A simple proof of one probabilistic inequality is presented. Let P and Q be two given probability measures on a measurable space (/spl Xscr/, /spl Ascr/). We consider testing of hypotheses P and Q using one observation. For an arbitrary decision rule, let /spl alpha/ and /spl beta/ denote the two kinds of error probabilities. If both error probabilities have equal costs (or we want to minimize the maximum of them) then it is natural to investigate the minimal possible sum inf{/spl alpha/+/spl beta/} for the best decision rule.
{"title":"On One Useful Inequality in Testing of Hypotheses","authors":"M. Burnashev","doi":"10.1109/18.681348","DOIUrl":"https://doi.org/10.1109/18.681348","url":null,"abstract":"A simple proof of one probabilistic inequality is presented. Let P and Q be two given probability measures on a measurable space (/spl Xscr/, /spl Ascr/). We consider testing of hypotheses P and Q using one observation. For an arbitrary decision rule, let /spl alpha/ and /spl beta/ denote the two kinds of error probabilities. If both error probabilities have equal costs (or we want to minimize the maximum of them) then it is natural to investigate the minimal possible sum inf{/spl alpha/+/spl beta/} for the best decision rule.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"395 1","pages":"1668-1670"},"PeriodicalIF":0.0,"publicationDate":"1998-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80009352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive a closed-form expression for the power spectral density of amplitude/phase-shift keyed bit sequences randomized through self-synchronizing scrambling when the source sequence is a stationary sequence of statistically independent bits. In addition to the dependence on the symbol pulse shape, duration, and the signal space values with which symbols are represented, we show that the power spectral density is dependent only on the probability of logic ones in the source bit stream, the period of the impulse response of the scrambling shift register, and the number of logic ones in this period. Our results confirm that optimum randomization results with use of primitive scrambling polynomials and poorest randomization occurs with "two-tap" polynomials of the form x/sup D/+1.
{"title":"On the Power Spectral Density of Self-Synchronizing Scrambled Sequences","authors":"I. Fair, V. Bhargava, Qiang Wang","doi":"10.1109/18.681352","DOIUrl":"https://doi.org/10.1109/18.681352","url":null,"abstract":"We derive a closed-form expression for the power spectral density of amplitude/phase-shift keyed bit sequences randomized through self-synchronizing scrambling when the source sequence is a stationary sequence of statistically independent bits. In addition to the dependence on the symbol pulse shape, duration, and the signal space values with which symbols are represented, we show that the power spectral density is dependent only on the probability of logic ones in the source bit stream, the period of the impulse response of the scrambling shift register, and the number of logic ones in this period. Our results confirm that optimum randomization results with use of primitive scrambling polynomials and poorest randomization occurs with \"two-tap\" polynomials of the form x/sup D/+1.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"7 1","pages":"1687-1693"},"PeriodicalIF":0.0,"publicationDate":"1998-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89653740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The generalized cyclotomic sequence of order two has several good randomness properties and behaves like the Legendre sequence in several aspects. We calculate the autocorrelation values of the generalized cyclotomic sequence of order two. Our result shows that this sequence could have very good autocorrelation property and pattern distributions of length two if the two primes are chosen properly.
{"title":"Autocorrelation Values of Generalized Cyclotomic Sequences of Order Two","authors":"C. Ding","doi":"10.1109/18.681354","DOIUrl":"https://doi.org/10.1109/18.681354","url":null,"abstract":"The generalized cyclotomic sequence of order two has several good randomness properties and behaves like the Legendre sequence in several aspects. We calculate the autocorrelation values of the generalized cyclotomic sequence of order two. Our result shows that this sequence could have very good autocorrelation property and pattern distributions of length two if the two primes are chosen properly.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"44 1","pages":"1699-1702"},"PeriodicalIF":0.0,"publicationDate":"1998-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90754055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Group codes over dihedral groups D/sub M/, with 2M elements, that are two-level constructible using a binary code and a code over Z/sub M/ residue class integer ring modulo M, as component codes are studied for arbitrary M. A set of necessary and sufficient conditions on the component codes for the two-level construction to result in a group code over D/sub M/ are obtained. The conditions differ for M odd and even. Using two-level group codes over D/sub M/ as label codes, the performance of a block-coded modulation scheme is discussed under all possible matched labelings of 2M-APSK and 2M-SPSK (asymmetric and symmetric PSK) signal sets in terms of the minimum squared Euclidean distance. Matched labelings that lead to automorphic Euclidean distance equivalent codes are identified. It is shown that depending upon the ratio of Hamming distances of the component codes some labelings perform better than others. The best labeling is identified under a set of restrictive conditions. Finally, conditions on the component codes for phase rotational invariance properties of the signal space codes are discussed.
{"title":"Block-Coded PSK Modulation Using Two-Level Group Codes Over Dihedral Groups","authors":"Jyoti Bali, B. Rajan","doi":"10.1109/18.681341","DOIUrl":"https://doi.org/10.1109/18.681341","url":null,"abstract":"A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Group codes over dihedral groups D/sub M/, with 2M elements, that are two-level constructible using a binary code and a code over Z/sub M/ residue class integer ring modulo M, as component codes are studied for arbitrary M. A set of necessary and sufficient conditions on the component codes for the two-level construction to result in a group code over D/sub M/ are obtained. The conditions differ for M odd and even. Using two-level group codes over D/sub M/ as label codes, the performance of a block-coded modulation scheme is discussed under all possible matched labelings of 2M-APSK and 2M-SPSK (asymmetric and symmetric PSK) signal sets in terms of the minimum squared Euclidean distance. Matched labelings that lead to automorphic Euclidean distance equivalent codes are identified. It is shown that depending upon the ratio of Hamming distances of the component codes some labelings perform better than others. The best labeling is identified under a set of restrictive conditions. Finally, conditions on the component codes for phase rotational invariance properties of the signal space codes are discussed.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"32 1","pages":"1620-1631"},"PeriodicalIF":0.0,"publicationDate":"1998-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75292677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of calculating minimum-redundancy codes for alphabets in which there is significant repetition of symbol weights. On a sorted-by-weight alphabet of, n symbols and r distinct symbol weights we show that a minimum-redundancy prefix code can be constructed in O(r+r log(n/r)) time, and that a minimum redundancy L-bit length-limited prefix code can be constructed in O(Lr+Lrlog(n/r)) time. When r is small relative to n-which is necessarily the case for most practical coding problems on large alphabets-these bounds represent a substantial improvement upon the best previous algorithms for these two problems, which consumed O(n) time and O(nL) time, respectively. The improved algorithms are also space-efficient.
{"title":"Efficient Construction of Minimum-Redundancy Codes for Large Alphabets","authors":"Alistair Moffat, A. Turpin","doi":"10.1109/18.681345","DOIUrl":"https://doi.org/10.1109/18.681345","url":null,"abstract":"We consider the problem of calculating minimum-redundancy codes for alphabets in which there is significant repetition of symbol weights. On a sorted-by-weight alphabet of, n symbols and r distinct symbol weights we show that a minimum-redundancy prefix code can be constructed in O(r+r log(n/r)) time, and that a minimum redundancy L-bit length-limited prefix code can be constructed in O(Lr+Lrlog(n/r)) time. When r is small relative to n-which is necessarily the case for most practical coding problems on large alphabets-these bounds represent a substantial improvement upon the best previous algorithms for these two problems, which consumed O(n) time and O(nL) time, respectively. The improved algorithms are also space-efficient.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"119 1","pages":"1650-1657"},"PeriodicalIF":0.0,"publicationDate":"1998-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77931983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q(2/sup m/), m/spl ges/3, are constructed using a binary code and a code over Z(2/sup m-1/), the ring of integers modulo 2/sup m-1/ as component codes and a mapping f from Z/sub 2//spl times/Z(2/sup m-1/)to Q(2/sup m/). A set of necessary and sufficient conditions on the component codes is derived which will give group codes over Q(2/sup m/). Given the generator matrices of the component codes, the computational effort involved in checking the necessary and sufficient conditions is discussed. Starting from a four-dimensional signal set matched to Q(2/sup m/), it is shown that the Euclidean space codes obtained from the group codes over Q(2/sup m/) have Euclidean distance profiles which are independent of the coset representative selection involved in f. A closed-form expression for the minimum Euclidean distance of the resulting group codes over Q(2/sup m/) is obtained in terms of the Euclidean distances of the component codes. Finally, it is shown that all four-dimensional signal sets matched to Q(2/sup m/) have the same Euclidean distance profile and hence the Euclidean space codes corresponding to each signal set for a given group code over Q(2/sup m/) are automorphic Euclidean-distance equivalent.
{"title":"Block-Coded Modulation Using Two-Level Group Codes Over Generalized Quaternion Groups","authors":"T. Selvakumaran, B. Rajan","doi":"10.1109/18.746847","DOIUrl":"https://doi.org/10.1109/18.746847","url":null,"abstract":"A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q(2/sup m/), m/spl ges/3, are constructed using a binary code and a code over Z(2/sup m-1/), the ring of integers modulo 2/sup m-1/ as component codes and a mapping f from Z/sub 2//spl times/Z(2/sup m-1/)to Q(2/sup m/). A set of necessary and sufficient conditions on the component codes is derived which will give group codes over Q(2/sup m/). Given the generator matrices of the component codes, the computational effort involved in checking the necessary and sufficient conditions is discussed. Starting from a four-dimensional signal set matched to Q(2/sup m/), it is shown that the Euclidean space codes obtained from the group codes over Q(2/sup m/) have Euclidean distance profiles which are independent of the coset representative selection involved in f. A closed-form expression for the minimum Euclidean distance of the resulting group codes over Q(2/sup m/) is obtained in terms of the Euclidean distances of the component codes. Finally, it is shown that all four-dimensional signal sets matched to Q(2/sup m/) have the same Euclidean distance profile and hence the Euclidean space codes corresponding to each signal set for a given group code over Q(2/sup m/) are automorphic Euclidean-distance equivalent.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"1 1","pages":"365-372"},"PeriodicalIF":0.0,"publicationDate":"1997-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83617451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shaping codes can be obtained from optimal unequal cost coding algorithms due to Lempel, Even, and Cohn (1973) in the variable-length-to-block case and to Varn (1971) in the block-to-variable-length case. The former is related to ad hoc approaches to shaping previously described while the latter is novel in the shaping context.
{"title":"Variable-Length Unequal Cost Parsing and Coding for Shaping","authors":"J. Abrahams","doi":"10.1109/18.681344","DOIUrl":"https://doi.org/10.1109/18.681344","url":null,"abstract":"Shaping codes can be obtained from optimal unequal cost coding algorithms due to Lempel, Even, and Cohn (1973) in the variable-length-to-block case and to Varn (1971) in the block-to-variable-length case. The former is related to ad hoc approaches to shaping previously described while the latter is novel in the shaping context.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"14 1","pages":"1648-1650"},"PeriodicalIF":0.0,"publicationDate":"1997-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84532380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze the quadrature amplitude modulation (QAM) version of multilevel modulation with multistage decoding using a suboptimal metric, when transmission takes place over a memoryless Gaussian channel. The upper bounds for decoding error probabilities are functions of the Chernoff bounding parameter Z. We argue that the conventional approximation of Z is not adequate; new values of Z that tightens the error bounds without causing them to lose their validity are given. The capacity for this system is also calculated, and we conclude that the use of a suboptimal metric in multistage decoding causes very little degradation in capacity compared to when the optimal metric is used in each decoding stage.
{"title":"On the Calculation of the Error Probability for a Multilevel Modulation Scheme Using QAM-Signaling","authors":"K. Engdahl, K. Zigangirov","doi":"10.1109/18.681340","DOIUrl":"https://doi.org/10.1109/18.681340","url":null,"abstract":"We analyze the quadrature amplitude modulation (QAM) version of multilevel modulation with multistage decoding using a suboptimal metric, when transmission takes place over a memoryless Gaussian channel. The upper bounds for decoding error probabilities are functions of the Chernoff bounding parameter Z. We argue that the conventional approximation of Z is not adequate; new values of Z that tightens the error bounds without causing them to lose their validity are given. The capacity for this system is also calculated, and we conclude that the use of a suboptimal metric in multistage decoding causes very little degradation in capacity compared to when the optimal metric is used in each decoding stage.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"11 1","pages":"1612-1620"},"PeriodicalIF":0.0,"publicationDate":"1997-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76698898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a 2D-symmetric lattice /spl Lambda/, it was conjectured by Forney (1989) that the projection of the Voronoi region R(/spl Lambda/) onto two coordinates equals the Voronoi region of the constituent 2D-sublattice /spl Lambda//spl nu//sub 2/. We present a three-dimensional counterexample. >
{"title":"A counterexample to a Voronoi region conjecture","authors":"R. Urbanke, D. Agrawal","doi":"10.1109/18.391270","DOIUrl":"https://doi.org/10.1109/18.391270","url":null,"abstract":"Given a 2D-symmetric lattice /spl Lambda/, it was conjectured by Forney (1989) that the projection of the Voronoi region R(/spl Lambda/) onto two coordinates equals the Voronoi region of the constituent 2D-sublattice /spl Lambda//spl nu//sub 2/. We present a three-dimensional counterexample. >","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"32 1","pages":"1195-1196"},"PeriodicalIF":0.0,"publicationDate":"1995-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89945911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The trellis complexity of composite-length cyclic codes (CLCC's) is addressed. We first investigate the trellis properties of concatenated and product codes in general. Known factoring of CLCC's into concatenated subcodes is thereby employed to derive upper bounds on the minimal trellis size and state-space profile. New decomposition of CLCC's into product subcodes is established and utilized to derive further upper hounds on the trellis parameters. The coordinate permutations that correspond to these bounds are exhibited. Additionally, new results on the generalized Hamming weights of CLCC's are obtained. The reduction in trellis complexity of many CLCC's leads to soft-decision decoders with relatively low complexity. >
{"title":"Trellis-oriented decomposition and trellis complexity of composite-length cyclic codes","authors":"Y. Berger, Yair Be’ery","doi":"10.1109/18.391268","DOIUrl":"https://doi.org/10.1109/18.391268","url":null,"abstract":"The trellis complexity of composite-length cyclic codes (CLCC's) is addressed. We first investigate the trellis properties of concatenated and product codes in general. Known factoring of CLCC's into concatenated subcodes is thereby employed to derive upper bounds on the minimal trellis size and state-space profile. New decomposition of CLCC's into product subcodes is established and utilized to derive further upper hounds on the trellis parameters. The coordinate permutations that correspond to these bounds are exhibited. Additionally, new results on the generalized Hamming weights of CLCC's are obtained. The reduction in trellis complexity of many CLCC's leads to soft-decision decoders with relatively low complexity. >","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"20 1","pages":"1185-1191"},"PeriodicalIF":0.0,"publicationDate":"1995-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88848806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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