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Constant envelope signals can go through class C amplifiers. Nonconstant types have to go through class B with a lot of backoff to keep the out of channel power down. There is a big difference in DC power input between these two methods for the same RF output power.
You can do the math to get a better understanding. The generalized signal can be described as
A(t) cos ( f(t) t + angle(t))
That is the amplitude, frequency, and phase are functions of time.
Approximate the amplifier as a polynomial a1 v(t) + a2 (v(t))^2 and so on. Make three signals by making just one of the variables in the first equation a variable and the rest constants. Put them individually through the polynomial. What you get out is the revealing thing.
When A(t) is a constant you get out just harmonics of f(t) which can be filtered. When A(t) is a variable you get extra sidebands on the carrier.
As far as real world amplifiers go, if the signal level is lower the a2 (v(t))^2 and higher terms are lower.
Since the DC power into an amplifier is voltage times current, swinging the output much less than the power supply level is inefficient.
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