DERIVATION OF BURST STATISTICS FOR HIDDEN MARKOV CHANNEL MODELS

In this paper we present an enumerative method to derive analytic expressions for burst error statistics of hid­ den Markov channel (HMC) models with an arbitrary number of states. It is demonstrated that the error weight distribution can be analytically described from the distributions of inter­ vals between errors. The study of the statistical description of the error process has immediate applications in the model­ ing of real channels, and in the designi of coding schemes for channels with memory.


INTRODUCTION
Two fundamental problems arise in the mathematical mod eling of communication systems: The problem of channel modeling and the problem of applying the model to further system analysis.The aim of channel modeling is to obtain as simple an analytical model as possible that accurately reflects the important statistical description of the real error process.Therefore, we can use the statistics of the model to design coding schemes and to evaluate the performance of commu nication systems.
An HMC model is characterized by an underlying non observable Markov chain [1].The discrete output symbol to the channel at the k th time interval, Yk, is a function of the input symbol x k and the state of the Markov chain S k .The channel is described statistically by the conditional probabil ity P(Yk, Sk I XI,., sk-d.The difference between the input and output sequence to the channel is defined as the error sequence.Thus, for example, in a binary system the error sequence will be a sequence of zeros and ones with a one indicating an error.The characteristics of the channel impair ments, are incorporated into the model in such a manner to produce a statistically similar error sequence as produced by the real channel.This HMC model is suitable when the chan nel has a set of properties constant for a certain period of time, and then, change sequentially to another set of properties.This approach has some interesting features: (i) The model is quite general and flexible.For a broad class of real com munication channels, the correlation structure of the error se quence can be accurately characterized by the proper choice of the model parameters.(ii) The model has a well known an alytical description.When its parameters are known, the mul tidimensional probability of the error sequence can be calcu lated.
Some burst error statistics have been used as a criterion for selecting models to represent real channels.One issue of con siderable importance is the investigation of the renewal na ture of the error process.Renewal channels possess the prop erty that the gap (sequences of zeros) lengths before and after an error are independently distributed.Examples of renewal models include Fritchman channels with one error state [1].The Gilbert-Elliott channel [2], for example, is non-renewal.Renewal models have gained some popularity mainly due to their simplicity of analysis.However, it can be demonstrated that many real communication systems cannot be accurately modeled as renewal models [1,3].The statistic called multi gap distribution has been used as a test of non-renewalness of the error process.Despite this important property, no analyt ical expression for the multigap distribution of HMC models has been given in the literature so far.Another application of burst error statistics is the design and performance evaluation of coded systems over HMC models.Some statistics of in terest for the encoder designer may include the probability of exactly m errors occurring in a block of length n, the average burst length, and many others.
In this work we follow the theory of enumeration of con strained sequences described in [4] to enumerate a particular subset of error sequences generated by the HMC model.We will show that the probability of this subset can be obtained by acting on the generating series with a linear mapping ex tended as a homomorphism to the whole of the ring of all formal power series.As an application, we specialize these general expressions for the cases of Gilbert-Elliott and Fritch man channel models.Moreover, we will also use the generat ing series to find recurrence formulas, which are convenient for computation.The techniques developed here are not lim ited to these particular models and can be readily applied to more general Markov models.
This work is organized in five sections.Section 2. de scribes the details of the generation of the error process with HMC models and its properties.In Section 3. we define the generating series in a ring of all formal power series in non commuting indeterminates.A method for determining the burst error statistics from the generating series is introduced in this section.In Section 4. we give explicit expressions for the burst error statistics considered in this work.Finally, con cluding remarks are given in Section 5..

MODEL DESCRIPTION
Consider {8 k } k= 1 an N -state Markov chain with a finite state spaceNN = {O,I, ... ,N-I}.LetPbeanNxNtran sition probability matrix, whose (i,j)th entry is Pi,j' The error sequence is generated by the HMC model as follows.At the k th time interval, the chain makes a transition from state 8 k = ito 8 k + 1 = j with probability Pi,j and generates an output symbol ek E N 2 (independent of j), with prob ability bi,e, = P(E k = ek I Sk = i).We are assuming that the distribution of the initial state is the stationary distri bution IT = [11"0,71"1, •.. ,11" N-1V (where the superscript [.]T indicates the transpose of a matrix).
The probability of an error sequence is calculated as fol lows.Define the N x N matrix P(ek) whose (i,j)th entry is Pi,j(ek) = P(Ek = ek, Sk+] = j I Sk = i) = Pi,j bi,e" which is the probability that the output symbol is ek when the chain makes a transition from state i to j.The probability of an error sequence oflength n, ell = e] e2 ... en, has a matrix form given by: where 1 is a column matrix with all entries ones.
The Gilbert-Elliott channel [2] is a two-state Markov chain composed of a good state, state 0, where errors occur with small probability, and a bad state, state 1, where errors occur with higher probability, as illustrated in Figure 1.When the chain is in the good state the error bit ek is zero (correct) with probability 1-g, or one (error) with probability g.Otherwise, when it is in a bad state, the error bit is zero with probability 1 -b, or one with probability b.The matrices P,P(O),P(I) and IT are given by: (2) q l -q ' ( Some error distributions for this channel can be calculated using Equation (1).As an example, the probability the bit error is a 1, P(I) ~ P(Ek = 1), and the probability of two consecutive ones, P(ll) ~ P(E k = 1, E HI = 1), are given by: (6)

P(ll)
An HMC model can also be described as a deterministic function of a Markov chain.In the where 1\00 and A]] are diagonal matrices.The block matrices Pk,1 represent the transition probabilities from the set A k to AI.A model with K good states and N -K bad states is denoted by (K, N -K)-EFC.In particular, the matrices P, P(O), P(I) and IT for the (2,1 )-EFC model are given by:

THE GENERATING SERIES
The problem we will address in this section is to find the prob ability of error events associated with HMC models.The er ror events of length n will be denoted generically by £11' It is well known that the probability of an error sequence is not preserved under commutation of its symbols.This prompts us to define the generating series in non-commuting indeter minates in order to keep all the information about the original sequence.So denote the generating series for £11 (error event) as: which is in R «xo, Xl », the ring of all power series in the non-commuting indeterminates xo, Xl.The indeterminates Xo and Xl mark an error bit equal to 0 or 1, respectively.No tice that P(£,J may be obtained from the generating series FE" simply by replacing X e , by P(ei) and the vec tor rrT around the front and 1 around the back.We can formalize this concept by defining the mapping: acting as a homomorphism to the whole of the ring.M N (R) is the ring of all N x N matrices with entries taken from R (the field of real numbers).The probability of the set £11 may be expressed very compactly as: (10) The main step to find P (£11) is to determine the generating series FE".The key point to solve problems of this type is to find a bijection that expresses the set £11 in terms of con catenation products of binary strings.More examples will be addressed in the next section.

DERIVING BURST STATISTICS FROM GENERATING SERIES
In this section we will derive expressions for two important statistics: • The error weight probability, P(m, n), the probability of exactly m errors occurring in a block of n bits.This measure is important to determining the performance of block codes and interleaving on HMC models.
• The multigap distribution, M(r, l), is the probability of r consecutive gaps in a sequence of length l.The multigap distribution has been used as a test of non-renewalness of the error process.
In all cases, we first obtain an expression for an N -state HMC in terms of the matrices rr, P(O), P(I).We now de fine some notation.If s and z are commutative indetermi nates, [sk zll] T(s, z) denotes the coefficient of sk z1l in the formal power series T( s, z).If A is a set of sequences, A * is the set of all sequences formed by concatenating any number of sequences in A. The identity matrix will be denoted by I.

The Error Weight Distribution P(m, n)
We wish to determine the probability of the set £n composed of sequences of Hamming weight m and length n.The gen erating series for £11 is obtained directly from the generating series for the set of all binary sequences, {O, I}* , by defining the indeterminate z to mark the length of the sequence and s to mark the number of I's.Since F{O,I}' = (1-Xo -xdl , it follows that FE" is: From Equations ( 10) and (II) the error weight distribution P(m, n) is given by: where = rrT(I -P(O)z -P(I)SZ)-l 1.
The generating series Hp(s, z) is a polynomial in s.Then, Hp(s, z) is a formal power series in z with a coefficient ring R[s].We now specialize the calculation of P(m, n) for the Gilbert-Elliott channel.An expression for H p (s, z) can be obtained upon substitution of Equations ( 3)-( 5) into ( 13).An explicit formula for P( m, n) can be found by carrying out the partial fraction technique to extract the coefficient of Equa tion (12).Alternatively, it is simple go from generating series to recurrence formulas, which provides a rapid computational scheme for the problem.From Equation (12) we can prove that P( m, n) for the Gilbert-Elliott channel satisfies the fol lowing 6-term recurrence formula:
A new recurrence fonnula for P(m, n) for the (2,1)-EFC Fritchman channel is given below: Let the random variable E" be the number of errors in a block of length n.It is obvious that P(E" = m) = P(m, n).
Moments of the random variable E" of any order, E{ (E") k}, An exact formula for ::' (1 -P(O)z -P(l)sZ)-l will be stated without proof in the next lemma.

Lemma 4..1 The ktl> partial derivative of the matrix
A(s, z) g (I -P(D)z -P(l)sz)-1 is: Using the result of the lemma, we are able to prove the following result: It is easy to see from Equation ( 14) that E{E"} = nP(l) where P(l) is defined in Equation (6).Moreover, the vari ance of E" , for a general HMC model can be found directly from (14) by setting n = 2.The final expression is: =E{(E")2} -(np(1))2; From the variance of E", we can readily calculate the corre lation coefficient between two blocks of length n.Consider a block of length kn formed as a concatenation of k blocks of length n.Let the random variable El! be the number of errors of the k th block.The correlation coefficient between EO' and E'k is given by [3]: -'----'----:::-:--'-:-=-7----....:......----' 2Var(En) The quantity Pk can be a useful indicator to select codes.For example, as mentioned in reference [3], the burst-trapping code works well for channels whose Ph' is negative.It is easy to show that for the Gilbert-Elliott channel, p~~ is positive for all values of k and n, as long as 0 < (1q -Q) < 1.
A positive correlation indicates that whenever a block i has more than the average number of errors, block i + k has the tendency to have more than average.

The Multigap Distribution M(r, l)
The length of a gap is the number of zeros between two er rors plus one (the last error is included).The error process {Ed~l can be regarded as a sequence of gaps {Gd~I' where GIis the length of the k f I> gap.The gap process is a convenient representation for the error sequence, since a large number of consecutive D's is expected to occur on chan nels with low bit error probability.Let the random variable G" = L::7~:-1 G be the sum of r consecutive gap lengths " G".The multigap length distribution, denoted as M(r, i), is defined as M(r, I) = P(Gl' = I).If the error process were renewal, this means that {GI-} 1:=.] are independent random variables, then the variance of G' is V areG") = r V are G1)' The problem of finding M(r, i) may be formulated as fol lows: Find the probability of the set E l , composed of binary sequences of length l such that the r th error will occur at the lti> time interval.Note that the set of all sequences that ends with a 1 may be expressed as {O* 1} *.Let the indeterminate z mark the length of the sequence and let s mark the occurrence of a 1.The generating series for the set El may be obtained from {O* 1} * by replacing: 0* by 1+xoZ+x6z2+ ... =(l-xoz)-J; 1 by X1SZ.
It follows that FE, is: The multigap distribution M (r, l) is the probability of the set El' conditioned on Eo = 1.Then  Therefore, the proof is complete.• The variance of Gr, denoted as Var (Gr), can be expressed as: Adoul [3] defined a quantity called variation coefficient K (r) for a channel as: where V ar BsdGr) = r(1 -P(I)) / P(I)2 is the variance of G' for the BSC channel with crossover probability P(I).
Adoul [3] shows experimental curves for K(r), as a function Since K(r) is constant with r, the process is renewal.Let us sample the process en to obtain the process Zn, or Zn = ed", for d = 1,2, .... Using the above theory, it can be shown that z" is a renewal process.Table 1 shows the values of K(I) versus d for the sampled process Zn when the HMC model is the (2,I)-EFC channel given by Equation (8).
Adoul [3] defined a process whose K(I) is greater than one as a more variable process, in the sense that the gap lengths spread widely from their mean value.In this case, errors have a trend to be clustered.The further K(I) is from 1, the more pronounced is this trend.As the value of d increases, the process z" tends to become memoryless and K(I) tends to 1.If we encompass the channel with an interleaving and an deinterleaving with finite interleaving depth d, we can regard the sampled process z" as the error sequence at each row of the deinterleaver.We can use the table above to investigate the minimum value of d that renders the channel memoryless.
The plots show that Gilbert-Elliott channels are able to model channels with gap lengths that spread widely from their mean value.
Variation coefficient 5 4.5 M2 4 of r, for troposcatter channels.This curve was used to com-K(r) 3.5 pare the spread of multigap lengths for a particular channel and its corresponding BSC.It is important to notice that for renewal processes K(r) = K(I), for all r, that is, the curve K(r) versus r is a constant for all r.An expression for K(r) for the renewal (2,1 )-EFC model is given below:

Figure 2 :
Figure 2: K(r) as a function ofr, for Gilbert-Elliott channels.

Revista da Sociedade Brasileira de Telecomunica~oes Volume 12, numero 2, dezembro 1997 where
the generating series HlII(s, z) is: The generating series H P ( s, z) and HAt{ s, z) are the ratio of two polynomials in s and z.The denominator polynomial is responsible for the recurrence relation, and the numerator polynomial defines the initial conditions.So, it suffices to prove that H P (s, z) and H AI (s, z) have the same denominator polynomial.Hp(s, z) is defined in Equation (13), and it follows from Cramer's rule that its denomina tor polynomial is det( 1 -P(O)z -P(I)sz).Using the same argument, we can conclude from Equation (18) that the denominator polynomial of H 1II (s, z) is the numerator of det( 1 -(1 -P(O)z)-lP(I)sz).But:

Table I :
K(I) versus d