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- Agrell, Erik, 1965, et al.
(författare)
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Gray coding for multilevel constellations in Gaussian noise
- 2007
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Ingår i: IEEE Transactions on Information Theory. - 0018-9448 .- 1557-9654. ; 53:1, s. 224-235
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Tidskriftsartikel (refereegranskat)abstract
- The problem of finding the optimal labeling (bit-to-symbol mapping) of multilevel coherent phase shift keying (PSK), pulse amplitude modulation (PAM), and quadrature amplitude modulation (QAM) constellations with respect to minimizing the bit-error probability (BEP) over a Gaussian channel is addressed. We show that using the binary reflected Gray code (BRGC) to label the signal constellation results in the lowest possible BEP for high enough signal energy-to-noise ratios and analyze what is "high enough"in this sense. It turns out that the BRGC is optimal for PSK and PAM systems whenever the target BEP is at most a few percent, which covers most systems of practical interest. New and simple closed-form expressions are presented for the BEP of PSK, PAM, and QAM using the BRGC. © 2007 IEEE.
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| 2. |
- Agrell, Erik, 1965, et al.
(författare)
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On the optimality of the binary reflected Gray code
- 2004
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Ingår i: IEEE Transactions on Information Theory. - 0018-9448 .- 1557-9654. ; 50:12, s. 3170-3182
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns the problem of selecting a binary labeling for the signal constellation in M-PSK, M-PAM, and M-QAM communication systems. Gray labelings are discussed and the original work by Frank Gray is analyzed. As is noted, the number of distinct Gray labelings that result in different bit-error probability grows rapidly with increasing constellation size. By introducing a recursive Gray labeling construction method called expansion, the paper answers the natural question of what labeling, among all possible constellation labelings, will give the lowest possible average probability of bit errors for the considered constellations. Under certain assumptions on the channel, the answer is that the labeling proposed by Gray, the binary reflected Gray code, is the optimal labeling for all three constellations, which has, surprisingly, never been proved before.
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| 3. |
- Agrell, Erik, 1965, et al.
(författare)
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The binary reflected Gray code is optimal for M-PSK
- 2004
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Ingår i: Proc. IEEE International Symposium on Information Theory, Chicago, Illinois, USA. - 0780382803 ; , s. 164-
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Konferensbidrag (refereegranskat)abstract
- This paper is concerned with the problem of selecting a binary labeling for the signal constellation in an M-PSK communication system. A good starting point is labelings having the Gray property, but this is not altogether enough, since the number of distinct Gray labelings that result in different bit error probability grows rapidly with increasing constellation size. By introducing a recursive Gray labeling construction method called expansion, the paper answers the natural question of what labeling, among all possible constellation labelings (not only Gray), that will give the lowest possible average probability of bit errors. Under certain assumptions on the channel, the answer is that the labeling originally proposed by Gray, the binary reflected Gray code, is the optimal labeling for M-PSK systems, which has, surprisingly, never been proved before.
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| 6. |
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| 7. |
- Lassing, Johan, 1973, et al.
(författare)
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Computation of the Exact Bit-Error Rate of Coherent M-ary PSK With Gray Code Bit Mapping
- 2003
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Ingår i: IEEE Transactions on Communications. - 0090-6778 .- 1558-0857. ; 51:11, s. 1758-1760
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Tidskriftsartikel (refereegranskat)abstract
- The problem of calculating the average bit-error probability (BEP) of coherent M-ary phase-shift keying (PSK) over a Gaussian channel has been studied previously in the literature. A solution to the problem for systems using a binary reflected Gray code (BRGC) to map bits to symbols was first presented by Lee. In this letter, we show that the results obtained by Lee are incorrect for M greater than or equal to 16. We show that the reason for this is an invalid assumption that the bit-error rate (BER) is independent of the transmitted symbols, an assumption which has also propagated to textbooks. We give a new expression for the BER of M-PSK systems using the BRGC and compare this with Lee's results.
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| 8. |
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| 9. |
- Lassing, Johan, 1973
(författare)
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On the labeling of Signal Constellations
- 2005
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The work in this thesis studies a simple model of a digital communication system in which the transmitter operates on a sequence of information bits by grouping them into blocks of m bits and mapping such blocks onto one out of M = 2m waveforms, which is transmitted over the channel. The receiver recovers the original message by demapping the transmitted signals disturbed by noise back to m-bit blocks using maximum-likelihood symbol detection. Whenever a bit produced by the source is not regenerated correctly at the destination, a bit error is said to have occurred. The presented work addresses certain methods to limit the occurrence of such events. A communication system operating as described above is commonly studied within a geometrical framework in which the transmitted waveforms are represented as vectors in a linear vector space and this set of vectors is referred to as the signal constellation. The labelling of the signal constellation, i.e., the mapping of m-bit blocks onto the signal constellation is of big importance for the resulting bit error probability (BEP). We study the constellation labelling problem for two-dimensional constellations and for systems using PSK, PAM, or QAM; three types of modulation methods that are commonly encountered in the existing literature and in practice. We present a new expression for the exact BEP of systems using coherent symbol detection of M-PSK, which rectifies a common misconception in previously published results found in the literature. We introduce the average distance spectrum (ADS) which allows for a convenient method to analyze the influence of the constellation labelling on the BEP, which leads to new results relating to the commonly encountered binary reflected Gray code (BRGC). In particular, we show that the BRGC is the optimal labelling with respect to minimizing the BEP over a Gaussian channel as long as the BEP is less than a few percent. This result, which has not been proved before, motivates and validates the extensive usage of the BRGC both in theoretical studies and in practice. Keywords: constellation labelling, bit-to-symbol mapping, optimal labelling, bit error probability (BEP), Gray code, binary reflected Gray code (BRGC), phase-shift keying (PSK), pulse amplitude modulation (PAM), quadrature amplitude modulation (QAM).
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