Zero-Error Capacity of a Class of Timing Channels
aa r X i v : . [ c s . I T ] A ug Zero-Error Capacity of a Class of Timing Channels
Mladen Kovaˇcevi´c,
Student Member, IEEE, and Petar Popovski,
Senior Member, IEEE
Abstract —We analyze the problem of zero-error communica-tion through timing channels that can be interpreted as discrete-time queues with bounded waiting times. The channel modelincludes the following assumptions: 1) Time is slotted, 2) at most N “particles” are sent in each time slot, 3) every particle isdelayed in the channel for a number of slots chosen randomlyfrom the set { , , . . . , K } , and 4) the particles are identical. It isshown that the zero-error capacity of this channel is log r , where r is the unique positive real root of the polynomial x K +1 − x K − N .Capacity-achieving codes are explicitly constructed, and a linear-time decoding algorithm for these codes devised. In the particularcase N = 1 , K = 1 , the capacity is equal to log φ , where φ = (1 + √ / is the golden ratio, and the constructed codesgive another interpretation of the Fibonacci sequence. Index Terms —Zero-error capacity, zero-error code, timingchannel, timing capacity, molecular communications, discrete-time queue, Fibonacci sequence.
I. P
RELIMINARIES T HE study of timing channels, i.e., channels that arisewhen the information is being encoded in the transmis-sion times of messages, has resulted in many interesting andrelevant models. Two important and relatively recent examplesare the models adopted from queuing theory [3], [4], [15]and those that arise in molecular communications [5]. Weanalyze here the problem of zero-error communication overcertain channels of this type. The study is partly motivatedby settings in which the communication is done with ratherunconventional physical carriers, such as particles, molecules,items, etc. These channels can also be viewed as discrete-timequeues with bounded waiting times, and the results can thus beseen as supplementing in a sense the work carried out in [4],[15] (see also [12], [10]); however, due to the combinatorialnature of zero-error information theory [14], [8], the methodsused are quite different from those in [4], [15].
A. The channel model
We assume that multiple transmissions can occur at thesame time instant without interfering with each other. In thisregard, we will use the term particle (instead of symbol or packet ) for the unit of transmission. We believe that thisconvention will make the discussion clearer. Date: February 16, 2018.M. Kovaˇcevi´c is with the Department of Electrical Engineering, Universityof Novi Sad, Serbia (e-mail: [email protected]).P. Popovski is with the Department of Electronic Systems, Aalborg Uni-versity, Denmark (e-mail: [email protected]).M. Kovaˇcevi´c was supported by the Ministry of Education, Science andTechnological Development of the Republic of Serbia (grants TR32040 andIII44003). P. Popovski was partially supported by the Danish Council forIndependent Research within the Sapere Aude Research Leader program,Grant No. 11-105159, ”Dependable wireless bits for M2M communication”.Part of the work was done while M. Kovaˇcevi´c was visiting AalborgUniversity, Denmark, under the support of the EU COST Action IC1104.
Let N denote the set of nonnegative integers { , , . . . } . Definition 1:
The Discrete-Time Particle Channel with pa-rameters
N, K ∈ N , denoted DTPC( N, K ), is the communi-cation channel described by the following assumptions:1) Time is slotted, meaning that the particles are sent andreceived in integer time instants;2) At most N particles are sent in each time slot;3) Every particle is delayed in the channel for a number ofslots chosen randomly from the set { , , . . . , K } ;4) The particles are indistinguishable, and hence the infor-mation is conveyed via timing only, or equivalently, viathe number of particles in each slot. N
12 3 7 8 2 3 14 74 5 6 86 5
Fig. 1. Illustration of the DTPC( , ). The particles are numbered only forthe purpose of illustration, they are assumed identical. We elaborate briefly on the definition of the DTPC. If theduration of the transmission is n slots, then the assumption 4)implies that the sequence of particles can be identified with an n -tuple of integers ( x , . . . , x n ) ∈ { , , . . . , N } n , where x i represents the number of particles in the i ’th slot. For example,Figure 1 illustrates a situation where the transmitted sequenceis (3 , , , , and the received sequence is (2 , , , , .Hence, the DTPC can be defined purely in terms of sequencesof nonnegative integers, and in the rest of the paper we willrely entirely on this representation.As for the assumption 3), observe that if the delays ofthe particles were unbounded (as is the case, e.g., in queueshaving service times with geometric distribution [4]), the zero-error capacity would be zero. Therefore, in order to obtaininteresting models, some restrictions on the delays have to beimposed. Similarly, if there is no restriction on the numberof particles sent in each slot, then the zero-error capacity isinfinite for any K ∈ N , which justifies the assumption 2).Note that we have not imposed a restriction on the numberof particles at the output of the DTPC( N, K ) in a single slot(though it is obviously bounded by ( K + 1) N ). It is not hardto argue that this does not affect the zero-error capacity of theabove channel, i.e., it would be the same if this number werealso bounded by N . This is proven in Appendix A.Let us also give several more concrete interpretations of theDTPC. Namely, the “particles” referred to in the definition ofthis channel can be interpreted in various ways depending onthe context, e.g., as: • “Molecules” in the so-called molecular communications,where the transmission of information via the number ofmolecules and their emission times is considered. Themolecules are usually assumed identical, and their arrival times are random due to their interaction with the fluidmedium. The codes described in the present paper arerelevant precisely for the channels of this type, at leastin discrete-time models [5]. • “Customers” in queuing systems, an important exampleof which are queues of “packets” formed in networkrouters (see the discussion in Remark 1 below). • “Packets” in channels introducing random delays (causedby effects different than queuing). Note that the packetsreferred to in this and the previous paragraph are notidentical in practice and usually carry information viatheir contents. In this paper we will be interested in thetransmission of information via timing only , similarly asin [4]. Alternatively, one can imagine a receiver that isnot processing the packets (e.g., a low power node ina wireless sensor network), but only infers their arrivalsthrough energy detection. • “Energy quanta” in a simultaneous transmission of energyand information [11]. Remark 1 (DTPC vs. Discrete-Time Queues):
We havepointed out already that the results of the paper apply also toqueuing systems of certain type. We introduce them here ina bit more detail. Denote by DTQR(
N, K ) the Discrete-TimeQueue with N servers/processors (meaning that N particlescan be processed simultaneously), with at most N arrivalsper slot, and with Residence times bounded by K slots (theresidence time of the particle is the total time that it spends inthe queue, either waiting to be processed or being processed).It is not difficult to argue that the DTPC( N, K ) and theDTQR(
N, K ) have identical zero-error codes and zero-errorcapacities. The key difference between these channels is thatthe delays of the particles in the DTPC are independent, whilein the DTQR they are not as they are affected also by theservice procedure (for example, in FIFO queues the particlescannot be reordered). The assumption that the particles areidentical, however, makes this difference irrelevant in thezero-error case. N B. Notation and definitions
By a “sequence” of length n over a nonempty alphabet A wemean an n -tuple from A n . When there is no risk of confusion,a sequence ( x , . . . , x n ) will also be written as x · · · x n . If,for a given channel, the sequence x at its input can producethe sequence y at its output with nonzero probability, thenwe write x y . For any two sequences x and y , theirconcatenation is denoted by x ◦ y , or sometimes simply by x y .Also, if Z is a set of sequences, we let x ◦ Z = (cid:8) x ◦ z : z ∈ Z (cid:9) and Z ◦ x = (cid:8) z ◦ x : z ∈ Z (cid:9) . We assume that x ◦∅ = ∅◦ x = ∅ ,and x ◦ ∅ = ∅ ◦ x = x , where ∅ denotes an empty setand ∅ an empty sequence. For a sequence x and a number k ∈ N , x k will denote the concatenation of k copies of x ,where it is assumed that x = ∅ . The weight of a sequence x = x · · · x n , x i ∈ N , is defined as wt ( x ) = P ni =1 x i .A code of length n for the DTPC( N, K ) is a subsetof { , , . . . , N } n . Codes will be denoted by calligraphic Service procedure, service time distribution, and interarrival distributionare irrelevant in this context and hence are not specified. letters C , D , etc., or C ( n ) , D ( n ) , if their length needs to beemphasized. The set of codewords of C having prefix u isdenoted by C u , and the code obtained by removing this prefixby C u = { v : u ◦ v ∈ C} . Clearly, C u = u ◦ C u . Definition 2: C is said to be a zero-error code for theDTPC if for any m ≥ and any two distinct sequences x = x · · · x m and y = y · · · y m , where x i , y i ∈ C , thereexists no sequence z such that both x z and y z . N In words, no two sequences of codewords of C can producethe same channel output, and hence there is no confusionabout which sequence was sent. Note that we demand thedistinguishability of sequences of codewords , rather that justof codewords . This is necessary in the delay channels. Toillustrate this, let C = { , , } be a code of length threefor the DTPC( , ), introducing delays of at most one slot.Then it is easy to check that no two codewords can produce thesame channel output, but on the other hand ,and hence the sequences of codewords , and , are confusable. C is therefore not a zero-error code.This problem can easily be circumvented by simply paddingeach codeword with K zeros (empty slots, in the originalterminology). Empty slots at the end of each codeword serve to“catch” the particles that are (potentially) sent in the precedingslots and are (potentially) delayed in the channel. In this waythese particles do not interfere with the following codeword. Definition 3:
A code C ( n ) for the DTPC( N, K ) is said tobe zero-padded if all of its codewords end with min { n, K } zeros. N Clearly, a zero-padded code is zero-error if and only if forevery two distinct codewords x , y , there exists no sequence z with x z and y z . Definition 4:
The rate of a code C ( n ) for the DTPC( N, K )is defined as n log |C ( n ) | . The zero-error capacity of the DTPCis the supremum of the rates of all zero-error codes for thischannel. The base of log is assumed to be and hence therates and capacities are expressed in bits per time slot. N It is easy to show that this supremum is equal to the lim sup of the rates of the largest zero-error codes for theDTPC. Furthermore, when considering the capacity of theDTPC(
N, K ), there is no loss in generality to restrict oneselfto zero-padded codes, because padding with a constant numberof zeros does not affect the code rate in the asymptotic sense.II. O
PTIMAL ZERO - ERROR CODES FOR THE
DTPCIn this section we give two constructions of optimal zero-padded zero-error codes for the DTPC. The results havesimilar flavor to those obtained for some other types ofcombinatorial channels, e.g., [1], [7], [18], [2].
A. Recursive construction
The claim that follows establishes a general property ofzero-padded zero-error codes for the DTPC(
N, K ), fromwhich the construction of optimal codes will follow in astraightforward way. It states that such codes can, withoutloss of generality, be assumed to contain only codewords withprefixes N and i ◦ K , i ∈ { , , . . . , N − } . Proposition 5:
Let C ( n ) , n > K , be a zero-padded zero-error code for the DTPC( N, K ). Then there exists a zero-padded zero-error code D ( n ) of the same size, and such that: D = D N ∪ N − [ i =0 D i ◦ K . (1) Proof:
Let C ( n ) be a zero-padded zero-error code forthe DTPC( N, K ). We will construct D ( n ) by removing thecodewords of C ( n ) that do not satisfy the desired form, andadd the corresponding codewords that do. For any codewordof C ( n ) of the form x = u ◦ v , where u = u · · · u K +1 is oflength K + 1 and weight wt ( u ) = q , we let the correspondingcodeword ˜ x of D ( n ) be specified as follows: If q < N , then ˜ x = q ◦ K ◦ v , while if q ≥ N , then ˜ x = N ◦ ˜ u ◦ v , where ˜ u = ˜ u · · · ˜ u K +1 is some sequence of length K and weight q − N satisfying ˜ u i ≤ u i , i = 2 , . . . , K +1 (in other words, theprefix of ˜ x is in the latter case constructed from u by removing N − u of its particles from slots , . . . , K + 1 , and placingthem in the first slot, together with the u particles that arealready there). Thus, we can write D ( n ) = { ˜ x : x ∈ C ( n ) } . Itis now not difficult to argue that |D ( n ) | = |C ( n ) | and that thefact that C ( n ) is a zero-padded zero-error code implies that D ( n ) is such a code too. The key observation is that C ( n ) cannot contain two distinct codewords of the form x = u ◦ v and x = u ◦ v , where the prefixes u , u are of length K +1 and have the same weight, that is wt ( u ) = wt ( u ) = q . Thisis because C ( n ) is zero-error, and clearly x K ◦ q ◦ v and x K ◦ q ◦ v . Lemma 6:
Let D ( n ) , n > K , be a zero-padded code of theform (1) for the DTPC( N, K ). Then D ( n ) is zero-error if andonly if D N ( n − and D i ◦ K ( n − K − , ≤ i < N , areall zero-error. Proof: If D N and D i ◦ K are zero-error, then so are D N and D i ◦ K . Observe also that no two sequences x , y , suchthat x has prefix i ◦ K and y either j ◦ K or N (where i, j ∈ { , , . . . , N − } , i = j ), can produce the same channeloutput, implying that D is zero-error. The opposite directionis also easy.The above claims imply that an optimal zero-padded zero-error code of length n for the DTPC( N, K ) can be constructedrecursively from the codes of length n − and n − K − .To start the recursion, optimal zero-padded zero-error codes oflength j ∈ { , . . . , K } are needed, which are trivially { j } . Theorem 7:
The largest zero-padded zero-error code for theDTPC(
N, K ), denoted C N,K , is given by: C N,K ( n ) = (cid:0) N ◦ C N,K ( n − (cid:1) ∪ N − [ i =0 (cid:0) i ◦ K ◦ C N,K ( n − K − (cid:1) , (2)for n > K , and C N,K ( n ) = { n } for ≤ n ≤ K .In the following subsection we will describe a different,perhaps more intuitive construction of the codes C N,K .Theorem 7 implies that the cardinalities of the codes C N,K satisfy the recurrence relation: |C N,K ( n ) | = |C N,K ( n − | + N |C N,K ( n − K − | , (3) with initial conditions |C N,K ( n ) | = 1 , ≤ n ≤ K , whichfurther implies that: |C N,K ( n ) | = K +1 X k =0 a k r nk , (4)where r k are the (complex) roots of the polynomial x K +1 − x K − N , and a k are (complex) constants. Remark 2:
In the particular case N = 1 , K = 1 , theanalysis of the channel amounts to analyzing binary sequenceswhose ’s are being shifted in the channel by at most oneposition to the right (hence, the DTPC( , ) can also be seenas a type of a “bit-shift” channel [13], [9]). In this case, thecodes C , satisfy the relation : C , ( n ) = (cid:0) ◦ C , ( n − (cid:1) ∪ (cid:0) ◦ C , ( n − (cid:1) , (5)with C , (0) = { ∅ } , C , (1) = { } , which implies that |C , ( n ) | = |C , ( n − | + |C , ( n − | , with |C , (0) | = |C , (1) | = 1 . In other words, ( |C , ( n ) | ) is the Fibonaccisequence ( F n ) . N B. Direct construction
Let D N,K ( n ) be the code defined by the following pro-cedure. First enumerate in the inverse lexicographic orderall sequences of length n over { , , . . . , N } ending with min { n, K } zeros (so that, for n > K , the first sequence on thelist is N n − K ◦ K , the second one is N n − K − ◦ ( N − ◦ K ,etc.; see Table I). Then repeat the following step until there areno more sequences to process: Select the first sequence on thelist that has not been processed, call it x , to be a codeword,and then exclude all sequences y such that x y . TableI illustrates the construction for N = 2 , K = 1 (only thecodewords are listed to save space). TABLE IZ
ERO - ERROR CODES OF LENGTH UP TO FOR THE
DTPC( , ).C ARDINALITY OF THE CODES IS SHOWN IN THE RIGHTMOST COLUMN .2 2 2 2 0 12 2 2 1 02 2 2 0 0 32 2 1 0 02 2 0 0 0 52 1 0 2 02 1 0 1 02 1 0 0 02 0 0 2 02 0 0 1 02 0 0 0 0 111 0 2 2 01 0 2 1 01 0 2 0 01 0 1 0 01 0 0 0 00 0 2 2 00 0 2 1 00 0 2 0 00 0 1 0 00 0 0 0 0 21 As one of the referees pointed out, this resembles a well known char-acterization of F n as the number of binary sequences of length n − withno consecutive ones. Such a set of sequences, S ( n ) , obeys the recursion S ( n ) = (cid:0) ◦S ( n − (cid:1) ∪ (cid:0) ◦S ( n − (cid:1) , with S (0) = { ∅ } , S (1) = { , } . The name Fibonacci code would thus be appropriate here, but it hasalready been used in some other contexts [6], [18].
Proposition 8: D N,K ( n ) = C N,K ( n ) for every n ∈ N . Proof:
Since D N,K ( n ) = C N,K ( n ) = { n } for ≤ n ≤ K , it is enough to show that D N,K ( n ) satisfy the relation(2). Observe that D NN,K ( n ) = N ◦ D N,K ( n − becauseadding a fixed prefix to a set of sequences does not affectthe process of construction and, moreover, the prefix N putsthe sequences on the top of the list. It is left to prove that D i N,K ( n ) = i ◦ K ◦ D N,K ( n − K − , for ≤ i < N . Firstconsider the case i = N − . Let x be a sequence with prefix ( N − ◦ u , where u is of length K and strictly positiveweight, and construct ˜ x as in the proof of Proposition 5. Now,if ˜ x is a codeword, then x is not because ˜ x x and so x would have been excluded in the process of construction. Onthe other hand, if ˜ x is not a codeword, then it has itself beenexcluded by some sequence y that precedes it in the inverselexicographic order ( y ˜ x ). But then we also have y x ,and therefore x is not a codeword either. We have shown that D N − N,K ( n ) does not contain codewords having prefix ( N − ◦ u ,where wt ( u ) > , and hence it can only contain codewordsstarting with ( N − ◦ K . Since none of the sequenceswith this prefix could have been excluded in the process ofconstruction by a codeword from D NN,K ( n ) , it follows that theyhave been processed independently of the rest of the list andso D N − N,K ( n ) = ( N − ◦ K ◦ D N,K ( n − K − . One can nowprove by induction that D i N,K ( n ) = i ◦ K ◦ D N,K ( n − K − for i = N − , N − , . . . , , ; the argument is very similarto the above and is omitted. C. Decoding algorithm
The structure of the codes C N,K , captured by the relation (2),suggests a very simple algorithm for recovering the transmittedsequence x = x · · · x n ∈ C N,K ( n ) from the received sequence y = y · · · y n . The procedure is as follows:Set y (1) = y , and observe the prefix of y (1) of length K +1 ,namely y · · · y K +1 , and its weight q .1. If q < N , conclude that x · · · x K +1 = q ◦ K (see(2)), and set y (2) = y K +2 · · · y n . Note that y (2) is the(possible) output of the DTPC( N, K ) when the input isthe codeword x K +2 · · · x n from C N,K ( n − K − .2. If q ≥ N , conclude that x = N . If also y < N , thismeans that some of the particles from the first slot havebeen delayed in the channel. In that case remove N − y of these particles from slots , . . . , K + 1 (first takingparticles from slot , then slot , etc., until N − y ofthem are collected) and put them in the first slot. Thenset y (2) = y ′ · · · y ′ K +1 ◦ y K +2 · · · y n , where y ′ · · · y ′ K +1 is obtained from y · · · y K +1 by removing the particles inthe above-described way, i.e., for some k ∈ { , . . . , K +1 } we have y ′ i = 0 for i ∈ { , . . . , k − } , y ′ k = P ki =1 y i − N ≥ , and y ′ i = y i for i ∈ { k + 1 , . . . , K + 1 } . Note that y (2) is the (possible) output of the DTPC( N, K ) when theinput is the codeword x · · · x n ∈ C N,K ( n − .The procedure is repeated with y (2) by considering its prefixof length K + 1 , and so on.Since at least one symbol of x is determined in everyiteration, the algorithm will terminate in at most n iterations(in fact, at most n − K due to the trailing zeros). III. Z ERO - ERROR CAPACITY OF THE
DTPCThe results of Section II-A imply that the capacity of theDTPC(
N, K ) can be simply found as lim n →∞ n |C N,K ( n ) | ,and by using the fact that the asymptotic behavior of |C N,K ( n ) | is determined by the largest (in modulus) root of the polyno-mial x K +1 − x K − N (see (4)). Lemma 9:
The largest (in modulus) root r of the polyno-mial x K +1 − x K − N is real and greater than . Moreover, if K → ∞ , then r → . Proof:
The following theorem is proven in [16, Ch. 3,Thm 2] (see also [17]): If p ( x ) = c m x m + c m − x m − + · · · + c x + c is an arbitrary polynomial with complex coefficients,and c · c m = 0 , then all roots of p ( x ) lie in the (complex)circle | x | ≤ r , where r is the unique positive real root of ˜ p ( x ) = | c m | x m − | c m − | x m − − · · · − | c | x − | c | . Since ourpolynomial is precisely of the form ˜ p ( x ) , we conclude that ithas a unique positive real root r , and that all other roots aresmaller in modulus than r . This root can be found as the pointof intersection of the curves x K and N ( x − − (viewed asreal functions). By analyzing these curves it follows easily that r > and that r → when K → ∞ . Theorem 10:
The zero-error capacity of the DTPC(
N, K )is equal to log r , where r is the unique positive real root ofthe polynomial x K +1 − x K − N .The zero-error capacity of the DTPC( , ) is therefore log φ ,where φ = (1+ √ / . More generally, the zero-error capacityof the DTPC( N, ) equals log (cid:0) (1 + √ N ) (cid:1) . Explicitexpressions can also be obtained in the following two cases,which are intuitively clear: The zero-error capacity of theDTPC( N, ) is log( N + 1) , while that of the DTPC( N, ∞ )(which allows arbitrarily large delays) is zero. K (maximum delay in the channel) Z e r o − e rr o r c apa c i t y (a) Dependence on K , for N = 1 , , , , , . N (maximum number of particles per slot) Z e r o − e rr o r c apa c i t y (b) Dependence on N , for K = 0 , , . . . , .Fig. 2. The behavior of the zero-error capacity of the DTPC( N, K ). The following proposition states some basic properties ofthe capacity, regarded as a function of the channel parameters N and K . This function is also illustrated in Figure 2. Proposition 11:
Both r and log r are monotonically in-creasing concave functions of N , for fixed K , and monotoni-cally decreasing convex functions of K , for fixed N . Proof:
The function r is defined implicitly by r K +1 − r K − N = 0 , r > , and the function c = log r by c ( K +1) − cK − N = 0 , c > . Note that r and c are well-defined forall N, K ∈ R + , not necessarily integers. One can thereforedifferentiate them with respect to N and K and verify that ˙ r N > , ¨ r N < , ˙ c K < , ¨ c K > . The remaining claimsfollow from the properties of the logarithm and the exponentialfunction. A CKNOWLEDGMENT
The authors would like to thank the Associate Editor J´anosK¨orner and the anonymous reviewers for providing construc-tive comments and significantly improving the quality of themanuscript. A
PPENDIX AR ESTRICTING THE CHANNEL OUTPUT
In this section we demonstrate that bounding the number ofparticles that can be received in a slot by N (or by N ′ ≥ N )does not change the zero-error capacity of the DTPC. For thepurpose of this argument we will refer to the channel with thisadditional restriction as the DTPC( N, K ; N ). To clarify whatis meant by the DTPC( N, K ; N ), we emphasize that there isno “limiter” in the channel that drops some of the particles iftheir number in a slot exceeds N . As in the DTPC( N, K ), allparticles must arrive at the destination, only now their delays,in addition to being ≤ K , have to be such that the number ofparticles at the channel output in every slot is ≤ N . One canperhaps imagine a “membrane” at the channel output allowingat most N particles per slot to pass through. Proposition 12:
Any zero-error code for the DTPC(
N, K )is a zero-error code for the DTPC(
N, K ; N ), and vice versa. Proof:
Let x and y be two sequences such that they canboth produce z = z · · · z l at the output of the DTPC( N, K ).Then there exists w = w · · · w l such that x w and y w in the DTPC( N, K ; N ), i.e., such that w i ≤ N . To see this,observe that if z i > N for some i ∈ { , . . . , l } , then someof these z i particles have not been delayed for a maximalnumber of slots ( K ) and could be further delayed. We cantherefore find the desired w by going through slots , . . . , l ,respectively, and whenever we find that z ′ i > N , we move z ′ i − N of these particles to slot i + 1 , where z ′ i is the sumof z i and the number of particles that were moved from slots , . . . , i − to slot i during this procedure. We conclude thatif a code is not a zero-error code for the DTPC( N, K ), thenit is not a zero-error code for the DTPC(
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