# Do changing sidebands have their own sidebands?

An amplitude changing carrier wave causes sidebands to appear next to it when viewed in the frequency domain. But if those sidebands themselves also change with time (because the modulating signal is not constant), shouldn't those themselves behave the same as a carrier wave and produce their own side bands? These secondary sidebands should then also have their own sidebands again, ad infinitum. Although the amplitude of those higher order sidebands will probably diminish exponentially. There are no laws of physics specific to carriers or sidebands, they all just follow the same laws of electromagnetic waves.

I read everywhere that the bandwidth of an AM modulated signal is twice the bandwidth of the modulating input. But according to the above logic, an AM signal should have an infinite bandwidth, in the same way that FM signals have. What am I missing here?

• In this case the hand-wavey explanation of the changing carrier having sidebands isn't helping you enough. But try this: Wikipedia has a short section that shows the basic maths, which ends "Using ... identities, y(t) can be shown to be the sum of three sine waves ... Therefore, the modulated signal has three components: the carrier wave c(t) which is unchanged in frequency, and two sidebands with frequencies slightly above and below the carrier frequency fc." I can't do that derivation but the result is pretty clear in the end. Jun 11, 2022 at 15:05
• I love this question but haven't had a chance to really puzzle it through (a.k.a. scour the web until I find someone else's explanation ;-). I'm not sure many of the current answers quite settle the situation at its core. One thing to consider though is that the rate of change of the "modulating input" tends to be much slower than any of the other frequencies involved. Perhaps some 0.5–3 Hz additional splatter simply isn't worth accounting for within a 3000–5000 Hz signal bandwidth? Jul 6, 2022 at 23:15

Part of your problem is in thinking that "a changing carrier causes sidebands to appear next to it" is a physical cause and effect. The carrier doesn't create the sidebands, the sidebands are the changes in the carrier. They're two different mathematical descriptions of the same thing. It's just a mathematical fact that any physical or mathematical object that responds to a 1000Hz sine wave will also respond to a 900Hz sine wave that changes its amplitude with a period of 100Hz.

These secondary sidebands should then also have their own sidebands again, ad infinitum. [...] But according to the above logic, an AM signal should have an infinite bandwidth

Nope. You could look at things the way you suggest, with the sidebands viewed as an infinite collection of individual time-varying signals, and calculate the sidebands for each one. Each one would have its own bandwidth, but if you added them all up perfectly, you would find something funny: they would all cancel out! You would get the original signal, with zero energy outside of the original bandwidth. The sidebands don't have "lives of their own" to vary in any other way; they have this strict mathematical relation because of how they're created — or rather what they are, which is a signal with a certain center frequency and bandwidth.

• Perhaps you can say, "A change can be made in a carrier's amplitude without causing distortion by adding in another (pure) sinusoid of a different frequency"? Apparently, you can't change frequency while keeping amplitude constant, without including a bevy of sidebands. Jun 11, 2022 at 19:08
• I love the question and this answer. Jul 25, 2022 at 12:38

For a linear modulator the frequency spectrum of the baseband modulating signal is replicated just once into each sideband. It doesn't keep being copied. So each sideband will have the same bandwidth as the modulating signal. A double sideband signal, which includes broadcast AM, has two sidebands and therefore has a bandwidth which is double the modulating signal bandwidth.

In the Americas broadcast AM stations need to fit into a 10 kHz bandwidth so are limited to a maximum of 5 kHz audio bandwidth. (Strictly that's a maximum modulating frequency of 5 kHz but where the minimum modulating frequency is much smaller than the maximum we usually assume it is zero when calculating the bandwidth, even though AM broadcast signals don't actually go down to 0 Hz.)

• That is indeed what I read everywhere, so what is wrong with the reasoning in the question? What makes a changing wave get sidebands when you label it 'carrier' but not when you label it 'sideband'? Jun 11, 2022 at 9:57

There are two sidebands: Upper sideband and Lower sideband. You don't get an upper upper sideband. :) The carrier is merely the center frequency, and is not part of the audio modulation or the sidebands. Normally on demodulation, the carrier would be "zero beated" (moved to frequency 0) so would not be part of the demodulated audio, but if you are not exactly on the center frequency, it may be demodulated as a whistle. This can be very annoying, and most radios have an option to suppress it from the audio.

The sidebands are a function of the inherent symmetry of the audio, not a function of the RF itself, so you don't get sidebands of sidebands. It doesn't have an infinite bandwidth, because the audio itself is filtered to have a fixed bandwidth. It is possible to have a wider audio bandwidth that causes a wider RF bandwidth, and this can be seen on the amateur spectrum with people using uncalibrated SDR radios. This is not strictly legal and there have been a number of observer reports of it; doing this risks your license. AM signals don't have an infinite bandwidth for the same reason FM signals do not have an infinite bandwidth -- it isn't legal. The modes in use have fixed defined maximum bandwidths dictated by FCC regulations.

While you only get two sidebands (which are mirrored), it is possible to get multiple harmonics, where the signal is repeated multiple times on either side of the center frequency. Harmonics are a function of how the RF is generated, and again, this is something that is regulated. Good engineering design and FCC rules dictate that we put filters on the RF generator output to eliminate (or at least attenuate sufficiently) any such harmonics.

Generation of signal outside the intended frequency range is sometimes called splattering, and harmonics is one of the causes of this. Other causes include AGC issues and insufficient filtering of the audio input.

AM has two sidebands corresponding to the symmetry of the audio signal. Single sideband (carrier suppressed) is a more efficient modulation method that involves a more complicated modulation and demodulation circuit. The SSB demodulator must reconstruct the missing sideband for the audio to sound correct. Because SSB has neither the carrier nor the symmetry of AM, it is much harder to align both the center frequency and the reconstructed sideband, so it sounds characteristically strange when it is off slightly.

The r-f spectrum of a double-sideband "AM" transmitter includes the carrier wave and the upper and lower sidebands produced by the modulation of that transmitter.

It is being assumed in this question that the amplitude of the carrier itself must vary with modulation. However when an AM transmitter is optimally adjusted, the amplitude of the carrier remains very nearly constant with modulation.

When the total modulated spectrum is evaluated in the frequency domain, it can be seen that the carrier amplitude remains ~constant as long as the transmitter is not "overmodulated," and that the r-f spectra occupied by the upper and lower sidebands vary only as functions of the frequency spectrum of the audio waveform modulating the transmitter — not the amplitudes of those r-f sidebands.

The clip below published in a broadcast trade journal shows the frequency domain spectrum of an AM transmitter, and discusses this in more detail.

The sidebands are just the audio frequency signal moved up to the RF carrier frequency.

If you turn the soundwaves into an electrical signal and do the Fourier transform of that signal you will two sidebands -- one is the reflection of the other, being reflected around zero Hz.

This is because (as Fourier analysis shows us) any signal (even non-periodic ones) can be turned into a superposition of sines and cosines. And since:

$$sin(x) = \frac{e^{ix} - e^{-ix}}{2i}\\ cos(x) = \frac{e^{ix} + e^{ix}}{2}$$

you end up with negative frequency components that match the positive frequency components. Those positive and negative frequencies are the sidebands.

When you then do an AM modulation onto the carrier those frequencies are simply shifted by the carrier frequency but otherwise their relationships stay the same. You have the exact same spectrum as you had before, except now it's reflected around the carrier frequency instead of around zero Hz. This is now the classic AM spectrum -- a big spike at the carrier frequency representing the carrier and the two sidebands centered on it. To get SSB from that, you use filters to take out the carrier and one of the sidebands. The filter is simply eliminating frequencies outside of some desired range. There are no "sidebands of sidebands" because (a) those frequencies were never there in the original audio signal and because (b) even if they were, the filter blocks the frequencies outside of what we want.

(Re: modulation. Just to be a completist, when the modulation onto the carrier happens there is a whole separate carrier spike and sidebands down at the negative of the carrier frequency (since the carrier, being a sinusoid, also is a superposition of one positive frequency and one negative frequency, but that whole thing is filtered out whether you're doing SSB or standard AM.)

After looking in to this some more, I think the answer I was looking for is the following:

If the baseband signal that gets modulated onto a carrier is not a constant sinusoid but e.g. itself a sinusoid that is varying in amplitude, that baseband signal already has its own side bands. If this baseband is modulated onto a carrier, the resulting signal will have its own side bands, each of which is composed of the baseband signal with its own sidebands. So in a sense the secondary (and higher order) side bands are there, but they already exist as part of the baseband signal before it is modulated onto the carrier. No additional side bands appear as a result of the modulation other than the two well known ones.

Secondly, the uncertainty principle as applied to the time domain and frequency domain views of a wave guarantees that any signal that is localized in time is spread out in the frequency domain, and includes arbitrarily high frequencies (though at vanishing amplitudes). But for the case of AM modulation, such arbitrarily high frequencies would be part of the baseband as it is generated (assuming the baseband is a type of signal that is not localized in the frequency domain). In practice the very high frequencies are filtered out either by explicit filters or because components don't have an unlimited passband, and you end up with some finite bandwidth signal that is modulated onto the carrier.
I do wonder now if filtering out these high frequencies makes the signal non-localized in time, but I'll leave that to a different question.