Received Frequencies and the Factor

Aj represents the amplitude of the th line in the sinusoidal case, Pj represents the peak value of the Gaussian spectral distribution and y is its 

Standard deviation, whereas Dj and F represent the height and width of the Lorentzian spectrum, respectively. The quantity 4 is the error function. It has been assumed for simplicity that all spectral widths are identical, i.e., yj=yj+i= y and rj=Fj+ i=F similar but more complex expressions are obtained when this is not the case. 

In the case where all such lines are of equal spacing, power, and width (Aj=Aj i = Dj=Dj+, Fj Fj+i Fl the braces in 

For fixed input radiation power, the best operation is clearly achieved forn = 2 (so that p,G,L = V4), since additional lines increase the (signal-by-noise contribution to the) total noise more than they do the signal. When increased radiation power becomes available by virtue of the additional lines, however (e.g., the detection of more than one Doppler feature), n 2 can be advantageous. 

We also consider the case in which n equal-power, equal-width lines are allowed through the broad bandpass filter, these not being equally spaced, however, so that only one pair of lines contributes to the output signal. In this case, the braces in (7.181) must be replaced by 

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