93:
4604:
36:
7039:
4231:
6592:
12599:
4599:{\displaystyle {\begin{aligned}\overbrace {\int _{-\infty }^{\infty }h(\tau )\,Ae^{s(t-\tau )}\,\mathrm {d} \tau } ^{{\mathcal {H}}f}&=\int _{-\infty }^{\infty }h(\tau )\,Ae^{st}e^{-s\tau }\,\mathrm {d} \tau \\&=Ae^{st}\int _{-\infty }^{\infty }h(\tau )\,e^{-s\tau }\,\mathrm {d} \tau \\&=\overbrace {\underbrace {Ae^{st}} _{\text{Input}}} ^{f}\overbrace {\underbrace {H(s)} _{\text{Scalar}}} ^{\lambda },\\\end{aligned}}}
12145:
7286:
3864:
7034:{\displaystyle {\begin{aligned}{\mathcal {A}}\{c_{1}x_{1}(t)+c_{2}x_{2}(t)\}&=\int _{t-a}^{t+a}(c_{1}x_{1}(\lambda )+c_{2}x_{2}(\lambda ))\,\mathrm {d} \lambda \\&=c_{1}\int _{t-a}^{t+a}x_{1}(\lambda )\,\mathrm {d} \lambda +c_{2}\int _{t-a}^{t+a}x_{2}(\lambda )\,\mathrm {d} \lambda \\&=c_{1}{\mathcal {A}}\{x_{1}(t)\}+c_{2}{\mathcal {A}}\{x_{2}(t)\},\end{aligned}}}
2806:
3512:
1155:
6061:
9591:
7044:
3634:
12852:
2983:
9880:
3302:
12893:. Two of the most important properties of a system are causality and stability. Non-causal (in time) systems can be defined and analyzed as above, but cannot be realized in real-time. Unstable systems can also be analyzed and built, but are only useful as part of a larger system whose overall transfer function
7609:
12594:{\displaystyle {\begin{aligned}{\mathcal {A}}\left\{c_{1}x_{1}+c_{2}x_{2}\right\}&=\sum _{k=n-a}^{n+a}\left(c_{1}x_{1}+c_{2}x_{2}\right)\\&=c_{1}\sum _{k=n-a}^{n+a}x_{1}+c_{2}\sum _{k=n-a}^{n+a}x_{2}\\&=c_{1}{\mathcal {A}}\left\{x_{1}\right\}+c_{2}{\mathcal {A}}\left\{x_{2}\right\},\end{aligned}}}
5595:
9178:
2602:
1187:
in the case of discrete-time systems). As a result of the properties of these transforms, the output of the system in the frequency domain is the product of the transfer function and the transform of the input. In other words, convolution in the time domain is equivalent to multiplication in the
7467:
1463:
Most LTI system concepts are similar between the continuous-time and discrete-time (linear shift-invariant) cases. In image processing, the time variable is replaced with two space variables, and the notion of time invariance is replaced by two-dimensional shift invariance. When analyzing
5871:
9348:
9001:
6587:
5847:
less than some value. Usually, this "start time" is set to zero, for convenience and without loss of generality, with the transform integral being taken from zero to infinity (the transform shown above with lower limit of integration of negative infinity is formally known as the
1675:
11803:
8166:
In many contexts, a discrete time (DT) system is really part of a larger continuous time (CT) system. For example, a digital recording system takes an analog sound, digitizes it, possibly processes the digital signals, and plays back an analog sound for people to listen to.
6297:
12604:
9712:
7472:
10728:
4706:
7281:{\displaystyle {\begin{aligned}{\mathcal {A}}\left\{x(t-\tau )\right\}&=\int _{t-a}^{t+a}x(\lambda -\tau )\,\mathrm {d} \lambda \\&=\int _{(t-\tau )-a}^{(t-\tau )+a}x(\xi )\,\mathrm {d} \xi \\&={\mathcal {A}}\{x\}(t-\tau ),\end{aligned}}}
2832:
13317:
12140:
5462:
3859:{\displaystyle {\begin{aligned}(x*h)(t)&=O_{t}\left\{\int _{-\infty }^{\infty }x(\tau )\cdot \delta (u-\tau )\,\mathrm {d} \tau ;\ u\right\}\\&=O_{t}\left\{x(u);\ u\right\}\\&\mathrel {\stackrel {\text{def}}{=}} y(t).\,\end{aligned}}}
6065:
One can use the system response directly to determine how any particular frequency component is handled by a system with that
Laplace transform. If we evaluate the system response (Laplace transform of the impulse response) at complex frequency
5860:. The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite signals via the
7620:
Some of the most important properties of a system are causality and stability. Causality is a necessity for a physical system whose independent variable is time, however this restriction is not present in other cases such as image processing.
8282:
1763:
7347:
10822:
6470:
8045:
11462:
2801:{\displaystyle O_{t}\left\{\int \limits _{-\infty }^{\infty }c_{\tau }\ x_{\tau }(u)\,\mathrm {d} \tau ;\ u\right\}=\int \limits _{-\infty }^{\infty }c_{\tau }\ \underbrace {y_{\tau }(t)} _{O_{t}\{x_{\tau }\}}\,\mathrm {d} \tau .}
9027:
3507:{\displaystyle {\begin{aligned}(x*h)(t)&=\int _{-\infty }^{\infty }x(\tau )\cdot h(t-\tau )\,\mathrm {d} \tau \\&=\int _{-\infty }^{\infty }x(\tau )\cdot O_{t}\{\delta (u-\tau );\ u\}\,\mathrm {d} \tau ,\,\end{aligned}}}
4222:
3209:
6479:
6379:
5867:
Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain, given signals for which the transforms exist
3088:
11009:
8803:
5855:
The
Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it cannot be directly applied to input and output signals that are not
2427:
9312:
8835:
5702:
8638:
4236:
3639:
9222:, characterizes the system completely. That is, for any input sequence, the output sequence can be calculated in terms of the input and the impulse response. To see how that is done, consider the identity:
1579:
11553:
11546:
8419:
8093:
is not BIBO stable, because the sinc function does not have a finite L norm. Thus, for some bounded input, the output of the ideal low-pass filter is unbounded. In particular, if the input is zero for
12028:
3307:
6056:{\displaystyle y(t)=(h*x)(t)\mathrel {\stackrel {\text{def}}{=}} \int _{-\infty }^{\infty }h(t-\tau )x(\tau )\,\mathrm {d} \tau \mathrel {\stackrel {\text{def}}{=}} {\mathcal {L}}^{-1}\{H(s)X(s)\}.}
10081:
9996:
9586:{\displaystyle {\begin{aligned}y=O_{n}\{x\}&=O_{n}\left\{\sum _{k=-\infty }^{\infty }x\cdot \delta ;\ m\right\}\\&=\sum _{k=-\infty }^{\infty }x\cdot O_{n}\{\delta ;\ m\},\,\end{aligned}}}
10569:
10559:
4611:
12609:
12150:
11206:
9717:
9353:
9032:
7049:
6597:
2837:
6125:
5765:
12911:
A discrete-time LTI system is causal if the current value of the output depends on only the current value and past values of the input. A necessary and sufficient condition for causality is
1425:
are a sum of complex exponentials with complex-conjugate frequencies, if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different
13209:
12035:
7845:
7792:
3280:
10415:
5413:
11298:
Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain. That is,
7684:
3622:
13111:
13061:
10206:
5005:
3954:
8210:
5309:
11291:
Like the one-sided
Laplace transform, the Z transform is usually used in the context of one-sided signals, i.e. signals that are zero for t<0. The discrete-time Fourier transform
11072:
5656:
5092:
8296:
which removes frequencies above the "folding frequency" 1/(2T); this guarantees that no information in the filtered signal will be lost. Without filtering, any frequency component
2978:{\displaystyle {\begin{aligned}O_{t}\{x(u-\tau );\ u\}&\mathrel {\stackrel {\quad }{=}} y(t-\tau )\\&\mathrel {\stackrel {\text{def}}{=}} O_{t-\tau }\{x\}.\,\end{aligned}}}
12962:
1855:
12847:{\displaystyle {\begin{aligned}{\mathcal {A}}\left\{x\right\}&=\sum _{k=n-a}^{n+a}x\\&=\sum _{k'=(n-m)-a}^{(n-m)+a}x\\&={\mathcal {A}}\left\{x\right\},\end{aligned}}}
11466:
Just as with the
Laplace transform transfer function in continuous-time system analysis, the Z transform makes it easier to analyze systems and gain insight into their behavior.
10733:
6392:
4747:
4055:
4938:
11939:
10356:
10316:
5446:
1133:
11274:
7947:
3562:
2160:
740:
691:
11301:
11135:
7310:
11044:
10288:
9875:{\displaystyle {\begin{aligned}O_{n}\{\delta ;\ m\}&\mathrel {\stackrel {\quad }{=}} O_{n-k}\{\delta ;\ m\}\\&\mathrel {\stackrel {\text{def}}{=}} h.\end{aligned}}}
9698:
5628:
5125:
1573:
4115:
5826:
5202:
1396:
1327:
1240:
253:; these terms are briefly defined in the overview below. These properties apply (exactly or approximately) to many important physical systems, in which case the response
8714:
8494:
8080:
3132:
827:
642:
608:
8364:
7604:{\displaystyle \Pi (t)\mathrel {\stackrel {\text{def}}{=}} {\begin{cases}1&{\text{if }}|t|<{\frac {1}{2}},\\0&{\text{if }}|t|>{\frac {1}{2}}.\end{cases}}}
4149:
2305:
10232:
3978:
10460:
9641:
7342:
4842:
4780:
4021:
2195:
8551:
8148:
8118:
1945:
1685:
973:
938:
13367:
11099:
10856:
10123:
9340:
9220:
8522:
8447:
2014:
1913:
1354:
1267:
574:
542:
11235:
10889:
8197:
7938:
7909:
7880:
7715:
5794:
4879:
4809:
4144:
1072:
1043:
1010:
903:
874:
792:
490:
461:
10905:
5335:
2334:
8723:
5590:{\displaystyle H(s)\mathrel {\stackrel {\text{def}}{=}} {\mathcal {L}}\{h(t)\}\mathrel {\stackrel {\text{def}}{=}} \int _{0}^{\infty }h(t)e^{-st}\,\mathrm {d} t}
3892:
2454:
2360:
2261:
2235:
2034:
1436:
LTI system theory is good at describing many important systems. Most LTI systems are considered "easy" to analyze, at least compared to the time-varying and/or
13204:
13175:
13146:
12991:
12891:
10489:
9918:
8825:
8571:
5222:
2503:
2063:
1884:
1807:
1416:
1287:
760:
510:
9173:{\displaystyle {\begin{aligned}O_{n}\{x;\ m\}&\mathrel {\stackrel {\quad }{=}} y\\&\mathrel {\stackrel {\text{def}}{=}} O_{n-k}\{x\}.\,\end{aligned}}}
2586:
2563:
2543:
2523:
2474:
2215:
2103:
2083:
1985:
1965:
12993:
is the impulse response. It is not possible in general to determine causality from the Z transform, because the inverse transform is not unique. When a
8308:
to a different frequency (thus distorting the original signal), since a DT signal can only support frequency components lower than the folding frequency.
2365:
328:), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any
317:
and ∗ represents convolution (not to be confused with multiplication). What's more, there are systematic methods for solving any such system (determining
9225:
5597:
is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids (i.e., exponential functions of the form
3138:
8576:
7462:{\displaystyle {\mathcal {A}}\left\{x(t)\right\}=\int _{-\infty }^{\infty }\Pi \left({\frac {\lambda -t}{2a}}\right)x(\lambda )\,\mathrm {d} \lambda ,}
13527:
3008:
11953:
7635:
A system is causal if the output depends only on present and past, but not future inputs. A necessary and sufficient condition for causality is
6304:
11013:
is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids; i.e. exponentials of the form
1433:, but always with the same frequency upon reaching steady-state. LTI systems cannot produce frequency components that are not in the input.
13617:
8996:{\displaystyle O_{n}\left\{\sum _{k=-\infty }^{\infty }c_{k}\cdot x_{k};\ m\right\}=\sum _{k=-\infty }^{\infty }c_{k}\cdot O_{n}\{x_{k}\}.}
10494:
979:
The fundamental result in LTI system theory is that any LTI system can be characterized entirely by a single function called the system's
6582:{\displaystyle {\mathcal {A}}\left\{x(t)\right\}\mathrel {\stackrel {\text{def}}{=}} \int _{t-a}^{t+a}x(\lambda )\,\mathrm {d} \lambda .}
10141:
changes, the weighting function emphasizes different parts of the input function. Equivalently, the system's response to an impulse at
5661:
1078:
system. Similarly, a discrete-time linear time-invariant (or, more generally, "shift-invariant") system is defined as one operating in
1670:{\displaystyle \mathrel {\stackrel {\mathrm {def} }{=}} \int \limits _{-\infty }^{\infty }x(t-\tau )\cdot h(\tau )\,\mathrm {d} \tau }
11798:{\displaystyle D\left(c_{1}\cdot x_{1}+c_{2}\cdot x_{2}\right)=c_{1}\cdot x_{1}+c_{2}\cdot x_{2}=c_{1}\cdot Dx_{1}+c_{2}\cdot Dx_{2}}
7799:
7746:
355:, where the systems have spatial dimensions instead of, or in addition to, a temporal dimension. These systems may be referred to as
5340:
3902:, combined in the same way. And the time-invariance property allows that combination to be represented by the convolution integral.
11479:
8369:
7638:
13068:
13018:
10172:
3920:
8553:
In general, every element of the output can depend on every element of the input. Representing the transformation operator by
376:
92:
13629:
13544:
13511:
10006:
3898:. The system's linearity property allows the system's response to be represented by the corresponding continuum of impulse
13688:
9924:
6292:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left(c_{1}x_{1}(t)+c_{2}x_{2}(t)\right)=c_{1}x'_{1}(t)+c_{2}x'_{2}(t)}
1460:(RLC circuits). Ideal spring–mass–damper systems are also LTI systems, and are mathematically equivalent to RLC circuits.
12914:
11144:
5456:
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The one-sided
6385:
When the
Laplace transform of the derivative is taken, it transforms to a simple multiplication by the Laplace variable
5843:
The
Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of
5710:
13736:
13716:
13726:
13399:
3219:
975:. Hence, the system is time invariant because the output does not depend on the particular time the input is applied.
79:
57:
10361:
2362:. And in general, every value of the output can depend on every value of the input. This concept is represented by:
103:
and time invariance for a deterministic continuous-time single-input single-output system. The system satisfies the
50:
13721:
11138:
4943:
3567:
13682:
12864:
The input-output characteristics of discrete-time LTI system are completely described by its impulse response
10899:
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The
10169:
is a function for which the output of the operator is the same function, scaled by some constant. In symbols,
408:
6098:. The relative phase shift between the output and input for that frequency component is likewise given by arg(
5230:
13741:
11049:
10723:{\displaystyle \sum _{m=-\infty }^{\infty }h\,z^{(n-m)}=z^{n}\sum _{m=-\infty }^{\infty }h\,z^{-m}=z^{n}H(z)}
5633:
4701:{\displaystyle H(s)\mathrel {\stackrel {\text{def}}{=}} \int _{-\infty }^{\infty }h(t)e^{-st}\,\mathrm {d} t}
7743:(BIBO stable) if, for every bounded input, the output is finite. Mathematically, if every input satisfying
5012:
13691: – Gives an intuitive explanation of the source of phase shift in two common electrical LTI systems.
13312:{\displaystyle \|h\|_{1}\mathrel {\stackrel {\text{def}}{=}} \sum _{n=-\infty }^{\infty }|h|<\infty .}
8204:
7718:
5861:
5849:
4718:
4026:
1816:
17:
13694:
12135:{\displaystyle {\mathcal {A}}\left\{x\right\}\mathrel {\stackrel {\text{def}}{=}} \sum _{k=n-a}^{n+a}x.}
4895:
3894:, can be represented by a continuum of time-shifted impulse functions, combined "linearly", as shown at
13731:
11810:
10333:
10293:
5418:
1085:
3917:
is a function for which the output of the operator is a scaled version of the same function. That is,
11240:
6473:
That the derivative has such a simple
Laplace transform partly explains the utility of the transform.
11104:
8170:
In practical systems, DT signals obtained are usually uniformly sampled versions of CT signals. If
7509:
7291:
3525:
2115:
11016:
10247:
9646:
8666:
5600:
5097:
1522:
44:
8277:{\displaystyle x_{n}\mathrel {\stackrel {\text{def}}{=}} x(nT)\qquad \forall \,n\in \mathbb {Z} ,}
4072:
5799:
5132:
1359:
1292:
1205:
830:
104:
100:
8672:
8452:
8056:
3102:
696:
647:
13641:"Role of anticausal inverses in multirate filter banks — Part I: system theoretic fundamentals"
13640:
13575:"Role of anticausal inverses in multirate filter banks — Part I: system theoretic fundamentals"
13574:
8319:
1758:{\displaystyle =\int \limits _{-\infty }^{\infty }x(\tau )\cdot h(t-\tau )\,\mathrm {d} \tau ,}
392:
61:
10217:
8646:, the output sequence is just constant, and the system is uninteresting. (Thus the subscript,
4889:
It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems.
3963:
2266:
13322:
12994:
10423:
9604:
8050:
7318:
4814:
4752:
3993:
1469:
1441:
250:
10817:{\displaystyle H(z)\mathrel {\stackrel {\text{def}}{=}} \sum _{m=-\infty }^{\infty }hz^{-m}}
8527:
8127:
8097:
6465:{\displaystyle {\mathcal {L}}\left\{{\frac {\mathrm {d} }{\mathrm {d} t}}x(t)\right\}=sX(s)}
2165:
1479:
A linear system that is not time-invariant can be solved using other approaches such as the
943:
908:
13655:
13589:
13336:
11077:
10834:
10242:
10101:
9319:
9199:
8501:
8426:
8293:
3988:
1918:
1810:
1473:
1332:
1245:
1199:
798:
613:
579:
547:
515:
400:
11211:
10865:
8173:
7914:
7885:
7856:
7691:
5770:
4855:
4785:
4120:
1990:
1889:
1151:
and the convolution, in discrete time, uses a discrete summation rather than an integral.
1048:
1019:
986:
879:
850:
768:
466:
437:
8:
13480:
13330:
10862:
of an LTI system because the system response is the same as the input times the constant
8158:
Almost everything in continuous-time systems has a counterpart in discrete-time systems.
5314:
2112:
To understand why the convolution produces the output of an LTI system, let the notation
1480:
372:
13659:
13593:
8040:{\displaystyle \|h(t)\|_{1}=\int _{-\infty }^{\infty }|h(t)|\,\mathrm {d} t<\infty .}
7717:
is the impulse response. It is not possible in general to determine causality from the
2310:
246:
13521:
13465:
13180:
13151:
13122:
12967:
12867:
11457:{\displaystyle y=(h*x)=\sum _{m=-\infty }^{\infty }hx={\mathcal {Z}}^{-1}\{H(z)X(z)\}.}
10465:
10087:
9894:
8810:
8556:
8289:
5207:
3871:
2432:
2339:
2240:
2220:
2019:
1987:
changes, the weighting function emphasizes different parts of the input function. When
1769:
1445:
1401:
1272:
745:
495:
329:
13177:
are finite), then the system is stable. A necessary and sufficient condition is that
2479:
2039:
1860:
1783:
13625:
13540:
13507:
13485:
13395:
13015:(BIBO stable) if, for every bounded input, the output is finite. Mathematically, if
8301:
8121:
7722:
5857:
5705:
5457:
1180:
1176:
380:
13663:
13597:
13470:
13460:
10125:
can therefore be interpreted as proportional to a weighted average of the function
2571:
2548:
2528:
2508:
2459:
2200:
2088:
2068:
1970:
1950:
1496:
The behavior of a linear, continuous-time, time-invariant system with input signal
1171:
980:
404:
396:
352:
345:
314:
245:
that produces an output signal from any input signal subject to the constraints of
13700:
4715:
So the system's response is a scaled version of the input. In particular, for any
4217:{\displaystyle \int _{-\infty }^{\infty }h(t-\tau )Ae^{s\tau }\,\mathrm {d} \tau }
3204:{\displaystyle \mathrel {\stackrel {\text{def}}{=}} O_{t-\tau }\{\delta (u);\ u\}}
13555:
13475:
8200:
8150:, then the output will be unbounded for all times other than the zero crossings.
8086:
7313:
1075:
364:
226:
13006:
11292:
10327:
7734:
4066:
1196:
388:
13685: – Short primer on the mathematical analysis of (electrical) LTI systems.
13497:
3083:{\textstyle h(t)\mathrel {\stackrel {\text{def}}{=}} O_{t}\{\delta (u);\ u\}.}
13710:
13116:
12906:
10859:
10319:
10166:
10154:
8090:
7850:
7630:
4845:
4058:
3914:
2106:
1430:
1192:
1079:
384:
360:
96:
11004:{\displaystyle H(z)={\mathcal {Z}}\{h\}=\sum _{n=-\infty }^{\infty }hz^{-n}}
7911:), then the system is stable. A necessary and sufficient condition is that
8798:{\displaystyle h\mathrel {\stackrel {\text{def}}{=}} O_{n}\{\delta ;\ m\}.}
6374:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}x(t-\tau )=x'(t-\tau )}
1444:
with constant coefficients is an LTI system. Examples of such systems are
13697:. An encapsulated course on LTI system theory. Adequate for self teaching.
843:
seconds from now, the output will be identical except for a time delay of
13326:
13115:(that is, if bounded input implies bounded output, in the sense that the
10900:
10564:
4225:
2565:, the output function is just constant, and the system is uninteresting.
1465:
1184:
1013:
276:
10145:=0 is a "time" reversed copy of the unshifted weighting function. When
8292:. Before sampling, the input signal is normally run through a so-called
4069:
operator. A simple proof illustrates this concept. Suppose the input is
13499:
10235:
6117:
4849:
3981:
359:
to give the terminology the most general reach. In the case of generic
13667:
13601:
5448:
as input will give a complex exponential of same frequency as output.
3905:
The mathematical operations above have a simple graphical simulation.
1886:
is therefore proportional to a weighted average of the input function
8161:
1457:
1437:
1426:
1422:
1148:
337:
11945:
The Z transform of the delay operator is a simple multiplication by
10563:
which is equivalent to the following by the commutative property of
2422:{\displaystyle y(t)\mathrel {\stackrel {\text{def}}{=}} O_{t}\{x\},}
9307:{\displaystyle x\equiv \sum _{k=-\infty }^{\infty }x\cdot \delta ,}
8305:
7941:
1453:
1449:
341:
333:
10098:
which is the familiar discrete convolution formula. The operator
7721:. However, when working in the time domain, one normally uses the
5697:{\displaystyle j\mathrel {\stackrel {\text{def}}{=}} {\sqrt {-1}}}
8633:{\displaystyle y\mathrel {\stackrel {\text{def}}{=}} O_{n}\{x\}.}
1289:, the output will be some complex constant times the input, say
10323:
4062:
242:
2307:. Then a continuous-time system transforms an input function,
1202:. This is, if the input to a system is the complex waveform
11541:{\displaystyle D\{x\}\mathrel {\stackrel {\text{def}}{=}} x}
8414:{\displaystyle \{x;{\text{ for all integer values of }}m\}.}
492:, both being regarded as functions, is a linear mapping: If
12023:{\displaystyle {\mathcal {Z}}\left\{Dx\right\}=z^{-1}X(z).}
7597:
371:
is the corresponding term. LTI system theory is an area of
11476:
A simple example of an LTI operator is the delay operator
5864:
even when
Fourier transforms of the signals do not exist.
5767:
gives the eigenvalues for pure complex sinusoids. Both of
839:
means that whether we apply an input to the system now or
27:
Mathematical model which is both linear and time-invariant
10894:
3299:
Substituting this result into the convolution integral:
12032:
Another simple LTI operator is the averaging operator
10076:{\displaystyle =\sum _{k=-\infty }^{\infty }x\cdot h,}
3874:
3570:
3528:
3011:
2574:
2551:
2531:
2511:
2482:
2462:
2435:
2342:
2313:
2269:
2243:
2223:
2203:
2168:
2118:
2091:
2071:
2042:
2022:
1993:
1973:
1953:
1921:
1892:
1863:
1819:
1786:
1154:
13339:
13212:
13183:
13154:
13125:
13071:
13021:
12970:
12917:
12870:
12607:
12148:
12038:
11956:
11813:
11556:
11482:
11304:
11243:
11214:
11147:
11107:
11080:
11052:
11019:
10908:
10868:
10837:
10736:
10572:
10497:
10468:
10426:
10364:
10336:
10296:
10250:
10220:
10175:
10104:
10009:
9991:{\displaystyle =\sum _{k=-\infty }^{\infty }x\cdot h}
9927:
9897:
9715:
9649:
9607:
9351:
9322:
9228:
9202:
9030:
8838:
8813:
8726:
8675:
8579:
8559:
8530:
8504:
8455:
8429:
8372:
8322:
8311:
8213:
8176:
8130:
8100:
8059:
7950:
7917:
7888:
7859:
7802:
7749:
7694:
7641:
7475:
7350:
7321:
7294:
7047:
6595:
6482:
6476:
Another simple LTI operator is an averaging operator
6395:
6307:
6128:
5874:
5802:
5773:
5713:
5664:
5636:
5603:
5465:
5421:
5343:
5317:
5233:
5210:
5135:
5100:
5015:
4946:
4898:
4858:
4817:
4788:
4755:
4721:
4614:
4234:
4152:
4123:
4075:
4029:
3996:
3966:
3923:
3637:
3305:
3222:
3141:
3105:
2835:
2605:
2368:
1688:
1582:
1525:
1404:
1362:
1335:
1295:
1275:
1248:
1208:
1088:
1051:
1022:
989:
946:
911:
882:
853:
801:
771:
748:
699:
650:
616:
582:
550:
518:
498:
469:
440:
10554:{\displaystyle \sum _{m=-\infty }^{\infty }h\,z^{m}}
351:
Linear time-invariant system theory is also used in
11201:{\displaystyle H(e^{j\omega })={\mathcal {F}}\{h\}}
10160:
8642:Note that unless the transform itself changes with
1491:
829:. The latter condition is often referred to as the
13361:
13311:
13198:
13169:
13140:
13105:
13055:
12985:
12956:
12885:
12846:
12593:
12134:
12022:
11933:
11797:
11540:
11456:
11268:
11229:
11208:gives the eigenvalues of pure sinusoids. Both of
11200:
11129:
11093:
11066:
11038:
11003:
10883:
10850:
10816:
10722:
10553:
10483:
10454:
10409:
10350:
10310:
10282:
10226:
10200:
10117:
10075:
9990:
9912:
9874:
9692:
9635:
9585:
9334:
9306:
9214:
9172:
8995:
8819:
8797:
8708:
8632:
8565:
8545:
8516:
8488:
8441:
8413:
8358:
8276:
8191:
8162:Discrete-time systems from continuous-time systems
8142:
8112:
8074:
8039:
7932:
7903:
7874:
7839:
7786:
7709:
7678:
7603:
7461:
7336:
7304:
7280:
7033:
6581:
6464:
6373:
6291:
6055:
5820:
5788:
5760:{\displaystyle H(j\omega )={\mathcal {F}}\{h(t)\}}
5759:
5696:
5650:
5622:
5589:
5440:
5407:
5329:
5303:
5216:
5196:
5119:
5086:
4999:
4932:
4873:
4836:
4803:
4774:
4741:
4700:
4598:
4216:
4138:
4109:
4049:
4015:
3972:
3948:
3886:
3858:
3616:
3556:
3506:
3274:
3203:
3126:
3082:
2977:
2800:
2580:
2557:
2537:
2517:
2497:
2468:
2448:
2421:
2354:
2328:
2299:
2255:
2229:
2209:
2189:
2154:
2097:
2077:
2057:
2028:
2008:
1979:
1959:
1939:
1907:
1878:
1849:
1801:
1757:
1669:
1567:
1440:case. Any system that can be modeled as a linear
1410:
1390:
1348:
1321:
1281:
1261:
1234:
1127:
1066:
1037:
1004:
967:
932:
897:
868:
821:
786:
754:
734:
685:
636:
602:
568:
536:
504:
484:
455:
407:, and many other technical areas where systems of
10462:. The output of the system with impulse response
5451:
4117:. The output of the system with impulse response
3908:
1195:, and the basis functions of the transforms, are
13708:
13638:
13572:
12997:is specified, then causality can be determined.
12859:
8498:A discrete system transforms an input sequence,
4749:, the system output is the product of the input
9342:in terms of a sum of weighted delta functions.
7840:{\displaystyle \ \|y(t)\|_{\infty }<\infty }
7787:{\displaystyle \ \|x(t)\|_{\infty }<\infty }
3275:{\displaystyle =O_{t}\{\delta (u-\tau );\ u\}.}
13534:
10410:{\displaystyle z=e^{sT},\ z,s\in \mathbb {C} }
7615:
5408:{\displaystyle H(\tau )=e^{i\omega \tau }H(0)}
434:means that the relationship between the input
419:The defining properties of any LTI system are
1169:LTI systems can also be characterized in the
847:seconds. That is, if the output due to input
13695:JHU 520.214 Signals and Systems course notes
13526:: CS1 maint: multiple names: authors list (
13229:
13213:
13088:
13072:
13038:
13022:
11913:
11907:
11838:
11817:
11501:
11486:
11448:
11421:
11295:may be used for analyzing periodic signals.
11195:
11180:
10946:
10931:
10417:. A simple proof illustrates this concept.
9821:
9797:
9760:
9730:
9572:
9542:
9387:
9381:
9329:
9323:
9209:
9203:
9159:
9153:
9075:
9045:
8987:
8974:
8789:
8765:
8624:
8618:
8537:
8531:
8511:
8505:
8480:
8456:
8436:
8430:
8405:
8373:
8353:
8323:
7967:
7951:
7822:
7806:
7769:
7753:
7253:
7247:
7021:
6999:
6976:
6954:
6671:
6607:
6047:
6020:
5754:
5739:
5516:
5501:
3881:
3875:
3484:
3454:
3266:
3236:
3198:
3174:
3074:
3050:
2964:
2958:
2880:
2850:
2781:
2768:
2413:
2407:
2349:
2343:
2320:
2314:
2294:
2270:
2250:
2244:
2149:
2119:
1512:) is described by the convolution integral:
13553:
7882:implies a finite maximum absolute value of
7679:{\displaystyle h(t)=0\quad \forall t<0,}
6116:A simple example of an LTI operator is the
3617:{\textstyle x_{\tau }(u)=\delta (u-\tau ).}
2545:. Unless the transform itself changes with
13683:ECE 209: Review of Circuits as LTI Systems
13639:Vaidyanathan, P. P.; Chen, T. (May 1995).
13573:Vaidyanathan, P. P.; Chen, T. (May 1995).
13383:
13106:{\displaystyle \|y\|_{\infty }<\infty }
13056:{\displaystyle \|x\|_{\infty }<\infty }
10201:{\displaystyle {\mathcal {H}}f=\lambda f,}
6094:)| which is the system gain for frequency
5094:by linearity with respect to the constant
5000:{\displaystyle v_{a}(t)=e^{i\omega (t+a)}}
3949:{\displaystyle {\mathcal {H}}f=\lambda f,}
1486:
11060:
10681:
10609:
10540:
10403:
10344:
10304:
9578:
9165:
8267:
8259:
8019:
7447:
7312:can be written as a convolution with the
7221:
7139:
6918:
6850:
6775:
6567:
5978:
5644:
5578:
4735:
4689:
4482:
4465:
4400:
4367:
4302:
4273:
4205:
4043:
3851:
3739:
3499:
3487:
3392:
2970:
2786:
2671:
1743:
1658:
80:Learn how and when to remove this message
9196:In such a system, the impulse response,
9019:And the time-invariance requirement is:
8654:depends most heavily on the elements of
8153:
5304:{\displaystyle H(t+a)=e^{i\omega a}H(t)}
4848:of an LTI system, and the corresponding
3516:which has the form of the right side of
2824:And the time-invariance requirement is:
2456:is the transformation operator for time
1472:systems, it is often useful to consider
1153:
512:is a constant then the system output to
91:
43:This article includes a list of general
13554:Crutchfield, Steve (October 12, 2010),
11067:{\displaystyle \omega \in \mathbb {R} }
5651:{\displaystyle \omega \in \mathbb {R} }
14:
13709:
13701:LTI system example: RC low-pass filter
13389:
10895:Z and discrete-time Fourier transforms
5087:{\displaystyle H(t)=e^{i\omega a}H(t)}
4224:which, by the commutative property of
2505:depends most heavily on the values of
1398:is the transfer function at frequency
610:is a further input with system output
377:electrical circuit analysis and design
13622:A Course in Digital Signal Processing
13616:
8399: for all integer values of
1183:of the system's impulse response (or
107:and is time-invariant if and only if
13013:bounded input, bounded output stable
12957:{\displaystyle h=0\ \forall n<0,}
9021:
8829:
7741:bounded-input, bounded-output stable
7041:it is linear. Additionally, because
2826:
2596:
1850:{\textstyle x(\tau )=\delta (\tau )}
264:of the system to an arbitrary input
29:
11941: (i.e., it is time invariant)
10824:is dependent only on the parameter
6381: (i.e., it is time invariant)
4742:{\displaystyle A,s\in \mathbb {C} }
4708:is dependent only on the parameter
4050:{\displaystyle A,s\in \mathbb {C} }
3001:In this notation, we can write the
1045:with the system's impulse response
742:, this applying for all choices of
24:
13609:
13498:Phillips, C.L., Parr, J.M., &
13303:
13273:
13268:
13206:, the impulse response, satisfies
13100:
13092:
13050:
13042:
12939:
12806:
12614:
12550:
12501:
12155:
12142:Because of the linearity of sums,
12041:
11959:
11407:
11366:
11361:
11175:
10971:
10966:
10926:
10784:
10779:
10664:
10659:
10592:
10587:
10517:
10512:
10178:
10032:
10027:
9950:
9945:
9512:
9507:
9431:
9426:
9263:
9258:
8946:
8941:
8873:
8868:
8256:
8207:will transform it to a DT signal:
8031:
8021:
7992:
7987:
7834:
7826:
7781:
7773:
7661:
7476:
7449:
7401:
7396:
7391:
7353:
7322:
7297:
7242:
7223:
7141:
7054:
6994:
6949:
6920:
6852:
6777:
6602:
6569:
6485:
6417:
6411:
6398:
6317:
6311:
6138:
6132:
6006:
5980:
5943:
5938:
5734:
5580:
5545:
5496:
4933:{\displaystyle v(t)=e^{i\omega t}}
4691:
4656:
4651:
4484:
4448:
4443:
4402:
4350:
4345:
4320:
4304:
4256:
4251:
4207:
4166:
4161:
3926:
3741:
3701:
3696:
3489:
3424:
3419:
3394:
3354:
3349:
2788:
2710:
2705:
2673:
2634:
2629:
1745:
1705:
1700:
1660:
1620:
1615:
1599:
1596:
1593:
49:it lacks sufficient corresponding
25:
13753:
13676:
11934:{\displaystyle D\{x\}=x=x=D\{x\}}
10351:{\displaystyle T\in \mathbb {R} }
10311:{\displaystyle n\in \mathbb {Z} }
8089:with impulse response equal to a
6589:By the linearity of integration,
5441:{\displaystyle e^{i\omega \tau }}
1128:{\displaystyle y_{i}=x_{i}*h_{i}}
644:then the output of the system to
375:which has direct applications in
229:, among other fields of study, a
11074:. These can also be written as
8312:Impulse response and convolution
8053:must contain the imaginary axis
5415:i.e. that a complex exponential
3868:In summary, the input function,
1492:Impulse response and convolution
34:
13703:. Amplitude and phase response.
13689:ECE 209: Sources of Phase Shift
13504:Signals, systems and Transforms
11269:{\displaystyle H(e^{j\omega })}
11139:discrete-time Fourier transform
9776:
9091:
8255:
8120:and equal to a sinusoid at the
7660:
7288:it is time invariant. In fact,
4884:
3631:then allows this continuation:
3557:{\textstyle c_{\tau }=x(\tau )}
2896:
2237:. And let the shorter notation
2155:{\textstyle \{x(u-\tau );\ u\}}
2105:, and the system is said to be
1809:is the system's response to an
1329:for some new complex amplitude
905:, then the output due to input
409:ordinary differential equations
13539:. Princeton university press.
13441:
13429:
13417:
13408:
13349:
13341:
13296:
13292:
13286:
13279:
13225:
13219:
13193:
13187:
13164:
13158:
13135:
13129:
13084:
13078:
13034:
13028:
13000:
12980:
12974:
12927:
12921:
12900:
12880:
12874:
12834:
12822:
12791:
12780:
12766:
12754:
12743:
12731:
12702:
12690:
12639:
12627:
12601:and so it is linear. Because,
12576:
12570:
12527:
12521:
12476:
12470:
12411:
12405:
12334:
12328:
12302:
12296:
12223:
12217:
12191:
12185:
12126:
12120:
12060:
12054:
12014:
12008:
11981:
11975:
11928:
11916:
11898:
11889:
11877:
11874:
11865:
11847:
11835:
11823:
11792:
11786:
11754:
11748:
11716:
11704:
11675:
11663:
11629:
11623:
11594:
11588:
11535:
11523:
11498:
11492:
11445:
11439:
11433:
11427:
11398:
11392:
11386:
11374:
11341:
11335:
11332:
11320:
11314:
11308:
11263:
11247:
11224:
11218:
11192:
11186:
11167:
11151:
11130:{\displaystyle z=e^{j\omega }}
10985:
10979:
10943:
10937:
10918:
10912:
10878:
10872:
10798:
10792:
10746:
10740:
10717:
10711:
10678:
10672:
10627:
10615:
10606:
10600:
10537:
10525:
10478:
10472:
10436:
10430:
10358:is the sampling interval, and
10161:Exponentials as eigenfunctions
10084:
10067:
10061:
10052:
10040:
9985:
9973:
9964:
9958:
9907:
9901:
9862:
9850:
9809:
9803:
9748:
9736:
9705:
9687:
9675:
9666:
9660:
9630:
9624:
9597:
9560:
9548:
9526:
9520:
9466:
9454:
9445:
9439:
9365:
9359:
9298:
9286:
9277:
9271:
9238:
9232:
9186:
9111:
9099:
9063:
9051:
9009:
8907:
8901:
8777:
8771:
8736:
8730:
8700:
8694:
8685:
8679:
8589:
8583:
8468:
8462:
8391:
8379:
8341:
8329:
8252:
8243:
8186:
8180:
8015:
8011:
8005:
7998:
7963:
7957:
7940:, the impulse response, is in
7927:
7921:
7898:
7892:
7869:
7863:
7818:
7812:
7796:leads to an output satisfying
7765:
7759:
7704:
7698:
7651:
7645:
7574:
7566:
7531:
7523:
7485:
7479:
7444:
7438:
7372:
7366:
7331:
7325:
7305:{\displaystyle {\mathcal {A}}}
7268:
7256:
7218:
7212:
7198:
7186:
7175:
7163:
7136:
7124:
7079:
7067:
7018:
7012:
6973:
6967:
6915:
6909:
6847:
6841:
6772:
6769:
6763:
6737:
6731:
6708:
6668:
6662:
6636:
6630:
6564:
6558:
6504:
6498:
6459:
6453:
6436:
6430:
6368:
6356:
6342:
6330:
6286:
6280:
6251:
6245:
6211:
6205:
6179:
6173:
6044:
6038:
6032:
6026:
5975:
5969:
5963:
5951:
5911:
5905:
5902:
5890:
5884:
5878:
5815:
5806:
5783:
5777:
5751:
5745:
5726:
5717:
5559:
5553:
5513:
5507:
5475:
5469:
5452:Fourier and Laplace transforms
5402:
5396:
5393:
5387:
5362:
5356:
5353:
5347:
5298:
5292:
5289:
5283:
5258:
5246:
5243:
5237:
5191:
5179:
5176:
5170:
5161:
5155:
5152:
5139:
5081:
5075:
5072:
5066:
5041:
5035:
5032:
5019:
5007:a time-shifted version of it.
4992:
4980:
4963:
4957:
4908:
4902:
4868:
4862:
4798:
4792:
4670:
4664:
4624:
4618:
4563:
4557:
4462:
4456:
4364:
4358:
4297:
4285:
4270:
4264:
4186:
4174:
4133:
4127:
4085:
4079:
3909:Exponentials as eigenfunctions
3895:
3845:
3839:
3796:
3790:
3736:
3724:
3715:
3709:
3663:
3657:
3654:
3642:
3627:
3608:
3596:
3587:
3581:
3551:
3545:
3518:
3472:
3460:
3438:
3432:
3389:
3377:
3368:
3362:
3331:
3325:
3322:
3310:
3287:
3283:
3254:
3242:
3186:
3180:
3121:
3109:
3062:
3056:
3021:
3015:
2991:
2916:
2904:
2868:
2856:
2814:
2748:
2742:
2668:
2662:
2590:
2492:
2486:
2378:
2372:
2282:
2276:
2184:
2172:
2137:
2125:
2052:
2046:
2003:
1997:
1934:
1925:
1902:
1896:
1873:
1867:
1844:
1838:
1829:
1823:
1796:
1790:
1766:
1740:
1728:
1719:
1713:
1655:
1649:
1640:
1628:
1562:
1556:
1553:
1541:
1535:
1529:
1061:
1055:
1032:
1026:
999:
993:
962:
950:
927:
915:
892:
886:
863:
857:
816:
810:
781:
775:
729:
723:
709:
703:
680:
674:
660:
654:
631:
625:
597:
591:
563:
557:
531:
525:
479:
473:
450:
444:
13:
1:
13491:
13321:In the frequency domain, the
11469:
11039:{\displaystyle e^{j\omega n}}
10283:{\displaystyle z^{n}=e^{sTn}}
9693:{\displaystyle x_{k}=\delta }
8423:And let the shorter notation
8049:In the frequency domain, the
5623:{\displaystyle e^{j\omega t}}
5120:{\displaystyle e^{i\omega a}}
4940:some complex exponential and
1568:{\displaystyle y(t)=(x*h)(t)}
13394:. Springer. pp. 27–28.
10129:. The weighting function is
8665:For the special case of the
7728:
7624:
4110:{\displaystyle x(t)=Ae^{st}}
1915:. The weighting function is
983:. The output of the system
357:linear translation-invariant
275:can be found directly using
7:
13454:
12860:Important system properties
11805: (i.e., it is linear)
10153:, the system is said to be
10133:, simply shifted by amount
8716:the output sequence is the
8205:analog-to-digital converter
7723:one-sided Laplace transform
7719:two-sided Laplace transform
7616:Important system properties
6299: (i.e., it is linear)
6109:
5850:bilateral Laplace transform
5821:{\displaystyle H(j\omega )}
5197:{\displaystyle H(t)=H(t+a)}
1947:, simply shifted by amount
1391:{\displaystyle B_{s}/A_{s}}
1322:{\displaystyle B_{s}e^{st}}
1242:for some complex amplitude
1235:{\displaystyle A_{s}e^{st}}
1016:of the input to the system
414:
221:. Click image to expand it.
10:
13758:
13648:IEEE Trans. Signal Process
13582:IEEE Trans. Signal Process
13004:
12904:
12854:it is also time invariant.
8709:{\displaystyle x=\delta ,}
8489:{\displaystyle \{x;\ m\}.}
8300:the folding frequency (or
8075:{\displaystyle s=j\omega }
7732:
7725:which requires causality.
7628:
7469:where the boxcar function
3127:{\displaystyle h(t-\tau )}
2065:depends only on values of
735:{\displaystyle y(t)+y'(t)}
686:{\displaystyle x(t)+x'(t)}
13737:Frequency-domain analysis
13717:Digital signal processing
13390:Bessai, Horst J. (2005).
10214:is the eigenfunction and
10149:is zero for all negative
8524:into an output sequence,
8359:{\displaystyle \{x;\ m\}}
8199:is a CT signal, then the
8085:As an example, the ideal
3960:is the eigenfunction and
2336:into an output function,
2300:{\textstyle \{x(u);\ u\}}
2016:is zero for all negative
1191:For all LTI systems, the
1158:Relationship between the
173:, for all real constants
13727:Classical control theory
13624:. New York: John Wiley.
13560:Johns Hopkins University
13556:"The Joy of Convolution"
13392:MIMO Signals and Systems
13376:
10227:{\displaystyle \lambda }
8667:Kronecker delta function
8650:.) In a typical system,
3973:{\displaystyle \lambda }
2525:that occurred near time
13535:Hespanha, J.P. (2009).
13117:maximum absolute values
10455:{\displaystyle x=z^{n}}
9636:{\displaystyle c_{k}=x}
8658:whose indices are near
8366:represent the sequence
7944:(has a finite L norm):
7337:{\displaystyle \Pi (t)}
5862:Wiener–Khinchin theorem
4837:{\displaystyle Ae^{st}}
4775:{\displaystyle Ae^{st}}
4016:{\displaystyle Ae^{st}}
2476:. In a typical system,
2190:{\textstyle x(u-\tau )}
2162:represent the function
1487:Continuous-time systems
831:superposition principle
313:is called the system's
105:superposition principle
101:superposition principle
64:more precise citations.
13722:Electrical engineering
13447:Phillips 2007, p. 508.
13363:
13313:
13277:
13200:
13171:
13142:
13107:
13057:
12987:
12958:
12887:
12848:
12776:
12686:
12595:
12459:
12394:
12270:
12136:
12116:
12024:
11935:
11799:
11542:
11458:
11370:
11270:
11231:
11202:
11131:
11095:
11068:
11040:
11005:
10975:
10885:
10852:
10818:
10788:
10724:
10668:
10596:
10555:
10521:
10485:
10456:
10411:
10352:
10312:
10284:
10228:
10202:
10119:
10077:
10036:
9992:
9954:
9914:
9876:
9694:
9637:
9595:where we have invoked
9587:
9516:
9435:
9336:
9308:
9267:
9216:
9174:
8997:
8950:
8877:
8821:
8799:
8710:
8634:
8567:
8547:
8546:{\displaystyle \{y\}.}
8518:
8490:
8443:
8415:
8360:
8278:
8193:
8144:
8143:{\displaystyle t>0}
8114:
8113:{\displaystyle t<0}
8076:
8041:
7934:
7905:
7876:
7851:maximum absolute value
7841:
7788:
7711:
7680:
7605:
7463:
7338:
7306:
7282:
7035:
6583:
6466:
6375:
6293:
6057:
5822:
5790:
5761:
5698:
5652:
5624:
5591:
5442:
5409:
5331:
5305:
5218:
5204:by time invariance of
5198:
5121:
5088:
5001:
4934:
4875:
4838:
4805:
4776:
4743:
4702:
4600:
4218:
4140:
4111:
4051:
4017:
3974:
3950:
3888:
3860:
3618:
3558:
3508:
3276:
3205:
3128:
3084:
2979:
2802:
2714:
2638:
2582:
2559:
2539:
2519:
2499:
2470:
2450:
2423:
2356:
2330:
2301:
2257:
2231:
2211:
2191:
2156:
2099:
2079:
2059:
2030:
2010:
1981:
1961:
1941:
1940:{\textstyle h(-\tau )}
1909:
1880:
1851:
1803:
1759:
1709:
1671:
1624:
1569:
1412:
1392:
1350:
1323:
1283:
1269:and complex frequency
1263:
1236:
1166:
1129:
1068:
1039:
1006:
969:
968:{\displaystyle y(t-T)}
934:
933:{\displaystyle x(t-T)}
899:
870:
823:
788:
756:
736:
687:
638:
604:
570:
538:
506:
486:
457:
393:mechanical engineering
369:linear shift-invariant
222:
13414:Hespanha 2009, p. 78.
13364:
13362:{\displaystyle |z|=1}
13323:region of convergence
13314:
13254:
13201:
13172:
13143:
13108:
13058:
12995:region of convergence
12988:
12959:
12888:
12849:
12715:
12654:
12596:
12427:
12362:
12238:
12137:
12084:
12025:
11936:
11800:
11543:
11459:
11347:
11271:
11232:
11203:
11132:
11096:
11094:{\displaystyle z^{n}}
11069:
11041:
11006:
10952:
10886:
10853:
10851:{\displaystyle z^{n}}
10819:
10765:
10725:
10645:
10573:
10556:
10498:
10486:
10457:
10420:Suppose the input is
10412:
10353:
10313:
10285:
10243:exponential functions
10229:
10203:
10120:
10118:{\displaystyle O_{n}}
10078:
10013:
9993:
9931:
9915:
9877:
9695:
9638:
9588:
9493:
9412:
9337:
9335:{\displaystyle \{x\}}
9309:
9244:
9217:
9215:{\displaystyle \{h\}}
9175:
8998:
8927:
8854:
8822:
8807:For a linear system,
8800:
8711:
8635:
8568:
8548:
8519:
8517:{\displaystyle \{x\}}
8491:
8444:
8442:{\displaystyle \{x\}}
8416:
8361:
8279:
8194:
8154:Discrete-time systems
8145:
8115:
8077:
8051:region of convergence
8042:
7935:
7906:
7877:
7842:
7789:
7712:
7681:
7606:
7464:
7339:
7307:
7283:
7036:
6584:
6467:
6376:
6294:
6058:
5823:
5791:
5762:
5699:
5653:
5625:
5592:
5443:
5410:
5337:and renaming we get:
5332:
5306:
5219:
5199:
5122:
5089:
5002:
4935:
4876:
4839:
4806:
4777:
4744:
4703:
4601:
4219:
4141:
4112:
4052:
4018:
3989:exponential functions
3975:
3951:
3889:
3861:
3619:
3559:
3509:
3277:
3206:
3129:
3085:
2980:
2803:
2697:
2621:
2583:
2568:For a linear system,
2560:
2540:
2520:
2500:
2471:
2451:
2424:
2357:
2331:
2302:
2258:
2232:
2212:
2192:
2157:
2100:
2080:
2060:
2031:
2011:
2009:{\textstyle h(\tau )}
1982:
1962:
1942:
1910:
1908:{\textstyle x(\tau )}
1881:
1852:
1804:
1760:
1692:
1672:
1607:
1570:
1442:differential equation
1413:
1393:
1351:
1349:{\displaystyle B_{s}}
1324:
1284:
1264:
1262:{\displaystyle A_{s}}
1237:
1157:
1130:
1069:
1040:
1007:
970:
935:
900:
871:
824:
822:{\displaystyle x'(t)}
789:
757:
737:
688:
639:
637:{\displaystyle y'(t)}
605:
603:{\displaystyle x'(t)}
571:
569:{\displaystyle ay(t)}
539:
537:{\displaystyle ax(t)}
507:
487:
458:
401:measuring instruments
231:linear time-invariant
95:
13742:Time domain analysis
13537:Linear System Theory
13337:
13210:
13181:
13152:
13123:
13069:
13019:
12968:
12915:
12868:
12605:
12146:
12036:
11954:
11811:
11554:
11480:
11302:
11241:
11230:{\displaystyle H(z)}
11212:
11145:
11105:
11078:
11050:
11017:
10906:
10884:{\displaystyle H(z)}
10866:
10835:
10734:
10570:
10495:
10466:
10424:
10362:
10334:
10294:
10248:
10218:
10173:
10102:
10007:
9925:
9895:
9713:
9647:
9605:
9349:
9320:
9226:
9200:
9028:
8836:
8811:
8724:
8673:
8577:
8557:
8528:
8502:
8453:
8427:
8370:
8320:
8211:
8192:{\displaystyle x(t)}
8174:
8128:
8098:
8057:
7948:
7933:{\displaystyle h(t)}
7915:
7904:{\displaystyle y(t)}
7886:
7875:{\displaystyle x(t)}
7857:
7800:
7747:
7710:{\displaystyle h(t)}
7692:
7639:
7473:
7348:
7319:
7292:
7045:
6593:
6480:
6393:
6305:
6126:
5872:
5800:
5789:{\displaystyle H(s)}
5771:
5711:
5662:
5634:
5601:
5463:
5419:
5341:
5315:
5231:
5208:
5133:
5098:
5013:
4944:
4896:
4874:{\displaystyle H(s)}
4856:
4815:
4804:{\displaystyle H(s)}
4786:
4753:
4719:
4612:
4232:
4150:
4139:{\displaystyle h(t)}
4121:
4073:
4027:
3994:
3964:
3921:
3872:
3635:
3568:
3526:
3303:
3220:
3139:
3103:
3009:
2833:
2603:
2572:
2549:
2529:
2509:
2480:
2460:
2433:
2366:
2340:
2311:
2267:
2241:
2221:
2201:
2166:
2116:
2089:
2069:
2040:
2020:
1991:
1971:
1951:
1919:
1890:
1861:
1817:
1784:
1686:
1580:
1523:
1504:) and output signal
1402:
1360:
1333:
1293:
1273:
1246:
1206:
1086:
1067:{\displaystyle h(t)}
1049:
1038:{\displaystyle x(t)}
1020:
1005:{\displaystyle y(t)}
987:
944:
909:
898:{\displaystyle y(t)}
880:
869:{\displaystyle x(t)}
851:
799:
787:{\displaystyle x(t)}
769:
746:
697:
648:
614:
580:
548:
516:
496:
485:{\displaystyle y(t)}
467:
456:{\displaystyle x(t)}
438:
411:present themselves.
13660:1995ITSP...43.1090V
13594:1995ITSP...43.1090V
13435:Crutchfield, p. 1.
13423:Crutchfield, p. 1.
7996:
7849:(that is, a finite
7400:
7208:
7120:
6898:
6830:
6707:
6554:
6279:
6244:
5947:
5549:
5330:{\displaystyle t=0}
4660:
4452:
4354:
4260:
4228:, is equivalent to
4170:
3705:
3428:
3358:
2329:{\textstyle \{x\},}
1446:electrical circuits
1074:. This is called a
373:applied mathematics
196:and for all inputs
13466:Frequency response
13359:
13309:
13196:
13167:
13138:
13103:
13053:
12983:
12954:
12883:
12844:
12842:
12591:
12589:
12132:
12020:
11931:
11795:
11538:
11454:
11266:
11227:
11198:
11127:
11091:
11064:
11036:
11001:
10881:
10848:
10814:
10720:
10551:
10481:
10452:
10407:
10348:
10308:
10280:
10224:
10198:
10115:
10073:
9988:
9910:
9872:
9870:
9690:
9633:
9583:
9581:
9332:
9304:
9212:
9170:
9168:
8993:
8817:
8795:
8706:
8630:
8563:
8543:
8514:
8486:
8439:
8411:
8356:
8274:
8189:
8140:
8110:
8072:
8037:
7979:
7930:
7901:
7872:
7837:
7784:
7707:
7676:
7601:
7596:
7459:
7383:
7334:
7302:
7278:
7276:
7158:
7094:
7031:
7029:
6872:
6804:
6681:
6579:
6528:
6462:
6371:
6289:
6267:
6232:
6053:
5930:
5818:
5786:
5757:
5694:
5648:
5620:
5587:
5535:
5438:
5405:
5327:
5301:
5214:
5194:
5117:
5084:
4997:
4930:
4871:
4834:
4801:
4772:
4739:
4698:
4643:
4596:
4594:
4577:
4570:
4535:
4528:
4435:
4337:
4243:
4214:
4153:
4136:
4107:
4047:
4013:
3970:
3946:
3887:{\textstyle \{x\}}
3884:
3856:
3854:
3688:
3614:
3554:
3504:
3502:
3411:
3341:
3272:
3201:
3124:
3080:
2975:
2973:
2798:
2785:
2755:
2578:
2555:
2535:
2515:
2495:
2466:
2449:{\textstyle O_{t}}
2446:
2419:
2355:{\textstyle \{y\}}
2352:
2326:
2297:
2256:{\textstyle \{x\}}
2253:
2230:{\textstyle \tau }
2227:
2207:
2187:
2152:
2095:
2075:
2055:
2029:{\textstyle \tau }
2026:
2006:
1977:
1957:
1937:
1905:
1876:
1847:
1799:
1755:
1667:
1565:
1408:
1388:
1346:
1319:
1279:
1259:
1232:
1188:frequency domain.
1167:
1125:
1064:
1035:
1002:
965:
930:
895:
866:
819:
784:
752:
732:
683:
634:
600:
566:
534:
502:
482:
453:
330:electrical circuit
223:
13732:Signal processing
13668:10.1109/78.382395
13631:978-0-471-14961-3
13602:10.1109/78.382395
13546:978-0-691-14021-6
13513:978-0-13-041207-2
13506:. Prentice Hall.
13486:Signal-flow graph
13325:must contain the
13251:
13249:
13199:{\displaystyle h}
13170:{\displaystyle y}
13141:{\displaystyle x}
12986:{\displaystyle h}
12938:
12886:{\displaystyle h}
12081:
12079:
11517:
11515:
11286:transfer function
10762:
10760:
10484:{\displaystyle h}
10389:
10094:
10093:
9913:{\displaystyle y}
9844:
9842:
9817:
9778:
9756:
9568:
9474:
9194:
9193:
9134:
9132:
9093:
9071:
9017:
9016:
8915:
8820:{\displaystyle O}
8785:
8752:
8750:
8605:
8603:
8566:{\displaystyle O}
8476:
8400:
8349:
8302:Nyquist frequency
8237:
8235:
8122:cut-off frequency
7805:
7752:
7589:
7563:
7546:
7520:
7501:
7499:
7429:
6525:
6523:
6425:
6325:
6146:
6000:
5998:
5927:
5925:
5858:square integrable
5838:transfer function
5706:Fourier transform
5692:
5681:
5679:
5532:
5530:
5491:
5489:
5458:Laplace transform
5217:{\displaystyle H}
4782:and the constant
4640:
4638:
4608:where the scalar
4588:
4581:
4575:
4552:
4550:
4546:
4539:
4533:
4506:
4504:
4329:
4315:
3833:
3831:
3804:
3753:
3480:
3295:
3294:
3262:
3194:
3155:
3153:
3070:
3037:
3035:
2999:
2998:
2939:
2937:
2898:
2876:
2822:
2821:
2730:
2728:
2727:
2685:
2651:
2498:{\textstyle y(t)}
2394:
2392:
2290:
2145:
2058:{\textstyle y(t)}
1879:{\textstyle y(t)}
1802:{\textstyle h(t)}
1776:
1775:
1604:
1411:{\displaystyle s}
1282:{\displaystyle s}
1181:Laplace transform
1177:transfer function
755:{\displaystyle a}
505:{\displaystyle a}
381:signal processing
346:linear amplifiers
99:illustrating the
90:
89:
82:
16:(Redirected from
13749:
13671:
13645:
13635:
13605:
13579:
13569:
13568:
13566:
13550:
13531:
13525:
13517:
13471:Impulse response
13461:Circulant matrix
13448:
13445:
13439:
13433:
13427:
13421:
13415:
13412:
13406:
13405:
13387:
13368:
13366:
13365:
13360:
13352:
13344:
13318:
13316:
13315:
13310:
13299:
13282:
13276:
13271:
13253:
13252:
13250:
13247:
13245:
13240:
13237:
13236:
13205:
13203:
13202:
13197:
13176:
13174:
13173:
13168:
13147:
13145:
13144:
13139:
13112:
13110:
13109:
13104:
13096:
13095:
13062:
13060:
13059:
13054:
13046:
13045:
12992:
12990:
12989:
12984:
12963:
12961:
12960:
12955:
12936:
12892:
12890:
12889:
12884:
12853:
12851:
12850:
12845:
12843:
12821:
12810:
12809:
12797:
12790:
12775:
12752:
12727:
12708:
12685:
12674:
12646:
12642:
12618:
12617:
12600:
12598:
12597:
12592:
12590:
12583:
12579:
12569:
12568:
12554:
12553:
12547:
12546:
12534:
12530:
12520:
12519:
12505:
12504:
12498:
12497:
12482:
12469:
12468:
12458:
12447:
12426:
12425:
12404:
12403:
12393:
12382:
12361:
12360:
12345:
12341:
12337:
12327:
12326:
12317:
12316:
12295:
12294:
12285:
12284:
12269:
12258:
12230:
12226:
12216:
12215:
12206:
12205:
12184:
12183:
12174:
12173:
12159:
12158:
12141:
12139:
12138:
12133:
12115:
12104:
12083:
12082:
12080:
12077:
12075:
12070:
12067:
12063:
12045:
12044:
12029:
12027:
12026:
12021:
12004:
12003:
11988:
11984:
11963:
11962:
11940:
11938:
11937:
11932:
11804:
11802:
11801:
11796:
11785:
11784:
11769:
11768:
11747:
11746:
11731:
11730:
11703:
11702:
11690:
11689:
11662:
11661:
11649:
11648:
11636:
11632:
11622:
11621:
11609:
11608:
11587:
11586:
11574:
11573:
11547:
11545:
11544:
11539:
11519:
11518:
11516:
11513:
11511:
11506:
11463:
11461:
11460:
11455:
11420:
11419:
11411:
11410:
11369:
11364:
11275:
11273:
11272:
11267:
11262:
11261:
11236:
11234:
11233:
11228:
11207:
11205:
11204:
11199:
11179:
11178:
11166:
11165:
11136:
11134:
11133:
11128:
11126:
11125:
11100:
11098:
11097:
11092:
11090:
11089:
11073:
11071:
11070:
11065:
11063:
11045:
11043:
11042:
11037:
11035:
11034:
11010:
11008:
11007:
11002:
11000:
10999:
10974:
10969:
10930:
10929:
10890:
10888:
10887:
10882:
10857:
10855:
10854:
10849:
10847:
10846:
10823:
10821:
10820:
10815:
10813:
10812:
10787:
10782:
10764:
10763:
10761:
10758:
10756:
10751:
10729:
10727:
10726:
10721:
10707:
10706:
10694:
10693:
10667:
10662:
10644:
10643:
10631:
10630:
10595:
10590:
10560:
10558:
10557:
10552:
10550:
10549:
10520:
10515:
10490:
10488:
10487:
10482:
10461:
10459:
10458:
10453:
10451:
10450:
10416:
10414:
10413:
10408:
10406:
10387:
10383:
10382:
10357:
10355:
10354:
10349:
10347:
10317:
10315:
10314:
10309:
10307:
10289:
10287:
10286:
10281:
10279:
10278:
10260:
10259:
10233:
10231:
10230:
10225:
10207:
10205:
10204:
10199:
10182:
10181:
10124:
10122:
10121:
10116:
10114:
10113:
10085:
10082:
10080:
10079:
10074:
10035:
10030:
9997:
9995:
9994:
9989:
9953:
9948:
9919:
9917:
9916:
9911:
9889:
9888:
9881:
9879:
9878:
9873:
9871:
9846:
9845:
9843:
9840:
9838:
9833:
9827:
9815:
9796:
9795:
9780:
9779:
9777:
9774:
9769:
9754:
9729:
9728:
9709:, we may write:
9699:
9697:
9696:
9691:
9659:
9658:
9642:
9640:
9639:
9634:
9617:
9616:
9592:
9590:
9589:
9584:
9582:
9566:
9541:
9540:
9515:
9510:
9486:
9482:
9478:
9472:
9434:
9429:
9406:
9405:
9380:
9379:
9341:
9339:
9338:
9333:
9316:which expresses
9313:
9311:
9310:
9305:
9266:
9261:
9221:
9219:
9218:
9213:
9188:
9179:
9177:
9176:
9171:
9169:
9152:
9151:
9136:
9135:
9133:
9130:
9128:
9123:
9117:
9095:
9094:
9092:
9089:
9084:
9069:
9044:
9043:
9022:
9011:
9002:
9000:
8999:
8994:
8986:
8985:
8973:
8972:
8960:
8959:
8949:
8944:
8923:
8919:
8913:
8900:
8899:
8887:
8886:
8876:
8871:
8848:
8847:
8830:
8826:
8824:
8823:
8818:
8804:
8802:
8801:
8796:
8783:
8764:
8763:
8754:
8753:
8751:
8748:
8746:
8741:
8718:impulse response
8715:
8713:
8712:
8707:
8639:
8637:
8636:
8631:
8617:
8616:
8607:
8606:
8604:
8601:
8599:
8594:
8573:, we can write:
8572:
8570:
8569:
8564:
8552:
8550:
8549:
8544:
8523:
8521:
8520:
8515:
8495:
8493:
8492:
8487:
8474:
8448:
8446:
8445:
8440:
8420:
8418:
8417:
8412:
8401:
8398:
8365:
8363:
8362:
8357:
8347:
8283:
8281:
8280:
8275:
8270:
8239:
8238:
8236:
8233:
8231:
8226:
8223:
8222:
8201:sampling circuit
8198:
8196:
8195:
8190:
8149:
8147:
8146:
8141:
8119:
8117:
8116:
8111:
8081:
8079:
8078:
8073:
8046:
8044:
8043:
8038:
8024:
8018:
8001:
7995:
7990:
7975:
7974:
7939:
7937:
7936:
7931:
7910:
7908:
7907:
7902:
7881:
7879:
7878:
7873:
7846:
7844:
7843:
7838:
7830:
7829:
7803:
7793:
7791:
7790:
7785:
7777:
7776:
7750:
7716:
7714:
7713:
7708:
7685:
7683:
7682:
7677:
7610:
7608:
7607:
7602:
7600:
7599:
7590:
7582:
7577:
7569:
7564:
7561:
7547:
7539:
7534:
7526:
7521:
7518:
7503:
7502:
7500:
7497:
7495:
7490:
7468:
7466:
7465:
7460:
7452:
7434:
7430:
7428:
7420:
7409:
7399:
7394:
7379:
7375:
7357:
7356:
7343:
7341:
7340:
7335:
7311:
7309:
7308:
7303:
7301:
7300:
7287:
7285:
7284:
7279:
7277:
7246:
7245:
7233:
7226:
7207:
7184:
7151:
7144:
7119:
7108:
7086:
7082:
7058:
7057:
7040:
7038:
7037:
7032:
7030:
7011:
7010:
6998:
6997:
6991:
6990:
6966:
6965:
6953:
6952:
6946:
6945:
6930:
6923:
6908:
6907:
6897:
6886:
6871:
6870:
6855:
6840:
6839:
6829:
6818:
6803:
6802:
6787:
6780:
6762:
6761:
6752:
6751:
6730:
6729:
6720:
6719:
6706:
6695:
6661:
6660:
6651:
6650:
6629:
6628:
6619:
6618:
6606:
6605:
6588:
6586:
6585:
6580:
6572:
6553:
6542:
6527:
6526:
6524:
6521:
6519:
6514:
6511:
6507:
6489:
6488:
6471:
6469:
6468:
6463:
6443:
6439:
6426:
6424:
6420:
6414:
6409:
6402:
6401:
6380:
6378:
6377:
6372:
6355:
6326:
6324:
6320:
6314:
6309:
6298:
6296:
6295:
6290:
6275:
6266:
6265:
6240:
6231:
6230:
6218:
6214:
6204:
6203:
6194:
6193:
6172:
6171:
6162:
6161:
6147:
6145:
6141:
6135:
6130:
6085:
6075:
6062:
6060:
6059:
6054:
6019:
6018:
6010:
6009:
6002:
6001:
5999:
5996:
5994:
5989:
5983:
5946:
5941:
5929:
5928:
5926:
5923:
5921:
5916:
5827:
5825:
5824:
5819:
5795:
5793:
5792:
5787:
5766:
5764:
5763:
5758:
5738:
5737:
5703:
5701:
5700:
5695:
5693:
5685:
5683:
5682:
5680:
5677:
5675:
5670:
5657:
5655:
5654:
5649:
5647:
5629:
5627:
5626:
5621:
5619:
5618:
5596:
5594:
5593:
5588:
5583:
5577:
5576:
5548:
5543:
5534:
5533:
5531:
5528:
5526:
5521:
5500:
5499:
5493:
5492:
5490:
5487:
5485:
5480:
5447:
5445:
5444:
5439:
5437:
5436:
5414:
5412:
5411:
5406:
5383:
5382:
5336:
5334:
5333:
5328:
5310:
5308:
5307:
5302:
5279:
5278:
5223:
5221:
5220:
5215:
5203:
5201:
5200:
5195:
5151:
5150:
5126:
5124:
5123:
5118:
5116:
5115:
5093:
5091:
5090:
5085:
5062:
5061:
5031:
5030:
5006:
5004:
5003:
4998:
4996:
4995:
4956:
4955:
4939:
4937:
4936:
4931:
4929:
4928:
4880:
4878:
4877:
4872:
4843:
4841:
4840:
4835:
4833:
4832:
4810:
4808:
4807:
4802:
4781:
4779:
4778:
4773:
4771:
4770:
4748:
4746:
4745:
4740:
4738:
4707:
4705:
4704:
4699:
4694:
4688:
4687:
4659:
4654:
4642:
4641:
4639:
4636:
4634:
4629:
4605:
4603:
4602:
4597:
4595:
4587:
4582:
4576:
4573:
4571:
4566:
4549:
4547:
4545:
4540:
4534:
4531:
4529:
4524:
4523:
4522:
4503:
4501:
4494:
4487:
4481:
4480:
4451:
4446:
4434:
4433:
4412:
4405:
4399:
4398:
4383:
4382:
4353:
4348:
4328:
4324:
4323:
4316:
4311:
4307:
4301:
4300:
4259:
4254:
4241:
4239:
4223:
4221:
4220:
4215:
4210:
4204:
4203:
4169:
4164:
4145:
4143:
4142:
4137:
4116:
4114:
4113:
4108:
4106:
4105:
4056:
4054:
4053:
4048:
4046:
4022:
4020:
4019:
4014:
4012:
4011:
3979:
3977:
3976:
3971:
3955:
3953:
3952:
3947:
3930:
3929:
3897:
3893:
3891:
3890:
3885:
3865:
3863:
3862:
3857:
3855:
3835:
3834:
3832:
3829:
3827:
3822:
3816:
3812:
3808:
3802:
3781:
3780:
3765:
3761:
3757:
3751:
3744:
3704:
3699:
3682:
3681:
3623:
3621:
3620:
3615:
3580:
3579:
3563:
3561:
3560:
3555:
3538:
3537:
3513:
3511:
3510:
3505:
3503:
3492:
3478:
3453:
3452:
3427:
3422:
3404:
3397:
3357:
3352:
3284:
3281:
3279:
3278:
3273:
3260:
3235:
3234:
3210:
3208:
3207:
3202:
3192:
3173:
3172:
3157:
3156:
3154:
3151:
3149:
3144:
3133:
3131:
3130:
3125:
3097:
3096:
3089:
3087:
3086:
3081:
3068:
3049:
3048:
3039:
3038:
3036:
3033:
3031:
3026:
3003:impulse response
2993:
2984:
2982:
2981:
2976:
2974:
2957:
2956:
2941:
2940:
2938:
2935:
2933:
2928:
2922:
2900:
2899:
2897:
2894:
2889:
2874:
2849:
2848:
2827:
2816:
2807:
2805:
2804:
2799:
2791:
2784:
2780:
2779:
2767:
2766:
2756:
2751:
2741:
2740:
2725:
2724:
2723:
2713:
2708:
2693:
2689:
2683:
2676:
2661:
2660:
2649:
2648:
2647:
2637:
2632:
2615:
2614:
2597:
2587:
2585:
2584:
2579:
2564:
2562:
2561:
2556:
2544:
2542:
2541:
2536:
2524:
2522:
2521:
2516:
2504:
2502:
2501:
2496:
2475:
2473:
2472:
2467:
2455:
2453:
2452:
2447:
2445:
2444:
2428:
2426:
2425:
2420:
2406:
2405:
2396:
2395:
2393:
2390:
2388:
2383:
2361:
2359:
2358:
2353:
2335:
2333:
2332:
2327:
2306:
2304:
2303:
2298:
2288:
2262:
2260:
2259:
2254:
2236:
2234:
2233:
2228:
2216:
2214:
2213:
2208:
2196:
2194:
2193:
2188:
2161:
2159:
2158:
2153:
2143:
2104:
2102:
2101:
2096:
2084:
2082:
2081:
2076:
2064:
2062:
2061:
2056:
2035:
2033:
2032:
2027:
2015:
2013:
2012:
2007:
1986:
1984:
1983:
1978:
1966:
1964:
1963:
1958:
1946:
1944:
1943:
1938:
1914:
1912:
1911:
1906:
1885:
1883:
1882:
1877:
1856:
1854:
1853:
1848:
1808:
1806:
1805:
1800:
1767:
1764:
1762:
1761:
1756:
1748:
1708:
1703:
1676:
1674:
1673:
1668:
1663:
1623:
1618:
1606:
1605:
1603:
1602:
1590:
1585:
1574:
1572:
1571:
1566:
1517:
1516:
1429:and a different
1417:
1415:
1414:
1409:
1397:
1395:
1394:
1389:
1387:
1386:
1377:
1372:
1371:
1355:
1353:
1352:
1347:
1345:
1344:
1328:
1326:
1325:
1320:
1318:
1317:
1305:
1304:
1288:
1286:
1285:
1280:
1268:
1266:
1265:
1260:
1258:
1257:
1241:
1239:
1238:
1233:
1231:
1230:
1218:
1217:
1175:by the system's
1172:frequency domain
1164:frequency domain
1134:
1132:
1131:
1126:
1124:
1123:
1111:
1110:
1098:
1097:
1073:
1071:
1070:
1065:
1044:
1042:
1041:
1036:
1011:
1009:
1008:
1003:
981:impulse response
974:
972:
971:
966:
939:
937:
936:
931:
904:
902:
901:
896:
875:
873:
872:
867:
828:
826:
825:
820:
809:
793:
791:
790:
785:
761:
759:
758:
753:
741:
739:
738:
733:
722:
692:
690:
689:
684:
673:
643:
641:
640:
635:
624:
609:
607:
606:
601:
590:
575:
573:
572:
567:
543:
541:
540:
535:
511:
509:
508:
503:
491:
489:
488:
483:
462:
460:
459:
454:
405:NMR spectroscopy
399:, the design of
397:image processing
353:image processing
327:
315:impulse response
312:
301:
274:
263:
220:
195:
172:
168:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
13757:
13756:
13752:
13751:
13750:
13748:
13747:
13746:
13707:
13706:
13679:
13674:
13643:
13632:
13612:
13610:Further reading
13577:
13564:
13562:
13547:
13519:
13518:
13514:
13494:
13476:System analysis
13457:
13452:
13451:
13446:
13442:
13434:
13430:
13422:
13418:
13413:
13409:
13402:
13388:
13384:
13379:
13348:
13340:
13338:
13335:
13334:
13295:
13278:
13272:
13258:
13246:
13241:
13239:
13238:
13232:
13228:
13211:
13208:
13207:
13182:
13179:
13178:
13153:
13150:
13149:
13124:
13121:
13120:
13091:
13087:
13070:
13067:
13066:
13041:
13037:
13020:
13017:
13016:
13009:
13003:
12969:
12966:
12965:
12916:
12913:
12912:
12909:
12903:
12869:
12866:
12865:
12862:
12857:
12841:
12840:
12811:
12805:
12804:
12795:
12794:
12783:
12753:
12720:
12719:
12706:
12705:
12675:
12658:
12647:
12623:
12619:
12613:
12612:
12608:
12606:
12603:
12602:
12588:
12587:
12564:
12560:
12559:
12555:
12549:
12548:
12542:
12538:
12515:
12511:
12510:
12506:
12500:
12499:
12493:
12489:
12480:
12479:
12464:
12460:
12448:
12431:
12421:
12417:
12399:
12395:
12383:
12366:
12356:
12352:
12343:
12342:
12322:
12318:
12312:
12308:
12290:
12286:
12280:
12276:
12275:
12271:
12259:
12242:
12231:
12211:
12207:
12201:
12197:
12179:
12175:
12169:
12165:
12164:
12160:
12154:
12153:
12149:
12147:
12144:
12143:
12105:
12088:
12076:
12071:
12069:
12068:
12050:
12046:
12040:
12039:
12037:
12034:
12033:
11996:
11992:
11968:
11964:
11958:
11957:
11955:
11952:
11951:
11812:
11809:
11808:
11780:
11776:
11764:
11760:
11742:
11738:
11726:
11722:
11698:
11694:
11685:
11681:
11657:
11653:
11644:
11640:
11617:
11613:
11604:
11600:
11582:
11578:
11569:
11565:
11564:
11560:
11555:
11552:
11551:
11512:
11507:
11505:
11504:
11481:
11478:
11477:
11472:
11412:
11406:
11405:
11404:
11365:
11351:
11303:
11300:
11299:
11282:system response
11278:system function
11276:are called the
11254:
11250:
11242:
11239:
11238:
11213:
11210:
11209:
11174:
11173:
11158:
11154:
11146:
11143:
11142:
11118:
11114:
11106:
11103:
11102:
11085:
11081:
11079:
11076:
11075:
11059:
11051:
11048:
11047:
11024:
11020:
11018:
11015:
11014:
10992:
10988:
10970:
10956:
10925:
10924:
10907:
10904:
10903:
10897:
10867:
10864:
10863:
10842:
10838:
10836:
10833:
10832:
10805:
10801:
10783:
10769:
10757:
10752:
10750:
10749:
10735:
10732:
10731:
10702:
10698:
10686:
10682:
10663:
10649:
10639:
10635:
10614:
10610:
10591:
10577:
10571:
10568:
10567:
10545:
10541:
10516:
10502:
10496:
10493:
10492:
10467:
10464:
10463:
10446:
10442:
10425:
10422:
10421:
10402:
10375:
10371:
10363:
10360:
10359:
10343:
10335:
10332:
10331:
10303:
10295:
10292:
10291:
10268:
10264:
10255:
10251:
10249:
10246:
10245:
10219:
10216:
10215:
10177:
10176:
10174:
10171:
10170:
10163:
10109:
10105:
10103:
10100:
10099:
10083:
10031:
10017:
10008:
10005:
10004:
9949:
9935:
9926:
9923:
9922:
9896:
9893:
9892:
9869:
9868:
9839:
9834:
9832:
9831:
9825:
9824:
9785:
9781:
9775:
9770:
9768:
9767:
9763:
9724:
9720:
9716:
9714:
9711:
9710:
9703:And because of
9654:
9650:
9648:
9645:
9644:
9612:
9608:
9606:
9603:
9602:
9580:
9579:
9536:
9532:
9511:
9497:
9484:
9483:
9430:
9416:
9411:
9407:
9401:
9397:
9390:
9375:
9371:
9352:
9350:
9347:
9346:
9321:
9318:
9317:
9262:
9248:
9227:
9224:
9223:
9201:
9198:
9197:
9167:
9166:
9141:
9137:
9129:
9124:
9122:
9121:
9115:
9114:
9090:
9085:
9083:
9082:
9078:
9039:
9035:
9031:
9029:
9026:
9025:
8981:
8977:
8968:
8964:
8955:
8951:
8945:
8931:
8895:
8891:
8882:
8878:
8872:
8858:
8853:
8849:
8843:
8839:
8837:
8834:
8833:
8812:
8809:
8808:
8759:
8755:
8747:
8742:
8740:
8739:
8725:
8722:
8721:
8674:
8671:
8670:
8612:
8608:
8600:
8595:
8593:
8592:
8578:
8575:
8574:
8558:
8555:
8554:
8529:
8526:
8525:
8503:
8500:
8499:
8454:
8451:
8450:
8428:
8425:
8424:
8397:
8371:
8368:
8367:
8321:
8318:
8317:
8314:
8290:sampling period
8266:
8232:
8227:
8225:
8224:
8218:
8214:
8212:
8209:
8208:
8203:used before an
8175:
8172:
8171:
8164:
8156:
8129:
8126:
8125:
8099:
8096:
8095:
8087:low-pass filter
8058:
8055:
8054:
8020:
8014:
7997:
7991:
7983:
7970:
7966:
7949:
7946:
7945:
7916:
7913:
7912:
7887:
7884:
7883:
7858:
7855:
7854:
7825:
7821:
7801:
7798:
7797:
7772:
7768:
7748:
7745:
7744:
7737:
7731:
7693:
7690:
7689:
7640:
7637:
7636:
7633:
7627:
7618:
7613:
7595:
7594:
7581:
7573:
7565:
7560:
7558:
7552:
7551:
7538:
7530:
7522:
7517:
7515:
7505:
7504:
7496:
7491:
7489:
7488:
7474:
7471:
7470:
7448:
7421:
7410:
7408:
7404:
7395:
7387:
7362:
7358:
7352:
7351:
7349:
7346:
7345:
7320:
7317:
7316:
7314:boxcar function
7296:
7295:
7293:
7290:
7289:
7275:
7274:
7241:
7240:
7231:
7230:
7222:
7185:
7162:
7149:
7148:
7140:
7109:
7098:
7087:
7063:
7059:
7053:
7052:
7048:
7046:
7043:
7042:
7028:
7027:
7006:
7002:
6993:
6992:
6986:
6982:
6961:
6957:
6948:
6947:
6941:
6937:
6928:
6927:
6919:
6903:
6899:
6887:
6876:
6866:
6862:
6851:
6835:
6831:
6819:
6808:
6798:
6794:
6785:
6784:
6776:
6757:
6753:
6747:
6743:
6725:
6721:
6715:
6711:
6696:
6685:
6674:
6656:
6652:
6646:
6642:
6624:
6620:
6614:
6610:
6601:
6600:
6596:
6594:
6591:
6590:
6568:
6543:
6532:
6520:
6515:
6513:
6512:
6494:
6490:
6484:
6483:
6481:
6478:
6477:
6416:
6415:
6410:
6408:
6407:
6403:
6397:
6396:
6394:
6391:
6390:
6348:
6316:
6315:
6310:
6308:
6306:
6303:
6302:
6271:
6261:
6257:
6236:
6226:
6222:
6199:
6195:
6189:
6185:
6167:
6163:
6157:
6153:
6152:
6148:
6137:
6136:
6131:
6129:
6127:
6124:
6123:
6112:
6077:
6067:
6011:
6005:
6004:
6003:
5995:
5990:
5988:
5987:
5979:
5942:
5934:
5922:
5917:
5915:
5914:
5873:
5870:
5869:
5834:system response
5830:system function
5828:are called the
5801:
5798:
5797:
5772:
5769:
5768:
5733:
5732:
5712:
5709:
5708:
5684:
5676:
5671:
5669:
5668:
5663:
5660:
5659:
5643:
5635:
5632:
5631:
5608:
5604:
5602:
5599:
5598:
5579:
5566:
5562:
5544:
5539:
5527:
5522:
5520:
5519:
5495:
5494:
5486:
5481:
5479:
5478:
5464:
5461:
5460:
5454:
5426:
5422:
5420:
5417:
5416:
5372:
5368:
5342:
5339:
5338:
5316:
5313:
5312:
5268:
5264:
5232:
5229:
5228:
5209:
5206:
5205:
5146:
5142:
5134:
5131:
5130:
5105:
5101:
5099:
5096:
5095:
5051:
5047:
5026:
5022:
5014:
5011:
5010:
4973:
4969:
4951:
4947:
4945:
4942:
4941:
4918:
4914:
4897:
4894:
4893:
4887:
4857:
4854:
4853:
4825:
4821:
4816:
4813:
4812:
4787:
4784:
4783:
4763:
4759:
4754:
4751:
4750:
4734:
4720:
4717:
4716:
4690:
4677:
4673:
4655:
4647:
4635:
4630:
4628:
4627:
4613:
4610:
4609:
4593:
4592:
4583:
4572:
4553:
4551:
4548:
4541:
4530:
4515:
4511:
4507:
4505:
4502:
4492:
4491:
4483:
4470:
4466:
4447:
4439:
4426:
4422:
4410:
4409:
4401:
4388:
4384:
4375:
4371:
4349:
4341:
4330:
4319:
4318:
4317:
4303:
4281:
4277:
4255:
4247:
4242:
4240:
4235:
4233:
4230:
4229:
4206:
4196:
4192:
4165:
4157:
4151:
4148:
4147:
4122:
4119:
4118:
4098:
4094:
4074:
4071:
4070:
4042:
4028:
4025:
4024:
4004:
4000:
3995:
3992:
3991:
3965:
3962:
3961:
3925:
3924:
3922:
3919:
3918:
3911:
3873:
3870:
3869:
3853:
3852:
3828:
3823:
3821:
3820:
3814:
3813:
3786:
3782:
3776:
3772:
3763:
3762:
3740:
3700:
3692:
3687:
3683:
3677:
3673:
3666:
3638:
3636:
3633:
3632:
3575:
3571:
3569:
3566:
3565:
3533:
3529:
3527:
3524:
3523:
3501:
3500:
3488:
3448:
3444:
3423:
3415:
3402:
3401:
3393:
3353:
3345:
3334:
3306:
3304:
3301:
3300:
3282:
3230:
3226:
3221:
3218:
3217:
3162:
3158:
3150:
3145:
3143:
3142:
3140:
3137:
3136:
3104:
3101:
3100:
3044:
3040:
3032:
3027:
3025:
3024:
3010:
3007:
3006:
2972:
2971:
2946:
2942:
2934:
2929:
2927:
2926:
2920:
2919:
2895:
2890:
2888:
2887:
2883:
2844:
2840:
2836:
2834:
2831:
2830:
2787:
2775:
2771:
2762:
2758:
2757:
2736:
2732:
2731:
2729:
2719:
2715:
2709:
2701:
2672:
2656:
2652:
2643:
2639:
2633:
2625:
2620:
2616:
2610:
2606:
2604:
2601:
2600:
2573:
2570:
2569:
2550:
2547:
2546:
2530:
2527:
2526:
2510:
2507:
2506:
2481:
2478:
2477:
2461:
2458:
2457:
2440:
2436:
2434:
2431:
2430:
2401:
2397:
2389:
2384:
2382:
2381:
2367:
2364:
2363:
2341:
2338:
2337:
2312:
2309:
2308:
2268:
2265:
2264:
2242:
2239:
2238:
2222:
2219:
2218:
2202:
2199:
2198:
2167:
2164:
2163:
2117:
2114:
2113:
2090:
2087:
2086:
2070:
2067:
2066:
2041:
2038:
2037:
2021:
2018:
2017:
1992:
1989:
1988:
1972:
1969:
1968:
1952:
1949:
1948:
1920:
1917:
1916:
1891:
1888:
1887:
1862:
1859:
1858:
1818:
1815:
1814:
1785:
1782:
1781:
1765:
1744:
1704:
1696:
1687:
1684:
1683:
1659:
1619:
1611:
1592:
1591:
1586:
1584:
1583:
1581:
1578:
1577:
1524:
1521:
1520:
1494:
1489:
1403:
1400:
1399:
1382:
1378:
1373:
1367:
1363:
1361:
1358:
1357:
1340:
1336:
1334:
1331:
1330:
1310:
1306:
1300:
1296:
1294:
1291:
1290:
1274:
1271:
1270:
1253:
1249:
1247:
1244:
1243:
1223:
1219:
1213:
1209:
1207:
1204:
1203:
1179:, which is the
1119:
1115:
1106:
1102:
1093:
1089:
1087:
1084:
1083:
1076:continuous time
1050:
1047:
1046:
1021:
1018:
1017:
988:
985:
984:
945:
942:
941:
910:
907:
906:
881:
878:
877:
852:
849:
848:
837:Time invariance
802:
800:
797:
796:
770:
767:
766:
747:
744:
743:
715:
698:
695:
694:
666:
649:
646:
645:
617:
615:
612:
611:
583:
581:
578:
577:
549:
546:
545:
517:
514:
513:
497:
494:
493:
468:
465:
464:
463:and the output
439:
436:
435:
425:time invariance
417:
403:of many sorts,
318:
303:
280:
265:
254:
251:time-invariance
227:system analysis
214:
203:
197:
194:
187:
180:
174:
170:
166:
155:
149:
142:
131:
125:
114:
108:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
13755:
13745:
13744:
13739:
13734:
13729:
13724:
13719:
13705:
13704:
13698:
13692:
13686:
13678:
13677:External links
13675:
13673:
13672:
13636:
13630:
13613:
13611:
13608:
13607:
13606:
13570:
13551:
13545:
13532:
13512:
13493:
13490:
13489:
13488:
13483:
13481:Green function
13478:
13473:
13468:
13463:
13456:
13453:
13450:
13449:
13440:
13428:
13416:
13407:
13400:
13381:
13380:
13378:
13375:
13358:
13355:
13351:
13347:
13343:
13308:
13305:
13302:
13298:
13294:
13291:
13288:
13285:
13281:
13275:
13270:
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13264:
13261:
13257:
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13128:
13102:
13099:
13094:
13090:
13086:
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13080:
13077:
13074:
13052:
13049:
13044:
13040:
13036:
13033:
13030:
13027:
13024:
13007:BIBO stability
13005:Main article:
13002:
12999:
12982:
12979:
12976:
12973:
12953:
12950:
12947:
12944:
12941:
12935:
12932:
12929:
12926:
12923:
12920:
12905:Main article:
12902:
12899:
12882:
12879:
12876:
12873:
12861:
12858:
12856:
12855:
12839:
12836:
12833:
12830:
12827:
12824:
12820:
12817:
12814:
12808:
12803:
12800:
12798:
12796:
12793:
12789:
12786:
12782:
12779:
12774:
12771:
12768:
12765:
12762:
12759:
12756:
12751:
12748:
12745:
12742:
12739:
12736:
12733:
12730:
12726:
12723:
12718:
12714:
12711:
12709:
12707:
12704:
12701:
12698:
12695:
12692:
12689:
12684:
12681:
12678:
12673:
12670:
12667:
12664:
12661:
12657:
12653:
12650:
12648:
12645:
12641:
12638:
12635:
12632:
12629:
12626:
12622:
12616:
12611:
12610:
12586:
12582:
12578:
12575:
12572:
12567:
12563:
12558:
12552:
12545:
12541:
12537:
12533:
12529:
12526:
12523:
12518:
12514:
12509:
12503:
12496:
12492:
12488:
12485:
12483:
12481:
12478:
12475:
12472:
12467:
12463:
12457:
12454:
12451:
12446:
12443:
12440:
12437:
12434:
12430:
12424:
12420:
12416:
12413:
12410:
12407:
12402:
12398:
12392:
12389:
12386:
12381:
12378:
12375:
12372:
12369:
12365:
12359:
12355:
12351:
12348:
12346:
12344:
12340:
12336:
12333:
12330:
12325:
12321:
12315:
12311:
12307:
12304:
12301:
12298:
12293:
12289:
12283:
12279:
12274:
12268:
12265:
12262:
12257:
12254:
12251:
12248:
12245:
12241:
12237:
12234:
12232:
12229:
12225:
12222:
12219:
12214:
12210:
12204:
12200:
12196:
12193:
12190:
12187:
12182:
12178:
12172:
12168:
12163:
12157:
12152:
12151:
12131:
12128:
12125:
12122:
12119:
12114:
12111:
12108:
12103:
12100:
12097:
12094:
12091:
12087:
12074:
12066:
12062:
12059:
12056:
12053:
12049:
12043:
12030:
12019:
12016:
12013:
12010:
12007:
12002:
11999:
11995:
11991:
11987:
11983:
11980:
11977:
11974:
11971:
11967:
11961:
11943:
11942:
11930:
11927:
11924:
11921:
11918:
11915:
11912:
11909:
11906:
11903:
11900:
11897:
11894:
11891:
11888:
11885:
11882:
11879:
11876:
11873:
11870:
11867:
11864:
11861:
11858:
11855:
11852:
11849:
11846:
11843:
11840:
11837:
11834:
11831:
11828:
11825:
11822:
11819:
11816:
11806:
11794:
11791:
11788:
11783:
11779:
11775:
11772:
11767:
11763:
11759:
11756:
11753:
11750:
11745:
11741:
11737:
11734:
11729:
11725:
11721:
11718:
11715:
11712:
11709:
11706:
11701:
11697:
11693:
11688:
11684:
11680:
11677:
11674:
11671:
11668:
11665:
11660:
11656:
11652:
11647:
11643:
11639:
11635:
11631:
11628:
11625:
11620:
11616:
11612:
11607:
11603:
11599:
11596:
11593:
11590:
11585:
11581:
11577:
11572:
11568:
11563:
11559:
11537:
11534:
11531:
11528:
11525:
11522:
11510:
11503:
11500:
11497:
11494:
11491:
11488:
11485:
11473:
11471:
11468:
11453:
11450:
11447:
11444:
11441:
11438:
11435:
11432:
11429:
11426:
11423:
11418:
11415:
11409:
11403:
11400:
11397:
11394:
11391:
11388:
11385:
11382:
11379:
11376:
11373:
11368:
11363:
11360:
11357:
11354:
11350:
11346:
11343:
11340:
11337:
11334:
11331:
11328:
11325:
11322:
11319:
11316:
11313:
11310:
11307:
11293:Fourier series
11265:
11260:
11257:
11253:
11249:
11246:
11226:
11223:
11220:
11217:
11197:
11194:
11191:
11188:
11185:
11182:
11177:
11172:
11169:
11164:
11161:
11157:
11153:
11150:
11124:
11121:
11117:
11113:
11110:
11088:
11084:
11062:
11058:
11055:
11033:
11030:
11027:
11023:
10998:
10995:
10991:
10987:
10984:
10981:
10978:
10973:
10968:
10965:
10962:
10959:
10955:
10951:
10948:
10945:
10942:
10939:
10936:
10933:
10928:
10923:
10920:
10917:
10914:
10911:
10896:
10893:
10880:
10877:
10874:
10871:
10845:
10841:
10811:
10808:
10804:
10800:
10797:
10794:
10791:
10786:
10781:
10778:
10775:
10772:
10768:
10755:
10748:
10745:
10742:
10739:
10719:
10716:
10713:
10710:
10705:
10701:
10697:
10692:
10689:
10685:
10680:
10677:
10674:
10671:
10666:
10661:
10658:
10655:
10652:
10648:
10642:
10638:
10634:
10629:
10626:
10623:
10620:
10617:
10613:
10608:
10605:
10602:
10599:
10594:
10589:
10586:
10583:
10580:
10576:
10548:
10544:
10539:
10536:
10533:
10530:
10527:
10524:
10519:
10514:
10511:
10508:
10505:
10501:
10480:
10477:
10474:
10471:
10449:
10445:
10441:
10438:
10435:
10432:
10429:
10405:
10401:
10398:
10395:
10392:
10386:
10381:
10378:
10374:
10370:
10367:
10346:
10342:
10339:
10328:time-invariant
10320:eigenfunctions
10306:
10302:
10299:
10277:
10274:
10271:
10267:
10263:
10258:
10254:
10238:, a constant.
10223:
10197:
10194:
10191:
10188:
10185:
10180:
10162:
10159:
10112:
10108:
10096:
10095:
10092:
10091:
10072:
10069:
10066:
10063:
10060:
10057:
10054:
10051:
10048:
10045:
10042:
10039:
10034:
10029:
10026:
10023:
10020:
10016:
10012:
10002:
9999:
9998:
9987:
9984:
9981:
9978:
9975:
9972:
9969:
9966:
9963:
9960:
9957:
9952:
9947:
9944:
9941:
9938:
9934:
9930:
9920:
9909:
9906:
9903:
9900:
9867:
9864:
9861:
9858:
9855:
9852:
9849:
9837:
9830:
9828:
9826:
9823:
9820:
9814:
9811:
9808:
9805:
9802:
9799:
9794:
9791:
9788:
9784:
9773:
9766:
9764:
9762:
9759:
9753:
9750:
9747:
9744:
9741:
9738:
9735:
9732:
9727:
9723:
9719:
9718:
9689:
9686:
9683:
9680:
9677:
9674:
9671:
9668:
9665:
9662:
9657:
9653:
9632:
9629:
9626:
9623:
9620:
9615:
9611:
9577:
9574:
9571:
9565:
9562:
9559:
9556:
9553:
9550:
9547:
9544:
9539:
9535:
9531:
9528:
9525:
9522:
9519:
9514:
9509:
9506:
9503:
9500:
9496:
9492:
9489:
9487:
9485:
9481:
9477:
9471:
9468:
9465:
9462:
9459:
9456:
9453:
9450:
9447:
9444:
9441:
9438:
9433:
9428:
9425:
9422:
9419:
9415:
9410:
9404:
9400:
9396:
9393:
9391:
9389:
9386:
9383:
9378:
9374:
9370:
9367:
9364:
9361:
9358:
9355:
9354:
9331:
9328:
9325:
9303:
9300:
9297:
9294:
9291:
9288:
9285:
9282:
9279:
9276:
9273:
9270:
9265:
9260:
9257:
9254:
9251:
9247:
9243:
9240:
9237:
9234:
9231:
9211:
9208:
9205:
9192:
9191:
9182:
9180:
9164:
9161:
9158:
9155:
9150:
9147:
9144:
9140:
9127:
9120:
9118:
9116:
9113:
9110:
9107:
9104:
9101:
9098:
9088:
9081:
9079:
9077:
9074:
9068:
9065:
9062:
9059:
9056:
9053:
9050:
9047:
9042:
9038:
9034:
9033:
9015:
9014:
9005:
9003:
8992:
8989:
8984:
8980:
8976:
8971:
8967:
8963:
8958:
8954:
8948:
8943:
8940:
8937:
8934:
8930:
8926:
8922:
8918:
8912:
8909:
8906:
8903:
8898:
8894:
8890:
8885:
8881:
8875:
8870:
8867:
8864:
8861:
8857:
8852:
8846:
8842:
8827:must satisfy:
8816:
8794:
8791:
8788:
8782:
8779:
8776:
8773:
8770:
8767:
8762:
8758:
8745:
8738:
8735:
8732:
8729:
8705:
8702:
8699:
8696:
8693:
8690:
8687:
8684:
8681:
8678:
8629:
8626:
8623:
8620:
8615:
8611:
8598:
8591:
8588:
8585:
8582:
8562:
8542:
8539:
8536:
8533:
8513:
8510:
8507:
8485:
8482:
8479:
8473:
8470:
8467:
8464:
8461:
8458:
8438:
8435:
8432:
8410:
8407:
8404:
8396:
8393:
8390:
8387:
8384:
8381:
8378:
8375:
8355:
8352:
8346:
8343:
8340:
8337:
8334:
8331:
8328:
8325:
8313:
8310:
8294:Nyquist filter
8273:
8269:
8265:
8262:
8258:
8254:
8251:
8248:
8245:
8242:
8230:
8221:
8217:
8188:
8185:
8182:
8179:
8163:
8160:
8155:
8152:
8139:
8136:
8133:
8109:
8106:
8103:
8071:
8068:
8065:
8062:
8036:
8033:
8030:
8027:
8023:
8017:
8013:
8010:
8007:
8004:
8000:
7994:
7989:
7986:
7982:
7978:
7973:
7969:
7965:
7962:
7959:
7956:
7953:
7929:
7926:
7923:
7920:
7900:
7897:
7894:
7891:
7871:
7868:
7865:
7862:
7836:
7833:
7828:
7824:
7820:
7817:
7814:
7811:
7808:
7783:
7780:
7775:
7771:
7767:
7764:
7761:
7758:
7755:
7735:BIBO stability
7733:Main article:
7730:
7727:
7706:
7703:
7700:
7697:
7675:
7672:
7669:
7666:
7663:
7659:
7656:
7653:
7650:
7647:
7644:
7629:Main article:
7626:
7623:
7617:
7614:
7612:
7611:
7598:
7593:
7588:
7585:
7580:
7576:
7572:
7568:
7559:
7557:
7554:
7553:
7550:
7545:
7542:
7537:
7533:
7529:
7525:
7516:
7514:
7511:
7510:
7508:
7494:
7487:
7484:
7481:
7478:
7458:
7455:
7451:
7446:
7443:
7440:
7437:
7433:
7427:
7424:
7419:
7416:
7413:
7407:
7403:
7398:
7393:
7390:
7386:
7382:
7378:
7374:
7371:
7368:
7365:
7361:
7355:
7333:
7330:
7327:
7324:
7299:
7273:
7270:
7267:
7264:
7261:
7258:
7255:
7252:
7249:
7244:
7239:
7236:
7234:
7232:
7229:
7225:
7220:
7217:
7214:
7211:
7206:
7203:
7200:
7197:
7194:
7191:
7188:
7183:
7180:
7177:
7174:
7171:
7168:
7165:
7161:
7157:
7154:
7152:
7150:
7147:
7143:
7138:
7135:
7132:
7129:
7126:
7123:
7118:
7115:
7112:
7107:
7104:
7101:
7097:
7093:
7090:
7088:
7085:
7081:
7078:
7075:
7072:
7069:
7066:
7062:
7056:
7051:
7050:
7026:
7023:
7020:
7017:
7014:
7009:
7005:
7001:
6996:
6989:
6985:
6981:
6978:
6975:
6972:
6969:
6964:
6960:
6956:
6951:
6944:
6940:
6936:
6933:
6931:
6929:
6926:
6922:
6917:
6914:
6911:
6906:
6902:
6896:
6893:
6890:
6885:
6882:
6879:
6875:
6869:
6865:
6861:
6858:
6854:
6849:
6846:
6843:
6838:
6834:
6828:
6825:
6822:
6817:
6814:
6811:
6807:
6801:
6797:
6793:
6790:
6788:
6786:
6783:
6779:
6774:
6771:
6768:
6765:
6760:
6756:
6750:
6746:
6742:
6739:
6736:
6733:
6728:
6724:
6718:
6714:
6710:
6705:
6702:
6699:
6694:
6691:
6688:
6684:
6680:
6677:
6675:
6673:
6670:
6667:
6664:
6659:
6655:
6649:
6645:
6641:
6638:
6635:
6632:
6627:
6623:
6617:
6613:
6609:
6604:
6599:
6598:
6578:
6575:
6571:
6566:
6563:
6560:
6557:
6552:
6549:
6546:
6541:
6538:
6535:
6531:
6518:
6510:
6506:
6503:
6500:
6497:
6493:
6487:
6474:
6461:
6458:
6455:
6452:
6449:
6446:
6442:
6438:
6435:
6432:
6429:
6423:
6419:
6413:
6406:
6400:
6383:
6382:
6370:
6367:
6364:
6361:
6358:
6354:
6351:
6347:
6344:
6341:
6338:
6335:
6332:
6329:
6323:
6319:
6313:
6300:
6288:
6285:
6282:
6278:
6274:
6270:
6264:
6260:
6256:
6253:
6250:
6247:
6243:
6239:
6235:
6229:
6225:
6221:
6217:
6213:
6210:
6207:
6202:
6198:
6192:
6188:
6184:
6181:
6178:
6175:
6170:
6166:
6160:
6156:
6151:
6144:
6140:
6134:
6113:
6111:
6108:
6052:
6049:
6046:
6043:
6040:
6037:
6034:
6031:
6028:
6025:
6022:
6017:
6014:
6008:
5993:
5986:
5982:
5977:
5974:
5971:
5968:
5965:
5962:
5959:
5956:
5953:
5950:
5945:
5940:
5937:
5933:
5920:
5913:
5910:
5907:
5904:
5901:
5898:
5895:
5892:
5889:
5886:
5883:
5880:
5877:
5817:
5814:
5811:
5808:
5805:
5785:
5782:
5779:
5776:
5756:
5753:
5750:
5747:
5744:
5741:
5736:
5731:
5728:
5725:
5722:
5719:
5716:
5691:
5688:
5674:
5667:
5646:
5642:
5639:
5617:
5614:
5611:
5607:
5586:
5582:
5575:
5572:
5569:
5565:
5561:
5558:
5555:
5552:
5547:
5542:
5538:
5525:
5518:
5515:
5512:
5509:
5506:
5503:
5498:
5484:
5477:
5474:
5471:
5468:
5453:
5450:
5435:
5432:
5429:
5425:
5404:
5401:
5398:
5395:
5392:
5389:
5386:
5381:
5378:
5375:
5371:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5326:
5323:
5320:
5300:
5297:
5294:
5291:
5288:
5285:
5282:
5277:
5274:
5271:
5267:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5236:
5213:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5160:
5157:
5154:
5149:
5145:
5141:
5138:
5114:
5111:
5108:
5104:
5083:
5080:
5077:
5074:
5071:
5068:
5065:
5060:
5057:
5054:
5050:
5046:
5043:
5040:
5037:
5034:
5029:
5025:
5021:
5018:
4994:
4991:
4988:
4985:
4982:
4979:
4976:
4972:
4968:
4965:
4962:
4959:
4954:
4950:
4927:
4924:
4921:
4917:
4913:
4910:
4907:
4904:
4901:
4886:
4883:
4870:
4867:
4864:
4861:
4831:
4828:
4824:
4820:
4800:
4797:
4794:
4791:
4769:
4766:
4762:
4758:
4737:
4733:
4730:
4727:
4724:
4697:
4693:
4686:
4683:
4680:
4676:
4672:
4669:
4666:
4663:
4658:
4653:
4650:
4646:
4633:
4626:
4623:
4620:
4617:
4591:
4586:
4580:
4569:
4565:
4562:
4559:
4556:
4544:
4538:
4527:
4521:
4518:
4514:
4510:
4500:
4497:
4495:
4493:
4490:
4486:
4479:
4476:
4473:
4469:
4464:
4461:
4458:
4455:
4450:
4445:
4442:
4438:
4432:
4429:
4425:
4421:
4418:
4415:
4413:
4411:
4408:
4404:
4397:
4394:
4391:
4387:
4381:
4378:
4374:
4370:
4366:
4363:
4360:
4357:
4352:
4347:
4344:
4340:
4336:
4333:
4331:
4327:
4322:
4314:
4310:
4306:
4299:
4296:
4293:
4290:
4287:
4284:
4280:
4276:
4272:
4269:
4266:
4263:
4258:
4253:
4250:
4246:
4238:
4237:
4213:
4209:
4202:
4199:
4195:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4168:
4163:
4160:
4156:
4135:
4132:
4129:
4126:
4104:
4101:
4097:
4093:
4090:
4087:
4084:
4081:
4078:
4067:time-invariant
4059:eigenfunctions
4045:
4041:
4038:
4035:
4032:
4010:
4007:
4003:
3999:
3984:, a constant.
3969:
3945:
3942:
3939:
3936:
3933:
3928:
3910:
3907:
3883:
3880:
3877:
3850:
3847:
3844:
3841:
3838:
3826:
3819:
3817:
3815:
3811:
3807:
3801:
3798:
3795:
3792:
3789:
3785:
3779:
3775:
3771:
3768:
3766:
3764:
3760:
3756:
3750:
3747:
3743:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3703:
3698:
3695:
3691:
3686:
3680:
3676:
3672:
3669:
3667:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3640:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3578:
3574:
3553:
3550:
3547:
3544:
3541:
3536:
3532:
3498:
3495:
3491:
3486:
3483:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3451:
3447:
3443:
3440:
3437:
3434:
3431:
3426:
3421:
3418:
3414:
3410:
3407:
3405:
3403:
3400:
3396:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3356:
3351:
3348:
3344:
3340:
3337:
3335:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3308:
3297:
3296:
3293:
3292:
3271:
3268:
3265:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3233:
3229:
3225:
3215:
3212:
3211:
3200:
3197:
3191:
3188:
3185:
3182:
3179:
3176:
3171:
3168:
3165:
3161:
3148:
3134:
3123:
3120:
3117:
3114:
3111:
3108:
3079:
3076:
3073:
3067:
3064:
3061:
3058:
3055:
3052:
3047:
3043:
3030:
3023:
3020:
3017:
3014:
2997:
2996:
2987:
2985:
2969:
2966:
2963:
2960:
2955:
2952:
2949:
2945:
2932:
2925:
2923:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2893:
2886:
2884:
2882:
2879:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2847:
2843:
2839:
2838:
2820:
2819:
2810:
2808:
2797:
2794:
2790:
2783:
2778:
2774:
2770:
2765:
2761:
2754:
2750:
2747:
2744:
2739:
2735:
2722:
2718:
2712:
2707:
2704:
2700:
2696:
2692:
2688:
2682:
2679:
2675:
2670:
2667:
2664:
2659:
2655:
2646:
2642:
2636:
2631:
2628:
2624:
2619:
2613:
2609:
2581:{\textstyle O}
2577:
2558:{\textstyle t}
2554:
2538:{\textstyle t}
2534:
2518:{\textstyle x}
2514:
2494:
2491:
2488:
2485:
2469:{\textstyle t}
2465:
2443:
2439:
2418:
2415:
2412:
2409:
2404:
2400:
2387:
2380:
2377:
2374:
2371:
2351:
2348:
2345:
2325:
2322:
2319:
2316:
2296:
2293:
2287:
2284:
2281:
2278:
2275:
2272:
2252:
2249:
2246:
2226:
2210:{\textstyle u}
2206:
2197:with variable
2186:
2183:
2180:
2177:
2174:
2171:
2151:
2148:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2098:{\textstyle t}
2094:
2085:prior to time
2078:{\textstyle x}
2074:
2054:
2051:
2048:
2045:
2025:
2005:
2002:
1999:
1996:
1980:{\textstyle t}
1976:
1960:{\textstyle t}
1956:
1936:
1933:
1930:
1927:
1924:
1904:
1901:
1898:
1895:
1875:
1872:
1869:
1866:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1798:
1795:
1792:
1789:
1778:
1777:
1774:
1773:
1754:
1751:
1747:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1707:
1702:
1699:
1695:
1691:
1681:
1678:
1677:
1666:
1662:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1622:
1617:
1614:
1610:
1601:
1598:
1595:
1589:
1575:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1493:
1490:
1488:
1485:
1481:Green function
1407:
1385:
1381:
1376:
1370:
1366:
1343:
1339:
1316:
1313:
1309:
1303:
1299:
1278:
1256:
1252:
1229:
1226:
1222:
1216:
1212:
1193:eigenfunctions
1122:
1118:
1114:
1109:
1105:
1101:
1096:
1092:
1063:
1060:
1057:
1054:
1034:
1031:
1028:
1025:
1012:is simply the
1001:
998:
995:
992:
977:
976:
964:
961:
958:
955:
952:
949:
929:
926:
923:
920:
917:
914:
894:
891:
888:
885:
865:
862:
859:
856:
834:
818:
815:
812:
808:
805:
783:
780:
777:
774:
751:
731:
728:
725:
721:
718:
714:
711:
708:
705:
702:
682:
679:
676:
672:
669:
665:
662:
659:
656:
653:
633:
630:
627:
623:
620:
599:
596:
593:
589:
586:
565:
562:
559:
556:
553:
533:
530:
527:
524:
521:
501:
481:
478:
475:
472:
452:
449:
446:
443:
416:
413:
389:control theory
332:consisting of
212:
201:
192:
185:
178:
164:
153:
147:
140:
129:
123:
112:
88:
87:
42:
40:
33:
26:
9:
6:
4:
3:
2:
13754:
13743:
13740:
13738:
13735:
13733:
13730:
13728:
13725:
13723:
13720:
13718:
13715:
13714:
13712:
13702:
13699:
13696:
13693:
13690:
13687:
13684:
13681:
13680:
13669:
13665:
13661:
13657:
13653:
13649:
13642:
13637:
13633:
13627:
13623:
13619:
13615:
13614:
13603:
13599:
13595:
13591:
13587:
13583:
13576:
13571:
13561:
13557:
13552:
13548:
13542:
13538:
13533:
13529:
13523:
13515:
13509:
13505:
13501:
13496:
13495:
13487:
13484:
13482:
13479:
13477:
13474:
13472:
13469:
13467:
13464:
13462:
13459:
13458:
13444:
13438:
13432:
13426:
13420:
13411:
13403:
13401:0-387-23488-8
13397:
13393:
13386:
13382:
13374:
13372:
13356:
13353:
13345:
13332:
13328:
13324:
13319:
13306:
13300:
13289:
13283:
13265:
13262:
13259:
13255:
13242:
13233:
13222:
13216:
13190:
13184:
13161:
13155:
13132:
13126:
13118:
13113:
13097:
13081:
13075:
13065:implies that
13063:
13047:
13031:
13025:
13014:
13008:
12998:
12996:
12977:
12971:
12951:
12948:
12945:
12942:
12933:
12930:
12924:
12918:
12908:
12907:Causal system
12898:
12896:
12877:
12871:
12837:
12831:
12828:
12825:
12818:
12815:
12812:
12801:
12799:
12787:
12784:
12777:
12772:
12769:
12763:
12760:
12757:
12749:
12746:
12740:
12737:
12734:
12728:
12724:
12721:
12716:
12712:
12710:
12699:
12696:
12693:
12687:
12682:
12679:
12676:
12671:
12668:
12665:
12662:
12659:
12655:
12651:
12649:
12643:
12636:
12633:
12630:
12624:
12620:
12584:
12580:
12573:
12565:
12561:
12556:
12543:
12539:
12535:
12531:
12524:
12516:
12512:
12507:
12494:
12490:
12486:
12484:
12473:
12465:
12461:
12455:
12452:
12449:
12444:
12441:
12438:
12435:
12432:
12428:
12422:
12418:
12414:
12408:
12400:
12396:
12390:
12387:
12384:
12379:
12376:
12373:
12370:
12367:
12363:
12357:
12353:
12349:
12347:
12338:
12331:
12323:
12319:
12313:
12309:
12305:
12299:
12291:
12287:
12281:
12277:
12272:
12266:
12263:
12260:
12255:
12252:
12249:
12246:
12243:
12239:
12235:
12233:
12227:
12220:
12212:
12208:
12202:
12198:
12194:
12188:
12180:
12176:
12170:
12166:
12161:
12129:
12123:
12117:
12112:
12109:
12106:
12101:
12098:
12095:
12092:
12089:
12085:
12072:
12064:
12057:
12051:
12047:
12031:
12017:
12011:
12005:
12000:
11997:
11993:
11989:
11985:
11978:
11972:
11969:
11965:
11950:
11948:
11925:
11922:
11919:
11910:
11904:
11901:
11895:
11892:
11886:
11883:
11880:
11871:
11868:
11862:
11859:
11856:
11853:
11850:
11844:
11841:
11832:
11829:
11826:
11820:
11814:
11807:
11789:
11781:
11777:
11773:
11770:
11765:
11761:
11757:
11751:
11743:
11739:
11735:
11732:
11727:
11723:
11719:
11713:
11710:
11707:
11699:
11695:
11691:
11686:
11682:
11678:
11672:
11669:
11666:
11658:
11654:
11650:
11645:
11641:
11637:
11633:
11626:
11618:
11614:
11610:
11605:
11601:
11597:
11591:
11583:
11579:
11575:
11570:
11566:
11561:
11557:
11550:
11549:
11532:
11529:
11526:
11520:
11508:
11495:
11489:
11483:
11475:
11474:
11467:
11464:
11451:
11442:
11436:
11430:
11424:
11416:
11413:
11401:
11395:
11389:
11383:
11380:
11377:
11371:
11358:
11355:
11352:
11348:
11344:
11338:
11329:
11326:
11323:
11317:
11311:
11305:
11296:
11294:
11289:
11287:
11283:
11279:
11258:
11255:
11251:
11244:
11221:
11215:
11189:
11183:
11170:
11162:
11159:
11155:
11148:
11140:
11122:
11119:
11115:
11111:
11108:
11086:
11082:
11056:
11053:
11031:
11028:
11025:
11021:
11011:
10996:
10993:
10989:
10982:
10976:
10963:
10960:
10957:
10953:
10949:
10940:
10934:
10921:
10915:
10909:
10902:
10892:
10875:
10869:
10861:
10860:eigenfunction
10843:
10839:
10829:
10827:
10809:
10806:
10802:
10795:
10789:
10776:
10773:
10770:
10766:
10753:
10743:
10737:
10714:
10708:
10703:
10699:
10695:
10690:
10687:
10683:
10675:
10669:
10656:
10653:
10650:
10646:
10640:
10636:
10632:
10624:
10621:
10618:
10611:
10603:
10597:
10584:
10581:
10578:
10574:
10566:
10561:
10546:
10542:
10534:
10531:
10528:
10522:
10509:
10506:
10503:
10499:
10475:
10469:
10447:
10443:
10439:
10433:
10427:
10418:
10399:
10396:
10393:
10390:
10384:
10379:
10376:
10372:
10368:
10365:
10340:
10337:
10329:
10325:
10321:
10300:
10297:
10275:
10272:
10269:
10265:
10261:
10256:
10252:
10244:
10239:
10237:
10221:
10213:
10208:
10195:
10192:
10189:
10186:
10183:
10168:
10167:eigenfunction
10158:
10156:
10152:
10148:
10144:
10140:
10136:
10132:
10128:
10110:
10106:
10089:
10088:commutativity
10070:
10064:
10058:
10055:
10049:
10046:
10043:
10037:
10024:
10021:
10018:
10014:
10010:
10003:
10001:
10000:
9982:
9979:
9976:
9970:
9967:
9961:
9955:
9942:
9939:
9936:
9932:
9928:
9921:
9904:
9898:
9891:
9890:
9887:
9886:
9885:
9882:
9865:
9859:
9856:
9853:
9847:
9835:
9829:
9818:
9812:
9806:
9800:
9792:
9789:
9786:
9782:
9771:
9765:
9757:
9751:
9745:
9742:
9739:
9733:
9725:
9721:
9708:
9707:
9701:
9684:
9681:
9678:
9672:
9669:
9663:
9655:
9651:
9627:
9621:
9618:
9613:
9609:
9601:for the case
9600:
9599:
9593:
9575:
9569:
9563:
9557:
9554:
9551:
9545:
9537:
9533:
9529:
9523:
9517:
9504:
9501:
9498:
9494:
9490:
9488:
9479:
9475:
9469:
9463:
9460:
9457:
9451:
9448:
9442:
9436:
9423:
9420:
9417:
9413:
9408:
9402:
9398:
9394:
9392:
9384:
9376:
9372:
9368:
9362:
9356:
9343:
9326:
9314:
9301:
9295:
9292:
9289:
9283:
9280:
9274:
9268:
9255:
9252:
9249:
9245:
9241:
9235:
9229:
9206:
9190:
9183:
9181:
9162:
9156:
9148:
9145:
9142:
9138:
9125:
9119:
9108:
9105:
9102:
9096:
9086:
9080:
9072:
9066:
9060:
9057:
9054:
9048:
9040:
9036:
9024:
9023:
9020:
9013:
9006:
9004:
8990:
8982:
8978:
8969:
8965:
8961:
8956:
8952:
8938:
8935:
8932:
8928:
8924:
8920:
8916:
8910:
8904:
8896:
8892:
8888:
8883:
8879:
8865:
8862:
8859:
8855:
8850:
8844:
8840:
8832:
8831:
8828:
8814:
8805:
8792:
8786:
8780:
8774:
8768:
8760:
8756:
8743:
8733:
8727:
8719:
8703:
8697:
8691:
8688:
8682:
8676:
8668:
8663:
8661:
8657:
8653:
8649:
8645:
8640:
8627:
8621:
8613:
8609:
8596:
8586:
8580:
8560:
8540:
8534:
8508:
8496:
8483:
8477:
8471:
8465:
8459:
8433:
8421:
8408:
8402:
8394:
8388:
8385:
8382:
8376:
8350:
8344:
8338:
8335:
8332:
8326:
8309:
8307:
8303:
8299:
8295:
8291:
8287:
8271:
8263:
8260:
8249:
8246:
8240:
8228:
8219:
8215:
8206:
8202:
8183:
8177:
8168:
8159:
8151:
8137:
8134:
8131:
8123:
8107:
8104:
8101:
8092:
8091:sinc function
8088:
8083:
8069:
8066:
8063:
8060:
8052:
8047:
8034:
8028:
8025:
8008:
8002:
7984:
7980:
7976:
7971:
7960:
7954:
7943:
7924:
7918:
7895:
7889:
7866:
7860:
7852:
7847:
7831:
7815:
7809:
7794:
7778:
7762:
7756:
7742:
7736:
7726:
7724:
7720:
7701:
7695:
7686:
7673:
7670:
7667:
7664:
7657:
7654:
7648:
7642:
7632:
7631:Causal system
7622:
7591:
7586:
7583:
7578:
7570:
7555:
7548:
7543:
7540:
7535:
7527:
7512:
7506:
7492:
7482:
7456:
7453:
7441:
7435:
7431:
7425:
7422:
7417:
7414:
7411:
7405:
7388:
7384:
7380:
7376:
7369:
7363:
7359:
7328:
7315:
7271:
7265:
7262:
7259:
7250:
7237:
7235:
7227:
7215:
7209:
7204:
7201:
7195:
7192:
7189:
7181:
7178:
7172:
7169:
7166:
7159:
7155:
7153:
7145:
7133:
7130:
7127:
7121:
7116:
7113:
7110:
7105:
7102:
7099:
7095:
7091:
7089:
7083:
7076:
7073:
7070:
7064:
7060:
7024:
7015:
7007:
7003:
6987:
6983:
6979:
6970:
6962:
6958:
6942:
6938:
6934:
6932:
6924:
6912:
6904:
6900:
6894:
6891:
6888:
6883:
6880:
6877:
6873:
6867:
6863:
6859:
6856:
6844:
6836:
6832:
6826:
6823:
6820:
6815:
6812:
6809:
6805:
6799:
6795:
6791:
6789:
6781:
6766:
6758:
6754:
6748:
6744:
6740:
6734:
6726:
6722:
6716:
6712:
6703:
6700:
6697:
6692:
6689:
6686:
6682:
6678:
6676:
6665:
6657:
6653:
6647:
6643:
6639:
6633:
6625:
6621:
6615:
6611:
6576:
6573:
6561:
6555:
6550:
6547:
6544:
6539:
6536:
6533:
6529:
6516:
6508:
6501:
6495:
6491:
6475:
6472:
6456:
6450:
6447:
6444:
6440:
6433:
6427:
6421:
6404:
6388:
6365:
6362:
6359:
6352:
6349:
6345:
6339:
6336:
6333:
6327:
6321:
6301:
6283:
6276:
6272:
6268:
6262:
6258:
6254:
6248:
6241:
6237:
6233:
6227:
6223:
6219:
6215:
6208:
6200:
6196:
6190:
6186:
6182:
6176:
6168:
6164:
6158:
6154:
6149:
6142:
6122:
6121:
6119:
6115:
6114:
6107:
6105:
6101:
6097:
6093:
6089:
6086:, we obtain |
6084:
6080:
6074:
6070:
6063:
6050:
6041:
6035:
6029:
6023:
6015:
6012:
5991:
5984:
5972:
5966:
5960:
5957:
5954:
5948:
5935:
5931:
5918:
5908:
5899:
5896:
5893:
5887:
5881:
5875:
5865:
5863:
5859:
5853:
5851:
5846:
5841:
5839:
5835:
5831:
5812:
5809:
5803:
5780:
5774:
5748:
5742:
5729:
5723:
5720:
5714:
5707:
5689:
5686:
5672:
5665:
5640:
5637:
5615:
5612:
5609:
5605:
5584:
5573:
5570:
5567:
5563:
5556:
5550:
5540:
5536:
5523:
5510:
5504:
5482:
5472:
5466:
5459:
5449:
5433:
5430:
5427:
5423:
5399:
5390:
5384:
5379:
5376:
5373:
5369:
5365:
5359:
5350:
5344:
5324:
5321:
5318:
5295:
5286:
5280:
5275:
5272:
5269:
5265:
5261:
5255:
5252:
5249:
5240:
5234:
5225:
5211:
5188:
5185:
5182:
5173:
5167:
5164:
5158:
5147:
5143:
5136:
5128:
5112:
5109:
5106:
5102:
5078:
5069:
5063:
5058:
5055:
5052:
5048:
5044:
5038:
5027:
5023:
5016:
5008:
4989:
4986:
4983:
4977:
4974:
4970:
4966:
4960:
4952:
4948:
4925:
4922:
4919:
4915:
4911:
4905:
4899:
4890:
4882:
4865:
4859:
4851:
4847:
4846:eigenfunction
4829:
4826:
4822:
4818:
4795:
4789:
4767:
4764:
4760:
4756:
4731:
4728:
4725:
4722:
4713:
4711:
4695:
4684:
4681:
4678:
4674:
4667:
4661:
4648:
4644:
4631:
4621:
4615:
4606:
4589:
4584:
4578:
4567:
4560:
4554:
4542:
4536:
4525:
4519:
4516:
4512:
4508:
4498:
4496:
4488:
4477:
4474:
4471:
4467:
4459:
4453:
4440:
4436:
4430:
4427:
4423:
4419:
4416:
4414:
4406:
4395:
4392:
4389:
4385:
4379:
4376:
4372:
4368:
4361:
4355:
4342:
4338:
4334:
4332:
4325:
4312:
4308:
4294:
4291:
4288:
4282:
4278:
4274:
4267:
4261:
4248:
4244:
4227:
4211:
4200:
4197:
4193:
4189:
4183:
4180:
4177:
4171:
4158:
4154:
4130:
4124:
4102:
4099:
4095:
4091:
4088:
4082:
4076:
4068:
4064:
4060:
4039:
4036:
4033:
4030:
4008:
4005:
4001:
3997:
3990:
3985:
3983:
3967:
3959:
3943:
3940:
3937:
3934:
3931:
3916:
3915:eigenfunction
3906:
3903:
3901:
3878:
3866:
3848:
3842:
3836:
3824:
3818:
3809:
3805:
3799:
3793:
3787:
3783:
3777:
3773:
3769:
3767:
3758:
3754:
3748:
3745:
3733:
3730:
3727:
3721:
3718:
3712:
3706:
3693:
3689:
3684:
3678:
3674:
3670:
3668:
3660:
3651:
3648:
3645:
3630:
3629:
3624:
3611:
3605:
3602:
3599:
3593:
3590:
3584:
3576:
3572:
3548:
3542:
3539:
3534:
3530:
3522:for the case
3521:
3520:
3514:
3496:
3493:
3481:
3475:
3469:
3466:
3463:
3457:
3449:
3445:
3441:
3435:
3429:
3416:
3412:
3408:
3406:
3398:
3386:
3383:
3380:
3374:
3371:
3365:
3359:
3346:
3342:
3338:
3336:
3328:
3319:
3316:
3313:
3290:
3289:
3269:
3263:
3257:
3251:
3248:
3245:
3239:
3231:
3227:
3223:
3216:
3214:
3213:
3195:
3189:
3183:
3177:
3169:
3166:
3163:
3159:
3146:
3135:
3118:
3115:
3112:
3106:
3099:
3098:
3095:
3094:
3093:
3090:
3077:
3071:
3065:
3059:
3053:
3045:
3041:
3028:
3018:
3012:
3004:
2995:
2988:
2986:
2967:
2961:
2953:
2950:
2947:
2943:
2930:
2924:
2913:
2910:
2907:
2901:
2891:
2885:
2877:
2871:
2865:
2862:
2859:
2853:
2845:
2841:
2829:
2828:
2825:
2818:
2811:
2809:
2795:
2792:
2776:
2772:
2763:
2759:
2752:
2745:
2737:
2733:
2720:
2716:
2702:
2698:
2694:
2690:
2686:
2680:
2677:
2665:
2657:
2653:
2644:
2640:
2626:
2622:
2617:
2611:
2607:
2599:
2598:
2595:
2593:
2592:
2588:must satisfy
2575:
2566:
2552:
2532:
2512:
2489:
2483:
2463:
2441:
2437:
2416:
2410:
2402:
2398:
2385:
2375:
2369:
2346:
2323:
2317:
2291:
2285:
2279:
2273:
2247:
2224:
2217:and constant
2204:
2181:
2178:
2175:
2169:
2146:
2140:
2134:
2131:
2128:
2122:
2110:
2108:
2092:
2072:
2049:
2043:
2023:
2000:
1994:
1974:
1954:
1931:
1928:
1922:
1899:
1893:
1870:
1864:
1841:
1835:
1832:
1826:
1820:
1812:
1793:
1787:
1771:
1770:commutativity
1752:
1749:
1737:
1734:
1731:
1725:
1722:
1716:
1710:
1697:
1693:
1689:
1682:
1680:
1679:
1664:
1652:
1646:
1643:
1637:
1634:
1631:
1625:
1612:
1608:
1587:
1576:
1559:
1550:
1547:
1544:
1538:
1532:
1526:
1519:
1518:
1515:
1514:
1513:
1511:
1507:
1503:
1499:
1484:
1482:
1477:
1475:
1471:
1467:
1461:
1459:
1455:
1451:
1447:
1443:
1439:
1434:
1432:
1428:
1424:
1419:
1405:
1383:
1379:
1374:
1368:
1364:
1356:. The ratio
1341:
1337:
1314:
1311:
1307:
1301:
1297:
1276:
1254:
1250:
1227:
1224:
1220:
1214:
1210:
1201:
1198:
1194:
1189:
1186:
1182:
1178:
1174:
1173:
1165:
1161:
1156:
1152:
1150:
1146:
1142:
1138:
1120:
1116:
1112:
1107:
1103:
1099:
1094:
1090:
1081:
1080:discrete time
1077:
1058:
1052:
1029:
1023:
1015:
996:
990:
982:
959:
956:
953:
947:
924:
921:
918:
912:
889:
883:
860:
854:
846:
842:
838:
835:
832:
813:
806:
803:
794:
778:
772:
764:
749:
726:
719:
716:
712:
706:
700:
677:
670:
667:
663:
657:
651:
628:
621:
618:
594:
587:
584:
560:
554:
551:
528:
522:
519:
499:
476:
470:
447:
441:
433:
430:
429:
428:
426:
422:
412:
410:
406:
402:
398:
394:
390:
386:
385:filter design
382:
378:
374:
370:
366:
362:
361:discrete-time
358:
354:
349:
347:
343:
339:
335:
331:
325:
321:
316:
310:
306:
299:
295:
291:
287:
283:
278:
272:
268:
261:
257:
252:
248:
244:
240:
236:
232:
228:
218:
211:
207:
200:
191:
184:
177:
169:for all time
163:
159:
152:
146:
139:
135:
128:
122:
118:
111:
106:
102:
98:
97:Block diagram
94:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
13651:
13647:
13621:
13585:
13581:
13565:November 21,
13563:, retrieved
13559:
13536:
13503:
13500:Riskin, E.A.
13443:
13436:
13431:
13424:
13419:
13410:
13391:
13385:
13370:
13369:for complex
13320:
13114:
13064:
13012:
13011:A system is
13010:
12910:
12894:
12863:
11946:
11944:
11465:
11297:
11290:
11285:
11281:
11277:
11012:
10898:
10830:
10825:
10562:
10419:
10240:
10211:
10209:
10164:
10150:
10146:
10142:
10138:
10134:
10130:
10126:
10097:
9883:
9704:
9702:
9596:
9594:
9344:
9315:
9195:
9184:
9018:
9007:
8806:
8717:
8664:
8659:
8655:
8651:
8647:
8643:
8641:
8497:
8422:
8315:
8297:
8285:
8169:
8165:
8157:
8084:
8048:
7848:
7795:
7740:
7739:A system is
7738:
7687:
7634:
7619:
6386:
6384:
6103:
6099:
6095:
6091:
6087:
6082:
6078:
6072:
6068:
6064:
5866:
5854:
5844:
5842:
5837:
5833:
5829:
5455:
5226:
5129:
5009:
4891:
4888:
4885:Direct proof
4714:
4709:
4607:
3986:
3957:
3912:
3904:
3899:
3867:
3626:
3625:
3517:
3515:
3298:
3286:
3091:
3002:
3000:
2989:
2823:
2812:
2589:
2567:
2111:
1779:
1509:
1505:
1501:
1497:
1495:
1478:
1476:of signals.
1466:filter banks
1462:
1435:
1420:
1200:exponentials
1190:
1170:
1168:
1163:
1159:
1144:
1140:
1136:
978:
844:
840:
836:
765:
762:
431:
424:
420:
418:
368:
356:
350:
323:
319:
308:
304:
297:
293:
289:
285:
281:
270:
266:
259:
255:
238:
234:
230:
224:
216:
209:
205:
198:
189:
182:
175:
161:
157:
150:
144:
137:
133:
126:
120:
116:
109:
76:
67:
48:
13654:(5): 1090.
13618:Porat, Boaz
13588:(6): 1090.
13333:satisfying
13329:(i.e., the
13327:unit circle
11949:. That is,
10901:Z transform
10565:convolution
9884:Therefore:
9345:Therefore:
7344:. That is,
4226:convolution
3092:Similarly:
1448:made up of
1185:Z transform
1160:time domain
1014:convolution
367:) systems,
277:convolution
62:introducing
18:LTI systems
13711:Categories
13492:References
10330:operator.
10236:eigenvalue
8449:represent
6118:derivative
5311:. Setting
4892:Let's set
4850:eigenvalue
3982:eigenvalue
2263:represent
1458:capacitors
338:capacitors
70:April 2009
45:references
13522:cite book
13437:Exercises
13304:∞
13274:∞
13269:∞
13266:−
13256:∑
13230:‖
13214:‖
13101:∞
13093:∞
13089:‖
13073:‖
13051:∞
13043:∞
13039:‖
13023:‖
13001:Stability
12940:∀
12901:Causality
12829:−
12761:−
12747:−
12738:−
12717:∑
12697:−
12669:−
12656:∑
12634:−
12442:−
12429:∑
12377:−
12364:∑
12253:−
12240:∑
12099:−
12086:∑
11998:−
11923:−
11893:−
11884:−
11860:−
11854:−
11830:−
11771:⋅
11733:⋅
11711:−
11692:⋅
11670:−
11651:⋅
11611:⋅
11576:⋅
11530:−
11414:−
11381:−
11367:∞
11362:∞
11359:−
11349:∑
11327:∗
11259:ω
11163:ω
11123:ω
11057:∈
11054:ω
11029:ω
10994:−
10972:∞
10967:∞
10964:−
10954:∑
10807:−
10785:∞
10780:∞
10777:−
10767:∑
10688:−
10665:∞
10660:∞
10657:−
10647:∑
10622:−
10593:∞
10588:∞
10585:−
10575:∑
10532:−
10518:∞
10513:∞
10510:−
10500:∑
10400:∈
10341:∈
10301:∈
10222:λ
10190:λ
10056:⋅
10047:−
10033:∞
10028:∞
10025:−
10015:∑
9980:−
9968:⋅
9951:∞
9946:∞
9943:−
9933:∑
9857:−
9801:δ
9790:−
9743:−
9734:δ
9682:−
9673:δ
9555:−
9546:δ
9530:⋅
9513:∞
9508:∞
9505:−
9495:∑
9461:−
9452:δ
9449:⋅
9432:∞
9427:∞
9424:−
9414:∑
9293:−
9284:δ
9281:⋅
9264:∞
9259:∞
9256:−
9246:∑
9242:≡
9146:−
9106:−
9058:−
8962:⋅
8947:∞
8942:∞
8939:−
8929:∑
8889:⋅
8874:∞
8869:∞
8866:−
8856:∑
8769:δ
8692:δ
8386:−
8336:−
8264:∈
8257:∀
8070:ω
8032:∞
7993:∞
7988:∞
7985:−
7981:∫
7968:‖
7952:‖
7835:∞
7827:∞
7823:‖
7807:‖
7782:∞
7774:∞
7770:‖
7754:‖
7729:Stability
7662:∀
7625:Causality
7477:Π
7454:λ
7442:λ
7415:−
7412:λ
7402:Π
7397:∞
7392:∞
7389:−
7385:∫
7323:Π
7266:τ
7263:−
7228:ξ
7216:ξ
7196:τ
7193:−
7179:−
7173:τ
7170:−
7160:∫
7146:λ
7134:τ
7131:−
7128:λ
7103:−
7096:∫
7077:τ
7074:−
6925:λ
6913:λ
6881:−
6874:∫
6857:λ
6845:λ
6813:−
6806:∫
6782:λ
6767:λ
6735:λ
6690:−
6683:∫
6574:λ
6562:λ
6537:−
6530:∫
6366:τ
6363:−
6340:τ
6337:−
6013:−
5985:τ
5973:τ
5961:τ
5958:−
5944:∞
5939:∞
5936:−
5932:∫
5897:∗
5813:ω
5724:ω
5687:−
5641:∈
5638:ω
5613:ω
5568:−
5546:∞
5537:∫
5434:τ
5431:ω
5380:τ
5377:ω
5360:τ
5273:ω
5110:ω
5056:ω
4978:ω
4923:ω
4811:. Hence,
4732:∈
4679:−
4657:∞
4652:∞
4649:−
4645:∫
4585:λ
4579:⏞
4568:⏟
4537:⏞
4526:⏟
4489:τ
4478:τ
4472:−
4460:τ
4449:∞
4444:∞
4441:−
4437:∫
4407:τ
4396:τ
4390:−
4362:τ
4351:∞
4346:∞
4343:−
4339:∫
4313:⏞
4309:τ
4295:τ
4292:−
4268:τ
4257:∞
4252:∞
4249:−
4245:∫
4212:τ
4201:τ
4184:τ
4181:−
4167:∞
4162:∞
4159:−
4155:∫
4040:∈
3968:λ
3938:λ
3900:responses
3746:τ
3734:τ
3731:−
3722:δ
3719:⋅
3713:τ
3702:∞
3697:∞
3694:−
3690:∫
3649:∗
3606:τ
3603:−
3594:δ
3577:τ
3549:τ
3535:τ
3494:τ
3470:τ
3467:−
3458:δ
3442:⋅
3436:τ
3425:∞
3420:∞
3417:−
3413:∫
3399:τ
3387:τ
3384:−
3372:⋅
3366:τ
3355:∞
3350:∞
3347:−
3343:∫
3317:∗
3252:τ
3249:−
3240:δ
3178:δ
3170:τ
3167:−
3119:τ
3116:−
3054:δ
2954:τ
2951:−
2914:τ
2911:−
2866:τ
2863:−
2793:τ
2777:τ
2753:⏟
2738:τ
2721:τ
2711:∞
2706:∞
2703:−
2699:∫
2678:τ
2658:τ
2645:τ
2635:∞
2630:∞
2627:−
2623:∫
2225:τ
2182:τ
2179:−
2135:τ
2132:−
2024:τ
2001:τ
1932:τ
1929:−
1900:τ
1842:τ
1836:δ
1827:τ
1750:τ
1738:τ
1735:−
1723:⋅
1717:τ
1706:∞
1701:∞
1698:−
1694:∫
1665:τ
1653:τ
1644:⋅
1638:τ
1635:−
1621:∞
1616:∞
1613:−
1609:∫
1548:∗
1454:inductors
1450:resistors
1438:nonlinear
1427:amplitude
1423:sinusoids
1149:sequences
1113:∗
957:−
922:−
432:Linearity
421:linearity
342:inductors
334:resistors
247:linearity
13620:(1997).
13502:(2007).
13455:See also
13425:Welcome!
12897:stable.
12788:′
12725:′
11470:Examples
11046:, where
10491:is then
10290:, where
7562:if
7519:if
6353:′
6277:′
6242:′
6110:Examples
6076:, where
4146:is then
4023:, where
1483:method.
1162:and the
807:′
720:′
671:′
622:′
588:′
415:Overview
13656:Bibcode
13590:Bibcode
11141:(DTFT)
10234:is the
8306:aliased
8288:is the
5704:). The
3980:is the
3285:(using
1811:impulse
1768:(using
1474:vectors
1197:complex
365:sampled
363:(i.e.,
58:improve
13628:
13543:
13510:
13398:
12964:where
12937:
11137:. The
10858:is an
10730:where
10388:
10324:linear
10318:, are
10210:where
10155:causal
9816:
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9567:
9473:
9070:
8914:
8784:
8475:
8348:
8284:where
7804:
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7688:where
5630:where
4844:is an
4574:Scalar
4063:linear
4057:, are
3956:where
3803:
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3479:
3261:
3193:
3069:
2875:
2726:
2684:
2650:
2429:where
2289:
2144:
2107:causal
1780:where
1456:, and
1421:Since
1143:, and
1135:where
302:where
243:system
239:system
47:, but
13644:(PDF)
13578:(PDF)
13377:Notes
13331:locus
11284:, or
11101:with
10322:of a
10137:. As
8304:) is
8298:above
5836:, or
4532:Input
4061:of a
1967:. As
1431:phase
576:; if
288:) = (
241:is a
13626:ISBN
13567:2010
13541:ISBN
13528:link
13508:ISBN
13396:ISBN
13301:<
13148:and
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13048:<
12946:<
11237:and
10241:The
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9643:and
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9187:Eq.5
9010:Eq.4
8316:Let
8135:>
8124:for
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6106:)).
5796:and
5658:and
3987:The
3896:Eq.1
3628:Eq.2
3564:and
3519:Eq.2
3288:Eq.3
2992:Eq.3
2815:Eq.2
2591:Eq.1
1470:MIMO
1468:and
1147:are
423:and
383:and
344:and
249:and
143:) +
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13664:doi
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