# frequency modulation?

#### suzuki

Joined Aug 10, 2011
119
Hi,

Not too familiar with communication theory myself, so i thought i would post this here.

when we talk about frequency modulation, we usually have an equation in the form of

y = sin(2*pi*(f+f_mod))

This is fine, but i am wondering if FM applies the same way to the harmonics of sinusoidal signals. For example, a square wave is composed of an infinite number of harmonics plus the fundamentals frequency. If i were to apply FM to a square wave, would the higher order harmonics be affected by the same way as the fundamental frequency, or would something else happen? i.e. would the frequency just change by f_mod?

tia

#### t_n_k

Joined Mar 6, 2009
5,455
Interesting problem. I'm not sure how to solve it mathematically.

Presumably you envisage a sinusoidal modulation of the base square wave frequency.

I took a 'simplistic' non-mathematical approach and generated a 1kHz square wave with a sinusoidal 200Hz frequency modulation superimposed. I then took the FFT of the generated data.

The partial amplitude spectrum is shown in the attached plot. Horizontal axis is the frequency in Hz.

One can see the 200Hz modulation components at each of the 1KHz square wave (odd) harmonics. Interestingly there are harmonics of the modulation signal appearing as well. I'm not sure if this is an artifact of the signal generating process, the applied FFT algorithm or a real phenomenon.

#### Attachments

• suzuki

#### suzuki

Joined Aug 10, 2011
119
thanks for the reply t_n_k. this is quite interesting. Just wondering what program you used to do this? This is a pretty strange result since it looks like the sidebands of the frequency modulation are even larger than the sidebands of the carrier signal, which to me implies that we cannot neglect some additional harmonics created due to the harmonics of the carrier signal!

i used the square wave as an example, but for my intention, it would probably be sufficient to approximate it as a 5th harmonic function. so far, i have also not been able to find any type of mathematical method, but i will post it if i do come across anything interesting!

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#### t_n_k

Joined Mar 6, 2009
5,455
I can generate and analyze the signals using Labview.

I suspect the particular manifestations of harmonics etc. is subject to a couple of factors including the modulation frequency and index. Interactions at certain carrier and modulation frequency harmonics would also be of interest.

I imagine one would have to investigate a range of values to gain a more complete picture.

#### suzuki

Joined Aug 10, 2011
119
hmm, I tried to do a quick simulation in matlab, but I'm not quite getting what I'm looking for. I plotted 2 frequency plots, one using a sinusoidal carrier, and one using a square carrier. the code i used is

Rich (BB code):
t = 1:1000;
f = 1:1000;

m = 0.1*cos((2*3.14*100)*t);
cosine_FM = sin((2*3.14*60)*t+m);
square_FM = square((2*3.14*60)*t+m);

fft_cos = fft(cosine_FM);
fft_square = fft(square_FM);
the resulting plots are: where blue is the sinusoidal carrier and the red is the square wave carrier.

it seems strange to me that both of these spectrums show the fundamental frequency somewhere that is around 33Hz/970Hz. I can't say I am sure why that is.

In the "red" plot, I think i can assume that the 2nd and 3rd largest spikes are the 3rd and 5th harmonics. i suppose the location of these make sense since the 3rd and 5th harmonics should be located at either 3 times or 5 times the fundamental.

I think you are correct in that one of the factors is the index. so, i also tried to change the amplitude of the message signal "m" (the modulation index). it seems like if the mod index is small enough, there is little to no effect on the location of the first third and fifth harmonic. I also expected the amplitudes to be either 1/3 or 1/5 of the fundamental amplitude, but unfortunately, this is not the case.

Does it seem like I am missing something here? sorry if this sounds like rambling but I am just trying to put alot of information together somehow #### t_n_k

Joined Mar 6, 2009
5,455
A couple of points re your code

1. You seem to have the modulation frequency higher than the carrier frequency - fm=100Hz and fc=60Hz ...???
2. The time step [1 sec] is probably too coarse
3. One would normally plot abs(fft ....) to give the magnitude plot.
4. Re the two peak frequencies in the fft ... remember the important part of the fft response is below the Nyquist frequency. The full fft is symmetrical about the Nyquist frequency - excluding the DC term.

• suzuki

#### t_n_k

Joined Mar 6, 2009
5,455
Had another look at the problem.

Attached is a spectrum plot for a square wave sinusoidally frequency modulated signal.

The modulation index is '1' with a carrier of 1Hz and modulating frequency of 0.1 Hz.

Interestingly the [odd only] harmonics of the 1Hz carrier have modulation harmonics distributed according to a modulating index of 'n' where n is the harmonic number. So the for the carrier 5th harmonic the modulation harmonic distribution is typical of a modulating index of 5. For the 3rd harmonic the distribution is typical of a modulating index of 3.

The result is probably not that surprising.

#### crutschow

Joined Mar 14, 2008
30,438
This is an interesting academic exercise, but I can think of no practical reason to FM modulate a square wave. #### t_n_k

Joined Mar 6, 2009
5,455
That's true. If it's really homework one wonders why a prof would set such an esoteric task. But the question was asked & it was a little more interesting than the usual circuit theory fare.
It was actually an interesting challenge to correctly generate the FM square wave. Maybe this should go in the math forum.

#### suzuki

Joined Aug 10, 2011
119
those are some pretty interesting results t_n_k. I do find it strange that even for a modulation index of 1, that the side bands of the harmonics are greater than the "main" component of the harmonics. I had thought this only really starts to happen when your modulation index becomes greater than 1.

typically (or what i have read so far), i also think we usually assume that this modulation index is much less than 1, which eliminates the need to consider higher order sidebands, but it looks like by doing so, we are missing a lot of information about the harmonics. i suppose this means that we need to be "smart" when we choose a modulation index and go with something small.

#### t_n_k

Joined Mar 6, 2009
5,455
So I would imagine setting the modulation index to 0.2 would result in a "better" outcome from your perspective.

#### t_n_k

Joined Mar 6, 2009
5,455
Here is a similar FFT plot but with the modulation index changed to 0.2.

#### t_n_k

Joined Mar 6, 2009
5,455
those are some pretty interesting results t_n_k. I do find it strange that even for a modulation index of 1, that the side bands of the harmonics are greater than the "main" component of the harmonics. I had thought this only really starts to happen when your modulation index becomes greater than 1.

typically (or what i have read so far), i also think we usually assume that this modulation index is much less than 1, which eliminates the need to consider higher order sidebands, but it looks like by doing so, we are missing a lot of information about the harmonics. i suppose this means that we need to be "smart" when we choose a modulation index and go with something small.
Keep in mind the original point of this discussion was in reference to a frequency modulated square wave rather than a typical FM sine wave. The comments you make with respect to the consequences of modulation index are also normally made with respect to a sinusoidal system. Would one ever comtemplate an FM transmission involving a square wave carrier? Highly unlikely. As crutschow points out in their earlier post, this is a somewhat academic discussion with little practical relevance..