Why the average signal power should be 1
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.