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I am looking for a formula to calculate the Average Power of a square wave used in APSK Modulation.

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In my textbook, the example above is given with no formula so I am a loss how the value for average power (4.5) is calculated in the bottom example.

I assumed from the top example that it was simply the square of the amplitude divided by the ON Time Period, but applying that logic to the bottom example gives a value no where near so I have a feeling some integration may be involved.

Any suggestions would be greatly appreciated.

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In both cases peak is just the square of the largest amplitude, so that is easy.

The pulses are square so there is no need for integration as such, just multiplying the peak power by the duty cycle gets you there.

In the top case, pulse with is 2, period is 4, amplitude is 3, so \$3^2 \times 2/4 = 4.5\$. In the lower one, pulse width is 0.5, period is 4, amplitude is 6, so \$6^2 \times 0.5/4 = 4.5\$

In a more general case you integrate \$x^2(t)\$ dt over the period of a cycle and get in this case exactly the same result.

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