A narrowband-signal-plus-Gaussian noise process is represented by the equation
where A cos (ωc t + θc) is a sinusoidal carrier and the remaining terms are the bandpass noise with independent Gaussian I and Q components. Let the noise have an RMS value of σ and a mean of 0. The signal-plus-noise process appears at the input to an envelope detector. Show that the PDF for the output of the envelope detector is
is the modified Bessel function of the first kind of zero order. f(R) is known as a Rician PDF in honor of the late S. O. Rice, who was an outstanding engineer at Bell Telephone Laboratories.