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232:(via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".
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Block diagram illustrating the time invariance for a deterministic continuous-time single-input single-output system. The system is time-invariant if and only if
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1921:{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} t}}={\frac {\partial f}{\partial t}}+{\frac {\partial f}{\partial x}}{\frac {\mathrm {d} x}{\mathrm {d} t}}.}
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A more formal proof of why systems A and B above differ is now presented. To perform this proof, the second definition will be used.
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In the context of a system schematic, this property can also be stated as follows, as shown in the figure to the right:
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2965:{\displaystyle \mathbb {T} _{r}\mathbb {H} {\tilde {x}}=\mathbb {T} _{r}{\tilde {y}}={\tilde {y}}_{r}}
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is "elapsed time", then "time-invariance" implies that the relationship between the input function
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below shows in more detail that while System B is a Shift-Invariant System as a function of time,
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2076:{\displaystyle f_{A}=tx(t)\qquad \implies \qquad {\frac {\partial f_{A}}{\partial t}}=x(t)\neq 0}
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of the system is not a direct function of time except as expressed by the input and output.
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2163:{\displaystyle f_{B}=10x(t)\qquad \implies \qquad {\frac {\partial f_{B}}{\partial t}}=0}
671:
To demonstrate how to determine if a system is time-invariant, consider the two systems:
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3044:{\displaystyle \mathbb {H} \mathbb {T} _{r}{\tilde {x}}=\mathbb {H} {\tilde {x}}_{r}}
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In contrast, system B's time-dependence is only a function of the time-varying input
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2675:{\displaystyle \mathbb {T} _{r}\mathbb {H} =\mathbb {H} \mathbb {T} _{r}\forall r}
235:
Mathematically speaking, "time-invariance" of a system is the following property:
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because the time-dependence is not explicitly a function of the input function.
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24:
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For time-invariant systems, the system properties remain constant with time,
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306:, the system will be considered time-invariant if a time-delay on the input
220:. The time-dependent system function is a function of the time-dependent
16:
Dynamical system whose system function is not directly dependent on time
1743:
More generally, the relationship between the input and output is
120:
3226:
Signals and
Systems: Analysis Using Transform Methods and MATLAB®
2544:{\displaystyle {\tilde {x}}_{1}=x(t+1)\forall t\in \mathbb {R} }
659:. Systems which lack the time-invariant property are studied as
651:
time-invariant systems lack a comprehensive, governing theory.
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then it is time-invariant if we can apply the system operator
18:
3113:{\displaystyle \mathbb {H} {\tilde {x}}_{r}={\tilde {y}}_{r}}
2374:{\displaystyle {\tilde {x}}_{1}=\mathbb {T} _{1}{\tilde {x}}}
1255:{\displaystyle y_{2}(t)=y(t+\delta )=(t+\delta )x(t+\delta )}
2467:{\displaystyle {\tilde {x}}=x(t)\forall t\in \mathbb {R} }
2238:
should be shifted. For example, the "advance-by-1" system
2871:, with the two computations yielding equivalent results.
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2730:{\displaystyle {\tilde {y}}=\mathbb {H} {\tilde {x}}}
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Given a system with a time-dependent output function
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is an operator that advances the input vector by 1.
1670:{\displaystyle y_{2}(t)=y(t+\delta )=10x(t+\delta )}
627:(linear time-invariant) with direct applications in
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If a system is time-invariant then the system block
1970:{\displaystyle {\frac {\partial f}{\partial t}}=0.}
216:are regarded as a class of systems in the field of
49:. Unsourced material may be challenged and removed.
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1516:{\displaystyle y_{1}(t)=10x_{d}(t)=10x(t+\delta )}
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2315:can be represented in this abstract notation by
1089:{\displaystyle y_{1}(t)=tx_{d}(t)=tx(t+\delta )}
345:directly equates to a time-delay of the output
1324:, therefore the system is not time-invariant.
8:
2477:with the system yielding the shifted output
3201:A Practical Approach to Signals and Systems
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2874:Applying the system operator first gives
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109:Learn how and when to remove this message
3228:(3 ed.). McGraw-Hill. p. 132.
2975:Applying the shift operator first gives
2305:{\displaystyle x(t+1)=\delta (t+1)*x(t)}
2083:in general, so it is not time-invariant,
598:, this property can be satisfied if the
3249:Oppenheim, Alan; Willsky, Alan (1997).
3166:
582:{\displaystyle y(t)=f(x(t),t)=f(x(t)).}
3054:If the system is time-invariant, then
273:, and a time-dependent input function
2820:, or we can apply the shift operator
1382:{\displaystyle x_{d}(t)=x(t+\delta )}
1317:{\displaystyle y_{1}(t)\neq y_{2}(t)}
955:{\displaystyle x_{d}(t)=x(t+\delta )}
7:
2586:Suppose we represent a system by an
655:time-invariant systems are known as
47:adding citations to reliable sources
2685:If our system equation is given by
806:for system A explicitly depends on
619:If a time-invariant system is also
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2527:
2450:
2234:is the amount by which a vector's
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2028:
1980:Applied to Systems A and B above:
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3253:(second ed.). Prentice Hall.
1732:{\displaystyle y_{1}(t)=y_{2}(t)}
474:is constant with respect to time
2849:followed by the system operator
2842:{\displaystyle \mathbb {T} _{r}}
2813:{\displaystyle \mathbb {T} _{r}}
2576:{\displaystyle \mathbb {T} _{1}}
2207:{\displaystyle \mathbb {T} _{r}}
1332:Start with a delay of the input
905:Start with a delay of the input
23:
3155:Autonomous system (mathematics)
2791:followed by the shift operator
2619:with the shift operator, i.e.,
2126:
2120:
2024:
2018:
1811:and its variation with time is
1801:{\displaystyle y(t)=f(x(t),t),}
384:function. For example, if time
34:needs additional citations for
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647:, and other technical areas.
3224:Roberts, Michael J. (2018).
2864:{\displaystyle \mathbb {H} }
2784:{\displaystyle {\tilde {x}}}
2755:{\displaystyle \mathbb {H} }
2604:{\displaystyle \mathbb {H} }
2406:{\displaystyle {\tilde {x}}}
625:linear time-invariant theory
375:{\displaystyle y(t+\delta )}
336:{\displaystyle x(t+\delta )}
1588:{\displaystyle y(t)=10x(t)}
1430:{\displaystyle y(t)=10x(t)}
764:{\displaystyle y(t)=10x(t)}
224:. If this function depends
184:. Click image to expand it.
3292:
1161:{\displaystyle y(t)=tx(t)}
1003:{\displaystyle y(t)=tx(t)}
715:{\displaystyle y(t)=tx(t)}
3199:Sundararajan, D. (2008).
3174:Bessai, Horst J. (2005).
3178:. Springer. p. 28.
3176:MIMO Signals and Systems
2170:so it is time-invariant.
1527:Now delay the output by
1100:Now delay the output by
614:with an arbitrary delay.
441:and the output function
161:, for all real constant
3130:Finite impulse response
2413:is a function given by
1540:{\displaystyle \delta }
1113:{\displaystyle \delta }
657:shift-invariant systems
623:, it is the subject of
58:"Time-invariant system"
3140:State space (controls)
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875:. This makes system B
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3203:. Wiley. p. 81.
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868:{\displaystyle x(t)}
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832:{\displaystyle x(t)}
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799:{\displaystyle y(t)}
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661:time-variant systems
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465:{\displaystyle y(t)}
447:
432:{\displaystyle x(t)}
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297:{\displaystyle x(t)}
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264:{\displaystyle y(t)}
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43:improve this article
3251:Signals and Systems
890:, System A is not.
594:In the language of
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2179:We can denote the
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492:{\displaystyle t:}
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228:indirectly on the
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170:and for all input
3276:Signal processing
3235:978-0-07-802812-0
3210:978-0-470-82353-8
3150:LTI system theory
3145:Signal-flow graph
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2611:. This system is
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2227:{\displaystyle r}
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641:signal processing
600:transfer function
596:signal processing
399:{\displaystyle t}
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3135:Sheffer sequence
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2175:Abstract example
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629:NMR spectroscopy
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383:
381:
379:
378:
373:
344:
342:
340:
339:
334:
305:
303:
301:
300:
295:
272:
270:
268:
267:
262:
183:
169:
160:
156:
114:
107:
103:
100:
94:
92:
51:
27:
19:
3291:
3290:
3286:
3285:
3284:
3282:
3281:
3280:
3261:
3260:
3259:
3258:
3248:
3247:
3243:
3236:
3223:
3222:
3218:
3211:
3198:
3197:
3193:
3186:
3173:
3172:
3168:
3163:
3126:
3091:
3069:
3059:
3058:
3022:
2990:
2980:
2979:
2943:
2916:
2884:
2879:
2878:
2851:
2850:
2827:
2822:
2821:
2798:
2793:
2792:
2764:
2763:
2742:
2741:
2690:
2689:
2654:
2629:
2624:
2623:
2591:
2590:
2561:
2556:
2555:
2487:
2482:
2481:
2418:
2417:
2386:
2385:
2347:
2325:
2320:
2319:
2243:
2242:
2216:
2215:
2192:
2187:
2186:
2177:
2144:
2133:
2129:
2092:
2087:
2086:
2042:
2031:
2027:
1990:
1985:
1984:
1951:
1943:
1936:
1935:
1903:
1893:
1881:
1873:
1858:
1850:
1833:
1823:
1816:
1815:
1748:
1747:
1710:
1688:
1683:
1682:
1603:
1598:
1597:
1550:
1549:
1529:
1528:
1470:
1445:
1440:
1439:
1392:
1391:
1339:
1334:
1333:
1295:
1273:
1268:
1267:
1176:
1171:
1170:
1123:
1122:
1102:
1101:
1043:
1018:
1013:
1012:
965:
964:
912:
907:
906:
896:
848:
847:
812:
811:
779:
778:
776:System Function
726:
725:
677:
676:
669:
505:
504:
478:
477:
475:
445:
444:
442:
412:
411:
409:
388:
387:
385:
349:
348:
346:
310:
309:
307:
277:
276:
274:
244:
243:
241:
218:system analysis
206:system function
177:
171:
168:
162:
158:
154:
143:
132:
126:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3289:
3287:
3279:
3278:
3273:
3271:Control theory
3263:
3262:
3257:
3256:
3241:
3234:
3216:
3209:
3191:
3184:
3165:
3164:
3162:
3159:
3158:
3157:
3152:
3147:
3142:
3137:
3132:
3125:
3122:
3121:
3120:
3107:
3100:
3097:
3090:
3085:
3078:
3075:
3067:
3052:
3051:
3038:
3031:
3028:
3020:
3016:
3010:
3007:
2999:
2994:
2988:
2973:
2972:
2959:
2952:
2949:
2942:
2936:
2933:
2925:
2920:
2915:
2909:
2906:
2899:
2893:
2888:
2859:
2836:
2831:
2807:
2802:
2777:
2774:
2750:
2738:
2737:
2723:
2720:
2713:
2709:
2703:
2700:
2683:
2682:
2671:
2668:
2663:
2658:
2652:
2648:
2644:
2638:
2633:
2613:time-invariant
2599:
2570:
2565:
2552:
2551:
2539:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2503:
2496:
2493:
2475:
2474:
2462:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2431:
2428:
2399:
2396:
2382:
2381:
2367:
2364:
2356:
2351:
2346:
2341:
2334:
2331:
2313:
2312:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2223:
2201:
2196:
2182:shift operator
2176:
2173:
2172:
2171:
2159:
2156:
2150:
2147:
2140:
2136:
2132:
2124:
2119:
2116:
2113:
2110:
2107:
2104:
2099:
2095:
2084:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2048:
2045:
2038:
2034:
2030:
2022:
2017:
2014:
2011:
2008:
2005:
2002:
1997:
1993:
1978:
1977:
1966:
1963:
1957:
1954:
1949:
1946:
1929:
1928:
1917:
1911:
1907:
1901:
1897:
1887:
1884:
1879:
1876:
1870:
1864:
1861:
1856:
1853:
1847:
1841:
1837:
1831:
1827:
1809:
1808:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1741:
1740:
1728:
1725:
1722:
1717:
1713:
1709:
1706:
1703:
1700:
1695:
1691:
1679:
1678:
1677:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1610:
1606:
1595:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1536:
1525:
1524:
1523:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1477:
1473:
1469:
1466:
1463:
1460:
1457:
1452:
1448:
1437:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1346:
1342:
1326:
1325:
1313:
1310:
1307:
1302:
1298:
1294:
1291:
1288:
1285:
1280:
1276:
1264:
1263:
1262:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1183:
1179:
1168:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1109:
1098:
1097:
1096:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1050:
1046:
1042:
1039:
1036:
1033:
1030:
1025:
1021:
1010:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
919:
915:
895:
894:Formal example
892:
884:Formal Example
877:time-invariant
864:
861:
858:
855:
841:time-invariant
828:
825:
822:
819:
795:
792:
789:
786:
772:
771:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
722:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
668:
667:Simple example
665:
645:control theory
617:
616:
592:
591:
590:
589:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
488:
485:
461:
458:
455:
452:
428:
425:
422:
419:
395:
371:
368:
365:
362:
359:
356:
332:
329:
326:
323:
320:
317:
293:
290:
287:
284:
260:
257:
254:
251:
222:input function
212:of time. Such
194:time-invariant
190:control theory
175:
166:
152:
141:
130:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3288:
3277:
3274:
3272:
3269:
3268:
3266:
3252:
3245:
3242:
3237:
3231:
3227:
3220:
3217:
3212:
3206:
3202:
3195:
3192:
3187:
3185:0-387-23488-8
3181:
3177:
3170:
3167:
3160:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3127:
3123:
3105:
3095:
3088:
3083:
3073:
3057:
3056:
3055:
3036:
3026:
3014:
3005:
2997:
2978:
2977:
2976:
2957:
2947:
2940:
2931:
2923:
2913:
2904:
2891:
2877:
2876:
2875:
2872:
2834:
2805:
2772:
2718:
2707:
2698:
2688:
2687:
2686:
2669:
2661:
2646:
2636:
2622:
2621:
2620:
2618:
2614:
2589:
2584:
2568:
2533:
2530:
2521:
2518:
2515:
2509:
2506:
2501:
2491:
2480:
2479:
2478:
2456:
2453:
2444:
2438:
2435:
2426:
2416:
2415:
2414:
2394:
2362:
2354:
2344:
2339:
2329:
2318:
2317:
2316:
2296:
2290:
2287:
2281:
2278:
2275:
2269:
2266:
2260:
2257:
2254:
2248:
2241:
2240:
2239:
2237:
2221:
2199:
2184:
2183:
2174:
2157:
2154:
2148:
2138:
2134:
2114:
2108:
2105:
2102:
2097:
2093:
2085:
2070:
2067:
2061:
2055:
2052:
2046:
2036:
2032:
2012:
2006:
2003:
2000:
1995:
1991:
1983:
1982:
1981:
1964:
1961:
1955:
1947:
1934:
1933:
1932:
1915:
1909:
1899:
1885:
1877:
1868:
1862:
1854:
1845:
1839:
1829:
1814:
1813:
1812:
1795:
1789:
1786:
1780:
1774:
1768:
1765:
1759:
1753:
1746:
1745:
1744:
1723:
1715:
1711:
1707:
1701:
1693:
1689:
1680:
1661:
1658:
1655:
1649:
1646:
1643:
1637:
1634:
1631:
1625:
1622:
1616:
1608:
1604:
1596:
1579:
1573:
1570:
1567:
1561:
1555:
1548:
1547:
1534:
1526:
1507:
1504:
1501:
1495:
1492:
1489:
1483:
1475:
1471:
1467:
1464:
1458:
1450:
1446:
1438:
1421:
1415:
1412:
1409:
1403:
1397:
1390:
1389:
1373:
1370:
1367:
1361:
1358:
1352:
1344:
1340:
1331:
1328:
1327:
1308:
1300:
1296:
1292:
1286:
1278:
1274:
1265:
1246:
1243:
1240:
1234:
1228:
1225:
1222:
1216:
1210:
1207:
1204:
1198:
1195:
1189:
1181:
1177:
1169:
1152:
1146:
1143:
1140:
1134:
1128:
1121:
1120:
1107:
1099:
1080:
1077:
1074:
1068:
1065:
1062:
1056:
1048:
1044:
1040:
1037:
1031:
1023:
1019:
1011:
994:
988:
985:
982:
976:
970:
963:
962:
946:
943:
940:
934:
931:
925:
917:
913:
904:
901:
900:
899:
893:
891:
889:
885:
880:
878:
859:
853:
844:
842:
823:
817:
809:
790:
784:
777:
755:
749:
746:
743:
737:
731:
723:
706:
700:
697:
694:
688:
682:
674:
673:
672:
666:
664:
662:
658:
654:
650:
646:
642:
638:
634:
630:
626:
622:
615:
613:
608:
607:
606:
603:
601:
597:
576:
567:
561:
555:
552:
546:
543:
537:
531:
525:
522:
516:
510:
503:
502:
501:
486:
483:
456:
450:
423:
417:
393:
366:
363:
360:
354:
327:
324:
321:
315:
288:
282:
255:
249:
238:
237:
236:
233:
231:
227:
223:
219:
215:
211:
207:
203:
199:
195:
191:
181:
174:
165:
157:for all time
151:
147:
140:
136:
129:
123:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3250:
3244:
3225:
3219:
3200:
3194:
3175:
3169:
3053:
2974:
2873:
2739:
2684:
2612:
2585:
2553:
2476:
2383:
2314:
2180:
2178:
1979:
1930:
1810:
1742:
1329:
902:
897:
887:
883:
881:
845:
839:, it is not
807:
775:
773:
670:
618:
609:
604:
593:
239:
234:
225:
221:
205:
201:
197:
193:
187:
179:
172:
163:
149:
145:
138:
134:
127:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
810:outside of
230:time-domain
3265:Categories
3161:References
774:Since the
724:System B:
675:System A:
633:seismology
69:newspapers
3099:~
3077:~
3030:~
3009:~
2951:~
2935:~
2908:~
2776:~
2722:~
2702:~
2667:∀
2534:∈
2528:∀
2495:~
2457:∈
2451:∀
2430:~
2398:~
2366:~
2333:~
2288:∗
2270:δ
2236:index set
2146:∂
2131:∂
2123:⟹
2068:≠
2044:∂
2029:∂
2021:⟹
1953:∂
1945:∂
1883:∂
1875:∂
1860:∂
1852:∂
1662:δ
1638:δ
1535:δ
1508:δ
1374:δ
1330:System B:
1293:≠
1247:δ
1229:δ
1211:δ
1108:δ
1081:δ
947:δ
903:System A:
649:Nonlinear
367:δ
328:δ
3124:See also
2617:commutes
2588:operator
1681:Clearly
1266:Clearly
653:Discrete
637:circuits
612:commutes
210:function
99:May 2018
499:
476:
472:
443:
439:
410:
406:
386:
382:
347:
343:
308:
304:
275:
271:
242:
214:systems
83:scholar
3232:
3207:
3182:
2615:if it
2384:where
2214:where
621:linear
202:system
85:
78:
71:
64:
56:
90:JSTOR
76:books
3230:ISBN
3205:ISBN
3180:ISBN
882:The
226:only
192:, a
137:) =
62:news
2762:on
2554:So
2185:by
188:In
45:by
3267::
2106:10
1965:0.
1647:10
1571:10
1493:10
1468:10
1413:10
879:.
747:10
663:.
643:,
639:,
635:,
631:,
200:)
198:TI
148:–
3238:.
3213:.
3188:.
3106:r
3096:y
3089:=
3084:r
3074:x
3066:H
3037:r
3027:x
3019:H
3015:=
3006:x
2998:r
2993:T
2987:H
2958:r
2948:y
2941:=
2932:y
2924:r
2919:T
2914:=
2905:x
2898:H
2892:r
2887:T
2858:H
2835:r
2830:T
2806:r
2801:T
2773:x
2749:H
2719:x
2712:H
2708:=
2699:y
2670:r
2662:r
2657:T
2651:H
2647:=
2643:H
2637:r
2632:T
2598:H
2569:1
2564:T
2538:R
2531:t
2525:)
2522:1
2519:+
2516:t
2513:(
2510:x
2507:=
2502:1
2492:x
2461:R
2454:t
2448:)
2445:t
2442:(
2439:x
2436:=
2427:x
2395:x
2363:x
2355:1
2350:T
2345:=
2340:1
2330:x
2300:)
2297:t
2294:(
2291:x
2285:)
2282:1
2279:+
2276:t
2273:(
2267:=
2264:)
2261:1
2258:+
2255:t
2252:(
2249:x
2222:r
2200:r
2195:T
2158:0
2155:=
2149:t
2139:B
2135:f
2118:)
2115:t
2112:(
2109:x
2103:=
2098:B
2094:f
2071:0
2065:)
2062:t
2059:(
2056:x
2053:=
2047:t
2037:A
2033:f
2016:)
2013:t
2010:(
2007:x
2004:t
2001:=
1996:A
1992:f
1962:=
1956:t
1948:f
1916:.
1910:t
1906:d
1900:x
1896:d
1886:x
1878:f
1869:+
1863:t
1855:f
1846:=
1840:t
1836:d
1830:y
1826:d
1796:,
1793:)
1790:t
1787:,
1784:)
1781:t
1778:(
1775:x
1772:(
1769:f
1766:=
1763:)
1760:t
1757:(
1754:y
1727:)
1724:t
1721:(
1716:2
1712:y
1708:=
1705:)
1702:t
1699:(
1694:1
1690:y
1665:)
1659:+
1656:t
1653:(
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577:.
574:)
571:)
568:t
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562:x
559:(
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547:t
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133:(
131:2
128:y
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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