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When I read some books or papers related into signal modulation, for example, frequency-domain OFDM signal, it's always mentioned that the average power of signal is assumed to be 1.

Why does that assumption is taken? what's if that average signal power is more or less than one?

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Assuming an average power of one for a signal, or for the noise, is a common assumption, mainly to simplify equations. It is assumed that for other powers, a scaling by the actual power can be done.

For instance, many denoising techniques set a threshold that is proportional to the noise power (the square root of it). In the theory, one often assumes (for computations) that $\sigma=1$, design an optimal $\lambda$ and set the actual threshold to $\lambda \sigma$.

The actual threshold is multiplied according to the estimated noise amplitude.

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    $\begingroup$ also, you have to use the same output power if you compare different methods, and using "1" is simply the most intuitive choice! $\endgroup$ – Marcus Müller Oct 15 '20 at 8:08
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    $\begingroup$ and demonstrating Laurent's point further: 10Log(1) = 0 dB, so this is most convenient for all log reported values that are referenced to the signal power such as SNR, dBc, etc, 1 is just the point of reference. $\endgroup$ – Dan Boschen Oct 15 '20 at 12:21
  • $\begingroup$ You mean the mean power of transmitted signal must be the same, is that right? OK what's about it was different? $\endgroup$ – Fatima_Ali Oct 16 '20 at 0:35

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