Application to Binary Communications and Pulsed Radar Lognormal Atmospheric Channel

9 Application to Binary Communications and Pulsed Radar (Lognormal Atmospheric Channel) 

Whereas the previous section (7.3,8) was concerned with the calculation of system performance for the vacuum channel, we now turn to the error probabilities for three-frequency nonlinear heterodyne detection for the atmospheric channel. The behavior of the clear-air turbulent atmosphere as a lognormal channel for optical radiation has been well documented both theoretically and experimentally [7.76-78, 80-82], We therefore choose the amplitudes /4i and A 2 to be lognormally distributed, and the phases j and (j 2 to be uniformly distributed over (0,27t). Since A ocAj and while 

Here Ox is the logarithmic-amplitude standard deviation which is related to the logarithmic-irradiance standard deviation a by the formula 4ffx=a [7.82]. Assuming energy is conserved and that there is no scattering of radiation out of the beam, we choose 

Since the quantities ,=ln and will both be normally distributed as 

We also consider the situation = b=which would arise if both incoming signals were sufficiently close in frequency and space such that they suffered precisely the same fluctuations at each instant of time [7.83]. This case is more likely to occur in a practical situation than the independent case. For dependent fluctuations, then. 

Assuming energy is conserved and that there is no scattering of radiation out of the beam, we choose 

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