Random processes, filtering, and telemetry problems in space communications theor

Beutler, F. J.

Rauch, L. L.

Root, W. L.

English

NASA Technical Reports Server

T H E  U N I V E R S I T Y  O F  M I C H I G A N  
Office of Research Administration 
Ann Arbor, Michigan 02905-18-P 
July 14, 1967 
SEMI-ANNUAL STATUS REPORT ON RESEARCH PROGRESS 
NASA Research Grant NsG-2-59 
October 1, 1966, to March 31, 1967 
During the subject reporting period, faculty members have derived 
support from NASA Research Grant NsG-2-59 a s  follows : 
1.  Professor Frederick J .  Beutler was supported on a 2570 basis.  
2 .  Professor Lawrence L. Rauch was supported on a 2570 basis.  
3 .  Professor William L. Root was supported on a 5070 basis.  
In addition, approximately half-time support was furnished Mr.  
Robert Bayma who is working on his doctoral dissertation research under 
the direction of Professor Rauch. 
In the subject reporting period, Professor Beutler has obtained 
sti l l  further new results treating applications related to the theory of s ta-  
tionary point processes. In collaboration with Dr.  0. Leneman, he has sub- 
mitted a paper, "The Spectral Analysis of Impulse Processes " for  the 
International Symposium on Information Theory, to be held in Athens, Greece 
in September 1967. This paper improves the results found in Dr. Lenemanf s 
1' Random Sampling of Random Processes: Impulse Processes,  ' I  Information 
and Control, 9 (1966, pp. 347-363. A summary of the new paper by Pro-  
fessor  Beutler and Dr.  Leneman follows: 
An expression for the spectral density of the impulse process 
co 
s ( t )  = C (Y 6(t - t ) is derived under the assumption that {cy ] is a 
-00 n n n 
stationary point process [B4] independent of {ant.  The spectral 
densit appears a s  an infinite se r ies  in te rms  of the correlation 
of {any and the interval statistics of {tn). The same result was 
obtained by Leneman [L2] by a different argument under consid- 
erably more restrictive conditbns of validity. 
Various models of impulse processes a r e  discussed relative 
to random sampling of random processes.  Random and system- 
atic loss of samples, separate read-in and read-out jitters, and 
correlated random scaling e r r o r s  can all be represented by ap- 
propriate assumptions on {on) and {tn]. 
? 
Finally, closed form expressions a r e  calculated for  the 
https://ntrs.nasa.gov/search.jsp?R=19670023205 2020-03-12T11:15:31+00:00Z
spectral density of s( t )  and the sampled process under combina- 
tions of the sampling e r ro r s  mentioned in the preceding paragraph. 
Professor Beutler has also been investigating the possibility of 
accomplishing alias-free sampling of random processes by sampling at ran- 
dom instants. A sampling sequence {tn) is said to be alias-free if  the spec- 
t rum of a random process x(t) can be recovered from the discrete parameter 
random process {x(tn)). It is already known that recovery is in some in- 
stances possible i f  sampling is at  a rate lower than the Nyquist ra te .  
fessor Beutler is  using a new approach to the problem to obtain additional 
results.  
Pro-  
The report" "The Use of Statistical Dependence Between Data 
Samples in Binary PCM Demodulation, 
Rauch with James K Strozier on optimum and suboptimum demodulation of 
uncoded PCM, was issued in November 1966. 
pears in the previous Semi-Annual Status Report. 
describing the work of Professor 
An abstract of the work ap- 
The work on improved analog demodulation with Professor Rauchf s 
doctoral student, Robert W .  Bayma, has continued. Professor Rauch and 
M r .  Bayma attended the special conference at  the Ames Research Center on 
March 30 and 31 and April 1, 1967. 
Professor Root completed a paper, "Estimation of €-Capacity for 
Certain Linear Communication Channels, I '  and submitted it for publication. 
Pa r t  of the work on this paper was  supported by the grant. This investiga- 
tion w a s  concerned with obtaining bounds on possible communication rates 
in environments where the signal is distorted by influences that cannot be 
treated probabilistically; a fuller discussion was given in the previous Pro-  
gress  Report. A briefer, more mathematically oriented paper on the same 
subject entitled, "The €-Entropy and E-Capacity of Certain Time-Invariant 
Channels, was co-authored by Root and Professor R .  T .  Prosser  (not sup- 
ported on the grant) of Dartmouth College. 
accepted for  publication in the Journal of Mathematical Analysis and Appli- 
cation. 
This second paper has been 
- 
A paper was given by Root, which also was co-authored by R .  T .  
Presser, at  the Princeton Conference on Information Sciences and Systems, 
March 30, 3 1, 1967. This paper entitled, "On the Identification of Unknown 
Systems, " will  appear in the Conference Proceedings. Root's work on this 
paper was supported by the grant. The summary of this paper is a s  follows: 
"A procedure is given for the identification from input-output 
data of systems which can be represented by integral operators, 
l inear o r  non-linear, time -invariant o r  time-varying . The pro- 
cedure is not a practical one, but it serves to classify those 
situations where there is sufficient - a priori  information to make 
Dcctoral Thesis of James K. Strozier. 
.I, *a* 
2 
I . 
an identification possible. The procedure is deterministic, but 
is mathematically somewhat analogous to a known stochastic 
technique, and a comparison between the two is made." 
Professor Root and Professor P. P. Varaiya (not supported by 
the grant) of the University of California have been working on the problem 
of finding the capacity (in the sense of Shannon) of channels with additive 
Gaussian noise when there is an unknown linear operation (belonging to a 
certain specified class) in the channel. 
Classes of Gaussian Channels: Par t  1, " has been completed and will  be sub- 
mitted for publication. 
grant. The abstract of the paper reads a s  follows: 
A report entitled "Capacity of 
Root's work on this report was supported by the 
"The usual definition of the capacity of a discrete-time, 
memoryless Gaussian channel is generalized to the case of a col- 
lection of such channels. 
fied by a pair (A ,  Q )  where A represents the deterministic trans- 
mission matrix, possibly infinite-dimensional, and Q is the 
covariance matrix of the additive Gaussian noise. The definition 
is justified by showing that the capacity is the supremum of the 
attainable rate.  1 1  
Each member of the collection is speci- 
Root and Varaiya a r e  continuing work in the same problem area .  
During the period covered by this report a paper, "Determinable 
Classes  of Channels, ' I  by Root and R .  T. P ras se r  appeared in the Journal 
of Mathematics and Mechanics, v 16, October 1966, pp 365-398. This 
paper has been reported earlier; Root's work was partly supported by the 
grant. 
3 


Investigations in space communications theory, including topics related to random processes, filtering, telemetry  Semiannual status report, 1 Oct. 1966 - 31 Mar. 1967

Investigations in space communications theory, including topics related to random processes, filtering, telemetry Semiannual status report, 1 Oct. 1966 - 31 Mar. 1967

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