An attractive application of lightwave technology is cable television (CATV) transmission. In an important lightwave application known as subcarrier multiplexing, a broadband analog signal, consisting of the sum of amplitude modulated subcarriers, is used to intensity modulate a laser to produce an optical signal for transmission by optical fiber. It has been determined that the large negative excursions of the broadband signal lead to nonlinear distortion of the optical signal which also distorts the information bearing modulation on the subcarriers. The purpose of this paper is to present a general result concerning the average power of a clipped Gaussian process and apply this result to analyze the nonlinear distortion resulting from intensity modulation of a laser. In particular, a general formula for computing the average power of a stationary Gaussian process that is clipped at two arbitrary levels is applied to develop an approximation for the carrier-to-nonlinear distortion ratio (C/NLD) per channel as a function of the dynamic range of the clipping levels of the broadband analog signal, and the total RMS modulation index /spl mu/. 2/spl mu//sup 2/ equals the product of the number of channels being multiplexed and the square of the optical modulation index per channel. The approximation stems from an exact result and is especially useful because the conservative version of the approximation is independent of the bandwidth or spectral location of the broadband analog signal. Results are presented in graphical form and they are in reasonable agreement with earlier theoretical and experimental works.>

The simplest and most economical method for transforming many channels of high quality video and audio analog signals into an optical signal makes use of subcarrier multiplexing. In this multiplexing scheme, a broadband analog signal consisting of the sum of many modulated subcarriers (one for each channel), is applied to amplitude modulate the intensity of a laser for transmission by optical fiber. Laser intensity modulation is also attractive for interconnecting broadband RF signals using optical fiber as, for example, between a cellular base station and its remote antenna sites, especially for implementing microcells. A fundamental, limiting transmission impairment is the nonlinear distortion of the clipped laser intensity resulting from occasional signal excursions below the laser threshold. The purpose of this paper is to apply noise theory along with numerical analysis, involving fast Fourier transforms, to determine some basic characteristics of the nonlinear distortion associated with a fully loaded system. In particular, the exact distortion spectrum and the carrier-to-nonlinear distortion ratio are determined as a function of the dynamic range and bandwidth of the multiplexed signal, and the total RMS modulation index /spl mu/. 2/spl mu//sup 2/ equals the product of the number of channels and the square of the optical modulation index per channel. The exact carrier-to-nonlinear distortion ratio per channel when all or specific orders of distortion are present is also treated. The results are presented in graphical form and they agree reasonably well with earlier theoretical and experimental works. All of the results apply to both a directly modulated laser or an externally modulated laser.>

The first and second passage times of a stationary Gaussian process crossing a time-varying boundary were studied in a recent work. The results apply to stationary Gaussian processes having a finite expected rate of level crossings, but they are restricted to time-varying boundaries having zero slope at the origin. Here, the earlier results are extended to include general time-varying boundaries. Some exact, asymptotic probability densities of the first passage time of a stationary Gaussian process crossing a ramp or linear boundaries are developed. Some approximate results concerning the initial behaviors of the probability densities of the first passage times associated with a linear boundary are presented.>

The origin of S.O. Rice's formula for the average rate of level crossings of a rather general random process is traced to his 1936 private notes entitled Singing Transmission Lines. The notes formulate an engineering problem in terms of the probability distribution of the absolute maximum of an envelope process in a closed interval. Rice showed that if the envelope of the reflected wave on a loaded transmission line were to equal or exceed the maximum value in the closed interval, then 'singing' or oscillation could be expected to occur and the analog transmission system would become unstable and useless for the transmission of information. The formula and its generalizations are now basic for the analysis of level crossings and first and second passage times of random processes.>

The first and second passage times of a stationary Rayleigh processR(t,a)are discussed.R(t,a)represents the envelope of a stationary random process consisting of a sinusoidal signal of amplitude and frequencyf_{0}plus stationary Gaussian noise of unit variance having a narrow-band power spectral density which is symmetrical aboutf_{0}. Approximate integral equations are developed whose solutions yield approximate probability densities concerning the first and second passage times ofR(t,a). The resulting probability functions are presented in graphs for the case when the power spectral density of the noise is Gaussian. Related results concerning the approximate distribution function of the absolute minimum or absolute maximum ofR(t,a)in the closed interval[0,\tau]are also presented. The exact probability densities are expressed in the form of an infinite series of multiple integrals.

For the case of an unmodulated carrier, a simple approximate result is developed for the power spectral density of frequency-modulated (FM) clicks at the output of an ideal FM discriminator. In the baseband region the result agrees well with known exact results obtained by numerical integration at the threshold carrier-to-noise power ratio of10. Simple results are also presented for the bandwidth of FM clicks, the baseband noise power resulting from the FM clicks, and the total noise power associated with FM clicks.

For the Case of an unmodulated carrier, the detection of an FM click is formulated as the detection of a non-Gaussian random signal embedded in colored Gaussian noise. This correspondence presents an approximation to the optimum detector of FM clicks along with its theoretical performance in terms of probability of false alarm and probability of error. Numerical results are presented in graphical form. The corresponding exact results for detection problems of this type are currently not available. The theoretical results help to explain the known practical result that FM threshold extenders are limited to threshold improvements of only a few dB. For the case of a modulated carrier, some comments concerning detector performance are presented. For comparison purposes, detector performance of a detector based on the heights of local maxima is also presented.

The exact initial behavior of the probability density of the duration of an FM click in the case of an unmodulated carrier plus narrowband stationary Gaussian noise is presented. A simple Maxwell probability density is developed to approximate the probability density of the duration. The development leads to the conjecture that the Maxwell probability density is the exact limiting probability density of the duration as the carrier-to-noise power ratio approaches infinity. Analogous results are presented for the amplitude of an FM click. Simple formulas are developed for the mean, variance, and most probable value of the duration and amplitude of an FM click. It is shown that these theoretical developments yield numerical results which are in good agreement with many experimental and numerical results reported in the literature.

This paper presents experimental results concerning the zero-crossing intervals of Gaussian processes. Probabilities and probability densities defined by the zero-crossing points of Gaussian processes having various power spectral densities are presented. At present none of the probabilities or probability densities can be derived explicitly by analytical methods. The results are compared with theoretical approximations. The standard deviations of successive zero-crossing intervals are compared with theoretical values resulting from an assumed wide sense Markov model. In some cases the comparisons are unsatisfactory.