Study of statistical techniques for obtaining estimates of true data parameters uses discrete measured quantities containing random error. These techniques develop estimation procedures as an iterative algorithm for digital computation in real time

Hung, J. C.

Irwin, J. D.

English

NASA Technical Reports Server

September 1968
	 Brief 68-10348 
NASA TECH BRIEF	
a, 
NASA Tech Briefs are issued to summarize specific innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Study of Optimum Discrete Estimators in Measurement Analysis 
A survey study, which is presented in a report, has 
been made of statistical techniques (statistical filtering 
and prediction) for obtaining the best estimate of the 
true data parameters from discrete measured quan-
tities containing random errors. The equations studied 
are developed in the time domain and lead to a digital 
computer realization of the optimal filter, as opposed 
to the frequency domain approach in which the op-
timal filter may be conveniently obtained through net-
work synthesis. The state variable notation is used 
throughout the study. Unknown parameters, such as 
position and velocity of a space vehicle, are grouped 
into a composite state vector which represents the in-
formation necessary to specify the process in phase 
space. Using these techniques, estimation procedures 
are developed as an iterative algorithm for digital 
computation in real time. 
Chapter 11 of the report contains the mathematical 
foundations which are used extensively in the sub-
sequent chapters. The basic concepts of probability 
theory and random processes are discussed. The state 
space formulation of dynamic systems is also intro-
duced. The concept of dynamic programming is pre-
sented and two very important matrix identities which 
are used extensively in the remaining chapters are in-
troduced. An expression for the gradient of a general 
quadratic form is also given. 
Chapter Ill considers the static problem in which 
the measurements of the process are linearly trans-
formed state variables corrupted by random measure-
ment noise. A modern sequential technique for proc-
essing the measurements in an optimal manner is 
presented. This modern technique is compared with a 
recursive version of the classical sample average ap-
proach. The conditions under which the two tech-
niquçs are identical are also discussed and examples
are given to illustrate the methods. The optimum es-
timation equations presented in this Chapter are de-
rived in Appendix A. 
In Chapter IV, the techniques of Chapter 111 are 
generalized to the case where the process dynamics 
are known and given by a vector difference equation. 
Recursive relationships for processing the measure-
ments in an optimal fashion are obtained. A dynamic 
version of the sample average technique is fabricated 
and compared with the modern approach. Generaliza-
tions of the approach to include problems of trajec-
tory estimation, correlated errors, etc. are discussed. 
The optimum estimation equations presented in this 
Chapter are obtained, using a variety of techniques, 
in Appendix B. The trajectory estimation equations 
are derived in Appendix C. 
In Chapter V, all the available techniques for esti-
mating a nonlinear process are reviewed. Both the 
discrete and continuous cases with known and un-
known process dynamics are considered. Some of the 
techniquewhich are closely related to those of the 
previous chapter are derived in detail in Appendix D. 
Note: 
Copies of the report may be obtained from: 
Technology Utilization Officer 
Marshall Space Flight Center 
Huntsville, Alabama 35812 
Reference: B68-10348 
Patent status: 
No patent action is contemplated by NASA. 
Source: J. D. Irwin and J. C. Hung 
of the University of Tennessee 
under contract to 
Marshall Space Flight Center 
(M FS- 149l5) 
Category 02 
This document was prepared under the sponsorship of the National 	 Government assumes any liability resulting from the use of the 
Aeronautics and Space Administration. Neither the United States 	 information contained in this document, or warrants that such use 
Government nor any person acting on behalf of the United States 	 will be free from privately owned rights.
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Study of optimum discrete estimators in measurement analysis

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