4175:
3672:
4170:{\displaystyle {\begin{matrix}\mathbf {x} &=&e^{\mathbf {A} T}\mathbf {x} -\left(\int _{v(kT)}^{v((k+1)T)}e^{\mathbf {A} v}dv\right)\mathbf {B} \mathbf {u} \\&=&e^{\mathbf {A} T}\mathbf {x} -\left(\int _{T}^{0}e^{\mathbf {A} v}dv\right)\mathbf {B} \mathbf {u} \\&=&e^{\mathbf {A} T}\mathbf {x} +\left(\int _{0}^{T}e^{\mathbf {A} v}dv\right)\mathbf {B} \mathbf {u} \\&=&e^{\mathbf {A} T}\mathbf {x} +\mathbf {A} ^{-1}\left(e^{\mathbf {A} T}-\mathbf {I} \right)\mathbf {B} \mathbf {u} \end{matrix}}}
4694:
31:
4352:
4689:{\displaystyle {\begin{matrix}\mathbf {x} &=&e^{{\mathbf {A} }T}\mathbf {x} +\left(\int _{0}^{T}e^{{\mathbf {A} }v}dv\right)\mathbf {B} \mathbf {u} \\&=&\left(\sum _{k=0}^{\infty }{\frac {1}{k!}}({\mathbf {A} }T)^{k}\right)\mathbf {x} +\left(\sum _{k=1}^{\infty }{\frac {1}{k!}}{\mathbf {A} }^{k-1}T^{k}\right)\mathbf {B} \mathbf {u} ,\end{matrix}}}
1620:
1928:
2098:
1138:
5056:
3248:
2436:
2695:
3063:
2829:
1493:
3519:
1803:
1342:
5243:
1001:
2552:
288:
2294:
381:
5178:
4861:
748:
641:
1809:
4341:
2191:
1947:
4939:
1007:
4944:
3069:
2920:
2305:
4707:
Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate discrete model, based on that for small timesteps
2563:
4757:
2926:
2701:
1615:{\displaystyle e^{{\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {0} &\mathbf {0} \end{bmatrix}}T}={\begin{bmatrix}\mathbf {A_{d}} &\mathbf {B_{d}} \\\mathbf {0} &\mathbf {I} \end{bmatrix}}}
3254:
1390:
1726:
877:
815:
511:
456:
1246:
1240:
1202:
5293:
5184:
888:
2447:
192:
5316:
5266:
1442:
3603:
2206:
4235:
1718:
1681:
1652:
5446:
294:
5119:
4765:
4203:
3660:
3553:
1464:
1160:
3638:
647:
534:
5062:, or Tustin transform. Each of these approximations has different stability properties. The bilinear transform preserves the instability of the continuous-time system.
1923:{\displaystyle \mathbf {G} =e^{\mathbf {F} }={\begin{bmatrix}\dots &\mathbf {A} _{d}^{-1}\mathbf {Q} _{d}\\\mathbf {0} &\mathbf {A} _{d}^{\top }\end{bmatrix}}.}
5470:
5387:
5363:
5084:
refers to the process of converting continuous features or variables to discretized or nominal features. This can be useful when creating probability mass functions.
4247:
2114:
2093:{\displaystyle \mathbf {Q} _{d}=(\mathbf {A} _{d}^{\top })^{\top }(\mathbf {A} _{d}^{-1}\mathbf {Q} _{d})=\mathbf {A} _{d}(\mathbf {A} _{d}^{-1}\mathbf {Q} _{d}).}
58:
counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers.
5497:
5418:
5340:
1413:
5616:
5556:
4873:
1720:
is a bit trickier due to the matrix exponential integral. It can, however, be computed by first constructing a matrix, and computing the exponential of it
1133:{\displaystyle \mathbf {B} _{d}=\left(\int _{\tau =0}^{T}e^{\mathbf {A} \tau }d\tau \right)\mathbf {B} =\mathbf {A} ^{-1}(\mathbf {A} _{d}-I)\mathbf {B} }
5051:{\displaystyle e^{\mathbf {A} T}\approx \left(\mathbf {I} +{\frac {1}{2}}\mathbf {A} T\right)\left(\mathbf {I} -{\frac {1}{2}}\mathbf {A} T\right)^{-1}}
3243:{\displaystyle \mathbf {x} =e^{\mathbf {A} (k+1)T}\mathbf {x} (0)+\int _{0}^{(k+1)T}e^{\mathbf {A} ((k+1)T-\tau )}\mathbf {B} \mathbf {u} (\tau )d\tau }
1466:. The equation for the discretized measurement noise is a consequence of the continuous measurement noise being defined with a power spectral density.
2847:
2431:{\displaystyle e^{-\mathbf {A} t}\mathbf {\dot {x}} (t)=e^{-\mathbf {A} t}\mathbf {A} \mathbf {x} (t)+e^{-\mathbf {A} t}\mathbf {B} \mathbf {u} (t)}
2690:{\displaystyle e^{-\mathbf {A} t}\mathbf {x} (t)-e^{0}\mathbf {x} (0)=\int _{0}^{t}e^{-\mathbf {A} \tau }\mathbf {B} \mathbf {u} (\tau )d\tau }
3058:{\displaystyle \mathbf {x} =e^{\mathbf {A} kT}\mathbf {x} (0)+\int _{0}^{kT}e^{\mathbf {A} (kT-\tau )}\mathbf {B} \mathbf {u} (\tau )d\tau }
2824:{\displaystyle \mathbf {x} (t)=e^{\mathbf {A} t}\mathbf {x} (0)+\int _{0}^{t}e^{\mathbf {A} (t-\tau )}\mathbf {B} \mathbf {u} (\tau )d\tau }
5394:
5111:
5093:
4711:
5107:
3514:{\displaystyle \mathbf {x} =e^{\mathbf {A} T}\left+\int _{kT}^{(k+1)T}e^{\mathbf {A} (kT+T-\tau )}\mathbf {B} \mathbf {u} (\tau )d\tau }
62:
is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a
1798:{\displaystyle \mathbf {F} ={\begin{bmatrix}-\mathbf {A} &\mathbf {Q} \\\mathbf {0} &\mathbf {A} ^{\top }\end{bmatrix}}T}
1348:
5647:
5071:
821:
759:
462:
407:
1337:{\displaystyle \mathbf {Q} _{d}=\int _{\tau =0}^{T}e^{\mathbf {A} \tau }\mathbf {Q} e^{\mathbf {A} ^{\top }\tau }d\tau }
5838:
5781:
5762:
5699:
5637:
1208:
1170:
125:
5238:{\displaystyle {\mathcal {F}}\{\alpha \cdot \operatorname {III} \}={\mathcal {F}}\{\alpha \}*\operatorname {III} }
996:{\displaystyle \mathbf {A} _{d}=e^{\mathbf {A} T}={\mathcal {L}}^{-1}\{(s\mathbf {I} -\mathbf {A} )^{-1}\}_{t=T}}
17:
5275:
2547:{\displaystyle {\frac {d}{dt}}(e^{-\mathbf {A} t}\mathbf {x} (t))=e^{-\mathbf {A} t}\mathbf {B} \mathbf {u} (t)}
283:{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)+\mathbf {w} (t)}
5855:
Discretization in
Geometry and Dynamics: research on the discretization of differential geometry and dynamics
1933:
The discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of
2289:{\displaystyle {\frac {d}{dt}}e^{\mathbf {A} t}=\mathbf {A} e^{\mathbf {A} t}=e^{\mathbf {A} t}\mathbf {A} }
5879:
5874:
5298:
5251:
1418:
5869:
3558:
5884:
4208:
376:{\displaystyle \mathbf {y} (t)=\mathbf {C} \mathbf {x} (t)+\mathbf {D} \mathbf {u} (t)+\mathbf {v} (t)}
5173:{\displaystyle {\mathcal {F}}\{f*\operatorname {III} \}={\mathcal {F}}\{f\}\cdot \operatorname {III} }
4856:{\displaystyle \mathbf {x} \approx (\mathbf {I} +\mathbf {A} T)\mathbf {x} +T\mathbf {B} \mathbf {u} }
1694:
1657:
1628:
5627:
153:
5427:
5642:
743:{\displaystyle \mathbf {y} =\mathbf {C} _{d}\mathbf {x} +\mathbf {D} _{d}\mathbf {u} +\mathbf {v} }
636:{\displaystyle \mathbf {x} =\mathbf {A} _{d}\mathbf {x} +\mathbf {B} _{d}\mathbf {u} +\mathbf {w} }
4186:
3643:
3527:
1447:
1143:
5889:
3608:
5792:
5691:
5397:
399:
184:
4336:{\displaystyle e^{{\mathbf {A} }T}=\sum _{k=0}^{\infty }{\frac {1}{k!}}({\mathbf {A} }T)^{k}.}
2186:{\displaystyle \mathbf {\dot {x}} (t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}
5657:
5455:
5372:
5366:
5348:
2838:
169:
75:
35:
5683:
97:, as when multiple discrete variables are aggregated or multiple discrete categories fused.
5563:
5343:
5099:
173:
142:
105:
82:
8:
5662:
51:
43:
5559:
5482:
5403:
5325:
5319:
5059:
2197:
1398:
177:
146:
86:
5577:
5505:
5834:
5777:
5758:
5705:
5695:
5684:
5449:
4934:{\displaystyle e^{\mathbf {A} T}\approx \left(\mathbf {I} -\mathbf {A} T\right)^{-1}}
4870:, which is also known as the forward Euler method. Other possible approximations are
1163:
5815:
5807:
113:
71:
517:
157:
63:
109:
5681:
5632:
2915:{\displaystyle \mathbf {x} \ {\stackrel {\mathrm {def} }{=}}\ \mathbf {x} (kT)}
138:
5863:
5811:
5752:
4238:
34:
A solution to a discretized partial differential equation, obtained with the
55:
5421:
4867:
134:
5709:
3555:, and the second term can be simplified by substituting with the function
30:
395:
5724:
Linear
Systems: A State Variable Approach with Numerical Implementation
5571:
5567:
5390:
5269:
5077:
168:
Discretization is also concerned with the transformation of continuous
130:
27:
Process of transferring continuous functions into discrete counterparts
5820:
5652:
5473:
67:
2837:
Now we want to discretise the above expression. We assume that u is
5500:
5094:
Distribution (mathematics) § Convolution_versus_Multiplication
4752:{\displaystyle e^{\mathbf {A} T}\approx \mathbf {I} +\mathbf {A} T}
3663:
4205:
is singular, the latter expression can still be used by replacing
5854:
5828:
5479:
As an example, discretization of the function that is constantly
5690:. Gelb, Arthur, 1937-. Cambridge, Mass.: M.I.T. Press. pp.
5739:, IEEE Transactions on Automatic Control. 23 (3): 395–404, 1978
163:
152:
Mathematical methods relating to discretization include the
5755:
Introduction to random signals and applied Kalman filtering
1385:{\displaystyle \mathbf {R} _{d}=\mathbf {R} {\frac {1}{T}}}
4180:
which is an exact solution to the discretization problem.
2834:
which is an analytical solution to the continuous model.
108:. The goal is to reduce the amount to a level considered
5682:
Analytic
Sciences Corporation. Technical Staff. (1974).
5065:
93:
may also refer to modification of variable or category
5793:"Computing integrals involving the matrix exponential"
5776:. Philadelphia, PA, USA: Saunders College Publishing.
5753:
Robert Grover Brown & Patrick Y. C. Hwang (1997).
5342:
is a rapidly decreasing tempered distribution (e.g. a
4357:
3677:
1841:
1743:
1559:
1507:
872:{\displaystyle \mathbf {v} \sim N(0,\mathbf {R} _{d})}
810:{\displaystyle \mathbf {w} \sim N(0,\mathbf {Q} _{d})}
5580:
5508:
5485:
5458:
5430:
5406:
5375:
5351:
5328:
5301:
5278:
5254:
5187:
5122:
5087:
4947:
4876:
4768:
4714:
4355:
4250:
4211:
4189:
3675:
3646:
3611:
3561:
3530:
3257:
3072:
2929:
2850:
2704:
2566:
2450:
2308:
2209:
2117:
1950:
1812:
1729:
1697:
1660:
1631:
1496:
1450:
1421:
1401:
1351:
1249:
1211:
1173:
1146:
1010:
891:
824:
762:
650:
537:
506:{\displaystyle \mathbf {v} (t)\sim N(0,\mathbf {R} )}
465:
451:{\displaystyle \mathbf {w} (t)\sim N(0,\mathbf {Q} )}
410:
297:
195:
5737:
Computing integrals involving the matrix exponential
1487:
in one step is by utilizing the following property:
4941:, otherwise known as the backward Euler method and
5831:Digital control and estimation: a unified approach
5610:
5550:
5491:
5464:
5440:
5412:
5381:
5357:
5334:
5310:
5287:
5260:
5237:
5172:
5050:
4933:
4855:
4751:
4688:
4335:
4229:
4197:
4169:
3654:
3632:
3597:
3547:
3513:
3242:
3057:
2914:
2823:
2689:
2546:
2430:
2288:
2185:
2092:
1922:
1797:
1712:
1686:
1675:
1646:
1614:
1458:
1436:
1407:
1384:
1336:
1234:
1196:
1154:
1132:
995:
871:
809:
742:
635:
505:
450:
375:
282:
5618:. They are discrete in both, time and frequency.
54:functions, models, variables, and equations into
5861:
5472:which are not smooth can be made smooth using a
5574:is applied, one obtains finite sequences, e.g.
5771:
1235:{\displaystyle \mathbf {D} _{d}=\mathbf {D} }
1197:{\displaystyle \mathbf {C} _{d}=\mathbf {C} }
5558:which, interpreted as the coefficients of a
5226:
5220:
5207:
5195:
5161:
5155:
5142:
5130:
978:
942:
164:Discretization of linear state space models
5790:
5829:R.H. Middleton & G.C. Goodwin (1990).
5288:{\displaystyle \cdot \operatorname {III} }
1683:are the discretized state-space matrices.
5819:
4759:. The approximate solution then becomes:
3524:We recognize the bracketed expression as
524:and continuous integration for the noise
29:
2299:and by premultiplying the model we get
137:. (Specifically, the two terms share a
14:
5862:
5800:IEEE Transactions on Automatic Control
5648:Finite volume method for unsteady flow
5400:(e.g. the function that is constantly
5448:is the (unitary, ordinary frequency)
5311:{\displaystyle *\operatorname {III} }
5072:Discretization of continuous features
5066:Discretization of continuous features
5261:{\displaystyle \operatorname {III} }
4699:which is the form used in practice.
1437:{\displaystyle \mathbf {A} ^{\top }}
5106:arises as a particular case of the
3598:{\displaystyle v(\tau )=kT+T-\tau }
2108:Starting with the continuous model
85:, and is an important component of
24:
5746:
5433:
5215:
5190:
5150:
5125:
5088:Discretization of smooth functions
4605:
4519:
4287:
2884:
2881:
2878:
1991:
1981:
1937:with the upper-right partition of
1904:
1779:
1429:
1318:
928:
81:Discretization is also related to
25:
5901:
5848:
5638:Discrete time and continuous time
4702:
4230:{\displaystyle e^{\mathbf {A} T}}
104:, there is always some amount of
5027:
5009:
4990:
4972:
4954:
4910:
4902:
4883:
4840:
4835:
4815:
4804:
4796:
4770:
4742:
4734:
4721:
4666:
4661:
4628:
4568:
4544:
4475:
4470:
4448:
4405:
4394:
4361:
4312:
4258:
4218:
4191:
4150:
4145:
4135:
4122:
4098:
4080:
4070:
4041:
4036:
4015:
3973:
3963:
3934:
3929:
3908:
3866:
3856:
3827:
3822:
3801:
3723:
3713:
3681:
3648:
3532:
3492:
3487:
3456:
3390:
3385:
3360:
3320:
3307:
3287:
3259:
3221:
3216:
3179:
3127:
3102:
3074:
3036:
3031:
3006:
2966:
2953:
2931:
2896:
2852:
2802:
2797:
2775:
2738:
2728:
2706:
2668:
2663:
2653:
2613:
2586:
2576:
2531:
2526:
2516:
2488:
2478:
2415:
2410:
2400:
2375:
2370:
2360:
2333:
2330:
2318:
2282:
2272:
2254:
2244:
2231:
2170:
2165:
2148:
2143:
2124:
2121:
2074:
2054:
2039:
2021:
2001:
1971:
1953:
1894:
1886:
1871:
1851:
1827:
1814:
1774:
1766:
1757:
1750:
1731:
1713:{\displaystyle \mathbf {Q} _{d}}
1700:
1676:{\displaystyle \mathbf {B} _{d}}
1663:
1647:{\displaystyle \mathbf {A} _{d}}
1634:
1600:
1593:
1582:
1578:
1568:
1564:
1534:
1527:
1518:
1511:
1452:
1424:
1368:
1354:
1313:
1302:
1292:
1252:
1228:
1214:
1190:
1176:
1148:
1126:
1106:
1088:
1079:
1058:
1013:
960:
952:
913:
894:
856:
826:
794:
764:
727:
710:
699:
681:
670:
652:
620:
603:
592:
574:
563:
539:
496:
467:
441:
412:
360:
343:
338:
321:
316:
299:
267:
250:
245:
228:
223:
200:
5774:Linear System Theory and Design
1687:Discretization of process noise
50:is the process of transferring
5833:. Prentice Hall. p. 33f.
5729:
5716:
5675:
5605:
5581:
5545:
5509:
5441:{\displaystyle {\mathcal {F}}}
4850:
4844:
4825:
4819:
4811:
4792:
4786:
4774:
4676:
4670:
4578:
4572:
4553:
4539:
4485:
4479:
4415:
4409:
4377:
4365:
4321:
4307:
4160:
4154:
4090:
4084:
4051:
4045:
3983:
3977:
3944:
3938:
3876:
3870:
3837:
3831:
3790:
3784:
3772:
3769:
3761:
3752:
3733:
3727:
3697:
3685:
3571:
3565:
3542:
3536:
3502:
3496:
3481:
3460:
3442:
3430:
3400:
3394:
3379:
3364:
3330:
3324:
3275:
3263:
3231:
3225:
3210:
3198:
3186:
3183:
3165:
3153:
3137:
3131:
3118:
3106:
3090:
3078:
3046:
3040:
3025:
3010:
2976:
2970:
2941:
2935:
2909:
2900:
2862:
2856:
2812:
2806:
2791:
2779:
2748:
2742:
2716:
2710:
2678:
2672:
2623:
2617:
2596:
2590:
2541:
2535:
2501:
2498:
2492:
2466:
2425:
2419:
2385:
2379:
2345:
2339:
2180:
2174:
2158:
2152:
2136:
2130:
2084:
2049:
2031:
1996:
1987:
1966:
1122:
1101:
965:
945:
866:
845:
836:
830:
804:
783:
774:
768:
737:
731:
720:
714:
691:
685:
662:
656:
630:
624:
613:
607:
584:
578:
555:
543:
500:
486:
477:
471:
445:
431:
422:
416:
370:
364:
353:
347:
331:
325:
309:
303:
277:
271:
260:
254:
238:
232:
216:
210:
183:The following continuous-time
13:
1:
5668:
2103:
1415:is the sample time, although
516:can be discretized, assuming
4198:{\displaystyle \mathbf {A} }
3655:{\displaystyle \mathbf {u} }
3548:{\displaystyle \mathbf {x} }
1459:{\displaystyle \mathbf {A} }
1444:is the transposed matrix of
1155:{\displaystyle \mathbf {A} }
100:Whenever continuous data is
7:
5621:
10:
5906:
5686:Applied optimal estimation
5091:
5069:
4866:This is also known as the
3633:{\displaystyle d\tau =-dv}
1469:A clever trick to compute
5726:, Prentice Hall, NJ, 1989
5628:Discrete event simulation
5476:prior to discretization.
394:are continuous zero-mean
133:but not always identical
5812:10.1109/TAC.1978.1101743
5791:C. Van Loan (Jun 1978).
5643:Finite difference method
5058:, which is known as the
1691:Numerical evaluation of
400:power spectral densities
5772:Chi-Tsong Chen (1984).
5465:{\displaystyle \alpha }
5382:{\displaystyle \alpha }
5358:{\displaystyle \delta }
3666:, which in turn yields
3662:is constant during the
141:.) The same is true of
5612:
5552:
5493:
5466:
5442:
5414:
5383:
5359:
5336:
5312:
5289:
5262:
5239:
5174:
5112:tempered distributions
5080:and machine learning,
5052:
4935:
4857:
4753:
4690:
4609:
4523:
4337:
4291:
4231:
4199:
4171:
3656:
3640:. We also assume that
3634:
3599:
3549:
3515:
3244:
3059:
2916:
2841:during each timestep.
2825:
2691:
2548:
2441:which we recognize as
2432:
2290:
2187:
2094:
1924:
1799:
1714:
1677:
1648:
1616:
1460:
1438:
1409:
1386:
1338:
1236:
1198:
1156:
1134:
997:
873:
811:
744:
637:
507:
452:
377:
284:
170:differential equations
39:
5658:Stochastic simulation
5613:
5564:Dirac delta functions
5553:
5494:
5467:
5443:
5415:
5384:
5360:
5337:
5313:
5290:
5263:
5240:
5175:
5100:generalized functions
5053:
4936:
4858:
4754:
4691:
4589:
4503:
4338:
4271:
4232:
4200:
4172:
3657:
3635:
3600:
3550:
3516:
3245:
3060:
2917:
2826:
2692:
2557:and by integrating..
2549:
2433:
2291:
2188:
2095:
1925:
1800:
1715:
1678:
1649:
1617:
1461:
1439:
1410:
1387:
1339:
1237:
1199:
1157:
1135:
998:
874:
812:
745:
638:
508:
453:
378:
285:
154:Euler–Maruyama method
76:binary classification
36:finite element method
33:
5578:
5506:
5483:
5456:
5428:
5404:
5373:
5349:
5344:Dirac delta function
5326:
5299:
5276:
5252:
5185:
5120:
4945:
4874:
4766:
4712:
4353:
4248:
4209:
4187:
3673:
3644:
3609:
3559:
3528:
3255:
3070:
2927:
2848:
2702:
2564:
2448:
2306:
2207:
2115:
1948:
1810:
1727:
1695:
1658:
1629:
1494:
1448:
1419:
1399:
1349:
1247:
1209:
1171:
1144:
1008:
889:
822:
760:
648:
535:
463:
408:
295:
193:
174:difference equations
143:discretization error
129:often have the same
106:discretization error
89:. In this context,
83:discrete mathematics
5880:Functional analysis
5875:Applied mathematics
5663:Time-scale calculus
5367:compactly supported
5295:is discretization,
5108:Convolution Theorem
4440:
4008:
3901:
3794:
3449:
3353:
3172:
2999:
2768:
2643:
2071:
2018:
1985:
1908:
1868:
1285:
1051:
178:numerical computing
44:applied mathematics
5870:Numerical analysis
5735:Charles Van Loan:
5608:
5570:. If additionally
5560:linear combination
5548:
5489:
5462:
5438:
5410:
5379:
5355:
5332:
5308:
5285:
5258:
5235:
5170:
5060:bilinear transform
5048:
4931:
4853:
4749:
4686:
4684:
4426:
4333:
4227:
4195:
4167:
4165:
3994:
3887:
3744:
3652:
3630:
3595:
3545:
3511:
3417:
3336:
3240:
3143:
3055:
2982:
2912:
2821:
2754:
2687:
2629:
2544:
2428:
2286:
2198:matrix exponential
2183:
2090:
2052:
1999:
1969:
1920:
1911:
1892:
1849:
1795:
1786:
1710:
1673:
1644:
1612:
1606:
1540:
1456:
1434:
1405:
1382:
1334:
1265:
1232:
1194:
1152:
1130:
1031:
993:
869:
807:
740:
633:
503:
448:
373:
280:
147:quantization error
116:purposes at hand.
87:granular computing
40:
5885:Iterative methods
5722:Raymond DeCarlo:
5492:{\displaystyle 1}
5450:Fourier transform
5413:{\displaystyle 1}
5398:ordinary function
5335:{\displaystyle f}
5024:
4987:
4623:
4537:
4305:
2894:
2889:
2867:
2464:
2336:
2223:
2196:we know that the
2127:
1408:{\displaystyle T}
1380:
753:with covariances
207:
185:state space model
16:(Redirected from
5897:
5844:
5825:
5823:
5797:
5787:
5768:
5757:(3rd ed.).
5740:
5733:
5727:
5720:
5714:
5713:
5689:
5679:
5617:
5615:
5614:
5611:{\displaystyle }
5609:
5557:
5555:
5554:
5551:{\displaystyle }
5549:
5498:
5496:
5495:
5490:
5471:
5469:
5468:
5463:
5447:
5445:
5444:
5439:
5437:
5436:
5419:
5417:
5416:
5411:
5388:
5386:
5385:
5380:
5364:
5362:
5361:
5356:
5341:
5339:
5338:
5333:
5317:
5315:
5314:
5309:
5294:
5292:
5291:
5286:
5267:
5265:
5264:
5259:
5244:
5242:
5241:
5236:
5219:
5218:
5194:
5193:
5179:
5177:
5176:
5171:
5154:
5153:
5129:
5128:
5057:
5055:
5054:
5049:
5047:
5046:
5038:
5034:
5030:
5025:
5017:
5012:
5001:
4997:
4993:
4988:
4980:
4975:
4962:
4961:
4957:
4940:
4938:
4937:
4932:
4930:
4929:
4921:
4917:
4913:
4905:
4891:
4890:
4886:
4862:
4860:
4859:
4854:
4843:
4838:
4818:
4807:
4799:
4773:
4758:
4756:
4755:
4750:
4745:
4737:
4729:
4728:
4724:
4695:
4693:
4692:
4687:
4685:
4669:
4664:
4659:
4655:
4654:
4653:
4644:
4643:
4632:
4631:
4624:
4622:
4611:
4608:
4603:
4571:
4566:
4562:
4561:
4560:
4548:
4547:
4538:
4536:
4525:
4522:
4517:
4491:
4478:
4473:
4468:
4464:
4457:
4456:
4452:
4451:
4439:
4434:
4408:
4403:
4402:
4398:
4397:
4364:
4342:
4340:
4339:
4334:
4329:
4328:
4316:
4315:
4306:
4304:
4293:
4290:
4285:
4267:
4266:
4262:
4261:
4239:Taylor expansion
4236:
4234:
4233:
4228:
4226:
4225:
4221:
4204:
4202:
4201:
4196:
4194:
4176:
4174:
4173:
4168:
4166:
4153:
4148:
4143:
4139:
4138:
4130:
4129:
4125:
4110:
4109:
4101:
4083:
4078:
4077:
4073:
4057:
4044:
4039:
4034:
4030:
4023:
4022:
4018:
4007:
4002:
3976:
3971:
3970:
3966:
3950:
3937:
3932:
3927:
3923:
3916:
3915:
3911:
3900:
3895:
3869:
3864:
3863:
3859:
3843:
3830:
3825:
3820:
3816:
3809:
3808:
3804:
3793:
3764:
3726:
3721:
3720:
3716:
3684:
3661:
3659:
3658:
3653:
3651:
3639:
3637:
3636:
3631:
3604:
3602:
3601:
3596:
3554:
3552:
3551:
3546:
3535:
3520:
3518:
3517:
3512:
3495:
3490:
3485:
3484:
3459:
3448:
3428:
3413:
3409:
3393:
3388:
3383:
3382:
3363:
3352:
3344:
3323:
3318:
3317:
3310:
3295:
3294:
3290:
3262:
3249:
3247:
3246:
3241:
3224:
3219:
3214:
3213:
3182:
3171:
3151:
3130:
3125:
3124:
3105:
3077:
3064:
3062:
3061:
3056:
3039:
3034:
3029:
3028:
3009:
2998:
2990:
2969:
2964:
2963:
2956:
2934:
2921:
2919:
2918:
2913:
2899:
2892:
2891:
2890:
2888:
2887:
2875:
2870:
2865:
2855:
2830:
2828:
2827:
2822:
2805:
2800:
2795:
2794:
2778:
2767:
2762:
2741:
2736:
2735:
2731:
2709:
2696:
2694:
2693:
2688:
2671:
2666:
2661:
2660:
2656:
2642:
2637:
2616:
2611:
2610:
2589:
2584:
2583:
2579:
2553:
2551:
2550:
2545:
2534:
2529:
2524:
2523:
2519:
2491:
2486:
2485:
2481:
2465:
2463:
2452:
2437:
2435:
2434:
2429:
2418:
2413:
2408:
2407:
2403:
2378:
2373:
2368:
2367:
2363:
2338:
2337:
2329:
2326:
2325:
2321:
2295:
2293:
2292:
2287:
2285:
2280:
2279:
2275:
2262:
2261:
2257:
2247:
2239:
2238:
2234:
2224:
2222:
2211:
2192:
2190:
2189:
2184:
2173:
2168:
2151:
2146:
2129:
2128:
2120:
2099:
2097:
2096:
2091:
2083:
2082:
2077:
2070:
2062:
2057:
2048:
2047:
2042:
2030:
2029:
2024:
2017:
2009:
2004:
1995:
1994:
1984:
1979:
1974:
1962:
1961:
1956:
1929:
1927:
1926:
1921:
1916:
1915:
1907:
1902:
1897:
1889:
1880:
1879:
1874:
1867:
1859:
1854:
1832:
1831:
1830:
1817:
1804:
1802:
1801:
1796:
1791:
1790:
1783:
1782:
1777:
1769:
1760:
1753:
1734:
1719:
1717:
1716:
1711:
1709:
1708:
1703:
1682:
1680:
1679:
1674:
1672:
1671:
1666:
1653:
1651:
1650:
1645:
1643:
1642:
1637:
1621:
1619:
1618:
1613:
1611:
1610:
1603:
1596:
1587:
1586:
1585:
1573:
1572:
1571:
1550:
1549:
1545:
1544:
1537:
1530:
1521:
1514:
1465:
1463:
1462:
1457:
1455:
1443:
1441:
1440:
1435:
1433:
1432:
1427:
1414:
1412:
1411:
1406:
1391:
1389:
1388:
1383:
1381:
1373:
1371:
1363:
1362:
1357:
1343:
1341:
1340:
1335:
1327:
1326:
1322:
1321:
1316:
1305:
1300:
1299:
1295:
1284:
1279:
1261:
1260:
1255:
1241:
1239:
1238:
1233:
1231:
1223:
1222:
1217:
1203:
1201:
1200:
1195:
1193:
1185:
1184:
1179:
1161:
1159:
1158:
1153:
1151:
1139:
1137:
1136:
1131:
1129:
1115:
1114:
1109:
1100:
1099:
1091:
1082:
1077:
1073:
1066:
1065:
1061:
1050:
1045:
1022:
1021:
1016:
1002:
1000:
999:
994:
992:
991:
976:
975:
963:
955:
941:
940:
932:
931:
921:
920:
916:
903:
902:
897:
878:
876:
875:
870:
865:
864:
859:
829:
816:
814:
813:
808:
803:
802:
797:
767:
749:
747:
746:
741:
730:
713:
708:
707:
702:
684:
679:
678:
673:
655:
642:
640:
639:
634:
623:
606:
601:
600:
595:
577:
572:
571:
566:
542:
512:
510:
509:
504:
499:
470:
457:
455:
454:
449:
444:
415:
382:
380:
379:
374:
363:
346:
341:
324:
319:
302:
289:
287:
286:
281:
270:
253:
248:
231:
226:
209:
208:
203:
198:
74:purposes, as in
21:
5905:
5904:
5900:
5899:
5898:
5896:
5895:
5894:
5860:
5859:
5851:
5841:
5795:
5784:
5765:
5749:
5747:Further reading
5744:
5743:
5734:
5730:
5721:
5717:
5702:
5680:
5676:
5671:
5624:
5579:
5576:
5575:
5507:
5504:
5503:
5484:
5481:
5480:
5457:
5454:
5453:
5432:
5431:
5429:
5426:
5425:
5405:
5402:
5401:
5374:
5371:
5370:
5350:
5347:
5346:
5327:
5324:
5323:
5300:
5297:
5296:
5277:
5274:
5273:
5253:
5250:
5249:
5214:
5213:
5189:
5188:
5186:
5183:
5182:
5149:
5148:
5124:
5123:
5121:
5118:
5117:
5096:
5090:
5074:
5068:
5039:
5026:
5016:
5008:
5007:
5003:
5002:
4989:
4979:
4971:
4970:
4966:
4953:
4952:
4948:
4946:
4943:
4942:
4922:
4909:
4901:
4900:
4896:
4895:
4882:
4881:
4877:
4875:
4872:
4871:
4839:
4834:
4814:
4803:
4795:
4769:
4767:
4764:
4763:
4741:
4733:
4720:
4719:
4715:
4713:
4710:
4709:
4705:
4683:
4682:
4665:
4660:
4649:
4645:
4633:
4627:
4626:
4625:
4615:
4610:
4604:
4593:
4588:
4584:
4567:
4556:
4552:
4543:
4542:
4529:
4524:
4518:
4507:
4502:
4498:
4496:
4489:
4488:
4474:
4469:
4447:
4446:
4445:
4441:
4435:
4430:
4425:
4421:
4404:
4393:
4392:
4391:
4387:
4385:
4380:
4360:
4356:
4354:
4351:
4350:
4324:
4320:
4311:
4310:
4297:
4292:
4286:
4275:
4257:
4256:
4255:
4251:
4249:
4246:
4245:
4217:
4216:
4212:
4210:
4207:
4206:
4190:
4188:
4185:
4184:
4164:
4163:
4149:
4144:
4134:
4121:
4120:
4116:
4115:
4111:
4102:
4097:
4096:
4079:
4069:
4068:
4064:
4062:
4055:
4054:
4040:
4035:
4014:
4013:
4009:
4003:
3998:
3993:
3989:
3972:
3962:
3961:
3957:
3955:
3948:
3947:
3933:
3928:
3907:
3906:
3902:
3896:
3891:
3886:
3882:
3865:
3855:
3854:
3850:
3848:
3841:
3840:
3826:
3821:
3800:
3799:
3795:
3765:
3748:
3743:
3739:
3722:
3712:
3711:
3707:
3705:
3700:
3680:
3676:
3674:
3671:
3670:
3647:
3645:
3642:
3641:
3610:
3607:
3606:
3560:
3557:
3556:
3531:
3529:
3526:
3525:
3491:
3486:
3455:
3454:
3450:
3429:
3421:
3389:
3384:
3359:
3358:
3354:
3345:
3340:
3319:
3306:
3305:
3301:
3300:
3296:
3286:
3285:
3281:
3258:
3256:
3253:
3252:
3220:
3215:
3178:
3177:
3173:
3152:
3147:
3126:
3101:
3100:
3096:
3073:
3071:
3068:
3067:
3035:
3030:
3005:
3004:
3000:
2991:
2986:
2965:
2952:
2951:
2947:
2930:
2928:
2925:
2924:
2895:
2877:
2876:
2871:
2869:
2868:
2851:
2849:
2846:
2845:
2801:
2796:
2774:
2773:
2769:
2763:
2758:
2737:
2727:
2726:
2722:
2705:
2703:
2700:
2699:
2667:
2662:
2652:
2648:
2644:
2638:
2633:
2612:
2606:
2602:
2585:
2575:
2571:
2567:
2565:
2562:
2561:
2530:
2525:
2515:
2511:
2507:
2487:
2477:
2473:
2469:
2456:
2451:
2449:
2446:
2445:
2414:
2409:
2399:
2395:
2391:
2374:
2369:
2359:
2355:
2351:
2328:
2327:
2317:
2313:
2309:
2307:
2304:
2303:
2281:
2271:
2270:
2266:
2253:
2252:
2248:
2243:
2230:
2229:
2225:
2215:
2210:
2208:
2205:
2204:
2169:
2164:
2147:
2142:
2119:
2118:
2116:
2113:
2112:
2106:
2078:
2073:
2072:
2063:
2058:
2053:
2043:
2038:
2037:
2025:
2020:
2019:
2010:
2005:
2000:
1990:
1986:
1980:
1975:
1970:
1957:
1952:
1951:
1949:
1946:
1945:
1910:
1909:
1903:
1898:
1893:
1890:
1885:
1882:
1881:
1875:
1870:
1869:
1860:
1855:
1850:
1847:
1837:
1836:
1826:
1825:
1821:
1813:
1811:
1808:
1807:
1785:
1784:
1778:
1773:
1772:
1770:
1765:
1762:
1761:
1756:
1754:
1749:
1739:
1738:
1730:
1728:
1725:
1724:
1704:
1699:
1698:
1696:
1693:
1692:
1689:
1667:
1662:
1661:
1659:
1656:
1655:
1638:
1633:
1632:
1630:
1627:
1626:
1605:
1604:
1599:
1597:
1592:
1589:
1588:
1581:
1577:
1576:
1574:
1567:
1563:
1562:
1555:
1554:
1539:
1538:
1533:
1531:
1526:
1523:
1522:
1517:
1515:
1510:
1503:
1502:
1501:
1497:
1495:
1492:
1491:
1486:
1477:
1451:
1449:
1446:
1445:
1428:
1423:
1422:
1420:
1417:
1416:
1400:
1397:
1396:
1372:
1367:
1358:
1353:
1352:
1350:
1347:
1346:
1317:
1312:
1311:
1310:
1306:
1301:
1291:
1290:
1286:
1280:
1269:
1256:
1251:
1250:
1248:
1245:
1244:
1227:
1218:
1213:
1212:
1210:
1207:
1206:
1189:
1180:
1175:
1174:
1172:
1169:
1168:
1147:
1145:
1142:
1141:
1125:
1110:
1105:
1104:
1092:
1087:
1086:
1078:
1057:
1056:
1052:
1046:
1035:
1030:
1026:
1017:
1012:
1011:
1009:
1006:
1005:
981:
977:
968:
964:
959:
951:
933:
927:
926:
925:
912:
911:
907:
898:
893:
892:
890:
887:
886:
860:
855:
854:
825:
823:
820:
819:
798:
793:
792:
763:
761:
758:
757:
726:
709:
703:
698:
697:
680:
674:
669:
668:
651:
649:
646:
645:
619:
602:
596:
591:
590:
573:
567:
562:
561:
538:
536:
533:
532:
518:zero-order hold
495:
466:
464:
461:
460:
440:
411:
409:
406:
405:
359:
342:
337:
320:
315:
298:
296:
293:
292:
266:
249:
244:
227:
222:
199:
197:
196:
194:
191:
190:
176:, suitable for
166:
158:zero-order hold
121:discretization
64:binary variable
60:Dichotomization
28:
23:
22:
18:Dichotomization
15:
12:
11:
5:
5903:
5893:
5892:
5890:Control theory
5887:
5882:
5877:
5872:
5858:
5857:
5850:
5849:External links
5847:
5846:
5845:
5840:978-0132116657
5839:
5826:
5806:(3): 395–404.
5788:
5783:978-0030716911
5782:
5769:
5764:978-0471128397
5763:
5748:
5745:
5742:
5741:
5728:
5715:
5700:
5673:
5672:
5670:
5667:
5666:
5665:
5660:
5655:
5650:
5645:
5640:
5635:
5633:Discrete space
5630:
5623:
5620:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5583:
5547:
5544:
5541:
5538:
5535:
5532:
5529:
5526:
5523:
5520:
5517:
5514:
5511:
5488:
5461:
5435:
5424:function) and
5409:
5395:slowly growing
5378:
5354:
5331:
5307:
5304:
5284:
5281:
5257:
5246:
5245:
5234:
5231:
5228:
5225:
5222:
5217:
5212:
5209:
5206:
5203:
5200:
5197:
5192:
5180:
5169:
5166:
5163:
5160:
5157:
5152:
5147:
5144:
5141:
5138:
5135:
5132:
5127:
5104:discretization
5092:Main article:
5089:
5086:
5082:discretization
5070:Main article:
5067:
5064:
5045:
5042:
5037:
5033:
5029:
5023:
5020:
5015:
5011:
5006:
5000:
4996:
4992:
4986:
4983:
4978:
4974:
4969:
4965:
4960:
4956:
4951:
4928:
4925:
4920:
4916:
4912:
4908:
4904:
4899:
4894:
4889:
4885:
4880:
4864:
4863:
4852:
4849:
4846:
4842:
4837:
4833:
4830:
4827:
4824:
4821:
4817:
4813:
4810:
4806:
4802:
4798:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4772:
4748:
4744:
4740:
4736:
4732:
4727:
4723:
4718:
4704:
4703:Approximations
4701:
4697:
4696:
4681:
4678:
4675:
4672:
4668:
4663:
4658:
4652:
4648:
4642:
4639:
4636:
4630:
4621:
4618:
4614:
4607:
4602:
4599:
4596:
4592:
4587:
4583:
4580:
4577:
4574:
4570:
4565:
4559:
4555:
4551:
4546:
4541:
4535:
4532:
4528:
4521:
4516:
4513:
4510:
4506:
4501:
4497:
4495:
4492:
4490:
4487:
4484:
4481:
4477:
4472:
4467:
4463:
4460:
4455:
4450:
4444:
4438:
4433:
4429:
4424:
4420:
4417:
4414:
4411:
4407:
4401:
4396:
4390:
4386:
4384:
4381:
4379:
4376:
4373:
4370:
4367:
4363:
4359:
4358:
4344:
4343:
4332:
4327:
4323:
4319:
4314:
4309:
4303:
4300:
4296:
4289:
4284:
4281:
4278:
4274:
4270:
4265:
4260:
4254:
4224:
4220:
4215:
4193:
4178:
4177:
4162:
4159:
4156:
4152:
4147:
4142:
4137:
4133:
4128:
4124:
4119:
4114:
4108:
4105:
4100:
4095:
4092:
4089:
4086:
4082:
4076:
4072:
4067:
4063:
4061:
4058:
4056:
4053:
4050:
4047:
4043:
4038:
4033:
4029:
4026:
4021:
4017:
4012:
4006:
4001:
3997:
3992:
3988:
3985:
3982:
3979:
3975:
3969:
3965:
3960:
3956:
3954:
3951:
3949:
3946:
3943:
3940:
3936:
3931:
3926:
3922:
3919:
3914:
3910:
3905:
3899:
3894:
3890:
3885:
3881:
3878:
3875:
3872:
3868:
3862:
3858:
3853:
3849:
3847:
3844:
3842:
3839:
3836:
3833:
3829:
3824:
3819:
3815:
3812:
3807:
3803:
3798:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3763:
3760:
3757:
3754:
3751:
3747:
3742:
3738:
3735:
3732:
3729:
3725:
3719:
3715:
3710:
3706:
3704:
3701:
3699:
3696:
3693:
3690:
3687:
3683:
3679:
3678:
3650:
3629:
3626:
3623:
3620:
3617:
3614:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3544:
3541:
3538:
3534:
3522:
3521:
3510:
3507:
3504:
3501:
3498:
3494:
3489:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3458:
3453:
3447:
3444:
3441:
3438:
3435:
3432:
3427:
3424:
3420:
3416:
3412:
3408:
3405:
3402:
3399:
3396:
3392:
3387:
3381:
3378:
3375:
3372:
3369:
3366:
3362:
3357:
3351:
3348:
3343:
3339:
3335:
3332:
3329:
3326:
3322:
3316:
3313:
3309:
3304:
3299:
3293:
3289:
3284:
3280:
3277:
3274:
3271:
3268:
3265:
3261:
3250:
3239:
3236:
3233:
3230:
3227:
3223:
3218:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3181:
3176:
3170:
3167:
3164:
3161:
3158:
3155:
3150:
3146:
3142:
3139:
3136:
3133:
3129:
3123:
3120:
3117:
3114:
3111:
3108:
3104:
3099:
3095:
3092:
3089:
3086:
3083:
3080:
3076:
3065:
3054:
3051:
3048:
3045:
3042:
3038:
3033:
3027:
3024:
3021:
3018:
3015:
3012:
3008:
3003:
2997:
2994:
2989:
2985:
2981:
2978:
2975:
2972:
2968:
2962:
2959:
2955:
2950:
2946:
2943:
2940:
2937:
2933:
2922:
2911:
2908:
2905:
2902:
2898:
2886:
2883:
2880:
2874:
2864:
2861:
2858:
2854:
2832:
2831:
2820:
2817:
2814:
2811:
2808:
2804:
2799:
2793:
2790:
2787:
2784:
2781:
2777:
2772:
2766:
2761:
2757:
2753:
2750:
2747:
2744:
2740:
2734:
2730:
2725:
2721:
2718:
2715:
2712:
2708:
2697:
2686:
2683:
2680:
2677:
2674:
2670:
2665:
2659:
2655:
2651:
2647:
2641:
2636:
2632:
2628:
2625:
2622:
2619:
2615:
2609:
2605:
2601:
2598:
2595:
2592:
2588:
2582:
2578:
2574:
2570:
2555:
2554:
2543:
2540:
2537:
2533:
2528:
2522:
2518:
2514:
2510:
2506:
2503:
2500:
2497:
2494:
2490:
2484:
2480:
2476:
2472:
2468:
2462:
2459:
2455:
2439:
2438:
2427:
2424:
2421:
2417:
2412:
2406:
2402:
2398:
2394:
2390:
2387:
2384:
2381:
2377:
2372:
2366:
2362:
2358:
2354:
2350:
2347:
2344:
2341:
2335:
2332:
2324:
2320:
2316:
2312:
2297:
2296:
2284:
2278:
2274:
2269:
2265:
2260:
2256:
2251:
2246:
2242:
2237:
2233:
2228:
2221:
2218:
2214:
2194:
2193:
2182:
2179:
2176:
2172:
2167:
2163:
2160:
2157:
2154:
2150:
2145:
2141:
2138:
2135:
2132:
2126:
2123:
2105:
2102:
2101:
2100:
2089:
2086:
2081:
2076:
2069:
2066:
2061:
2056:
2051:
2046:
2041:
2036:
2033:
2028:
2023:
2016:
2013:
2008:
2003:
1998:
1993:
1989:
1983:
1978:
1973:
1968:
1965:
1960:
1955:
1931:
1930:
1919:
1914:
1906:
1901:
1896:
1891:
1888:
1884:
1883:
1878:
1873:
1866:
1863:
1858:
1853:
1848:
1846:
1843:
1842:
1840:
1835:
1829:
1824:
1820:
1816:
1805:
1794:
1789:
1781:
1776:
1771:
1768:
1764:
1763:
1759:
1755:
1752:
1748:
1745:
1744:
1742:
1737:
1733:
1707:
1702:
1688:
1685:
1670:
1665:
1641:
1636:
1623:
1622:
1609:
1602:
1598:
1595:
1591:
1590:
1584:
1580:
1575:
1570:
1566:
1561:
1560:
1558:
1553:
1548:
1543:
1536:
1532:
1529:
1525:
1524:
1520:
1516:
1513:
1509:
1508:
1506:
1500:
1482:
1473:
1454:
1431:
1426:
1404:
1393:
1392:
1379:
1376:
1370:
1366:
1361:
1356:
1344:
1333:
1330:
1325:
1320:
1315:
1309:
1304:
1298:
1294:
1289:
1283:
1278:
1275:
1272:
1268:
1264:
1259:
1254:
1242:
1230:
1226:
1221:
1216:
1204:
1192:
1188:
1183:
1178:
1166:
1150:
1128:
1124:
1121:
1118:
1113:
1108:
1103:
1098:
1095:
1090:
1085:
1081:
1076:
1072:
1069:
1064:
1060:
1055:
1049:
1044:
1041:
1038:
1034:
1029:
1025:
1020:
1015:
1003:
990:
987:
984:
980:
974:
971:
967:
962:
958:
954:
950:
947:
944:
939:
936:
930:
924:
919:
915:
910:
906:
901:
896:
880:
879:
868:
863:
858:
853:
850:
847:
844:
841:
838:
835:
832:
828:
817:
806:
801:
796:
791:
788:
785:
782:
779:
776:
773:
770:
766:
751:
750:
739:
736:
733:
729:
725:
722:
719:
716:
712:
706:
701:
696:
693:
690:
687:
683:
677:
672:
667:
664:
661:
658:
654:
643:
632:
629:
626:
622:
618:
615:
612:
609:
605:
599:
594:
589:
586:
583:
580:
576:
570:
565:
560:
557:
554:
551:
548:
545:
541:
520:for the input
514:
513:
502:
498:
494:
491:
488:
485:
482:
479:
476:
473:
469:
458:
447:
443:
439:
436:
433:
430:
427:
424:
421:
418:
414:
384:
383:
372:
369:
366:
362:
358:
355:
352:
349:
345:
340:
336:
333:
330:
327:
323:
318:
314:
311:
308:
305:
301:
290:
279:
276:
273:
269:
265:
262:
259:
256:
252:
247:
243:
240:
237:
234:
230:
225:
221:
218:
215:
212:
206:
202:
172:into discrete
165:
162:
139:semantic field
91:discretization
48:discretization
26:
9:
6:
4:
3:
2:
5902:
5891:
5888:
5886:
5883:
5881:
5878:
5876:
5873:
5871:
5868:
5867:
5865:
5856:
5853:
5852:
5842:
5836:
5832:
5827:
5822:
5817:
5813:
5809:
5805:
5801:
5794:
5789:
5785:
5779:
5775:
5770:
5766:
5760:
5756:
5751:
5750:
5738:
5732:
5725:
5719:
5711:
5707:
5703:
5701:0-262-20027-9
5697:
5693:
5688:
5687:
5678:
5674:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5636:
5634:
5631:
5629:
5626:
5625:
5619:
5602:
5599:
5596:
5593:
5590:
5587:
5584:
5573:
5569:
5565:
5561:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5502:
5486:
5477:
5475:
5459:
5451:
5423:
5420:or any other
5407:
5399:
5396:
5392:
5376:
5368:
5365:or any other
5352:
5345:
5329:
5321:
5320:periodization
5305:
5302:
5282:
5279:
5271:
5255:
5232:
5229:
5223:
5210:
5204:
5201:
5198:
5181:
5167:
5164:
5158:
5145:
5139:
5136:
5133:
5116:
5115:
5114:
5113:
5109:
5105:
5101:
5095:
5085:
5083:
5079:
5073:
5063:
5061:
5043:
5040:
5035:
5031:
5021:
5018:
5013:
5004:
4998:
4994:
4984:
4981:
4976:
4967:
4963:
4958:
4949:
4926:
4923:
4918:
4914:
4906:
4897:
4892:
4887:
4878:
4869:
4847:
4831:
4828:
4822:
4808:
4800:
4789:
4783:
4780:
4777:
4762:
4761:
4760:
4746:
4738:
4730:
4725:
4716:
4700:
4679:
4673:
4656:
4650:
4646:
4640:
4637:
4634:
4619:
4616:
4612:
4600:
4597:
4594:
4590:
4585:
4581:
4575:
4563:
4557:
4549:
4533:
4530:
4526:
4514:
4511:
4508:
4504:
4499:
4493:
4482:
4465:
4461:
4458:
4453:
4442:
4436:
4431:
4427:
4422:
4418:
4412:
4399:
4388:
4382:
4374:
4371:
4368:
4349:
4348:
4347:
4330:
4325:
4317:
4301:
4298:
4294:
4282:
4279:
4276:
4272:
4268:
4263:
4252:
4244:
4243:
4242:
4240:
4222:
4213:
4181:
4157:
4140:
4131:
4126:
4117:
4112:
4106:
4103:
4093:
4087:
4074:
4065:
4059:
4048:
4031:
4027:
4024:
4019:
4010:
4004:
3999:
3995:
3990:
3986:
3980:
3967:
3958:
3952:
3941:
3924:
3920:
3917:
3912:
3903:
3897:
3892:
3888:
3883:
3879:
3873:
3860:
3851:
3845:
3834:
3817:
3813:
3810:
3805:
3796:
3787:
3781:
3778:
3775:
3766:
3758:
3755:
3749:
3745:
3740:
3736:
3730:
3717:
3708:
3702:
3694:
3691:
3688:
3669:
3668:
3667:
3665:
3627:
3624:
3621:
3618:
3615:
3612:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3568:
3562:
3539:
3508:
3505:
3499:
3478:
3475:
3472:
3469:
3466:
3463:
3451:
3445:
3439:
3436:
3433:
3425:
3422:
3418:
3414:
3410:
3406:
3403:
3397:
3376:
3373:
3370:
3367:
3355:
3349:
3346:
3341:
3337:
3333:
3327:
3314:
3311:
3302:
3297:
3291:
3282:
3278:
3272:
3269:
3266:
3251:
3237:
3234:
3228:
3207:
3204:
3201:
3195:
3192:
3189:
3174:
3168:
3162:
3159:
3156:
3148:
3144:
3140:
3134:
3121:
3115:
3112:
3109:
3097:
3093:
3087:
3084:
3081:
3066:
3052:
3049:
3043:
3022:
3019:
3016:
3013:
3001:
2995:
2992:
2987:
2983:
2979:
2973:
2960:
2957:
2948:
2944:
2938:
2923:
2906:
2903:
2872:
2859:
2844:
2843:
2842:
2840:
2835:
2818:
2815:
2809:
2788:
2785:
2782:
2770:
2764:
2759:
2755:
2751:
2745:
2732:
2723:
2719:
2713:
2698:
2684:
2681:
2675:
2657:
2649:
2645:
2639:
2634:
2630:
2626:
2620:
2607:
2603:
2599:
2593:
2580:
2572:
2568:
2560:
2559:
2558:
2538:
2520:
2512:
2508:
2504:
2495:
2482:
2474:
2470:
2460:
2457:
2453:
2444:
2443:
2442:
2422:
2404:
2396:
2392:
2388:
2382:
2364:
2356:
2352:
2348:
2342:
2322:
2314:
2310:
2302:
2301:
2300:
2276:
2267:
2263:
2258:
2249:
2240:
2235:
2226:
2219:
2216:
2212:
2203:
2202:
2201:
2199:
2177:
2161:
2155:
2139:
2133:
2111:
2110:
2109:
2087:
2079:
2067:
2064:
2059:
2044:
2034:
2026:
2014:
2011:
2006:
1976:
1963:
1958:
1944:
1943:
1942:
1940:
1936:
1917:
1912:
1899:
1876:
1864:
1861:
1856:
1844:
1838:
1833:
1822:
1818:
1806:
1792:
1787:
1746:
1740:
1735:
1723:
1722:
1721:
1705:
1684:
1668:
1639:
1607:
1556:
1551:
1546:
1541:
1504:
1498:
1490:
1489:
1488:
1485:
1481:
1476:
1472:
1467:
1402:
1377:
1374:
1364:
1359:
1345:
1331:
1328:
1323:
1307:
1296:
1287:
1281:
1276:
1273:
1270:
1266:
1262:
1257:
1243:
1224:
1219:
1205:
1186:
1181:
1167:
1165:
1119:
1116:
1111:
1096:
1093:
1083:
1074:
1070:
1067:
1062:
1053:
1047:
1042:
1039:
1036:
1032:
1027:
1023:
1018:
1004:
988:
985:
982:
972:
969:
956:
948:
937:
934:
922:
917:
908:
904:
899:
885:
884:
883:
861:
851:
848:
842:
839:
833:
818:
799:
789:
786:
780:
777:
771:
756:
755:
754:
734:
723:
717:
704:
694:
688:
675:
665:
659:
644:
627:
616:
610:
597:
587:
581:
568:
558:
552:
549:
546:
531:
530:
529:
527:
523:
519:
492:
489:
483:
480:
474:
459:
437:
434:
428:
425:
419:
404:
403:
402:
401:
398:sources with
397:
393:
389:
367:
356:
350:
334:
328:
312:
306:
291:
274:
263:
257:
241:
235:
219:
213:
204:
189:
188:
187:
186:
181:
179:
175:
171:
161:
159:
155:
150:
148:
144:
140:
136:
132:
128:
127:
122:
117:
115:
111:
107:
103:
98:
96:
92:
88:
84:
79:
77:
73:
69:
65:
61:
57:
53:
49:
45:
37:
32:
19:
5830:
5803:
5799:
5773:
5754:
5736:
5731:
5723:
5718:
5685:
5677:
5478:
5452:. Functions
5422:band-limited
5247:
5103:
5097:
5081:
5075:
4868:Euler method
4865:
4706:
4698:
4346:This yields
4345:
4182:
4179:
3605:. Note that
3523:
2836:
2833:
2556:
2440:
2298:
2195:
2107:
1938:
1934:
1932:
1690:
1624:
1483:
1479:
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1470:
1468:
1394:
881:
752:
525:
521:
515:
391:
387:
385:
182:
167:
151:
135:connotations
126:quantization
124:
120:
118:
101:
99:
94:
90:
80:
66:(creating a
59:
47:
41:
5499:yields the
5369:function),
1164:nonsingular
396:white noise
102:discretized
95:granularity
5864:Categories
5669:References
5572:truncation
5568:Dirac comb
5566:, forms a
5270:Dirac comb
5078:statistics
2104:Derivation
131:denotation
119:The terms
110:negligible
52:continuous
5821:1813/7095
5653:Smoothing
5474:mollifier
5460:α
5377:α
5353:δ
5303:∗
5280:⋅
5230:∗
5224:α
5202:⋅
5199:α
5165:⋅
5137:∗
5041:−
5014:−
4964:≈
4924:−
4907:−
4893:≈
4790:≈
4731:≈
4638:−
4606:∞
4591:∑
4520:∞
4505:∑
4428:∫
4288:∞
4273:∑
4132:−
4104:−
3996:∫
3889:∫
3880:−
3746:∫
3737:−
3622:−
3616:τ
3593:τ
3590:−
3569:τ
3509:τ
3500:τ
3479:τ
3476:−
3419:∫
3407:τ
3398:τ
3377:τ
3374:−
3338:∫
3238:τ
3229:τ
3208:τ
3205:−
3145:∫
3053:τ
3044:τ
3023:τ
3020:−
2984:∫
2819:τ
2810:τ
2789:τ
2786:−
2756:∫
2685:τ
2676:τ
2658:τ
2650:−
2631:∫
2600:−
2573:−
2513:−
2475:−
2397:−
2357:−
2334:˙
2315:−
2125:˙
2065:−
2012:−
1992:⊤
1982:⊤
1905:⊤
1862:−
1845:…
1780:⊤
1747:−
1430:⊤
1332:τ
1324:τ
1319:⊤
1297:τ
1271:τ
1267:∫
1117:−
1094:−
1071:τ
1063:τ
1037:τ
1033:∫
970:−
957:−
935:−
840:∼
778:∼
481:∼
426:∼
205:˙
68:dichotomy
5622:See also
5501:sequence
5102:theory,
3664:integral
2839:constant
156:and the
114:modeling
112:for the
72:modeling
56:discrete
5268:is the
4237:by its
5837:
5780:
5761:
5710:960061
5708:
5698:
5391:smooth
5248:where
2893:
2866:
1625:Where
882:where
386:where
5796:(PDF)
5389:is a
4183:When
1140:, if
528:, to
5835:ISBN
5778:ISBN
5759:ISBN
5706:OCLC
5696:ISBN
1654:and
1478:and
1395:and
390:and
145:and
123:and
70:for
5816:hdl
5808:doi
5692:121
5562:of
5318:is
5306:III
5283:III
5256:III
5233:III
5205:III
5168:III
5140:III
5110:on
5098:In
5076:In
2200:is
1162:is
78:).
42:In
5866::
5814:.
5804:23
5802:.
5798:.
5704:.
5694:.
5393:,
5322:,
5272:,
4241:,
1941::
180:.
160:.
149:.
46:,
5843:.
5824:.
5818::
5810::
5786:.
5767:.
5712:.
5606:]
5603:1
5600:,
5597:1
5594:,
5591:1
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5585:1
5582:[
5546:]
5543:.
5540:.
5537:,
5534:1
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5528:1
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5522:1
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5516:.
5513:.
5510:[
5487:1
5434:F
5408:1
5330:f
5227:}
5221:{
5216:F
5211:=
5208:}
5196:{
5191:F
5162:}
5159:f
5156:{
5151:F
5146:=
5143:}
5134:f
5131:{
5126:F
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5032:T
5028:A
5022:2
5019:1
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5005:(
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4787:]
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4674:k
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4586:(
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4378:]
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4161:]
4158:k
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4107:1
4099:A
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4088:k
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4081:x
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4060:=
4052:]
4049:k
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4032:)
4028:v
4025:d
4020:v
4016:A
4011:e
4005:T
4000:0
3991:(
3987:+
3984:]
3981:k
3978:[
3974:x
3968:T
3964:A
3959:e
3953:=
3945:]
3942:k
3939:[
3935:u
3930:B
3925:)
3921:v
3918:d
3913:v
3909:A
3904:e
3898:0
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3884:(
3877:]
3874:k
3871:[
3867:x
3861:T
3857:A
3852:e
3846:=
3838:]
3835:k
3832:[
3828:u
3823:B
3818:)
3814:v
3811:d
3806:v
3802:A
3797:e
3791:)
3788:T
3785:)
3782:1
3779:+
3776:k
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3770:(
3767:v
3762:)
3759:T
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3753:(
3750:v
3741:(
3734:]
3731:k
3728:[
3724:x
3718:T
3714:A
3709:e
3703:=
3698:]
3695:1
3692:+
3689:k
3686:[
3682:x
3649:u
3628:v
3625:d
3619:=
3613:d
3587:T
3584:+
3581:T
3578:k
3575:=
3572:)
3566:(
3563:v
3543:]
3540:k
3537:[
3533:x
3506:d
3503:)
3497:(
3493:u
3488:B
3482:)
3473:T
3470:+
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3464:k
3461:(
3457:A
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3440:1
3437:+
3434:k
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3426:T
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3325:(
3321:x
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3298:[
3292:T
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3279:=
3276:]
3273:1
3270:+
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3260:x
3235:d
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3226:(
3222:u
3217:B
3211:)
3202:T
3199:)
3196:1
3193:+
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3187:(
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3103:A
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3094:=
3091:]
3088:1
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3041:(
3037:u
3032:B
3026:)
3017:T
3014:k
3011:(
3007:A
3002:e
2996:T
2993:k
2988:0
2980:+
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2974:0
2971:(
2967:x
2961:T
2958:k
2954:A
2949:e
2945:=
2942:]
2939:k
2936:[
2932:x
2910:)
2907:T
2904:k
2901:(
2897:x
2885:f
2882:e
2879:d
2873:=
2863:]
2860:k
2857:[
2853:x
2816:d
2813:)
2807:(
2803:u
2798:B
2792:)
2783:t
2780:(
2776:A
2771:e
2765:t
2760:0
2752:+
2749:)
2746:0
2743:(
2739:x
2733:t
2729:A
2724:e
2720:=
2717:)
2714:t
2711:(
2707:x
2682:d
2679:)
2673:(
2669:u
2664:B
2654:A
2646:e
2640:t
2635:0
2627:=
2624:)
2621:0
2618:(
2614:x
2608:0
2604:e
2597:)
2594:t
2591:(
2587:x
2581:t
2577:A
2569:e
2542:)
2539:t
2536:(
2532:u
2527:B
2521:t
2517:A
2509:e
2505:=
2502:)
2499:)
2496:t
2493:(
2489:x
2483:t
2479:A
2471:e
2467:(
2461:t
2458:d
2454:d
2426:)
2423:t
2420:(
2416:u
2411:B
2405:t
2401:A
2393:e
2389:+
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2383:t
2380:(
2376:x
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2349:=
2346:)
2343:t
2340:(
2331:x
2323:t
2319:A
2311:e
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2277:t
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2268:e
2264:=
2259:t
2255:A
2250:e
2245:A
2241:=
2236:t
2232:A
2227:e
2220:t
2217:d
2213:d
2181:)
2178:t
2175:(
2171:u
2166:B
2162:+
2159:)
2156:t
2153:(
2149:x
2144:A
2140:=
2137:)
2134:t
2131:(
2122:x
2088:.
2085:)
2080:d
2075:Q
2068:1
2060:d
2055:A
2050:(
2045:d
2040:A
2035:=
2032:)
2027:d
2022:Q
2015:1
2007:d
2002:A
1997:(
1988:)
1977:d
1972:A
1967:(
1964:=
1959:d
1954:Q
1939:G
1935:G
1918:.
1913:]
1900:d
1895:A
1887:0
1877:d
1872:Q
1865:1
1857:d
1852:A
1839:[
1834:=
1828:F
1823:e
1819:=
1815:G
1793:T
1788:]
1775:A
1767:0
1758:Q
1751:A
1741:[
1736:=
1732:F
1706:d
1701:Q
1669:d
1664:B
1640:d
1635:A
1608:]
1601:I
1594:0
1583:d
1579:B
1569:d
1565:A
1557:[
1552:=
1547:T
1542:]
1535:0
1528:0
1519:B
1512:A
1505:[
1499:e
1484:d
1480:B
1475:d
1471:A
1453:A
1425:A
1403:T
1378:T
1375:1
1369:R
1365:=
1360:d
1355:R
1329:d
1314:A
1308:e
1303:Q
1293:A
1288:e
1282:T
1277:0
1274:=
1263:=
1258:d
1253:Q
1229:D
1225:=
1220:d
1215:D
1191:C
1187:=
1182:d
1177:C
1149:A
1127:B
1123:)
1120:I
1112:d
1107:A
1102:(
1097:1
1089:A
1084:=
1080:B
1075:)
1068:d
1059:A
1054:e
1048:T
1043:0
1040:=
1028:(
1024:=
1019:d
1014:B
989:T
986:=
983:t
979:}
973:1
966:)
961:A
953:I
949:s
946:(
943:{
938:1
929:L
923:=
918:T
914:A
909:e
905:=
900:d
895:A
867:)
862:d
857:R
852:,
849:0
846:(
843:N
837:]
834:k
831:[
827:v
805:)
800:d
795:Q
790:,
787:0
784:(
781:N
775:]
772:k
769:[
765:w
738:]
735:k
732:[
728:v
724:+
721:]
718:k
715:[
711:u
705:d
700:D
695:+
692:]
689:k
686:[
682:x
676:d
671:C
666:=
663:]
660:k
657:[
653:y
631:]
628:k
625:[
621:w
617:+
614:]
611:k
608:[
604:u
598:d
593:B
588:+
585:]
582:k
579:[
575:x
569:d
564:A
559:=
556:]
553:1
550:+
547:k
544:[
540:x
526:v
522:u
501:)
497:R
493:,
490:0
487:(
484:N
478:)
475:t
472:(
468:v
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442:Q
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429:N
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417:(
413:w
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361:v
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344:u
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335:+
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329:t
326:(
322:x
317:C
313:=
310:)
307:t
304:(
300:y
278:)
275:t
272:(
268:w
264:+
261:)
258:t
255:(
251:u
246:B
242:+
239:)
236:t
233:(
229:x
224:A
220:=
217:)
214:t
211:(
201:x
38:.
20:)
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