507:
361:
181:
40:
3690:
816:
In a system that satisfies the homogeneity property, scaling the input always results in scaling the zero-state response by the same factor. In a system that satisfies the additivity property, adding two inputs always results in adding the corresponding two zero-state responses due to the individual
1287:
The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation. This mathematical property makes the solution of modelling equations simpler than many nonlinear systems. For
3411:
3361:
1030:
2582:). It is linear because it satisfies the superposition principle. However, when the input is a sinusoid, the output is also a sinusoid, and so its output-input plot is an ellipse centered at the origin rather than a straight line passing through the origin.
812:
The superposition principle means that a linear combination of inputs to the system produces a linear combination of the individual zero-state outputs (that is, outputs setting the initial conditions to zero) corresponding to the individual inputs.
1157:
4060:
4210:
3906:
2135:
890:
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2061:
885:
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610:
895:
828:
3685:{\displaystyle y(t)=\int _{-\infty }^{t}h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(\tau )x(t-\tau )d\tau =\int _{0}^{\infty }h(\tau )x(t-\tau )d\tau }
1779:
3181:
1371:
The previous definition of a linear system is applicable to SISO (single-input single-output) systems. For MIMO (multiple-input multiple-output) systems, input and output signal vectors (
510:
Block diagram illustrating the superposition principle for a deterministic continuous-time SISO system. The system satisfies the superposition principle and is thus linear if and only if
2661:
262:
3193:
3075:
2502:, and a system with odd-symmetry output consisting of a linear region and a saturation (constant) region, are non-linear because they don't always satisfy the superposition principle.
3190:
The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition:
2975:
1529:
1489:
1449:
1409:
364:
Block diagram illustrating the homogeneity property for a deterministic continuous-time SISO system. The system satisfies the homogeneity property or is homogeneous if and only if
1035:
424:
2231:
2713:
2446:
184:
Block diagram illustrating the additivity property for a deterministic continuous-time SISO system. The system satisfies the additivity property or is additive if and only if
2398:
2347:
2289:
1992:
2500:
1673:
1637:
1601:
1565:
756:
720:
500:
354:
318:
2170:
4242:
684:
657:
3918:
4071:
3821:
2716:
630:
464:
444:
282:
823:
809:, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.)
1786:
3701:
2505:
The output versus input graph of a linear system need not be a straight line through the origin. For example, consider a system described by
2066:
2589:(and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by
2722:
2508:
1997:
1709:
3105:
3694:
Linear time-invariant systems are most commonly characterized by the
Laplace transform of the impulse response function called the
3003:
3098:
is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following
513:
4557:
4532:
4482:
4457:
4427:
4402:
4372:
4319:
2842:
of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is
1025:{\displaystyle {\begin{aligned}y_{1}(t)&=H\left\{x_{1}(t)\right\}\\y_{2}(t)&=H\left\{x_{2}(t)\right\}\end{aligned}}}
104:
2923:
76:
763:
4507:
4347:
123:
83:
2663:. It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form
2592:
172:. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.
187:
4589:
90:
61:
57:
4594:
4271:
4215:
3915:
The output of any discrete time linear system is related to the input by the time-varying convolution sum:
1679:
1494:
1454:
1414:
1374:
506:
367:
72:
3365:
If the properties of the system do not depend on the time at which it is operated then it is said to be
2175:
4584:
4286:
1703:
360:
20:
2666:
180:
2403:
24:
2352:
160:
case. As a mathematical abstraction or idealization, linear systems find important applications in
3356:{\displaystyle y(t)=\int _{-\infty }^{t}h(t,t')x(t')dt'=\int _{-\infty }^{\infty }h(t,t')x(t')dt'}
2294:
2236:
1949:
4579:
2451:
806:
50:
4261:
1354:
1642:
1606:
1570:
1534:
725:
689:
469:
323:
287:
2839:
1331:
1327:
1289:
1152:{\displaystyle \alpha y_{1}(t)+\beta y_{2}(t)=H\left\{\alpha x_{1}(t)+\beta x_{2}(t)\right\}}
2140:
2986:
2137:, and any system described by ordinary linear differential equations. Systems described by
1347:
1160:
662:
635:
97:
156:. Linear systems typically exhibit features and properties that are much simpler than the
8:
1339:
1320:
799:
3408:
it is then possible to write the input-output relation equivalently in any of the ways,
2826:, that is, the output doesn't consist only of sinusoids of same frequency as the input (
1358:
1297:
615:
449:
429:
267:
169:
145:
19:
This article is about the systems theory concept. For the linear algebra concept, see
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4528:
4503:
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4368:
4343:
4315:
2586:
1335:
1301:
165:
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4276:
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1293:
28:
3804:, the integral may equally be written over the doubly infinite range and putting
4281:
153:
4266:
1691:
137:
4573:
4055:{\displaystyle y=\sum _{m=-\infty }^{n}{hx}=\sum _{m=-\infty }^{\infty }{hx}}
1365:
1353:
Another perspective is that solutions to linear systems comprise a system of
4205:{\displaystyle y=\sum _{k=0}^{\infty }{hx}=\sum _{k=-\infty }^{\infty }{hx}}
1316:
3901:{\displaystyle H(i\omega )=\int _{-\infty }^{\infty }h(t)e^{-i\omega t}dt}
1343:
820:
Mathematically, for a continuous-time system, given two arbitrary inputs
4307:
1687:
1678:
This definition of a linear system is analogous to the definition of a
2575:
2130:{\displaystyle y(t)=k\,\int _{-\infty }^{t}x(\tau )\mathrm {d} \tau }
157:
2878:
of a linear system is defined as the response of the system at time
39:
2579:
1683:
1364:
A common use of linear models is to describe a nonlinear system by
1943:
which shows that a simple harmonic oscillator is a linear system.
4450:
Signals and
Systems: Analysis Using Transform Methods and MATLAB®
3785:
In applications this is usually a rational algebraic function of
2819:{\displaystyle y(t)=1.5\cos {(3t)}+0.5\cos {(2t)}+0.5\cos {(4t)}}
2567:{\displaystyle y(t)=k\,{\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}
2056:{\displaystyle y(t)=k\,{\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}
880:{\displaystyle {\begin{aligned}x_{1}(t)\\x_{2}(t)\end{aligned}}}
149:
1890:{\displaystyle H(x(t))=m{\frac {d^{2}(x(t))}{dt^{2}}}+kx(t),}
1946:
Other examples of linear systems include those described by
4062:
or equivalently for a time-invariant system on redefining
3776:{\displaystyle H(s)=\int _{0}^{\infty }h(t)e^{-st}\,dt.}
605:{\displaystyle y_{3}(t)=a_{1}\,y_{1}(t)+a_{2}\,y_{2}(t)}
4502:(3 ed.). Oxford University Press. pp. 55–57.
2406:
1531:) are considered instead of input and output signals (
1368:. This is usually done for mathematical convenience.
16:
Physical system satisfying the superposition principle
4244:
represents the lag time between the stimulus at time
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4074:
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3704:
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3196:
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2239:
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2143:
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1952:
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1038:
893:
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728:
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665:
638:
618:
516:
472:
452:
432:
370:
326:
290:
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190:
805:A system is linear if and only if it satisfies the
64:. Unsourced material may be challenged and removed.
4452:(3 ed.). McGraw-Hill. pp. 131, 133–134.
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4236:
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1024:
879:
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348:
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256:
4365:Signals and Systems: A MATLAB Integrated Approach
4571:
4550:Signals and Systems: Principles and Applications
4411:
2849:
2830:), but instead also of sinusoids of frequencies
2585:Also, the output of a linear system can contain
1774:{\displaystyle m{\frac {d^{2}(x)}{dt^{2}}}=-kx.}
3176:{\displaystyle h(t_{2},t_{1})=0,t_{2}<t_{1}}
1350:case (especially in computer implementations).
1334:systems are well adapted to analysis using the
887:as well as their respective zero-state outputs
4516:
4466:
4443:
4441:
4439:
4333:
4331:
2846:, which is different than that of the input.
4417:
4388:
4386:
4384:
4299:
1917:we can rewrite the differential equation as
4552:. Cambridge University Press. p. 187.
4523:ElAli, Taan S.; Karim, Mohammad A. (2008).
4472:
4436:
4420:A Practical Approach to Signals and Systems
4397:. McGraw-Hill. pp. 162–164, 166, 183.
3185:
2656:{\displaystyle y(t)=(1.5+\cos {(t)})\,x(t)}
1304:), which describe a general input function
1206:The system is then defined by the equation
4525:Continuous Signals and Systems with MATLAB
4522:
4328:
3373:is a function only of the time difference
2719:it can be easily shown that the output is
257:{\displaystyle y_{3}(t)=y_{1}(t)+y_{2}(t)}
23:. For the algebraic geometry concept, see
4541:
4381:
4356:
3763:
3070:{\displaystyle y(t=t_{2})=h(t_{2},t_{1})}
2640:
2530:
2197:
2088:
2019:
1971:
582:
549:
396:
124:Learn how and when to remove this message
4491:
3910:
1240:is some arbitrary function of time, and
505:
359:
179:
4447:
2717:product-to-sum trigonometric identities
4572:
4337:
4527:(2 ed.). CRC Press. p. 53.
4392:
4362:
4306:Phillips, Charles L.; Parr, John M.;
2970:{\displaystyle x(t)=\delta (t-t_{1})}
1524:{\displaystyle {\mathbf {y} }_{2}(t)}
1484:{\displaystyle {\mathbf {y} }_{1}(t)}
1444:{\displaystyle {\mathbf {x} }_{2}(t)}
1404:{\displaystyle {\mathbf {x} }_{1}(t)}
4547:
4497:
419:{\displaystyle y_{2}(t)=a\,y_{1}(t)}
62:adding citations to reliable sources
33:
4314:(4 ed.). Pearson. p. 74.
13:
4165:
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4106:
4015:
4010:
3951:
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3851:
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3641:
3587:
3582:
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3510:
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2554:
2535:
2226:{\displaystyle y(t)=k\,x(t)+k_{0}}
2120:
2097:
2043:
2024:
1032:then a linear system must satisfy
27:. For the tactical formation, see
14:
4606:
2989:, and the corresponding response
1706:obeys the differential equation:
1292:systems this is the basis of the
766:can be described by an operator,
4312:Signals, Systems, and Transforms
2708:{\displaystyle x(t)=\cos {(3t)}}
2574:(such as a constant-capacitance
2441:{\textstyle y(t)={\sqrt {x(t)}}}
1501:
1461:
1421:
1381:
38:
1903:is a linear operator. Letting
49:needs additional citations for
4198:
4186:
4180:
4174:
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4127:
4121:
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4084:
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3619:
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3595:
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3546:
3540:
3523:
3485:
3474:
3468:
3451:
3424:
3418:
3339:
3328:
3322:
3305:
3267:
3256:
3250:
3233:
3206:
3200:
3138:
3112:
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3038:
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3010:
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2760:
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2735:
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2701:
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2673:
2650:
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2637:
2633:
2627:
2611:
2605:
2599:
2548:
2542:
2521:
2515:
2488:
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2471:
2464:
2458:
2433:
2427:
2416:
2410:
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2201:
2188:
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2153:
2147:
2116:
2110:
2079:
2073:
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2010:
2004:
1981:
1975:
1962:
1956:
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1875:
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1808:
1805:
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1729:
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1626:
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1590:
1584:
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1392:
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201:
1:
4292:
2909:In other words, if the input
2856:time-varying impulse response
2850:Time-varying impulse response
2393:{\displaystyle y(t)=x^{2}(t)}
1271:the system can be solved for
175:
4548:Apte, Shaila Dinkar (2016).
4477:. Springer. pp. 43–44.
4448:Roberts, Michael J. (2018).
4342:. Springer. pp. 27–28.
4272:Linear time-invariant system
3814:follows the formula for the
2342:{\displaystyle y(t)=\cos {}}
2284:{\displaystyle y(t)=\sin {}}
1987:{\displaystyle y(t)=k\,x(t)}
1680:linear differential equation
1251:is the system state. Given
7:
4255:
3816:frequency response function
2495:{\displaystyle y(t)=|x(t)|}
1697:
758:. Click image to expand it.
502:. Click image to expand it.
356:. Click image to expand it.
10:
4611:
4287:System of linear equations
2838:; furthermore, taking the
1704:simple harmonic oscillator
21:System of linear equations
18:
4418:Sundararajan, D. (2008).
4367:. CRC Press. p. 99.
4338:Bessai, Horst J. (2005).
4248:and the response at time
2578:or a constant-inductance
632:, for all real constants
25:Linear system of divisors
4498:Chen, Chi-Tsong (2004).
4473:Deergha Rao, K. (2018).
4340:MIMO Signals and Systems
3186:The convolution integral
1668:{\displaystyle y_{2}(t)}
1632:{\displaystyle y_{1}(t)}
1596:{\displaystyle x_{2}(t)}
1560:{\displaystyle x_{1}(t)}
1361:in the geometric sense.
1171:, for any input signals
751:{\displaystyle x_{2}(t)}
715:{\displaystyle x_{1}(t)}
495:{\displaystyle x_{1}(t)}
446:, for all real constant
349:{\displaystyle x_{2}(t)}
313:{\displaystyle x_{1}(t)}
4393:Nahvi, Mahmood (2014).
2920:to a linear system is
807:superposition principle
4262:Shift invariant system
4238:
4206:
4169:
4110:
4056:
4019:
3960:
3902:
3777:
3686:
3404:). By redefinition of
3357:
3177:
3071:
2971:
2820:
2709:
2657:
2568:
2496:
2442:
2394:
2343:
2285:
2227:
2166:
2165:{\displaystyle y(t)=k}
2131:
2057:
1988:
1891:
1775:
1669:
1633:
1597:
1561:
1525:
1485:
1445:
1405:
1328:differential equations
1153:
1026:
881:
772:, that maps an input,
759:
752:
716:
680:
653:
626:
606:
503:
496:
460:
440:
420:
357:
350:
314:
278:
258:
152:based on the use of a
4590:Mathematical modeling
4422:. Wiley. p. 80.
4363:Alkin, Oktay (2014).
4239:
4237:{\displaystyle k=n-m}
4207:
4146:
4090:
4057:
3996:
3937:
3911:Discrete-time systems
3903:
3800:is zero for negative
3778:
3687:
3358:
3178:
3072:
2972:
2840:least common multiple
2821:
2710:
2658:
2569:
2497:
2443:
2395:
2344:
2286:
2228:
2167:
2132:
2058:
1989:
1892:
1776:
1688:linear transformation
1670:
1634:
1598:
1562:
1526:
1486:
1446:
1406:
1154:
1027:
882:
753:
717:
681:
679:{\displaystyle a_{2}}
654:
652:{\displaystyle a_{1}}
627:
607:
509:
497:
461:
441:
421:
363:
351:
315:
279:
259:
183:
4216:
4072:
3919:
3822:
3702:
3412:
3194:
3106:
3004:
2987:Dirac delta function
2924:
2723:
2667:
2593:
2509:
2452:
2404:
2353:
2295:
2237:
2176:
2141:
2067:
1998:
1950:
1787:
1710:
1643:
1607:
1571:
1535:
1495:
1455:
1415:
1375:
1321:frequency components
1036:
891:
824:
764:deterministic system
726:
690:
663:
636:
616:
514:
470:
450:
430:
368:
324:
288:
268:
188:
58:improve this article
4595:Concepts in physics
4500:Signals and systems
4475:Signals and Systems
4395:Signals and Systems
3860:
3734:
3645:
3591:
3519:
3447:
3301:
3229:
3102:must be satisfied:
3100:causality condition
2106:
1199:, and for all time
783:, as a function of
686:and for all inputs
284:and for all inputs
4234:
4202:
4052:
3898:
3843:
3773:
3720:
3682:
3631:
3574:
3502:
3430:
3387:which is zero for
3353:
3284:
3212:
3173:
3077:then the function
3067:
2967:
2816:
2705:
2653:
2564:
2492:
2438:
2390:
2339:
2281:
2223:
2162:
2127:
2089:
2053:
1984:
1887:
1771:
1665:
1629:
1593:
1557:
1521:
1481:
1441:
1401:
1298:frequency response
1149:
1022:
1020:
877:
875:
760:
748:
712:
676:
649:
622:
602:
504:
492:
466:and for all input
456:
436:
416:
358:
346:
310:
274:
254:
170:telecommunications
146:mathematical model
4585:Dynamical systems
4559:978-1-107-14624-2
4534:978-1-4200-5475-0
4484:978-3-319-68674-5
4459:978-0-07-802812-0
4429:978-0-470-82353-8
4404:978-0-07-338070-4
4374:978-1-4665-9854-6
4321:978-0-13-198923-8
3696:transfer function
3000:of the system is
2562:
2436:
2051:
1864:
1754:
1336:Laplace transform
1302:LTI system theory
625:{\displaystyle t}
459:{\displaystyle a}
439:{\displaystyle t}
277:{\displaystyle t}
166:signal processing
162:automatic control
134:
133:
126:
108:
4602:
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4563:
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4470:
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4415:
4409:
4408:
4390:
4379:
4378:
4360:
4354:
4353:
4335:
4326:
4325:
4303:
4277:Nonlinear system
4243:
4241:
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4211:
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3689:
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3441:
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3403:
3393:
3386:
3372:
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3352:
3338:
3321:
3300:
3295:
3280:
3266:
3249:
3228:
3223:
3182:
3180:
3179:
3174:
3172:
3171:
3159:
3158:
3137:
3136:
3124:
3123:
3097:
3076:
3074:
3073:
3068:
3063:
3062:
3050:
3049:
3028:
3027:
2999:
2984:
2976:
2974:
2973:
2968:
2963:
2962:
2919:
2908:
2906:
2893:applied at time
2877:
2845:
2837:
2833:
2829:
2825:
2823:
2822:
2817:
2815:
2789:
2763:
2714:
2712:
2711:
2706:
2704:
2662:
2660:
2659:
2654:
2636:
2573:
2571:
2570:
2565:
2563:
2561:
2557:
2551:
2538:
2532:
2501:
2499:
2498:
2493:
2491:
2474:
2447:
2445:
2444:
2439:
2437:
2423:
2399:
2397:
2396:
2391:
2380:
2379:
2348:
2346:
2345:
2340:
2338:
2290:
2288:
2287:
2282:
2280:
2232:
2230:
2229:
2224:
2222:
2221:
2171:
2169:
2168:
2163:
2136:
2134:
2133:
2128:
2123:
2105:
2100:
2062:
2060:
2059:
2054:
2052:
2050:
2046:
2040:
2027:
2021:
1993:
1991:
1990:
1985:
1942:
1940:
1916:
1914:
1902:
1896:
1894:
1893:
1888:
1865:
1863:
1862:
1861:
1848:
1829:
1828:
1818:
1780:
1778:
1777:
1772:
1755:
1753:
1752:
1751:
1738:
1728:
1727:
1717:
1674:
1672:
1671:
1666:
1655:
1654:
1638:
1636:
1635:
1630:
1619:
1618:
1602:
1600:
1599:
1594:
1583:
1582:
1566:
1564:
1563:
1558:
1547:
1546:
1530:
1528:
1527:
1522:
1511:
1510:
1505:
1504:
1490:
1488:
1487:
1482:
1471:
1470:
1465:
1464:
1450:
1448:
1447:
1442:
1431:
1430:
1425:
1424:
1410:
1408:
1407:
1402:
1391:
1390:
1385:
1384:
1314:
1294:impulse response
1284:
1282:
1270:
1268:
1261:
1250:
1239:
1228:
1202:
1198:
1184:
1170:
1166:
1158:
1156:
1155:
1150:
1148:
1144:
1134:
1133:
1109:
1108:
1076:
1075:
1051:
1050:
1031:
1029:
1028:
1023:
1021:
1017:
1013:
1003:
1002:
969:
968:
955:
951:
941:
940:
907:
906:
886:
884:
883:
878:
876:
863:
862:
840:
839:
797:
786:
782:
771:
757:
755:
754:
749:
738:
737:
721:
719:
718:
713:
702:
701:
685:
683:
682:
677:
675:
674:
658:
656:
655:
650:
648:
647:
631:
629:
628:
623:
611:
609:
608:
603:
592:
591:
581:
580:
559:
558:
548:
547:
526:
525:
501:
499:
498:
493:
482:
481:
465:
463:
462:
457:
445:
443:
442:
437:
425:
423:
422:
417:
406:
405:
380:
379:
355:
353:
352:
347:
336:
335:
319:
317:
316:
311:
300:
299:
283:
281:
280:
275:
263:
261:
260:
255:
244:
243:
222:
221:
200:
199:
129:
122:
118:
115:
109:
107:
66:
42:
34:
29:Line (formation)
4610:
4609:
4605:
4604:
4603:
4601:
4600:
4599:
4570:
4569:
4568:
4567:
4560:
4546:
4542:
4535:
4521:
4517:
4510:
4496:
4492:
4485:
4471:
4467:
4460:
4446:
4437:
4430:
4416:
4412:
4405:
4391:
4382:
4375:
4361:
4357:
4350:
4336:
4329:
4322:
4304:
4300:
4295:
4282:System analysis
4258:
4217:
4214:
4213:
4170:
4164:
4150:
4111:
4105:
4094:
4073:
4070:
4069:
4063:
4020:
4014:
4000:
3961:
3955:
3941:
3920:
3917:
3916:
3913:
3877:
3873:
3855:
3847:
3823:
3820:
3819:
3805:
3801:
3790:
3786:
3751:
3747:
3729:
3724:
3703:
3700:
3699:
3640:
3635:
3586:
3578:
3563:
3549:
3532:
3514:
3506:
3491:
3477:
3460:
3442:
3434:
3413:
3410:
3409:
3405:
3395:
3388:
3374:
3370:
3345:
3331:
3314:
3296:
3288:
3273:
3259:
3242:
3224:
3216:
3195:
3192:
3191:
3188:
3167:
3163:
3154:
3150:
3132:
3128:
3119:
3115:
3107:
3104:
3103:
3095:
3088:
3078:
3058:
3054:
3045:
3041:
3023:
3019:
3005:
3002:
3001:
2990:
2985:represents the
2978:
2958:
2954:
2925:
2922:
2921:
2910:
2905:
2895:
2894:
2888:
2875:
2868:
2858:
2852:
2843:
2835:
2831:
2827:
2802:
2776:
2750:
2724:
2721:
2720:
2691:
2668:
2665:
2664:
2626:
2594:
2591:
2590:
2553:
2552:
2534:
2533:
2531:
2510:
2507:
2506:
2487:
2470:
2453:
2450:
2449:
2422:
2405:
2402:
2401:
2375:
2371:
2354:
2351:
2350:
2319:
2296:
2293:
2292:
2261:
2238:
2235:
2234:
2217:
2213:
2177:
2174:
2173:
2142:
2139:
2138:
2119:
2101:
2093:
2068:
2065:
2064:
2042:
2041:
2023:
2022:
2020:
1999:
1996:
1995:
1951:
1948:
1947:
1919:
1918:
1905:
1904:
1898:
1857:
1853:
1849:
1824:
1820:
1819:
1817:
1788:
1785:
1784:
1747:
1743:
1739:
1723:
1719:
1718:
1716:
1711:
1708:
1707:
1700:
1650:
1646:
1644:
1641:
1640:
1614:
1610:
1608:
1605:
1604:
1578:
1574:
1572:
1569:
1568:
1542:
1538:
1536:
1533:
1532:
1506:
1500:
1499:
1498:
1496:
1493:
1492:
1466:
1460:
1459:
1458:
1456:
1453:
1452:
1426:
1420:
1419:
1418:
1416:
1413:
1412:
1386:
1380:
1379:
1378:
1376:
1373:
1372:
1357:which act like
1305:
1273:
1272:
1264:
1263:
1252:
1241:
1230:
1207:
1200:
1192:
1186:
1178:
1172:
1168:
1164:
1129:
1125:
1104:
1100:
1096:
1092:
1071:
1067:
1046:
1042:
1037:
1034:
1033:
1019:
1018:
998:
994:
993:
989:
979:
964:
960:
957:
956:
936:
932:
931:
927:
917:
902:
898:
894:
892:
889:
888:
874:
873:
858:
854:
851:
850:
835:
831:
827:
825:
822:
821:
788:
784:
773:
767:
733:
729:
727:
724:
723:
697:
693:
691:
688:
687:
670:
666:
664:
661:
660:
643:
639:
637:
634:
633:
617:
614:
613:
587:
583:
576:
572:
554:
550:
543:
539:
521:
517:
515:
512:
511:
477:
473:
471:
468:
467:
451:
448:
447:
431:
428:
427:
401:
397:
375:
371:
369:
366:
365:
331:
327:
325:
322:
321:
295:
291:
289:
286:
285:
269:
266:
265:
239:
235:
217:
213:
195:
191:
189:
186:
185:
178:
154:linear operator
130:
119:
113:
110:
73:"Linear system"
67:
65:
55:
43:
32:
17:
12:
11:
5:
4608:
4598:
4597:
4592:
4587:
4582:
4580:Systems theory
4566:
4565:
4558:
4540:
4533:
4515:
4508:
4490:
4483:
4465:
4458:
4435:
4428:
4410:
4403:
4380:
4373:
4355:
4348:
4327:
4320:
4308:Riskin, Eve A.
4297:
4296:
4294:
4291:
4290:
4289:
4284:
4279:
4274:
4269:
4267:Linear control
4264:
4257:
4254:
4233:
4230:
4227:
4224:
4221:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4167:
4162:
4159:
4156:
4153:
4149:
4145:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4108:
4103:
4100:
4097:
4093:
4089:
4086:
4083:
4080:
4077:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4017:
4012:
4009:
4006:
4003:
3999:
3995:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3958:
3953:
3950:
3947:
3944:
3940:
3936:
3933:
3930:
3927:
3924:
3912:
3909:
3897:
3894:
3889:
3886:
3883:
3880:
3876:
3872:
3869:
3866:
3863:
3858:
3853:
3850:
3846:
3842:
3839:
3836:
3833:
3830:
3827:
3772:
3769:
3766:
3760:
3757:
3754:
3750:
3746:
3743:
3740:
3737:
3732:
3727:
3723:
3719:
3716:
3713:
3710:
3707:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3643:
3638:
3634:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3589:
3584:
3581:
3577:
3573:
3569:
3566:
3562:
3559:
3555:
3552:
3548:
3545:
3542:
3538:
3535:
3531:
3528:
3525:
3522:
3517:
3512:
3509:
3505:
3501:
3497:
3494:
3490:
3487:
3483:
3480:
3476:
3473:
3470:
3466:
3463:
3459:
3456:
3453:
3450:
3445:
3440:
3437:
3433:
3429:
3426:
3423:
3420:
3417:
3367:time-invariant
3351:
3348:
3344:
3341:
3337:
3334:
3330:
3327:
3324:
3320:
3317:
3313:
3310:
3307:
3304:
3299:
3294:
3291:
3287:
3283:
3279:
3276:
3272:
3269:
3265:
3262:
3258:
3255:
3252:
3248:
3245:
3241:
3238:
3235:
3232:
3227:
3222:
3219:
3215:
3211:
3208:
3205:
3202:
3199:
3187:
3184:
3170:
3166:
3162:
3157:
3153:
3149:
3146:
3143:
3140:
3135:
3131:
3127:
3122:
3118:
3114:
3111:
3093:
3086:
3066:
3061:
3057:
3053:
3048:
3044:
3040:
3037:
3034:
3031:
3026:
3022:
3018:
3015:
3012:
3009:
2966:
2961:
2957:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2903:
2886:
2873:
2866:
2851:
2848:
2814:
2811:
2808:
2805:
2801:
2798:
2795:
2792:
2788:
2785:
2782:
2779:
2775:
2772:
2769:
2766:
2762:
2759:
2756:
2753:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2703:
2700:
2697:
2694:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2652:
2649:
2646:
2643:
2639:
2635:
2632:
2629:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2560:
2556:
2550:
2547:
2544:
2541:
2537:
2529:
2526:
2523:
2520:
2517:
2514:
2490:
2486:
2483:
2480:
2477:
2473:
2469:
2466:
2463:
2460:
2457:
2435:
2432:
2429:
2426:
2421:
2418:
2415:
2412:
2409:
2389:
2386:
2383:
2378:
2374:
2370:
2367:
2364:
2361:
2358:
2337:
2334:
2331:
2328:
2325:
2322:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2279:
2276:
2273:
2270:
2267:
2264:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2220:
2216:
2212:
2209:
2206:
2203:
2200:
2196:
2193:
2190:
2187:
2184:
2181:
2161:
2158:
2155:
2152:
2149:
2146:
2126:
2122:
2118:
2115:
2112:
2109:
2104:
2099:
2096:
2092:
2087:
2084:
2081:
2078:
2075:
2072:
2049:
2045:
2039:
2036:
2033:
2030:
2026:
2018:
2015:
2012:
2009:
2006:
2003:
1983:
1980:
1977:
1974:
1970:
1967:
1964:
1961:
1958:
1955:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1860:
1856:
1852:
1847:
1844:
1841:
1838:
1835:
1832:
1827:
1823:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1770:
1767:
1764:
1761:
1758:
1750:
1746:
1742:
1737:
1734:
1731:
1726:
1722:
1715:
1699:
1696:
1692:linear algebra
1664:
1661:
1658:
1653:
1649:
1628:
1625:
1622:
1617:
1613:
1592:
1589:
1586:
1581:
1577:
1556:
1553:
1550:
1545:
1541:
1520:
1517:
1514:
1509:
1503:
1480:
1477:
1474:
1469:
1463:
1440:
1437:
1434:
1429:
1423:
1400:
1397:
1394:
1389:
1383:
1342:case, and the
1332:time-invariant
1290:time-invariant
1190:
1176:
1147:
1143:
1140:
1137:
1132:
1128:
1124:
1121:
1118:
1115:
1112:
1107:
1103:
1099:
1095:
1091:
1088:
1085:
1082:
1079:
1074:
1070:
1066:
1063:
1060:
1057:
1054:
1049:
1045:
1041:
1016:
1012:
1009:
1006:
1001:
997:
992:
988:
985:
982:
980:
978:
975:
972:
967:
963:
959:
958:
954:
950:
947:
944:
939:
935:
930:
926:
923:
920:
918:
916:
913:
910:
905:
901:
897:
896:
872:
869:
866:
861:
857:
853:
852:
849:
846:
843:
838:
834:
830:
829:
787:to an output,
747:
744:
741:
736:
732:
711:
708:
705:
700:
696:
673:
669:
646:
642:
621:
601:
598:
595:
590:
586:
579:
575:
571:
568:
565:
562:
557:
553:
546:
542:
538:
535:
532:
529:
524:
520:
491:
488:
485:
480:
476:
455:
435:
415:
412:
409:
404:
400:
395:
392:
389:
386:
383:
378:
374:
345:
342:
339:
334:
330:
309:
306:
303:
298:
294:
273:
253:
250:
247:
242:
238:
234:
231:
228:
225:
220:
216:
212:
209:
206:
203:
198:
194:
177:
174:
138:systems theory
132:
131:
46:
44:
37:
15:
9:
6:
4:
3:
2:
4607:
4596:
4593:
4591:
4588:
4586:
4583:
4581:
4578:
4577:
4575:
4561:
4555:
4551:
4544:
4536:
4530:
4526:
4519:
4511:
4509:0-19-515661-7
4505:
4501:
4494:
4486:
4480:
4476:
4469:
4461:
4455:
4451:
4444:
4442:
4440:
4431:
4425:
4421:
4414:
4406:
4400:
4396:
4389:
4387:
4385:
4376:
4370:
4366:
4359:
4351:
4349:0-387-23488-8
4345:
4341:
4334:
4332:
4323:
4317:
4313:
4309:
4302:
4298:
4288:
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4280:
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4275:
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3325:
3318:
3315:
3311:
3308:
3302:
3289:
3285:
3281:
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3270:
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3253:
3246:
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3213:
3209:
3203:
3197:
3183:
3168:
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3147:
3144:
3141:
3133:
3129:
3125:
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3109:
3101:
3092:
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3046:
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3013:
3007:
2997:
2993:
2988:
2982:
2959:
2955:
2951:
2948:
2942:
2939:
2933:
2927:
2917:
2913:
2902:
2898:
2892:
2885:
2881:
2872:
2865:
2861:
2857:
2847:
2841:
2809:
2806:
2799:
2796:
2793:
2790:
2783:
2780:
2773:
2770:
2767:
2764:
2757:
2754:
2747:
2744:
2741:
2738:
2732:
2726:
2718:
2698:
2695:
2688:
2685:
2682:
2676:
2670:
2647:
2641:
2630:
2623:
2620:
2617:
2614:
2608:
2602:
2596:
2588:
2583:
2581:
2577:
2558:
2545:
2539:
2527:
2524:
2518:
2512:
2503:
2481:
2475:
2467:
2461:
2455:
2430:
2424:
2419:
2413:
2407:
2384:
2376:
2372:
2368:
2362:
2356:
2329:
2323:
2316:
2313:
2310:
2304:
2298:
2271:
2265:
2258:
2255:
2252:
2246:
2240:
2218:
2214:
2210:
2204:
2198:
2194:
2191:
2185:
2179:
2159:
2156:
2150:
2144:
2124:
2113:
2107:
2102:
2094:
2090:
2085:
2082:
2076:
2070:
2047:
2034:
2028:
2016:
2013:
2007:
2001:
1978:
1972:
1968:
1965:
1959:
1953:
1944:
1938:
1934:
1930:
1926:
1922:
1912:
1908:
1901:
1884:
1878:
1872:
1869:
1866:
1858:
1854:
1850:
1839:
1833:
1825:
1821:
1814:
1811:
1802:
1796:
1790:
1781:
1768:
1765:
1762:
1759:
1756:
1748:
1744:
1740:
1732:
1724:
1720:
1713:
1705:
1695:
1693:
1689:
1685:
1681:
1676:
1659:
1651:
1647:
1623:
1615:
1611:
1587:
1579:
1575:
1551:
1543:
1539:
1515:
1507:
1475:
1467:
1435:
1427:
1395:
1387:
1369:
1367:
1366:linearization
1362:
1360:
1356:
1351:
1349:
1345:
1341:
1337:
1333:
1329:
1324:
1322:
1318:
1317:unit impulses
1312:
1308:
1303:
1300:methods (see
1299:
1295:
1291:
1285:
1280:
1276:
1267:
1259:
1255:
1248:
1244:
1237:
1233:
1226:
1222:
1218:
1214:
1210:
1204:
1196:
1189:
1182:
1175:
1162:
1145:
1138:
1130:
1126:
1122:
1119:
1113:
1105:
1101:
1097:
1093:
1089:
1086:
1080:
1072:
1068:
1064:
1061:
1055:
1047:
1043:
1039:
1014:
1007:
999:
995:
990:
986:
983:
981:
973:
965:
961:
952:
945:
937:
933:
928:
924:
921:
919:
911:
903:
899:
867:
859:
855:
844:
836:
832:
818:
814:
810:
808:
803:
802:description.
801:
795:
791:
780:
776:
770:
765:
742:
734:
730:
706:
698:
694:
671:
667:
644:
640:
619:
612:for all time
596:
588:
584:
577:
573:
569:
563:
555:
551:
544:
540:
536:
530:
522:
518:
508:
486:
478:
474:
453:
433:
426:for all time
410:
402:
398:
393:
390:
384:
376:
372:
362:
340:
332:
328:
304:
296:
292:
271:
264:for all time
248:
240:
236:
232:
226:
218:
214:
210:
204:
196:
192:
182:
173:
171:
167:
163:
159:
155:
151:
147:
143:
142:linear system
139:
128:
125:
117:
106:
103:
99:
96:
92:
89:
85:
82:
78:
75: –
74:
70:
69:Find sources:
63:
59:
53:
52:
47:This article
45:
41:
36:
35:
30:
26:
22:
4549:
4543:
4524:
4518:
4499:
4493:
4474:
4468:
4449:
4419:
4413:
4394:
4364:
4358:
4339:
4311:
4301:
4249:
4245:
4064:
3914:
3815:
3810:
3806:
3795:
3791:
3784:
3695:
3693:
3400:
3396:
3389:
3383:
3379:
3375:
3366:
3364:
3189:
3099:
3090:
3083:
3079:
2995:
2991:
2980:
2915:
2911:
2900:
2896:
2889:to a single
2883:
2879:
2870:
2863:
2859:
2855:
2853:
2584:
2504:
1945:
1936:
1932:
1928:
1924:
1920:
1910:
1906:
1899:
1782:
1701:
1677:
1370:
1363:
1352:
1325:
1315:in terms of
1310:
1306:
1286:
1278:
1274:
1265:
1257:
1253:
1246:
1242:
1235:
1231:
1224:
1220:
1216:
1212:
1208:
1205:
1194:
1187:
1180:
1173:
819:
815:
811:
804:
798:, a type of
793:
789:
778:
774:
768:
761:
141:
135:
120:
111:
101:
94:
87:
80:
68:
56:Please help
51:verification
48:
1344:Z-transform
4574:Categories
4293:References
3789:. Because
3698:which is:
1340:continuous
1330:of linear
762:A general
176:Definition
84:newspapers
4229:−
4193:−
4166:∞
4161:∞
4158:−
4148:∑
4134:−
4107:∞
4092:∑
4016:∞
4011:∞
4008:−
3998:∑
3952:∞
3949:−
3939:∑
3885:ω
3879:−
3857:∞
3852:∞
3849:−
3845:∫
3835:ω
3753:−
3731:∞
3722:∫
3680:τ
3671:τ
3668:−
3653:τ
3642:∞
3633:∫
3626:τ
3617:τ
3614:−
3599:τ
3588:∞
3583:∞
3580:−
3576:∫
3530:−
3516:∞
3511:∞
3508:−
3504:∫
3458:−
3439:∞
3436:−
3432:∫
3298:∞
3293:∞
3290:−
3286:∫
3221:∞
3218:−
3214:∫
2952:−
2943:δ
2800:
2774:
2748:
2689:
2624:
2587:harmonics
2576:capacitor
2317:
2259:
2125:τ
2114:τ
2098:∞
2095:−
2091:∫
1760:−
1355:functions
1123:β
1098:α
1065:β
1040:α
800:black box
158:nonlinear
114:June 2021
4310:(2008).
4256:See also
3568:′
3554:′
3537:′
3496:′
3482:′
3465:′
3394:(namely
3350:′
3336:′
3319:′
3278:′
3264:′
3247:′
2715:, using
2580:inductor
1698:Examples
1686:, and a
1684:calculus
1348:discrete
1326:Typical
1229:, where
1159:for any
817:inputs.
164:theory,
2891:impulse
2844:1 rad/s
2836:4 rad/s
2832:2 rad/s
2828:3 rad/s
1359:vectors
1346:in the
1338:in the
1296:or the
1163:values
98:scholar
4556:
4531:
4506:
4481:
4456:
4426:
4401:
4371:
4346:
4318:
4212:where
3392:< 0
2977:where
1161:scalar
168:, and
150:system
100:
93:
86:
79:
71:
3399:<
1931:)) =
1913:) = 0
1897:then
1219:)) =
148:of a
144:is a
105:JSTOR
91:books
4554:ISBN
4529:ISBN
4504:ISBN
4479:ISBN
4454:ISBN
4424:ISBN
4399:ISBN
4369:ISBN
4344:ISBN
4316:ISBN
3369:and
3161:<
2854:The
2834:and
1262:and
1185:and
1167:and
722:and
659:and
320:and
140:, a
77:news
3401:t'
3384:t'
2797:cos
2794:0.5
2771:cos
2768:0.5
2745:cos
2742:1.5
2686:cos
2621:cos
2615:1.5
2314:cos
2256:sin
1783:If
1690:in
1682:in
1675:.)
1319:or
136:In
60:by
4576::
4438:^
4383:^
4330:^
4252:.
4068:,
3818::
3811:iω
3809:=
3382:−
3378:=
3089:,
2979:δ(
2899:=
2882:=
2869:,
2448:,
2400:,
2349:,
2291:,
2233:,
2172:,
2063:,
1994:,
1702:A
1694:.
1639:,
1603:,
1567:,
1491:,
1451:,
1411:,
1323:.
1203:.
4562:.
4537:.
4512:.
4487:.
4462:.
4432:.
4407:.
4377:.
4352:.
4324:.
4250:n
4246:m
4232:m
4226:n
4223:=
4220:k
4199:]
4196:k
4190:n
4187:[
4184:x
4181:]
4178:k
4175:[
4172:h
4155:=
4152:k
4144:=
4140:]
4137:k
4131:n
4128:[
4125:x
4122:]
4119:k
4116:[
4113:h
4102:0
4099:=
4096:k
4088:=
4085:]
4082:n
4079:[
4076:y
4065:h
4049:]
4046:m
4043:[
4040:x
4037:]
4034:m
4031:,
4028:n
4025:[
4022:h
4005:=
4002:m
3994:=
3990:]
3987:m
3984:[
3981:x
3978:]
3975:m
3972:,
3969:n
3966:[
3963:h
3957:n
3946:=
3943:m
3935:=
3932:]
3929:n
3926:[
3923:y
3896:t
3893:d
3888:t
3882:i
3875:e
3871:)
3868:t
3865:(
3862:h
3841:=
3838:)
3832:i
3829:(
3826:H
3807:s
3802:t
3798:)
3796:t
3794:(
3792:h
3787:s
3771:.
3768:t
3765:d
3759:t
3756:s
3749:e
3745:)
3742:t
3739:(
3736:h
3726:0
3718:=
3715:)
3712:s
3709:(
3706:H
3677:d
3674:)
3665:t
3662:(
3659:x
3656:)
3650:(
3647:h
3637:0
3629:=
3623:d
3620:)
3611:t
3608:(
3605:x
3602:)
3596:(
3593:h
3572:=
3565:t
3561:d
3558:)
3551:t
3547:(
3544:x
3541:)
3534:t
3527:t
3524:(
3521:h
3500:=
3493:t
3489:d
3486:)
3479:t
3475:(
3472:x
3469:)
3462:t
3455:t
3452:(
3449:h
3444:t
3428:=
3425:)
3422:t
3419:(
3416:y
3406:h
3397:t
3390:τ
3380:t
3376:τ
3371:h
3347:t
3343:d
3340:)
3333:t
3329:(
3326:x
3323:)
3316:t
3312:,
3309:t
3306:(
3303:h
3282:=
3275:t
3271:d
3268:)
3261:t
3257:(
3254:x
3251:)
3244:t
3240:,
3237:t
3234:(
3231:h
3226:t
3210:=
3207:)
3204:t
3201:(
3198:y
3169:1
3165:t
3156:2
3152:t
3148:,
3145:0
3142:=
3139:)
3134:1
3130:t
3126:,
3121:2
3117:t
3113:(
3110:h
3096:)
3094:1
3091:t
3087:2
3084:t
3082:(
3080:h
3065:)
3060:1
3056:t
3052:,
3047:2
3043:t
3039:(
3036:h
3033:=
3030:)
3025:2
3021:t
3017:=
3014:t
3011:(
3008:y
2998:)
2996:t
2994:(
2992:y
2983:)
2981:t
2965:)
2960:1
2956:t
2949:t
2946:(
2940:=
2937:)
2934:t
2931:(
2928:x
2918:)
2916:t
2914:(
2912:x
2907:.
2904:1
2901:t
2897:t
2887:2
2884:t
2880:t
2876:)
2874:1
2871:t
2867:2
2864:t
2862:(
2860:h
2813:)
2810:t
2807:4
2804:(
2791:+
2787:)
2784:t
2781:2
2778:(
2765:+
2761:)
2758:t
2755:3
2752:(
2739:=
2736:)
2733:t
2730:(
2727:y
2702:)
2699:t
2696:3
2693:(
2683:=
2680:)
2677:t
2674:(
2671:x
2651:)
2648:t
2645:(
2642:x
2638:)
2634:)
2631:t
2628:(
2618:+
2612:(
2609:=
2606:)
2603:t
2600:(
2597:y
2559:t
2555:d
2549:)
2546:t
2543:(
2540:x
2536:d
2528:k
2525:=
2522:)
2519:t
2516:(
2513:y
2489:|
2485:)
2482:t
2479:(
2476:x
2472:|
2468:=
2465:)
2462:t
2459:(
2456:y
2434:)
2431:t
2428:(
2425:x
2420:=
2417:)
2414:t
2411:(
2408:y
2388:)
2385:t
2382:(
2377:2
2373:x
2369:=
2366:)
2363:t
2360:(
2357:y
2336:]
2333:)
2330:t
2327:(
2324:x
2321:[
2311:=
2308:)
2305:t
2302:(
2299:y
2278:]
2275:)
2272:t
2269:(
2266:x
2263:[
2253:=
2250:)
2247:t
2244:(
2241:y
2219:0
2215:k
2211:+
2208:)
2205:t
2202:(
2199:x
2195:k
2192:=
2189:)
2186:t
2183:(
2180:y
2160:k
2157:=
2154:)
2151:t
2148:(
2145:y
2121:d
2117:)
2111:(
2108:x
2103:t
2086:k
2083:=
2080:)
2077:t
2074:(
2071:y
2048:t
2044:d
2038:)
2035:t
2032:(
2029:x
2025:d
2017:k
2014:=
2011:)
2008:t
2005:(
2002:y
1982:)
1979:t
1976:(
1973:x
1969:k
1966:=
1963:)
1960:t
1957:(
1954:y
1941:,
1939:)
1937:t
1935:(
1933:y
1929:t
1927:(
1925:x
1923:(
1921:H
1915:,
1911:t
1909:(
1907:y
1900:H
1885:,
1882:)
1879:t
1876:(
1873:x
1870:k
1867:+
1859:2
1855:t
1851:d
1846:)
1843:)
1840:t
1837:(
1834:x
1831:(
1826:2
1822:d
1815:m
1812:=
1809:)
1806:)
1803:t
1800:(
1797:x
1794:(
1791:H
1769:.
1766:x
1763:k
1757:=
1749:2
1745:t
1741:d
1736:)
1733:x
1730:(
1725:2
1721:d
1714:m
1663:)
1660:t
1657:(
1652:2
1648:y
1627:)
1624:t
1621:(
1616:1
1612:y
1591:)
1588:t
1585:(
1580:2
1576:x
1555:)
1552:t
1549:(
1544:1
1540:x
1519:)
1516:t
1513:(
1508:2
1502:y
1479:)
1476:t
1473:(
1468:1
1462:y
1439:)
1436:t
1433:(
1428:2
1422:x
1399:)
1396:t
1393:(
1388:1
1382:x
1313:)
1311:t
1309:(
1307:x
1283:.
1281:)
1279:t
1277:(
1275:x
1269:,
1266:H
1260:)
1258:t
1256:(
1254:y
1249:)
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205:t
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197:3
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127:)
121:(
116:)
112:(
102:·
95:·
88:·
81:·
54:.
31:.
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