Figure a, shows a non-coherent receiver using a matched filter

Figure a, shows a non-coherent receiver using a matched filter for the detection of a sinusoidal signal of known frequency but random phase, in the presence o additive white Gaussian noise. An alternative implementation of this receiver is its mechanization in the frequency domain as a spectrum analyzer receiver, as in Figure b, where the correlator computes the finite time auto correlation function Rx(t) defined by show that the square-law envelope detector output sampled at time t = T in Figure a is twice the spectral output of the Fourier transformer sampled at frequency f = fc in Figure b.