:29 pm, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
> wrote:
>
>> It is much more complicated. AFAIK there is no closed form solution for
>> this case.
>
> Very true.  The expression for the spectrum of the
> general FM signal is messy because of the need
> to take various special cases (e.g. narrowband FM
> versus wideband FM) into account.  But some special
> cases can be obtained in "closed form".  For example,
> if the modulator signal is
>
> x(t) = M_1 sin(2 pi f_1t) + M_2 sin(2 pi f_2 t)
>
> where |M_1| + |M_2| < 1 (so that |x(t)| < 1), then
> sin(2 pi f_c t - pi x(t)), a **phase-modulated** signal at
> carrier frequency f_c  with a maximum phase deviation
> of plus/minus pi can be expressed as an unmodulated
> carrier signal (2/pi) sin(2pi f_c t) together with a sum
> of signals
>
> J_m(pi M_1)J_n(pi m_2)sin(2 pi [f_c - mf_1 - nf_2]t)
>
> at frequencies f_c - mf_1 - nf_2 where m and n range from
> -infinity to +infinity, and J_m, J_n denote ordinary Bessel
> functions of orders m and n respectively.  Since frequency
> modulation is a special case of phase modulation, this result
> does give the spectrum of at least one FM signal.  Which
> one is left as an exercise for the reader.

Is there a difference between phase-and frequency modulation? As far as
I know, a differentiator in the modulator converts one to the other.

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;
```
```Allan Herriman wrote:
> Jerry Avins <jya@ieee.org> wrote in
>
>> 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.

I haven't gone through my old texts, but unless it has changed since the
1950s, yes, really.

> 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.

Note that NBFM is essentially double-sideband AM with the carrier
shifted 90 degree. Armstrong's early crystal-stabilized FM modulators
used this.

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;
```
```On Dec 20, 10:06&#2013266080;pm, "dvsarw...@yahoo.com" <dvsarw...@gmail.com>
averred:
>.......
>.......
> then
> sin(2 pi f_c t - pi x(t)), a **phase-modulated** signal at
> carrier frequency f_c &#2013266080;with a maximum phase deviation
> of plus/minus pi can be expressed as an unmodulated
> carrier signal (2/pi) sin(2pi f_c t) together with a sum
> of signals
>
> J_m(pi M_1)J_n(pi M_2)sin(2 pi [f_c - mf_1 - nf_2]t)
>
> at frequencies f_c - mf_1 - nf_2 where m and n range from
> -infinity to +infinity, and J_m, J_n denote ordinary Bessel
> functions of orders m and n respectively.
>&#2013266080;......

The phrase beginning "an unmodulated carrier..." and
ending ".... together with" should be deleted from this
assertion.  There *can* be a tone at the carrier frequency
but it arises from terms such as

J_0(pi M1)J_0(pi M_2)sin(2 pi f_c t)

and

J_m(pi M_1)J_n(pi M_2)sin(2 pi [f_c - mf_1 - nf_2]t)

where mf_1 + nf_2 = 0, etc.

--Dilip Sarwate

```
```On Dec 20, 9:57&#2013266080;pm, makol...@yahoo.com wrote:
> > > The math becomes intractable. The sidebands are not symmetric about
> > > the carrier.
>
> > > Jerry
>
> > Really? &#2013266080;I thought the spectra would be symmetric as long as the
> > modulating waveform was real. &#2013266080;I haven't looked at a spectrum analyser in
> > years though.
>
> Yes I was thinking about that too.. &#2013266080; I thought the sidebands would
> always be symmetrical as long as we were talking about pure FM with no
> AM combined &#2013266080;and the modulating waveform was symmetrical around 0.
> But what if the modulating wavform is f + 2f such that it's not
> symmetrical around 0 and the + deviation is not equal to the -
> deviation, then the sidebands might be asyymetrical..
>
> I think I'm going to hook &#2013266080;up a generator to the spectrum analyzer and
> see.

I did the experiment...
with two tones, if the tones are unrelated and therefore the combined
modulating waveform is symmetrical around zero, then the sidebands are
symmetrical around the carrier.   But for the speical case where the
two tones are harmonically related, then the phase realtionship
controls the symmetry of the modulating  waveform around zero and also
the symmetry of the sidebands around the carrier.

Conclusion: a symmetrical modulating waveform yields symmetrical
sidebands.

Mark

```
```Please clarify "... the combined modulating waveform is symmetrical around
zero ..."  Symmetrical in what sense?  Waveform amplitude above and below
zero, where zero input voltage represents the nominal modulator carrier
output frequency?

I have in mind two audio generators summed at the modulator input
capacitively coupled with zero DC offset.  Would that be what you mean by
symmetrical?

<makolber@yahoo.com> wrote in message
On Dec 20, 9:57 pm, makol...@yahoo.com wrote:
> > > 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.
>
> Yes I was thinking about that too.. I thought the sidebands would
> always be symmetrical as long as we were talking about pure FM with no
> AM combined and the modulating waveform was symmetrical around 0.
> But what if the modulating wavform is f + 2f such that it's not
> symmetrical around 0 and the + deviation is not equal to the -
> deviation, then the sidebands might be asyymetrical..
>
> I think I'm going to hook up a generator to the spectrum analyzer and
> see.

I did the experiment...
with two tones, if the tones are unrelated and therefore the combined
modulating waveform is symmetrical around zero, then the sidebands are
symmetrical around the carrier.   But for the speical case where the
two tones are harmonically related, then the phase realtionship
controls the symmetry of the modulating  waveform around zero and also
the symmetry of the sidebands around the carrier.

Conclusion: a symmetrical modulating waveform yields symmetrical
sidebands.

Mark

```
```> Please clarify "... the combined modulating waveform is symmetrical around
> zero ..." &#2013266080;Symmetrical in what sense? &#2013266080;Waveform amplitude above and below
> zero, where zero input voltage represents the nominal modulator carrier
> output frequency?

Yes that is what I meant.  Symmetrical above and below zero Volts
where zero Volts into the modulator results in zero freq deviation
from the nominal carrier frequency.

>
> I have in mind two audio generators summed at the modulator input
> capacitively coupled with zero DC offset. &#2013266080;Would that be what you mean by
> symmetrical?
>

In the typical case YES. But not just the AVERAGWE voltage above and
below zero, but also the peak and the shape are symmetrical above and
below zero Volts.   This will be true for the combination of two
typical unrelated sine tones.

But... I also tested and spoke about a special pathological case where
one generator frequency was set to exactly  2x the other generator
frequency (I used 5 kHz and 10 kHz) so that we have F and 2F. Then I
could vary the phase relationship between them.  This results in a
tone with a  second harmonic which is NOT a symmetrical signal around
zero volts. (If you AC couple, the average voltage above and below
zero may (must) be the same, but not the peak.  The waveform is
asymmetrical in voltage.)   In this special pathological case the
sidebands also are not symmetrical and I could change the upper
sideband re lower sideband by adjusting the phase between the two
tones and that also effects the asymmetry of the voltage waveform.

If you can, I strongly suggest connecting a generator to a spectum
analyzer and playing.

P.S. sorry for any spelling errors.

Mark
```
```Very helpful.  Thank you.  I plan to do some tests here as you suggest.

<makolber@yahoo.com> wrote in message

> Please clarify "... the combined modulating waveform is symmetrical around
> zero ..." Symmetrical in what sense? Waveform amplitude above and below
> zero, where zero input voltage represents the nominal modulator carrier
> output frequency?

Yes that is what I meant.  Symmetrical above and below zero Volts
where zero Volts into the modulator results in zero freq deviation
from the nominal carrier frequency.

>
> I have in mind two audio generators summed at the modulator input
> capacitively coupled with zero DC offset. Would that be what you mean by
> symmetrical?
>

In the typical case YES. But not just the AVERAGWE voltage above and
below zero, but also the peak and the shape are symmetrical above and
below zero Volts.   This will be true for the combination of two
typical unrelated sine tones.

But... I also tested and spoke about a special pathological case where
one generator frequency was set to exactly  2x the other generator
frequency (I used 5 kHz and 10 kHz) so that we have F and 2F. Then I
could vary the phase relationship between them.  This results in a
tone with a  second harmonic which is NOT a symmetrical signal around
zero volts. (If you AC couple, the average voltage above and below
zero may (must) be the same, but not the peak.  The waveform is
asymmetrical in voltage.)   In this special pathological case the
sidebands also are not symmetrical and I could change the upper
sideband re lower sideband by adjusting the phase between the two
tones and that also effects the asymmetry of the voltage waveform.

If you can, I strongly suggest connecting a generator to a spectum
analyzer and playing.

P.S. sorry for any spelling errors.

Mark

```