Information theory in improving digital data guidance syste

Chang, S.-H.

English

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... 
i. kvestigatioas Being Un$ert&en arnd Planaed 
-. :Le c3jectives of the reseazch work supsorted by this g r a t  are 
tvo-fold. 
~ c * L i , r ~ n t  o  t%e a;?plica%ion i n  the dig i t& data system used for  
zerospme gc5dznce. (2) To perform theoretical analysis necessary 
t o  Ge;aor,s-Lrzte how the range, capacity, re l iabi l i ty ,  and efficiency 
of a d i g i t d  data transnission system can be improved. 
(1) To conduct a r e v i e w  of informa-lion theory l i terature  
> a i r 2  S3i.s p e r i d ,  eqhasis is OD iten (2).  i t e n  (1) %ill be 
CG'.T^ -.- izl core d e t a l  d u b g  the next period when a graduate research 
zz;ls,tmt joins this groupe He will be assigned the i n i t i a l  phase of 
uA~v-c; work and the liasson trork with the Electronic Resemch Center. -- 
?our i t e m  of vork, in t'ae area of (i) Data Rate versus Intersyiiibol 
- -_ - - ~ , ~ ~ ~ ~ _ p z c e ;  ( i i )  &sign of Orthogonal Codes; d (iii) E r r o r  Control 
i"c;ec.; & ( iv)  A technique for bandwidth coqpression in telemetry are 
i+z>G>vLsd belm. 
52 ass-= a i i igi ta l  transmission system as i n  Figure 1 i n  which a 
s l ~ d  Si(".; is  t r a s x i t t e d  over the c-1 every T seconds With a 
p~u; o r  XXLZ-S pola-ity. 
.ad then d d s  white gaussian noise. \7e assume that  the problem is 
sc&d such t'nat s ( t ) ,  the? d-lstorted si- befoE noise &clition, 
izs-Ls Tor exactbj one second and such that the noise power i s  one w a t t .  
be detected by mtched f i l ter  tec'aniques i n  the data processor. 
Tne channel dis tor ts  s l ( t )  into a kn0r.m s( t )  
T- <.-? - - o ~ ~ t l o n  b i t  strea, represented by the polarity of s ( t ) ,  i s  t o  
Fib.;ure 1 Mgita l  Transmission System 
T.f -- cz3 x 2 e s  t o  t r d t  at a faster  rate, t L a t  i s  T < I, then he 
See Ngum 2, -her, Pe i s  increased and 
. .  - -,n--- - .- Li-b--uLuces iZ%erSpl’~Ok interference aad the individual b i t s  are no 
Lsx;~ eF-.l.j rsco,-cized. 
i n  bits-per-channel-use is decreased. The increased 
7, however, more than coqensate f o r  this prelimfnary 
* *  ._ --?&ation of C, the capacity i n  bits-per-secoxl. 
.- 1 ----, ..,c: -_ _ _ _  *is investigzte if  there exists an optisam choicz of T 
bu .--JIEL~ 2 i’or a known s ( t ) .  ---. -,‘-.* 
fs(-L) when T 2 1 i 
I 
I 1 s ( t )  when T = - 
2 
1 . 5 ~ ~  2 Received Signal &fore Noise Addition 
Cest-:: tke problem i n  a different light, suppose we a-2 t r axx i t t i ng  
a$ a s m z ~  :=-be grezter than one sp3ol per secoad, yielding a poor 
* Per-syzbol 1 because of intersymbol interference . We conteaslate sladng 
the source LZte -Lo cut dam the interference, hoping t o  i q z 0 v i n . g  our 
overzli syste2 perfomace. Cleazly this i s  identic& to- the  problem 
stated abovz i f  the criterion i s  to maximize the overall systen capacity C. 
Let us l-Mt the para;neter T to the range 5 T S 1 t o  repI=sent, 2 
m s t ,  z davibling of the transmission ?ate. (This also limits intersyDbo1 
-2- 
3- ,,e rzs-d+s of t'2ese c&cfiztions f o r  a general s ( t )  with =to- 
c o m L a t i o 2  I21-ction p ( ~ )  25e presented in Figure 3. 
z3scissa p is the si& energy on 0 c t c T: 
I n  that F i g i  %he 
0 
<;le pzsz!!';cr r represents the correlation of s ( t )  on 0 c t < T with the 
r a d d e r  32 s( t ) :  T 
J 
r =  0 - m; 
T P n 
0"8 1 
E L w z  3 ELts-per-chanwl-use VS. Usable S i p &  Energy 
--s 8 ;.-cy sizzle exwle of the proced-are, we assme the s ( t )  of 
f 
1.- Eeceived 
- - c  ,n -,Jx~ i,:; ::s plo t  p(T) aril i n  Figure 4c we have 
5 L d  0" ?%&me 1 we axe nm able t o  plot C vs. T as sham i n  Figure 5. 
..L ZL? Yaf;-L t'ae o2%imxi T is  approxis?a-i;ely T = -70, and that this choice 
U- - :??resel?-s a 18 bit-per-second increase of the capacity mer T = 1 
r ( T )  = l-T . With 
T _-  ~ 
c_-- 
r .  
- .- ;i 
. - , - * A  
ii - b U--b.. 
I C  
1.01 
- 1  
i 
1 1 
I 
i- 1 
I I 
I 
0; I 
t 
-41- 1 I 
I 
"I 1 1 
10 02 04 .6 .8 1 T  
ZLgure 5 Capacity VS. T 
-4- 
r.? --e ~ C X W ~ C L T ~  i i lus t ra tes  that  i n  soae cases It is profitabie t o  
-5ce tzasixtssion rate t o  the point where scxe inters-j&ol - - - , . -~se   < ,-, . 
L z e ~ f e r e i c e  i s  created i f  one L s s  t'ae rzceiver t o  cope with fkis 
ixxyfzrezce. 
c;?zael capacity is presented. 
A s L ? l e  mthod 0.C i:v&uatiag tha t  rate which optkdzes 
(ii)--Iksi,gn of Orthogonal Codes 
I n  digl5al cczmmication sys t e rq i f  a sequence of n digl ts  i s  used 
t o  r2Sresen-L a pa.r%icul2.r code word, %hen one c a  construct (n-1) addi t ioaal  
cc2.e rrords, d1 or' wkich a e  orthogonal i n  the n-dir?ecsior?al space. 
or;rs&ozdlky i s  expressed by the f a c t  that the inner product of each code 
t -ec -~or  bj i%seiT is a c o n s t a t  aad the i m e r  procuct be$veen 4x0 difrlerent 
cc% V ~ C ~ G ~ S  i s  zero. 
s-x: :mer ~ZO&LC+S a w h s t i c a l l y  by means of d ig i ta l  mtched f i l t e r s ,  
22 L: 0zt'ilogona.l se t  of vectors f o r  code words. 
T'.e 
. _  
if the detection of e a h  code vector i s  by coqut ing  - .  
- - - - -  _ h  L e  ckz-zce 02 xzaking a w r o q  identification is  ?nininized by the use 
. 7 . ~  teckLque of constrat ing binary orthogonal code words i s  - _  - . .- --- -imzi. TLe mtrk descr ib iq  an ort'nogonal b i n ~ ~ ~  vector se t  i s  - -  e ~ - &  the Z&,w,rd matrix, an exa;llple of which is  as follows: 
l i i i i i i i  
1 -1 1 -1 1 -1 1 -1 
1 1 -1 -1 1 1 -1 -1  
I H =  I 1 1  11-1-1-1-1 1 -1 -1 1 1 -1 -1 1 
1-1 -1 1-1 1 1-1 I .  1 -1 1 -1 -1 1 -1 1 1 1-1 -1 -1 -1 1 1 
-5 - 
n I 0 = x;r, X = n .  
It i s  ZcsLraSle thzt siiilar orthogonal xatrices be constructed f o r  
---- - -. ~ - 
u--L L,je v~z l -e   ore thm two elements t-1 and -1 are used. 
-:v c : - ; ~ c g ~ x , l  vectors consisting 02 digits vhich are e l eEn t s  of more t u  
-::a 2-rels. (1) construc- 
This w i l l  lead 
Tkxe =%hods are explored dming this period: 
-yen ->y 1 3 c -  ,L-s o_" pseucio-radon sequence; (2) coastruction by inspectioa; 
( 3 )  ccl=stnc$iort by recursion. 
-2- pse~Lo-rzrZo-r? sequence c m  be generated by a shift register c i rcui t  
md-lil.0 5 S X . 2  CS ~ G ~ ~ G l i S o  i- 4 LJJ I 2 
A t ~ i c z l  s;<:;ace of 21: dig i t s  
1 0  2 -1 2 2 -2 0 1 2  
U t 1  
is 
1 1 -1 0 -2 1 -2 -2 2 0 -1 -2 -1 -1. 
'This seqceme hes s a  radorn properties. 
0 (4 ix xzbe r ) ,  the other eleneats il, k2 each has a;n equal nwber of 
occu--:enc~s , m d y  5. 
CLc r ~ y  Est tce  original sequence zs the first row together w i t 3  i t s  
__ s.x:cessivc ~ M I X ~  as the succeeding rows. 
- A - 3 u  12 c o l m s  of the array. 
First ,  except f o r  the elemnt 
Secondly, tkie correlations between t h i s  seGecce 
~ 3 6  22- us cyclic shifts are all zero exce?t the s e a n c e  with I 2  SMIXS. 
- -  Then retain only the 
3.i -,.^ A The result is  a 12 X l2 matr ix ,  as follows: 
- 
1 0  2 - 1  2 2 - 2  0 1 2  1 1  
-1 1 0  2 - 1  2 2 - 2  0 1 2  1 
-1-1 1 0  2 - 1  2 2 - 2  0 1 2  
- 2 - 1 - 1  1 0  2 - 1  2 2 - 2  0 1 
'-1 -2 -1 -1 1 0 2 -1 2 2 -2 0 
0 -1 -2 -1 -1 1 0 2 -1 2 2 -2 
-2 2 0 -1 -2 -1 -1 1 0 2 -1 2 
-2 -2 2 0 -1 -2 -1 -1 1 0 2 -1 
1 -2 -2 2 ' 0 -1 -2 -1 -1 1 0 2 
-2 1 -2 -2 2 0 -1 -2 -1 -1 1 0 
* =  2 0 -1 -2 -1 -1 1 0 2 -1 2 2 L 0 -2 1 -2 -2 2 0 -1 -2 -1 -1 1 . 
I . I  
-b a d c - f  e - h  g 
-d a b -Q h e -f 
.%is t m e  of construction has been found applicable t o  other primes 
scc2 1s 3 ,  7 a d  11 nmber of elements. 
x&kcd r e q a x s  certain elenents i n  the f i e ld  of i r r a t i o n a l  numbers. 
For pri;nes higher tban 11, the 
( 2 ) c onst,-_-: - -.- 4 ~ m  -. by InsFection 
';Le 2sZariri.g ort%ogonal matrices are obtdned by inspection 
- 
2;- x - x % i t u t i n s  a suitable nmber f o r  each le t te r ,  orthogonal matrices 
olc' S-LZLWS zabsr 02 elemnts a,re obtained. 
-- AT A z .5  B are n X n orthogonal mtr ices ,  tiien 
L - ~  lJ l@A is a 2n X 2n orthogonal matrix with the s a x  elements cis A; 
(b 1 
t ' a "1 @ A  i s  a 2n X 2n orthogonal matrix v i th  different 
elements as A; !-3 a 
i s  a 2n X 2n orthogonal mtrix with the SEIZE 
elements as A and B provided AB = EA. 
. .  
El= 0 1 0 1 1 1 0  
I o  0 1 0 1 1 1 
__ 
:=lis co2e c a  either (1) c o m c t  a single randon error, o r  (2) cletect -. 7 -  -..-=J L..J 
LUCS%T&LGZ, 7 7 2  w i l l  use this co5e f o r  both C1 zd C 2 .  
& - p 5 r t ,  a L'TZJXCCS su3block mong 7 subblocks. Thus, for a t o t a l  
c:-;:--ed c c h  -,.-2l locate ~ t s y  single subblock that 'has two or  less  d ig i t s  
L ~ ~ ~ ~ .  
L A ~  is  z t e s o r  product of two cmponent check matrices. 
e"rozs in a 3 b c k  s i ze  of seven units. For simplicity of 
A s  C2,  it w i l l  
cz<ec< ";.;.:Q -fl.-F --a e x o r s  i n  a subblock of '7 a g i t s .  2s c1J it w i l l  
--. .,. 
7->* --LdL- - --* of 7 X. 7 = 49 binary d ig i t s ,  divided into 7 subblocks, the 
I .  
- -.. -- , c-- =..--3-j-. . .?->me. 49 digits contain 40 i n f o m t i o n  d ig i t s  aad 9 check 
- .  . - -._ _- 3 . z  ckc?&ng e p t i o n s  are contained i n  the f o l l a d n g  check matrix 
- *  _ _ - -  
into the expression, it becomes a 9 X 49 nsLtr ix .  - 
- 1  ;:.,:- enco5.i~; 2 decc&bng can be per fomd by cmwting the inner products 
02 -;:e ~2ssz;e seauence of 49 digits with the 9 rars of the check mtrix. 
T0-F- C O Z ~ ~ Z ~ ~ G J S  yield the values of t3.e 9 check digits i n  enc-ng or 
t t e  vdxes of the 9 s y i i r a  d i g i t s  i n  decoding, 
suck C C ~ L - ~ Z % ~ G ~ S  c a  be irrrpiemnted i n  the fora of shift registers. 
c<7m,2-'- r,b-u~ f o r  encoder and deccder a r e  akaost identiczl. 
. -  
In  case of cyclic codes, 
The 
-8- 
Xgcre 1 SLOWS sach a c i rmi t .  &perding OIL t'ae rbctions and tinings 
-.-a 2- 
u- a e  gztes, t h i s  circui-l can be used ei ther  as an encoder or decoder. 
Fib--? 1 A.Ci&t  fo r  the Znccder and Decaer of Code 
-9- 
-... 
~ 2 2  :sei -2 a eccoder the gates are synchronized according t o  Table 1. 
1 
1 
1 
0 
1 
0 
0 
0 
0 
0 
0 
1 
1 
1 
0 
0 
1 
0 
0 
0 
1 
1 
0 
1 
0 
0 
0 
0 
1 
0 
0 1  
0 1  
0 1  
0 1  
0 1  
1 0  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
1 0  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
0 1  
1 0  
Table 1 Encoder Timing fo r  Gates of Figure 1 
(1 denotes gate closed and 0 gate open) 
-10- 
.. 
T&le 2 &coder Timing f o r  Gates of Figure 1 
3 2 r z  c;"" "9 dig i t s  i n  each received bloc!; ~.rhic'n are fed successively 
2:~ %:e L.._L;';~. 32 t'ile decoder. 
,Z-J zzzaL:&: a% e x h  stage of the shif t  register. 
z2;.csy Tqe +o "c$'-' ZG e:rror i n  the block Jus% received. Otherwise, the= 
z r s  s o x  e z - c n e c x  digi ts  i n  it. The procedure of locating the subblock 
22 e,--rcr I s  8s ~ o l l o v s :  
A t  the end of ezch block, we investigate 
-- - - If all of tzez axe? 
22vL&e the block of 49 into 7 subblocks of 
7 Clgits each. 
C r - m >  the 9 shift register stages in to  3 grou>s. 
X&eI the Erst t ' m e  f,mx r i g h t  Si, the second 
2zm r i g h t  +, and the last  %&e S3. 
Assign a value 0 t o  Si, f o r  i = 1, 2, or  3, i f  
3 s3f-L register stages i n  that group all have 
zero iwm.irders. Otherwise, assign a value 1 
ZQ si. 
m e  follarLng matrix coatains a l l  possible 
coxbimtion of Si which is  the s w  as the 
pz-ity check matrix of code C1, Le., HI. 
,rie&nder Iiepre seatation 
S ~ O l O O l O l l  
s , , 0 0 1 0 1 1 1 0  
s 3 0 0 0 1 0 1 1 1  
- 
-A S ~ C X E  zzse of x(t )  being a Z i r s t  order ~ a r k o v  process i s  being 
7.: r_ ,. - _  - --- LLX- --&d%ic tec'nniqies which have been used are as fo l lc i s .  
(I) ?or the L i s t i t i n g  case of At 3 0, n 3 03 such tha t  
-sa + d t  -~t, given %hzt x ( t )  i s  sta'iionary 
C G ~ Z - L ~ ~ ~ L S ,  orre can study the baaiwidth conpzession 
%ec-3xLap as a bo-mdazy value problem associated 
Ik 'hwe foad the solution f o r  the clean time betveen 
trms~ttcd saqples for the specific case of x ( t )  
bel?< a Gz~ss-Xzrkov process. 
15t22 the solu-bion of the  t v o  Kolnogorov equatiolls. 2 
. .  I 
:?kt ca t  be s2id of the tecWC;ue wken the input 
is a hi&er-order Ykrkov process. 
Zov c a  zero-cross- a z l y s i s  techniques be exten5ed 
-Lo o r  q p l i e d  t o  t h e  solution of this problem f o r  nore 
general izsct processes. 
Eov does one analyze an interpolztive, as opposed t o  
a 2redictive, compression kchzlique. 
-13- 
r .  ' I 
I i * 


Utilization of information theory in digital data guidance systems Quarterly progress report no. 1, 1 Sep. - 30 Nov. 1965

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