Frequency and phase of fluctuation are strictly connected among themselves. Frequency is equal derivative of a phase. In the opposite direction the integral from frequency follows. The simplest of tasks which arise at the same time – measurement of a constant initial phase of harmonic oscillation. The decision will be made by methods of statistical radio engineering.

## Instruction

1. Consider the simplest mathematical model of a harmonious signal of S (t, ψ) =Acos(ωt-ψ). Consider that this representation of a radio impulse with duration determined by amplitude And yes frequency ω and also set τ. The unknown (but a constant) an initial phase ψ is subject to measurement. Optimum measurement (or estimation) signal parameter in statistical radio engineering is based on criterion of a maximum of a logarithm of functionality of credibility of F(ψ) (see fig. 1).

2. Analyze structure of the subintegral function given on fig. 1. (ξ-S) ^2=ξ^2-2ξS+S^2. The integral from a square of the accepted realization ξ(t) does not contain a phase in an explicit form and cannot affect condition of a maximum of F(ψ). The phase belongs to not power parameters. Therefore integral from a signal square, equal to its energy, a constant (it can be carried to coefficient). Therefore, all data on a maximum concentrate in cross-correlation integral of y(ψ) (fig. 1). Pay also attention to the accepted model of a hindrance n(t). It is normal white noise by a zero average and mathematical spectral density of N/2. Besides, the true value of a phase is designated φ.

3. Remember that it is about definition of assessment, that is approximate value of a phase of the closest to true, contained in the accepted realization. It means a random variable. The size which allows to choose rather precisely estimated value of a phase ψ* is a cross-correlation y(ψ) function. Such device can already be realized technically by means of the correlation receiver (see fig. 2). At its exit pinnacled tension is formed (that increases noise stability).

4. ψ* carry out the choice by use of several correlation receivers connected in parallel. Send signals from their exits to the scheme of comparison in which it will be established, from what "ruler" the maximum tension arrived and the decision on the "measured" phase assessment size, as is made on its value in a basic signal (see fig. 3).