Optical Communications, 2nd Edition

3.8 SPECTRAL DENSITY OF SHOT NOISE

The power spectral density of a random process is important in specifying the power distribution of the process as a function of frequency. In this section, we derive the power spectral density by computing the time averaged power at any radian frequency ω. That is, we consider the current power spectrum of i(t) as

where X_T(ω) is the Fourier transform of a sample function of the current process i(t) restricted to the interval (− T, T). [Equation (3.8.1) is the classical definition of a power spectrum.] The function |X_T(ω)|² is the energy density at each ω for the sample function, and therefore is a random variable at each ω. It must, therefore, be averaged over the statistics of the process. Thus S_i(ω) is effectively the time average of the ensemble average of the energy density at each ω. Now if we condition upon {z_i} and k, we have

where H_T (ω) is the Fourier transform of h(t), − T < t < T Thus,

The ensemble average over {z_j}, conditioned upon k, can now be taken. Because the exponent is unity when q = i, we have

The subsequent averaging over