You, too, can build your own AM radio station. All you need are a few parts and a broadcast license from the FCC.
The AM (amplitude modulated) signal is a combination of two signals:
The carrier wave: c(t) = A sin(ωt)
The sound wave: s(t)
The total wave is given by [A + s(t)] sin(ωt)
In other words, the sound signal is coded into the amplitude of the carrier wave.
An AM radio receiver is relatively straightforward. Use an antenna to supply a voltage to a resonating RLC circuit. The resonance frequency of the circuit matches the carrier wave frequency of the radio station you want to listen to, so the circuit picks that signal out from all the other stations.
Once you've isolated the one radio signal you want you don't need to do much more than to pass the signal through a low-pass filter. This keeps the sound signal, which has a frequency between 20 Hz and 20 kHz, and removes the carrier wave, which has a frequency of hundreds of kHz. Send the signal through an amplifier and out through a speaker and you're done.
FM (frequency modulation) just modifies the frequency of the carrier wave rather than the amplitude. The amplitude of the sound signal determines the frequency shift of the carrier wave.
The carrier wave: c(t) = A sin(ωt)
The sound wave: s(t)
The total wave is given by A sin[(ω + s(t)) * t]
The first stage of an FM radio receiver is the same as for the AM receiver. Use an antenna to supply a voltage to a resonating RLC circuit. The resonance frequency of the circuit matches the carrier wave frequency of the radio station you want to listen to, so the circuit picks that signal out from all the other stations.
Because the sound signal is coded into the frequency of the carrier rather than the amplitude, now you need to do things a little differently. The main idea is the same, though - separate the sound signal from the carrier wave and then work just with the sound signal. Send the signal through an amplifier and out through a speaker and you're done.