1826:
561:
1648:
564:
Fig 1: A sequence of five plots depicts one cycle of the overlap-add convolution algorithm. The first plot is a long sequence of data to be processed with a lowpass FIR filter. The 2nd plot is one segment of the data to be processed in piecewise fashion. The 3rd plot is the filtered segment,
1180:
2626:. The two methods are also compared in Figure 3, created by Matlab simulation. The contours are lines of constant ratio of the times it takes to perform both methods. When the overlap-add method is faster, the ratio exceeds 1, and ratios as high as 3 are seen.
815:
1496:
300:
565:
including the filter rise and fall transients. The 4th plot indicates where the new data will be added with the result of previous segments. The 5th plot is the updated output stream. The FIR filter is a boxcar lowpass with
2498:
961:
1904:
2014:
966:
2624:
2398:
2561:
1263:
531:
687:
1643:{\displaystyle y_{k}\ =\ \scriptstyle {\text{IDFT}}_{N}\displaystyle (\ \scriptstyle {\text{DFT}}_{N}\displaystyle (x_{k})\cdot \ \scriptstyle {\text{DFT}}_{N}\displaystyle (h)\ ),}
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918:
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2842:
Senobari, Nader
Shakibay; Funning, Gareth J.; Keogh, Eamonn; Zhu, Yan; Yeh, Chin-Chia Michael; Zimmerman, Zachary; Mueen, Abdullah (2019).
2409:
1175:{\displaystyle {\begin{aligned}y=\left(\sum _{k}x_{k}\right)*h&=\sum _{k}\left(x_{k}*h\right)\\&=\sum _{k}y_{k},\end{aligned}}}
28:
1832:
1480: The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the
2633:
Fig 3: Gain of the overlap-add method compared to a single, large circular convolution. The axes show values of signal length N
1938:
1814:
y(position+(1:N)) = y(position+(1:N)) + IDFT(DFT(x(position+(1:step_size)), N) × H) position = position + step_size
2566:
2329:
533:
in which it is understood that the functions should be thought of in their totality, rather than at specific instants
2832:
2813:
2784:
2510:
1188:
1800:(8 times the smallest power of two bigger than filter length M. See next section for a slightly better choice.)
810:{\displaystyle x_{k}\ \triangleq \ {\begin{cases}x,&n=1,2,\ldots ,L\\0,&{\text{otherwise}},\end{cases}}}
476:
2742:
2902:
2844:"Super-Efficient Cross-Correlation (SEC-C): A Fast Matched Filtering Code Suitable for Desktop Computers"
849:
17:
1825:
27:
This article is about the convolution method. For the "Weight, OverLap, Add" channelization method, see
2887:
1685:
2897:
2843:
1315:
724:
2656:
2651:
414:
295:{\displaystyle y=x*h\ \triangleq \ \sum _{m=-\infty }^{\infty }h\cdot x=\sum _{m=1}^{M}h\cdot x,}
76:
1909:
When the DFT and IDFT are implemented by the FFT algorithm, the pseudocode above requires about
1769:
2403:
Comparatively, the number of complex multiplications required by the pseudocode algorithm is:
2776:
2767:
2064:
1733:
1381:
2098:
2678:
2299:
2038:
1919:
complex multiplications for the FFT, product of arrays, and IFFT. Each iteration produces
1375:
594:
1829:
Fig 2: A graph of the values of N (an integer power of 2) that minimize the cost function
568:
8:
2892:
2705:
2127:
1418:
324:
2215:
2187:
2728:
appended zeros, which prevents circular overlap of the output rise and fall transients.
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2147:
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624:
379:
359:
82:
49:
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2828:
2809:
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35:
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1925:
output samples, so the number of complex multiplications per output sample is about
536:
554:
2881:
2870:
1798:
h = FIR_filter M = length(h) Nx = length(x) N = 8 × 2^ceiling( log2(M) )
2144:
complex multiplications per output sample, the worst case being when both
43:
560:
2862:
1782:
1768:
is an integer power-of-2, and the transforms are implemented with the
2803:
2493:{\displaystyle N_{x}\cdot (\log _{2}(N)+1)\cdot {\frac {N}{N-M+1}}.}
621:
The concept is to divide the problem into multiple convolutions of
2764:
2629:
2563:
while the cost of a single, large circular convolution is almost
2323:
samples. The total number of complex multiplications would be:
1899:{\displaystyle {\tfrac {N\left(\log _{2}N+1\right)}{N-M+1}}}
411: This article uses common abstract notations, such as
803:
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1837:
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2009:{\displaystyle {\frac {N(\log _{2}(N)+1)}{N-M+1}}.\,}
1941:
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85:
52:
1806:
H = DFT(h, N) position = 0 y(1 : Nx + M-1) = 0
2773:
Theory and application of digital signal processing
2827:. Schaum's Outline Series. New York: McGraw Hill.
2775:. Englewood Cliffs, N.J.: Prentice-Hall. pp.
2766:
2737:Cooley–Tukey FFT algorithm for N=2 needs (N/2) log
2720:
2694:
2618:
2555:
2492:
2392:
2315:
2273:
2247:
2227:
2199:
2184:are complex-valued. Also note that for any given
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2156:
2136:
2110:
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2053:
2008:
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671:
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100:
67:
2619:{\displaystyle O\left(N_{x}\log _{2}N_{x}\right)}
2879:
2822:
2393:{\displaystyle N_{x}\cdot (\log _{2}(N_{x})+1).}
2804:Oppenheim, Alan V.; Schafer, Ronald W. (1975).
2556:{\displaystyle O\left(N_{x}\log _{2}N\right)}
1258:{\displaystyle y_{k}\ \triangleq \ x_{k}*h\,}
952:can be written as a sum of short convolutions
42:is an efficient way to evaluate the discrete
2765:Rabiner, Lawrence R.; Gold, Bernard (1975).
1820:
1794:Overlap-add algorithm for linear convolution
517:
502:
2507:of the overlap–add method scales almost as
2808:. Englewood Cliffs, N.J.: Prentice-Hall.
2005:
1434:
1404:
1343:
1254:
909:
526:{\textstyle y(t)={\mathcal {H}}\{x(t)\},}
2628:
1824:
559:
591:samples, the length of the segments is
14:
2880:
2758:
617:samples and the overlap is 15 samples.
2235:Figure 2 is a graph of the values of
840:is an arbitrary segment length. Then
1931:
1489:
113:
913:{\displaystyle x=\sum _{k}x_{k},\,}
24:
2797:
497:
194:
189:
25:
2914:
2675:This condition implies that the
2290:, we can also consider applying
1730:is customarily chosen such that
1688:and its inverse, evaluated over
2261:for a range of filter lengths (
2851:Seismological Research Letters
2731:
2669:
2457:
2448:
2442:
2426:
2384:
2375:
2362:
2346:
2292:
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2257:
2212:has a minimum with respect to
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2091:
2024:
1979:
1970:
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1948:
1658:
1628:
1622:
1619:
1613:
1607:
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1581:
1575:
1562:
1542:
1516:
1510:
1464:
1452:
1431:
1425:
1401:
1395:
1347:{\displaystyle N\geq L+M-1,\,}
1296:
1272:
1251:
1245:
1236:
1230:
1208:
1202:
1162:
1147:
1109:
1103:
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95:
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62:
56:
13:
1:
2751:
2296:to a long sequence of length
2118:whereas direct evaluation of
1776:
1185:where the linear convolution
2657:Circular_convolution#Example
1482:circular convolution theorem
466:{\textstyle y(t)=x(t)*h(t),}
7:
2645:
1265:is zero outside the region
10:
2919:
2743:FFT – Definition and speed
1810:position + step_size ≤ Nx
1772:algorithm, for efficiency.
1686:Discrete Fourier transform
26:
2825:Digital Signal Processing
2823:Hayes, M. Horace (1999).
2806:Digital signal processing
1821:Efficiency considerations
2662:
1354:it is equivalent to the
2083:{\displaystyle N=1024,}
1802:step_size = N - (M-1)
1761:{\displaystyle N=L+M-1}
1408:{\displaystyle x_{k}\,}
650:with short segments of
77:finite impulse response
2722:
2696:
2642:
2620:
2557:
2494:
2394:
2317:
2275:
2249:
2229:
2201:
2178:
2158:
2138:
2112:
2111:{\displaystyle 13.67,}
2084:
2055:
2010:
1906:
1900:
1762:
1724:
1702:
1644:
1474:
1439:
1409:
1368:
1348:
1312:And for any parameter
1306:
1259:
1176:
946:
914:
834:
811:
673:
644:
618:
611:
585:
547:
527:
467:
405:
370:
350:
296:
255:
198:
102:
69:
46:of a very long signal
2723:
2702:segment has at least
2697:
2695:{\displaystyle x_{k}}
2632:
2621:
2558:
2495:
2395:
2318:
2316:{\displaystyle N_{x}}
2276:
2250:
2230:
2202:
2179:
2159:
2139:
2113:
2085:
2056:
2054:{\displaystyle M=201}
2011:
1901:
1828:
1804:(L in the text above)
1763:
1725:
1703:
1645:
1475:
1440:
1410:
1369:
1349:
1307:
1260:
1177:
947:
915:
835:
812:
674:
645:
612:
610:{\displaystyle L=100}
586:
563:
548:
528:
468:
406:
371:
351:
297:
235:
175:
103:
70:
2706:
2679:
2567:
2511:
2410:
2330:
2300:
2265:
2239:
2216:
2188:
2168:
2148:
2128:
2124:would require up to
2099:
2065:
2039:
1939:
1833:
1734:
1714:
1708:discrete points, and
1692:
1497:
1449:
1419:
1382:
1376:circular convolution
1358:
1316:
1269:
1189:
962:
927:
850:
824:
688:
654:
625:
595:
584:{\displaystyle M=16}
569:
555:Convolution#Notation
537:
477:
415:
380:
360:
325:
121:
83:
50:
2721:{\displaystyle M-1}
2652:Overlap–save method
2637:and filter length N
2137:{\displaystyle 201}
1781:The following is a
1438:{\displaystyle h\,}
376:outside the region
349:{\displaystyle h=0}
2903:Numerical analysis
2863:10.1785/0220180122
2718:
2692:
2643:
2616:
2553:
2490:
2390:
2313:
2271:
2245:
2228:{\displaystyle N.}
2225:
2200:{\displaystyle M,}
2197:
2174:
2154:
2134:
2108:
2080:
2051:
2035:For example, when
2006:
1907:
1896:
1894:
1758:
1720:
1698:
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619:
607:
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543:
523:
463:
401:
366:
346:
292:
98:
65:
40:overlap–add method
2888:Signal processing
2485:
2274:{\displaystyle M}
2248:{\displaystyle N}
2177:{\displaystyle h}
2157:{\displaystyle x}
2032:
2031:
2000:
1893:
1723:{\displaystyle L}
1701:{\displaystyle N}
1666:
1665:
1627:
1598:
1592:
1553:
1547:
1533:
1527:
1521:
1473:{\displaystyle .}
1367:{\displaystyle N}
1305:{\displaystyle .}
1219:
1213:
1127:
1054:
989:
945:{\displaystyle y}
868:
833:{\displaystyle L}
795:
718:
712:
672:{\displaystyle x}
643:{\displaystyle h}
404:{\displaystyle .}
369:{\displaystyle m}
318:
317:
174:
168:
101:{\displaystyle h}
68:{\displaystyle x}
36:signal processing
29:Sampling the DTFT
16:(Redirected from
2910:
2898:Fourier analysis
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2819:
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2015:
2013:
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1875:
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1805:
1801:
1797:
1785:of the algorithm
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2798:Further reading
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2033:
1983:
1955:
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84:
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15:
12:
11:
5:
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2875:
2857:(1): 322–334.
2839:
2833:
2820:
2814:
2799:
2796:
2793:
2792:
2785:
2756:
2755:
2753:
2750:
2747:
2746:
2738:
2730:
2717:
2714:
2711:
2689:
2685:
2667:
2666:
2664:
2661:
2660:
2659:
2654:
2647:
2644:
2638:
2634:
2614:
2608:
2604:
2600:
2595:
2591:
2585:
2581:
2576:
2572:
2551:
2547:
2544:
2539:
2535:
2529:
2525:
2520:
2516:
2501:
2500:
2489:
2483:
2480:
2477:
2474:
2471:
2467:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2436:
2432:
2428:
2425:
2420:
2416:
2401:
2400:
2389:
2386:
2383:
2380:
2377:
2372:
2368:
2364:
2361:
2356:
2352:
2348:
2345:
2340:
2336:
2310:
2306:
2270:
2255:that minimize
2244:
2224:
2221:
2196:
2193:
2173:
2153:
2133:
2107:
2104:
2079:
2076:
2073:
2070:
2050:
2047:
2044:
2030:
2029:
2020:
2018:
2016:
2004:
1998:
1995:
1992:
1989:
1986:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1958:
1954:
1950:
1947:
1930:
1913:
1891:
1888:
1885:
1882:
1879:
1873:
1869:
1866:
1863:
1860:
1855:
1851:
1846:
1842:
1822:
1819:
1790:
1778:
1775:
1774:
1773:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1719:
1709:
1697:
1681:
1677:
1664:
1663:
1654:
1652:
1650:
1633:
1630:
1624:
1621:
1618:
1615:
1612:
1609:
1603:
1589:
1586:
1583:
1580:
1577:
1572:
1568:
1564:
1558:
1544:
1538:
1524:
1518:
1515:
1512:
1507:
1503:
1488:
1469:
1466:
1463:
1460:
1457:
1454:
1445:in the region
1433:
1430:
1427:
1424:
1403:
1400:
1397:
1392:
1388:
1363:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1227:
1223:
1216:
1210:
1207:
1204:
1199:
1195:
1183:
1182:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1144:
1140:
1134:
1130:
1126:
1123:
1121:
1119:
1115:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1076:
1072:
1067:
1061:
1057:
1053:
1050:
1048:
1046:
1043:
1040:
1037:
1034:
1030:
1026:
1023:
1020:
1017:
1014:
1011:
1006:
1002:
996:
992:
987:
983:
980:
977:
974:
971:
968:
967:
941:
938:
935:
932:
921:
920:
908:
905:
902:
899:
896:
893:
890:
885:
881:
875:
871:
867:
864:
861:
858:
855:
829:
818:
817:
804:
799:
791:
789:
786:
783:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
750:
747:
744:
741:
738:
735:
732:
729:
726:
725:
723:
715:
709:
706:
703:
698:
694:
668:
665:
662:
659:
639:
636:
633:
630:
606:
603:
600:
580:
577:
574:
546:{\textstyle t}
542:
522:
519:
516:
513:
510:
507:
504:
499:
494:
491:
488:
485:
482:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
400:
397:
394:
391:
388:
385:
365:
345:
342:
339:
336:
333:
330:
316:
315:
306:
304:
302:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
261:
258:
253:
248:
245:
242:
238:
234:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
196:
191:
188:
185:
182:
178:
171:
165:
162:
159:
156:
153:
150:
147:
144:
141:
138:
135:
132:
129:
126:
112:
97:
94:
91:
88:
64:
61:
58:
55:
9:
6:
4:
3:
2:
2915:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2885:
2883:
2872:
2868:
2864:
2860:
2856:
2852:
2845:
2840:
2836:
2834:0-07-027389-8
2830:
2826:
2821:
2817:
2815:0-13-214635-5
2811:
2807:
2802:
2801:
2788:
2786:0-13-914101-4
2782:
2778:
2774:
2769:
2761:
2757:
2744:
2734:
2715:
2712:
2709:
2687:
2683:
2672:
2668:
2658:
2655:
2653:
2650:
2649:
2631:
2627:
2612:
2606:
2602:
2598:
2593:
2589:
2583:
2579:
2574:
2570:
2549:
2545:
2542:
2537:
2533:
2527:
2523:
2518:
2514:
2506:
2487:
2481:
2478:
2475:
2472:
2469:
2465:
2460:
2454:
2451:
2445:
2439:
2434:
2430:
2423:
2418:
2414:
2406:
2405:
2404:
2387:
2381:
2378:
2370:
2366:
2359:
2354:
2350:
2343:
2338:
2334:
2326:
2325:
2324:
2308:
2304:
2295:
2294:
2289:
2288:
2282:
2268:
2260:
2259:
2242:
2222:
2219:
2211:
2210:
2194:
2191:
2171:
2151:
2131:
2123:
2122:
2105:
2102:
2094:
2093:
2077:
2074:
2071:
2068:
2048:
2045:
2042:
2028:
2021:
2019:
2017:
2002:
1996:
1993:
1990:
1987:
1984:
1976:
1973:
1967:
1961:
1956:
1952:
1945:
1934:
1933:
1929:
1928:
1923:
1917:
1889:
1886:
1883:
1880:
1877:
1871:
1867:
1864:
1861:
1858:
1853:
1849:
1844:
1840:
1827:
1817:
1813:
1809:
1795:
1789:
1788:
1784:
1771:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1717:
1710:
1695:
1687:
1684:refer to the
1675:
1674:
1673:
1672:
1662:
1655:
1653:
1651:
1631:
1616:
1610:
1601:
1587:
1578:
1570:
1566:
1556:
1536:
1522:
1513:
1505:
1501:
1492:
1491:
1487:
1486:
1483:
1467:
1461:
1458:
1455:
1428:
1422:
1398:
1390:
1386:
1377:
1361:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1299:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1248:
1242:
1239:
1233:
1225:
1221:
1214:
1205:
1197:
1193:
1165:
1159:
1156:
1153:
1150:
1142:
1138:
1132:
1128:
1124:
1122:
1113:
1106:
1100:
1097:
1091:
1088:
1085:
1082:
1074:
1070:
1065:
1059:
1055:
1051:
1049:
1041:
1035:
1032:
1028:
1021:
1018:
1015:
1012:
1004:
1000:
994:
990:
985:
981:
975:
969:
958:
957:
956:
955:
936:
930:
906:
900:
897:
894:
891:
883:
879:
873:
869:
865:
859:
853:
846:
845:
844:
843:
827:
797:
787:
784:
777:
774:
771:
768:
765:
762:
759:
756:
753:
748:
742:
739:
736:
733:
727:
721:
713:
704:
696:
692:
684:
683:
682:
681:
663:
657:
634:
628:
604:
601:
598:
578:
575:
572:
562:
558:
556:
540:
520:
511:
505:
492:
486:
480:
460:
454:
448:
445:
439:
433:
430:
424:
418:
398:
392:
389:
386:
363:
343:
340:
334:
328:
314:
307:
305:
303:
289:
283:
280:
277:
271:
268:
262:
256:
251:
246:
243:
240:
236:
232:
226:
223:
220:
214:
211:
205:
199:
186:
183:
180:
176:
169:
160:
154:
151:
145:
139:
136:
130:
124:
116:
115:
111:
110:
92:
86:
79:(FIR) filter
78:
59:
53:
45:
41:
37:
30:
19:
2854:
2850:
2824:
2805:
2772:
2760:
2733:
2671:
2504:
2502:
2402:
2291:
2285:
2283:
2256:
2207:
2119:
2090:
2034:
2022:
1935:
1926:
1921:
1911:
1908:
1815:
1811:
1807:
1793:
1786:
1780:
1670:
1668:
1656:
1493:
1484:
1184:
953:
922:
841:
819:
679:
620:
320:
308:
117:
108:
39:
33:
2284:Instead of
44:convolution
18:Overlap add
2893:Transforms
2882:Categories
2752:References
2741:(N) – see
2503:Hence the
1783:pseudocode
1777:Pseudocode
2871:0895-0695
2713:−
2599:
2543:
2473:−
2461:⋅
2440:
2424:⋅
2360:
2344:⋅
1988:−
1962:
1881:−
1859:
1753:−
1588:⋅
1335:−
1323:≥
1291:−
1240:∗
1215:≜
1154:−
1129:∑
1098:∗
1086:−
1056:∑
1033:∗
1016:−
991:∑
895:−
870:∑
794:otherwise
772:…
714:≜
446:∗
281:−
269:⋅
237:∑
224:−
212:⋅
195:∞
190:∞
187:−
177:∑
170:≜
152:∗
2646:See also
1916:(N) + 1)
1680:and IDFT
2095:equals
1374:-point
75:with a
2869:
2831:
2812:
2783:
2768:"2.25"
1912:N (log
1626:
1591:
1546:
1526:
1520:
1218:
1212:
820:where
717:
711:
321:where
173:
167:
38:, the
2847:(PDF)
2777:63–65
2663:Notes
2103:13.67
1922:N-M+1
1808:while
1669:where
1415:with
553:(see
2867:ISSN
2829:ISBN
2810:ISBN
2781:ISBN
2505:cost
2293:Eq.2
2287:Eq.1
2258:Eq.3
2209:Eq.3
2164:and
2121:Eq.1
2092:Eq.3
2075:1024
2061:and
2025:Eq.3
1659:Eq.2
1532:IDFT
923:and
356:for
311:Eq.1
2859:doi
2590:log
2534:log
2431:log
2351:log
2281:).
2132:201
2049:201
1953:log
1850:log
1816:end
1770:FFT
1676:DFT
1597:DFT
1552:DFT
1378:of
605:100
557:).
473:or
34:In
2884::
2865:.
2855:90
2853:.
2849:.
2779:.
2771:.
1812:do
579:16
2873:.
2861::
2837:.
2818:.
2789:.
2739:2
2716:1
2710:M
2688:k
2684:x
2641:.
2639:h
2635:x
2613:)
2607:x
2603:N
2594:2
2584:x
2580:N
2575:(
2571:O
2550:)
2546:N
2538:2
2528:x
2524:N
2519:(
2515:O
2488:.
2482:1
2479:+
2476:M
2470:N
2466:N
2458:)
2455:1
2452:+
2449:)
2446:N
2443:(
2435:2
2427:(
2419:x
2415:N
2388:.
2385:)
2382:1
2379:+
2376:)
2371:x
2367:N
2363:(
2355:2
2347:(
2339:x
2335:N
2309:x
2305:N
2269:M
2243:N
2223:.
2220:N
2195:,
2192:M
2172:h
2152:x
2106:,
2078:,
2072:=
2069:N
2046:=
2043:M
2027:)
2023:(
2003:.
1997:1
1994:+
1991:M
1985:N
1980:)
1977:1
1974:+
1971:)
1968:N
1965:(
1957:2
1949:(
1946:N
1927::
1914:2
1890:1
1887:+
1884:M
1878:N
1872:)
1868:1
1865:+
1862:N
1854:2
1845:(
1841:N
1796:)
1792:(
1787::
1756:1
1750:M
1747:+
1744:L
1741:=
1738:N
1718:L
1696:N
1682:N
1678:N
1671::
1661:)
1657:(
1632:,
1629:)
1623:)
1620:]
1617:n
1614:[
1611:h
1608:(
1602:N
1585:)
1582:]
1579:n
1576:[
1571:k
1567:x
1563:(
1557:N
1543:(
1537:N
1523:=
1517:]
1514:n
1511:[
1506:k
1502:y
1485::
1468:.
1465:]
1462:N
1459:,
1456:1
1453:[
1432:]
1429:n
1426:[
1423:h
1402:]
1399:n
1396:[
1391:k
1387:x
1362:N
1341:,
1338:1
1332:M
1329:+
1326:L
1320:N
1300:.
1297:]
1294:1
1288:M
1285:+
1282:L
1279:,
1276:1
1273:[
1252:]
1249:n
1246:[
1243:h
1237:]
1234:n
1231:[
1226:k
1222:x
1209:]
1206:n
1203:[
1198:k
1194:y
1166:,
1163:]
1160:L
1157:k
1151:n
1148:[
1143:k
1139:y
1133:k
1125:=
1114:)
1110:]
1107:n
1104:[
1101:h
1095:]
1092:L
1089:k
1083:n
1080:[
1075:k
1071:x
1066:(
1060:k
1052:=
1045:]
1042:n
1039:[
1036:h
1029:)
1025:]
1022:L
1019:k
1013:n
1010:[
1005:k
1001:x
995:k
986:(
982:=
979:]
976:n
973:[
970:y
954::
940:]
937:n
934:[
931:y
907:,
904:]
901:L
898:k
892:n
889:[
884:k
880:x
874:k
866:=
863:]
860:n
857:[
854:x
842::
828:L
798:,
788:,
785:0
778:L
775:,
769:,
766:2
763:,
760:1
757:=
754:n
749:,
746:]
743:L
740:k
737:+
734:n
731:[
728:x
722:{
708:]
705:n
702:[
697:k
693:x
680::
667:]
664:n
661:[
658:x
638:]
635:n
632:[
629:h
602:=
599:L
576:=
573:M
541:t
521:,
518:}
515:)
512:t
509:(
506:x
503:{
498:H
493:=
490:)
487:t
484:(
481:y
461:,
458:)
455:t
452:(
449:h
443:)
440:t
437:(
434:x
431:=
428:)
425:t
422:(
419:y
399:.
396:]
393:M
390:,
387:1
384:[
364:m
344:0
341:=
338:]
335:m
332:[
329:h
313:)
309:(
290:,
287:]
284:m
278:n
275:[
272:x
266:]
263:m
260:[
257:h
252:M
247:1
244:=
241:m
233:=
230:]
227:m
221:n
218:[
215:x
209:]
206:m
203:[
200:h
184:=
181:m
164:]
161:n
158:[
155:h
149:]
146:n
143:[
140:x
137:=
134:]
131:n
128:[
125:y
109::
96:]
93:n
90:[
87:h
63:]
60:n
57:[
54:x
31:.
20:)
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