573:
1624:
monitoring. In this context a key characteristic of such applications is that nearby sensor nodes monitoring an environmental feature typically register similar values. This kind of data redundancy due to the spatial correlation between sensor observations inspires the techniques for in-network data
121:
decreases the spatial correlation by spreading the signal such that multipath components are received from many different spatial directions. Short antenna separations increase the spatial correlation as adjacent antennas will receive similar signal components. The existence of spatial correlation
793:
When modeling spatial correlation it is useful to employ the
Kronecker model, where the correlation between transmit antennas and receive antennas are assumed independent and separable. This model is reasonable when the main scattering appears close to the antenna arrays and has been validated by
106:
channel between base stations and users, describing how the signal is reflected at different obstacles on its way from the transmitter to the receiver. However, the received signal may still have a strong spatial signature in the sense that stronger average signal gains are received from certain
1625:
aggregation and mining. By measuring the spatial correlation between data sampled by different sensors, a wide class of specialized algorithms can be developed to develop more efficient spatial data mining algorithms as well as more efficient routing strategies.
680:
1115:
902:
258:
503:
1591:
1560:
1484:
1453:
1414:
1383:
1346:
1315:
1278:
1247:
1218:. Each eigenvector represents a spatial direction of the channel and its corresponding eigenvalue describes the average channel/signal gain in this direction. For the transmit-side matrix
1216:
1185:
1033:
1002:
971:
936:
586:
414:
379:
344:
527:
438:
309:
285:
784:
1044:
1138:
807:
751:
718:
208:
181:
461:
567:
547:
86:). In practice, the channels between different antennas are often correlated and therefore the potential multi antenna gains may not always be obtainable.
1759:
Paulraj, Arogyaswami, Rohit Nabar, and
Dhananjay Gore. Introduction to space-time wireless communications. Cambridge university press, 2003. - p.40
98:
between the transmitter and receiver that represents clear spatial channel characteristics. In urban cellular systems, this is seldom the case as
1693:
1153:
102:
are located on rooftops while many users are located either indoors or at streets far from base stations. Thus, there is a non-line-of-sight
1782:
Ma, Y.; Guo, Y.; Tian, X.; Ghanem, M. (2011). "Distributed
Clustering-Based Aggregation Algorithm for Spatial Correlated Sensor Networks".
220:
1846:
1732:
Optimal transmission strategies and impact of correlation in multi-antenna systems with different types of channel state information
63:
54:
and the average received signal gain. Theoretically, the performance of wireless communication systems can be improved by having
1513:
as it decreases the number of (strong) spatial directions that the signal is received from. This makes it harder to perform
1604:
measures the theoretical performance, but similar results have been proved for more practical performance measures as the
939:
721:
675:{\displaystyle \mathbf {R} =E\left\{vec\left(\mathbf {H} \right)\left(vec\left(\mathbf {H} \right)\right)^{H}\right\}}
1527:. If the transmitter is perfectly informed or is uninformed, then the more spatial correlation there is the less the
466:
1565:
1534:
1458:
1427:
1388:
1357:
1320:
1289:
1252:
1221:
1190:
1159:
1007:
976:
945:
910:
133:). This seems intuitive as spatial correlation decreases the number of independent channels that can be created by
1249:
it describes the average gain in a spatial transmit direction, while it describes a spatial receive direction for
1523:
138:
95:
384:
349:
314:
508:
419:
290:
266:
59:
32:
67:
1799:
39:
28:
1644:
118:
103:
1110:{\displaystyle \mathbf {H} \sim {\mathcal {CN}}(\mathbf {0} ,\mathbf {R} _{T}\otimes \mathbf {R} _{R})}
110:
Spatial correlation means that there is a correlation between the received average signal gain and the
760:
1791:
1654:
129:
and put a limit on the number of antennas that can be effectively squeezed into a small device (as a
75:
1804:
1690:
1639:
1514:
787:
79:
1817:
1674:
1498:
1731:
1769:
897:{\displaystyle \mathbf {H} =\mathbf {R} _{R}^{1/2}\mathbf {H} _{w}(\mathbf {R} _{T}^{1/2})^{T}}
1841:
1141:
1123:
51:
1809:
1601:
1594:
1528:
1506:
1502:
727:
688:
111:
1713:
186:
159:
1697:
798:
1795:
1505:
represents the amount of information that can be transmitted reliably. Intuitively, the
1416:, meaning that almost the same signal gain can be expected from all spatial directions.
443:
1747:
1743:
1605:
754:
552:
532:
83:
70:, then multiple independent channels with identical characteristics can be created by
1835:
1621:
1821:
1491:
130:
99:
17:
572:
1675:
Fading
Correlation and Its Effect on the Capacity of Multielement Antenna Systems
153:
47:
1348:, meaning that some spatial directions are statistically stronger than others.
1813:
150:
1700:, IEEE Journal on Selected Areas Communications, vol 20, pp. 1211-1226, 2002.
1649:
134:
71:
1772:, IEEE Transactions on Wireless Communications, vol 9, pp. 3578-3589, 2010.
1714:
Modeling of Wide-Band MIMO Radio
Channels Based on NLoS Indoor Measurements
1152:
Under the
Kronecker model, the spatial correlation depends directly on the
942:
with zero-mean and unit variance. The important part of the model is that
801:, the Kronecker model means that the channel matrix can be factorized as
1770:
Impact of
Spatial Correlation and Precoding Design in OSTBC MIMO Systems
1750:, IEEE Transactions on Information Theory, vol 51, pp. 2491-2509, 2005.
1716:, IEEE Transactions on Vehicular Technology, vol 53, pp. 655-665, 2004.
1712:
K. Yu, M. Bengtsson, B. Ottersten, D. McNamara, P. Karlsson, M. Beach,
1689:
J. Kermoal, L. Schumacher, K.I. Pedersen, P. Mogensen, F. Frederiksen,
1597:
since the dominating effect is that the channel uncertainty decreases.
1748:
Impact of antenna correlation on the capacity of multiantenna channels
1734:, IEEE Transactions on Signal Processing, vol 52, pp. 3440-3453, 2004.
27:"Spatial correlation" redirects here. For the concept in ecology, see
1617:
1531:. However, if the transmitter has statistical knowledge (i.e., knows
1691:
A Stochastic MIMO Radio
Channel Model With Experimental Validation
571:
253:{\displaystyle \mathbf {y} =\mathbf {H} \mathbf {x} +\mathbf {n} }
1677:, IEEE Transactions on Communications, vol 48, pp. 502-513, 2000.
973:
is pre-multiplied by the receive-side spatial correlation matrix
1634:
1593:) it is the other way around – spatial correlation improves the
1487:
1004:
and post-multiplied by transmit-side spatial correlation matrix
211:
126:
125:
Spatial correlation is often said to degrade the performance of
55:
1616:
Spatial correlation can have another meaning in the context of
576:
A short illustration of the correlation matrix common formula.
58:
at the transmitter and the receiver. The idea is that if the
1061:
1058:
1522:
impact of transmit-side spatial correlation depends on the
1620:
data in the context of a variety of applications such as
1424:
The spatial correlation (i.e., the eigenvalue spread in
62:
between each pair of transmit and receive antennas are
1569:
1538:
1490:
system. This effect can be analyzed mathematically by
1462:
1431:
1392:
1361:
1324:
1293:
1256:
1225:
1194:
1163:
1011:
980:
949:
914:
512:
470:
423:
388:
353:
318:
294:
270:
1568:
1537:
1461:
1430:
1391:
1360:
1323:
1292:
1255:
1224:
1193:
1162:
1126:
1047:
1038:
Equivalently, the channel matrix can be expressed as
1010:
979:
948:
913:
810:
763:
730:
691:
589:
555:
535:
511:
469:
446:
422:
387:
352:
317:
293:
269:
223:
189:
162:
1585:
1554:
1478:
1447:
1408:
1377:
1340:
1309:
1272:
1241:
1210:
1179:
1132:
1109:
1027:
996:
965:
930:
896:
778:
745:
712:
674:
580:The common formula for the correlation matrix is:
561:
541:
521:
497:
455:
432:
408:
373:
338:
303:
279:
252:
202:
175:
938:are independent and identically distributed as
94:In an ideal communication scenario, there is a
1673:D. Shiu, G.J. Foschini, M.J. Gans, J.M. Kahn,
498:{\displaystyle \scriptstyle N_{r}\times N_{t}}
1586:{\displaystyle \scriptstyle \mathbf {R} _{R}}
1555:{\displaystyle \scriptstyle \mathbf {R} _{T}}
1479:{\displaystyle \scriptstyle \mathbf {R} _{R}}
1448:{\displaystyle \scriptstyle \mathbf {R} _{T}}
1409:{\displaystyle \scriptstyle \mathbf {R} _{R}}
1378:{\displaystyle \scriptstyle \mathbf {R} _{T}}
1354:is represented by small eigenvalue spread in
1341:{\displaystyle \scriptstyle \mathbf {R} _{R}}
1310:{\displaystyle \scriptstyle \mathbf {R} _{T}}
1286:is represented by large eigenvalue spread in
1273:{\displaystyle \scriptstyle \mathbf {R} _{R}}
1242:{\displaystyle \scriptstyle \mathbf {R} _{T}}
1211:{\displaystyle \scriptstyle \mathbf {R} _{R}}
1180:{\displaystyle \scriptstyle \mathbf {R} _{T}}
1028:{\displaystyle \scriptstyle \mathbf {R} _{T}}
997:{\displaystyle \scriptstyle \mathbf {R} _{R}}
966:{\displaystyle \scriptstyle \mathbf {H} _{w}}
931:{\displaystyle \scriptstyle \mathbf {H} _{w}}
8:
1511:degraded by receive-side spatial correlation
740:
734:
1726:
1724:
1722:
409:{\displaystyle \scriptstyle N_{r}\times 1}
374:{\displaystyle \scriptstyle N_{t}\times 1}
339:{\displaystyle \scriptstyle N_{r}\times 1}
1803:
1768:E. Björnson, E. Jorswieck, B. Ottersten,
1685:
1683:
1576:
1571:
1567:
1545:
1540:
1536:
1469:
1464:
1460:
1438:
1433:
1429:
1399:
1394:
1390:
1368:
1363:
1359:
1331:
1326:
1322:
1300:
1295:
1291:
1263:
1258:
1254:
1232:
1227:
1223:
1201:
1196:
1192:
1170:
1165:
1161:
1125:
1098:
1093:
1083:
1078:
1069:
1057:
1056:
1048:
1046:
1018:
1013:
1009:
987:
982:
978:
956:
951:
947:
921:
916:
912:
888:
874:
870:
865:
860:
850:
845:
834:
830:
825:
820:
811:
809:
770:
765:
762:
729:
690:
661:
647:
619:
590:
588:
554:
534:
522:{\displaystyle \scriptstyle \mathbf {H} }
513:
510:
488:
475:
468:
445:
433:{\displaystyle \scriptstyle \mathbf {n} }
424:
421:
393:
386:
358:
351:
323:
316:
304:{\displaystyle \scriptstyle \mathbf {x} }
295:
292:
280:{\displaystyle \scriptstyle \mathbf {y} }
271:
268:
245:
237:
232:
224:
222:
214:), the propagation channel is modeled as
194:
188:
167:
161:
1666:
794:both outdoor and indoor measurements.
7:
1708:
1706:
381:transmit vectors, respectively. The
74:and be used for either transmitting
31:. For the concept in geography, see
940:circular symmetric complex Gaussian
137:, but is not true for all kinds of
122:has been experimentally validated.
25:
1572:
1541:
1465:
1434:
1395:
1364:
1327:
1296:
1259:
1228:
1197:
1166:
1094:
1079:
1070:
1049:
1014:
983:
952:
917:
861:
846:
821:
812:
779:{\displaystyle \mathbf {A} ^{H}}
766:
648:
620:
591:
514:
425:
296:
272:
246:
238:
233:
225:
1486:) affects the performance of a
529:describes the channel from the
1104:
1066:
885:
856:
707:
701:
1:
1494:of vectors with eigenvalues.
1156:of the correlation matrices
1148:Spatial correlation matrices
549:th transmit antenna to the
1863:
1730:E.A. Jorswieck, H. Boche,
26:
1814:10.1109/JSEN.2010.2056916
64:statistically independent
1847:Telecommunication theory
1284:High spatial correlation
1154:eigenvalue distributions
1133:{\displaystyle \otimes }
416:noise vector is denoted
145:Mathematical description
33:Spatial auto-correlation
1746:, A. Lozano, S. Verdú,
1352:Low spatial correlation
68:identically distributed
1587:
1556:
1480:
1449:
1410:
1379:
1342:
1311:
1274:
1243:
1212:
1181:
1134:
1111:
1029:
998:
967:
932:
907:where the elements of
898:
780:
747:
746:{\displaystyle E\{*\}}
714:
713:{\displaystyle vec(*)}
676:
577:
563:
543:
523:
499:
457:
434:
410:
375:
340:
305:
281:
254:
204:
183:transmit antennas and
177:
40:wireless communication
29:Ecological correlation
1645:Multipath propagation
1588:
1557:
1481:
1450:
1420:Impact on performance
1411:
1380:
1343:
1312:
1275:
1244:
1213:
1182:
1135:
1112:
1030:
999:
968:
933:
899:
781:
748:
715:
677:
575:
564:
544:
524:
500:
458:
435:
411:
376:
341:
306:
282:
255:
205:
203:{\displaystyle N_{r}}
178:
176:{\displaystyle N_{t}}
127:multi antenna systems
119:multipath propagation
104:multipath propagation
76:multiple data streams
1784:IEEE Sensors Journal
1655:Spatial multiplexing
1566:
1535:
1459:
1428:
1389:
1358:
1321:
1290:
1253:
1222:
1191:
1160:
1124:
1045:
1008:
977:
946:
911:
808:
761:
728:
689:
587:
569:th receive antenna.
553:
533:
509:
467:
444:
420:
385:
350:
315:
291:
267:
221:
187:
160:
141:as described below.
107:spatial directions.
60:propagation channels
1796:2011ISenJ..11..641M
1640:Diversity combining
1612:Sensor measurements
1515:diversity combining
883:
843:
50:between a signal's
44:spatial correlation
18:Spatial correlation
1696:2009-12-29 at the
1583:
1582:
1552:
1551:
1499:information theory
1476:
1475:
1445:
1444:
1406:
1405:
1375:
1374:
1338:
1337:
1307:
1306:
1270:
1269:
1239:
1238:
1208:
1207:
1177:
1176:
1130:
1107:
1025:
1024:
994:
993:
963:
962:
928:
927:
894:
859:
819:
776:
743:
710:
672:
578:
559:
539:
519:
518:
495:
494:
463:th element of the
456:{\displaystyle ij}
453:
430:
429:
406:
405:
371:
370:
336:
335:
301:
300:
277:
276:
250:
210:receive antennas (
200:
173:
96:line-of-sight path
78:or increasing the
1524:channel knowledge
1142:Kronecker product
562:{\displaystyle i}
542:{\displaystyle j}
139:channel knowledge
56:multiple antennas
52:spatial direction
16:(Redirected from
1854:
1826:
1825:
1807:
1779:
1773:
1766:
1760:
1757:
1751:
1741:
1735:
1728:
1717:
1710:
1701:
1687:
1678:
1671:
1602:channel capacity
1595:channel capacity
1592:
1590:
1589:
1584:
1581:
1580:
1575:
1561:
1559:
1558:
1553:
1550:
1549:
1544:
1529:channel capacity
1507:channel capacity
1503:channel capacity
1485:
1483:
1482:
1477:
1474:
1473:
1468:
1454:
1452:
1451:
1446:
1443:
1442:
1437:
1415:
1413:
1412:
1407:
1404:
1403:
1398:
1384:
1382:
1381:
1376:
1373:
1372:
1367:
1347:
1345:
1344:
1339:
1336:
1335:
1330:
1316:
1314:
1313:
1308:
1305:
1304:
1299:
1279:
1277:
1276:
1271:
1268:
1267:
1262:
1248:
1246:
1245:
1240:
1237:
1236:
1231:
1217:
1215:
1214:
1209:
1206:
1205:
1200:
1186:
1184:
1183:
1178:
1175:
1174:
1169:
1139:
1137:
1136:
1131:
1116:
1114:
1113:
1108:
1103:
1102:
1097:
1088:
1087:
1082:
1073:
1065:
1064:
1052:
1034:
1032:
1031:
1026:
1023:
1022:
1017:
1003:
1001:
1000:
995:
992:
991:
986:
972:
970:
969:
964:
961:
960:
955:
937:
935:
934:
929:
926:
925:
920:
903:
901:
900:
895:
893:
892:
882:
878:
869:
864:
855:
854:
849:
842:
838:
829:
824:
815:
785:
783:
782:
777:
775:
774:
769:
752:
750:
749:
744:
719:
717:
716:
711:
681:
679:
678:
673:
671:
667:
666:
665:
660:
656:
655:
651:
627:
623:
594:
568:
566:
565:
560:
548:
546:
545:
540:
528:
526:
525:
520:
517:
504:
502:
501:
496:
493:
492:
480:
479:
462:
460:
459:
454:
439:
437:
436:
431:
428:
415:
413:
412:
407:
398:
397:
380:
378:
377:
372:
363:
362:
345:
343:
342:
337:
328:
327:
310:
308:
307:
302:
299:
286:
284:
283:
278:
275:
259:
257:
256:
251:
249:
241:
236:
228:
209:
207:
206:
201:
199:
198:
182:
180:
179:
174:
172:
171:
112:angle of arrival
21:
1862:
1861:
1857:
1856:
1855:
1853:
1852:
1851:
1832:
1831:
1830:
1829:
1805:10.1.1.724.1158
1781:
1780:
1776:
1767:
1763:
1758:
1754:
1742:
1738:
1729:
1720:
1711:
1704:
1698:Wayback Machine
1688:
1681:
1672:
1668:
1663:
1631:
1614:
1570:
1564:
1563:
1539:
1533:
1532:
1463:
1457:
1456:
1432:
1426:
1425:
1422:
1393:
1387:
1386:
1362:
1356:
1355:
1325:
1319:
1318:
1294:
1288:
1287:
1257:
1251:
1250:
1226:
1220:
1219:
1195:
1189:
1188:
1164:
1158:
1157:
1150:
1122:
1121:
1092:
1077:
1043:
1042:
1012:
1006:
1005:
981:
975:
974:
950:
944:
943:
915:
909:
908:
884:
844:
806:
805:
799:Rayleigh fading
764:
759:
758:
726:
725:
687:
686:
643:
633:
629:
628:
615:
605:
601:
585:
584:
551:
550:
531:
530:
507:
506:
505:channel matrix
484:
471:
465:
464:
442:
441:
418:
417:
389:
383:
382:
354:
348:
347:
319:
313:
312:
289:
288:
265:
264:
219:
218:
190:
185:
184:
163:
158:
157:
147:
92:
36:
23:
22:
15:
12:
11:
5:
1860:
1858:
1850:
1849:
1844:
1834:
1833:
1828:
1827:
1774:
1761:
1752:
1736:
1718:
1702:
1679:
1665:
1664:
1662:
1659:
1658:
1657:
1652:
1647:
1642:
1637:
1630:
1627:
1613:
1610:
1579:
1574:
1548:
1543:
1501:, the ergodic
1472:
1467:
1441:
1436:
1421:
1418:
1402:
1397:
1371:
1366:
1334:
1329:
1303:
1298:
1266:
1261:
1235:
1230:
1204:
1199:
1173:
1168:
1149:
1146:
1129:
1118:
1117:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1072:
1068:
1063:
1060:
1055:
1051:
1021:
1016:
990:
985:
959:
954:
924:
919:
905:
904:
891:
887:
881:
877:
873:
868:
863:
858:
853:
848:
841:
837:
833:
828:
823:
818:
814:
773:
768:
755:expected value
742:
739:
736:
733:
709:
706:
703:
700:
697:
694:
683:
682:
670:
664:
659:
654:
650:
646:
642:
639:
636:
632:
626:
622:
618:
614:
611:
608:
604:
600:
597:
593:
558:
538:
516:
491:
487:
483:
478:
474:
452:
449:
427:
404:
401:
396:
392:
369:
366:
361:
357:
334:
331:
326:
322:
298:
274:
261:
260:
248:
244:
240:
235:
231:
227:
197:
193:
170:
166:
146:
143:
91:
88:
84:bit error rate
24:
14:
13:
10:
9:
6:
4:
3:
2:
1859:
1848:
1845:
1843:
1840:
1839:
1837:
1823:
1819:
1815:
1811:
1806:
1801:
1797:
1793:
1789:
1785:
1778:
1775:
1771:
1765:
1762:
1756:
1753:
1749:
1745:
1740:
1737:
1733:
1727:
1725:
1723:
1719:
1715:
1709:
1707:
1703:
1699:
1695:
1692:
1686:
1684:
1680:
1676:
1670:
1667:
1660:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1633:
1632:
1628:
1626:
1623:
1622:air pollution
1619:
1611:
1609:
1607:
1603:
1598:
1596:
1577:
1546:
1530:
1526:
1525:
1518:
1516:
1512:
1508:
1504:
1500:
1495:
1493:
1489:
1470:
1439:
1419:
1417:
1400:
1369:
1353:
1349:
1332:
1301:
1285:
1281:
1264:
1233:
1202:
1171:
1155:
1147:
1145:
1143:
1127:
1099:
1089:
1084:
1074:
1053:
1041:
1040:
1039:
1036:
1019:
988:
957:
941:
922:
889:
879:
875:
871:
866:
851:
839:
835:
831:
826:
816:
804:
803:
802:
800:
795:
791:
789:
771:
756:
737:
731:
723:
722:vectorization
704:
698:
695:
692:
668:
662:
657:
652:
644:
640:
637:
634:
630:
624:
616:
612:
609:
606:
602:
598:
595:
583:
582:
581:
574:
570:
556:
536:
489:
485:
481:
476:
472:
450:
447:
402:
399:
394:
390:
367:
364:
359:
355:
332:
329:
324:
320:
242:
229:
217:
216:
215:
213:
195:
191:
168:
164:
156:channel with
155:
152:
144:
142:
140:
136:
132:
128:
123:
120:
115:
114:of a signal.
113:
108:
105:
101:
100:base stations
97:
89:
87:
85:
82:(in terms of
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
34:
30:
19:
1787:
1783:
1777:
1764:
1755:
1739:
1669:
1615:
1600:The ergodic
1599:
1521:
1519:
1510:
1496:
1492:majorization
1488:multiantenna
1423:
1351:
1350:
1283:
1282:
1151:
1140:denotes the
1119:
1037:
906:
796:
792:
684:
579:
346:receive and
262:
148:
131:mobile phone
124:
116:
109:
93:
43:
37:
154:flat-fading
80:reliability
48:correlation
1836:Categories
1790:(3): 641.
1661:References
1606:error rate
1509:is always
151:narrowband
1800:CiteSeerX
1744:A. Tulino
1650:Precoding
1128:⊗
1090:⊗
1054:∼
788:Hermitian
738:∗
705:∗
482:×
400:×
365:×
330:×
135:precoding
90:Existence
72:precoding
1842:Wireless
1694:Archived
1629:See also
753:denotes
720:denotes
311:are the
1822:1639100
1792:Bibcode
46:is the
1820:
1802:
1618:sensor
1120:where
786:means
685:where
440:. The
263:where
1818:S2CID
797:With
149:In a
117:Rich
1635:MIMO
1562:and
1520:The
1187:and
757:and
287:and
212:MIMO
66:and
1810:doi
1497:In
1455:or
1385:or
1317:or
38:In
1838::
1816:.
1808:.
1798:.
1788:11
1786:.
1721:^
1705:^
1682:^
1608:.
1517:.
1280:.
1144:.
1035:.
790:.
724:,
42:,
1824:.
1812::
1794::
1578:R
1573:R
1547:T
1542:R
1471:R
1466:R
1440:T
1435:R
1401:R
1396:R
1370:T
1365:R
1333:R
1328:R
1302:T
1297:R
1265:R
1260:R
1234:T
1229:R
1203:R
1198:R
1172:T
1167:R
1105:)
1100:R
1095:R
1085:T
1080:R
1075:,
1071:0
1067:(
1062:N
1059:C
1050:H
1020:T
1015:R
989:R
984:R
958:w
953:H
923:w
918:H
890:T
886:)
880:2
876:/
872:1
867:T
862:R
857:(
852:w
847:H
840:2
836:/
832:1
827:R
822:R
817:=
813:H
772:H
767:A
741:}
735:{
732:E
708:)
702:(
699:c
696:e
693:v
669:}
663:H
658:)
653:)
649:H
645:(
641:c
638:e
635:v
631:(
625:)
621:H
617:(
613:c
610:e
607:v
603:{
599:E
596:=
592:R
557:i
537:j
515:H
490:t
486:N
477:r
473:N
451:j
448:i
426:n
403:1
395:r
391:N
368:1
360:t
356:N
333:1
325:r
321:N
297:x
273:y
247:n
243:+
239:x
234:H
230:=
226:y
196:r
192:N
169:t
165:N
35:.
20:)
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