518:
372:
192:
51:
3701:
827:
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
1298:
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
3422:
3372:
1041:
2593:). 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.
823:
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.
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906:
839:
3696:{\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 }
1790:
3192:
1382:
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 (
521:
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
2672:
273:
3204:
3086:
2513:, 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.
3201:
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:
2986:
1540:
1500:
1460:
1420:
375:
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
1046:
435:
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2724:
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195:
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
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2358:
2300:
2003:
2511:
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1648:
1612:
1576:
767:
731:
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329:
2181:
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695:
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3929:
4082:
3832:
641:
475:
455:
293:
834:
820:, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.)
1797:
3712:
4568:
4543:
4493:
4468:
4438:
4413:
4383:
4330:
2516:
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
2077:
2600:(and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by
2733:
2519:
2008:
1720:
774:
3116:
4518:
4358:
3705:
Linear time-invariant systems are most commonly characterized by the
Laplace transform of the impulse response function called the
134:
517:
2727:
3014:
371:
3109:
is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following
524:
191:
4600:
2853:
of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is
72:
68:
1036:{\displaystyle {\begin{aligned}y_{1}(t)&=H\left\{x_{1}(t)\right\}\\y_{2}(t)&=H\left\{x_{2}(t)\right\}\end{aligned}}}
115:
2934:
87:
4605:
4282:
1690:
94:
4595:
4297:
2674:. It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form
1714:
31:
61:
2603:
183:. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.
35:
198:
101:
4590:
817:
4226:
3926:
The output of any discrete time linear system is related to the input by the time-varying convolution sum:
1505:
1465:
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1385:
4272:
1365:
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83:
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If the properties of the system do not depend on the time at which it is operated then it is said to be
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1300:
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2414:
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171:
case. As a mathematical abstraction or idealization, linear systems find important applications in
3367:{\displaystyle y(t)=\int _{-\infty }^{t}h(t,t')x(t')dt'=\int _{-\infty }^{\infty }h(t,t')x(t')dt'}
2305:
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1960:
2462:
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1308:
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176:
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1163:{\displaystyle \alpha y_{1}(t)+\beta y_{2}(t)=H\left\{\alpha x_{1}(t)+\beta x_{2}(t)\right\}}
4287:
2901:
2151:
1304:
39:
2148:, and any system described by ordinary linear differential equations. Systems described by
673:
646:
108:
4292:
167:. Linear systems typically exhibit features and properties that are much simpler than the
164:
3419:
it is then possible to write the input-output relation equivalently in any of the ways,
2837:, that is, the output doesn't consist only of sinusoids of same frequency as the input (
4277:
1702:
626:
460:
440:
278:
148:
30:
This article is about the systems theory concept. For the linear algebra concept, see
4584:
1376:
1327:
3815:, the integral may equally be written over the doubly infinite range and putting
1354:
50:
4318:
1698:
4066:{\displaystyle y=\sum _{m=-\infty }^{n}{hx}=\sum _{m=-\infty }^{\infty }{hx}}
1364:
Another perspective is that solutions to linear systems comprise a system of
17:
4216:{\displaystyle y=\sum _{k=0}^{\infty }{hx}=\sum _{k=-\infty }^{\infty }{hx}}
2586:
168:
3912:{\displaystyle H(i\omega )=\int _{-\infty }^{\infty }h(t)e^{-i\omega t}dt}
2590:
1694:
831:
Mathematically, for a continuous-time system, given two arbitrary inputs
1689:
This definition of a linear system is analogous to the definition of a
2141:{\displaystyle y(t)=k\,\int _{-\infty }^{t}x(\tau )\mathrm {d} \tau }
160:
2889:
of a linear system is defined as the response of the system at time
1375:
A common use of linear models is to describe a nonlinear system by
1954:
which shows that a simple harmonic oscillator is a linear system.
516:
370:
190:
4461:
Signals and
Systems: Analysis Using Transform Methods and MATLAB®
3796:
In applications this is usually a rational algebraic function of
2830:{\displaystyle y(t)=1.5\cos {(3t)}+0.5\cos {(2t)}+0.5\cos {(4t)}}
2578:{\displaystyle y(t)=k\,{\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}
2067:{\displaystyle y(t)=k\,{\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}
891:{\displaystyle {\begin{aligned}x_{1}(t)\\x_{2}(t)\end{aligned}}}
44:
1901:{\displaystyle H(x(t))=m{\frac {d^{2}(x(t))}{dt^{2}}}+kx(t),}
1957:
Other examples of linear systems include those described by
4073:
or equivalently for a time-invariant system on redefining
3787:{\displaystyle H(s)=\int _{0}^{\infty }h(t)e^{-st}\,dt.}
616:{\displaystyle y_{3}(t)=a_{1}\,y_{1}(t)+a_{2}\,y_{2}(t)}
4513:(3 ed.). Oxford University Press. pp. 55–57.
2417:
1542:) are considered instead of input and output signals (
1379:. This is usually done for mathematical convenience.
27:
Physical system satisfying the superposition principle
4255:
represents the lag time between the stimulus at time
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201:
816:A system is linear if and only if it satisfies the
75:. Unsourced material may be challenged and removed.
4463:(3 ed.). McGraw-Hill. pp. 131, 133–134.
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4376:Signals and Systems: A MATLAB Integrated Approach
4561:Signals and Systems: Principles and Applications
2841:), but instead also of sinusoids of frequencies
2596:Also, the output of a linear system can contain
1785:{\displaystyle m{\frac {d^{2}(x)}{dt^{2}}}=-kx.}
3187:{\displaystyle h(t_{2},t_{1})=0,t_{2}<t_{1}}
1361:case (especially in computer implementations).
1345:systems are well adapted to analysis using the
898:as well as their respective zero-state outputs
4454:
4452:
4450:
4344:
4342:
2857:, which is different than that of the input.
8:
4399:
4397:
4395:
1928:we can rewrite the differential equation as
4563:. Cambridge University Press. p. 187.
4534:ElAli, Taan S.; Karim, Mohammad A. (2008).
4431:A Practical Approach to Signals and Systems
4408:. McGraw-Hill. pp. 162–164, 166, 183.
2667:{\displaystyle y(t)=(1.5+\cos {(t)})\,x(t)}
1315:), which describe a general input function
1217:The system is then defined by the equation
4536:Continuous Signals and Systems with MATLAB
3384:is a function only of the time difference
2730:it can be easily shown that the output is
268:{\displaystyle y_{3}(t)=y_{1}(t)+y_{2}(t)}
34:. For the algebraic geometry concept, see
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4181:
4175:
4161:
4122:
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4105:
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4031:
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4011:
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3206:
3178:
3165:
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3130:
3118:
3081:{\displaystyle y(t=t_{2})=h(t_{2},t_{1})}
3069:
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3034:
3016:
2969:
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2813:
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228:
206:
200:
135:Learn how and when to remove this message
1251:is some arbitrary function of time, and
4309:
2728:product-to-sum trigonometric identities
4538:(2 ed.). CRC Press. p. 53.
4317:Phillips, Charles L.; Parr, John M.;
2981:{\displaystyle x(t)=\delta (t-t_{1})}
1535:{\displaystyle {\mathbf {y} }_{2}(t)}
1495:{\displaystyle {\mathbf {y} }_{1}(t)}
1455:{\displaystyle {\mathbf {x} }_{2}(t)}
1415:{\displaystyle {\mathbf {x} }_{1}(t)}
7:
430:{\displaystyle y_{2}(t)=a\,y_{1}(t)}
73:adding citations to reliable sources
4325:(4 ed.). Pearson. p. 74.
4176:
4171:
4117:
4026:
4021:
3962:
3867:
3862:
3741:
3652:
3598:
3593:
3526:
3521:
3449:
3308:
3303:
3231:
2565:
2546:
2237:{\displaystyle y(t)=k\,x(t)+k_{0}}
2131:
2108:
2054:
2035:
1043:then a linear system must satisfy
38:. For the tactical formation, see
25:
3000:, and the corresponding response
1717:obeys the differential equation:
1303:systems this is the basis of the
777:can be described by an operator,
4323:Signals, Systems, and Transforms
2719:{\displaystyle x(t)=\cos {(3t)}}
2585:(such as a constant-capacitance
2452:{\textstyle y(t)={\sqrt {x(t)}}}
1512:
1472:
1432:
1392:
49:
1914:is a linear operator. Letting
60:needs additional citations for
4209:
4197:
4191:
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4150:
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4132:
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4059:
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3333:
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2427:
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2218:
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2199:
2193:
2164:
2158:
2127:
2121:
2090:
2084:
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2042:
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2015:
1992:
1986:
1973:
1967:
1892:
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1841:
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1816:
1810:
1804:
1746:
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1601:
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1565:
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1529:
1523:
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1483:
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1403:
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1021:
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348:
318:
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218:
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1:
2920:In other words, if the input
2867:time-varying impulse response
2861:Time-varying impulse response
2404:{\displaystyle y(t)=x^{2}(t)}
1282:the system can be solved for
4559:Apte, Shaila Dinkar (2016).
4488:. Springer. pp. 43–44.
4459:Roberts, Michael J. (2018).
4353:. Springer. pp. 27–28.
4283:Linear time-invariant system
3825:follows the formula for the
2353:{\displaystyle y(t)=\cos {}}
2295:{\displaystyle y(t)=\sin {}}
1998:{\displaystyle y(t)=k\,x(t)}
1691:linear differential equation
1262:is the system state. Given
3827:frequency response function
2506:{\displaystyle y(t)=|x(t)|}
769:. Click image to expand it.
513:. Click image to expand it.
367:. Click image to expand it.
4622:
4298:System of linear equations
2849:; furthermore, taking the
1715:simple harmonic oscillator
32:System of linear equations
29:
4429:Sundararajan, D. (2008).
4378:. CRC Press. p. 99.
4349:Bessai, Horst J. (2005).
4259:and the response at time
2589:or a constant-inductance
643:, for all real constants
36:Linear system of divisors
4509:Chen, Chi-Tsong (2004).
4484:Deergha Rao, K. (2018).
4351:MIMO Signals and Systems
3197:The convolution integral
1679:{\displaystyle y_{2}(t)}
1643:{\displaystyle y_{1}(t)}
1607:{\displaystyle x_{2}(t)}
1571:{\displaystyle x_{1}(t)}
1372:in the geometric sense.
1182:, for any input signals
762:{\displaystyle x_{2}(t)}
726:{\displaystyle x_{1}(t)}
506:{\displaystyle x_{1}(t)}
457:, for all real constant
360:{\displaystyle x_{2}(t)}
324:{\displaystyle x_{1}(t)}
4404:Nahvi, Mahmood (2014).
2931:to a linear system is
818:superposition principle
4273:Shift invariant system
4249:
4217:
4180:
4121:
4067:
4030:
3971:
3913:
3788:
3697:
3415:). By redefinition of
3368:
3188:
3082:
2982:
2831:
2720:
2668:
2579:
2507:
2453:
2405:
2354:
2296:
2238:
2177:
2176:{\displaystyle y(t)=k}
2142:
2068:
1999:
1902:
1786:
1680:
1644:
1608:
1572:
1536:
1496:
1456:
1416:
1339:differential equations
1164:
1037:
892:
783:, that maps an input,
770:
763:
727:
691:
664:
637:
617:
514:
507:
471:
451:
431:
368:
361:
325:
289:
269:
163:based on the use of a
4601:Mathematical modeling
4433:. Wiley. p. 80.
4374:Alkin, Oktay (2014).
4250:
4248:{\displaystyle k=n-m}
4218:
4157:
4101:
4068:
4007:
3948:
3922:Discrete-time systems
3914:
3811:is zero for negative
3789:
3698:
3369:
3189:
3083:
2983:
2851:least common multiple
2832:
2721:
2669:
2580:
2508:
2454:
2406:
2355:
2297:
2239:
2178:
2143:
2069:
2000:
1903:
1787:
1699:linear transformation
1681:
1645:
1609:
1573:
1537:
1497:
1457:
1417:
1165:
1038:
893:
764:
728:
692:
690:{\displaystyle a_{2}}
665:
663:{\displaystyle a_{1}}
638:
618:
520:
508:
472:
452:
432:
374:
362:
326:
290:
270:
194:
4227:
4083:
3930:
3833:
3713:
3423:
3205:
3117:
3015:
2998:Dirac delta function
2935:
2734:
2678:
2604:
2520:
2463:
2415:
2364:
2306:
2248:
2187:
2152:
2078:
2009:
1961:
1798:
1721:
1654:
1618:
1582:
1546:
1506:
1466:
1426:
1386:
1332:frequency components
1047:
902:
835:
775:deterministic system
737:
701:
674:
647:
627:
525:
481:
461:
441:
379:
335:
299:
279:
199:
69:improve this article
4606:Concepts in physics
4511:Signals and systems
4486:Signals and Systems
4406:Signals and Systems
3871:
3745:
3656:
3602:
3530:
3458:
3312:
3240:
3113:must be satisfied:
3111:causality condition
2117:
1210:, and for all time
794:, as a function of
697:and for all inputs
295:and for all inputs
4245:
4213:
4063:
3909:
3854:
3784:
3731:
3693:
3642:
3585:
3513:
3441:
3398:which is zero for
3364:
3295:
3223:
3184:
3088:then the function
3078:
2978:
2827:
2716:
2664:
2575:
2503:
2449:
2401:
2350:
2292:
2234:
2173:
2138:
2100:
2064:
1995:
1898:
1782:
1676:
1640:
1604:
1568:
1532:
1492:
1452:
1412:
1309:frequency response
1160:
1033:
1031:
888:
886:
771:
759:
723:
687:
660:
633:
613:
515:
503:
477:and for all input
467:
447:
427:
369:
357:
321:
285:
265:
181:telecommunications
157:mathematical model
4596:Dynamical systems
4570:978-1-107-14624-2
4545:978-1-4200-5475-0
4495:978-3-319-68674-5
4470:978-0-07-802812-0
4440:978-0-470-82353-8
4415:978-0-07-338070-4
4385:978-1-4665-9854-6
4332:978-0-13-198923-8
3707:transfer function
3011:of the system is
2573:
2447:
2062:
1875:
1765:
1347:Laplace transform
1313:LTI system theory
636:{\displaystyle t}
470:{\displaystyle a}
450:{\displaystyle t}
288:{\displaystyle t}
177:signal processing
173:automatic control
145:
144:
137:
119:
16:(Redirected from
4613:
4575:
4574:
4556:
4550:
4549:
4531:
4525:
4524:
4506:
4500:
4499:
4481:
4475:
4474:
4456:
4445:
4444:
4426:
4420:
4419:
4401:
4390:
4389:
4371:
4365:
4364:
4346:
4337:
4336:
4314:
4288:Nonlinear system
4254:
4252:
4251:
4246:
4222:
4220:
4219:
4214:
4212:
4179:
4174:
4153:
4120:
4115:
4078:
4072:
4070:
4069:
4064:
4062:
4029:
4024:
4003:
3970:
3965:
3918:
3916:
3915:
3910:
3902:
3901:
3870:
3865:
3824:
3814:
3810:
3799:
3793:
3791:
3790:
3785:
3773:
3772:
3744:
3739:
3702:
3700:
3699:
3694:
3655:
3650:
3601:
3596:
3581:
3567:
3550:
3529:
3524:
3509:
3495:
3478:
3457:
3452:
3418:
3414:
3404:
3397:
3383:
3373:
3371:
3370:
3365:
3363:
3349:
3332:
3311:
3306:
3291:
3277:
3260:
3239:
3234:
3193:
3191:
3190:
3185:
3183:
3182:
3170:
3169:
3148:
3147:
3135:
3134:
3108:
3087:
3085:
3084:
3079:
3074:
3073:
3061:
3060:
3039:
3038:
3010:
2995:
2987:
2985:
2984:
2979:
2974:
2973:
2930:
2919:
2917:
2904:applied at time
2888:
2856:
2848:
2844:
2840:
2836:
2834:
2833:
2828:
2826:
2800:
2774:
2725:
2723:
2722:
2717:
2715:
2673:
2671:
2670:
2665:
2647:
2584:
2582:
2581:
2576:
2574:
2572:
2568:
2562:
2549:
2543:
2512:
2510:
2509:
2504:
2502:
2485:
2458:
2456:
2455:
2450:
2448:
2434:
2410:
2408:
2407:
2402:
2391:
2390:
2359:
2357:
2356:
2351:
2349:
2301:
2299:
2298:
2293:
2291:
2243:
2241:
2240:
2235:
2233:
2232:
2182:
2180:
2179:
2174:
2147:
2145:
2144:
2139:
2134:
2116:
2111:
2073:
2071:
2070:
2065:
2063:
2061:
2057:
2051:
2038:
2032:
2004:
2002:
2001:
1996:
1953:
1951:
1927:
1925:
1913:
1907:
1905:
1904:
1899:
1876:
1874:
1873:
1872:
1859:
1840:
1839:
1829:
1791:
1789:
1788:
1783:
1766:
1764:
1763:
1762:
1749:
1739:
1738:
1728:
1685:
1683:
1682:
1677:
1666:
1665:
1649:
1647:
1646:
1641:
1630:
1629:
1613:
1611:
1610:
1605:
1594:
1593:
1577:
1575:
1574:
1569:
1558:
1557:
1541:
1539:
1538:
1533:
1522:
1521:
1516:
1515:
1501:
1499:
1498:
1493:
1482:
1481:
1476:
1475:
1461:
1459:
1458:
1453:
1442:
1441:
1436:
1435:
1421:
1419:
1418:
1413:
1402:
1401:
1396:
1395:
1325:
1305:impulse response
1295:
1293:
1281:
1279:
1272:
1261:
1250:
1239:
1213:
1209:
1195:
1181:
1177:
1169:
1167:
1166:
1161:
1159:
1155:
1145:
1144:
1120:
1119:
1087:
1086:
1062:
1061:
1042:
1040:
1039:
1034:
1032:
1028:
1024:
1014:
1013:
980:
979:
966:
962:
952:
951:
918:
917:
897:
895:
894:
889:
887:
874:
873:
851:
850:
808:
797:
793:
782:
768:
766:
765:
760:
749:
748:
732:
730:
729:
724:
713:
712:
696:
694:
693:
688:
686:
685:
669:
667:
666:
661:
659:
658:
642:
640:
639:
634:
622:
620:
619:
614:
603:
602:
592:
591:
570:
569:
559:
558:
537:
536:
512:
510:
509:
504:
493:
492:
476:
474:
473:
468:
456:
454:
453:
448:
436:
434:
433:
428:
417:
416:
391:
390:
366:
364:
363:
358:
347:
346:
330:
328:
327:
322:
311:
310:
294:
292:
291:
286:
274:
272:
271:
266:
255:
254:
233:
232:
211:
210:
140:
133:
129:
126:
120:
118:
77:
53:
45:
40:Line (formation)
21:
4621:
4620:
4616:
4615:
4614:
4612:
4611:
4610:
4581:
4580:
4579:
4578:
4571:
4558:
4557:
4553:
4546:
4533:
4532:
4528:
4521:
4508:
4507:
4503:
4496:
4483:
4482:
4478:
4471:
4458:
4457:
4448:
4441:
4428:
4427:
4423:
4416:
4403:
4402:
4393:
4386:
4373:
4372:
4368:
4361:
4348:
4347:
4340:
4333:
4316:
4315:
4311:
4306:
4293:System analysis
4269:
4225:
4224:
4081:
4080:
4074:
3928:
3927:
3924:
3884:
3831:
3830:
3816:
3812:
3801:
3797:
3758:
3711:
3710:
3574:
3560:
3543:
3502:
3488:
3471:
3421:
3420:
3416:
3406:
3399:
3385:
3381:
3356:
3342:
3325:
3284:
3270:
3253:
3203:
3202:
3199:
3174:
3161:
3139:
3126:
3115:
3114:
3106:
3099:
3089:
3065:
3052:
3030:
3013:
3012:
3001:
2996:represents the
2989:
2965:
2933:
2932:
2921:
2916:
2906:
2905:
2899:
2886:
2879:
2869:
2863:
2854:
2846:
2842:
2838:
2732:
2731:
2676:
2675:
2602:
2601:
2563:
2544:
2518:
2517:
2461:
2460:
2413:
2412:
2382:
2362:
2361:
2304:
2303:
2246:
2245:
2224:
2185:
2184:
2150:
2149:
2076:
2075:
2052:
2033:
2007:
2006:
1959:
1958:
1930:
1929:
1916:
1915:
1909:
1864:
1860:
1831:
1830:
1796:
1795:
1754:
1750:
1730:
1729:
1719:
1718:
1711:
1657:
1652:
1651:
1621:
1616:
1615:
1585:
1580:
1579:
1549:
1544:
1543:
1509:
1504:
1503:
1469:
1464:
1463:
1429:
1424:
1423:
1389:
1384:
1383:
1368:which act like
1316:
1284:
1283:
1275:
1274:
1263:
1252:
1241:
1218:
1211:
1203:
1197:
1189:
1183:
1179:
1175:
1136:
1111:
1107:
1103:
1078:
1053:
1045:
1044:
1030:
1029:
1005:
1004:
1000:
990:
971:
968:
967:
943:
942:
938:
928:
909:
900:
899:
885:
884:
865:
862:
861:
842:
833:
832:
799:
795:
784:
778:
740:
735:
734:
704:
699:
698:
677:
672:
671:
650:
645:
644:
625:
624:
594:
583:
561:
550:
528:
523:
522:
484:
479:
478:
459:
458:
439:
438:
408:
382:
377:
376:
338:
333:
332:
302:
297:
296:
277:
276:
246:
224:
202:
197:
196:
189:
165:linear operator
141:
130:
124:
121:
84:"Linear system"
78:
76:
66:
54:
43:
28:
23:
22:
15:
12:
11:
5:
4619:
4617:
4609:
4608:
4603:
4598:
4593:
4591:Systems theory
4583:
4582:
4577:
4576:
4569:
4551:
4544:
4526:
4519:
4501:
4494:
4476:
4469:
4446:
4439:
4421:
4414:
4391:
4384:
4366:
4359:
4338:
4331:
4319:Riskin, Eve A.
4308:
4307:
4305:
4302:
4301:
4300:
4295:
4290:
4285:
4280:
4278:Linear control
4275:
4268:
4265:
4244:
4241:
4238:
4235:
4232:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4178:
4173:
4170:
4167:
4164:
4160:
4156:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4119:
4114:
4111:
4108:
4104:
4100:
4097:
4094:
4091:
4088:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4028:
4023:
4020:
4017:
4014:
4010:
4006:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3969:
3964:
3961:
3958:
3955:
3951:
3947:
3944:
3941:
3938:
3935:
3923:
3920:
3908:
3905:
3900:
3897:
3894:
3891:
3887:
3883:
3880:
3877:
3874:
3869:
3864:
3861:
3857:
3853:
3850:
3847:
3844:
3841:
3838:
3783:
3780:
3777:
3771:
3768:
3765:
3761:
3757:
3754:
3751:
3748:
3743:
3738:
3734:
3730:
3727:
3724:
3721:
3718:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3654:
3649:
3645:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3600:
3595:
3592:
3588:
3584:
3580:
3577:
3573:
3570:
3566:
3563:
3559:
3556:
3553:
3549:
3546:
3542:
3539:
3536:
3533:
3528:
3523:
3520:
3516:
3512:
3508:
3505:
3501:
3498:
3494:
3491:
3487:
3484:
3481:
3477:
3474:
3470:
3467:
3464:
3461:
3456:
3451:
3448:
3444:
3440:
3437:
3434:
3431:
3428:
3378:time-invariant
3362:
3359:
3355:
3352:
3348:
3345:
3341:
3338:
3335:
3331:
3328:
3324:
3321:
3318:
3315:
3310:
3305:
3302:
3298:
3294:
3290:
3287:
3283:
3280:
3276:
3273:
3269:
3266:
3263:
3259:
3256:
3252:
3249:
3246:
3243:
3238:
3233:
3230:
3226:
3222:
3219:
3216:
3213:
3210:
3198:
3195:
3181:
3177:
3173:
3168:
3164:
3160:
3157:
3154:
3151:
3146:
3142:
3138:
3133:
3129:
3125:
3122:
3104:
3097:
3077:
3072:
3068:
3064:
3059:
3055:
3051:
3048:
3045:
3042:
3037:
3033:
3029:
3026:
3023:
3020:
2977:
2972:
2968:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2914:
2897:
2884:
2877:
2862:
2859:
2825:
2822:
2819:
2816:
2812:
2809:
2806:
2803:
2799:
2796:
2793:
2790:
2786:
2783:
2780:
2777:
2773:
2770:
2767:
2764:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2714:
2711:
2708:
2705:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2663:
2660:
2657:
2654:
2650:
2646:
2643:
2640:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2571:
2567:
2561:
2558:
2555:
2552:
2548:
2540:
2537:
2534:
2531:
2528:
2525:
2501:
2497:
2494:
2491:
2488:
2484:
2480:
2477:
2474:
2471:
2468:
2446:
2443:
2440:
2437:
2432:
2429:
2426:
2423:
2420:
2400:
2397:
2394:
2389:
2385:
2381:
2378:
2375:
2372:
2369:
2348:
2345:
2342:
2339:
2336:
2333:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2290:
2287:
2284:
2281:
2278:
2275:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2231:
2227:
2223:
2220:
2217:
2214:
2211:
2207:
2204:
2201:
2198:
2195:
2192:
2172:
2169:
2166:
2163:
2160:
2157:
2137:
2133:
2129:
2126:
2123:
2120:
2115:
2110:
2107:
2103:
2098:
2095:
2092:
2089:
2086:
2083:
2060:
2056:
2050:
2047:
2044:
2041:
2037:
2029:
2026:
2023:
2020:
2017:
2014:
1994:
1991:
1988:
1985:
1981:
1978:
1975:
1972:
1969:
1966:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1871:
1867:
1863:
1858:
1855:
1852:
1849:
1846:
1843:
1838:
1834:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1781:
1778:
1775:
1772:
1769:
1761:
1757:
1753:
1748:
1745:
1742:
1737:
1733:
1726:
1710:
1707:
1703:linear algebra
1675:
1672:
1669:
1664:
1660:
1639:
1636:
1633:
1628:
1624:
1603:
1600:
1597:
1592:
1588:
1567:
1564:
1561:
1556:
1552:
1531:
1528:
1525:
1520:
1514:
1491:
1488:
1485:
1480:
1474:
1451:
1448:
1445:
1440:
1434:
1411:
1408:
1405:
1400:
1394:
1353:case, and the
1343:time-invariant
1301:time-invariant
1201:
1187:
1158:
1154:
1151:
1148:
1143:
1139:
1135:
1132:
1129:
1126:
1123:
1118:
1114:
1110:
1106:
1102:
1099:
1096:
1093:
1090:
1085:
1081:
1077:
1074:
1071:
1068:
1065:
1060:
1056:
1052:
1027:
1023:
1020:
1017:
1012:
1008:
1003:
999:
996:
993:
991:
989:
986:
983:
978:
974:
970:
969:
965:
961:
958:
955:
950:
946:
941:
937:
934:
931:
929:
927:
924:
921:
916:
912:
908:
907:
883:
880:
877:
872:
868:
864:
863:
860:
857:
854:
849:
845:
841:
840:
798:to an output,
758:
755:
752:
747:
743:
722:
719:
716:
711:
707:
684:
680:
657:
653:
632:
612:
609:
606:
601:
597:
590:
586:
582:
579:
576:
573:
568:
564:
557:
553:
549:
546:
543:
540:
535:
531:
502:
499:
496:
491:
487:
466:
446:
426:
423:
420:
415:
411:
406:
403:
400:
397:
394:
389:
385:
356:
353:
350:
345:
341:
320:
317:
314:
309:
305:
284:
264:
261:
258:
253:
249:
245:
242:
239:
236:
231:
227:
223:
220:
217:
214:
209:
205:
188:
185:
149:systems theory
143:
142:
57:
55:
48:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4618:
4607:
4604:
4602:
4599:
4597:
4594:
4592:
4589:
4588:
4586:
4572:
4566:
4562:
4555:
4552:
4547:
4541:
4537:
4530:
4527:
4522:
4520:0-19-515661-7
4516:
4512:
4505:
4502:
4497:
4491:
4487:
4480:
4477:
4472:
4466:
4462:
4455:
4453:
4451:
4447:
4442:
4436:
4432:
4425:
4422:
4417:
4411:
4407:
4400:
4398:
4396:
4392:
4387:
4381:
4377:
4370:
4367:
4362:
4360:0-387-23488-8
4356:
4352:
4345:
4343:
4339:
4334:
4328:
4324:
4320:
4313:
4310:
4303:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4270:
4266:
4264:
4262:
4258:
4242:
4239:
4236:
4233:
4230:
4206:
4203:
4200:
4194:
4188:
4182:
4168:
4165:
4162:
4158:
4154:
4147:
4144:
4141:
4135:
4129:
4123:
4112:
4109:
4106:
4102:
4098:
4092:
4086:
4077:
4056:
4050:
4044:
4041:
4038:
4032:
4018:
4015:
4012:
4008:
4004:
3997:
3991:
3985:
3982:
3979:
3973:
3967:
3959:
3956:
3953:
3949:
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3898:
3895:
3892:
3889:
3885:
3878:
3872:
3859:
3855:
3851:
3845:
3842:
3836:
3828:
3823:
3819:
3808:
3804:
3794:
3781:
3778:
3775:
3769:
3766:
3763:
3759:
3752:
3746:
3736:
3732:
3728:
3722:
3716:
3708:
3703:
3690:
3687:
3681:
3678:
3675:
3669:
3663:
3657:
3647:
3643:
3639:
3636:
3633:
3627:
3624:
3621:
3615:
3609:
3603:
3590:
3586:
3582:
3578:
3575:
3571:
3564:
3561:
3554:
3547:
3544:
3540:
3537:
3531:
3518:
3514:
3510:
3506:
3503:
3499:
3492:
3489:
3482:
3475:
3472:
3468:
3465:
3459:
3454:
3446:
3442:
3438:
3432:
3426:
3413:
3409:
3402:
3396:
3392:
3388:
3379:
3374:
3360:
3357:
3353:
3346:
3343:
3336:
3329:
3326:
3322:
3319:
3313:
3300:
3296:
3292:
3288:
3285:
3281:
3274:
3271:
3264:
3257:
3254:
3250:
3247:
3241:
3236:
3228:
3224:
3220:
3214:
3208:
3196:
3194:
3179:
3175:
3171:
3166:
3162:
3158:
3155:
3152:
3144:
3140:
3136:
3131:
3127:
3120:
3112:
3103:
3096:
3092:
3070:
3066:
3062:
3057:
3053:
3046:
3043:
3035:
3031:
3027:
3024:
3018:
3008:
3004:
2999:
2993:
2970:
2966:
2962:
2959:
2953:
2950:
2944:
2938:
2928:
2924:
2913:
2909:
2903:
2896:
2892:
2883:
2876:
2872:
2868:
2860:
2858:
2852:
2820:
2817:
2810:
2807:
2804:
2801:
2794:
2791:
2784:
2781:
2778:
2775:
2768:
2765:
2758:
2755:
2752:
2749:
2743:
2737:
2729:
2709:
2706:
2699:
2696:
2693:
2687:
2681:
2658:
2652:
2641:
2634:
2631:
2628:
2625:
2619:
2613:
2607:
2599:
2594:
2592:
2588:
2569:
2556:
2550:
2538:
2535:
2529:
2523:
2514:
2492:
2486:
2478:
2472:
2466:
2441:
2435:
2430:
2424:
2418:
2395:
2387:
2383:
2379:
2373:
2367:
2340:
2334:
2327:
2324:
2321:
2315:
2309:
2282:
2276:
2269:
2266:
2263:
2257:
2251:
2229:
2225:
2221:
2215:
2209:
2205:
2202:
2196:
2190:
2170:
2167:
2161:
2155:
2135:
2124:
2118:
2113:
2105:
2101:
2096:
2093:
2087:
2081:
2058:
2045:
2039:
2027:
2024:
2018:
2012:
1989:
1983:
1979:
1976:
1970:
1964:
1955:
1949:
1945:
1941:
1937:
1933:
1923:
1919:
1912:
1895:
1889:
1883:
1880:
1877:
1869:
1865:
1861:
1850:
1844:
1836:
1832:
1825:
1822:
1813:
1807:
1801:
1792:
1779:
1776:
1773:
1770:
1767:
1759:
1755:
1751:
1743:
1735:
1731:
1724:
1716:
1708:
1706:
1704:
1700:
1696:
1692:
1687:
1670:
1662:
1658:
1634:
1626:
1622:
1598:
1590:
1586:
1562:
1554:
1550:
1526:
1518:
1486:
1478:
1446:
1438:
1406:
1398:
1380:
1378:
1377:linearization
1373:
1371:
1367:
1362:
1360:
1356:
1352:
1348:
1344:
1340:
1335:
1333:
1329:
1328:unit impulses
1323:
1319:
1314:
1311:methods (see
1310:
1306:
1302:
1296:
1291:
1287:
1278:
1270:
1266:
1259:
1255:
1248:
1244:
1237:
1233:
1229:
1225:
1221:
1215:
1207:
1200:
1193:
1186:
1173:
1156:
1149:
1141:
1137:
1133:
1130:
1124:
1116:
1112:
1108:
1104:
1100:
1097:
1091:
1083:
1079:
1075:
1072:
1066:
1058:
1054:
1050:
1025:
1018:
1010:
1006:
1001:
997:
994:
992:
984:
976:
972:
963:
956:
948:
944:
939:
935:
932:
930:
922:
914:
910:
878:
870:
866:
855:
847:
843:
829:
825:
821:
819:
814:
813:description.
812:
806:
802:
791:
787:
781:
776:
753:
745:
741:
717:
709:
705:
682:
678:
655:
651:
630:
623:for all time
607:
599:
595:
588:
584:
580:
574:
566:
562:
555:
551:
547:
541:
533:
529:
519:
497:
489:
485:
464:
444:
437:for all time
421:
413:
409:
404:
401:
395:
387:
383:
373:
351:
343:
339:
315:
307:
303:
282:
275:for all time
259:
251:
247:
243:
237:
229:
225:
221:
215:
207:
203:
193:
186:
184:
182:
178:
174:
170:
166:
162:
158:
154:
153:linear system
150:
139:
136:
128:
117:
114:
110:
107:
103:
100:
96:
93:
89:
86: –
85:
81:
80:Find sources:
74:
70:
64:
63:
58:This article
56:
52:
47:
46:
41:
37:
33:
19:
18:Linear theory
4560:
4554:
4535:
4529:
4510:
4504:
4485:
4479:
4460:
4430:
4424:
4405:
4375:
4369:
4350:
4322:
4312:
4260:
4256:
4075:
3925:
3826:
3821:
3817:
3806:
3802:
3795:
3706:
3704:
3411:
3407:
3400:
3394:
3390:
3386:
3377:
3375:
3200:
3110:
3101:
3094:
3090:
3006:
3002:
2991:
2926:
2922:
2911:
2907:
2900:to a single
2894:
2890:
2881:
2874:
2870:
2866:
2864:
2595:
2515:
1956:
1947:
1943:
1939:
1935:
1931:
1921:
1917:
1910:
1793:
1712:
1688:
1381:
1374:
1363:
1336:
1326:in terms of
1321:
1317:
1297:
1289:
1285:
1276:
1268:
1264:
1257:
1253:
1246:
1242:
1235:
1231:
1227:
1223:
1219:
1216:
1205:
1198:
1191:
1184:
830:
826:
822:
815:
809:, a type of
804:
800:
789:
785:
779:
772:
152:
146:
131:
122:
112:
105:
98:
91:
79:
67:Please help
62:verification
59:
1355:Z-transform
4585:Categories
4304:References
3800:. Because
3709:which is:
1351:continuous
1341:of linear
773:A general
187:Definition
95:newspapers
4240:−
4204:−
4177:∞
4172:∞
4169:−
4159:∑
4145:−
4118:∞
4103:∑
4027:∞
4022:∞
4019:−
4009:∑
3963:∞
3960:−
3950:∑
3896:ω
3890:−
3868:∞
3863:∞
3860:−
3856:∫
3846:ω
3764:−
3742:∞
3733:∫
3691:τ
3682:τ
3679:−
3664:τ
3653:∞
3644:∫
3637:τ
3628:τ
3625:−
3610:τ
3599:∞
3594:∞
3591:−
3587:∫
3541:−
3527:∞
3522:∞
3519:−
3515:∫
3469:−
3450:∞
3447:−
3443:∫
3309:∞
3304:∞
3301:−
3297:∫
3232:∞
3229:−
3225:∫
2963:−
2954:δ
2811:
2785:
2759:
2700:
2635:
2598:harmonics
2587:capacitor
2328:
2270:
2136:τ
2125:τ
2109:∞
2106:−
2102:∫
1771:−
1366:functions
1134:β
1109:α
1076:β
1051:α
811:black box
169:nonlinear
125:June 2021
4321:(2008).
4267:See also
3579:′
3565:′
3548:′
3507:′
3493:′
3476:′
3405:(namely
3361:′
3347:′
3330:′
3289:′
3275:′
3258:′
2726:, using
2591:inductor
1709:Examples
1697:, and a
1695:calculus
1359:discrete
1337:Typical
1240:, where
1170:for any
828:inputs.
175:theory,
2902:impulse
2855:1 rad/s
2847:4 rad/s
2843:2 rad/s
2839:3 rad/s
1370:vectors
1357:in the
1349:in the
1307:or the
1174:values
109:scholar
4567:
4542:
4517:
4492:
4467:
4437:
4412:
4382:
4357:
4329:
4223:where
3403:< 0
2988:where
1172:scalar
179:, and
161:system
111:
104:
97:
90:
82:
3410:<
1942:)) =
1924:) = 0
1908:then
1230:)) =
159:of a
155:is a
116:JSTOR
102:books
4565:ISBN
4540:ISBN
4515:ISBN
4490:ISBN
4465:ISBN
4435:ISBN
4410:ISBN
4380:ISBN
4355:ISBN
4327:ISBN
3380:and
3172:<
2865:The
2845:and
1273:and
1196:and
1178:and
733:and
670:and
331:and
151:, a
88:news
3412:t'
3395:t'
2808:cos
2805:0.5
2782:cos
2779:0.5
2756:cos
2753:1.5
2697:cos
2632:cos
2626:1.5
2325:cos
2267:sin
1794:If
1701:in
1693:in
1686:.)
1330:or
147:In
71:by
4587::
4449:^
4394:^
4341:^
4263:.
4079:,
3829::
3822:iω
3820:=
3393:−
3389:=
3100:,
2990:δ(
2910:=
2893:=
2880:,
2459:,
2411:,
2360:,
2302:,
2244:,
2183:,
2074:,
2005:,
1713:A
1705:.
1650:,
1614:,
1578:,
1502:,
1462:,
1422:,
1334:.
1214:.
4573:.
4548:.
4523:.
4498:.
4473:.
4443:.
4418:.
4388:.
4363:.
4335:.
4261:n
4257:m
4243:m
4237:n
4234:=
4231:k
4210:]
4207:k
4201:n
4198:[
4195:x
4192:]
4189:k
4186:[
4183:h
4166:=
4163:k
4155:=
4151:]
4148:k
4142:n
4139:[
4136:x
4133:]
4130:k
4127:[
4124:h
4113:0
4110:=
4107:k
4099:=
4096:]
4093:n
4090:[
4087:y
4076:h
4060:]
4057:m
4054:[
4051:x
4048:]
4045:m
4042:,
4039:n
4036:[
4033:h
4016:=
4013:m
4005:=
4001:]
3998:m
3995:[
3992:x
3989:]
3986:m
3983:,
3980:n
3977:[
3974:h
3968:n
3957:=
3954:m
3946:=
3943:]
3940:n
3937:[
3934:y
3907:t
3904:d
3899:t
3893:i
3886:e
3882:)
3879:t
3876:(
3873:h
3852:=
3849:)
3843:i
3840:(
3837:H
3818:s
3813:t
3809:)
3807:t
3805:(
3803:h
3798:s
3782:.
3779:t
3776:d
3770:t
3767:s
3760:e
3756:)
3753:t
3750:(
3747:h
3737:0
3729:=
3726:)
3723:s
3720:(
3717:H
3688:d
3685:)
3676:t
3673:(
3670:x
3667:)
3661:(
3658:h
3648:0
3640:=
3634:d
3631:)
3622:t
3619:(
3616:x
3613:)
3607:(
3604:h
3583:=
3576:t
3572:d
3569:)
3562:t
3558:(
3555:x
3552:)
3545:t
3538:t
3535:(
3532:h
3511:=
3504:t
3500:d
3497:)
3490:t
3486:(
3483:x
3480:)
3473:t
3466:t
3463:(
3460:h
3455:t
3439:=
3436:)
3433:t
3430:(
3427:y
3417:h
3408:t
3401:τ
3391:t
3387:τ
3382:h
3358:t
3354:d
3351:)
3344:t
3340:(
3337:x
3334:)
3327:t
3323:,
3320:t
3317:(
3314:h
3293:=
3286:t
3282:d
3279:)
3272:t
3268:(
3265:x
3262:)
3255:t
3251:,
3248:t
3245:(
3242:h
3237:t
3221:=
3218:)
3215:t
3212:(
3209:y
3180:1
3176:t
3167:2
3163:t
3159:,
3156:0
3153:=
3150:)
3145:1
3141:t
3137:,
3132:2
3128:t
3124:(
3121:h
3107:)
3105:1
3102:t
3098:2
3095:t
3093:(
3091:h
3076:)
3071:1
3067:t
3063:,
3058:2
3054:t
3050:(
3047:h
3044:=
3041:)
3036:2
3032:t
3028:=
3025:t
3022:(
3019:y
3009:)
3007:t
3005:(
3003:y
2994:)
2992:t
2976:)
2971:1
2967:t
2960:t
2957:(
2951:=
2948:)
2945:t
2942:(
2939:x
2929:)
2927:t
2925:(
2923:x
2918:.
2915:1
2912:t
2908:t
2898:2
2895:t
2891:t
2887:)
2885:1
2882:t
2878:2
2875:t
2873:(
2871:h
2824:)
2821:t
2818:4
2815:(
2802:+
2798:)
2795:t
2792:2
2789:(
2776:+
2772:)
2769:t
2766:3
2763:(
2750:=
2747:)
2744:t
2741:(
2738:y
2713:)
2710:t
2707:3
2704:(
2694:=
2691:)
2688:t
2685:(
2682:x
2662:)
2659:t
2656:(
2653:x
2649:)
2645:)
2642:t
2639:(
2629:+
2623:(
2620:=
2617:)
2614:t
2611:(
2608:y
2570:t
2566:d
2560:)
2557:t
2554:(
2551:x
2547:d
2539:k
2536:=
2533:)
2530:t
2527:(
2524:y
2500:|
2496:)
2493:t
2490:(
2487:x
2483:|
2479:=
2476:)
2473:t
2470:(
2467:y
2445:)
2442:t
2439:(
2436:x
2431:=
2428:)
2425:t
2422:(
2419:y
2399:)
2396:t
2393:(
2388:2
2384:x
2380:=
2377:)
2374:t
2371:(
2368:y
2347:]
2344:)
2341:t
2338:(
2335:x
2332:[
2322:=
2319:)
2316:t
2313:(
2310:y
2289:]
2286:)
2283:t
2280:(
2277:x
2274:[
2264:=
2261:)
2258:t
2255:(
2252:y
2230:0
2226:k
2222:+
2219:)
2216:t
2213:(
2210:x
2206:k
2203:=
2200:)
2197:t
2194:(
2191:y
2171:k
2168:=
2165:)
2162:t
2159:(
2156:y
2132:d
2128:)
2122:(
2119:x
2114:t
2097:k
2094:=
2091:)
2088:t
2085:(
2082:y
2059:t
2055:d
2049:)
2046:t
2043:(
2040:x
2036:d
2028:k
2025:=
2022:)
2019:t
2016:(
2013:y
1993:)
1990:t
1987:(
1984:x
1980:k
1977:=
1974:)
1971:t
1968:(
1965:y
1952:,
1950:)
1948:t
1946:(
1944:y
1940:t
1938:(
1936:x
1934:(
1932:H
1926:,
1922:t
1920:(
1918:y
1911:H
1896:,
1893:)
1890:t
1887:(
1884:x
1881:k
1878:+
1870:2
1866:t
1862:d
1857:)
1854:)
1851:t
1848:(
1845:x
1842:(
1837:2
1833:d
1826:m
1823:=
1820:)
1817:)
1814:t
1811:(
1808:x
1805:(
1802:H
1780:.
1777:x
1774:k
1768:=
1760:2
1756:t
1752:d
1747:)
1744:x
1741:(
1736:2
1732:d
1725:m
1674:)
1671:t
1668:(
1663:2
1659:y
1638:)
1635:t
1632:(
1627:1
1623:y
1602:)
1599:t
1596:(
1591:2
1587:x
1566:)
1563:t
1560:(
1555:1
1551:x
1530:)
1527:t
1524:(
1519:2
1513:y
1490:)
1487:t
1484:(
1479:1
1473:y
1450:)
1447:t
1444:(
1439:2
1433:x
1410:)
1407:t
1404:(
1399:1
1393:x
1324:)
1322:t
1320:(
1318:x
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