The phase error probability density of the first-order phase-locked loop is determined when the signal amplitude has a wideband noise component. The problem is solved using the Fokker-Planck equation. Use is made of the "correction term" for conversion from ordinary calculus to the Ito calculus model. Serious errors in previously published works are pointed out.

We consider the problem of simultaneously estimating the phase and polarization angles of a linearly polarized electromagnetic wave. The solution of this problem has applications to space communications and ionospheric probing. The nonlinear stochastic characteristics of a cross-coupled double-loop implementation for the tracker are evaluated using the multidimensional Fokker-Planck equation. The exact stationary probability density is found for a special case of equal signal-to-noise ratio (SNR) in the two loops. The stationary probability density is obtained for a general class of double loops when the two loop bandwidths differ greatly.

The exact input-output relationship is derived for a firstorder automatic gain control loop wherein the variable gain is an exponential function of the gain control voltage. The exact solution is compared to the linearized solution, and the condition for valid linearization is given.

The transient behavior of the phase-error probability density function of the first-order phase-locked loop (PLL) in the presence of additive white Gaussian noise is determined using numerical techniques. In addition, the modulo-2π probability density function and cycle-slipping density functions are found. The algorithm used in these calculations involves numerical integration of a modified form of the Fokker-Planck (F-P)equation. Results are shown for cases involving various signal-to-noise ratios (SNR's and initial conditions.

A first-order phase-locked loop with detuning is considered in the presence of white Gaussian noise and random amplitude impulsive noise with Poisson times. The stochastic equation for the phase error density is of infinite order, but when the stationary mod-2π phase density is represented by a Fourier series, a linear second-order difference equation is the Fourier coefficients results. The difference equation is solved numerically, and the phase error density is generated from the Fourier series. This method uses no approximations and is valid for any impulsive amplitude probability density.