20:
2394:
2008:
75:
of arbitrary form. Schmidt, in particular, accomplished this by first deriving a complete geometric solution in the absence of noise, then cleverly extending the geometric concepts to obtain a reasonable approximate solution in the presence of noise. The resulting algorithm was called MUSIC (MUltiple
54:
In many practical signal processing problems, the objective is to estimate from measurements a set of constant parameters upon which the received signals depend. There have been several approaches to such problems including the so-called maximum likelihood (ML) method of Capon (1969) and Burg's
3057:
Its chief disadvantage is that it requires the number of components to be known in advance, so the original method cannot be used in more general cases. Methods exist for estimating the number of source components purely from statistical properties of the autocorrelation matrix. See, e.g. In
2732:
coefficients, whose zeros can be found analytically or with polynomial root finding algorithms. In contrast, MUSIC assumes that several such functions have been added together, so zeros may not be present. Instead there are local minima, which can be located by computationally searching the
79:
In a detailed evaluation based on thousands of simulations, the
Massachusetts Institute of Technology's Lincoln Laboratory concluded in 1998 that, among currently accepted high-resolution algorithms, MUSIC was the most promising and a leading candidate for further study and actual hardware
2548:
2200:
1828:
724:
3046:
MUSIC outperforms simple methods such as picking peaks of DFT spectra in the presence of noise, when the number of components is known in advance, because it exploits knowledge of this number to ignore the noise in its final report.
55:
maximum entropy (ME) method. Although often successful and widely used, these methods have certain fundamental limitations (especially bias and sensitivity in parameter estimates), largely because they use an incorrect model (e.g.,
982:
2687:
3076:
A modified version of MUSIC, denoted as Time-Reversal MUSIC (TR-MUSIC) has been recently applied to computational time-reversal imaging. MUSIC algorithm has also been implemented for fast detection of the DTMF frequencies
1246:
2193:
as implied by the orthogonality condition. Taking a reciprocal of the squared norm expression creates sharp peaks at the signal frequencies. The frequency estimation function for MUSIC (or the pseudo-spectrum) is
70:
in additive noise using a covariance approach. Schmidt (1977), while working at
Northrop Grumman and independently Bienvenu and Kopp (1979) were the first to correctly exploit the measurement model in the case of
219:
1331:
80:
implementation. However, although the performance advantages of MUSIC are substantial, they are achieved at a cost in computation (searching over parameter space) and storage (of array calibration data).
2434:
3200:
1514:
3050:
Unlike DFT, it is able to estimate frequencies with accuracy higher than one sample, because its estimation function can be evaluated for any frequency, not just those of DFT bins. This is a form of
1820:
1656:
2389:{\displaystyle {\hat {P}}_{MU}(e^{j\omega })={\frac {1}{\mathbf {e} ^{H}\mathbf {U} _{N}\mathbf {U} _{N}^{H}\mathbf {e} }}={\frac {1}{\sum _{i=p+1}^{M}|\mathbf {e} ^{H}\mathbf {v} _{i}|^{2}}},}
2158:
1693:
1617:
1752:
455:
2003:{\displaystyle d^{2}=\|\mathbf {U} _{N}^{H}\mathbf {e} \|^{2}=\mathbf {e} ^{H}\mathbf {U} _{N}\mathbf {U} _{N}^{H}\mathbf {e} =\sum _{i=p+1}^{M}|\mathbf {e} ^{H}\mathbf {v} _{i}|^{2}}
3032:
2978:
2852:
2119:
1466:
1382:
2088:
522:
2772:
2426:
1275:
1155:
1119:
911:
831:
1090:
647:
304:
1774:
1578:
879:
776:
639:
170:
108:
1435:
754:
2728:. In Pisarenko's method, only a single eigenvector is used to form the denominator of the frequency estimation function; and the eigenvector is interpreted as a set of
857:
802:
617:
330:
3392:
Zhang, Qilin; Abeida, Habti; Xue, Ming; Rowe, William; Li, Jian (2012). "Fast implementation of sparse iterative covariance-based estimation for source localization".
2901:
2191:
148:
2821:
1011:
588:
2726:
1549:
1408:
3001:
2924:
2947:
2872:
2792:
2591:
2571:
1351:
1175:
562:
542:
128:
919:
2599:
1555:. The general idea behind MUSIC method is to use all the eigenvectors that span the noise subspace to improve the performance of the Pisarenko estimator.
3331:
Abeida, Habti; Zhang, Qilin; Li, Jian; Merabtine, Nadjim (2013). "Iterative Sparse
Asymptotic Minimum Variance Based Approaches for Array Processing".
1180:
178:
66:
Pisarenko (1973) was one of the first to exploit the structure of the data model, doing so in the context of estimation of parameters of
2929:
This fundamental result, although often skipped in spectral analysis books, is a reason why the input signal can be distributed into
2543:{\displaystyle \mathbf {e} ={\begin{bmatrix}1&e^{j\omega }&e^{j2\omega }&\cdots &e^{j(M-1)\omega }\end{bmatrix}}^{T}}
1280:
60:
1471:
1779:
3271:
1622:
3078:
2124:
1661:
1583:
3588:
1698:
3157:
2693:
1552:
338:
3291:," in IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 7, pp. 984–995, Jul 1989.
3196:"Super-Resolution Spectral Approach for the Accuracy Enhancement of Biomedical Resonant Microwave Sensors"
3174:
3090:
3058:
addition, MUSIC assumes coexistent sources to be uncorrelated, which limits its practical applications.
3006:
2952:
2826:
2093:
1440:
1356:
3130:
2016:
460:
2748:
2402:
1251:
1131:
1095:
887:
807:
719:{\displaystyle \mathbf {R} _{x}=\mathbf {A} \mathbf {R} _{s}\mathbf {A} ^{H}+\sigma ^{2}\mathbf {I} ,}
3125:
2741:
The fundamental observation MUSIC and other subspace decomposition methods are based on is about the
3486:
Ciuonzo, D.; Romano, G.; Solimene, R. (2015-05-01). "Performance
Analysis of Time-Reversal MUSIC".
3062:
3051:
1016:
227:
3034:. It is based on signal embedding theory and can also be explained by the topological theory of
1757:
1561:
862:
759:
622:
153:
91:
3120:
1413:
732:
43:
24:
67:
3435:
Devaney, A.J. (2005-05-01). "Time reversal imaging of obscured targets from multistatic data".
836:
781:
596:
309:
2877:
2742:
2163:
133:
3575:
The estimation and tracking of frequency, Quinn and Hannan, Cambridge
University Press 2001.
2800:
990:
567:
3495:
3444:
3401:
3350:
3319:
Estimation of the number of sources in unbalanced arrays via information theoretic criteria
3302:
3110:
2729:
2699:
1522:
56:
39:
8:
3246:
1387:
1121:, MUSIC estimates the frequency content of the signal or autocorrelation matrix using an
3499:
3448:
3405:
3354:
2983:
2906:
977:{\displaystyle {\widehat {\mathbf {R} }}_{x}={\frac {1}{N}}\mathbf {X} \mathbf {X} ^{H}}
3519:
3468:
3374:
3340:
3288:
3225:
2932:
2857:
2777:
2576:
2556:
1336:
1160:
547:
527:
332:
113:
1353:
largest eigenvalues (i.e. directions of largest variability) span the signal subspace
3511:
3460:
3417:
3366:
3229:
3217:
3153:
3105:
2682:{\displaystyle {\hat {\omega }}=\arg \max _{\omega }\;{\hat {P}}_{MU}(e^{j\omega }).}
3472:
3378:
3523:
3503:
3452:
3409:
3358:
3209:
3115:
3066:
1754:
that spans the noise subspace. In order to measure the degree of orthogonality of
3555:
3261:"Performance Comparison of Superresolution Array Processing Algorithms. Revised"
3260:
3249:," IEEE Trans. Antennas Propagation, Vol. AP-34 (March 1986), pp. 276–280.
3195:
3100:
3537:
3213:
2573:
largest peaks of the estimation function give the frequency estimates for the
1241:{\displaystyle \{\mathbf {v} _{1},\mathbf {v} _{2},\ldots ,\mathbf {v} _{M}\}}
3582:
3515:
3507:
3464:
3370:
3362:
3221:
3081:) in the form of C library - libmusic (including for MATLAB implementation).
3456:
3421:
3289:
ESPRIT-estimation of signal parameters via rotational invariance techniques
3201:
72:
3095:
524:
is the amplitude vector. A crucial assumption is that number of sources,
3339:(4). Institute of Electrical and Electronics Engineers (IEEE): 933–944.
1122:
3413:
3035:
3345:
3318:
214:{\displaystyle \mathbf {x} =\mathbf {A} \mathbf {s} +\mathbf {n} .}
16:
Algorithm used for frequency estimation and radio direction finding
3307:, University College London, Lecture notes 1999–2000 academic year
3003:
for real valued signals) and noise subspace eigenvectors spanning
3321:." IEEE Transactions on Signal Processing 53.9 (2005): 3543–3553.
3194:
Costanzo, Sandra; Buonanno, Giovanni; Solimene, Raffaele (2022).
544:, is less than the number of elements in the measurement vector,
1326:{\displaystyle \{\mathbf {v} _{1},\ldots ,\mathbf {v} _{p}\}}
19:
3183:] (Thesis) (in Polish). Warsaw University of Technology.
2926:, i.e. each real sinusoid is generated by two base vectors.
913:
is traditionally estimated using sample correlation matrix
2090:
is the matrix of eigenvectors that span the noise subspace
1509:{\displaystyle {\mathcal {U}}_{S}\perp {\mathcal {U}}_{N}}
3247:
Multiple
Emitter Location and Signal Parameter Estimation
3193:
2553:
is the candidate steering vector. The locations of the
3538:"libmusic: A powerful C library for spectral analysis"
3330:
3061:
Recent iterative semi-parametric methods offer robust
2452:
1815:{\displaystyle \mathbf {v} _{i}\in {\mathcal {U}}_{N}}
150:
are unknown, in the presence of
Gaussian white noise,
3485:
3009:
2986:
2955:
2935:
2909:
2880:
2860:
2829:
2803:
2780:
2751:
2702:
2602:
2579:
2559:
2437:
2405:
2203:
2166:
2127:
2096:
2019:
1831:
1782:
1760:
1701:
1664:
1625:
1586:
1564:
1525:
1474:
1443:
1416:
1390:
1359:
1339:
1283:
1254:
1183:
1163:
1134:
1098:
1019:
993:
922:
890:
865:
839:
810:
784:
762:
735:
650:
625:
599:
570:
550:
530:
463:
341:
312:
230:
181:
156:
136:
116:
94:
76:
1651:{\displaystyle \mathbf {e} \perp {\mathcal {U}}_{N}}
1248:
are orthogonal to each other. If the eigenvalues of
3391:
3150:Statistical Digital Signal Processing and Modeling
3026:
2995:
2972:
2941:
2918:
2895:
2866:
2846:
2815:
2786:
2766:
2720:
2681:
2585:
2565:
2542:
2420:
2388:
2185:
2153:{\displaystyle \mathbf {e} \in {\mathcal {U}}_{S}}
2152:
2113:
2082:
2002:
1814:
1768:
1746:
1688:{\displaystyle \mathbf {e} \perp \mathbf {v} _{i}}
1687:
1650:
1612:{\displaystyle \mathbf {e} \in {\mathcal {U}}_{S}}
1611:
1572:
1543:
1508:
1460:
1429:
1402:
1376:
1345:
1325:
1269:
1240:
1169:
1149:
1113:
1084:
1005:
976:
905:
873:
851:
825:
796:
770:
748:
718:
633:
611:
582:
556:
536:
516:
449:
324:
298:
213:
164:
142:
122:
102:
1277:are sorted in decreasing order, the eigenvectors
3580:
3394:The Journal of the Acoustical Society of America
3268:Massachusetts Inst of Tech Lexington Lincoln Lab
3181:Application of MUSIC algorithm to DTMF detection
2625:
1747:{\displaystyle \{\mathbf {v} _{i}\}_{i=p+1}^{M}}
3176:Zastosowanie algorytmu MUSIC do wykrywania DTMF
1410:eigenvectors correspond to eigenvalue equal to
3041:
1822:, the MUSIC algorithm defines a squared norm
1468:, which is orthogonal to the signal subspace,
3437:IEEE Transactions on Antennas and Propagation
3168:
3166:
2774:which is related to number of signal sources
2736:
2696:, and it reduces to Pisarenko's method when
1871:
1845:
1718:
1702:
1320:
1284:
1235:
1184:
88:MUSIC method assumes that a signal vector,
3324:
3163:
2634:
1619:must be orthogonal to the noise subspace,
450:{\displaystyle \mathbf {a} (\omega )=^{T}}
3556:"libmusic_m : MATLAB implementation"
3344:
3241:
3239:
3065:despite highly correlated sources, e.g.,
2823:and the dimension of the signal subspace
1013:is the number of vector observations and
3258:
2903:and dimension of the signal subspace is
130:complex exponentials, whose frequencies
18:
3434:
3581:
3488:IEEE Transactions on Signal Processing
3333:IEEE Transactions on Signal Processing
3236:
3172:
2949:signal subspace eigenvectors spanning
3317:Fishler, Eran, and H. Vincent Poor. "
3300:
3152:, John Wiley & Sons, Inc., 1996.
3071:
1580:that resides in the signal subspace
3079:Dual-tone multi-frequency signaling
13:
3569:
3385:
3277:from the original on May 25, 2021.
3027:{\displaystyle {\mathcal {U}}_{N}}
3013:
2973:{\displaystyle {\mathcal {U}}_{S}}
2959:
2847:{\displaystyle {\mathcal {U}}_{S}}
2833:
2139:
2114:{\displaystyle {\mathcal {U}}_{N}}
2100:
1801:
1637:
1598:
1495:
1478:
1461:{\displaystyle {\mathcal {U}}_{N}}
1447:
1377:{\displaystyle {\mathcal {U}}_{S}}
1363:
1157:is a Hermitian matrix, all of its
14:
3600:
2797:If the sources are complex, then
2083:{\displaystyle \mathbf {U} _{N}=}
517:{\displaystyle \mathbf {s} =^{T}}
2767:{\displaystyle \mathbf {R} _{x}}
2754:
2439:
2421:{\displaystyle \mathbf {v} _{i}}
2408:
2358:
2346:
2296:
2280:
2268:
2256:
2129:
2067:
2040:
2022:
1978:
1966:
1925:
1909:
1897:
1885:
1866:
1850:
1785:
1762:
1707:
1675:
1666:
1627:
1588:
1566:
1553:Pisarenko harmonic decomposition
1310:
1289:
1270:{\displaystyle \mathbf {R} _{x}}
1257:
1225:
1204:
1189:
1150:{\displaystyle \mathbf {R} _{x}}
1137:
1114:{\displaystyle \mathbf {R} _{x}}
1101:
1069:
1048:
1033:
1021:
964:
958:
928:
906:{\displaystyle \mathbf {R} _{x}}
893:
867:
826:{\displaystyle \mathbf {R} _{s}}
813:
764:
709:
685:
673:
667:
653:
627:
465:
343:
273:
243:
232:
204:
196:
191:
183:
158:
96:
3548:
3530:
3479:
3428:
2733:estimation function for peaks.
2428:are the noise eigenvectors and
172:, as given by the linear model
3311:
3294:
3281:
3252:
3187:
3142:
2745:of the autocorrelation matrix
2673:
2657:
2642:
2609:
2518:
2506:
2370:
2340:
2242:
2226:
2211:
2077:
2035:
1990:
1960:
1079:
1028:
505:
472:
438:
429:
417:
359:
353:
347:
293:
290:
277:
260:
247:
239:
36:MUltiple SIgnal Classification
1:
3136:
2692:MUSIC is a generalization of
1085:{\displaystyle \mathbf {X} =}
299:{\displaystyle \mathbf {A} =}
2874:. If sources are real, then
1769:{\displaystyle \mathbf {e} }
1573:{\displaystyle \mathbf {e} }
1437:and span the noise subspace
874:{\displaystyle \mathbf {s} }
771:{\displaystyle \mathbf {I} }
634:{\displaystyle \mathbf {x} }
165:{\displaystyle \mathbf {n} }
103:{\displaystyle \mathbf {x} }
7:
3091:Spectral density estimation
3084:
3042:Comparison to other methods
1430:{\displaystyle \sigma ^{2}}
884:The autocorrelation matrix
749:{\displaystyle \sigma ^{2}}
38:) is an algorithm used for
10:
3605:
3131:High-resolution microscopy
859:autocorrelation matrix of
619:autocorrelation matrix of
49:
3589:Digital signal processing
3214:10.1109/JERM.2022.3210457
3126:Pitch detection algorithm
2737:Dimension of signal space
1695:for all the eigenvectors
852:{\displaystyle p\times p}
797:{\displaystyle M\times M}
612:{\displaystyle M\times M}
325:{\displaystyle M\times p}
83:
3508:10.1109/TSP.2015.2417507
3363:10.1109/tsp.2012.2231676
3304:Signal Processing Course
3287:R. Roy and T. Kailath, "
3259:Barabell, A. J. (1998).
1776:with respect to all the
1558:Since any signal vector
1551:, MUSIC is identical to
1092:. Given the estimate of
63:) of the measurements.
3457:10.1109/TAP.2005.846723
3121:Radio direction finding
2896:{\displaystyle M>2p}
2186:{\displaystyle d^{2}=0}
756:is the noise variance,
143:{\displaystyle \omega }
44:radio direction finding
25:radio direction finding
3173:Gregor, Piotr (2022).
3028:
2997:
2974:
2943:
2920:
2897:
2868:
2848:
2817:
2816:{\displaystyle M>p}
2788:
2768:
2722:
2683:
2587:
2567:
2544:
2422:
2390:
2338:
2187:
2154:
2115:
2084:
2004:
1958:
1816:
1770:
1748:
1689:
1652:
1613:
1574:
1545:
1510:
1462:
1431:
1404:
1378:
1347:
1327:
1271:
1242:
1171:
1151:
1115:
1086:
1007:
1006:{\displaystyle N>M}
978:
907:
875:
853:
827:
798:
772:
750:
720:
635:
613:
584:
583:{\displaystyle p<M}
558:
538:
518:
451:
326:
300:
215:
166:
144:
124:
104:
28:
27:by the MUSIC algorithm
3301:Penny, W. D. (2009),
3029:
2998:
2975:
2944:
2921:
2898:
2869:
2849:
2818:
2789:
2769:
2723:
2721:{\displaystyle M=p+1}
2684:
2588:
2568:
2545:
2423:
2391:
2312:
2188:
2155:
2116:
2085:
2005:
1932:
1817:
1771:
1749:
1690:
1653:
1614:
1575:
1546:
1544:{\displaystyle M=p+1}
1511:
1463:
1432:
1405:
1379:
1348:
1333:corresponding to the
1328:
1272:
1243:
1172:
1152:
1116:
1087:
1008:
979:
908:
876:
854:
828:
804:identity matrix, and
799:
773:
751:
721:
636:
614:
585:
559:
539:
519:
452:
327:
301:
216:
167:
145:
125:
105:
22:
3007:
2984:
2953:
2933:
2907:
2878:
2858:
2827:
2801:
2778:
2749:
2700:
2600:
2577:
2557:
2435:
2403:
2201:
2164:
2125:
2094:
2017:
1829:
1780:
1758:
1699:
1662:
1623:
1584:
1562:
1523:
1472:
1441:
1414:
1388:
1357:
1337:
1281:
1252:
1181:
1161:
1132:
1096:
1017:
991:
920:
888:
863:
837:
808:
782:
760:
733:
648:
623:
597:
568:
548:
528:
461:
339:
335:of steering vectors
310:
228:
179:
154:
134:
114:
92:
59:rather than special
40:frequency estimation
3500:2015ITSP...63.2650C
3449:2005ITAP...53.1600D
3406:2012ASAJ..131.1249Z
3355:2013ITSP...61..933A
2294:
1923:
1864:
1743:
1403:{\displaystyle M-p}
3562:. 2023. MathWorks.
3148:Hayes, Monson H.,
3072:Other applications
3024:
2996:{\displaystyle 2p}
2993:
2970:
2939:
2919:{\displaystyle 2p}
2916:
2893:
2864:
2844:
2813:
2784:
2764:
2718:
2694:Pisarenko's method
2679:
2633:
2593:signal components
2583:
2563:
2540:
2528:
2418:
2386:
2278:
2183:
2150:
2111:
2080:
2000:
1907:
1848:
1812:
1766:
1744:
1717:
1685:
1658:, it must be that
1648:
1609:
1570:
1541:
1506:
1458:
1427:
1400:
1374:
1343:
1323:
1267:
1238:
1167:
1147:
1111:
1082:
1003:
974:
903:
871:
849:
823:
794:
768:
746:
716:
631:
609:
580:
554:
534:
514:
447:
333:Vandermonde matrix
322:
296:
211:
162:
140:
120:
100:
29:
3494:(10): 2650–2662.
3414:10.1121/1.3672656
3111:Bartlett's method
2942:{\displaystyle p}
2867:{\displaystyle p}
2787:{\displaystyle p}
2645:
2624:
2612:
2586:{\displaystyle p}
2566:{\displaystyle p}
2381:
2301:
2214:
2013:where the matrix
1346:{\displaystyle p}
1170:{\displaystyle M}
955:
935:
641:is then given by
557:{\displaystyle M}
537:{\displaystyle p}
123:{\displaystyle p}
68:complex sinusoids
3596:
3564:
3563:
3552:
3546:
3545:
3534:
3528:
3527:
3483:
3477:
3476:
3443:(5): 1600–1610.
3432:
3426:
3425:
3400:(2): 1249–1259.
3389:
3383:
3382:
3348:
3328:
3322:
3315:
3309:
3308:
3298:
3292:
3285:
3279:
3278:
3276:
3265:
3256:
3250:
3243:
3234:
3233:
3191:
3185:
3184:
3170:
3161:
3146:
3116:SAMV (algorithm)
3033:
3031:
3030:
3025:
3023:
3022:
3017:
3016:
3002:
3000:
2999:
2994:
2979:
2977:
2976:
2971:
2969:
2968:
2963:
2962:
2948:
2946:
2945:
2940:
2925:
2923:
2922:
2917:
2902:
2900:
2899:
2894:
2873:
2871:
2870:
2865:
2853:
2851:
2850:
2845:
2843:
2842:
2837:
2836:
2822:
2820:
2819:
2814:
2793:
2791:
2790:
2785:
2773:
2771:
2770:
2765:
2763:
2762:
2757:
2727:
2725:
2724:
2719:
2688:
2686:
2685:
2680:
2672:
2671:
2656:
2655:
2647:
2646:
2638:
2632:
2614:
2613:
2605:
2592:
2590:
2589:
2584:
2572:
2570:
2569:
2564:
2549:
2547:
2546:
2541:
2539:
2538:
2533:
2532:
2525:
2524:
2490:
2489:
2472:
2471:
2442:
2427:
2425:
2424:
2419:
2417:
2416:
2411:
2395:
2393:
2392:
2387:
2382:
2380:
2379:
2378:
2373:
2367:
2366:
2361:
2355:
2354:
2349:
2343:
2337:
2332:
2307:
2302:
2300:
2299:
2293:
2288:
2283:
2277:
2276:
2271:
2265:
2264:
2259:
2249:
2241:
2240:
2225:
2224:
2216:
2215:
2207:
2192:
2190:
2189:
2184:
2176:
2175:
2159:
2157:
2156:
2151:
2149:
2148:
2143:
2142:
2132:
2120:
2118:
2117:
2112:
2110:
2109:
2104:
2103:
2089:
2087:
2086:
2081:
2076:
2075:
2070:
2055:
2054:
2043:
2031:
2030:
2025:
2009:
2007:
2006:
2001:
1999:
1998:
1993:
1987:
1986:
1981:
1975:
1974:
1969:
1963:
1957:
1952:
1928:
1922:
1917:
1912:
1906:
1905:
1900:
1894:
1893:
1888:
1879:
1878:
1869:
1863:
1858:
1853:
1841:
1840:
1821:
1819:
1818:
1813:
1811:
1810:
1805:
1804:
1794:
1793:
1788:
1775:
1773:
1772:
1767:
1765:
1753:
1751:
1750:
1745:
1742:
1737:
1716:
1715:
1710:
1694:
1692:
1691:
1686:
1684:
1683:
1678:
1669:
1657:
1655:
1654:
1649:
1647:
1646:
1641:
1640:
1630:
1618:
1616:
1615:
1610:
1608:
1607:
1602:
1601:
1591:
1579:
1577:
1576:
1571:
1569:
1550:
1548:
1547:
1542:
1515:
1513:
1512:
1507:
1505:
1504:
1499:
1498:
1488:
1487:
1482:
1481:
1467:
1465:
1464:
1459:
1457:
1456:
1451:
1450:
1436:
1434:
1433:
1428:
1426:
1425:
1409:
1407:
1406:
1401:
1384:. The remaining
1383:
1381:
1380:
1375:
1373:
1372:
1367:
1366:
1352:
1350:
1349:
1344:
1332:
1330:
1329:
1324:
1319:
1318:
1313:
1298:
1297:
1292:
1276:
1274:
1273:
1268:
1266:
1265:
1260:
1247:
1245:
1244:
1239:
1234:
1233:
1228:
1213:
1212:
1207:
1198:
1197:
1192:
1176:
1174:
1173:
1168:
1156:
1154:
1153:
1148:
1146:
1145:
1140:
1120:
1118:
1117:
1112:
1110:
1109:
1104:
1091:
1089:
1088:
1083:
1078:
1077:
1072:
1057:
1056:
1051:
1042:
1041:
1036:
1024:
1012:
1010:
1009:
1004:
983:
981:
980:
975:
973:
972:
967:
961:
956:
948:
943:
942:
937:
936:
931:
926:
912:
910:
909:
904:
902:
901:
896:
880:
878:
877:
872:
870:
858:
856:
855:
850:
832:
830:
829:
824:
822:
821:
816:
803:
801:
800:
795:
777:
775:
774:
769:
767:
755:
753:
752:
747:
745:
744:
725:
723:
722:
717:
712:
707:
706:
694:
693:
688:
682:
681:
676:
670:
662:
661:
656:
640:
638:
637:
632:
630:
618:
616:
615:
610:
589:
587:
586:
581:
563:
561:
560:
555:
543:
541:
540:
535:
523:
521:
520:
515:
513:
512:
503:
502:
484:
483:
468:
456:
454:
453:
448:
446:
445:
436:
435:
399:
398:
380:
379:
346:
331:
329:
328:
323:
305:
303:
302:
297:
289:
288:
276:
259:
258:
246:
235:
220:
218:
217:
212:
207:
199:
194:
186:
171:
169:
168:
163:
161:
149:
147:
146:
141:
129:
127:
126:
121:
109:
107:
106:
101:
99:
3604:
3603:
3599:
3598:
3597:
3595:
3594:
3593:
3579:
3578:
3572:
3570:Further reading
3567:
3560:Data and Signal
3554:
3553:
3549:
3542:Data and Signal
3536:
3535:
3531:
3484:
3480:
3433:
3429:
3390:
3386:
3329:
3325:
3316:
3312:
3299:
3295:
3286:
3282:
3274:
3263:
3257:
3253:
3245:Schmidt, R.O, "
3244:
3237:
3192:
3188:
3171:
3164:
3147:
3143:
3139:
3087:
3074:
3063:superresolution
3052:superresolution
3044:
3018:
3012:
3011:
3010:
3008:
3005:
3004:
2985:
2982:
2981:
2964:
2958:
2957:
2956:
2954:
2951:
2950:
2934:
2931:
2930:
2908:
2905:
2904:
2879:
2876:
2875:
2859:
2856:
2855:
2838:
2832:
2831:
2830:
2828:
2825:
2824:
2802:
2799:
2798:
2779:
2776:
2775:
2758:
2753:
2752:
2750:
2747:
2746:
2739:
2701:
2698:
2697:
2664:
2660:
2648:
2637:
2636:
2635:
2628:
2604:
2603:
2601:
2598:
2597:
2578:
2575:
2574:
2558:
2555:
2554:
2534:
2527:
2526:
2502:
2498:
2496:
2491:
2479:
2475:
2473:
2464:
2460:
2458:
2448:
2447:
2446:
2438:
2436:
2433:
2432:
2412:
2407:
2406:
2404:
2401:
2400:
2374:
2369:
2368:
2362:
2357:
2356:
2350:
2345:
2344:
2339:
2333:
2316:
2311:
2306:
2295:
2289:
2284:
2279:
2272:
2267:
2266:
2260:
2255:
2254:
2253:
2248:
2233:
2229:
2217:
2206:
2205:
2204:
2202:
2199:
2198:
2171:
2167:
2165:
2162:
2161:
2144:
2138:
2137:
2136:
2128:
2126:
2123:
2122:
2105:
2099:
2098:
2097:
2095:
2092:
2091:
2071:
2066:
2065:
2044:
2039:
2038:
2026:
2021:
2020:
2018:
2015:
2014:
1994:
1989:
1988:
1982:
1977:
1976:
1970:
1965:
1964:
1959:
1953:
1936:
1924:
1918:
1913:
1908:
1901:
1896:
1895:
1889:
1884:
1883:
1874:
1870:
1865:
1859:
1854:
1849:
1836:
1832:
1830:
1827:
1826:
1806:
1800:
1799:
1798:
1789:
1784:
1783:
1781:
1778:
1777:
1761:
1759:
1756:
1755:
1738:
1721:
1711:
1706:
1705:
1700:
1697:
1696:
1679:
1674:
1673:
1665:
1663:
1660:
1659:
1642:
1636:
1635:
1634:
1626:
1624:
1621:
1620:
1603:
1597:
1596:
1595:
1587:
1585:
1582:
1581:
1565:
1563:
1560:
1559:
1524:
1521:
1520:
1500:
1494:
1493:
1492:
1483:
1477:
1476:
1475:
1473:
1470:
1469:
1452:
1446:
1445:
1444:
1442:
1439:
1438:
1421:
1417:
1415:
1412:
1411:
1389:
1386:
1385:
1368:
1362:
1361:
1360:
1358:
1355:
1354:
1338:
1335:
1334:
1314:
1309:
1308:
1293:
1288:
1287:
1282:
1279:
1278:
1261:
1256:
1255:
1253:
1250:
1249:
1229:
1224:
1223:
1208:
1203:
1202:
1193:
1188:
1187:
1182:
1179:
1178:
1162:
1159:
1158:
1141:
1136:
1135:
1133:
1130:
1129:
1105:
1100:
1099:
1097:
1094:
1093:
1073:
1068:
1067:
1052:
1047:
1046:
1037:
1032:
1031:
1020:
1018:
1015:
1014:
992:
989:
988:
968:
963:
962:
957:
947:
938:
927:
925:
924:
923:
921:
918:
917:
897:
892:
891:
889:
886:
885:
866:
864:
861:
860:
838:
835:
834:
817:
812:
811:
809:
806:
805:
783:
780:
779:
763:
761:
758:
757:
740:
736:
734:
731:
730:
708:
702:
698:
689:
684:
683:
677:
672:
671:
666:
657:
652:
651:
649:
646:
645:
626:
624:
621:
620:
598:
595:
594:
569:
566:
565:
549:
546:
545:
529:
526:
525:
508:
504:
498:
494:
479:
475:
464:
462:
459:
458:
441:
437:
413:
409:
388:
384:
372:
368:
342:
340:
337:
336:
311:
308:
307:
284:
280:
272:
254:
250:
242:
231:
229:
226:
225:
203:
195:
190:
182:
180:
177:
176:
157:
155:
152:
151:
135:
132:
131:
115:
112:
111:
95:
93:
90:
89:
86:
52:
17:
12:
11:
5:
3602:
3592:
3591:
3577:
3576:
3571:
3568:
3566:
3565:
3547:
3529:
3478:
3427:
3384:
3323:
3310:
3293:
3280:
3251:
3235:
3208:(4): 539–545.
3186:
3162:
3140:
3138:
3135:
3134:
3133:
3128:
3123:
3118:
3113:
3108:
3106:Welch's method
3103:
3101:Matched filter
3098:
3093:
3086:
3083:
3073:
3070:
3043:
3040:
3021:
3015:
2992:
2989:
2967:
2961:
2938:
2915:
2912:
2892:
2889:
2886:
2883:
2863:
2841:
2835:
2812:
2809:
2806:
2783:
2761:
2756:
2738:
2735:
2730:autoregressive
2717:
2714:
2711:
2708:
2705:
2690:
2689:
2678:
2675:
2670:
2667:
2663:
2659:
2654:
2651:
2644:
2641:
2631:
2627:
2623:
2620:
2617:
2611:
2608:
2582:
2562:
2551:
2550:
2537:
2531:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2501:
2497:
2495:
2492:
2488:
2485:
2482:
2478:
2474:
2470:
2467:
2463:
2459:
2457:
2454:
2453:
2451:
2445:
2441:
2415:
2410:
2397:
2396:
2385:
2377:
2372:
2365:
2360:
2353:
2348:
2342:
2336:
2331:
2328:
2325:
2322:
2319:
2315:
2310:
2305:
2298:
2292:
2287:
2282:
2275:
2270:
2263:
2258:
2252:
2247:
2244:
2239:
2236:
2232:
2228:
2223:
2220:
2213:
2210:
2182:
2179:
2174:
2170:
2147:
2141:
2135:
2131:
2108:
2102:
2079:
2074:
2069:
2064:
2061:
2058:
2053:
2050:
2047:
2042:
2037:
2034:
2029:
2024:
2011:
2010:
1997:
1992:
1985:
1980:
1973:
1968:
1962:
1956:
1951:
1948:
1945:
1942:
1939:
1935:
1931:
1927:
1921:
1916:
1911:
1904:
1899:
1892:
1887:
1882:
1877:
1873:
1868:
1862:
1857:
1852:
1847:
1844:
1839:
1835:
1809:
1803:
1797:
1792:
1787:
1764:
1741:
1736:
1733:
1730:
1727:
1724:
1720:
1714:
1709:
1704:
1682:
1677:
1672:
1668:
1645:
1639:
1633:
1629:
1606:
1600:
1594:
1590:
1568:
1540:
1537:
1534:
1531:
1528:
1519:Note that for
1503:
1497:
1491:
1486:
1480:
1455:
1449:
1424:
1420:
1399:
1396:
1393:
1371:
1365:
1342:
1322:
1317:
1312:
1307:
1304:
1301:
1296:
1291:
1286:
1264:
1259:
1237:
1232:
1227:
1222:
1219:
1216:
1211:
1206:
1201:
1196:
1191:
1186:
1166:
1144:
1139:
1108:
1103:
1081:
1076:
1071:
1066:
1063:
1060:
1055:
1050:
1045:
1040:
1035:
1030:
1027:
1023:
1002:
999:
996:
985:
984:
971:
966:
960:
954:
951:
946:
941:
934:
930:
900:
895:
869:
848:
845:
842:
820:
815:
793:
790:
787:
766:
743:
739:
727:
726:
715:
711:
705:
701:
697:
692:
687:
680:
675:
669:
665:
660:
655:
629:
608:
605:
602:
579:
576:
573:
553:
533:
511:
507:
501:
497:
493:
490:
487:
482:
478:
474:
471:
467:
444:
440:
434:
431:
428:
425:
422:
419:
416:
412:
408:
405:
402:
397:
394:
391:
387:
383:
378:
375:
371:
367:
364:
361:
358:
355:
352:
349:
345:
321:
318:
315:
295:
292:
287:
283:
279:
275:
271:
268:
265:
262:
257:
253:
249:
245:
241:
238:
234:
222:
221:
210:
206:
202:
198:
193:
189:
185:
160:
139:
119:
110:, consists of
98:
85:
82:
51:
48:
15:
9:
6:
4:
3:
2:
3601:
3590:
3587:
3586:
3584:
3574:
3573:
3561:
3557:
3551:
3543:
3539:
3533:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3497:
3493:
3489:
3482:
3474:
3470:
3466:
3462:
3458:
3454:
3450:
3446:
3442:
3438:
3431:
3423:
3419:
3415:
3411:
3407:
3403:
3399:
3395:
3388:
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3347:
3342:
3338:
3334:
3327:
3320:
3314:
3306:
3305:
3297:
3290:
3284:
3273:
3269:
3262:
3255:
3248:
3242:
3240:
3231:
3227:
3223:
3219:
3215:
3211:
3207:
3203:
3202:
3197:
3190:
3182:
3178:
3177:
3169:
3167:
3159:
3158:0-471-59431-8
3155:
3151:
3145:
3141:
3132:
3129:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3089:
3088:
3082:
3080:
3069:
3068:
3064:
3059:
3055:
3053:
3048:
3039:
3037:
3019:
2990:
2987:
2965:
2936:
2927:
2913:
2910:
2890:
2887:
2884:
2881:
2861:
2839:
2810:
2807:
2804:
2795:
2781:
2759:
2744:
2734:
2731:
2715:
2712:
2709:
2706:
2703:
2695:
2676:
2668:
2665:
2661:
2652:
2649:
2639:
2629:
2621:
2618:
2615:
2606:
2596:
2595:
2594:
2580:
2560:
2535:
2529:
2521:
2515:
2512:
2509:
2503:
2499:
2493:
2486:
2483:
2480:
2476:
2468:
2465:
2461:
2455:
2449:
2443:
2431:
2430:
2429:
2413:
2383:
2375:
2363:
2351:
2334:
2329:
2326:
2323:
2320:
2317:
2313:
2308:
2303:
2290:
2285:
2273:
2261:
2250:
2245:
2237:
2234:
2230:
2221:
2218:
2208:
2197:
2196:
2195:
2180:
2177:
2172:
2168:
2145:
2133:
2106:
2072:
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2059:
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2027:
1995:
1983:
1971:
1954:
1949:
1946:
1943:
1940:
1937:
1933:
1929:
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1914:
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1890:
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1177:eigenvectors
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3096:Periodogram
3346:1802.03070
3137:References
1123:eigenspace
3516:1053-587X
3465:0018-926X
3371:1053-587X
3230:252792474
3222:2469-7249
3036:manifolds
2669:ω
2643:^
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3375:S2CID
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3275:(PDF)
3264:(PDF)
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