Convolutional coding techniques for data protectio

Massey, J. L.

English

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https://ntrs.nasa.gov/search.jsp?R=19690021049 2020-03-12T03:39:26+00:00Z
June 5, 11069
QUAKILY FROGR; S REFOR
CORVOLUIIOML CGDDr, ` I:C=' T;--q.
FOR DATA PRO7LETECTILON
NASA GRr",: ' i'^.s11-15-004-L25
Submitted to:	 Flight Data Syster Rr° . h
_ - NAoA Goddard S_eace F i&a Center
Greenbelt., 4d. 20'7
	v
AM: Dr. Roh- `rat . Rochelle (Code 710)
and:	 National A°ro iutics and S;.-ace Ace-driistr tier_
Washington, D. C._20546
AT%: Miss::in: =- M. cn-gm-n ,Code - LSI)	 -
Technical Reports Ass,- ant
Principal investiE,t^r: Dr. James T_ ! "a.^.=.e
Professor of Rlec -r_cal a~.z:ms^_nom
Univ.
 . of iiotre Dee
	
- e
V
Notre Dame, ind. 46556
Research gssist rts:	 J. CIL._ng
D. Colstello
J. Geist
Research ri: d
 Reported: Feb. 16, 196 to
	 1„ N.6?	 w
X69.x427
1ACC[f+ION NVM ERt n""'
=[ 1
: aPAC[SI aCl. [I
s I
INAfA	 R OR t 1I OR AL tiYM ^CkI tCAT[GORYI
1. Deve V'-A _r:L of a Po aft]., as' iy-T ie _nteu, er_ :
Binary C„,, ^^ts
	
	j t	 s= -onvcluL ILonal Ccde of Rate 1/2 Sui table  f b*+ - ;7 1j"^
t
(a' Descript-fan of the Code
	
The
	•»
	 is - f t'-.,.e -7 17 1 2 ,.- Data 	at t	 # si J
-	 1116 1'eQu1y e:'^e2"1	 -.^ ^ L..^. j .^. L -S:+ 	^ _ s~
have itridicated t",-,e me£'d for i.hort 	 -
convolutional codes that will y1eld 	 e.: P.M*
3	 s
de- nded by seque_,t ?! -eccj r, f'  ^r ^ *_ ^_	 -s 3
search for a gmod -n-orsysterratic cede sLnce tim
straint ler;Sth of a non-syste-.^tic =_ 112 ode 	 ar._ t^^: 
that of tre rn r° usual Sy$tE'^ %ic code. r1 Lowther ++• a ^ ^ ^ r ^S
that	 ercode„ sh. -Uld be Si. wle--that is, ;,jmt -here sI^- d 'La
l	 a smli nix-e'r of ?r"-pet.' 1­0 the M0dUI -t:^3 adders used- 1= vim'
?nC ?*'.	 ^1^'^S '_'`'+="u=Z'^' w-'zc`  i,e- .	 Gi??t`..^2 2C't t!'a .` `he ^CnCt°r
is a ha3't+wavre dE is =Ce in ° Space reh c le ? tself . 7 s search
led to the f=t"tE:il'sgl i;f the '.'.tom° veSC='itC." a.. CelC:•: : LyCtl crGv des
extr l le d^ rode^ e_r r procab M ty and can L e e_ e dea b	 _
device of Z -_—.m_'kable' _`•.^11i c t;: , ='e! ^L _ °c::eT' _ i^ t i'_C-k' -o adu'':'3
than the- pZ"ese t ly	 S;; Ste'^:ii cc-de _f he Same cons"L Omt
lenEt::. -11 "l-le code has 1ne- flit'-.: _. 1 ., ^" :i ^_ fe_.,r. 'e ^^ at, alt? Ct:^}
	
-L	 e 1,n om' -- on ._ .`:o_'_ :-qr- t ea	 otzsl ined °±'o
the emcde a ;Ls 'v:- +.:Inca. the use	 a V4- . d . mi in ite-r feature
	
D2i?iLi'.S	 ,: ir _:^^': in"	 a=r`w^ ':r- ^a^^ f ^' r^i.L* r.` 3 ` 2t rC :s^^
For a ene,,._1 ^ to +!7^ `nar^^ cC.^.^'v'lutirn`'_ cG7e^ the
--I-
together, one obtains the infor: ration sequence wrialtered except
for a delay of one tir;.e instant.
It has been observed fro experience, that generators (i.e.
the coefficients in the pcl^mc mials G1 (D) ^nd G2 (i ) ) -with a hi g:-.
density of "ones" generall y result in loa error probability. As
will be see: liter, a density of "ores'' well above one-half also
leads to a s.*rple encoder. For these reasons, a search was :*ode
to find a good code using- the 'cll.-)vP_?9 algorithT:
Aigoritym: (1) Set the first t:;o coefficients in Gl(D)
aqual .tu	 and set k = 3.
(2) Set the yarn dJ git in G1 (D) equal. to "one"
-2.^-
unless setting to "zero"-gives a greater
mintm n distance over the first k branches
of the code tree.
(3) Increase k by 1 and go to (2).
Application of this algor ahm in a cuter pro^-n up to k = 48
yielded the following generators (coefficients shown in the usual
octal form):
G1
 = (733, 533, 6fi6, 737, 355, 3)8
G2
 = (533, 533, 67", 737, 355, 3)3
fie ffinimut* distance of tine f :,l codes is 16._ Since the algorithms
_
-	 is `nested", tuncatSon	 the two generators at any k, k = ^,
will yield a good code at t! tt cors;,raint length.
For purposes of t^cb^* ,! the code was truncated at k = 36
sinm tl is Is a like?;► ricure to be used in some applicat^cn.
Hence, t ine Lu -_ 4tors used 1.n the test care:
Gl
 = (733, 533, 676, 731	
(6)
G2 = (533, 533, 076, 737)3
By computer search, it was determined that this code ',ad a
distance" (miez_^ured over the constraint lenEth of 36 branches) of 1l
and a "free distance" (mdnim-m distance over the full code tree)
of at least 17. The fie distance has proved to be a better
predictor of error pro:abiiity that the mani.m, zn  distancE. are__1. this
code has a h.ij-, valuE. of this parameter. The exact free dis tanCe.
however, is not yet kmr)tin. It sLould be nosed that G, co,-taitnZ 28 "Oneslr
out of 36 digits, an exceptionally h1rh dens ty of"ones."
(b) Irplementation of the Encode.-
The "trick" used to reduce modulo Iwo adder corrections
in the encoder when. the- generators have- a high density of
"ones" is to implement the corplenent of the generator plus
adding 9. circuit whose ei f ect is to cowple^rient the transfer
- - -functior- T)receding it. This -latter circuit can be-sirTly built
as-shown in Fig. 1. At the output of the adder vrhere Y(D) is_
formed, we haste the equation
Y(D) = I(D) + DLY(D) + U I(D)]
Y(D) = l + 
DM+l
	 2	 1
or	 I(D)	 l+D	 1 + D + D + .... +D^_.	 (7)
^ience, if G(D) is a polynomial transfer _flznction of degree MI, a
circuit whose transfer function is fine co:plerert of G(D), i.e.
G(D)+1+D+D2+.,.+DM
can be obtained by addirg the output of the circuit In	 I	 -
the output of the circuit whose transfer function is G(D).
M memory cells used to realize G(D) can be the sa..r.Pe as thos e ::; ,-^i
in the circuit of Fio. 1 so that the total circuit can be built
simply as shown  in Fig . 21.
These considerations ca—n now be used to develop an encoder -'C'r
the code urhose generators a-re given in (o'). Taking G(D) as the com-
gla ent of G,(D), we have in octal form
G = (o44, 244, 101)a4o18
or	 G(D) = D3 + D6 + D10 + D12 + D15 + D20 + D26 + D-0.
-4-
Upon taking; t i = 35 and. t aki r g G (D) a s in (E) , it follows
from tine analysis of Fib,. 2 that Gl (D) as in (6) . is the transfer
function rel3tir^ Tl (D) to i(D) in Fib. 3. ^?oreover, the transfer -
function re?_atlrg T.,(D) to I(D) in Fib. 3 is Just
D + i(D) =G2 (D)
so that the circuit in Fig,. 3 is a cc.^lete encoder for t1le tirary,
R = 1/2,-code with. coast-alit lenEth 36 branches as-specified by
the gerrtors Win. (6) .
-
	
.	 The corplete encoder uses only ll ti•o-input modulo-two adders,
compe-_ ,ed .;o 21 tiro-input modulo-t`wc adders required for a tapped
shift-register to irplement the -syste-.a tic code .•rith the same .
	
'	 constraint length that is presently utilized in the NIP.SA GSFG
convolutional coding ::yste^s. This code has the generators:
	
f	
G1 = (4OO,000,000,CCO),
G2 = (715, 473, 701, 317)6•
Since G2 has 22 "ones" arinni; its 36 cceffici ,nts, the encoder for
the syster tic code could profitably to instru..ented in the r;anner
shoi%.a in Fig. 2. T 1is t•ruld lead toan encoder with 16 two-input
modulo-two adc:.^. 4 a considerable s^vi_gs over the single tapnFd
s'nift-r2ois;	 i° .._.;tat ion but	 considerably more than the
11 adders r--,qu 	 4 " .	 t. n-s,yste-.-. ti(- cods. This advantage of
the non-syste-.^.'c Cade is quite sua-^^isin? -ince one "intuitively"
expects t .at a	 _ :3 non-sysf-_:rat i c cod=- •^ould be harder to encode
than a good systematic code with the same constr?i r t len_ th.
(b) Performance of the Code
The error probability and con&,utation perfornance of t1he non-
systematic code of (6) relative to the sysier-atis code of (9) is
given in Tables I to IV. Table 
.
I gives the perfor-monce of the
codes on tine additive Gaussian noise channel :•;ith an b/1 o
(Energy per infcrr„ation bit to single-side noise power per Hertz
ratio) of 2.0 (3 d'c) . The perfon-once is nearby identical for the
tv-- ucdes,	 with the systematic code having a
very slight ca-,putstional advantage. As Thrill be seen, this
advantage deril ves front -!:e fact that the systematic code often
decodes a frame in reasonably furl corrnutations when "prudence"
demands a closer e^jmination, i.e. the systematic code is cob:-
sideraNly rrore prone to decoding errGr:; .
This '.latter fact is bro t out clearly in Tables 1T, IlI.l tt rw
and N .4hich sho,•.,s perfoznnax ce on successively v:orse binaury s yi-,r-_ietric
channels (ESL's.) These BSC's are chosen so that the code rate
R = 2 represen`-.s 0, 05 , 100:5 and 110;0 respectively of the cor:N. ta-
tion.al cutoff rate (R corm ) of the channel. For the worst charnel
(Table IY), there were no dec ud ynE errors over 1000 decode-d fr,,neZ
whereaS rear,, 10;^ of ^ne sar.e f--a,--,es ttiere deco,._u _Ir-c_.,rr,_:..;t._y
v:hen the systematic code bras used.
AllovinC, a rat:_er iarze (SG,NO cc-^puca,;icns--a cc_^ ^ut^t cr:
being defined as a "for>.?T^ci look" and r ecL i_Y'ir  aUGUt 100 Uo` C on
t: ' UINf'AC 11W conmate, )r-	 C" i =r + . before	 at '''.^	 tt"'. %ta,._^r^	 c^..,,uta,i^r	 o.
to decode each fra:-ne ,)1 256 int'Oii" i:lor'_ tI.t5 iz at—n do.`1e' , it is
reti;a.rka!,' e that no deco{? i r? r error haZ yet 't'--en 	 i n a. ;y of the
-6-
sequential deccdizv_ sim:alaticrs
	 the non-syste: at c code of (U) .
A good qualitative coriparison of the non-systematic code of (6)
to the systematic code of (9) can be obtained from Table N which
gives their perfor:rance on a very noisy BSC. Tnere are 141 r-nore
frames out of 1000 frames wrhich fail to decode (in 1;0,000 co::pu-
tations or less) fcr the non-syste:.atic code. However, 87 frames
are erroneously decoded wnth_th syste,,,atic code compared to none
for the non-syste.,,atic code. The conclusion is that the decoding
terrrnated on the extra 141 frames ith the systematic code by
"decoding" when the decoded frar-e error probability was near 50".
Wi thout trying to be flippant, cne could tern this a "Pools rash
in where angels fear to tread".phenomenon that accounts for the
computational advantage of the systematic cafe as a . consequence of
its ?xeater proneness to decoding ;.nor. The non-systeratic.code
emerSes a clear .^rinner in system perf o'_"mance as :•:ell as system
complexity..
2. Free Distance of Convolutional C-odes.
Prior worn: done 1 :nder this 'ant has estatlished the :L-portan,e
of the free: distance, dfree, of	 ^convolutional cedes hen used with
sequential deccd!nE_
	
a deter _Lr r of the decoder error probability.
Recent v.or'k	 _ . Costello has res'.iltetd in a "0ilbert-lit•.e" lo:rer
bound on `.	 're,^:aY:._ a`tait:a ie ..:tr periodic, time-ve_yir_
_n_;. 4:ork. ^C... that surprisinc,ly larc-e free
w:r__, eL	 _ 1 12, a free distanc
t0 ^'CC"._t2'3;::' 1='I. `:. rate-^. Cf at leas'. 0.39 Ca :e obtained.
d'_s.w.ce to constraint ienC" h rat o
-7-
of at least 0.11 guaranteed by the usual rilb 'e.Mt sound. As
R	 1, the ratio between these 	 bounds beco :cv infinite.
Costello t s loner bound on 
dtree has also been used to.clbtain
an asyffptotically ^i.ht bound on the-error - probability attainable
with low rate codes on the RSC.
A technical report, now in prepar2ti-m trill g ive come lete
details of this work.
3. Simlation of the Jelinek Sepuential_Decoding alcorithm
J. Geist has just completed the programdng of the UNEVAC 11u'
computer in the Univ. of Notre Dare _COr^
	 -Iutirc Center to s ulate
r	 _
a sequential decoder employir the Jelinek decoding algcrit'lm.
This facility will be used in the next quarter to ctt?i, detailed
perfor .:ice comparisons with the Fano al c-Or':	 ., ^r 1 ^*n^ r dry
M
results indicate that:
(a) The t;;o a1='-ri-Ti s require at.;ut t- 1-.e sane 3ecodir- time ^'?:en
the code rat-- R is about 90 15 of R?or 10:•:er rates, thecorp
Jelinek algOrit: ,m is :'1ic t;Ly supericr. , For hiyr_er rates, t'r_e
algori L im becon.es much superior.
(b) The time p-r conq .eta`icn of the JeliY.ek a!,-,;T ^ t ,:,.^ ^':d
quadratically Aith ti:e total rn.mher of co-m.utat ors rc u_'_r-"? t
decoct the fr?;`?. The t—I'me oer	 is 1'1;:-^d both t :e Far—.0
a-1 gOri t i"m .
r" ST-
I(D)
90, _ '' ^i	 9M
T(:j
Fig. 2 Binary Linear Se quential Circuit with
Transfer Function GD) + 1 + D +	 + DM
^(n1	 r1where ,D, - 'o + 91 D + ... + g1,D"
c
14
v
0
U
C)
14
v
a!
E to r;
L7 C1	 C!1
^ ^ N
G1
I	 w"	 w
C ^v Vi
G 1 :^
w w
r C-% cn
C1 ch
LCt L'i
L..C w w
Cy m C%
C1 !n
^: v v
a-
^ Y
C r! CV
0
43
G
ri
C.E
CU
C'1
L
rl
A
ti
0TANJE' 1:
u
256
No. of
or	 es	 _	 :, ,,an
500'A
600 (	 ^;2
700 523
850	 I 44_- 404
1009 3. S ?2;
18
;Iow !	 60
jr, 'I l9
2' ,?CC 9
i
P]o. of er^.^ :
-- 50'C,00	 - --.^
ac of
n decod? 
"r
-11-
iI
.2
ccu:
0-
r.
firs
'iDC
code of
	
 
n^ =	
Ii
	
670
	
- ^	 i	 h r-	 3	 •L;,J
	7 p;	 2'f	 105
	
850 (	 ^3,-;	 105
	
1000	 99	 69
	
1500	 47	 2?
	20.:	 17
	
25CO	 -18	 13
	
5,C .	 6
10,
	
2	 2
	
D	 0
ir
II
50,
No. Of f:
dencxL 0
Tysteratic Code
Nb. of erazel fry--.^
50,E
i
4
2
,1-
TABLE N:	 'e. ,forr ancc ;r. tl.e
with Crossover Fro'ta"11Ity 0.045
ccr::^
Resulta cf ,eccd?:x- 1300 .:'a: s cf 2jF Tnfor_
;..-.uL_cn bits : ach
Non-Systematic	 tic
Code of E	 (c)	 of Eq. ( )
No. of f "c:..	 with	 292 I 1000 1000
corputat_en >>v31 	 400 1000 1000
to or ^:r--titer ts_rn	 550 94; 932
the nl.L	 .er st.c:•.: In	 70u 863 0-17
the first column	 350 ^	 802 734
1000 753 673
1500 64 5 32
2000 585 455
2500 543 412
5000 440 319
lgcoc 35d 237
20,rK)0 1R1^0- I--
 era_ ^^. fry: ^^ _7No. of I
—	 -
( r -,M .	 °	 r.	 v	 n.- . • r .-... 1
0,00,1) 249 108
No. of fremes m3ultlr-^
III dec^_dirAE '':'vorz	 I	 C	 v (
-14-


Convolutional coding techniques for data protection Quarterly progress report, 16 Feb. - 15 May 1969

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