1519:
Viterbi algorithm, in which the branch-metric computation involves data-dependent noise prediction. Because the predictor coefficients and prediction error both depend on the local data pattern, the resulting structure has been called a data-dependent NPML detector. Reduced-state sequence detection schemes can be applied to data-dependent NPML, reducing implementation complexity.
197:
it correlated. A close match between the desired target polynomial and the physical channel can minimize losses. An effective way to achieve near optimal performance independently of the operating point—in terms of linear recording density—and the noise conditions is via noise prediction. In particular, the power of a stationary noise sequence
1571:(ARMA) stationary noise processes The concept was extended to include a variety of non-stationary noise sources, such as head, transition jitter and media noise; it was applied to various post-processing schemes. Noise prediction became an integral part of the metric computation in a wide variety of iterative detection/decoding schemes.
1503:
render the data-dependent medium noise a significant component of the total noise affecting performance. Because medium noise is correlated and data-dependent, information about noise and data patterns in past samples can provide information about noise in other samples. Thus, the concept of noise prediction for stationary
196:
In the absence of noise enhancement or noise correlation, the PRML sequence detector performs maximum-likelihood sequence estimation. As the operating point moves to higher linear recording densities, optimality declines with linear partial-response (PR) equalization, which enhances noise and renders
1535:
error-correcting code. Moreover, the soft information computed by the decoder can be fed back again to the soft detector to improve detection performance. In this way it is possible to iteratively improve the error-rate performance at the decoder output in successive soft detection/decoding rounds.
81:
on which the
Viterbi algorithm operates as well as tentative decisions corresponding to the path memory associated with each trellis state. NPML detectors can thus be viewed as reduced-state sequence-estimation detectors offering a range of implementation complexities. The complexity is governed by
1518:
for channels with ISI has been derived. In particular, it when the data-dependent noise is conditionally Gauss–Markov, the branch metrics can be computed from the conditional second-order statistics of the noise process. In other words, the optimum MLSE can be implemented efficiently by using the
1502:
Depending on the surface roughness and particle size, particulate media might exhibit nonstationary data-dependent transition or medium noise rather than colored stationary medium noise. Improvements o\in the quality of the readback head as well as the incorporation of low-noise preamplifiers may
1534:
algorithm then NPML and NPML-like detection allow the computation of soft reliability information on individual code symbols, while retaining all the performance advantages associated with noise-predictive techniques. The soft information generated in this manner is used for soft decoding of the
1551:
techniques were introduced into disk drives to improve the drive error-rate performance for operation at higher areal densities and for reducing manufacturing and servicing costs. In the early 1990s, partial-response class-4 (PR4) signal shaping in conjunction with maximum-likelihood sequence
53:
and coding established themselves as cost-efficient techniques for enabling additional increases in areal density while preserving reliability. Accordingly, the deployment of sophisticated detection schemes based on the concept of noise prediction are of paramount importance in the disk drive
1498:
can be viewed as a family of reduced-state detectors with embedded feedback. These detectors exist in a form in which the decision-feedback path can be realized by simple table look-up operations, whereby the contents of these tables can be updated as a function of the operating conditions.
1351:
920:
499:
678:
An infinitely long predictor filter would lead to a sequence detector structure that requires an unbounded number of states. Therefore, finite-length predictors that render the noise at the input of the sequence detector approximately white are of interest.
1499:
Analytical and experimental studies have shown that a judicious tradeoff between performance and state complexity leads to practical schemes with considerable performance gains. Thus, reduced-state approaches are promising for increasing linear density.
1095:
1494:
747:
1590:
line of HDD products in the late 1990s. Eventually, noise-predictive detection became a de facto standard and in its various instantiations became the core technology of the read channel module in HDD systems.
1212:
1428:
Reduced-state sequence-detection schemes have been studied for application in the magnetic-recording channel and the references therein. For example, the NPML detectors with generalized PR target polynomials
1522:
NPML and its various forms represent the core read-channel and detection technology used in recording systems employing advanced error-correcting codes that lend themselves to soft decoding, such as
578:
637:
383:
376:
979:
254:
operator corresponds to a delay of one bit interval, at the output of a PR equalizer can be minimized by using an infinitely long predictor. A linear predictor with coefficients
1578:(PRML) and noise-predictive maximum-likelihood (NPML) detection and its impact on the industry were recognized in 2005 by the European Eduard Rhein Foundation Technology Award.
303:
143:
169:
denoting the maximum number of controlled ISI terms introduced by the combination of a partial-response shaping equalizer and the noise predictor. By judiciously choosing
1563:
NPML detection was first described in 1996 and eventually found wide application in HDD read channel design. The “noise predictive” concept was later extended to handle
1169:
912:
1132:
874:
109:
1421:
1384:
1200:
1012:
814:
781:
671:
334:
226:
838:
250:
189:
165:
1560:)-constrained coding. This development paved the way for future applications of advanced coding and signal-processing techniques in magnetic data storage.
919:
1789:
Eleftheriou, E.; S. Ölçer; R. A. Hutchins (2010). "Adaptive Noise-Predictive
Maximum-Likelihood (NPML) Data Detection for Magnetic Tape Storage Systems".
193:, practical NPML detectors can be devised that improve performance over PRML and EPRML detectors in terms of error rate and/or linear recording density.
62:
The NPML family of sequence-estimation data detectors arise by embedding a noise prediction/whitening process into the branch metric computation of the
1021:
49:
Although advances in head and media technologies historically have been the driving forces behind the increases in the areal recording density,
1434:
1346:{\displaystyle {\hat {a}}\left(D\right)=arg\ min_{a\left(D\right)}\parallel z\left(D\right)-a\left(D\right)G\left(D\right)\parallel {^{2}}}
687:
38:
signal. NPML aims at minimizing the influence of noise in the detection process. Successfully applied, it allows recording data at higher
2318:
1908:
Chevillat, P. R.; E. Eleftheriou (1989). "Decoding of
Trellis-encoded Signals in the Presence of Intersymbol Interference and Noise".
1507:
noise sources developed in can be naturally extended to the case where noise characteristics depend highly on local data patterns.
2222:
Cideciyan, R. D.; J. D. Coker; E. Eleftheriou; R. L. Galbraith (2001). "NPML Detection
Combined with Parity-Based Postprocessing".
1940:
Caroselli, J.; S. A. Altekar; P. McEwen; J. K. Wolf (1997). "Improved
Detection for Magnetic Recording Systems with Media Noise".
1568:
842:, give rise to NPML systems when combined with sequence detection In this case, the effective memory of the system is limited to
2207:
Sonntag, J. L.; B. Vasic (2000). "Implementation and Bench
Characterization of a Read Channel with Parity Check Postprocessor".
1207:
The NPML detector is efficiently implemented via the
Viterbi algorithm, which recursively computes the estimated data sequence.
1515:
1135:
2274:
1769:
Eleftheriou, E.; W. Hirt (1996). "Noise-Predictive
Maximum-Likelihood (NPML) Detection for the Magnetic Recording Channel".
508:
2059:
Bahl, L. R.; J. Cocke; F. Jelinek; J. Raviv (1974). "Optimal
Decoding of Linear Codes for Minimizing Symbol Error Rate".
1612:
1575:
1553:
43:
1838:
Eyuboglu, V. M.; S. U. Qureshi (1998). "Reduced-state
Sequence Estimation with Set Partitioning and Decision Feedback".
494:{\displaystyle e\left(D\right)=n\left(D\right)-{\acute {n}}\left(D\right)=n\left(D\right)\left(1-P\left(D\right)\right)}
2429:
1681:
Coker, J. D.; E. Eleftheriou; R. L. Galbraith; W. Hirt (1998). "Noise-Predictive Maximum Likelihood (NPML) Detection".
588:
2257:
Feng, W.; A. Vityaev; G. Burd; N. Nazari (2000). "On the performance of parity codes in magnetic recording systems".
1823:
Chevillat, P.R.; E. Eleftheriou; D. Maiwald (1992). "Noise Predictive Partial-Response Equalizers and Applications".
2149:
Cideciyan, R. D.; F. Dolivo; R. Hermann; W. Hirt; W. Schott (1992). "A PRML System for Digital Magnetic Recording".
2405:
2089:
Kobayashi, H.; D. T. Tang (1970). "Application of Partial-Response Channel Coding to Magnetic Recording Systems".
341:
932:
2424:
1645:
Eleftheriou, E. (2003). John G., Proakis (ed.). "Signal Processing for Magnetic-Recording Channels".
257:
50:
2304:
2391:
2036:
116:
67:
2176:
Eleftheriou, E.; W. Hirt (1996). "Improving Performance of PRML/EPRML through Noise Prediction".
2259:
Globecom '00 - IEEE. Global Telecommunications Conference. Conference Record (Cat. No.00CH37137)
1725:
Eleftheriou, E; W. Hirt (1996). "Improving Performance of PRML/EPRML through Noise Prediction".
2031:
2322:
2119:
Kobayashi, H. (1971). "Application of Probabilistic Decoding to Digital Magnetic Recording".
1141:
884:
24:
1526:(LDPC) codes. For example, if noise-predictive detection is performed in conjunction with a
2231:
2181:
1949:
1882:
1847:
1730:
1690:
1564:
1527:
1105:
847:
87:
2022:
Moon, J.; J. Park (2001). "Pattern-Dependent Noise Prediction in Signal-Dependent Noise".
1397:
1360:
1176:
988:
790:
757:
647:
310:
202:
8:
2235:
2185:
1953:
1886:
1851:
1734:
1694:
2292:
2280:
1965:
1607:
823:
235:
174:
150:
1873:
Duell-Hallen, A.; C. Heegard (1989). "Delayed Decision-feedback Sequence Estimation".
2270:
1617:
1504:
63:
20:
2368:
2284:
1990:
Kavcic, A.; J. M. F. Moura (2000). "The Viterbi Algorithm and Markov Noise Memory".
1969:
983:
then this corresponds to the classical PR4 signal shaping. Using a whitening filter
2262:
2239:
2189:
2158:
2128:
2098:
2068:
2041:
1999:
1957:
1917:
1890:
1855:
1798:
1738:
1698:
1595:
66:. The latter is a data detection technique for communication channels that exhibit
35:
28:
1556:
technique replaced the peak detection systems that used run-length-limited (RLL) (
2221:
1548:
78:
2266:
1544:
2418:
2072:
1802:
1090:{\displaystyle G\left(D\right)=\left(1-{D^{2}}\right)\times W\left(D\right)}
46:(PRML), and extended partial-response maximum likelihood (EPRML) detection.
1511:
74:
39:
1939:
1680:
2132:
2102:
31:
densities. It is used for retrieval of data recorded on magnetic media.
2343:
2243:
2193:
2162:
2045:
2003:
1961:
1742:
1702:
1921:
1894:
1489:{\displaystyle G\left(D\right)=F\left(D\right)\times W\left(D\right)}
742:{\displaystyle G\left(D\right)=F\left(D\right)\times W\left(D\right)}
1859:
2148:
1822:
1788:
1392:
the signal sequence at the output of the noise whitening filter
916:-state NPML detector if no reduced-state detection is employed.
34:
Data are read back by the read head, producing a weak and noisy
2256:
2058:
1134:
symbols. In this case, the full-state NMPL detector performs
923:
a diagram of a Magnetic-recording system with NPML detection
1531:
1523:
785:
is a polynomial of order S and the noise-whitening filter
2392:"Marvell Contributes to Read-Rite's Areal Density Record"
1587:
1100:
and the effective ISI memory of the system is limited to
642:
is the one that minimizes the prediction error sequence
1907:
1510:
By modeling the data-dependent noise as a finite-order
73:
Reliable operation of the process is achieved by using
1872:
1388:
denotes the binary sequence of recorded data bits and
1837:
1437:
1400:
1363:
1215:
1179:
1144:
1108:
1024:
991:
935:
887:
850:
826:
793:
760:
690:
650:
591:
511:
386:
344:
313:
260:
238:
205:
177:
153:
119:
90:
1989:
573:{\displaystyle P\left(D\right)=p_{1}D+p_{2}{D^{2}}+}
2088:
1598:and in 2011 in IBM's enterprise-class tape drives.
503:is white with minimum power. The optimum predictor
2175:
1768:
1724:
1488:
1415:
1378:
1345:
1194:
1163:
1126:
1089:
1006:
973:
906:
868:
832:
808:
775:
741:
665:
631:
572:
493:
370:
328:
297:
244:
220:
183:
159:
137:
103:
632:{\displaystyle W\left(D\right)=1-P\left(D\right)}
82:the number of detector states, which is equal to
2416:
2206:
1784:
1782:
1780:
1764:
1762:
1760:
1758:
1756:
1754:
1752:
1720:
1718:
1716:
1714:
1712:
682:Generalized PR shaping polynomials of the form
378:. Then, the prediction-error sequence given by
2114:
2112:
77:decisions associated with the branches of the
2209:Digest of the Magnetic Recording Conf. (TMRC)
1640:
1638:
1636:
1634:
1632:
2144:
2142:
2084:
2082:
1935:
1933:
1931:
1777:
1749:
1709:
274:
261:
2406:"Samsung SV0802N hard drive specifications"
2109:
2021:
1818:
1816:
1814:
1812:
1644:
1586:NPML technology were first introduced into
2369:"Hitachi to Buy IBM's Hard Drive Business"
1629:
1596:Linear Tape Open (LTO) tape drive products
371:{\displaystyle {\acute {n}}\left(D\right)}
17:Noise-Predictive Maximum-Likelihood (NPML)
2139:
2118:
2079:
2035:
1985:
1983:
1981:
1979:
1928:
1653:. John Wiley & Sons, Inc.: 2247–2268.
974:{\displaystyle F\left(D\right)=1-{D^{2}}}
1809:
1647:Wiley Encyclopedia of Telecommunications
1594:In 2010, NPML was introduced into IBM's
2017:
2015:
2013:
1676:
1674:
1672:
1670:
1668:
1666:
1664:
1662:
1660:
42:. Alternatives include peak detection,
2417:
1976:
1530:(MAP) detection algorithm such as the
1136:maximum likelihood sequence estimation
583:or the optimum noise-whitening filter
338:produces the estimated noise sequence
305:,..., operating on the noise sequence
2366:
2010:
1657:
1016:, the generalized PR target becomes
2261:. Vol. 3. pp. 1877–1881.
1613:Partial-response maximum-likelihood
1576:partial-response maximum-likelihood
44:partial-response maximum-likelihood
13:
918:
14:
2441:
1574:The pioneering research work on
1171:-state trellis corresponding to
2398:
2389:
2383:
2360:
2336:
2311:
2250:
2215:
2200:
2169:
2052:
1581:
1552:detection, eventually known as
1543:Beginning in the 1980s several
298:{\displaystyle \{p_{l}\},l=1,2}
1901:
1866:
1831:
1410:
1404:
1373:
1367:
1222:
1189:
1183:
1001:
995:
803:
797:
770:
764:
660:
654:
323:
317:
215:
209:
1:
1825:Proc. IEEE Int. Conf. Commun.
1771:Proc. IEEE Int. Conf. Commun.
1623:
1569:autoregressive moving-average
138:{\displaystyle 0\leq K\leq M}
57:
25:magnetic data storage systems
2348:www.eduard-rhein-stiftung.de
2180:. 32 part 1 (5): 3968–3970.
1729:. 32 part 1 (5): 3968–3970.
7:
1601:
10:
2446:
2267:10.1109/GLOCOM.2000.891959
1538:
70:(ISI) with finite memory.
2430:Digital signal processing
2151:IEEE J. Sel. Areas Commun
2024:IEEE J. Sel. Areas Commun
1567:(AR) noise processes and
51:digital signal processing
21:digital signal-processing
2073:10.1109/TIT.1974.1055186
1803:10.1147/JRD.2010.2041034
1524:low-density parity check
68:intersymbol interference
2344:"Eduard Rhein Stiftung"
2319:"Technologiepreis 2005"
2061:IEEE Trans. Inf. Theory
1992:IEEE Trans. Inf. Theory
1164:{\displaystyle 2^{L+2}}
907:{\displaystyle 2^{L}+S}
675:in a mean-square sense
1490:
1417:
1380:
1347:
1196:
1165:
1128:
1091:
1008:
975:
924:
908:
870:
834:
818:has a finite order of
810:
777:
743:
667:
633:
574:
495:
372:
330:
299:
246:
222:
185:
161:
139:
105:
1491:
1418:
1381:
1348:
1197:
1166:
1129:
1127:{\displaystyle M=L+2}
1092:
1009:
976:
922:
909:
871:
869:{\displaystyle M=L+S}
835:
811:
778:
744:
668:
634:
575:
496:
373:
331:
300:
247:
223:
186:
162:
140:
106:
104:{\displaystyle 2^{K}}
27:that operate at high
23:methods suitable for
1528:maximum a posteriori
1435:
1416:{\displaystyle W(D)}
1398:
1379:{\displaystyle a(D)}
1361:
1213:
1195:{\displaystyle G(D)}
1177:
1142:
1106:
1022:
1007:{\displaystyle W(D)}
989:
933:
885:
848:
824:
809:{\displaystyle W(D)}
791:
776:{\displaystyle F(D)}
758:
688:
666:{\displaystyle e(D)}
648:
589:
509:
384:
342:
329:{\displaystyle n(D)}
311:
258:
236:
221:{\displaystyle n(D)}
203:
175:
151:
117:
88:
2236:2001ITM....37..714C
2186:1996ITM....32.3968E
2133:10.1147/rd.151.0064
2103:10.1147/rd.144.0368
1954:1997ITM....33.2779C
1887:1989ITCom..37..428D
1852:1988ITCom..36...13E
1797:(2, paper 7): 7:1.
1735:1996ITM....32.3968E
1695:1998ITM....34..110C
1910:IEEE Trans. Commun
1875:IEEE Trans. Commun
1840:IEEE Trans. Commun
1608:Maximum likelihood
1486:
1413:
1376:
1343:
1192:
1161:
1124:
1087:
1004:
971:
927:As an example, if
925:
904:
866:
830:
806:
773:
739:
663:
629:
570:
491:
368:
326:
295:
242:
218:
181:
157:
135:
101:
2276:978-0-7803-6451-6
2244:10.1109/20.917606
2194:10.1109/20.539233
2163:10.1109/49.124468
2046:10.1109/49.920181
2004:10.1109/18.817531
1962:10.1109/20.617728
1743:10.1109/20.539233
1703:10.1109/20.663468
1618:Viterbi algorithm
1253:
1225:
1138:(MLSE) using the
833:{\displaystyle L}
430:
354:
245:{\displaystyle D}
184:{\displaystyle K}
160:{\displaystyle M}
64:Viterbi algorithm
2437:
2425:Hard disk drives
2410:
2409:
2402:
2396:
2395:
2387:
2381:
2380:
2378:
2376:
2364:
2358:
2357:
2355:
2354:
2340:
2334:
2333:
2331:
2330:
2321:. Archived from
2315:
2309:
2308:
2302:
2298:
2296:
2288:
2254:
2248:
2247:
2224:IEEE Trans. Magn
2219:
2213:
2212:
2204:
2198:
2197:
2178:IEEE Trans. Magn
2173:
2167:
2166:
2146:
2137:
2136:
2116:
2107:
2106:
2086:
2077:
2076:
2056:
2050:
2049:
2039:
2019:
2008:
2007:
1987:
1974:
1973:
1948:(5): 2779–2781.
1942:IEEE Trans. Magn
1937:
1926:
1925:
1922:10.1109/26.31158
1905:
1899:
1898:
1895:10.1109/26.24594
1870:
1864:
1863:
1835:
1829:
1828:
1820:
1807:
1806:
1786:
1775:
1774:
1766:
1747:
1746:
1727:IEEE Trans. Magn
1722:
1707:
1706:
1683:IEEE Trans. Magn
1678:
1655:
1654:
1642:
1547:-processing and
1495:
1493:
1492:
1487:
1485:
1468:
1451:
1424:
1422:
1420:
1419:
1414:
1387:
1385:
1383:
1382:
1377:
1352:
1350:
1349:
1344:
1342:
1341:
1340:
1328:
1314:
1297:
1280:
1279:
1278:
1251:
1238:
1227:
1226:
1218:
1203:
1201:
1199:
1198:
1193:
1170:
1168:
1167:
1162:
1160:
1159:
1133:
1131:
1130:
1125:
1096:
1094:
1093:
1088:
1086:
1069:
1065:
1064:
1063:
1062:
1038:
1015:
1013:
1011:
1010:
1005:
980:
978:
977:
972:
970:
969:
968:
949:
915:
913:
911:
910:
905:
897:
896:
875:
873:
872:
867:
841:
839:
837:
836:
831:
817:
815:
813:
812:
807:
784:
782:
780:
779:
774:
748:
746:
745:
740:
738:
721:
704:
674:
672:
670:
669:
664:
638:
636:
635:
630:
628:
605:
579:
577:
576:
571:
566:
565:
564:
554:
553:
538:
537:
525:
500:
498:
497:
492:
490:
486:
485:
460:
443:
432:
431:
423:
417:
400:
377:
375:
374:
369:
367:
356:
355:
347:
337:
335:
333:
332:
327:
304:
302:
301:
296:
273:
272:
253:
251:
249:
248:
243:
229:
227:
225:
224:
219:
192:
190:
188:
187:
182:
168:
166:
164:
163:
158:
144:
142:
141:
136:
112:
110:
108:
107:
102:
100:
99:
29:linear recording
2445:
2444:
2440:
2439:
2438:
2436:
2435:
2434:
2415:
2414:
2413:
2404:
2403:
2399:
2388:
2384:
2374:
2372:
2367:Popovich, Ken.
2365:
2361:
2352:
2350:
2342:
2341:
2337:
2328:
2326:
2317:
2316:
2312:
2300:
2299:
2290:
2289:
2277:
2255:
2251:
2220:
2216:
2205:
2201:
2174:
2170:
2147:
2140:
2121:IBM J. Res. Dev
2117:
2110:
2091:IBM J. Res. Dev
2087:
2080:
2057:
2053:
2020:
2011:
1988:
1977:
1938:
1929:
1906:
1902:
1871:
1867:
1860:10.1109/26.2724
1836:
1832:
1821:
1810:
1791:IBM J. Res. Dev
1787:
1778:
1767:
1750:
1723:
1710:
1679:
1658:
1643:
1630:
1626:
1604:
1584:
1541:
1475:
1458:
1441:
1436:
1433:
1432:
1399:
1396:
1395:
1393:
1362:
1359:
1358:
1356:
1336:
1333:
1332:
1318:
1304:
1287:
1268:
1264:
1260:
1228:
1217:
1216:
1214:
1211:
1210:
1178:
1175:
1174:
1172:
1149:
1145:
1143:
1140:
1139:
1107:
1104:
1103:
1076:
1058:
1054:
1053:
1046:
1042:
1028:
1023:
1020:
1019:
990:
987:
986:
984:
964:
960:
959:
939:
934:
931:
930:
892:
888:
886:
883:
882:
880:
849:
846:
845:
825:
822:
821:
819:
792:
789:
788:
786:
759:
756:
755:
753:
728:
711:
694:
689:
686:
685:
649:
646:
645:
643:
618:
595:
590:
587:
586:
560:
556:
555:
549:
545:
533:
529:
515:
510:
507:
506:
475:
465:
461:
450:
433:
422:
421:
407:
390:
385:
382:
381:
357:
346:
345:
343:
340:
339:
312:
309:
308:
306:
268:
264:
259:
256:
255:
237:
234:
233:
231:
204:
201:
200:
198:
176:
173:
172:
170:
152:
149:
148:
146:
118:
115:
114:
95:
91:
89:
86:
85:
83:
60:
40:areal densities
19:is a class of
12:
11:
5:
2443:
2433:
2432:
2427:
2412:
2411:
2397:
2382:
2359:
2335:
2310:
2301:|journal=
2275:
2249:
2230:(2): 714–720.
2214:
2199:
2168:
2138:
2108:
2097:(4): 368–375.
2078:
2067:(2): 284–287.
2051:
2037:10.1.1.16.6310
2030:(4): 730–743.
2009:
1975:
1927:
1916:(7): 669–676.
1900:
1881:(5): 428–436.
1865:
1830:
1808:
1776:
1748:
1708:
1689:(1): 110–117.
1656:
1627:
1625:
1622:
1621:
1620:
1615:
1610:
1603:
1600:
1583:
1580:
1565:autoregressive
1545:digital signal
1540:
1537:
1514:, the optimum
1512:Markov process
1484:
1481:
1478:
1474:
1471:
1467:
1464:
1461:
1457:
1454:
1450:
1447:
1444:
1440:
1412:
1409:
1406:
1403:
1375:
1372:
1369:
1366:
1339:
1335:
1331:
1327:
1324:
1321:
1317:
1313:
1310:
1307:
1303:
1300:
1296:
1293:
1290:
1286:
1283:
1277:
1274:
1271:
1267:
1263:
1259:
1256:
1250:
1247:
1244:
1241:
1237:
1234:
1231:
1224:
1221:
1191:
1188:
1185:
1182:
1158:
1155:
1152:
1148:
1123:
1120:
1117:
1114:
1111:
1085:
1082:
1079:
1075:
1072:
1068:
1061:
1057:
1052:
1049:
1045:
1041:
1037:
1034:
1031:
1027:
1003:
1000:
997:
994:
967:
963:
958:
955:
952:
948:
945:
942:
938:
903:
900:
895:
891:
865:
862:
859:
856:
853:
829:
805:
802:
799:
796:
772:
769:
766:
763:
737:
734:
731:
727:
724:
720:
717:
714:
710:
707:
703:
700:
697:
693:
662:
659:
656:
653:
627:
624:
621:
617:
614:
611:
608:
604:
601:
598:
594:
569:
563:
559:
552:
548:
544:
541:
536:
532:
528:
524:
521:
518:
514:
489:
484:
481:
478:
474:
471:
468:
464:
459:
456:
453:
449:
446:
442:
439:
436:
429:
426:
420:
416:
413:
410:
406:
403:
399:
396:
393:
389:
366:
363:
360:
353:
350:
325:
322:
319:
316:
294:
291:
288:
285:
282:
279:
276:
271:
267:
263:
241:
217:
214:
211:
208:
180:
156:
134:
131:
128:
125:
122:
98:
94:
59:
56:
9:
6:
4:
3:
2:
2442:
2431:
2428:
2426:
2423:
2422:
2420:
2407:
2401:
2393:
2390:Yoo, Daniel.
2386:
2371:. PC Magazine
2370:
2363:
2349:
2345:
2339:
2325:on 2011-07-18
2324:
2320:
2314:
2306:
2294:
2286:
2282:
2278:
2272:
2268:
2264:
2260:
2253:
2245:
2241:
2237:
2233:
2229:
2225:
2218:
2210:
2203:
2195:
2191:
2187:
2183:
2179:
2172:
2164:
2160:
2156:
2152:
2145:
2143:
2134:
2130:
2126:
2122:
2115:
2113:
2104:
2100:
2096:
2092:
2085:
2083:
2074:
2070:
2066:
2062:
2055:
2047:
2043:
2038:
2033:
2029:
2025:
2018:
2016:
2014:
2005:
2001:
1997:
1993:
1986:
1984:
1982:
1980:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1936:
1934:
1932:
1923:
1919:
1915:
1911:
1904:
1896:
1892:
1888:
1884:
1880:
1876:
1869:
1861:
1857:
1853:
1849:
1845:
1841:
1834:
1826:
1819:
1817:
1815:
1813:
1804:
1800:
1796:
1792:
1785:
1783:
1781:
1772:
1765:
1763:
1761:
1759:
1757:
1755:
1753:
1744:
1740:
1736:
1732:
1728:
1721:
1719:
1717:
1715:
1713:
1704:
1700:
1696:
1692:
1688:
1684:
1677:
1675:
1673:
1671:
1669:
1667:
1665:
1663:
1661:
1652:
1648:
1641:
1639:
1637:
1635:
1633:
1628:
1619:
1616:
1614:
1611:
1609:
1606:
1605:
1599:
1597:
1592:
1589:
1579:
1577:
1572:
1570:
1566:
1561:
1559:
1555:
1550:
1546:
1536:
1533:
1529:
1525:
1520:
1517:
1513:
1508:
1506:
1500:
1496:
1482:
1479:
1476:
1472:
1469:
1465:
1462:
1459:
1455:
1452:
1448:
1445:
1442:
1438:
1430:
1426:
1407:
1401:
1391:
1370:
1364:
1353:
1337:
1334:
1329:
1325:
1322:
1319:
1315:
1311:
1308:
1305:
1301:
1298:
1294:
1291:
1288:
1284:
1281:
1275:
1272:
1269:
1265:
1261:
1257:
1254:
1248:
1245:
1242:
1239:
1235:
1232:
1229:
1219:
1208:
1205:
1186:
1180:
1156:
1153:
1150:
1146:
1137:
1121:
1118:
1115:
1112:
1109:
1101:
1098:
1083:
1080:
1077:
1073:
1070:
1066:
1059:
1055:
1050:
1047:
1043:
1039:
1035:
1032:
1029:
1025:
1017:
998:
992:
981:
965:
961:
956:
953:
950:
946:
943:
940:
936:
928:
921:
917:
901:
898:
893:
889:
877:
863:
860:
857:
854:
851:
843:
827:
800:
794:
767:
761:
750:
735:
732:
729:
725:
722:
718:
715:
712:
708:
705:
701:
698:
695:
691:
683:
680:
676:
657:
651:
640:
625:
622:
619:
615:
612:
609:
606:
602:
599:
596:
592:
584:
581:
567:
561:
557:
550:
546:
542:
539:
534:
530:
526:
522:
519:
516:
512:
504:
501:
487:
482:
479:
476:
472:
469:
466:
462:
457:
454:
451:
447:
444:
440:
437:
434:
427:
424:
418:
414:
411:
408:
404:
401:
397:
394:
391:
387:
379:
364:
361:
358:
351:
348:
320:
314:
292:
289:
286:
283:
280:
277:
269:
265:
239:
212:
206:
194:
178:
154:
132:
129:
126:
123:
120:
96:
92:
80:
76:
71:
69:
65:
55:
52:
47:
45:
41:
37:
32:
30:
26:
22:
18:
2400:
2385:
2373:. Retrieved
2362:
2351:. Retrieved
2347:
2338:
2327:. Retrieved
2323:the original
2313:
2258:
2252:
2227:
2223:
2217:
2208:
2202:
2177:
2171:
2154:
2150:
2124:
2120:
2094:
2090:
2064:
2060:
2054:
2027:
2023:
1995:
1991:
1945:
1941:
1913:
1909:
1903:
1878:
1874:
1868:
1843:
1839:
1833:
1824:
1794:
1790:
1770:
1726:
1686:
1682:
1650:
1646:
1593:
1585:
1582:Applications
1573:
1562:
1557:
1542:
1521:
1509:
1501:
1497:
1431:
1427:
1389:
1354:
1209:
1206:
1102:
1099:
1018:
982:
929:
926:
879:requiring a
878:
844:
751:
684:
681:
677:
641:
585:
582:
505:
502:
380:
230:, where the
195:
75:hypothesized
72:
61:
48:
33:
16:
15:
1998:: 291–301.
2419:Categories
2394:. Marvell.
2353:2017-07-04
2329:2012-07-26
1827:: 942–947.
1773:: 556–560.
1624:References
58:Principles
54:industry.
2303:ignored (
2293:cite book
2157:: 38–56.
2127:: 65–74.
2032:CiteSeerX
1846:: 13–20.
1470:×
1330:∥
1299:−
1282:∥
1223:^
1071:×
1051:−
957:−
723:×
613:−
470:−
428:´
419:−
352:´
130:≤
124:≤
2285:42438542
1970:42451727
1602:See also
1505:Gaussian
2375:June 5,
2232:Bibcode
2182:Bibcode
1950:Bibcode
1883:Bibcode
1848:Bibcode
1731:Bibcode
1691:Bibcode
1539:History
1423:
1394:
1386:
1357:
1202:
1173:
1014:
985:
914:
881:
840:
820:
816:
787:
783:
754:
673:
644:
336:
307:
252:
232:
228:
199:
191:
171:
167:
147:
145:, with
111:
84:
79:trellis
2283:
2273:
2034:
1968:
1549:coding
1355:where
1252:
752:where
36:analog
2281:S2CID
1966:S2CID
1588:IBM's
2377:2002
2305:help
2271:ISBN
1554:PRML
1532:BCJR
1516:MLSE
1390:z(D)
580:...
2263:doi
2240:doi
2190:doi
2159:doi
2129:doi
2099:doi
2069:doi
2042:doi
2000:doi
1958:doi
1918:doi
1891:doi
1856:doi
1799:doi
1739:doi
1699:doi
1558:d,k
2421::
2346:.
2297::
2295:}}
2291:{{
2279:.
2269:.
2238:.
2228:37
2226:.
2188:.
2155:10
2153:.
2141:^
2125:15
2123:.
2111:^
2095:14
2093:.
2081:^
2065:20
2063:.
2040:.
2028:19
2026:.
2012:^
1996:46
1994:.
1978:^
1964:.
1956:.
1946:33
1944:.
1930:^
1914:37
1912:.
1889:.
1879:37
1877:.
1854:.
1844:36
1842:.
1811:^
1795:54
1793:.
1779:^
1751:^
1737:.
1711:^
1697:.
1687:34
1685:.
1659:^
1649:.
1631:^
1425:.
1204:.
1097:,
876:,
749:,
639:,
113:,
2408:.
2379:.
2356:.
2332:.
2307:)
2287:.
2265::
2246:.
2242::
2234::
2211:.
2196:.
2192::
2184::
2165:.
2161::
2135:.
2131::
2105:.
2101::
2075:.
2071::
2048:.
2044::
2006:.
2002::
1972:.
1960::
1952::
1924:.
1920::
1897:.
1893::
1885::
1862:.
1858::
1850::
1805:.
1801::
1745:.
1741::
1733::
1705:.
1701::
1693::
1651:4
1483:)
1480:D
1477:(
1473:W
1466:)
1463:D
1460:(
1456:F
1453:=
1449:)
1446:D
1443:(
1439:G
1411:)
1408:D
1405:(
1402:W
1374:)
1371:D
1368:(
1365:a
1338:2
1326:)
1323:D
1320:(
1316:G
1312:)
1309:D
1306:(
1302:a
1295:)
1292:D
1289:(
1285:z
1276:)
1273:D
1270:(
1266:a
1262:n
1258:i
1255:m
1249:g
1246:r
1243:a
1240:=
1236:)
1233:D
1230:(
1220:a
1190:)
1187:D
1184:(
1181:G
1157:2
1154:+
1151:L
1147:2
1122:2
1119:+
1116:L
1113:=
1110:M
1084:)
1081:D
1078:(
1074:W
1067:)
1060:2
1056:D
1048:1
1044:(
1040:=
1036:)
1033:D
1030:(
1026:G
1002:)
999:D
996:(
993:W
966:2
962:D
954:1
951:=
947:)
944:D
941:(
937:F
902:S
899:+
894:L
890:2
864:S
861:+
858:L
855:=
852:M
828:L
804:)
801:D
798:(
795:W
771:)
768:D
765:(
762:F
736:)
733:D
730:(
726:W
719:)
716:D
713:(
709:F
706:=
702:)
699:D
696:(
692:G
661:)
658:D
655:(
652:e
626:)
623:D
620:(
616:P
610:1
607:=
603:)
600:D
597:(
593:W
568:+
562:2
558:D
551:2
547:p
543:+
540:D
535:1
531:p
527:=
523:)
520:D
517:(
513:P
488:)
483:)
480:D
477:(
473:P
467:1
463:(
458:)
455:D
452:(
448:n
445:=
441:)
438:D
435:(
425:n
415:)
412:D
409:(
405:n
402:=
398:)
395:D
392:(
388:e
365:)
362:D
359:(
349:n
324:)
321:D
318:(
315:n
293:2
290:,
287:1
284:=
281:l
278:,
275:}
270:l
266:p
262:{
240:D
216:)
213:D
210:(
207:n
179:K
155:M
133:M
127:K
121:0
97:K
93:2
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
