Assume that we have a single sinusoidal component at f = k/N. The model, as given by z[n] = a*cos(2*pi*f*n) + b*sin(2*pi*f*n) + w[n], where n = 0 to N-1. Using the identity Acos(w) + Bsin(w) = A*cos(w - arctan(B/A)), we can rewrite the model as z[n] = a + b*cos(2*pi*f*n - phi) + w[n]. An MVU estimator is used for a and b so that the estimated power of the sinusoid is p = a^2 + b^2.
A measure of detectability is E[p]/var(p). Compare the measure when a sinusoid is present to the case when only noise is present or a = b = 0. Could you use the estimated power to decide if a signal is present?