# Why do you get sideband frequencies for amplitude modulation (AM)?

by CraigH
Tags: amplitude, band, frequencies, frequency, modulation, side bands, sideband, signal
 P: 185 I thought that the frequency was constant for AM modulation, and just the amplitude was modulated. So why are there a range of frequency's (side bands around the baseband) when the signal is plotted on a frequency domain graph?
 P: 3 The AM carrier frequency is constant but the modulation signal can change. If you look at the case of modulating the AM carrier frequency (a sine wave) with a pure sine wave, the resulting AM waveform is close to what you would get if you multiplied the two sine waves. AM mixing is like multiplication. In the realm of trigonometry, the product of two sine waves can be simplified to the sum of three pure sine waves: The carrier and two side bands. So if you multiply two sine waves (carrier and modulation), you get the same result as adding three sine waves (carrier and two side band sine waves). That result, the AM waveform, can be viewed in two different ways:through mixing (multiplication) or as a sum of three sine wave (which is the frequency domain way of looking at things).
P: 3
 Quote by kevlat The AM carrier frequency is constant but the modulation signal can change. If you look at the case of modulating the AM carrier frequency (a sine wave) with a pure sine wave, the resulting AM waveform is close to what you would get if you multiplied the two sine waves. AM mixing is like multiplication. In the realm of trigonometry, the product of two sine waves can be simplified to the sum of three pure sine waves: The carrier and two side bands. So if you multiply two sine waves (carrier and modulation), you get the same result as adding three sine waves (carrier and two side band sine waves). That result, the AM waveform, can be viewed in two different ways:through mixing (multiplication) or as a sum of three sine wave (which is the frequency domain way of looking at things).
I'd like to amend that. Trig can be used to simplify the product of two sine waves to a sum of the carrier and two sidebands. The sidebands are really cosines, which are phase-shifted sines.

P: 578

## Why do you get sideband frequencies for amplitude modulation (AM)?

Craig,
As kevlat indicates, the sidebands fall out of the trig. However, you are right in that it seems counterintuitive at first. In order to get a sense of where the other frequencies come from consider that the amplitude modulation is in the form of periodic instantaneous shifts in amplitude. Say we have a 1MHz RF carrier 1Vpeak. It periodically shifts in amplitude to 100Vpeak, and the shift occurs at the crest of the sine wave. So we have a 1MHz sinusoidal signal that at some point reaches 1V and shifts instantaneously to 100V then continues on as a sinusoid. That vertical 1V to 100V step in voltage is where the non-1MHz frequency components are introduced. We will create the same frequencies that you would get from a square wave with instantaneous edges.
 P: 2,432 A good rule to remember is that you cannot do ANYTHING to a sine wave without creating new frequencies.
 P: 4,664 See The product of a small amplitude modulation on a carrier can be represented by $\left(1+A\cos\omega_1t \right)\cos\omega_2t$. This can be rewritten as $$\left(1+A\cos\omega_2t \right)\cos\omega_1t=\cos\omega_1t +A\cos\omega_2t\cos\omega_1t =\cos\omega_1t + \frac{A}{2} \cos\left(\omega_1-\omega_2 \right)t+\frac{A}{2} \cos\left(\omega_1+\omega_2 \right)t$$ So the AM modulation produces sum and difference sidebands at any amplitude modulation frequency.
 P: 1 Hi, I'm new to the forum and also to the topic and so pardon me for maybe not doing the proper protocol. I have a follow-up related question. Do the sidebands frequencies exist during transmission? I mean does the transmitter actually transmits three signals at a time, the lower, upper and the carrier?
PF Gold
P: 11,129
 Quote by jettenazas Hi, I'm new to the forum and also to the topic and so pardon me for maybe not doing the proper protocol. I have a follow-up related question. Do the sidebands frequencies exist during transmission? I mean does the transmitter actually transmits three signals at a time, the lower, upper and the carrier?
Absolutely. The sidebands are part of the signal. If you pass the AM signal through a very narrow band-pass filter, you will just see the carrier and there will be no modulation on it. A.M. transmitters need to be designed with enough bandwidth to accommodate the carrier and all the sidebands (+and- the highest audio frequency about the carrier). The spectrum when a real audio programme signal is transmitted looks like a single carrier frequency with sidebands, each of which look like the audio signal spectrum. (Mirror images, aamof).

The process of Modulation is Non Linear - it's a sort of Multiplication - and it produces those extra frequency components (if you choose to think of it in the frequency domain). In the time domain, it looks like a carrier with a varying amplitude (the 'envelope'). That's the 'scope picture, where the frequency domain is the Spectrum Analyser picture of the same signal. Mentally hopping between time and frequency domains can be very handy.
 P: 863 This is my weird intuitive view, and can't stand up to any rigorous scrutiny. (be gentle, folks) When you measure the spectrum of a modulated sinewave you are measuring the average energy over a period of time. That makes it look like there are three distinct frequencies with related phases. Well, there are 3 distinct frequencies by any measurement methods, but if you could track instantaneous frequency (a debatable concept) you would sort of see a continuum of frequencies. But the energy between sidebands sort of cancels out when you average over an interval. Think of a sinewave. If you lower the amplitude, there is a frequency/phase discontinuity that occurs. That is, there has to be a transition that isn't exactly a sinewave. That is, or results in, "spurious" energy at different frequencies. This get more complicated when you think of FM modulation, where a continuous sweeping frequency somehow unintuitively becomes bessel sidebands. Again, it is "sort of" an "artifact" of the measurement method caused by the fact that it is an average over an interval. Don't get me wrong. This average is very real and very useful. Instantaneous frequency is tricky because a single point can't have frequency without knowledge of past and future. But, in contrast, most people don't really think about the "averaging" occuring in a spectrum analysis filter (be it FFT, DFT, Fourier based or whatever).
 PF Gold P: 202 Here is a good site that explains things. http://www.expertsmind.com/topic/dat...and-91056.aspx
PF Gold
P: 11,129
 Quote by sas3 Here is a good site that explains things. http://www.expertsmind.com/topic/dat...and-91056.aspx
Couldn't we just sort out DSBAM first, before launching into SSB? The whole idea of modulation is hard enough as it is.
PF Gold
P: 11,129
 Quote by meBigGuy This is my weird intuitive view, and can't stand up to any rigorous scrutiny. (be gentle, folks) When you measure the spectrum of a modulated sinewave you are measuring the average energy over a period of time. That makes it look like there are three distinct frequencies with related phases. Well, there are 3 distinct frequencies by any measurement methods, but if you could track instantaneous frequency (a debatable concept) you would sort of see a continuum of frequencies. But the energy between sidebands sort of cancels out when you average over an interval. Think of a sinewave. If you lower the amplitude, there is a frequency/phase discontinuity that occurs. That is, there has to be a transition that isn't exactly a sinewave. That is, or results in, "spurious" energy at different frequencies. This get more complicated when you think of FM modulation, where a continuous sweeping frequency somehow unintuitively becomes bessel sidebands. Again, it is "sort of" an "artifact" of the measurement method caused by the fact that it is an average over an interval. Don't get me wrong. This average is very real and very useful. Instantaneous frequency is tricky because a single point can't have frequency without knowledge of past and future. But, in contrast, most people don't really think about the "averaging" occuring in a spectrum analysis filter (be it FFT, DFT, Fourier based or whatever).
I think that is rather too much of a 'personal' view of things to help anyone else. You have mixed up so many concepts in your soup of buzzwords that the uninitiated will just get further confused, I'm afraid.
(I love the way that the "Bessel sidebands" just pop out of 'intuition'. They emerge from a pretty difficult bit of Fourier Transform and I don't think that's very intuitive at all)
 P: 863 Well, your reading skills could use some honing. I absolutely did say unintuitive. And, what I am saying is actually not so far off as you seem to think. It is not a "soup of buzzwords" as you insultingly state (you do that sort of thing a lot, it seems). There is actually some real stuff there regarding instantaneous frequency that you obviously do not understand. Think about narrowband FM and why it appears to have 2 sidebands, yet is made by a continuously sweeping VCO.
PF Gold
P: 11,129
 Quote by meBigGuy Well, your reading skills could use some honing. I absolutely did say unintuitive. And, what I am saying is actually not so far off as you seem to think. It is not a "soup of buzzwords" as you insultingly state (you do that sort of thing a lot, it seems). There is actually some real stuff there regarding instantaneous frequency that you obviously do not understand. Think about narrowband FM and why it appears to have 2 sidebands, yet is made by a continuously sweeping VCO.
"unintuitive" -sorry about that but I was already into buzz-word fatigue before I got to it.
My problem with your post is that it contains a lot of correct words and phrases 'but not in the right order' - to quote Eric Morcambe, in his sketch with Andre Previn.

You could look at the 'Rules and Guidelines', to be found in the 'site info' at the top of this page. Greg Bernhardt makes it quite clear what the purpose of the site is and it is not to promote personal and non-standard views. Your initial post starts with an excuse for 'weird and intuitive' views and that is just what they are. I merely confirmed that. Many of the ideas in your post are, at best confusing and certainly cannot be found in mainstream text books (in that order).
Mr Fourier went to a lot of trouble to provide a way of relating time and frequency domains (defining them properly). You don't need to "think about" FM. You just write out the time domain formula and do the transform and you get the spectrum. We are lucky that Mr Bessel sorted out the integration for us. It may be difficult to accept that some things are not explicable by waving arms.
I do understand that "instantaneous frequency" is, in fact, an oxymoron and the term, whilst in common enough use, needs a lot of qualification before it can be used in a valid way.
 P: 863 Good response. I'll look around and see if I can find somthing more rigourous that addresses this. I've seen stuff in the past, but it didn't pop out when I did my search. You are obviously quite happy not trying to understand how a sweeping frequency becomes 2 sidebands. It just is.
 Sci Advisor PF Gold P: 11,129 Just two?
PF Gold
P: 11,129
 Quote by meBigGuy Good response. I'll look around and see if I can find somthing more rigourous that addresses this. I've seen stuff in the past, but it didn't pop out when I did my search. You are obviously quite happy not trying to understand how a sweeping frequency becomes 2 sidebands. It just is.
When I first studied forms of modulation, I read a few text books and they treated it formally, obtaining expressions for the spectra in each case and describing methods of achieving the various types. It all made perfect sense, except for how the Bessel Function was actually derived and I left it to Mr Bessel. I really don't see how there would be any rigorous way of describing time and frequency domain models and the mechanics of moving from one to the other in a valid way (windowing etc.), if you don't use the Maths. Without the maths it's just a 'black box'. That's fine as long as you don't try to discuss the workings inside as if you understand it and then try to tell someone else the personal theory. That was the reason I was a tad grumpy about your first post.

Anything you need to 'explain' all this to me is available in a dozen well know books on communications theory and signal analysis. Google tends to throw up a lot of less rigorous stuff - which is why PF can be a bit huffy about what's available on the Web.
P: 1
 Quote by sophiecentaur Mr Fourier went to a lot of trouble to provide a way of relating time and frequency domains (defining them properly). You don't need to "think about" FM. You just write out the time domain formula and do the transform and you get the spectrum. We are lucky that Mr Bessel sorted out the integration for us. It may be difficult to accept that some things are not explicable by waving arms.
So what you saying is "just write down this formula and accept it and shut-up!" That would be like telling a child to accept "2+2=4" without ever actually showing them with blocks. A math formula can ALWAYS be explained by other means as it is a language describing something observed.

meBigGuy actually provided a very good visual of what is actually going on. I've always had trouble understanding why AM produced sidebands, and accepting the math just wasn't cutting it. I'm a visual person. This quote right here:
 Quote by meBigGuy Think of a sinewave. If you lower the amplitude, there is a frequency/phase discontinuity that occurs. That is, there has to be a transition that isn't exactly a sinewave. That is, or results in, "spurious" energy at different frequencies.
I absolutely loved. Very intuitive. This goes well with other explanations I've read where even spending days to increase the amplitude of a signal would still produce finite sidebands.

As for the "soup of buzzwords" jab, I fail to see what you're talking about. Thanks meBigGuy, very simple and to the point.

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