Why does an AM radio channel require any bandwidth at all? Why can’t it just transmit on a single, precise frequency?
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A single, precise, continuous signal contains no information or, at best, a single bit of information. Even morse code, which is just turning the carrier signal on and off at short intervals, uses a small amount of bandwidth. The reason is that, as a signal transitions from zero to a pure sine wave, it consists of other frequencies; an instant transition means an infinite range of frequencies are generated.
For morse, it’s necessary to limit the rate at which the signal goes from off to on to limit the spread of frequencies used. Faster morse needs faster transitions which uses more bandwidth. In the same way, the higher frequency audio you mix with a carrier wave, the more bandwidth the signal will occupy.
AM is the multiplication of the carrier wave with the audio signal. One piece of maths that can help understand what happens is the formula for multiplying two different frequencies:
sin(*a*)×sin(*b*)=0.5[cos(*a*−*b*)−cos(*a*+*b*)].
Now cosine is just sine with a phase change of 90° so this formula tells you that multiplying two frequencies generates a signal with both the sum and difference of the original frequencies. You send a 1 kHz tone over a 1 MHz AM transmitter and you’ll generate both a 999 kHz and 1001 kHz signal.
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