TABLE OF CONTENTS Why are random processes an important consideration in optical signal processing? Because there ' s always noise present in real systems, whether in the form of fluctuations in the photon arrival time (photon shot noise), fluctuations of the current in the load resistor following the detector (Johnson noise), or additional noise in the preamplifiers that contributes to an increased noise figure for the overall system.
From a purely theoretical standpoint, noise also presents a fundamental problem. Noise signals are not periodic; hence we would need infinite time to give them a complete description. But infinite signals don ' t have finite energy, hence the Fourier transform will not converge. What can we do about this? We can truncate all realizations of X ( t) to a finite interval T, which is equivalent to what happens spatially when we use finite apertures in optical architectures. Then we can take time averages whenever appropriate, which is what we do after detection in an optical processor. (In time-domain signal processing we then may take the limit as T approaches infinity.)
In addition to characterizing noise in terms of its magnitude relative to the signal [expressed as the signal-to-noise ratio (SNR)], higher-order statistics are needed to characterize the spatial and or temporal structure of noise-like (or stochastic) processes. This is provided by the correlation function introduced earlier. To arrive at a complete understanding of correlation, we need to understand the basics of statistics. For...
