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# Performance assessment of DC-free multimode codes

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We report on a class of high-rate de-free codes, called multimode codes, where each source word can be represented by a codeword taken from a selection set of codeword alternatives. Conventional multimode codes will be analyzed using a simple mathematical model. The criterion used to select the "best" codeword from the selection set available has a significant bearing on the performance. Various selection criteria are introduced and their effect on the performance of multimode codes will be examined.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997 293
Transactions Papers
Performance Assessment of DC-Free
Multimode Codes
Kees A. Schouhamer Immink, Fellow, IEEE, and Levente P´
atrovics
Abstract—We report on a class of high-rate dc-free codes, called
multimode codes, where each source word can be represented by
a codeword taken from a selection set of codeword alternatives.
Conventional multimode codes will be analyzed using a simple
mathematical model. The criterion used to select the “best”
codeword from the selection set available has a signiﬁcant bearing
on the performance. Various selection criteria are introduced
and their effect on the performance of multimode codes will be
examined.
I. INTRODUCTION
BINARY sequences with spectral nulls at zero frequency
have found widespread application in optical and mag-
netic recording systems. The dc-balanced or dc-free codes, as
they are often called, have a long history and their application
is certainly not conﬁned to recording practice. Since the early
days of digital communication over cable, dc-balanced codes
have been employed to counter the effects of low-frequency
cut-off due to coupling components, isolating transformers,
etc. In optical recording, dc-balanced codes are employed to
circumvent or reduce interaction between the data written on
the disc and the servo systems that follow the track [1]. In
the literature, code implementations have been concentrated
on byte-oriented dc-free codes of rate 8/10 or 8/9. (see, for
example, [2]–[4]). For certain application it is desirable that
the code rate is much higher than 8/9. The construction of
such high-rate codes is far from obvious, as table look-up
for encoding and decoding is an engineering impracticality.
Two methods for high-rate code design have been described
in the literature [5], [6]. Both methods utilize the idea that the
correspondence between source words and the codewords is
as simple as possible. A serious drawback of both methods
is that the performance, in terms of suppression of low-
frequency components, is far from what could be obtained
according to the tenets of information theory [1], [7], but up
till now attempts to improve the performance failed. Recently,
however, the publications by Fair et al. on “guided scrambling”
Paper approved by E. Ayanoglu, the editor for Communication Theory
and Coding Application of the IEEE Communications Society. Manuscript
received May 7, 1996; revised August 2, 1996. This paper was presented
in part at the IEEE 3rd Symposium on Communications and Vehicular
Technology, Eindenhoven, The Netherlands, October 25–26, 1995.
The authors are with the Philips Research Laboratories, 5656 AA Eind-
hoven, The Netherlands.
Publisher Item Identiﬁer: S 0090-6778(97)01970-3.
stimulated us to investigate the performance of their method
and its varieties. In our context, guided scrambling is a
member of a larger class of related coding schemes called
multimode code. In multimode codes, each source word can
be represented by a member of a selection set consisting of
codewords. The encoder opts for transmitting that codeword
that minimizes, according to a criterion to be deﬁned, the low-
frequency spectral contents of the encoded sequence. There are
two key elements which need to be chosen judiciously: 1) the
mapping between the source words and their corresponding
selection sets and 2) the criterion used to select the “best”
word. The spectral performance of the code greatly depends
a section providing the state of the art. Thereafter, we will
outline the new multimode schemes and analyze their spectral
performance.
II. PRELIMINARIES
The running digital sum of a sequence (RDS) plays a
signiﬁcant role in the analysis and synthesis of codes whose
spectrum vanishes at the low-frequency end. Let
be a bipolar se-
quence. Note that in the sequel, we will denote the value of
by its logical equivalents ‘0’ or “1.” The (running) digital
sum is deﬁned as
It is an elementary exercise to show that if is bounded, the
spectral density vanishes at zero frequency [1]. The number
of RDS values that the sequence assumes is often called the
digital sum variation and denoted by . The value of
should be as small as possible as it has a direct bearing on
the amount of power at the low-frequency end. Given the
parameter it is possible to compute the maximum value
of the rate, , of any code, irrespective of its complexity,
that translates arbitrary source input into sequences obeying
the given constraint. Results of computation, taken from [1],
are listed in Table I. It can be seen that the sum constraint is not
very expensive in terms of rate loss when is relatively large.
For instance, a sequence that takes at maximum sum
values has a capacity , which implies a rate loss
0090–6778/97$10.00 1997 IEEE 294 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997 TABLE I CAPACITY AND SUM VARIANCE OF MAXENTROPIC SEQUENCES TAKING AT MOST RDS VALUES VERSUS DIGITAL SUM VARIATION of less than 6%. The quantity called sum variance plays an important role in the evaluation of the spectral properties of a code. Before explaining the relevance of the parameter , a few words are in order regarding the low-frequency properties of dc-free codes. If is ﬁnite, the spectral density is zero at zero frequency, but it is more relevant to note that there is a region of frequencies, close to the zero frequency, where the spectral density is low. The width of this region, termed the notch width, is of great engineering relevance. The width of the spectral notch, can be quantiﬁed by a parameter called the cutoff frequency,. According to the work by Justesen [8] there is a simple approximate relationship between and the sum variance, , of the encoded sequence, namely (1) Sequences of maximum entropy assuming at most RDS values obey the following fundamental relationship between the sum variance and the redundancy [1] (2) The above relationship can be employed to derive a simple yard stick for measuring the performance of implemented codes. The encoder efﬁciency is deﬁned as (3) The encoder efﬁciency , as deﬁned in (3), compares the “redundancy-sum variance products” of the implemented code and the maxentropic sequence with the same digital sum variation as the implemented code. The efﬁciency will be used in the sequel to measure the performance of dc-free codes. III. PRIOR ART Essentially, there are three basic methods for generating dc- free sequences which are relevant to the ensuing discussion. These methods are brieﬂy reviewed below. A. Monomode Codes In monomode codes, there is a one-to-one relationship between source words and codewords. By necessity, the code- words have equal numbers of 1’s and 1’s. There are two methods available for translating source words into codewords. The ﬁrst method uses an algebraic technique, called enumera- tion [1], and in the second method, devised by Knuth [6], -bit source words are translated into -bit codewords. The translation is achieved by selecting a bit position within the -bit word which deﬁnes two segments, each having one half of the total disparity of the -bit word, where the disparity of a codeword is deﬁned as the difference between the numbers of 1’s and 1’s in that codeword. A zero-disparity codeword, i.e., a codeword with an equal number of 1’s and 1’s, is now generated by the inversion of all the bits within one segment. The position information which deﬁnes the two segments is encoded in the bits. B. Bimode Codes Bimode codes ensure balanced transmission by providing for each source word two alternative channel representations. From the alternatives available, that codeword is transmitted that minimizes the absolute value of the RDS after transmis- sion of the new word. This selection criterion will be termed MRDS selection criterion. An archetypical example of a bi- mode code is the polarity switch code [5]. The encoder and decoder circuits of the polarity switch code are very simple as no look-up tables are required. Under polarity switch rules, source symbols are supplemented by one symbol called the polarity bit. The encoder has the option to transmit the - bit words without modiﬁcation or to invert all symbols. The choice of a speciﬁc translation is made in such a way 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, and can easily be reconstituted. Properties of the polarity bit code have been described in [1]. The performance of the polarity switch code can be summarized as follows. The rate of the polarity bit code is The sum variance of the code [1] is so that the efﬁciency is (4) From the above, we conclude that polarity switch codes are a far cry from the optimal situation. It is not difﬁcult to generalize the above principle of bimode codes to multimode codes, which, as the name already suggests, cater for more than two channel representations. C. Multimode Codes In multimode codes, each source word can be represented by a member of a selection set, denoted by , consisting of codewords. The MRDS selection criterion can be used to select the “best” codeword. More sophisticated selection criteria will be described in Section V. It should be appreciated that the usage of multimode codes is not conﬁned to the generation of dc-free sequences. Provided that is large enough and the selection sets contain sufﬁciently different codewords, IMMINK AND P ´ ATROVICS: DC-FREE MULTIMODE CODES 295 multimode codes can also be used to satisfy almost any channel constraint with a suitably chosen selection method. A basic element of multimode codes is the one-to- in- vertible mapping between the source and its selection set . Examples of such mappings are the guided scrambling algorithm presented by Fair et al. [9], the dc-free coset codes of Deng and Herro [10], and the scrambling using a Reed–Solomon code by Kunisa et al. [11]. In our context, a mapping is considered to be “good” if the sets contain suf- ﬁciently distinct codewords. The guided scrambling algorithm is brieﬂy described below. 1) Guided Scrambling: The guided scrambling algorithm uses selection sets of size , where is the number of redundant bits. Guided scrambling is summarized below. 1) In the ﬁrst step, called augmenting, the source word is preceded by all the possible binary sequences of length to produce the set . Hence 2) The selection set is obtained by scrambling all vectors in . Let the scrambler poly- nomial be denoted by where denotes the register length of the scrambler. The scrambler translates each vector into using the recursion (5) 3) The “best” codeword in is selected for transmission. 4) At the receiver’s site, the inverse operation is The source word is found by deleting the ﬁrst bits. In the guided scrambling algorithm described above, trans- lation of source words into random-like channel represen- tations is done in a fairly simple way. This basic algorithm is, however, prone to worst case situations since there is a probability that consecutive source words have representation sets whose members all have the same polarity of the disparity. In this vexatious situation, the RDS cannot be controlled, and long-term low-frequency components can build up. This ﬂaw can be solved by a construction where each selection set consists of pairs of words of opposite disparity. As a result, there is always a codeword in the selection set that can control the RDS. A simple method embodying this idea combines the features of guided scrambling and the polarity bit code. The improved algorithm using redundant bits is executed in six steps. In Steps 1), 2), and 5) the original guided scrambling principle is executed while Steps 3) and 4) embody the polarity bit code. 1) The source word is preceded by all the possible binary sequences of length to produce the elements of the set . Hence: 2) The selection set is obtained by scrambling all vectors in . 3) By preceding the vectors in with both a ‘one’ and a “zero,” we get the set , with elements. 4) The selection set is obtained by scrambling (pre- coding) the vectors in using the scrambler with polynomial . This embodies the polarity bit prin- ciple. 5) The “best” codeword in is selected. 6) At the receiver end, the codeword is ﬁrst descrambled using the polynomial, then after removing the ﬁrst bit, it is descrambled. The original source word is eventually reconstituted by removing the ﬁrst bits. All simulations and analyses discussed below assume the above structure where the selection set consists of pairs of words of opposite disparity. IV. ANALYSIS OF MULTI-MODE CODES A precise mathematical analysis of the performance of multimode codes is, considering the complexity of the code, out of the question. We can either rely on computer simulation to facilitate an understanding of the operation of the coding system or try to deﬁne a simple mathematical model, which embraces the essential characteristics of the code and is also analytically tractable. We followed both approaches, and we commence by describing the mathematical model. A. The Random Drawing Model The key characteristic of a multimode code is that each source word can be represented by a codeword taken from a set containing “random” alternatives. As the precise structure of the encoder is extremely difﬁcult to analyze, we assume, in our mathematical model, that for each source block the channel set is obtained by randomly drawing -bit words plus their complementary -bit words. The precise structure of the scrambler is ignored in our model. The “best” word in the set, according to the MRDS criterion, is transmitted. The MRDS criterion ensures that the state space of the encoder, that is, the number of possible word-end running digital sum (WRDS) values the encoded sequence may take, is ﬁnite. However, if the codewords are relatively long, the number of states and the resulting transition matrix are still too large for a simple mathematical analysis. We therefore truncated the state space by omitting those states that do not contribute signiﬁcantly to the sum variance. 296 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997 B. Transition Probabilities of the Finite-State Machine The implemented encoder schemes can be simply treated in terms of Markov models. The set of values that WRDS take prior to transmission of a codeword deﬁnes a set of states of a ﬁnite-state machine. We will use the shorthand notation to denote both the WRDS at the start of the th codeword and to refer to the encoder state itself. We commence our analysis with a computation of the state transition probabilities. Assume the th codeword starts with RDS . Then the multimode code can be cast into a Markov chain model whose state transition probabilities matrix, , is given by We make the following remarks concerning the state transition matrix. 1) For the sake of simplicity, only codes using codewords of even length are considered. 2) It is assumed that at the start of the transmission WRDS is set to . As a result, since the codeword length is even, . 3) For reasons of symmetry, only the probabilities for need to be calculated. 4) To reduce the computational load, we truncated the state space. Only those states are considered that can be reached from the , or the , state with probability greater than , where is chosen suitably small, say 10 6. Other values of have been tried without, however, causing signiﬁcant differences in the results obtained. The remaining states will be termed principal states. We now introduce several notations. If WRDS is positive, then, according to the simple MRDS criterion, the next code- word will be of zero or negative disparity. Therefore, assuming that the encoder occupies state , the set of possible next states is . Let denote the probability of a codeword pair having disparity and . The probability of the next-state candidate in a draw being is if otherwise. (6) The next-state candidate in the th draw is denoted by . According to the MRDS criterion, if the next state is , then for all . The probability that during a draw the next-state candidate is “worse” than , denoted by , is given by Now, the expression for the transition matrix is given by (7) The transition probabilities for each pair of WRDS states can be numerically determined by invoking (7). In order to make the analysis more tractable, those states are removed that can be reached from the , or the , state Fig. 1. Efﬁciency of random drawing algorithm using the MRDS selection criterion. with probability less than . The remaining set of states, the principal states, denoted by , and the truncated transition probability matrix with elements can easily be found. Thereafter, vector of the stationary probabilities, with elements ,is found by solving . The calculation of the variance of the digital sum at the start of the codewords is now straightforward. The computation of the sum variance within the codewords is more complex, and therefore given in the Appendix. C. Computational Results Using (13), we calculated the efﬁciency of the random drawing algorithm for selected values of the codeword length and redundancy. Fig. 1 shows the results. The connected points have the same redundancy , and the th point on a curve corresponds to a code having redundant bits, codeword length , and selection sets of size . For comparison purposes, we also plotted the efﬁciency of the polarity bit code [see (4)]. By comparing the efﬁciency values at the th point on each curve, we can see that these values are approximately the same. The efﬁciency of the random coding algorithm is practically independent of the codeword length and is essentially determined by the number of redundant bits used. It can be seen that codes with two or three redundant bits are clearly more efﬁcient than the polarity bit code. With an increasing number of redundant bits, however, the efﬁciency decreases. The decrease in performance, as will be explained in the next section, is due to the shortcomings of the MRDS criterion. V. ALTERNATIVE SELECTION CRITERIA The results, plotted in Fig. 1, reveal that using more than two redundant bits does not lead to improved performance. The reason that performance decreases with an increasing number of redundant bits can easily be understood. A quick calculation will make it clear that a large selection set contains with great probability at least one zero-disparity word. On the basis of IMMINK AND P ´ ATROVICS: DC-FREE MULTIMODE CODES 297 Fig. 2. Simulation results for the random drawing algorithm with ﬁxed redundancy 1/128 with different selection criteria (a) MRDS, (b) MMRDS, and (c) MSW. the simple MRDS criterion one of the zero-disparity words is randomly chosen and transmitted. As the sum variance of zero-disparity codewords equals , [1] irrespective of the rate of the code, we conclude that the efﬁciency will asymptotically approach zero. More sophisticated selection criteria, which take account of the running digital sum within the codeword, and not only at the end of the word, may result in increased performance. In order to describe these more sophisticated selection criteria, we introduce the squared weight,, of a codeword, deﬁned as the sum of the squared RDS values at each bit position of the codeword. The two selection criteria examined are as follows: 1) modiﬁed MRDS (MMRDS) criterion: from the code- words with minimal WRDS , the one with minimum is selected; 2) minimum squared weight (MSW) criterion: the code- word of minimal is selected from the selection set, irrespective of the WRDS of the codeword. Fig. 2 shows the simulation results obtained for redundancy 1/128. Simulations of codes with other values of the redun- dancy produced similar results. From the curves, we infer the following. The MRDS method wastes the opportunity offered by the broader selection sets. By properly selecting the codeword from the ones with minimal WRDS , the efﬁciency of the MMRDS scheme tends to unity. As indicated by the curve of the MSW criterion, the best codewords do not necessarily minimize the WRDS . Selecting the codeword with minimal squared weight clearly results in more efﬁcient codes. Based on the above observations, we searched for a criterion that is simple to implement while its efﬁciency approaches that of the MSW criterion. The outcome is described in the next section. A. The Minimum Threshold Overrun Criterion Our objective, in this section, is to construct a selection criterion which takes into account the RDS values within Fig. 3. Simulation results for the random drawing algorithm having ﬁxed redundancy 1/128 with (a) the MSW criterion and (b) the MTO criterion. The dotted line shows the results obtained for the implemented encoding scheme using a scramblers with polynomial . the codeword while having a structure that is also easy to implement. The proposed selection scheme, termed minimum threshold overrun (MTO) criterion, utilizes the parameter “RDS threshold,” denoted by . The MTO penalty is simply the number of times the absolute value of the running digital sum within a word is larger than . As the squaring operation needed for the MSW criterion is avoided, the implementation of the MTO criterion is not more complex than the MRDS method. The codeword with minimum penalty is transmitted. If two or more codewords have the same penalty, one of them is chosen randomly and transmitted. This procedure does not seriously deteriorate the performance as it is fairly improbable that two or more codewords in the selection set have the same penalty value. Fig. 3, curve (b), shows simulation results obtained with the MTO criterion. Optimal values of the threshold were found by trial and error. We can see that the MTO criterion is only slightly less efﬁcient than the MSW criterion. All results shown so far have been obtained by a simulation program of the random drawing algorithm. As a ﬁnal check we also conducted simulations with a full-ﬂedged implementation using a scramblers with polynomial . Experiments with other scrambler polynomials did not reveal signiﬁcant differences. The dotted curve, Fig. 3, gives results on the basis of the MTO criterion. The curve shows a nice agreement with results obtained with the random drawing algorithm. As the proof of the pudding we have computed the power spectral density (PSD) of two typical examples. The results are displayed in Fig. 4. VI. CONCLUSION Multimode codes have been mathematically analyzed by introducing a simple random drawing model. We have pre- sented alternative selection criteria and examined their effect on the spectral efﬁciency. Multimode codes are excellent candidate dc-free codes when both low-spectral content at the low-frequency end and high rate are at a premium. For given rate and proper selection criteria, the spectral content of multimode codes is very close to the minimal content promised 298 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997 Fig. 4. Spectra of encoded sequences generated by the polarity switch code (upper curve) and multimode code (lower curve). The redundancy is in both cases 1/128. The multimode code has six redundant bits (codeword length is ) and it uses the MTO selection criterion. by information theory. APPENDIX In this Appendix, we will compute the sum variance of se- quences encoded with the random drawing model. A codeword with binary elements is translated into the -tuple where . Suppose the th codeword in the sequence, , starts with initial RDS . The RDS at the th symbol position of , denoted by , equals The running sum variance at the th position, given , equals where the operator averages over all codewords that start with an initial RDS . As the source population of codewords is the full set of vectors of nonpositive (nonneg- ative) disparity, the expectations and , are independent of the symbol positions and . For the sake of convenience, we use the shorthand notation and . Substitution yields the running sum variance at the th symbol position (8) The sum variance of a codeword starting with initial RDS , designated by , is found by averaging the running digital sum variance over all symbol positions of the codeword or The probability that a codeword starts with an RDS equals the stationary probability , so that by taking the probability into account that a codeword starts with RDS and averaging over all initial states in , the following expression is found for the sum variance : (9) The variance of the initial sum values, , equals The quantity can be estimated by noting the periodicity, i.e., . Evaluating (8) yields and after averaging, where the the probability of starting with an initial RDS is taken into account, we obtain so that with we ﬁnd Substitution in (9) yields (10) 1) Computation of the Correlation: We next calculate the correlation of the symbols at the th and the th symbol position within the same codeword. It is IMMINK AND P ´ ATROVICS: DC-FREE MULTIMODE CODES 299 obvious that .If , some more work is needed. In that case, (11) Assume a codeword to be of disparity . Then the prob- ability that a symbol at position in the codeword equals 1is The probability that another symbol at position within the same codeword equals 1is Hence and (11) yields the correlation for codewords of disparity Using the above, we ﬁnd that (12) The sum variance can be determined using (10) and (12) (13) ACKNOWLEDGMENT The authors wish to thank I. Fair for his valuable remarks that helped improve the contents of this paper. REFERENCES [1] K. A. S. Immink, Coding Techniques for Digital Recorders. Engle- wood Cliffs, NJ: Prentice-Hall International, 1991. [2] S. Fukuda, Y. Kojima, Y. Shimpuku, and K. Odaka, “8/10 modulation codes for digital magnetic recording,” IEEE Trans. Magn., vol. MAG-22, pp. 1194–1196, Sept. 1986. [3] A. X. Widmer and P. A. Franaszek, “A dc-balanced, partitioned-block, 8b/10b transmission code,” IBM J. Res. Develop., vol. 27, no. 5, pp. 440–451, Sept. 1983. [4] H. Yoshida, T. Shimada, and Y. Hashimoto, “8-9 block code: A dc-free channel code for digital magnetic recording,” SMPTE J., vol. 92, pp. 918–922, Sept. 1983. [5] F. K. Bowers, U.S. Patent 2 957947, 1960. [6] D. E. Knuth, “Efﬁcient balanced codes,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 51–53, Jan. 1986. See also P. S. Henry, “Zero disparity coding system,” U.S. Patent 4 309694, Jan. 1982. [7] H. Hollmann and K. A. S. Immink, “Performance of efﬁcient balanced codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 913–918, May 1991. [8] J. Justesen, “Information rates and power spectra of digital codes,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 457–472, May 1982. [9] 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. 39, pp. 289–297, Feb. 1991. [10] R. H. Deng and M. A. Herro, “DC-free coset codes,” IEEE Trans. Inform. Theory, vol. 34, pp. 786–792, July 1988. [11] A. Kunisa, S. Takahashi, and N. Itoh, “Digital modulation method for recordable digital video disc,” in Proc. 1996 IEEE Int. Conf. Consumer Electron., June 1996, pp. 418–419. Kees A. Schouhamer Immink (M’81–SM’86– F’90) received the M.S. and Ph.D. degrees from the Eindhoven University of Technology, Eindenhoven, The Netherlands. He joined the Philips Research Laboratories, Eindhoven, in 1968, where he currently holds the position of Research Fellow. He has contributed to the design and development of a wide variety of digital consumer-type audio and video-recorders such as the compact disc, compact disc video, R- DAT, DCC, and DVD. He holds 32 U.S. patents and has written numerous papers in the ﬁeld of coding techniques for optical and magnetic recorders. Dr. Immink is the Chairman of the IEEE Benelux Chapter on Consumer Electronics, a Governor of the Audio Engineering Society (AES), and a Governor of the IEEE Information Theory Society. He was named a Fellow of the AES, SMPTE, and IEE. Furthermore, he was recognized and awarded with the AES Silver Medal in 1992, the IEE Sir Thomson Medal in 1993, the SMPTE Poniatoff Gold Medal in 1994, and the IEEE Ibuka Consumer Electronics Award in 1996. He is a member of the Royal Netherlands Academy of Arts and Sciences. Levente P´ atrovics was born in Budapest, Hungary. He received the M.Sc. degree in electrical engi- neering and computer science in 1994 from the Technical University of Budapest, Hungary. From May to December 1995, he was working towards the Ph.D. degree at Philips Research Laboratories, Eindhoven, The Netherlands. His research interests are in information theory, particular construction, and analysis of constrained codes. Currently, he is with Lufthansa Systems, Hungaria Kft. ... Line codes for these applications are simpler than T x -constrained and RLL codes, since streams of codewords are only required to be balanced and to support self-clocking. Examples include the 8b/10b code [16], the 64b/66b code [17], and the 128b/132b code. We note that constrained codes preserving parity are studied in [18], and that constrained codes for deoxyribonucleic acid (DNA) storage are studied in [19]. ... ... A critical additional requirement in line codes, which appears in applications like optical recording, Flash memories, in addition to USB and PCIe standards, is balancing [10], [14], [22]. Examples of balanced line codes are the 8b/10b [16] and the 64b/66b [17] codes. Balanced line codes have zero average power at frequency zero, i.e., no DC power component, when the signal levels are −A and +A. ... Preprint Full-text available Line codes make it possible to mitigate interference, to prevent short pulses, and to generate streams of bipolar signals with no direct-current (DC) power content through balancing. Thus, they find applications in magnetic recording (MR) devices, in Flash devices, in optical recording devices, in addition to some computer standards. This paper introduces a new family of fixed-length, binary constrained codes, namely, lexicographically-ordered constrained codes (LOCO codes) for bipolar non-return-to-zero signaling. LOCO codes are capacity achieving, the lexicographic indexing enables simple, practical encoding and decoding, and this simplicity is demonstrated through analysis of circuit complexity. LOCO codes are easy to balance, and their inherent symmetry minimizes the rate loss with respect to unbalanced codes having the same constraints. Furthermore, LOCO codes that forbid certain patterns can be used to alleviate inter-symbol interference in MR systems and inter-cell interference in Flash systems. Experimental results demonstrate a gain of up to 10% in rate achieved by LOCO codes compared with practical run-length limited codes designed for the same purpose. Simulation results suggest that it is possible to achieve channel density gains of about 20% in MR systems by using a LOCO code to encode only the parity bits of a low-density parity-check code before writing. ... Line codes for these applications are simpler than T x -constrained and RLL codes, since streams of codewords are only required to be balanced and to support self-clocking. Examples include the 8b/10b code [18], the 64b/66b code [19], and the 128b/132b code [20]. We note that constrained codes preserving parity are studied in [21], and that constrained codes for deoxyribonucleic acid (DNA) storage are studied in [22]. ... ... A critical additional requirement in line codes, which appears in applications like optical recording, Flash memories, in addition to USB and PCIe standards, is balancing [12], [16], [25]. Examples of balanced line codes are the 8b/10b [18] and the 64b/66b [19] codes (the latter is not strictly DC-free). Balanced line codes have zero average power at frequency zero, i.e., no DC power component, when the signal levels are −A and +A. ... Article Full-text available Line codes make it possible to mitigate interference, to prevent short pulses, and to generate streams of bipolar signals with no direct-current (DC) power content through balancing. They find application in magnetic recording (MR) devices, in Flash devices, in optical recording devices, and in some computer standards. This paper introduces a new family of fixed-length, binary constrained codes, named lexicographically-ordered constrained codes (LOCO codes), for bipolar non-return-to-zero signaling. LOCO codes are capacity-achieving, the lexicographic indexing enables simple, practical encoding and decoding, and this simplicity is demonstrated through analysis of circuit complexity. LOCO codes are easy to balance, and their inherent symmetry minimizes the rate loss with respect to unbalanced codes having the same constraints. Furthermore, LOCO codes that forbid certain patterns can be used to alleviate inter-symbol interference in MR systems and inter-cell interference in Flash systems. Numerical results demonstrate a gain of up to 10% in rate achieved by LOCO codes with respect to other practical constrained codes, including run-length-limited codes, designed for the same purpose. Simulation results suggest that it is possible to achieve a channel density gain of about 20% in MR systems by using a LOCO code to encode only the parity bits, limiting the rate loss, of a low-density parity-check code before writing. ... After obtaining the compressed output F comp , we perform randomization to make the GC balance ratio between 0.5 ± α. In the coding theory, there is a scheme called guided scrambling [17], which is similar to the randomization step. It is applied with the verification step to satisfy the GC balance ratio. ... Preprint Full-text available In this paper, we propose a novel iterative encoding algorithm for DNA storage to satisfy both the GC balance and run-length constraints using a greedy algorithm. DNA strands with run-length more than three and the GC balance ratio far from 50\% are known to be prone to errors. The proposed encoding algorithm stores data at high information density with high flexibility of run-length at most$m$and GC balance between$0.5\pm\alpha$for arbitrary$m$and$\alpha$. More importantly, we propose a novel mapping method to reduce the average bit error compared to the randomly generated mapping method, using a greedy algorithm. The proposed algorithm is implemented through iterative encoding, consisting of three main steps: randomization, M-ary mapping, and verification. It has an information density of 1.8616 bits/nt in the case of$m=3$, which approaches the theoretical upper bound of 1.98 bits/nt, while satisfying two constraints. Also, the average bit error caused by the one nt error is 2.3455 bits, which is reduced by$20.5\%\$, compared to the randomized mapping.
... A signal is called DC-free when its mean amplitude is zero. Non-DC-free signals can cause transmission errors for electrotechnical reasons [91]. ...
Thesis
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The Internet of Things (IoT) brings comfort into the life of users. It is convenient to control the lights at home with an app without leaving the couch or open the front door with a remote control. This comfort, however, comes with security risks as the wireless communication between components often relies on proprietary protocols. Such protocols are designed under size and energy constraints whereby security is often only a secondary factor. Moreover, even when a default protocol such as IEEE 802.11 WLAN with enabled encryption is used, mobile devices such as smartphones can be located threatening the location privacy of users. This thesis is divided into two main parts. In the first part, we demonstrate how to passively locate a smartphone indoors using IEEE 802.11 WLAN and contribute a geolocation system with a mean accuracy of 0.58m. Subsequently, we analyze how a company can incentivize users with different levels of privacy-awareness to connect to a provided WLAN and give up their location privacy in exchange for certain benefits such as shopping discounts. We model this situation as a Bayesian Stackelberg game to find the company's best strategy. In the second part, we showcase the challenges that arise for security researchers when investigating proprietary wireless protocols. Software Defined Radios (SDRs) propose a generic way to analyze such protocols operating on frequencies like 433.92 MHz or 868.3 MHz where no default hardware such as a WLAN stick is available. SDRs, however, deliver raw signals that have to be demodulated and decoded before researchers can reverse-engineer the protocol format. Our main contribution to this process is an open source software called Universal Radio Hacker (URH) which is, to the best of our knowledge, the first complete suite for wireless protocol investigation with SDRs. URH splits down the protocol investigation process into the phases Interpretation, Analysis, Generation and Simulation. The goal of Interpretation phase is identifying the transmitted bits and bytes by demodulating the signal. Apart from letting users manually adjust demodulation parameters, we contribute a set of algorithms to automatically find these parameters and integrate them into URH. In Analysis phase, the protocol format is reverse-engineered from the demodulated bits. This is a time-consuming manual process that slows down a security analysis. To address this problem, we design and implement a modular system that automatically finds protocol fields such as addresses and checksums. In combination with the automatic modulation parameter detection this speeds up the security analysis of unknown wireless protocols. URH enables researchers to perform attacks on stateless and stateful protocols in the Generation and Simulation phase, respectively. In Generation, users can apply a fuzzing to arbitrary data ranges while the Simulation component of URH models protocol state machines and dynamically reacts to incoming messages from investigated devices. In both phases, the software automatically applies modulation and encoding to the bits that should be sent. We demonstrate three attacks on IoT devices that were found and executed with URH. The most complex attack involves opening an AES protected wireless door lock in real-time.
... Among various methodologies proposed for the design of constrained codes, the GS [16], [17] has been found an effective statistical coding technique that is suitable for the design of dc-free codes as well as the weakly constrained codes [18]. In this work, we propose to combine the GS scheme with the k constrained code design methods presented in Section II, to design high-rate spectrum shaping k constrained codes. ...
Article
This paper proposes systematic code design methods for constructing efficient spectrum shaping codes with the maximum runlength limited constraint k, which are widely used in data storage systems for digital consumer electronics products. Through shaping the spectrum of the input user data sequence, the codes can effectively circumvent the interaction between the data signal and servo signal in high-density data storage systems. In particular, we first propose novel methods to design high-rate k constrained codes in the non-return-to-zero (NRZ) format, which can not only facilitate timing recovery of the storage system, but also avoid error propagation during decoding and reduce the system complexity. We further propose to combine the Guided Scrambling (GS) technique with the k constrained code design methods to construct highly efficient spectrum shaping k constrained codes. Simulation results demonstrate that the designed codes can achieve significant spectrum shaping effect with only around 1% code rate loss and reasonable computational complexity.
Conference Paper
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Abstract— WiGig stands for Wireless Gigabit Alliance, whichwas founded to encourage the adoption of IEEE 802.11ad, anupdate to the network of IEEE 802.11 wireless that was intendedfor enabling a Multiple Gigabit Wireless System (MGWS) atbandwidth of 60 GHz., In this paper illustrating IEEE 802.11 adDirectional multi-Gigabits performance in multi-impairmentenvironments with MATLAB Simulink where AWGN phase,frequency offset, phase noise, DC offset, IQ imbalance andmemory less cubic nonlinearity impairment effect on the IEEE802.11 ad spectrum and constellation performance, modulationscheme used DBPSK design and implemented in MATLABSimulink tested under AWGN and Rayleigh fading channels (PDF) WiGig WiFi performance with multi-impairment environments in MATLAB Simulink. Available from: https://www.researchgate.net/publication/361694493_WiGig_WiFi_performance_with_multi-impairment_environments_in_MATLAB_Simulink [accessed Jul 04 2022].
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Preprint
Visual receptive fields are characterised by their centre-surround organisation and are typically modelled by Difference-of-Gaussians (DoGs). The DoG captures the effect of surround modulation, where the central receptive field can be modulated by simultaneous stimulation of a surrounding area. Although it is well-established that this centre-surround organisation is crucial for extracting spatial information from visual scenes, the underlying law binding the organisation has remained hidden. Indeed, previous studies have reported a wide range of size and gain ratios of the DoG used to model the receptive fields. Here, we present an equation that describes a principle for receptive field organisation, and we demonstrate that functional Magnetic Resonance Imaging (fMRI) population Receptive Field (pRF) maps of human V1 adhere to this principle. We formulate and understand the equation through consideration of the concept of Direct-Current-free (DC-free) filtering from electrical engineering, and we show how this particular type of filtering effectively makes the DoG process frequencies of interest without misallocation of bandwidth to redundant frequencies. Taken together, our results reveal how this organisational principle enables the visual system to adapt its sampling strategy to optimally cover the stimulus-space relevant to the organism, restricted only by Heisenberg's uncertainty principle that imposes a lower bound on the simultaneous precision in spatial position and frequency. Since surround modulation has been observed in all sensory modalities, we expect these results will become a corner stone in our understanding of how biological systems in general achieve their high information processing capacity.
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We present a systematic variable-to-fixed (VF) length scheme encoding binary information sequences into binary balanced sequences. The redundancy of the proposed scheme is larger than the redundancy of the best fixed-to-fixed (FF) length schemes in case of long codes, but it is smaller in case of short codes. The biggest advantage comes from the simplicity of the scheme: encoding only requires one to keep track of the sequence weight, while decoding requires only one extremely simple step, irrespective of the sequence length.
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The technique introduced has relatively simple encoding and decoding procedures which can be implemented at the high bit rates used in optical fiber communication systems. Because it is similar to the established technique of self-synchronizing scrambling but is also capable of guiding the scrambling process to produce a balanced encoded bit stream, the technique is called guided scrambling, (GS). The concept of GS coding is explained, and design parameters which ensure good line code characteristics are discussed. The performance of a number of guided scrambling configurations is reported in terms of maximum consecutive like-encoded bits, encoded stream disparity, decoder error extension, and power spectral density of the encoded signal. Comparison of guided scrambling with conventional line code techniques indicates a performance which approaches that of alphabetic lookup table codes with an implementation complexity similar to that of current nonalphabetic coding techniques.