In this specific case, the probabilities of the outcomes do not actually matter; the only reason that I asked is that the probabilities do sometimes matter in the general case.
The reason they don't matter here is that the complex magnitude ( abs ) of each of the two outcomes in this specific case is always the same, which is exactly 1. So regardless of the individual probabilities, the average of the complex magnitudes across all of the outcomes is the magnitude of any one of the outcomes, which is exactly 1.
In other words, we can compute this average distance metric in MATLAB as follows:
% Alphabet of possible symbols:
a = [ +1 ; -1 ];
% Probability of each symbol:
p = [ 0.5 ; 0.5 ];
% Average distance from the origin in the complex plane:
A = (p.')*abs(a);
At this point, the value of A should be exactly 1:
disp(A);
The only other thing we need to find is the power of the carrier wave. That is why I asked for the amplitude of the carrier. Let's represent the carrier amplitude in MATLAB as:
% Amplitude of the carrier:
V = 1;
so that the carrier itself is:
% Carrier signal:
x = V*cos(2*pi*Fc*t);
where t is some appropriate discrete representation of the time domain (in seconds), and Fc is the carrier frequency (in hertz).
So, how do we compute the power in the carrier signal, which is simply a pure-tone cosine wave? Well, power in any signal is simply the time average of the square of the signal. In the case of any unity amplitude pure tone sine or cosine wave, the time average of the square of the signal is simply 1/2. So for a carrier with amplitude of V, the time average is:
% Carrier power:
Pc = V^2/2;
So the overall signal power is:
% Signal power:
Ps = A*Pc;
HTH.
Rick
