Convolutional coding techniques for data protectio

Massey, J. L.

English

NASA Technical Reports Server

F3NA.L REPORT FOR SECOND YEAR 
1 CONVO~&TIONAL CODING TECHNIQIBS 
F8E DATA TXOTECTION' 
Dt~tZrt~efft 01
ELECTRICAL ENGl N EERING 
H u ~ ~ ~ ~ ~ ~ ~ ~  OF NOTRE DAME, NOTRE DAME, INDIANA 
https://ntrs.nasa.gov/search.jsp?R=19700027056 2020-03-11T23:33:22+00:00Z
Submitted to t  
FmAL REPORT FOR SECOHD B A R  
(septa  16, 1968 t o  Sept. If;, 1969) 
I 1 CONVOI;UTIONAL CODING TECHWIQ~TXS 
F9i DATA ?20T%CTIClJ~ 
National Aero~iautics and Space Adg15nistrat9.on 
Washington, D, C,  20546 
At-tentioli:: Miss l?5.nnie M, McrgW, Code-US1 
Technical Reports Assj-stant 
I ~ o .  Copy to: NASA Goddard Space Flight Center 
Greenbelt, Md, 20771 
Attention: D r ,  Robert W, Rochelle, Code 750 
Submitted by: Dr.  James L, Massey 
Professor of Eleckrical  Engineering 
Univ, of Notre Dame 
Notre Dane, Indiana 46556 
ABSTRACT 
This repor t  Jescribes the princi.pal r e s i l t s  obtained during 
the  second year of research u ~ d e s  NASA Grant NGL 15-004-026, These 
r e s u l t s  are compiled i n t o  t h e  following six categories: 
(1) Co~ t r ibu t i ons  t o  %he general theory of convoluLional codes, 
(2) Development of powerful quick-look-in, nonsysLerna.tic 
convolu.tional. code ss 
( 3 ) Con briby~.-kions to secpentj-a1 decodi ng, 
( 4) UnriSS-ed a r ~ ; ? l ~ s i s  of decoders f o r  convolutional codes, 
(5') Iiiverses of l b e a r  systems, and 
( 6 )  SUb-ba~rd coding . 
This repdrt a,l.so gives a l i s t  of t he  pul~%ical;ions issuirlg 
from t h i s  research Logether wi th  a 2.is-b of personnel p e r f o r ~ i ~ g  
research under P;h.i.s grant;, 
I. SILV~!EZ;:L=; i'f !,lain Research Kesu3.t~ 
I~ "1- L , ~ C  -. follciai-ng sub-sections, we l i s t  t he  1r~aj.11 r e s u l t s  obtcii.ned from 
the res-c-z:x'il pe~.fon-oed. ~ ~ n d e r  this g r a n t  clu.ring t:ie peri.od from Sept .  16 ,  
l96e t o  Sept ,  l!:, 1 9 6 9 :  
( 1  Cori-tri.huti.ons t o  t h e  General  Theory of Convolut ional  Codes 
During t h i s  pc r iod ,  D. C o s t e l l c  ccixplcted a a a j o r  s tudy  of the  capa-. 
b i l i  t 5 . e ~  ;!nd lir : :  I, "-  ati ions OF convolu t iona l  codes, t o g e t h e r  with t h e  devel-  
0pmen.t of severa l  techniques  f o r  syni-hesi-zing good codes. This  \ . io~k is 
.. . f u l l y  desc.x-~j.:>e~ m !?is t e c h n i c a l  r e p o r t  (See Sec t ion  I1 ( 2 )  f o r  a coinpl.~-te 
r e f c ~ r r ~ c e )  7:liicii: was a l s o  s u h n i t t e d  a s  h i s  6oc.toral 2 i s s e r - t a t i o n ,  i''ri!o~?~ 
n;enti oil: 
( a ?  'rh, ne-,.r co::ccpts of the o2.de.r j  1.o~: distal:ce act: t h e  or6er. 
----- --- - 
j ,-olrn;~c d<.s-tsnce o f  a  convolu.ti.ona:l code, and t k c  r+ela4ii.on of' tl-tese 
-.--- --------- 
dlsta!:ce ceas\irc;s t o  the f ree  distance. of a convolu-ticnal cc;dc. 
(b) The n e w  coi:ce?t of c a u s a l  dornin;:nce of convo.t~!-tio~ial codes. 
--.- 
~: illis j d e z ;  i s  ceri-tpal t o  Coostello's irriportant new 1-esu1.t t i i a t  systc:l:atic 
C O ~ C S  L?TC as gocd as nori-c~rste~o.atic - J  odes (bctl-I i n  eryrol.-corri..cti11g por,ier 
an2 ease  of ii~;p.'.cnentc:.'ii.cii~) if onc i s  rqiJ.ling t o  pemi i t  tile use  of feed- 
back il) the e ~ l c o d e r ,  
-- 
(c )  The general. raetho6. fort syn-tliesizing the  syndpone f'oryning 
ci^rcuj-t f o r  ;in a r b i t r a r y  convolu.tional code . 
( d l  !lew z n d t i . g h t  l.ower bcunds oii t he  f r e e  di.star!ce and t h e  
def i i l i i ; ~ .  c~ccodini; mi.nii::ur;, d i s t znce  a t  ta!.n;J:le w i t 1 1  cofiuolu-tional codes . 
( 2 ) Developr:,erlt of Poi:eriul Quick- iook-in i lonsystcmaric Convol.utiona1 
Codes 
A s  was r epo r t ed  jn t h e  Qua r t e r ly  P r3o~wss  Zepor t  of Jvnc 5 ,  1969,  f o r  
t h e  p e r i o d  Feb. 2 6 ,  1.969, t o  E4ay 3.5, 1959, thc p r i n c i p a l  i i 2ves t i ga to r  p ro-  
posed a c l a s s  o f  non-systematic convolut ional  codes t h a t  pe rmi t  quick 
r.ecovery o f  t h e  inforiilatiori b i t s  (witl iout decoding) f o r  synch ron iza t ion  
and o t h e r  engineer ing  purposes and which perfa~-!rt b e t t e r  than  t h e  b e s t  
sy s t e i ca t i c  codes of t he  same constr la int  lengJih. D. C o s t e l l o  c a r r i e d  ou t  
a c o i ~ p u t e i ~  search  f o r  t he  b e s t  such codes arid coaducted cc~nparj.son sirnu- 
lc?tions.  These r e s u 3 . t ~  confi.rned t h e  su .pe r io r i t y  of  t h e  new c l a s s  of 
. . 
non-sys- tenialc  codes vihich a r e  expecte4 t o  fj.nd widespr>eacl a p p l i c a t i o n  i.11 
space channels .  
( 3 )  Ccm-t:i?ibutions t o  Sr-qcentiz.1 1)ecoding 
During .?.his p e ~ i o d ,  t h e  p r inc i2z l  ir!vcstig?.tc.r an5 ti. Sai.n repor-iied 
on their si.nip.l.rlfi.ed approach t o  zna lyz ing  the  search  perforired ; y  -the Fano 
sequent ia l .  decoding algcwi-thr:~ (See I I ( 3 )  for coinpletc reference)  and a l s o  
r epo r t ed  on t h e i r  nethod f o r  a s p e c i a l  c l a s s  of code:; t o  o b t a i n  t h e  e,:act: 
di.stiqibu.tion o f  t h e  n:e-tri.c a:.ong t h e  C O Y > ~ - C C ~ -  p a t h  ir! t h e  code t r e e  (See  
I J ( L t ) ) .  These i i e a s  v e r e  incor.porated by J .  Cc i s  t i n t o  an in-dcp th stucLy 
of t h e  a l g o ~ . i t n i i c  a spec t s  o f  sequer1-t-j.ai decoding on which he  er;iba&ed 
dur ing  t h i s  p e r i c d .  The TIarc'i-t. 4 ,  1969, Qu?r ' icr ly  Progzless Report f o r  t h c  
pe r iod  Ifov. 1 6 ,  1968  t o  F'eh. 15 ,  1969, de sc r ibes  one o f  t h c  e a r l y  r e s u l t s  
ob ta ined  E r m  G e j s t ' s  s t u d y ,  nciinely a geor:ii:tric pr30ceclur~ 
-20 iie-tei-inine the 
"ccriiputationa.1 b~r .s . t ' !  r e s ~ l t i n g  fror!i c l o s e l y  sp iced  e r r o r s  on t h e  c:~znnel. 
( 4 )  Un.iii.ed knz lys i s  of Decoders i o ~ . )  C ~ l i ' ~ ~ l i l t i ~ l l & ~  Codes 
During t h i s  per iod  , -the r e p o r t  of Ilv~:>ri.sscy ' s novel  axid powerful. 
2 
s t o c h a s t i c  s eque i l t i a l  il:achine a p p ~ o z c h  t o  t h e  decoding of  c o n v ~ l u t i o n a l  
codes was r e l e a s e d  (See I I ( 5 )  f o r  complete r e f e rence . )  This  work was 
descrLbed i n  d e t a i l  i n  the  r e p o r t  f o r  t he  f i r s t . y e a r  of  t h i s  g r a n t .  
( 5 )  I n v e r s w  of Linear* Systems 
During t h i s  pe r iod ,  t h e  r e p o r t  of  t he  f ~ n d w i e n t a l  work by t h e  p r i n -  
c i p a l  invest.i.gator and 11l.K. Sain on inve r se  l i n e a r  systems b r a s  publ i shed  
(See I I ( 6 )  f o r  caxp le t e  r e f e r e n c e . )  D.  Cos t e l lo  (See I I ( 2 ) )  found t h i s  
work and t h e i ~  e a r l i e r  work on feed-forward inve r ses  t o  be of fundamental 
importance 5-11 mriy phases of convolutional. coding. In  p a r t i c u l a r ,  he 
s h o w n  t h a t  t h e  ex i s t ence  of a feed-f'orvrard inverse  f o r  t he  encoder  guall- 
&?tees t h a t  +I.-. cllc r ; p d e ~  j ra;.; co].c~fin <'--"---;-- --------I- A"- '---- 33 a L a i i i ' c s  ~ ~ ~ > ~ . I L ' c J ~ ~ c I I  L I C  I I'CG 
d i s t a n c e  as j i n c r e a s e s  and a t ta i .n  th5.s li.il:i-t f o r  -j no g r e a t e r  than a 
srcall. cons tan t  tiriles t h e  square .of t h e  niernory order of  the  code. FIore-- 
over ,  C o s t e l L ~  sho7f:ed 'chat t h e  ex i s t ence  of a zeu>o-c'?~t_l.ay iiivepsc foi- 
the  encoder i s  necessary  and s u f f i c j e n t  f o r  t he  code t o  be c a u s a l l y  
equi-valent  t o  a systematic code (possj-bly vii-th encoc',cr feedback,) 6r1d i t  
i s  pi?eci.sely thc. case where -?-he.enco.'er I-ias no zero-,clelay inve r se  whi.ch 11ciii 
d e f i ed  past a-tJiexpts t o  foriil c a u s s l l y  e ~ c i v z l - e n t  systeiiLa-i:ic codes - a 
dif-:ficul.tjr ~..<qic'n Costel. lo obviated by i n t ~ ? o c l u c i ~ ? ~  the  concepts of  c z u s a l  
d~ i :~ inance  r a t h e r  than causa l  equiva lence .  S inply  p u t ,  one encoder i s  
causa l ly  d m i n a n t  t o  alo-tt ier if t h e  s e t  OF a l l  outpu-t sequences which i t  
pryoduces f r c a  caus a l  i n g u t s  prope~1.y inc lucks  t h e  1 i . k ~  s e t  f o r  the oti lcr  
encoder.  Costel3.o shoxed t h a t  t h e  czusa l ly  daninant  enco&cr albls,ys h s s  
g rca t ep  colu:nr: di.s.ta::ces (hence b e t - t e r  errii:? pr,o:~cd.~ j . l i ty  216 l e s s  cojr1pu- 
L> 
t a t i o n  i:rith scquenCJal. dcc :~ . ' in~)  z:l~d sho;!a2 -that ail encoder without s, 
zero-iclajr  ii-L.i.ers.z i s  causzliy- dgnirlztecl, by a sy;tpn.atic code, 
( G )  Skb-Eiau;: C0C;;6iL 
A s  p a r t  of h i s  e f f w t  cni!er t h i s  g r a n t ,  t i le principcil  i n v e s t i s a t o r  
p a r t i c i p a t e d  i n  a stui!;; 35 t h e  a ~ u l - t i p a t l ~  and niulti-user inter.fer~er,ce 
problems a s s o c i ~ t e d  wi th  t h e  proposed T e l e t y p  i ja ta  Keiay S a t e l l i t e  s y s  tea. 
His c o i l t r i b u i t o ~ ~  t o  Chts s tudy xias a -thorough i n v e s t i g a t i o n  o f  ~ub-bii:~:d 
coding (See Chapter I V  of t h c  Zocunent c i t e d  i n  II(7) . )  Sub-baud codirlg 
was t & e n  t o  mean t h e  assignr ,~ent  of an n - b i t  b ina ry  codebiord. t o  each of  
the  u s e r s  t o  spec i fy  forr7 each u s e r  t h e  p o l a r i t i e s  of n a d j a c e l ~ t  pu l se s  
which fonn t h e  baud sig:ial for t h &  u s e r ,  i.e., e i t h e r  t l i i s  corl:posite sig- 
n a l  o r  i t s  nega t ive  i s  transiaittcc! dpeending on the  value of t!ie ;.:;fomia- 
t ior]  b i t  t o  be s e n t  i n  t h a t  baud, The object i .ve i s  t o  ninimize t h e  c u t p u t  
of t i le i~iatclled f i l t e r  fm one user3 trhen t h e  j.12pu-t is the s i g ~ ~ a l  of another  
u se r  o r  a t ime-delay 05 t h e  sane u s e r ' s  s ignal . .  The p r i n c i p a l  j-nvestigatoi? 
shor.red t h a t  t h e  p t = , ~ f o ~ , ~ 1 ~ i 3 c k  of such suL-baud <:odes 2s  de-Lep-;i)iilec; by tll? 
s e t  of even and zdd corn-,c-].ation func-Lions f'or t h e  h i ~ - ; a r y  cnde words. 
. . T.ihi1.e tile ever, c o r r e l a t j  on fiulc-tions a r e  f s :n~ . l la r  i n  cor~municatior! 2;-tudj es , 
t he  necessi t j r  t o  consj.dcr tile ocld c o r r e l a t i o n  f u ~ l c t j - ~ g s  s~c i i l s  slot "co h2ve 
beer1 no t i ced  prcvi.ou:;ly. He a l s o  si?o.i:ed t h a t  bihereas tile even co r r~c  1ati.o;; 
functio:rs are unaffec te -  by which phase of an n d i g i t  sequciice i s  chosen 
f o r  a codeword, t h e  odd c o r r z l a t i o n  function:: are c r u c i a l l y  phase-dependent . 
k:e a l s o  showed hov c y c l i c  er~ror-cori .ect ing codes can be used t o  synt 'ncsize 
sub-baud codes w i t 2 1  bounds on both the even and oclc! co rpe l a t ion  fur~c? ions  
determined by t h e  r~iiniinti:~ dl.s-cance of t he  code, F i n a l l y ,  he shovled t h a t  
c o n v o l u t i o n a l - t y p  of  s-ii:!-brrcc3 codes appea-r. t c !  have l . j . t t te pract--.cal 
11. L i s t  o f  P u 3 l i c a t i o n s  
The iollov!ing tec 'r inical r e p ~ r t s  and jcurrial  a r t i c l e s  disclosi .ng 
r e sea rch  performed unber t h i s  g r a n t ' w e r e  rnade p u b l i c  o r  subrriitted f o r  pub- 
l i c a t i o n  during t h e  pe r iod  covered by t h i s  r e r o r t  (Sept .  1 6 ,  1968 t o  Sep t .  
15 ,  1969.1 i i e p r ) i ~ t s  and/or  prepr l in t s  o f  a l l  t h e s e  docu~~~en-irs were s u p p l i e d  
t o  t h e  Idat;-onal Ae170nautics and Space Adminis t ra t ion  and t o  p r i v a t e  i n -  
v e s t i g a t o r s  exp re s s ing  an i n t e r e s t  i n  r ece fv ing  t h e s e  publ ica . t ions .  
( 1 )  D. J .  Cos-Cello, Jr. , "A Cons t ruc t ion  Technique f o r  Random E r r o r  
Cor3rec-ting Convolut ional  Codes ," IEEE T~sns. I n f o .  TI]. , IT-3.5, pp . 621.-636, 
--------- -- 
Sept .  1969, ( t i lso r e l e a s d  as Ileno. EE-682,  i!cpi. of E lec .  Engr . ,  Uiil'v. 
( 2 )  D . J .  C o s t c l l o ,  J r . ,  "Construct ion of  ConvoltiCional Codcs f o r  
Seq~i.ential .  D e c o d i n ~ , ~ '  Tech. Rept.  EE-692, Dept,  o f  E l s c .  Engr . ,  U .  of' 
Notre Dcrrie, Notr-e &me, Ind . ,  August 1563. 
( 3 )  J. L .  "Iassey anc? ';.I. I<. Sain , "Ti'rui~k and Tr.ee Searcl7ing Propertj-rss 
of t h e  Fanc Sequent ia l .  Decoding Aigor~i thrns , !~ Froc. 6 t h  Ai le r to i l  Conf. c ~ n  
---.-- .---- +- 
C.i.rlcuit and Systeifi Th., U I I ~ V .  of I l l . ,  U r b a n ~ ,  I3.1., pp,  153--160, O c t .  
-----PA----- 
1968. ( a l s o  r e l c z s c d  as I4ei1:o. EE-6817, Univ. o f  :Totre. Dane, FTotre Dar!!e , 
Ind . ,  Oct,  1, 1.9E18. 
l a t i v e  RIetric f o r  Szqucnt ia l  Ticcoding ,'I Cabs crsct on ly)  prcsc; l tcd a t  ?EEL 
( 5 )  T .N. tbio?:rissey, Jr. , "A Un.i.J"Iiec1 l:~r.kovlar! Ana1ysi.s of Eecodei-s 
f o r  Con7~ol~t<0i ;21.  Codes , '' Tech . R p t  . EE-G P7 , I:.p?. of Elec . E~igr. , U. of 
1 , lo t~e  Dzne , I:otr-'e I);:r,;e, In.!. , Octci'itej:> ? I ; ,  195g. (To appear- i i i  IE:EE 
- -- 
. - r ,L \ 
.Lll.I.O, I l l .  J 
Dynac~ical  Systems,'' ICEE T r a n s .  - Au-to. C o n t . ,  i?C-14, pp.  lltl-3.4Y, April 1969 
( a l s o  r e l e a s e d  as : ! e n ~  FE-687r.1 Univ.  o f  ;To t r e  Dame, lu'otre Dci~ie, I n d . ,  
Aug. 8 ,  1368) .  
( 7 )  J . L .  Massey and J .  J .  Ullran, J r , ,  " F i n a l  Report  f o r  M u l t i p a t h  
Study (NASA C o n t r a c t  NAS5-10786) ,'? J u l y  1, 1969 t o  Aug. 31,  1-969, Dept.  
o f  E l e c .  Engr . ,  U .  of f i o t ~  Dame, Notre  Dane, I n d . ,  (The t j n e  devo ted  by 
J. I8Iassey t o  t l l i s  p r o j e c t  rras suppoiqted. under  t l i j  s g r a n t )  . 
Tc7&1fi I 0 i~e-s a complete J i s t  of all persoiinel en=q,e2 i n  rese; i rdi  
u.x?2~1' i I L j s  g r a t  dur ing  the  pe r iod  (Sep t .  1 6 ,  1369 to Sep t .  1 5 ,  1969) 
covere? h:i 1 2 r i s  I-?port. The source  of suppor t  for. gracli~a-te s t u d e n r s  
~ ~ r f o r i ~ : n z .  ~ s e ~ 1 7 c h  wider t !~e  d i r e c t i o n  of t h e  PI- incipal  i n v e s t i g a t o r  i s  
sl-.o\:.~-I en2y i f  d i f  ri-iirent fro;!? t h i s  g r a n t .  
Me a r e  p leased  t o  r e p o r t  t h a t  M r .  Daniel  J .  Cos t e l l o ,  JF. ,  c o ~ p l e t e d  
the reqlrirerlcritc f o r  t h e  Ph,D. degree i~ e l e c - t r i c a l  e+.neering wi th  t h e  
si!ppos7t 3f 'i1i.i~ gz-act i n  September 1969 and is  now afi a s s i s t a l i :  professor .  
of e lec tp ic .~ . J .  el?ti.rieer.ing a t  t h e  I l l i n o i s  Ins.titu.t;i of Tecii,n<jlc;:y, 
el.-'--,.r- T1 3 :,,:, 
I jLLCl .&,C . ,>  ,.i . ' . l l l C I . L 3  . 
i 
i 
Dabs  o f  ~oui7c:e 05 ~ t ; p p o y t  j 
N a!ne 
-. Categor:y - Aff i l i a . t i on  .- --- i f  n c t  t h i s  - G r m t  .-- : 
( 1 )  Dr. J .  L, Massey PI-iricipal. Lnves-t-i.gaCo:. 3-15 -6 8 t o  9-15 -63 
( 3 )  I ~ P ~  C, J. C o s t e l l o ,  Jr. Receai-dl Assistmt 9-1F-€8 t o  3-15--69 


Convolutional coding techniques for data protection Final report, 16 Sep. 1968 - 15 Sep. 1969

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server