Suppose you produce a pulse
x in discrete samples of duration
dt each (e.g. in Matlab) according to a function.
I was given some notes in which it shows that if I want it to let it have a specific power I have to first normalize it by doing
energy = sum((x.^2).*dt) %first calculate the pulse energy x = x./sqrt(energy) %normalize its energy to 1
then you calculate the target energy
power = 0 %target power in dBm power = (10^(power/10))/1000 %target power in W energy = power*len(x)*dt %target energy of the pulse
and finally you scale the pulse so that it has the target power
x = x .* sqrt(energy) %the pulse at the target power
My question is, how does that really work? Why dividing
sqrt(E) normalize it to unit energy? And why multiplying the resulting normalized
x by the new
sqrt(E) brings it to the target power? Why not just divide by
E for example? and then multiply by
E? Does it have any relation to RMS values? And finally is it something that can be used in any arbitrary pulses, e.g. in sinusoids as well?