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I am doing a course in Communication Engineering and on reading Amplitude Modulation in my textbook I found that a typical AM is defined by the following equation;

$$ s(t) = (1+k_{a}m(t))A_{m}cos(\omega t) $$ The reasoning which the book gives is that $$-1<k_{a}m(t)<1$$, so that $$1+k_{a}m(t) > 0$$ But why do this and instead just make $$0<k_{a}m(t)<1$$, because including a 1 would only make things worst by increasing transmission power and it also does not supress the carrier.

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It doesn't matter what equation you use to describe AM, the actual physical modulation itself has an unsuppressed carrier, and a certain amount of power. It's a very simple modulation, one of the earliest, and as such is inefficient, but easy to use.

If you want a suppressed carrier, that's a different type of modulation, which needs a more complicated demodulator to read it.

You could make \$k_a\$ vary in the range 0 to 1, if you also modified \$A_m\$, and interpretted \$k_a\$ as something other than the modulation that you put on the carrier with a conventional modulator, and recovered from the carrier with a conventional detector.

So you see it's easier to interpret \$k_a\$ conventionally, because then it maps onto what's typically happening as we use the system.

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\$k_a m(t)\$ is defined to be in the range -1..1 because in practice the signal \$m(t)\$ used to modulate the carrier \$A_m \cos(\omega t)\$ actually is an AC signal centered at 0V (e.g. an audio signal).

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