Spectral analysis of maximum entropy multitrack modulation codes

1574 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 4, JULY 1998

We considered magnetic data storage and showed that for twoexamples relevant to PRML in hard disk drives, modified concate-nation performs better than standard concatenation. For a rate16

17

modulation code and 8-bit Reed–Solomon code, modified concatena-tion permits the use of three interleaves per sector, whereas standardconcatenation requires at least four interleaves for good performance.Also, we showed how modified concatenation allows the practicalimplementation of a dc-free block code of length40.

In general, this modified concatenation technique allows the use ofmodulation codes whose rates approach the capacity of the desiredmodulation constraint, without a loss in performance. The ability touse sophisticated, high-rate modulation codes may prove useful formany data storage applications, such as hard disk drives, optical discs,and digital video-tape recorders [15]. The codesign of modulationcodes and block codes for lossless compression poses an interestingchallenge. In addition, the benefits of localized erasure and reliabilityinformation, which are facilitated by modified concatenation, deserveto be explored.

ACKNOWLEDGMENT

The authors wish to thank E. A. Gelblum, J. T. Gill, K. A. S.Immink, A. Patapoutian, and E. Soljanin for discussions relatedto this topic. They also wish to thank J. Sonntag, N. Sayinerand AT&T Microelectronics (now Lucent) for their support. Thehelpful comments from the anonymous reviewers were also greatlyappreciated.

REFERENCES

[1] K. A. S. Abdel-Ghaffar, M. Blaum, and J. Weber, “Analysis of codingschemes for modulation and error control,”IEEE Trans. Inform. Theory,vol. 41, pp. 1955–1968, Nov. 1995.

[2] A. Bassalygo, “Correcting codes with an additional property,”Probl.Inform. Transm., vol. 4, no. 1, pp. 1–5, Spring 1968.

[3] M. Blaum, “Combining ECC with modulation: Performance compar-isons,” IEEE Trans. Inform. Theory, vol. 37, pp. 945–949, May 1991.

[4] W. G. Bliss, “Circuitry for performing error correction calculations onbaseband encoded data to eliminate error propagation,”IBM Tech. Discl.Bull., vol. 23, pp. 4633–4634, 1981.

[5] R. D. Cideciyan, F. Dolivo, R. Hermann, W. Hirt, and W. Schott, “APRML system for digital magnetic recording,”IEEE J. Select. AreasCommun., vol. 10, pp. 38–56, Jan. 1992.

[6] A. B. Cooper III, “Soft decision decoding of reed-solomon codes,” inReed–Solomon Codes and Their Applications, S. B. Wicker and V. K.Bhargava, Eds. New York: IEEE Press, 1994.

[7] E. A. Gelblum and A. R. Calderbank, “A forbidden rate region forgeneralized cross constellations,”IEEE Trans. Inform. Theory, vol. 43,pp. 335–341, Jan. 1997.

[8] J. Hagenauer and P. Hoeher, “A viterbi algorithm with soft-decisionoutputs and its applications,” inProc. Globecom’89(Dallas, TX, Nov.1989), vol. 3, pp. 47.1.1–47.1.7.

[9] H. M. Hilden, D. G. Howe, and E. J. Weldon Jr., “Shift error correctingmodulation codes,”IEEE Trans. Magn., vol. 27, pp. 4600–4605, Nov.1991.

[10] K. A. S. Immink, Coding Techniques for Digital Recorders. Engle-wood Cliffs, NJ: Prentice Hall, 1991.

[11] K. A. S. Immink, “A practical method for approaching the channelcapacity of constrained channels,”IEEE Trans. Inform. Theory, vol. 43,pp. 1389–1399, Sept. 1997.

[12] R. Karabed and P. H. Siegel, “Matched spectral-null codes for partial-response channels,”IEEE Trans. Inform. Theory, vol. 37, pp. 818–855,May 1991.

[13] W. H. Kautz, “Fibonacci codes for synchronization control,”IEEETrans. Inform. Theory, vol. IT-11, pp. 284–292, Apr. 1965.

[14] K. Knudsen, J. Wolf, and L. Milstein, “A concatenated decoding schemefor (1-D) partial response with matched spectral-null coding,” inProc.

Globecom’93(Houston, TX, Nov. 1993), pp. 1960–1964.[15] J. C. Mallinson,The Foundations of Magnetic Recording, 2nd ed. San

Diego, CA: Academic, 1993.[16] M. Mansuripur, “Enumerative modulation coding with arbitrary con-

straints and post-modulation error correction coding and data storagesystems,”Proc. SPIE, vol. 1499, pp. 72–86, 1991.

[17] R. J. McEliece and L. Swanson, “On the decoder error probability forreed-solomon codes,”IEEE Trans. Inform. Theory, vol. 32, pp. 701–703,Sept. 1986.

[18] A. Patapoutian and P. V. Kumar, “The(d; k) subcode of a linear blockcode,” IEEE Trans. Inform. Theory, vol. 38, pp. 1375–1382, July 1992.

[19] P. N. Perry, “Runlength-limited codes for single error detection in themagnetic recording channel,”IEEE Trans. Inform. Theory, vol. 41, pp.809–814, May 1995.

[20] E. Soljanin, “A coding scheme for generating bipolar dc-free se-quences,”IEEE Trans. Magn., vol. 33, pp. 2755–2757, Sept. 1997.

[21] , “Decoding techniques for some specially constructed dc-freecodes,” inProc. 1997 IEEE Int. Conf. Communications (ICC’97)(Mon-treal, Que., Canada, June 1997).

[22] J. Sonntaget al., “A high speed, low power PRML read channel device,”IEEE Trans. Magn., vol. 31, pp. 1186–1195, Mar. 1995.

[23] L. M. G. M. Tolhuizen, K. A. S. Immink, and H. D. L. Hollmann, “Con-structions and properties of block codes for partial-response channels,”IEEE Trans. Inform. Theory, vol. 41, pp. 2019–2026, Nov. 1995.

[24] A. J. van Wijngaarden and K. A. S. Immink, “Combinatorial construc-tion of high rate runlength-limited codes,” inProc. IEEE Globecom(London, U.K., 1996), pp. 343–347.

[25] E. J. Weldon, “Coding gain of reed-solomon codes on disk memories,”in Proc. IEEE Int. Conf. Communications (ICC’92)(Chicago, IL, June1992), pp. 1369–1374.

[26] S. B. Wicker,Error Control Systems for Digital Communications andStorage. Englewood Cliffs, NJ: Prentice Hall, 1995.

[27] J. K. Wolf, “A survey of codes for partial response,”IEEE Trans. Magn.,vol. 27, pp. 4585–4589, Nov. 1991.

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Spectral Analysis of MaximumEntropy Multitrack Modulation Codes

Bane V. Vasic, Member, IEEE

Abstract—In this correspondence the problem of calculating the powerspectral density of a constrained maxentropic vector sequencefaaa(n)g,aaa(n) = [a

(n)i

]0�i�N�1, is considered. The constraints of the constituent

sequencesfa(n)i

g are defined by the sofic systemsSi presented by thedirected graphs Gi, 0 � i � N � 1, but the vector sequence itselfis constrained additionally, and given by a function � of constituentgraphs (G = �(G0; � � � ; GN�1)). This class of vector constraints ismet in parallel multitrack recording. The general case of a simultaneousrecording on N tracks is considered, assuming that the common vectorconstraint is track-invariant.

Index Terms—Input-constrained discrete channels, maxentropic se-quences spectral analysis, sofic systems.

I. INTRODUCTION

Let us consider a system for parallel recording of digital data.The information streams fromN sources enter the modulationencoder, which transforms them so that the streams at the output,

Manuscript received October 17, 1996; revised December 22, 1997.The author was with Kodak Research Laboratories, Rochester, NY 14623

USA. He is now with Bell Laboratories, Lucent Technologies, Allentown, PA18103 USA

Publisher Item Identifier S 0018-9448(98)03875-9.

0018–9448/98$10.00 1998 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 4, JULY 1998 1575

fa(n)i gn2Z(= fa

(n)i g); i = 0; � � � ; N � 1; satisfy the channel con-

straints (Z is a set of all integers). Then, these streams are recordedin parallel as a vector streamfaaa(n)g, aaa(n) = [fa

(n)i g]0�i�N�1.

Besides the constraints that must be satisfied in every single track, itis assumed that the vector sequencefaaa(n)g, as a whole, is constrainedtoo, and that the vector constraint depends on in-track constraints. Asusual, the channel constraint is defined by a sofic system, i.e., by aset of all bi-infinite sequences generated by walks on a finite-directedgraph, whose edges are labeled by symbols in a finite alphabet [1],[13], [16], [17]. The in-track constraints are defined by a sofic systemsSi, presented by the graphsGi, 0 � i � N � 1, and the constraintof the whole vector sequence is given by sofic systemS, the graphof which is a function of the graphsGi, i.e.,G = �(G0; � � �GN�1).It is assumed that the function� is symmetric. The symmetry of thefunction� appears in the practically important case when the commonor intertrack constraint is track-invariant. There are no privilegedtracks, but all tracks participate in the common constraint in thesame fashion. The introduced sofic system represents a good generalmodel of the system for two-dimensional optical recording [14], [15,pp. 219–224], where the data are recorded in parallel onN track atinformation surface of the disk. The model can also be applied to therecording in three dimensions [2], [18].

In this correspondence the problem of calculating the capacityof the sofic systemS, and power spectral density (spectrum) ofthe maxentropic [21] vector sequencefaaa(n)g 2 S is considered.The spectrum of sequencefaaa(n)g is a vector function�(f) =[�i(f)]0�i�N�1, wheref is frequency, and theith component of�(f) is the spectrum inith track. As we know, the spectrum is auseful characteristic of the recorded stream, showing how well therecorded signal is matched to the transfer function of the recordedchannel, which has a direct implication on the error performance,tracking, automatic gain control, etc. In high-density recording sys-tems, where the space between tracks is small and interferencebetween track is severe, parallel readback with multitrack equalization[25] reduces both intertrack and intratrack interference. The importantparameter in designing multitrack equilizers is the cross-spectrummatrix �(f) = [�i;j(f)]0�i;j�N�1 (�i;j(f) is calculated on thebasis of a cross-correlation functionc(k)i;j of the sequences in tracksi and j, i.e., �(f) = diag (�(f)). The approach presented in thiscorrespondence is applicable both to intratrack and to cross-spectralanalysis, however, for ease of explanation of the presentated materialwe will focus to the intratrack spectra only.

The spectrum of the vector sequence generated by walks on graphG can be expressed in the closed matrix form according to thewell-known approach of Cariolaroet al. [7] and Bilardi et al. [4]for spectral analysis of Markov chain driven signals. The Markovsource generating the maxentropic vector sequence is assigned tothe graphG by setting the transition probabilities according toShannon’s noiseless channel theory [19]. However, if we want touse this method, we have to deal with the graphG with jV j =jV0j � jV1j � � � � � jVN�1j vertices, wherejVij are the vertex numbers ofgraphsGi. Even for smallN the total number of vertices can be verylarge, in practice above a several thousand, so that inverse matricesappearing in expressions for the spectrum [7] are of high dimensionand are hard to invert. The same holds when we use the numericallyefficient method for the spectrum calculation given in [22]. So it isof essential importance to find some way to reduce the number ofvertices of the graphs we are dealing with.

Since the common constraint is track-invariant, if the in-trackconstraint in different tracks are the same, the sequences recordedin these tracks will have equal spectra. So, it is not necessary tocalculate the spectra of allN sequencesfa(n)i g, 0 � i � N � 1,but only the spectra of sequences given by the different constraints.

For the spectrum calculation not all elements of the edge label vectorare of interest, so we transform the graphG into the graphH byextracting the elements of edge vector label that are of interest, intoa new vector label (this extraction is discussed in Section III). Eachedgee of the obtained graphH has a two-component label: the firstcomponent is the extracted vector label�(e) and the second�(e)is referred to as an edge weight and represents the number of edgesbetween two vertices of the graphG having the same extracted vectorlabel.

It is easy to note that the extracted graphs carry all the spectralinformation about the sofic systems as their original graphs, butalso possess the important property of symmetry, which allowsconsiderable reduction in the number of vertices with respect to thatof the original graph. The graph symmetry can be simply explained asthe existence of subsets of vertices with equal numbers of outgoingedges having the same labels�, �, and having the same terminalvertices, so that the vertices of each such subset can be merged.Since the original graphsG contain a large number of vertices, thesymmetry and reduction ofH (explained in Section IV) is veryimportant for spectral analysis of the sequences generated by thecomposition of the graphs (given in Section V). As a simple exampleof application of this approach, in Section VI, we consider a two-dimensional runlength-limited constraint, for which we are able toconstruct the extracted graph directly.

II. DEFINITIONS

This section gives some notions and the graph operations thatmake it easier to formally describe the multitrack constraints. Thegraph addition and Kronecker multiplication operations are defined.The Kronecker product of graphs describes a graph generatingthe vector-valued sequence�(k) = [�i]0�i�N�1, wherein eachcomponent sequence�(k)

i , 0 � i � N � 1, is independent andimposes a constraint given by the graphGi. The graph subtractionand graph with negative edges is introduced as a formal way ofthe edge deleting, which is dictated by the common, intertrackconstraint.

To calculate the power spectrum of a sequence generated by somegraphG, it is necessary to define probabilities of edges inG (or statetransition probabilities). A graphG with the assigned probabilities oftransition between states defines a Markov source (Section II-B). Thedefinitions introduced in Sections II-B and -C are used in Section Vin the formulas for spectrum of multitrack sequences.

A. Sofic Systems and Graphs

A sofic systemS is the set of all bi-infinite sequences generatedby walks on a directed graphG = G(S) whose edges are labeled bysymbols in a finite alphabetA. The graphG = (V;E; �) is givenby a finite set of vertices (or states)V , a finite set of directed edgesE, and a labeling�: E ! A. So, for a given sequence of edgesfe(k)g(e(k) 2 E), we have the output sequencefa(k) = �(e(k))g.A graphG is strongly connected if for every two verticesu; v 2 V

there exists a path (sequence of edges) fromu to v. A graphG isdeterministic if for each statev 2 V , the outgoing edges fromv,E(v), are distinctly labeled.

The adjacency matrix (or vertex transition matrix)

DDD(G) = DDD = [D(u; v)]u;v2V

of graphG is a jV j � jV j matrix where entryD(u; v) is the numberof edges from vertexu to vertexv, andjV j is the number of verticesof the graphG.


(a) (b)

Fig. 1. (a) Positive edge. (b) Negative edge.

Fig. 2. Addition of the graphs.

The capacityC of the sofic systemS presented by the deterministicgraph G is defined asC = log

2(�0), where �0 = �0(G) is

the spectral radius (i.e., the largest of the absolute values of theeigenvalues [10, pp. 354–355]) of adjacency matrixDDD(G).

For two graphsG0 = (V0; E0; �0) and G1 = (V1; E1; �1),with alphabetsA0 andA1, we define their sum as the graphG =(V;E; �), for which V = V0 [ V1, E = E0 [ E1, A = A0 [ A1,and�(e) = �0(e), whene 2 E0, and�(e) = �1(e), whene 2 E1.

Let (u0; v0) and (u1; v1) be two pairs of equaly labeled verticescorresponding to the setsV0 andV1, respectively(u0 = u1 andv0 =v1). If an edgee emanates fromu0 and terminates atv0 in graphG0, and edgef does so in the graphG1, and if�0(e) = �1(f), thenwe assume thate = f . A zero graph, denoted byO, is the graphwith E = ;.

A negative graph, denoted by�G, is defined as a graph for whichthere exists the graphG, andG + (�G) = O. Consequently, theedges of the negative graph will be called negative edges (the arrowsof negative edges will be drawn white (Fig. 1)). The negative graphis simply a formal way of deleting edges in another graph.

When both the positive and the negative edge with equal labelsemanate from some vertexu and terminate at some vertexv, wetraverse no edge fromu to v. An example of the addition of graphswith positive and negative edges is shown in Fig. 2.

The sum of two graphsG0 and G1 will be deterministic if fortwo equal (i.e., equally labeled) verticesv0 and v1 (v0 2 V0; v1 2V1; v0 = v1) there are no outgoing edges with the same labels. It isalso obvious that adding new edges in the deterministic graphG(S)increases, and removing edges decreases, the capacity of the soficsystemS.

The Kronecker product of the graphsG0 = (V0; E0; �0) andG1 = (V1; E1; �1), G = G0 G1, is the graphG = (V;E; �) forwhich V = V0 � V1 (� denotes Cartesian product of the sets), andfor every edgee0 from u0 to v0 in G0 and every edgee1 from u1 tov1 in G1, there esists an edgee(e = (e0; e1)) in G, emanating fromthe vertexu = (u0; u1) 2 V and terminating atv = (v0; v1) 2 V

with the vector label

�(eee) = �(e0; e1) = [�0(e0)�1(e1)]:

So, the graphG = G0 G1 generates vector sequencesfaaa(k)g =

f[a(k)0 a

(k)1 ]g. If adjacency matrices of the graphsG0 andG1 areDDD0

andDDD1, then the adjacency matrix of the graphG = G0 G1, DDD,is DDD = DDD0 DDD1, where now denotes the Kronecker product of

(a)

(b)

Fig. 3. (a) An example of two graphsG0 and G1(V0 = f1; 2g;V1 = fI; IIg;A0 = fa; b; cg; A1 = fA;B; Cg). (b) Their Kroneckerproduct.

matrices [8, pp. 50–53].

DDD = [D(u; v)]u;v2V

= [D(u = (u0; u1); v = (�0; �1))]u ;v 2V ;u v 2V

= DDD0(u0; u1)DDD1(u1; v1)

= [D0(u0; v0) �D1(u1; v1)]u ;v 2V ;u ;v 2V : (2.1)

As an example, Fig. 3 shows two deterministic graphsG0 andG1

and their Kronecker productG.The Kronecker product of the graphs represents the vector sofic

system in which the component subsequences of the sofic systems aregenerated independently. So the vector sequence carries an averageamount of information equal to the sum of average amounts ofinformation of the constituent sequences. Consequently, the capacityof the Kronecker productG = 0�i�N�1Gi = G0 G1 � � � GN�1 is

C(G) =0�i�N�1

C(Gi): (2.2)

The proof of this statement follows directly from the fact that the setof eigenvalues ofDDD(G) is the set of all products of eigenvalues ofthe factor graph adjacency matrices [8, Ch. 7].

However, the practical multitrack constraints given by graphsGi,0 � i � N � 1, have an additional, intertrack constraint so thatsequences in different tracks are not independent (consequently, theresulting capacity is lower). It will be shown that the graph of thewhole constraint can be expressed as a difference of the Kroneckerproduct of intratrack graphsGi, 0 � i � N � 1, and a graphdescribing intertrack constraint.

B. Markov Sources

Associated with a sofic systemS is a Markov source definedby the probabilities of the transitions between states (vertices) over


different edges, i.e., the valuesP (e(k) = e j v(k) = v) [5,pp. 270–292]. We assume that these probabilities are time-invariant(P (e(k) = e j v(k) = v) = P (e j v)) and are nonzero(8v 2 V;

8e 2 E(v); P (e j v) > 0). Since we consider only stronglyconnected graphs, the assigned Markov chains are homogenous,ergodic, and fully regular [9]. Thus the transition probabilities providea complete characterization of the sequencesfe(k)g(e(k) 2 E), andfa(k) = �(e(k))g. We are especially interested in Markov sourcesgenerating the streamfa(k)g whose symbols carry the greatestaverage amount of information, called maxentropic sources [12], [19],[21]. The entropy of a homogenous, ergodic Markov source, assignedto the sofic systemS, is maximized by transition probabilities

P (e j v) = 1

�0

p(�(e))

p(v)(2.3)

(where�(e) denotes the terminal vertex of the edgee (analogously,��1(e) = v is the initial vertex ofe) [19]. In this case, the stationaryprobabilities of the states are

s(v) = p(v) � q(v); v 2 V (2.4)

wherep(v) and q(v), v 2 V , are the elements of the right and lefteigenvectors, respectively, corresponding to the spectral radius�0,i.e., the elements of the (column) vectors satisfying

ppp = ��10 �DDD � ppp

qqq = ��10 �DDDT � qqq

(2.5)

and normalized so thatpppT qqq = 1. The maximal entropy of theMarkov source is equal to the capacity of the sofic systemS, i.e.,Hmax(S) = C(S) = log2(�0).

The above expressions forP (eju) show that each edge of theexisting D(u; v) edges emanating fromu and terminating atvhave the same transition probability in the maxentropic Markovsource. Transition probabilities between the statesu and v, for themaxentropic source, according to [19], are

P (ujv) = 1

�0�e2E(�;u)

p(u)

p(v)= D(v; u) � 1

�0� p(u)p(v)

(2.6)

whereE(u; v) is a set of edges with initial stateu and terminalstatev.

C. Statistics of Sequences Generated by Markov Sources

The autocorrelation function of the streamfa(k)g (a(k) 2 A � R),generated by ergodic Markov source, is given by

r(k) = Efa(n) � a(n+k)g

whereEfg denotes operator of averaging over the ensemble. For twostreamsa(n)i anda(n+k)j the crosscorrelation function is defined as

c(k)i;j = Efa(n)i � a(n+k)j g

(the same deffinitions holds for vector sequences but the productsare componentwise, i.e.,r(k) = Efaaa(n) � aaa0(n+k)g, where 0 meansvector transponse).

The power spectral density of the streamfa(k)g, (a(k) 2 A � R),generated by the ergodic Markov source, is given by

�(f) =k2Z

r(k) � e�j2�kf (2.7)

where j =p�1 [4]. The definition of cross-spectrum is straight-

forward.

(a) (b)

Fig. 4. An example of the two-component edge labeling. (a) Nondetermin-istic graph. (b) Graph with two-component edge labels.

III. EXTRACTION OF EDGE LABEL ELEMENTS

The vector streamfaaa(n)g is recorded onN parallel tracks. Sincethe statistics of the sequences in the different tracks may be the same,it may not be necessary to evaluate the spectra of recorded sequencesin all N tracks. Let the vector stream whose spectrum we need be�(n) = EM(aaa(n)) = [a

(n)m ]m2M , whereM is the set of theL selected

tracks of interest. Assuming that the streamfaaa(n)g is generated bywalks on a graph, we will show how to extract the vector elementsof interest from the edge labels, by perceiving the information aboutthe graph structure as relevant to the statistical analysis.

Another step in graph reduction is edge merging. The merging isperformed on edges that have the same label elements on the positionsgiven by the setM . As a result of merging we get edges with two-component labels. The first edge label component is the extractedvector, and the second, referred to as the edge weight, is the numberof edges merged. The edge weights have an important role in graphreduction, since they enable transforming a nondeterministic graph,obtained after the label extraction, into a deterministic one, that isused for spectrum calculation. The graph addition and product canbe defined for these graphs as well.

A. Graphs with Variable Weight Edges

Let the vector sequencefaaa(n)g be generated by walks on the graphG, G = (V;E; �), where�: E ! A

A = X0�i�N�1(Ai) = A0 � � � � � AN�1

andAi, 0 � i � N � 1, are some alphabets. For a given set

M = fmj j0 � j � L� 1; 0 � mj � N � 1gwe define the function� = �M : E ! �m2M (Am), and definethe graphH = (V;E; �). The graphG is mapped into the graphH(H = HM(G)) in such a way that from the vector labels�(e) = [�i]0�i�N�1, of the edgese, e 2 E, in the graphG, theLelements at positions given byM are extracted. TheL-dimensionalvectors with these elements are the edge labels of the graphH. IfL = 0, i.e.,M = ;, the extracted graphH is unlabeled.

The extracted graphH is not deterministic, and in practice thereis a large number of equally labeled edges with the same initial andterminal vertices. Dealing with such graphs can be made easier byusing the following representation.

Let H = (W;E; �) be a nondeterministic graph. Denote theequally labeled edges, with the same initial and terminal vertices,by f , the number of such edges by�(f), and their label by�(f).Then, the graphH can be defined byH = (W;F; (�; �)). In graphH the edges fromu to v(u; v 2 W ) with equal labels are joined,and drawn as a single edgef with two-component label(�(f); �(f)),wherein�(f) is referred to as the weight of the edgef . Fig. 4 showsan example of a variable-weight edge graph described in the abovemanner. For example, instead of drawing two edges labeled byb,from w = 1 to w = 2 (in Fig. 4(a)), a single edge with label(b; 2)is drawn (Fig. 4(b)).

As an example consider the sequence of the first elements of thevector sequencef�(k)g, i.e., f�(k)0 g (�0 2 A) generated by the


(a)

(b)

Fig. 5. (a) A graph with vector labels. (b) Extracted graph of the graph (a).

graphG = (W;E; �) shown in Fig. 5(a). If we are not interestedin the second element of the vector sequencef�(k)g, we will mapgraphG by the functionHM = f0g and obtain the extracted graphH = (W;F; (�; �)) shown in Fig. 5(b).

The Kronecker product of two graphsH0 = (W0; F0; (�0; �0))andH1 = (W1; F1; (�1; �1)), given by the two-component labels, isthe graphH = (W;F; (�; �)) whereW = W0 �W1 and for everyedgef0 from u0 to v0 in theH0 and every edgef1 from u1 to v1in G1, there exists an edgee (f = (f0; f1)) in H, emanating fromvertexu = (u0; u1) 2W and terminating atv = (v0; v1) 2W with(two-component) label

�(f) = [�0(f0) � �1(f1)] �(f) = �0(f0) � �1(f1):

B. Spectrum Calculation by Using Variable-Weight Edge Graphs

To calculate the spectrum of sequencef�(n)g extracted (byM )from the maxentropic sequencefaaa(n)g generated by graphG, we firstmake the extracted graphH = HM(G) = (V; F; �). The vertex setof HM(G) as well as the adjacency matrixDDD(HM(G)) are the sameas those of graphG. Since�(n) = EM (aaa(n)), the vector spectrum ofthe sequencef�(n)g will be ��(f) = EM(�aaa(f)) if the transitionprobabilities between states of the assigned Markov source are givenby (2.6)

(P (ujv) = (D(u; v)=�0) � (p(u)=p(v))):

Such a Markov source will be called a maxentropic source assignedto the extracted graphH. The probability of an edgef in extractedgraphH is p(f ju) = (1=�0)�(f)=D(u;v).

When the Markov source is defined, the autocorrelation functionand spectrum can be obtained by the difference equation method-procedure for spectral analysis of signals generated by Markovsources given in [4]. However, more computational savings arepossible, as will be explained in the following.

Transform the graphH into the graphsQ0 andQ1, referred to asthe graph with averaged edge labels, by replacing all edges between

Fig. 6. GraphQ1 with averaged edge labels for the graph from Fig. 5(b).

u andv by one edgei with the label�(i) = [�j(i)]0�j�L�1 wherefor all j = 0; � � �L � 1

�(i) =1

D(u; v)�f2F (u;v)

(�(f)) � �(f): (3.1)

When is a square function( (�) = � � diag (�)), the transformedgraphQ0 will help us to calculate the average power of the recordedstreamfaaa(n)g, i.e., r(k), for k = 0. When is linear function( (�) = �), the obained graphQ1 will be used for calculatingr(k), for k > 0. The Markov sources�0, and�1, assigned toQ0

and Q1, used for calculatingr(k) have the transition probabilitiesgiven by (2.6).

For calculating the autocorrelation function we use the computa-tionaly efficient difference eqauation method given in [22]. So, wecalculate the autocorrelation (or crosscorrelation) functionr(k) in twosteps: fork = 0 and fork > 0. If the power of the recorded stream isknown (for example, in binary caseA = f�1;+1g, so the averagepower isr(0) = 1) the first step is not necessary. The purpose of thisgraph transformation is formalized trough the following lemma.

Lemma 1: The autocorrelation functionr(k) of the stream�(n) =EM (aaa(n)), generated by the maximum entropy ergodic Markovsource, assigned to the graphG, is equal to the autocorrelationfunction of the stream generated by Markov source�1 for k 6= 0.

Proof: The proof is given in Appendix A.

Remark 1: The same statement holds forG andQ0, whenk = 0.For example, for the graph from Fig. 5(b) the graphQ1 is shownin Fig. 6.

Remark 2: If we want to use this approach to calculate thecrosscorrelation functionc(k)i;j of the streamsfa(n)i g andfa(n)ij g, then,during this transformation, we should preserve the number of edgeswith the same pairs of elements at positionsi andj, so the extractingsetM should be at leastM = fi; jg.

For example, if we want to extract the first two components ofedge labels in graphG3

0, whereG0 is the graph shown in Fig. 3(a)in order to calculate the cross-spectrum, we can proceed as in theprevious example and form the graphsQ0 andQ1. The edge labelsof the graphQ1 are given in Table I.

IV. SYMMETRIC GRAPHS

Edge merging and edge averaging help to obtain a more compactrepresentation of the graph with extracted labels. These graphs areused to calculate the spectrum of a multitrack sequence. However,the high symmetry of the intertrack constraint enables much morecompact representation of the original graph.

As an example let us consider a case of extracting the firstcomponent of edge labels in graphG3

0, whereG0 is the graph shown


TABLE IAVERAGED LABELS OF THE GRAPH G3

0 WITH EXTRACTED FIRST TWO COMPONENTS

in Fig. 2. SinceV0 = f1; 2g, the vertex set ofG30 is

V = f(1; 1; 1); (1; 1; 2); (1; 2; 1); (1; 2; 2);

(2; 1; 1); (2; 1; 2); (2; 2; 1); (2; 2; 2)g:

Consider now only vertices(1; 1; 2) and (1; 2; 1). The local picturearound vertices these vertices is shown in Fig. 7(a). This part ofa graph exhibits a symmetry in the sense that the labels of thesymmetric parts (the axis of symmetry is dashed and dotted lineXX0)are identical when the second and third elements of the labels arepermuted (the first component is always the same). In the extractedgraph the first component of edge labels is preserved, and the pairs ofsymmetric labels have equals weights, so symmetric edges have equaltwo-component labels. In other words, if we consider the stream ofthe first component labels, then from the statistical analysis standpointit is immaterial if the Markov source is in statev(k) = (1; 1; 2) orv(k) = (1; 2; 1). We can consider these states equivalent and mergethem. Generally speaking, if we imagine the vertices of the originalgraph as points in someN -dimensional space, then in the case ofextractingjM j labels from the original graph, the explained mergingcan be understood as a projecting originalN -dimensional vertexspace to thejM j-dimensional vertex space. It is important to notethat by this merging, due to its homomorphic nature, informationabout the constraint in otherN � jM j dimensions is lost.

The following theorems explain the graph symmetry in moreformal way.

Let the sofic systemS be represented by the deterministic graphG = (V;E; �). Consider a partitionP of the vertex setV , V =V1[ � � � [VL, into disjoint subsets. Denote byDj(vi) the number ofall outgoing edges from vertexvi, vi 2 Vi, towards the vertices ofthe subsetVj , and byDT

j (vi) the number of all edges incoming tovi from the vertices of the subsetVj , i; j = 1; � � � ; L.

If there exists a partitionV1 [ � � � [ VL of the vertex setV for thegraphG such that for any two verticesui and vi from a subsetViwe haveDj(vi) = Dj(ui) = Dij andDT

j (vi) = DTj (ui) = DT

ij .That is, the graphG is called symmetric (with respect to the givenpartition), and vertices from the same subset are called equivalentwith respect to the given partition if for any two vertices from thesame subset the numbers of edges going towardVj /coming from thevertices ofVj /are equal. An example of symmetric graph is shownin Fig. 7(b).

If we write the right and left eigenvector of graphG in the form

ppp = [ppp1 ppp2 � � � pppL]; qqq = [qqq1 qqq2 � � � qqqL]

(a)

(b)

Fig. 7. (a) Outgoing edges from(1; 1; 2) and(1;2; 1) in graphG30 (G0 is

from Fig. 2). (b) An example of the symmetric graph withL = 3 subsetsof equivalent states.

where the vector elementspppi = [p(vi)]v 2V andqqqi = [q(vi)]v 2V

correspond to the eigenvector elements of the verticesvi from thesubsetVi, then the following proposition holds.

Proposition 1: If the graphG is symmetric with respect to thegiven partition, then all verticesv inside the same subsetVi have thesame eigenvector elements, i.e.,

8i; 1 � i � L and8v 2 Vi ) p(v) = pi andq(v) = qi: (4.1)

Proof: See Appendix B.

Now, the eigenvectors can be written in the form

ppp = p1 � 111jV j p2 � 111jV j � � � pL � 111jV jT

qqq = q1 � 111jV j q2 � 111jV j � � � qL � 111jV jT

(4.2)


Fig. 8. The graph representative of the graph form Fig. 7 (b).

wherejVij represents the cardinality of the setVi and111t is the rowvector, of the lengtht, with t ones(1). In fact, it is convenient todefine new eigenvectorspppR = [p1 � � � pL]

T and qqqR = [q1 � � � qL]T

and a matrixDDDR = [Dij ]0�i;j�L. The matrixDDDR as an adjacencymatrix defines the graphR, R = R(G), or graph representative,which is obtained from the graphG in the following manner. First,from every vertex subsetVi, i = 1; � � � ; L, one vertex representativeis chosen, which will be labeled byi, and after that, all edges outgoingfrom this vertex toward the vertices of the subsetVj should beterminated at the representative of the subsetVj , labeled byj. For thegraph in Fig. 7(b) with adjacency matrixDDD = [D(u; v)]u;v2V (4.3)

DDD =

1 2 21

11 1 11 1 1

DDDR =1 2

11 1 1

(4.3)

the graph representativeR = (I; F; �) shown at Fig. 8 is obtained,with the adjacency matrixDDDR = [Dij ]1�i;j�L (also given in (4.3)).

If the labels of the edges outgoing from the vertex representativesremain as in the graphG, then the graphR will be deterministicif the original graph is deterministic. So, as a direct consequence ofProposition 1, the following lemmas hold.

Lemma 2: Capacities of the sofic systems represented by the graphG with L subsets of equivalent vertices, and one represented by thegraph representatives of the graphG, R(G), are equal.

Lemma 3: If two graphs are symmetric with respect to a givenpartition of their vertex sets, then the sum of these two graphs willbe symmetric with respect to the same partition.

Let the vertices of the graphG = (V;E; �) beV = f1; � � � ; jV jg.The vertices of theN th Kronecker power of the graphG, GN , aredenoted by orderedN -tuples vvv = (v0; v1; � � � ; vN�1), vi 2 V ,0 � i � N � 1. Form the partition (denoted byPN ) of thevertex set of the graphGN = (V N ; F; �) in the following way.The subset denoted byV (v0; v1; � � � ; vN�1), wherevi;� vj for all0 � i � j � N � 1, embraces all vertices of the graphGN , whoseN -tuple label can be obtained by some permutation of theN -tuplevvv = (v0; v1; � � � ; vN�1).

Lemma 4: The graphGN is symmetric with respect to the formedpartition PN .

Proof: The proof is given in the Appendix C.

The number of subsets of the graphGN formed by the partitionPN is equal to the number of all combinations with repetition ofjV jelements takenN at a time (this number is denoted byCjV j�N ). So,the number of vertices of the representativeR = (I; F; �) is

jIj = CjV j�N =jV j+N � 1

N(4.4)

which is much smaller then the number of vertices in the graphGN ,jV jN , especially for largejV j andN . Moreover, the representative

of GN can be obtained by the following recursive procedure, whichcan be easily programmed on a computer:

R(G) = G

R(Gi) = R(GR(Gi�1)); i = 2; � � � ; N � 1:(4.5)

V. SPECTRAL ANALYSIS OF SEQUENCES

GENERATED BY THE GRAPH COMPOSITION

As can be seen, the constraint of the vector sequencefaaa(k)g,where the constituent sequencesfa(k)i g, i = 0; � � � ; N � 1, areindependent, can be modeled by the Kronecker product of the graphsgenerating these sequencesG = 0�i�N�1(Gi). The capacity of thevector sequence is equal to the sum of capacities of the constituentsequences. However, much more important and practical is the casewhen the generation of constituent sequences is not independent. Sucha case can be modeled by adding or by removing some edges in thegraphG. The addition of new edges will increase the capacity, whilethe removal of any edge will decrease the capacity with respect tothat of the graphG. In both cases the composite graph can be writtenasG = 0�i�N�1(Gi)+J , where the graphJ defines the commonconstraint, and may contain both positive and negative edges. ThegraphG = (V;E; �) has jV j = 0�i�N�1 jVij vertices and thealphabet ofN -dimensional vectorsA = f[ai]0�i�N�1jai 2 Aig,whereAi are the alphabets of sofic systemsSi, given by the graphsGi = (Vi; Ei; �i). If the graphG is symmetric with respect to somepartition of setV , then, according to Lemma 3, the graphJ is alsosymmetric with respect to the same partition. Consequently, the graphJ can be represented in the formJ = 0�i�N�1(Ji). Moreover,because of the independence of the constraint on the track ordering,it will be J0 = J1 = � � � = JN�1, i.e., J = JN0 .

Suppose, now, that all graphsGi, 0 � i � N � 1, participating inthe constraint are identical. The spectra of the sequences on all tracksare identical too, so it is sufficient to extract the first component ofthe vector label. The resulting graph of interest is

H = Hf0g(G0 GN�10 + J0 J

N�10 ):

According to Lemma 4,GN�10 andJN�1

0 are symmetric, and sincetheir labels are not of interest they can be replaced by their unlabeledrepresentativesH;(R(GN�1

0 )) andH;(R(JN�10 )). In this way we

considerably decrease the number of vertices. So, for the spectrumevaluation we need the graph

H = G0 H;(R(GN�10 )) + J0 H;(R(JN�1

0 )):

Let us explain the procedure of obtaining the graphH by thefollowing example. The constituent graphs are shown in Fig. 9(a).

Besides the graphG0 there exist only two graphsG1 (thealphabets of sofic systems should not be the same), whileJ

possesses only one negative edge. The Kronecker powerG21

is a symmetric graph with respect to the partition shown inFig. 6(b), and in the composition withG0, instead ofG2

1, itsgraph representativeR = R(G2

1) can be written. In fact, sinceonly the first label component is of interest, the unlabeledgraph R, K = H;(R) can be taken (Fig. 9(c)). The graphH(Fig. 9(d)) represents the composition ofG0 and K, and containsthe edge ofJ , but with a positive sign. The resulting graph will beobtained simply by removing the loop at vertex(2; 2; 2).

As another example consider calculation of the cross-spectrumbetween sequences in the first and second track of a four-track


(a)

(b)

(c)

(d)

Fig. 9. An example of obtaining the extracted graph from the composite symmetric graph.

constraint given by the graphH = H = G0G31�J0J1J

3. Allthe steps explained in the previous example remain the same, exceptthat the extracted graph needed to be fond in the third step is

H = G0 G1 H;(R(G4�20 ))� J0 J1 H;(R(J

4�20 )):

For the general case, when there areL groups of different graphs,the procedure for the spectrum evaluation is given by the followingtheorem.

Theorem: Let the Markov source assigned to the graphG of theform

G =

L�1

l=0

Gt

l+ J =

L�1

l= 0

Gt

l+

L�1

l=0

Jt

l;

L�1

l=0

tl = N (5.1)

generate the maxentropic vector sequencefaaa(n)g, with spectrum�aaa(f), and let the vector sequencef�(n)g be generated by themaxentropic Markov source assigned to the graphQ with averaged


Fig. 10. Reduction of the vertex numbers ofG0 H;(R(GN�10 )) with

respect toGN0 (L = 1).

edge labels of the extracted graphH

H =

L�1

l=0

Gl H; R Gt �1l +

L�1

l= 0

Jl H; R Jt �1l

(5.2)where the representative of each graphGt �1 is formed usingpartition P t �1 of its vertex setj = 0; � � � ; L � 1. If f�(n)g is thevector sequence extracted fromfaaa(n)g by the set

(d)M = fmj j0 � j � L� 1g

= f0; t0; t0 + t1; � � � ; t0 + t1 + � � �+ tL�2g

(i.e. �(n) = EMMM(aaa(n))), then

��(f) = ��(f) = EMMM(�aaa(f)):

Proof: The proof follows from the the given lemmas straight-forwardly, and therefore is omitted.

The number of the vertices of the graphH = (W; I; (�; �)) is

jW j =

L�1

l=0

jVlj �jVlj + tl � 2

tl � 1(5.3)

which is much less then the number of the graphG, which is

jV j =0�j�L�1

(jVj jt ):

For example, Fig. 10 plots the ratio between vertex numbers of theextracted and original graphs versus the vertex number of constituentgraph forL = 1 and different values ofN . As can be seen, thesuggested method becomes more efficient with increasingN andjV j,i.e., in the cases of complex constituent graphs and many recordedtracks. This reduction can be even greater, because the additionalreduction of the extracted graph vertex number is possible by vertexmerging [11].

VI. A PPLICATION TO TWO-DIMENSIONAL RLL CODES

Two-dimensional run-length-limited (RLL) codes have been in-troduced recently by Marcellin and Weber in [14], and suggestedfor systems with peak detectors [20] and parallel recording toN

tracks [15, pp. 219–224]. It is known that the output of the(d; k)modulation encoder enters the precoder where every1 followed bym zeros (0) (this sequence is called the phrase of lengthm + 1)causes the occurrence of the sequence ofm + 1 consecutive likesymbols (from alphabetf�1;+1g) in the channel stream at theoutput of the precoder [17], [24]. The(d; k;N) constraint means that

Fig. 11. A representation of thed-constrained RLL graphs.

Fig. 12. GraphJ for (d; k; N) constraint.

the phrase lengths are less than of equal tod + 1, while parameterk+ 1 determines the maximal interval between the beginning of thephrases, i.e., transitions in different channel streams. So, we haveN equal sofic systems with constrained minimal allowable phraselengths in every track, and the common constraint of the currentlyshortest phrase maximal length. Precisely, if we define the streamfs(n)g as

s(n) =

N�1

i=0

a(n)i (6.1)

where_ represents the logical “or” function, andfa(n)i g is the streamin the tracki, 0 � i � N � 1, then the commonk-constraint givesk + 1 as the maximal allowable length of the phrases infs(n)g.

A. Graph of(d; k;N) Constraint

If the d-constrained RLL graph,G0 is given in the form as inFig. 11 then the graphG of the whole(d; k;N) constraint can beobtained asG = GN

0 +J , whereJ = JN0 is a simple negative graphshown in Fig. 12.

Since the spectra are equal in all tracks, we will make the graphHf0g(G

N0 +JN0 ), and apply the procedure for spectral analysis of the

extracted graphs. The total number of vertices of the representativeR(GN�1

0 + JN�10 ) is

jIj =

d+1

i=1

Ci�N�2 =

d+1

i=1

N � 2 + i� 1N � 2

+ k � d

=N � 1 + d

N � 1+ k � d (6.2)

and the number of vertex of the variable-thicknes graph

G0 H;(R(GN�10 ))

jW j is

jW j = (k + 1) �N � 1 + d

N � 1+ k � d (6.3)

The derivation of these numbers is given in Appendix D.From (6.2) and (6.3) we can see that our approach leads to the

graphs of significantly smaller number of vertices then the graphsobtained by Marcellin and Weber [14], that have(d+1)N +(k� d)vertices.


TABLE IITRANSITION TABLE OF THE GRAPH REPRESENTATIVE FOR(2,4,3) CONSTRAINT

Fig. 13. Graph ofd-constrained sofic system.

Moreover, our approach is more general than one in [14] because

1) it is applicable not only to the(d; k;N) constraint, but to anytrack-invariant vector constraint;

2) it enables the calculation of both the channel capacities andspectra, while [14] allows to calculate only the channel ca-pacity. This is because the graph compression we performpreserves the information on edge labels, while the the ap-proach in [14] is based on the graph adjacency matrices.

To illustrate the importance of the reduction based on graphsymmetry, let us take an example when the sequence constraints onall of N = 8 tracks are the same(L = 1) and given by parametersd = 1, k = 3. The graphGN

0 + J has jV j = (k + 1)8 = 67536vertices. The analysis of such a huge graph is impossible. Afterreduction using the graph symmetry the vertex number isjIj = 840,but after a series of consecutive vertex merging, it can be shownthat we obtain a graph with onlyjW j = 40 vertices. For example,Table II gives the transition table of the extracted graph (i.e., graphG0 R(G2

0) + J0 R(J20 )) for (2; 4; 3) constraint, and (6.4) givesthe corresponding adjacency matrix:

D =

11

1 11

1 11 2 1

11 13 1

1 3 3 11 3 3 11 3 3

(6.4)

In (d; k;N) constraintk-constraint can be strengthened so thatk < d. The composite graph can be analogously obtained as for

Fig. 14. Critical difference of lengths of the longest and shortest phrases.

d < k case, starting from the constituent graphG0 of the form shownin Fig. 13, and removing certain vertices from the vertex set of thegraphG0 GN�1

0. The strongly connected part of the remainder

represents the graphG.Now we will explain briefly how to select the vertices of the graph

representativeR(GN

0 +JN0 ) that should be removed, by consideringthe bounds of current phrase lengthst (Fig. 14). The vertex set ofR(GN

0 + JN0 ) is

V = fvvv j vvv = [vi]0�i�N�1 ^ 8i) vi � d+ 1g

and vi corresponds to the current phrase length in some track. Allphrases are shorter thand+1(tmax � d+1), but always at least onephrase length must be less than or equal tok+ 1. Denote the lengthof this phrase bytmin. The condition for starting a new phrase istmax = d+1. Fig. 14 shows the moment when there are no conditionsfor starting a new phrase. If the moment when the condition forstarting the new phrase to be satisfied is too far, it may happen thateven the shortest phrase becomes longer thank + 1, which violatesthe k-constraint. Thus the longest phrase should achieve the lengthd + 1 before the shortest phrase achievesk + 1, or according toFig. 14, the conditiony � x must always be satisfied.

Since the constituent graph vertex labels represent the currentphrase lengths in the given track, for graph representativeR(GN

0 )

it will be v0 = tmin and vN�1 = tmax, and for the elements ofN -tuple labels it holds that

vN�1 � v0 + (d� k): (6.5)

The vertices whose labels do not satisfy this condition should beremoved fromV . The strongly connected part of the remainder is


TABLE IIIGRAPH TRANSITION TABLE OF (3; 2; 3) CONSTRAINT

the graphR. As an example, Table III gives the extracted graphtransitions and edge labels for(d = 3; k = 2; N = 3) constrainedsystem.

B. Examples of Spectra

When the graph describing the generation of the binary(d; k;N)-constrained sequencefaaa(n)g is constructed, we can easily obtain thegraph of the precoded sequencefbbb(n)g. The recorded signal is

xxx(t) =n2Z

bbb(n) � g(t� nT )

whereT is a clock period, andg(t) is a modulated waveform pulseof durationT . The power spectral density of the signalxxx(t + �),�X(f) is [3]

�x(f) =jG(jf)j2

T� �(f) (6.6)

whereG(jf) is the Fourier transform ofg(t), �(f) is the spectrum ofthe sequencefb(k)g, and� is a random variable uniformly distributedover the interval[�T=2; T=2) (which is introduced because of thecyclostationarity of the signalxxx(t)).

In the Figs. 15 and 16 the spectra of the recorded signalx0(t) (thespectra are the same for all tracks) are shown for several values ofparametersd, k, andN [23, Ch. VIII]. The per-track capacities arealso given. It is assumed that the isolated pulses are rectangular (sothat G(jf) = sin (�Tf)=(�Tf)), and that the recorded sequencesare maxentropic.

The first property of the maxentropic(d; k;N) sequence spectrum(that can be seen in Fig. 15) is that increasing the number of parallelywritten tracks increases the low-frequency content of the recordedsignal. This increase is more pronounced for sequences withd � k.Further, the increase of the number of tracks decreases the amplitudeof the peak characteristic for(d; k) constraint, and shifts it towardthe lower frequencies.

Since in recording channels large power concentrated at low fre-quencies cannot be tolerated, a possible solution can be a multitrackconstraint composed of intertrackk-constraint and the compositedand dc-free intratrack constraint.

Fig. 16(a) illustrates the behavior of the spectra for fixedd, whenk changes. Smallerk means a more pronounced spectral peakand narrower spectrum. This is a consequence of the fact that forsmall k, the channel capacity is small and that deterministic part

in the recorded sequence is dominant. However, when the numberof parallely written tracks is larger it does not allow the rapidnarrowing the spectra withk decreases. Similar behavior with thespectrum narrowing can be also observed whenk is fixed anddgrows (Fig. 16(b)).

VII. SUMMARY

A procedure for the spectral analysis of the constrained maxen-tropic vector sequences used in multitrack recording systems is givenin this correspondence. We have started from the vector labeled finitegraphG = (V;E; �) of the form

G =0�j�L�1

(Gjtj) + J(t0 + t1 + � � �+ tL�1 = N)

where graphs

Gj = (Vj ; Ej ; �j); j = 0; � � � ; L� 1

give the constraints on isolated tracks, and

J = G =0�j�L�1

(Jj)t

is the graph describing their mutual dependence.Assuming that the common constraint is track-invariant, it has been

shown that the vertex set of each Kronecker’s powerGt

j (alsoJt

j )can be divided into classes of equivalent vertices. This feature ofsymmetry allows the use, instead ofG

t

j , of the graphs

Hj = Gj R(Gt �1

j )

whereR(Gt �1

j ) is the so-called representative of the graphGt �1

j .In the representative, all vertices from the same subset of equivalentvertices are merged into one vertex, the�-labels of the edges areignored and replaced by�-labels that point out the numbers ofoutgoing edges from an initial subset of equivalent vertices towardthe vertices from the same terminal subset of equivalent vertices.Through a series of lemmas and theorems, it has been shown that forspectral analysis, instead of the graphG, we can use the extractedgraphH = (W; I; (�; �)) of the form

H =0�j�L�1

(Hj) +0�j�L�1

(Ij)


(a)

(b)

Fig. 15. Spectra of recorded signal modulated by maxentropic(D; k; N)-constrained sequences for different track numberN . (a) d < k: (b) d > k:

whereIj = Jj R(Jt �1

j ). Each edgef = (f0; � � � ; fL�1) of thegraphH has the label of the form

(�(f) = [�0(f) � � � �L�1(f)]; �(f)):

The parameter�(f) determines the transition probability over theedgef in the assigned maxentropic Markov source. Further compu-tational saving has been achieved by using the difference equationmethod for autocorrelation function calculation [22] applied onthe graphQ1 with averaged edge labels. All labels of the edgesf1; f2; � � � fD(u;v) from u to v, in the graphH, are replaced by theiraverage�-label, i.e., this edges are merged into one single edgeiwith the label

�(i) = (�(f1)�(f1)+ �(f2)�(f2)+ �(fD(u;v))�(fD(u;v))=D(u; v):

It has been proved that the maxentropic Markov sources assignedto the graphH and to the obtained graphQ1, with averaged edge

(a)

(b)

Fig. 16. Spectra of recorded signal modulated by maxentropic(d; k; 4)-constrained sequences. (a) Dependence onk: (b) Dependence ond:

labels, will generate sequences with equal autocorrelation functionsfor k 6= 0.

We have demonstrated that the extracted graphH fully determinesthe spectrum of the sequence generated by the graphG, and that thevertex number ofH is considerably smaller than that ofG. For thegivenL and tl the ratiojW j=jV j decreases with increasing numberof vertices of constituent graphsGj . Using the extracted graph is alsoconvenient because the procedure of obtaining the representatives hasa recursive character and can be programmed on a computer.

Finally, for the sake of the illustration, we apply the suggestedmethod to the two-dimensional runlength-limited sequences.

APPENDIX APROOF OF LEMMA 1

Here, we will prove the lemma for graphs with scalar labels, andfor k 6= 0. The remainder of the proof, fork = 0 and for vectorlabels, is straightforward and left to the reader.


Let us denote bys(u), u 2 V , the stationary probabilities ofthe ergodic Markov source assigned to the nondeterministic graphG = (V;E; �) and byE(u; v) the set of edges fromu to v u; v 2 V .Then

Efa(n)a(n+k)g

=u2V

s(u)a2A b2A

a � bPr fa(n) = a; a(n+k) = b j v(n) = ug

=u2V

s(u)a2A

a

v2V

Prfa(n) = a; v(n+1) = v j v(n) = ug

�w2V

Pr (v(n+k) = w j v(n+1) = v)

�b2A

b

y2V

Prfa(n+k) = b; v(n+k+1) = y j v(n+k) = wg

(A.1)

whereA is a subset of real numbers. Because of stationarity we willomit the time indices in the expressions for transition probabilities,and accept the notation

Prfv(n+k) = wjv(n+1) = vg = P(k�1)(wjv):

Then the expression for the autocorrelation function becomes

Efa(n)a(n+k)g =u2V

s(u)v2V e2E(u;v)

�(e)P (eju)

�w2V

P(k�1)(wjv)

y2V f2E(w;y)

�(f)P (f jw):

(A.2)

Multiplying and dividing at the same time byP (vju), we have

Efa(n)a(n+k)g

=u2V

s(u)v2V

e2E(u;v)

�(e)P(eju)

e2E(u;v)

P (eju)P (vju)

�w2V

P(k�1)(wjv)

y2V

f2E(w;y)

�(f)P (f jw)

f2E(w;y)

P (f jw)P (yjw)

(A.3)

The last equation holds since for everyu andv, u; v 2 V

P (�ju) =e2E(u;v)

:

Let us now replace all edgese, from E(u; v) with a single egdefwith a two-component label(�(f); �(f)), where�(f) is number ofedgese with label�(e) (as a result, we get a two-component labeledgraphH = (V; F; �)). In the maxentropic case, all edges betweenu

and v have the same probabilities (2.3), so that

e2E(u;v)

�(e)P (eju)

e2E(u;v)

P (eju)=

e2E(u;v)

�(e) 1�

p(v)p(u)

D(u; v) 1�

p(v)p(u)

=f2F (u;v)

�(f)�(f) 1�

p(v)p(u)

D(u; v) 1�

p(v)p(u)

=1

D(u; v)f2F(u;v)

�(f)�(f) (A.4)

whereF (u; v) is a set of edges fromu to v in the graphH.

On the other hand, since in graphQ1 there is only one edge from

u to v, the autocorrleation function can be expressed as

Efa(n)a(n+k)g =u2V

s(u)v2V

�(i) � P (vju)

�w2V

P(k�1)(wjv)

y2V

�(j) � P (yjw) (A.5)

where i is this single edge fromu to v (i.e., fig = E(u; v) and

fjg = E(w;y)).

Comparing (A.3) and (A.5) we conclude that edge labels of the

graphQ1 are given by the righthsnf side of (A.4), which completes

the proof.

APPENDIX B

PROOF OF PROPOSITION 1

Let the graphG(S) be symmetric with respect to the givenpartition P to L subsets of vertex setV , and v denote vertices inthe subsetVi, i = 1; � � � ; L. Show that the elements of the righteigenvector corresponding to the verticesvi are equal, i.e., that

8i; i = 1; � � � ; L ^ 8v 2 V ) p(v) = p ^ q(v) = q: (B.1)

Denote byN (k)(v) the number of distinct subsequences (ofS) ofthe lengthk, emanating from vertexv. If DDD = [D(u; v)]u;v2V is theadjacency matrix of the graphG, then

N(k)(v) =

u2V

D(u; v) �N (k�1)(u): (B.2)

This is a system ofjV j linear homogenous difference equationswith constant coefficients, and therefore the solution is a linearcombination of exponentials�k. To solve it, a solution in the formN (k)(v) = �� p(v) is assumed, to obtain

�k � p(v) = �

k�1

u2V

D(u; v) � p(u) (B.3)

or ppp = ��1DpDpDp. So, the number of subsequences of the lengthk,emanating from vertexv is proportional just to the element of theright eigenvector corresponding tov. Prove, now, that the numbers ofsubsequences of the lengthk, N (k)(v), emanating from the verticesv, are equal for allv 2 Vi, i = 1; � � � ; L. By using mathematicalinduction we have the following.

—From the definition of the symmetric sofic systems it followsthat for every two-vertexu andv from the same subsetVi andfor every subsetVj it holds that the numbers of outgoing edgesfrom these vertices incoming to the vertices of the subsetsVjare equal, i.e.,Dj(v) = Dj(u) = Dij . The total number of thepaths of length1 (i.e., edges) outgoing from vertexv is

N(1)(v) =

L

j =1

Dj(v) =

L

j = 1

Dij = N(1)i (B.4)

and it is equal for allv 2 Vi. So, the proposition is true fork = 1.

—Suppose now that the proposition is true for superscript equalsk, i.e., that for every two verticesu andv from Vi N

(k)(u) =

N (k)(v) = N(k)i , and prove that from this follows that the

proposition is true for superscript equalsk + 1. Since

N(k+1)(vi) =

w2V

D(viw) �N(k)(w)

=

L

j =1w2V

D(�i; w) �N(k)(w) (B.5)

and since on the base of the assumption for thek = k, it holdsthatN (k)(wj) = N

(k)j . Since for any two verticesv 2 Vi and

w 2 Vj of a symmetric graph it holds thatD(v;w) = Dij , it


follows that

N(k+1)(v) =

L

j = 1

Dij �N(k)j = N

(k+1)i (B.6)

so N (k+1)(v) is independent onv, and equal for all verticesinside the same subset. SinceN (k)(v) = N

(k)i andN (k)(v) =

�� p(v), it follows thatp(v) = pi = ��1N(k)i for all v 2 Vi.

Analogously we can prove the equality of the left eigenvectors.

APPENDIX CPROOF OF LEMMA 4

Here, we will prove that the graphGN is symmetric with respectto the partitionPN given in Section IV.

Proof: Since the adjacency matrix of the graphGN is

DDD(GN) = [D(uuu; vvv)]u;vvv2V

=

N�1

i=0

D(ui; vi)(C.1)

the number of edges emanating from the vertex

uuu = (u0; u1; � � � ; uN�1)

and terminating at vertex

vvv = (v0; v1; � � � ; vN�1)

is

D(u0; v0) �D(u1; v1) � � � �D(ui; vi) � � � �D(uj; vj) � � �

�D(uN�1; vN�1):

Obviously, this product will remain unchanged if any two elements,ui and uj or vi and vj , are permuted. So the numbers of edgesemanating from the vertices whoseN -tuple labels can be obtainedby permutation of the fixed vectoruuu are equal. Analogously, we canprove the equality of the numbers terminating at some vertexvvv. Fromthis it follows thatGN is symmetric.

APPENDIX DDERIVATION OF THE NUMBER OF VERTICES

IN THE TWO-DIMENSIONAL RLL GRAPHS

Since the verticesd+1; � � � ; k+1 of the graphG0 are equivalentand can be merged, and because of graphGN�1

0 symmetry, theelements of vertex labels of the reduced unlabeled graph

R(GN�10 + J

N�10 )vvv = (v1; � � � ; vN�1)

can be reduced to the values1 � v1 � k+1 and1 � vi � d+1 fori = 2; � � � ; N � 1. For the vertex labels of the representative it holds

8i; j; 2 � i; j � N � 1 ^ i � j ) vi � vj

and v1 � v2 whenv1 � d. First, we find the vertex number of thereduced representativeR(GN�1

0 +JN�10 ). Since1 � vi � d+1 fori = 2; � � � ; N � 1, the number of vertices with leading one(v1 = 1)

in the (N � 1)-tuple label is equal to the number of a combinationwith repetition ofd+1 elements takenN�2 at a time(Cd+1�N�2).Sincev1 � v2 � � � � vN�1, the number of vertices withv1 = 2 isequal to the number of the combination with repetition ofd elementstaken N � 2 at a time (Cd�N�2). The number of vertices withv1 = 3; � � � ; d+ 1 can be obtained analogously. Finally, for verticeswith v1 � d+ 1 the other elements of the label arevi = d+ 1, for2 = 1; � � � ; N �1, and number of such vertices isk�d. So, the total

number of vertices of the representativeR(GN�10 + JN�10 ) is

jIj =

d+1

i= 1

Ci�N�2 + k � d =

d+1

i

N � 2 + i� 1N � 2

+ k � d

=N � 1 + d

N � 1+ k � d (D.1)

and the number of vertex of the extracted graphG0H;(R(GN�10 )),

jW j is

jW j = (k + 1) �N � 1 + d

N � 1+ k � d : (D.2)

ACKNOWLEDGMENT

The author would like to thank Olgica Milenkovic and StevenMcLaughlin for their helpful comments.

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