Amplitude modulation
From Academic Kids

Amplitude modulation (AM) is a form of modulation in which the amplitude of a carrier wave is varied in direct proportion to that of a modulating signal. (Contrast this with frequency modulation, in which the frequency of the carrier is varied and phase modulation in which the phase is varied.)
AM is commonly used at radio frequencies and was the first method used to broadcast commercial radio. The term "AM" is sometimes used generically to refer to the AM broadcast (mediumwave) band (see AM radio).
The inventor of amplitudemodulated radio was Reginald Fessenden.
Contents 
Applications in radio
Amplitudemodulation.png
A basic AM radio transmitter works by first DCshifting the modulating signal, then multiplying it with the carrier wave using a frequency mixer. The output of this process is a signal with the same frequency as the carrier but with peaks and troughs that vary in proportion to the strength of the modulating signal. This is amplified and fed to an antenna.
An AM receiver consists primarily of a tunable filter and an envelope detector, which in simpler sets is a single diode. Its output is a signal at the carrier frequency, with peaks that trace the amplitude of the unmodulated signal. Amazingly, this is all that is needed to recover the original audio! In practice, a capacitor is used to undo the DC shift introduced by the transmitter and to eliminate the carrier frequency by connecting the signal peaks. The output is then fed to an audio amplifier.
The fact that signals can be decoded using very simple equipment is one of the primary advantages of amplitude modulation. This was especially important in the early days of commercial radio, when electronic components were still quite expensive. This simplicity and affordability helped make AM one of the most popular methods for sending voice and music over radio during the 20th century.
AM radio's main limitation is its susceptibility to atmospheric interference, which is heard as static from the receiver. The narrow bandwidth traditionally used for AM broadcasts further limits the quality of sound that can be received. Nowadays, wideband FM is preferred for musical broadcasts, due to its high audio fidelity and noisesuppression characteristics.
Forms of AM
In its basic form, amplitude modulation produces a signal with power concentrated at the carrier frequency and in two adjacent sidebands. Each sideband is equal in bandwidth to that of the modulating signal and is a mirror image of the other. Thus, most of the power output by an AM transmitter is effectively wasted: half the power is concentrated at the carrier frequency, which carries no useful information (beyond the fact that a signal is present); the remaining power is split between two identical sidebands, only one of which is needed.
To increase transmitter efficiency, the carrier can be removed (suppressed) from the AM signal. This produces a doublesideband suppressed carrier (DSSC) signal. If the carrier is only partially suppressed, a doublesideband reduced carrier (DSRC) signal results. DSSC and DSRC signals need their carrier to be regenerated (by a beat frequency oscillator, for instance) to be demodulated using conventional techniques.
Even greater efficiency is achieved—at the expense of increased transmitter and receiver complexity—by completely suppressing both the carrier and one of the sidebands. This is singlesideband modulation, widely used in amateur radio due to its efficient use of both power and bandwidth.
A simple form of AM often used for digital communications is onoff keying, a type of amplitudeshift keying by which binary data is represented as the presence or absence of a carrier wave. This is commonly used at radio frequencies to transmit Morse code, referred to as continuous wave (CW) operation.
Example
Suppose we wish to modulate a simple sine wave on a carrier wave. The equation for the carrier wave of frequency <math>\omega_c<math>, taking its phase to be a reference phase of zero, is
 <math>c(t) = C \sin(\omega_c t)<math>.
The equation for the simple sine wave of frequency <math>\omega_m<math> (the signal we wish to broadcast) is
 <math>m(t) = M \sin(\omega_m t + \phi)<math>,
with <math>\phi<math> its phase offset relative to <math>c(t)<math>.
Amplitude modulation is performed simply by adding <math>m(t)<math> to <math>C<math>. The amplitudemodulated signal is then
 <math>y(t) = (C + M \sin(\omega_m t + \phi)) \sin(\omega_c t)<math>
The formula for <math>m(t)<math> above may be written
 <math>y(t) = C \sin(\omega_c t) + M \frac{\cos(\phi  (\omega_m  \omega_c) t)}{2}  M \frac{\cos(\phi + (\omega_m + \omega_c) t)}{2}<math>
The broadcast signal consists of the carrier wave plus two sinusoidal waves each with a frequency slightly different from <math>\omega_c<math>, known as sidebands. For the sinusoidal signals used here, these are at <math>\omega_c + \omega_m<math> and <math>\omega_c  \omega_m<math>. As long as the broadcast (carrier wave) frequencies are sufficiently spaced out so that these side bands do not overlap, stations will not interfere with one another.
A more general example
 This relies on knowledge of the Fourier Transform. The discussion of the figure may prove more useful for a quicker understanding.
Consider a general modulating signal <math>m(t)<math>, which can now be anything at all. The same basic rules apply:
 <math>\,y(t) = [C + m(t)]\sin(\omega_c t)<math>.
Or, in complex form:
 <math>y(t) = [C + m(t)]\frac{e^{j\omega_c t}  e^{j\omega_c t}}{2}<math>
Taking Fourier Transforms, we get:
 <math>Y(\omega) = \pi{}C\delta(\omega  \omega_c) + \frac{1}{2}M(\omega  \omega_c) + \pi{}C\delta(\omega + \omega_c) + \frac{1}{2}M(\omega + \omega_c)<math>,
where <math>\delta(x)<math> is the Dirac delta function — a unit impulse at <math>x<math>, and capital functions indicate Fourier Transforms.
This has two components: one at positive frequencies (centered on <math>+\omega_c<math>) and one at negative frequencies (centered on <math>\omega_c<math>). There is nothing mathematically wrong with negative frequencies, and they need to be considered here — otherwise one of the sidebands will be missing. Shown below is a graphical representation of the above equation. It shows the modulating signal's spectrum on top, followed by the full spectrum of the modulated signal.
AM_spectrum.png
This makes clear the two sidebands that this modulation method yields, as well as the carrier signals that go with them. The carrier signals are the impulses. Clearly, an AM signal's spectrum consists of its original (2sided) spectrum shifted up to the carrier frequency. The negative frequencies are a mathematical nicety, but are essential since otherwise we would be missing the lower sideband in the original spectrum!
As already mentioned, if multiple signals are to be transmitted in this way (by frequency division multiplexing), then their carrier signals must be sufficiently separated that their spectra do not overlap. This analysis also shows that the transmission bandwidth of AM is twice the signal's original (baseband) bandwidth  since both the positive and negative sidebands are 'copied' up to the carrier frequency, but only the positive sideband is present originally. Thus, doublesideband AM is spectrally inefficient. The various suppression methods in Forms of AM, can be seen clearly in the figure  with the carrier suppressed there will be no impulses and with a sideband suppressed, the transmission bandwidth is reduced back to the original, baseband, bandwidth — a significant improvement in spectrum usage.
An analysis of the power consumption of AM reveals that DSAM with its carrier has an efficiency of about 33% — very poor. The forms of AM with suppressed carriers are found to be 100% power efficient, since no power is wasted on the carrier signal which conveys no information.
Modulation Index
As with other modulation indices, in AM, this quantity indicates by how much the modulated variable varies around its 'original' level. For AM, it relates to the variations in the carrier amplitude and is defined as:
 <math>h = \frac{\mathrm{peak\ value\ of\ } m(t)}{C}<math>.
So if <math>h=0.5<math>, the carrier amplitude varies by 50% above and below its unmodulated level, and for <math>h=1.0<math> it varies by 100%.
See also
 AM radio
 Mediumwave
 Modulation, for a list of other modulation techniques
References
 Newkirk, David and Karlquist, Rick (2004). Mixers, modulators and demodulators. In D. G. Reed (ed.), The ARRL Handbook for Radio Communications (81st ed.), pp. 15.1–15.36. Newington: ARRL. ISBN 0872591964.de:Amplitudenmodulation
fr:Modulation d'amplitude nl:Amplitudemodulatie ja:振幅変調 pl:Modulacja amplitudy pt:Amplitude modulada fi:AM