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The maximum power an amateur is allowed to generate at the output of the transmitter or amplifier is 1500 watts of peak envelope power (PEP). 

Definition: PEP is the average power during one RF cycle of the radio signal at the very peak of a modulating waveform. 

So if we look at an Amplitude Modulated wave, you see that each RF cycle of the carrier wave has a different peak.  All of those different peaks are a result of the "squishing effect" of the modulating envelope/signal.  Simply put - PEP is the power released/transmitted at the peak of the modulating envelope.

What confused me most was the phrase "average power".  What values are being averaged and where are they coming from?  My thoughts are that average power is defined as the energy transfer rate average over many cycles/periods of the RF waveform.  For an amplitude modulated wave, power varies in amplitude over many RF cycles, and must be averaged over many cycles/periods of the modulating waveform to get a representative average.  The averaging period for RF power meters may range from several hundredths of a second up to several seconds.

So then I wonder if PEP is looking at the peaks of the modulating envelope and averaging those out over a specified amount of time?  Any suggestions?

So for a CW signal, PEP is measured during the key-down period in which the transmitter is ON

FM is a constant-power mode (no amplitude changes), so it doesn't matter whether you are speaking or not when measuring PEP.

Either way, amateurs are expected to use the minimum power required to carry out the desired communication.

Transmitter output power is measured at the output of the last amplifier, whether internal to the transmitter or an external piece of equipment, at the input to the antenna feed line.  The 1500 watt limit works except:

In Technician sub-bands on 80, 40, 15 meters, all are limited to 200 watts PEP

Technician licensees are limited to 200 watts PEP in 10 meter allocation between 28.0-28.5MHz.

All amateurs are limited to 200 watts PEP on the 30meter band

All amateurs are limited to 50 watts PEP on 219-220MHz segment of 1.25 meter band.

Beacon stations are limited to 100 watts PEP

Stations operating in the 70 cm band near certain military installations may be limited to 50 watts PEP or less.

Other restrictions exist for novice class licensees and for stations operating on the 60 meter band.

But most rarely use or run more than a few hundred watts on the VHF/UHF bands.  Exceptions would be while pursuing very weak signal methods such as Earth-Moon-Earth, or tropospheric propagation where high power is required to establish and maintain contact.

http://www.radio-electronics.com/info/t_and_m/rf-microwave-power-meter/peak-envelope-average-pulse-power.php

http://cp.literature.agilent.com/litweb/pdf/5988-9213EN.pdf