When the main thing – obtaining the size possessing the minimum error is about measurements. From the mathematical point of view – the certain parameter having the maximum accuracy. For this purpose use criteria for selection of assessment.
Instruction
1. Explanations are given in a basis of optimum measurement of amplitude of a radio impulse which well keeps within a framework of mathematical approach to the solution of a task and was considered in statistical radio engineering.
2. All data on the measured parameter contain in its a posteriori density of probability which is proportional to the function of credibility increased by a priori density. If the priori density of probability is unknown, then instead of a posteriori density the function of credibility is used.
3. Assume that on reception came realization of a look x (t) = S (t, λ) +n(t) where S (t, λ) the determined function of time of t, and λ parameter. n(t) Gaussian white noise with a zero average and the known characteristics. On the reception party λ it is perceived as a random variable. The credibility equation for definition of assessment of parameters of a signal by a method of a maximum of functionality of credibility has an appearance d/dλ\• {∫ (0, T) • [x (t) - S (t, λ)]^2•dt } =0. (1) There is an integral, undertakes from zero to T (T is observation time).
4. Work out the equation of credibility (1), having put duration of a radio impulse of T equal to time of observation, and S (t, λ)=λcosωt (radio impulse). d/dλ\• {∫ (0, T) [x (t) - λcosωt)] ^2•dt] } =0. Find roots of this equation and take them for estimated values of amplitude: d/dλ\• {∫ (0, T) [x (t) - λ\• cosωt)] ^2dt } =-2•{∫ (0, T) • [x (t) - λ\• cosωt)] • cosωt • dt] } =-2 • ∫ (0, T) [x (t) • cosωt)] dt+2λ\• ∫ (0, T)(cosωt) ^2•dt=0.
5. Then assessment λ *= (1/E1) • ∫ (0, T) [x (t) • cosωt)] • dt, where E1= ∫ (0, T)(cosωt) ^2•dt – energy of a radio impulse with a single amplitude. On the basis of this expression construct the block diagram optimum (to the maximum credibility) the radio impulse amplitude measuring instrument (see fig. 1).
6. Finally to make sure of correctness of the choice of assessment, check it for not shift. For this purpose find its expected value and make sure that coincides with true value of parameter. M [λ *] = M [*= (1/E1) • ∫ (0, T) [x (t) • cosωt)] dt=(1/E1)•M {∫ (0, T) [λ\• cosωt+n(t)] cosωt • dt } = = (1/E1) • ∫ (0, T) [λ\• (cosωt) ^2+0] dt =λ.Оценка not displaced.