Moment Balancing Templates for Spectral Null Codes - IEEE Xplore

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Abstract—The generalized moment balancing templates for spectral null codes are investigated in this paper. A new approach based on the insertion of a ...
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 8, AUGUST 2010

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Moment Balancing Templates for Spectral Null Codes Ling Cheng, Hendrik C. Ferreira, and Khmaies Ouahada

Abstract—The generalized moment balancing templates for spectral null codes are investigated in this paper. A new approach based on the insertion of a determined number of vectors at determinate indices of a spectral null code word, is found to balance the moment of the code word. Moment balanced code words preserve the spectral null properties and can guarantee the correction of one insertion or one deletion error. As compared to the universal template for an uncoded sequence, the redundancy is determined by the structure of the original spectral null code, however, it can be kept small. The new approach involves the applications of the optimized moment balancing template and nonbinary enumeration of the balancing moment values.

II. MOMENT BALANCING TEMPLATE OF SPECTRAL NULL CODES A. Preliminaries A spectral null code has zero power at rational submultiples of the symbol frequency [4]. Let a spectral null code word

.. .

.. .

.. .

Index Terms—Extended codes, insertions/deletions, moment balancing template, number theoretic codes, spectral null codes.

have a length equal to the product of two positive integers and , where is a prime. Let denote the symbol frequency. If

I. INTRODUCTION

(1)

I

N [1], the authors presented a single insertion/deletion correcting scheme, named the moment balancing template, to encode an arbitrary sequence in order to obtain a code with a single insertion/deletion error correcting capability. As shown in [2], these types of synchronization errors occur in magnetic recording systems. It prompted our further study on moment balancing templates for different types of constrained codes. For a constrained sequence, nevertheless, in most cases it demands a specific template to encode the sequence in order to preserve the original channel constraint properties. In [1], the authors pre-free sequence and the template sented the template for a for a ( )-sequence with and . In this paper, we present the moment balancing template for further spectral null codes [3]. A new approach, which involves the optimized template and nonbinary enumeration is introduced. The new template is in a vector format based on the vector representation of the spectral null code as introduced by Gorog in [4]. The paper is organized as follows. The generalized moment balancing template for spectral null codes is presented in Section II. The redundancy of the new templates is discussed in Section III, and we also show the relationship between the moment spectra distribution of a code and the optimized moment balancing template. We conclude the paper with Section IV.

Manuscript received June 10, 2009; revised March 28, 2010. Date of current version July 14, 2010. This paper was presented in part at the IEEE Information Theory Workshop, Porto, Portugal, May 2008. The authors are with the Department of Electrical and Electronic Engineering Science, University of Johannesburg, Auckland Park, 2006, South Africa (e-mail: [email protected]; [email protected]; [email protected]). Communicated by H.-A. Loeliger, Associate Editor for Coding Techniques. Digital Object Identifier 10.1109/TIT.2010.2050926

the power spectral density of this code has spectral zeros at the , where and frequencies . For a binary sequence , the first-order moment function is defined as follows: (2) The Varshamov-Tenengolts codes [5] consist of all binary -vectors satisfying (3) are fixed nonnegative integers and where and . These codes were introduced to correct one single asymmetrical error. In [6], Levenshtein noted that by setting , these codes are single insertion or deletion error correcting codes. Therefore, for a binary code of length , the moment balancing template is used to map each code word into a distinct sequence whose moment is congruent to some fixed integer modulo and some number . Clearly, we assume that . For a convenient implementation in practice, we only allow the symbols of to appear in at determinate positions. For the code having a random moment distribution, a universal moment balancing template is defined in [1]. The number of the is lower bounded by . moment balancing bits For the code to have a certain narrower range of the moment , we can implement an optimized moment balancing value

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 8, AUGUST 2010

template, which introduces less redundancy than that of the universal template. For a universal template, the balancing symbols are independently making a contribution to the moment value. However, in some cases to maintain the property of a constrained code word, the redundancy of the moment balancing template may need to be larger than that of the universal template.

Each vector of the balancing bits has the format of or its cyclic shifts. denote the first-order moment value of the sequence Let with and . Let denote the first-order moment value of the sequence , which is the same as except that is replaced by . Here satisfies (8) but with and . The following lemma describes the relationship of and . Lemma 2:

B. Template According to (1), it follows naturally that a spectral null code has a vector format as follows. For , that satisfies (1), where we have . We consider to add bits of balancing bits to each vector of . The balancing bits can also be rep, resented in a vector format such as . Therefore, the moment balwhere anced sequence can be represented as , . If represents the where in , we have indices of the symbols from for some . For the redundant symbols in , named as the moment balancing and bits, we have . Clearly, and . Let

(9) Proof: When is replaced by changed. According to (5), we have

, in (6), only

is

(10)

The following lemma shows an interesting property of spectral null codes. Lemma

3:

For

a

spectral null code word , which has length equal to the product of two positive integers and , we have for for

(4)

is odd or is even is even and is odd (11)

(5)

where

and indicate the contribution of the code Then, bits and the balancing bits, respectively, to the moment of . In particular,

(12)

(6) We assume that columns.

vectors of the balancing bits are inserted at

Proof: For a spectral null code word, given column, we have

1’s at each

Lemma 1: If the balancing bits satisfy (7) then the balanced code preserves the spectral null property as in the original code. Proof: According to (1), the proof is straightforward.

(13) This proves Lemma 3. The indices of the balancing bits in the template have an explicit representation as follows:

To simplify the calculation, in this paper, we only consider (8)

(14) for bits

. The absolute index values of the balancing in a serial moment balanced sequence can be

CHENG et al.: MOMENT BALANCING TEMPLATES FOR SPECTRAL NULL CODES

represented as follows:

prime. Therefore, according to (17), by choosing the by ancing bit vectors, we can alter the value of .. . .. .

.. .

.. .

.. . .. . .. . .. . .. . .. . .. . .. . .. .

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.. . .. . .. . .. . .. . .. .

.. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

.. . .. . .. . .. . .. . .. . .. .

(18) and (19) is greater than the length of , it becomes a single If insertion/deletion correcting code. According to , we have (20) In (13), if length in

.. . .. . .. .

Theorem 1: Let and denote two positive integers and is prime. Let denote a spectral null code with length . If can be moment-balanced by inserting vectors of balancing bits whose values satisfy (8) at the indices in the final code word specified by (14). Proof: Let and denote two different moment balanced sequences from the same , respectively. According to Lemma 1, the balanced code word preserves the property of a spectral null code. From Lemma 2, and by only choosing different vectors of the balancing bits, the difference of the moment values is

and is odd, we only need to insert a prefix of to compensate the deficiency of .

of is taken by a balancing Since the index bit, when the length of is too short to fill the template, such , we can insert dummy 0’s in at as the end of the sequence, which obviously will introduce new redundancy. An alternative approach to reduce the redundancy in this case is to fill more than one spectral null code word into one template. Let denote the number of the spectral null code words of length each filled into one template. According to (18), we have (21) , let dummy 0’s fill the remaining unoccuIf . By this approach, pied indices before the index spectral null code words are protected from one single insertion/deletion error with one moment balancing template. , one solution is to insert Note that, if length another vectors of balancing bits which can satisfy (8) and have indices in as follows: .. . .. .

(15) .. . where We set

and

are the indices of 1’s in and respectively. for all . From (8), we have

.. . .. .

where

. Therefore, we can

(17)

Moreover, according to Lemma 3, we have , except for the case of , since

is

.. .

.. . .. . .. . .. . .. . .. . .. .

(16) Let denote write (15) as

bal-

where

.. . .. . .. . .. . .. . .. . .

.. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

.. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

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Example 1: Here (22) shows the indices of moment balancing . bits for

.. .

.. . .. .

.. . .. .

.. .

.. .

where Let

the

. original

B.

(22)

.. . .. . .. .

.. . .. . .. .

, the sequence, where redundancy of the template for a spectral null code is of that of the universal template. Note that this approach is a typical optimized moment balancing template. The number of the moment balancing bits in the template is reduced, since we apply the property of spectral null codes illustrated by Lemma 3.

.. . spectral

null codeword be with length 22. The length of this codeword is the product of and . As a spectral null codeword having the spectral null at , it satisfies (23) Let underlined bits denote the balancing runs. Inserting the with indices as described in balancing sequence (22), we obtain with length 30, which is the product of and . , we have . The aim of Taking redistributing 0’s in the balancing runs is to obtain . To make up the deficiency , which is , we swap the zero and the one at the indices of equal to 2 and 6, 8, and 24, respectively. As a result of changing into , we have . At the same time, maintains the spectral-null as the original codeword , since property at (24)

III. ANALYSIS AND DISCUSSION A. Redundancy According to (20), for , the redundancy of this moment balancing template on the spectral null codes is lower bounded by (25) . Therefore, as compared In view of (18), we have to the lower bound of the universal template for an uncoded

-Balanced Codes

In [1], the authors proposed the optimized moment balancing template, and in this paper we present two examples of the optimized template, the template for spectral null codes and the one -balancedcodes.The -balancedcodesaresubsetsof for the Levenshtein codes [7]. They satisfy (3) for fixed nonnegative integers and . According to the discussions of the optimized moment balancing template and the applications on the spectral -balnull codes presented in this paper, it is clear that the anced codes are just extreme examples of the optimized moment -balanced codes gives balanced codes. The property of the rise to the reduced number of the balancing bits to 0. IV. CONCLUSION In this paper, we have extended the investigation of the moment balancing template [1] by applying it to spectral null codes in order to implement the systematic encoding of certain number theoretic codes. We have also shown how to use the nonbinary base to enumerate the contributed moment values, as well as how to implement an optimized moment balancing template. We have shown that the redundancy added by our moment balancing templates may be kept small. It makes the application to longer codes useful and attractive for certain practical applications. REFERENCES [1] H. C. Ferreira, K. A. S. Abdel-Ghaffar, L. Cheng, T. G. Swart, and K. Ouahada, “Moment balancing templates: Constructions to add insertion/deletion correction capability to error correcting or constrained codes.,” IEEE Trans. Inf. Theory, vol. 55, no. 8, pp. 3494–3500, Aug. 2009. [2] J. Hu, T. M. Duman, E. M. Kurtas, and M. F. Erden, “Bit-patterned media with written-in errors: Modeling, detection, and theoretical limits,” IEEE Trans. Magn., vol. 43, no. 8, pp. 3517–3524, Aug. 2007. [3] K. A. S. Immink, “Spectral null codes,” IEEE Trans. Magn., vol. 26, no. 2, pp. 1130–1135, Mar. 1990. [4] E. Gorog, “Redundant alphabets with desirable frequency spectrum properties.,” IBM J. Res. Develop., vol. 12, pp. 234–241, May 1968. [5] R. P. Varshamov and G. M. Tenengol’ts, “Correction code for single asymmetric errors.,” Automat. Telemekh., vol. 26, pp. 288–292, 1965. [6] V. I. Levenshtein, “Binary codes capable of correcting deletions, insertions and substitutions of symbols.,” Doklady Acad. Sci. USSR, vol. 163, no. 4, pp. 845–848, 1965. [7] H. C. Ferreira, W. A. Clarke, A. S. J. Helberg, K. A. S. Abdel-Ghaffar, and A. J. H. Vinck, “Insertion/deletion correction with spectral nulls.,” IEEE Trans. Inf. Theory, vol. 43, no. 2, pp. 722–732, Mar. 1997. Ling Cheng received the B.Eng. degree in electrical and electronic engineering from the Huazhong University of Science and Technology, Wuhan, Hubei, China, in 1995. He received the M.Eng. degree in electrical and electronic engineering from the University of Johannesburg, South Africa, in 2005. From 1995 to 2002, he was with the China Mobile Communications Corporation. From 2003 to 2004, he was with Telematic Technologies. He is currently studying toward the D.Eng. degree at the University of Johannesburg. His research interests include power-line communications, error correction coding, and spectral shaping techniques.

CHENG et al.: MOMENT BALANCING TEMPLATES FOR SPECTRAL NULL CODES

Hendrik C. Ferreira was born and educated in South Africa where he received the D.Sc. (Eng.) degree from the University of Pretoria in 1980. From 1980 to 1981, he was a Postdoctoral Researcher with the Linkabit Corporation, San Diego, CA. In 1983, he joined the Rand Afrikaans University, Johannesburg, South Africa, where he was promoted to Professor in 1989 and served two terms as Chairman of the Department of Electrical and Electronic Engineering from 1994 to 1999. He is currently a Research Professor with the University of Johannesburg. His research interests are in digital communications and information theory, especially coding techniques, as well as in power line communications. From 1989 until 1993, he held a “Presidential Award for Young Investigators” Research Grant from the South African Foundation for Research Development. Since 1984, he has been a Visiting Researcher at eight different institutions in the USA and Europe. He has been Principle Advisor for more than 30 postgraduate students. Together with his students, he published more than 200 research papers in international journals and conferences. Dr. Ferreira is a past chairman of the Communications and Signal Processing Chapter of the IEEE South Africa section, and from 1997 to 2006, he was Editor-in-Chief of the Transactions of the South African Institute of Electrical Engineers. He has served as chairman of several conferences, including the International 1999 IEEE Information Theory Workshop in the Kruger National Park, South Africa, as well as the 2010 IEEE African Winter School on Information Theory and Communications.

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Khmaies Ouahada received the B.Eng. degree in electric and electronic engineering from the University of Khartoum, Sudan, in 1995, the M.Eng. degree in electric and electronic engineering from the Rand Afrikaans University, South Africa, in 2002, and the D.Eng. degree from the University of Johannesburg, South Africa, 2009. His research interests include digital communications, error correction coding, and spectral shaping techniques.