low-frequency end. Let fx i g = f::: x;1 x0 ::: x i :::g, x i 2 f;1 1g be a bipolar sequence. The running digital sum z i is dened by z i = ix j=;1 x

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1 Construction of DC-free Codes Using the Fast Hadamard Transform Kees A. Schouhamer Immink, November 7, 2001 Abstract We report on new class of dc-free codes that use the Fast Hadamard Transform (FHT) to evaluate a selection set of candidate codewords. The codeword with the least contribution to the low-frequency content is transmitted. The FHT makes it possible to eciently evaluate the selection set of candidate codewords. Spectral properties of the new codes have been evaluated by computer simulation. 1 Introduction Binary sequences with spectral nulls at zero frequency have found widespread application in optical and magnetic recording systems [1]. Dc-balanced codes, as they are often called, have a long history and their application is certainly not conned to recording practice. Since the early days of digital communication over cable, dc-balanced codes have been employed to counter the eects of low-frequency cut-o due to coupling components, isolating transformers, etc. The running digital sum of a sequence, in short, RDS, plays a signicant role in the analysis and synthesis of codes whose spectrum vanishes at the Kees A. Schouhamer Immink is with Turing Machines Inc, 15 W. Alexanderlaan, 5664 AN Geldrop, The Netherlands. immink@turing-machines.com. 1

2 low-frequency end. Let fx i g = f::: x;1 x0 ::: x i :::g, x i 2 f;1 1g be a bipolar sequence. The running digital sum z i is dened by z i = ix j=;1 x j = z i;1 + x i : It is an elementary exercise to show thatifz i is bounded, the spectral density vanishes at dc [1]. There is a variety of prior art methods for constructing dc-free codes. One simple construction method, called polarity switch, is based on a classical principle, and was rst described by Bowers [2] in Under polarity switch rules, n source symbols are supplemented by one symbol called the polarity bit. The encoder has the option to transmit the (n + 1)-bit word without modication or to invert all (n + 1)symbols. The selection between the two alternative translations is made in such away that the running digital sum after transmission of the new word is as close to zero as possible. The polarity bit is used at the decoder site to identify whether the transmitted codeword has been inverted or not. Spectral properties of the polarity bit code are described in [1, Chapter 10]. The publications by Fair et al. [3] and Immink & Patrovics [4] on guided scrambling stimulated new research in this area. Guided scrambling is a member of a larger class of related coding schemes called multi-mode codes. In multi-mode codes, each source word may be represented by amember of a selection set consisting of L codewords. The encoder transmits that codeword having a minimum contribution to the low-frequency content. Clearly, the computational load for evaluating the selection criterion puts a practical limit to the size of the selection set. Note that the decoder does not search or evaluate, so that this scheme is of particular interest to broadcasting or optical recording on mass-replicated discs. Nevertheless, a saving in the number of evaluations would be most helpful. Copeland & Tezcan [5] described a new method that oers a signicant saving in the complexity of the quality evaluation process. Their newly proposed scheme exploits the Fast (Walsh-)Hadamard Transform (FHT). The 2

3 FHT eciently generates the disparity, (sum of the n symbols in the codeword) of all codewords in the selection set. Then, as described above, the codeword with the minimum disparity value chosen for transmission. The entries of the n n Hadamard matrix, H n, can be expressed as h n [i j] =(;1) P k i kj k where i k 2f0 1g and j k 2f0 1g represent the k-th bit in the binary representation of the integers i and j, respectively. This structure leads to very ecient methods for computing the Hadamard matrix. Let the codeword length be n = 4. Then H4 = ;1 1 ;1 1 1 ;1 ;1 1 ;1 ;1 1 Let the n-bit source word be denoted by x. The selection set consists of the n (candidate) codewords that can be formed by the transformation (scrambling) of the source word x with the n columns of the Hadamard matrix. Then the entries v i of the vector v given by v = H n x equal the disparity of the candidate codewords obtained by inverting the bits in x on the positions, where the entries of the ith column of the Hadamard matrix equals -1. The attractiveness of the FHT stems from the fact that v can be evaluated with a number of computations that is proportional with logn instead of n. The encoder transmits that codeword from the selection set having an RDS as close to zero as possible. In addition, the encoder transmits the binary representation of the corresponding column index i to make it possible for the decoder to reconstitute the original source word. We require log 2 n bits for representing the index so that the rate of the code is R = n=(n +log 2 n): 3 75 : 3

4 2 Evaluation of the spectral performance In order to assess the quality of the low-frequency suppression, we employ a simple, but reliable, criterion [1, Chapter 10]. The criterion compares the rate, R, and sum variance, s 2 = Efz 2 i g, of the encoded sequences with the capacity and sum variance of a maxentropic dc-free sequence. The encoder eciency, E, is dened by E = 0:2326 s 2 (1 ; R) : We have written a computer program to simulate the performance of the encoder. Figure 1 depicts the encoder eciency of the FHT-based encoder as a function of the codeword length n. We mayobserve, that for n = 16 the eciency is approximately 40% decreasing to an eciency of approximately 15% for a codeword length n =512. The simple polarity switchcodeachieves an encoder eciency of 40% [1] irrespective of the codeword length, and we therefore conclude that the new FHT-based dc-free code is inferior to the simple polarity switch code. We have investigated the reasons that underlie the disappointing spectral performance of the new code. We found that the codeword disparity criterion used by the FHT fails to nd that codeword having the "best" low-frequency content. This drawback of the disparity criterion was reported before in [4]. A computer program was written to test this nding. It showed that if the disparity criterion is changed to a sum variance criterion (or other adequate criteria described in [4]) that the encoder performs much better reaching an encoder eciency of unity. Unfortunately, it is not clear how to eciently evaluate these more sophisticated selection criteria. 3 Conclusions We reported on a new class of multi-mode dc-free codes. In multi-mode codes, a source word can be represented by a member of a selection set of codewords. The member having the least contribution to the low-frequency 4

5 content is selected. The Fast Hadamard Transform (FHT) is employed to eciently evaluate the low-frequency properties of all codewords in the selection set. Spectral properties of the new codes have been evaluated by computer simulation. We have found that, for a given rate, the suppression of the new FHT based codes is inferior to the much simpler polarity switch code. References [1] K.A.S. Immink, Codes for Mass Data Storage Systems, ISBN X, Shannon Foundation Publishers, Netherlands, [2] F.K. Bowers, US Patent No. 2,957,947, [3] I.J. Fair, W.D. Gover, W.A. Krzymien, and R.I. MacDonald, 'Guided Scrambling: A New Line Coding Technique for High Bit Rate Fiber Optic Transmission Systems', IEEE Trans. Commun., vol. COM-39, no. 2, pp , Feb [4] K.A.S. Immink and L. Patrovics, 'Performance Assessment of DC-Free Multimode Codes', IEEE Trans. Commun., vol. COM-45, no. 3, March [5] G. Copeland and B. Tezcan, 'Disparity and Transition Density Control System and Method', US Patent 6,304,196, Oct

6 Encoder efficiency Codeword length Figure 1: Encoder eciency E versus codeword length n. 6

1 Introduction Codes for the noiseless recording channel, often called recording codes, on which we will concentrate, convert the input stream to a si

1 Introduction Codes for the noiseless recording channel, often called recording codes, on which we will concentrate, convert the input stream to a si A Survey of High-Rate Constrained Codes Kees A. Schouhamer Immink Institute for Experimental Mathematics, Ellernstrasse 29-31, D-45326 Essen, Germany. Phone +31 402 851 854 Fax +31 402 860852. E-mail: