Rectification of a Sinusoidally Modulated Carrier in the Presence of Noise

The low-frequency output, signal and noise, is determined following a general νth-law (ν/>0) half-wave rectification of a sinusoidally modulated carrier. The spectral width of the noise before rectification is assumed to be much narrower than its mean frequency f
0
, the normal condition for reception. Attention is focused on one of the harmonics of the rectified signal, generally the first, whose modulation frequency is small compared to the if frequency f
0
and to the width of the noise band. Special attention is given to the important cases of linear (ν=1) and quadratic (ν=2) detection, and audio signal-to-noise ratios s
a
/n
a
are calculated for them. The noise passed by the audio filter is found to depend on the spectral shape of the if; for the three types of if filter specifically considered, namely, rectangular, gaussian, and "optical," the optical yields the least, the rectangular the most noise; provided, of course, that the mean input signal-to-noise power p is of the order of unity or less, the customary region of interest. When p<<1, all half-wave detectors (ν>0) behave like the quadratic device. Modulation (O≦λ≦1) does not radically affect the output noise power, particularly when (p>1), and for very large signals, p→∞, s
a
/n
a
is directly proportional to the input carrier amplitude, is independent of if filter width, and is only slightly dependent on filter shape. The best if filter is one whose response is the modulus of the Fourier transform of the signal, as in the radar case of pulsed signals.