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

by CraigH
Tags: amplitude, frequency, modulation, side bands, signal
 P: 169 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: 532

## Why do you get side band 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: 2,377 A good rule to remember is that you cannot do ANYTHING to a sine wave without creating new frequencies.
 P: 4,667 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.

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