Analysis of the decoding process of T-codes

A.C.M. Fong, G.R. Higgie, B. Fong and G.Y. Hong

Abstract: T-codes are variable-length codes -(VLC) generated by an iterative construction algorithm known as T-augmentation. All T-codes possess strong self-synchronising properties by virtue of T-augnentation, which spreads synchronisation information throughout the code construction. A number of applications have been reported that reference the T-codes for their desirable properties. However, further theoretical development is needed to realise their full benefits. The authors highlight important applications of self-synchronising codes and survey reported research on the decoding process of T-codes. Building on previous work, the authors present a thorough treatment of this process. Understanding the decoding process provides iusight into the self-synchronisation process of T-codes. It also leads to an efficient method of determining the average synchronisation delay associated with each T-code, which is important for practical applications.

1 Introduction

Following years of development, a number of self- synchronising efficient codes have been reported e.g. [IM]. T-codes are families of variable-length codes (VLC) that exhibit extraordinarily strong tendency towards self-syn- chronisation [7, 81. They have been referenced in a number of practical applications, e.g. [9-121 and have been used for real-time channel evaluation [13]. In [9], the authors describe an apparatus for parallel transmission of data over fibres where each fibre carries self-clocking coded signals. This reportedly overcomes the problems of data skew due to different transmission times over the fibres. In [IO], a method for finding the length of a variable-length instruc- tion in a continuous stream without differentiation is specified to facilitate parallel processing. In [ll], a coded moving-picture signal that consists of segments is pre- scribed. Each segment includes picture data coded according to a self-synchronising rulc for attairiing resyn- chronisation after an error. In [12], the authors investigate the application of T-codes to text messaging. In [13], the authors propose and evaluate a statistical real-time channel evaluation (SRTCE) technique that employs variable-rate T-codes for source coding over an AWGN channel. SRTCE is reportedly most suitable when channel noise is severe.

Although T-codes may appear similar to other self- synchronising codes, e.g. [14], there are important differ- ences between their respective code construction algorithms [IS]. Following the generalisation of T-codes [16], the subset as proposed in [7j is now referred to as simple T-codes. It has been rcported that simple T-codes that exhibit the

IEE. 2002 IEE Proceedingr online no. 20020314 DOI: i n . i ~ 9 / i ~ ~ ~ ~ : 2 ~ n 2 0 3 1 4 Paper fin1 meived 29th May and in revised form 16th November 2001 A.C.M. Fong and G.Y. Hong are with the Institute of Information and Mathematical Scicncer (IIMS). Maisey University - Alhsny Campua..Ptivate Bag I O ? 904, North Shore Mail Centre. Auckland. New Zealand G.R. Higgie is with the Departmen1 of Electtical & Electronic Engineering. Univcrsitg or Auckland. New Zealand

B. Fong is with the Auckland University of Technology. Ptiwte Bag 92006, Auckland. New Lealand

202

strongest synchronising performance are also among the most efficient for any given information source [17]. Furthermore, it will become apparent in Section 2 that the T-generalisation process does not affect the relative synchronisiiig performance. Thus, coding efficiency does not necessarily need to be sacrificed for synchronising performance. The search for the best overall T-code set for any given source is therefore limited to the subgroup of T- codes whosc average codeword length (ACL) distribution is optimal for the source.

By understanding the decoding process of T-codes, one can measure the T-codes relative synchronising perfor- mance. This aspect is of practical importance, as the best T-code set can be selected for any given information source. Although criteria for identification of minimal synchronisa- tion delay T-codes based on construction of bit sequences have been proposed [la], more research is needed to relate them to source entropy. The process of T-code synchroni- sation is viewed from a different perspective in this paper by considering synchronisation as a natural phenomenon of the decoding process.

2 Decoding generalised T-codes

Since decoding amounts to finding codeword boundaries, the generalisation of T-codes does not affect the underlying decoding process. To illustrate, suppose a bit sequence y results in a valid decode at augmentation level i Li. Further, let y he a concatenation of bit streams a and 0 = ED), where a is the current level prefix pi. Then, the effmt of expansion due to T-generalisation is illustrated in Fig. I, where a partial list of valid codewords is shown for Lj (p; = n).

total k, expansions

Clearly. the suffix portion /I gives the location of the codeword boundary and the expansion due to T-general- isation only affects the prefix portion with multiple instances of n preceding /I. For example, let the base alphabet set S = (O,l}, the prefixes p = (0.1,OOO) and the expansion parameters k = (2,l.l). By iteratively applying the pre- scribed T-augmentation algorithm [16], the valid codewords in the various augmentation levels are

si;; = {l,0l,000,001)

si2;; = {01,000,001: 11; 101; 1000,1001)

s;;:::&) = { O l : O O l ; II,101: 1000,1001,00001,000000,

000001 ~ 0001 I , 0001 01 : 0001 000,0001 00 I }

Fig. 2 summarises the decoding process for this example.

0 oon

1 wn

01.m1

3 T-code decoding process

In the first reported treatment of this subject, the T-decoding process is modelled as a discrete Markov chain and the decoder is modelled as a state machine char- acterised by a probability transition matrix [19]. The model leads to the detennindtion of average synchronisation delay (ASD) in terms of the number of hits required to re- establish synchronisation following a lock loss. However, it does not apply to T-codes with the suffix condition occurring during the decoding process.

Monte Carlo simulations have hem successfully applied to compute ASD values for T-codes [17], but they do not explain the underlying decoding process. Thus, there is no means of finding a better code if a near-optimal solution is found. An analytical treatment of ASD calculation for all T-codes is given in [20]. That method is inefticient whcnever the suffix condition is encountered as it entails a search for special bit sequences among all possible bit sequences up to a specified length. In [21] the authors take a somewhat different approach by considering 'blocking prccouditions' instead of suffix and prefix conditions.

This research is based on the model presented in [20] for two reasons. First. the prefix/suffix model more naturally describes the T-decoding process. Also. many T-codes are prefix-only codes and these can be processed much more efficiently by bypassing the treatment for suffix conditions. Thus, this paper can be considered an extension of [20], analysing the decoding process of all T-codes, including generalised T-codes and with an emphasis on suffix conditions that occur during T-decoding.

In general, the number of bits used in the decoding process gives a measure of synchronising performance associated with each code set. In [20], the probability of blocking in an augmentation level during decoding is given by (1) and the probability of decode in a level as the consequence of a blockage is given by (2). Further, the numbers of valid hit sequences NVE,, NVEEh, NVDh and NVDEh are obtained b,y checking all possible h hits long sequences (given by 2 ' ) up to h,,,, when convergence

IEE Proc Convmm Vo/. 14Y. No 4, A u g m 2W2

Number of h hits long valid entry sequences (NVEIJ = number of h hits long sequences that cause the synchronising process to enter the required levcl and make a decision. IVVEBh = number of VEb sequences which also cause blockage in the required level. NVDh refers to hit sequences that cause a valid decode and NVDE, refers to VD8 bit sequences that block the decoding process.

Suppose NVE,, NVEBI,. NVD, and NVDB, are collectively represented by .Y_ then . ~ / 2 ~ are diminishing terms because s varies quasilinearly with increasing 6. Since NVEBh and NVDBt, are subsets of NVEh and NVD,,_ respectively, the search for the limiting case narrows down to NVE, and NVDh. From [20]_ a valid entry always precedes a valid decode. Consequently, when increasing h from unity, the temi 42" first becomes nonzero for NVE, berore NVDb. For example, NVE? and NVDS are the first nonzero .~/2*-tenns for the set S;,: l / u l , . respectively. Thus, (3) holds for all T-codes at any given decoding level.

NlE,,/2h > N D b / 2 h m m

h=l h= I w m

> N E B h / 2 h > N!JWBI,/~~ > o ( 3 ) 6=1 *= I

Since NlEh/2b sums to unity [22], other sums in (3) fall between the range 0 and I (exclusive).

Thus, the NVEb term represents a limiting case. In fact, by considering the VE tree (see Section 3.1) or otherwise, (4) is always valid. Simulation results indicate that the left-hand side of (4) reduces in significance compared to the right- hand side for increasing h.

NE~+I 5 2 m h (4) A h,,,, value of 30 was found to he sufficient for all sixth degree simple T-code sets [20]. Although there is no direct link between the degree of augmentation y and b,,,, h,,,;,, tends to increase with increasing q. The number of possible hit sequences increases exponentially with increasing hit length h and each step has to be applied to every code set in any given levcl. Thus, the application of the original method to high-level code sets can become computationally prohibitive.

Fig. 3 exemplifies the advantage of considering only valid bit sequences. It is a logarithmic plot of NVEb and the

3 8 13 18 23 bit Stream length b

Fig. 3 Jir the se1 S ~ ~ : ~ : ~ ~ ~ o l l

Plot of NVE, find toro/ nurribcr qfhit xquences ugyoina h

203

number of all possible bit sequences against h for the code set S(o,l,m:l!,l), I t is therefore much more efficient to characterise the valid bit sequences than to search for them from among all possible sequences.

( 1 . 1 . 1 , l )

3.1 sequences

Tree construction analysis (TCAI for valid

Although developed indcpendently, a valid-sequence tree can be considered an extension of the synchronising sequences theorem [8]. Each valid-sequence tree is con- posed of extensions and nodes. Following the notation in [21], a general VEjVEB tree may be formulated thus: let S::. be the set of all Lj valid codewords and q be the required degree_ then for cach i in O<i<q_ extension = and nodes = Ssf except pi+ TCA is then an iterative process starting from Lfi to L,l-I. At any L , the extension and nodes are constructed from the valid codewords, foiming a tree that charactcrises all valid sequences. To illustrate, consider Table I, which is a T-construction table for the code set

From Table I , a bit sequence must have a ’I’ to get from Loto L , . It may be preceded by any number of zeros (the L, prcfix), e.g. 001, I or 000 ... 001 can all pass from L,, to L, . Next, to get from L, to Lz a valid sequence can have none or more ‘1’s followed by either a ‘00 or ‘01.. These are the only two L, - Lz transitions. If ‘01’ is chosen, the bit sequence will enter the required levcl causing a blockage. Othenvise. since ‘00 is the L3 prefix, any number of ‘00’s can precede any of the remaining codewords in Lz. namely ‘Ol’> ‘11’. ‘ I00 and ‘101’. to enter the required level. A tree can now bc constructed to sumniarise the above statenients. Any valid entry sequence must be of the form shown in Fig. 4. Furthcr. two construction rules have been fomiu- lated. Cum/ruction ride I : To obtain i+ I bit variants from an i bit base. one can add a ’0 or ‘ I ‘ to precede a bit sequence beginning with a ‘I,? but can only add a ‘0’ to a bit sequence beginning with a ‘0, e.%. 0 0101 and 0/l 1101 are all the

(l:l.l.l] S(0. I .m. 101 ).

Table 1: T-code set S{:;:.,i& with prefix and suffix block- ing conditions

DPL Level 0 Level 1 L, Level 2 L, Level 3 L3 (prefix 01 (prefix 1) (prefix 00)

- - 0 0- ~

1 1 1- ~

2 00 00 - ~

3 01 01 01 suffix of L,

4

5 11 11

6 100 100

7 101 101-Lr prefix

8

9

-

prefix ~ ~

- -

10 0000

11 0001

12

13 001 1

14 00100

15 00101

~

204

none o r m r e none or mre none or m r e 01 L , prellx Of L, prefix 01 L , piellx

1 1 f. /,

/ 1 ~ ( 1 ... 1) T;o’ ‘\P VEB ,~

(0

00 ,’- (00 ... 00) a T only route /-’ \ 1 !:l I 100 5

L o i O L , routes L, IO L ,

VE

VE

VE

VEB

valid five-bit sequences. This applies to code sets with

Cons/wctiun ride 2: A weaker condition holds if p = (0_ J, ..) where y may not bc ‘1’. I f an i bit sequcnce can be derived from one of the i- 1 bit sequences then such an i bit sequence must begin with a ‘ 0 by adding the ‘ 0 extension. Otherwise. an i bit sequence begiiining with a ‘ I ’ must be a ‘new‘ sequence. Rule 2 could be extended for universal application simply by reversing the ’ 0 s and ’1‘s in this formulation.

Thus, suppose the only VE hit sequence when h = 5 is 10001. Ifthecodeset hasprefixesoftheformp = (0.y, ...’ ), J’ # I , rule 2 states that Ol000l is a six-hit VE sequence. Otherwise, i f y = I . then rule I states that both 010001 and 110001 are six-bit VE sequences. In summary_ the procedure for collecting NVE,, and NVE& counts is illustrated in Fig. 5 .

Thus the incrementing NVEh count when b increases from i to i + 1 is given by

IVI/E,+~ =2+(no.of ‘O . . . ’ ) j + ( n o . o f ‘ l . . . ‘ ) ;

where ‘0 ...’ denotes a sequence begnning with a ‘ 0 . etc. Thc last ‘2’ in ( 5 ) is due to the two new sequences created

whenever h is incremented: (u’)b(c’ )4. (a’)h(c’)r and (rr’)b(c’))v, (rr’)h(c’))t: forb k = 0, ._ . m. In fact. a simple noniterative formula caii he derived for NVE,:

N E h = ( h - 3)’+ I, forb = 3,4:5: ...a3 (6)

Assignment of labels to thc various branches and extensions is arbitrary, although a convention is followed in theoretical aiialysis (a> b, c, . . . foi- extensions; p: q, r, . . . for branches). In practice. computer implementation involves manipula- tion of arrays and makes no such reference to labels. Bit sequences derived from any fully labelled tree are unique because T-codes are uniquely decodable. Specifically, the decision at each node is never ambiguous.

p = (0, I; ...).

+ 2(no. of ‘1 . . . ’ + 2); ( 5 )

3.2 Application examples Two examples are presented. The first illustrates the applicationbf TCA to a code set with the sulfix condition occurring, the second to an expanded code set.

E.sul?i/~/e I (snffi\- condition): The VE tree for code set ~ ~ , ~ l , l ~ , l o l i is shown in Fig. 6. The shortest VE sequence is

IOlp. So the smallcst h that givcs a nonzero NVEh count is three. The four-bit valid bit sequences are 0 101 ap and 1 101 bp. Then, ‘new’ bit sequences may be derived from the tree for each successive b. New bit sequenccs are those that cannot be derived successively using rule I or rule 2. These always been with a ‘1’. followed by a ‘0 for code sets having prehes p = (0, I, . . .). The other VE sequences can be obtained using rule I as shown in Fig. 7.

An important result is that it is only necessary to deduce. for each incrcasing h, the number of‘new’ VE bit sequences

. p l . l )

I t X /’roc. C h ~ ~ ~ w . , Vu/. 14Y. ,Vo 4, Atriii,rI ZlM2

b = 5

b = 6

b=7

b = 8

b = 9

b=10

NVE = 2 2newsequences 10001 &loo11

NVE = 2 2 new sequences i w m a iwioi

NVE = 2

NVE = 2

NVE = 2

NVE = 2

Fig. 5

( 0 ... 01- 1 - (1 ... 1)

Derir:mion c r j NVE, coum

a b 7 O1

+ 1

+

W l l O l ) o i l 10011 011 11101)

+

~

from the tree. The rest can he obtained hv annlication of

+ (17+9) 2(9+2)

difference = 5 I

10 - -

I

difference = 7

difference = 9 I

= 26

I dinerence = 11

- - 37

I dilference= 13

I 50 - -

~ I .

rule I. Also, only bit sequences ending with the a 1-00) branch are VEB. the others are all <non-hlockit&) V E

Exuiiiple 2 (expmded code ret): As a further example, the expanded code set desciikd i n Section 2 has a VE tree

O0 ' sequcnces.

11 s suffix at L2

i o1 t shown in Fig. 8, Fig. 6 P" w e f b r r/w ,sc~r S ~ ~ : ~ & ~ l o l l

t 6- valid entry bif sequences O i l - 1 ..., namely01 l o l a b p a n d l 1101 b2p 0 ~ o.... namely 0 0 101 a2p

-1 011 ~ l . . . : - O l ~ l O l ab2p.010000aq,010001ar. 0 1001 1 as: 1 11101b3p. 1 IOOOObq, 1 10001 br. 1 10011 bs !

0 - 0 ... :-001101 azbp.OOO1O1 a3p -1 ",, PLUS one new bit sequence: 1 00 101 t L , , : , .,A'. '. i .- I ' ,,+

7 i9 ~ 4 i l ~ . ~ x Z '; 011 - l . . :~0111101 ab3p,0110000abq,0110001abr. , ., ~

0110011 abs.O1OO1O1 at; 1111101 b4p, 1110000 b2q. 1 110001b2r. 1110011 b2s.

-1 \ 1100101 bt " >, A,,, "

0-0.:-0011101 a2b2p.0010000azq.0010001 azr. 0 ~010011a2s.0001101 a3bp.0000101 a4p

I 14+2),L.. I . .,

\

8 i (5+0) x 2 O i l - 1 ..., i.e. a'bp. ab2q,abzr. ab% abl: I -1 b5p, b3q. b3r. b3s. b2t

'"(5t6) A' 0 - 0 ..., i.e. azb3p, a2bq. azbr. a2bs. $1, a3b2p. a3q. $1, a35. d b p , a5p

PLUS three new bit sequences: 1 00 100 00 cq. 1 00 100 01 C', 1 00 100 11 cs

and so on ...

Fig. 7 Deriwtion I ... and O... signily hi1 sequences beginning with it '1' and TI'. rcspcctively. f l - n mciins append [J (/I = 0 or I) to preccde a hit seq~trncz o

IEE Pmc. Chiaiiai., V d 149, N,,. 4, Auqrml XK12

NV& munts/i,r. SC&Y conrlirion ev-umpk (Section 3.2. I )

208

01 r 001 s

a 000- (000 ... 000) 11 1

101 " 1000 " c lOOi w

(0 ... 0)- 1 - (1 ... 1) b

Fig. 8 Y E n.eeej;,r lhe PI S:;;;;,!&

From Fig. 8, the shortest VE sequence is 101 p. So, h = 3 givcs the first nonzero term. Then, four bit long sequences are ap, bp and q. The rest of the VE scqtiences may be deduced as before.

3.3 Discussion VDjVDB trees are simply extended versions of their corresponding VEjVEB trees. The extension is brought about by the inclusion of the highest-level codewords. The tree model has been applied to ASD determination. Executed on identical machines, the improved algorithm using characterisation of valid sequences produces ASD values at 114 of the time required by the orignal method for precision of 0.99 or better. For a higher precision of 0.9999 or better; thc improved method takes 1/64 of the time required by the original method.

4 Conclusions

Self-synchronisiug variable-length codes possess desirable properties in terms of optimal efficiency and automatic synchronisation for many practical applications. The authors have highlighted a few applications, and outlined the motivation for understanding the T-decoding process. This paper contributes towards a fuller understanding of the phenomenon of self-synchronising codes. Following a brief survey of existing approaches_ t h s paper has presented a thorough treatment of T-decoding based on Higgie's analysis [20].

The contributions of this paper may be summarised as follows. Firstly, the paper has reaffirmed that T-general- isation preserves synchronising performance previously reported for simple T-codes. Thus. the best T-code for any given source can still be identified by finding the set with the best ASD value in the subgroup having an optimal ACL distribution for the source. Secondly, a thorough treatment of T-decoding has been presented including generalised T-codcs and those with the suffix condition occurring in the decoding process. Thirdly, the analysis has led to a tree-based model to characterise valid hit sequences that previously had to be searched froin among all possible sequences. Finally, the capability of chardcterising valid

sequences has led to significant savings in the computational time for determining ASD values. These facilitate the identification of the best T-code for any source.

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