The distribution of the amplitude and continuous phase of a sinusoid in noise

A derivation of the joint distribution of the amplitude and angle of a sinusoid in Gaussian noise is given. No assumptions about the structure of the noise at the output of the post-detection filter are required. However, it is assumed that the Gaussian noise at the input is generated by passing white noise through the bandpass equivalent of a single-pole low-pass filter. Hence, the noise is a two-dimensional Ornstein-Uhlenbeck or Gauss-Markov process. In practice, a higher-order bandpass filter would be encountered in FM reception. However, the approach is intended as a first step towards the goal of an understanding of the phase process. The method used is to derive and solve the Fokker-Planck partial differential equation that governs the joint distribution. An explicit integral formula for the solution is obtained. The integral is, in general, rather difficult to evaluate. A further integration over amplitude is required if the angle distribution rather than the joint distribution is required. For some special-cases, where the filtering time is large and the signal-to-noise power ratio is very large or very small, explicit approximate expressions are given.<
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