1562:) or over the actual realization of the input (i.e. the pseudo-random, bounded, almost-white version of gaussian white noise, or any other stimulus). The latter methods, despite their lack of mathematical elegance, have been shown to be more flexible (as arbitrary inputs can be easily accommodated) and precise (due to the effect that the idealized version of the input signal is not always realizable).
2907:
110:. He used the series to make an approximate analysis of the effect of radar noise in a nonlinear receiver circuit. The report became public after the war. As a general method of analysis of nonlinear systems, the Volterra series came into use after about 1957 as the result of a series of reports, at first privately circulated, from MIT and elsewhere. The name itself,
27:. It differs from the Taylor series in its ability to capture "memory" effects. The Taylor series can be used for approximating the response of a nonlinear system to a given input if the output of the system depends strictly on the input at that particular time. In the Volterra series, the output of the nonlinear system depends on the input to the system at
6160:
4952:
6428:
6453:) is computationally equivalent to the Volterra series and therefore contains the kernels hidden in its architecture. After such a network has been trained to successfully predict the output based on the current state and memory of the system, the kernels can then be computed from the weights and biases of that network.
4084:
2636:
5404:
1539:
3381:
1554:, and then recomputing the coefficients of the original Volterra series. The Volterra series main appeal over the orthogonalized series lies in its intuitive, canonical structure, i.e. all interactions of the input have one fixed degree. The orthogonalized basis functionals will generally be quite complicated.
4307:
6734:
are the weights from the input layer to the non-linear hidden layer. It is important to note that this method allows kernel extraction up until the number of input delays in the architecture of the network. Furthermore, it is vital to carefully construct the size of the network input layer so that it
1051:
1549:
Estimating the
Volterra coefficients individually is complicated, since the basis functionals of the Volterra series are correlated. This leads to the problem of simultaneously solving a set of integral equations for the coefficients. Hence, estimation of Volterra coefficients is generally performed
4408:
of truncated
Volterra series, can be revealed by calculating the output error of a series as a function of different variances of input. This test can be repeated with series identified with different input variances, obtaining different curves, each with a minimum in correspondence of the variance
4439:
value should be used for the lower-order kernel and gradually increased for higher-order kernels. This is not a theoretical problem in Wiener kernel identification, since the Wiener functional are orthogonal to each other, but an appropriate normalization is needed in Wiener-to-Volterra conversion
6743:
This method and its more efficient version (fast orthogonal algorithm) were invented by
Korenberg. In this method the orthogonalization is performed empirically over the actual input. It has been shown to perform more precisely than the crosscorrelation method. Another advantage is that arbitrary
779:. This theorem states that a time-invariant functional relation (satisfying certain very general conditions) can be approximated uniformly and to an arbitrary degree of precision by a sufficiently high finite-order Volterra series. Among other conditions, the set of admissible input functions
384:
6755:
is a standard tool from linear analysis. Hence, one of its main advantages is the widespread existence of standard tools for solving linear regressions efficiently. It has some educational value, since it highlights the basic property of
Volterra series: linear combination of non-linear
1871:
2524:
5875:
6622:
3143:
4686:
5701:
6168:
5409:
In the above formulas the impulse functions are introduced for the identification of diagonal kernel points. If the Wiener kernels are extracted with the new formulas, the following Wiener-to-Volterra formulas (explicited up the fifth order) are needed:
5867:
2902:{\displaystyle E\left\{y(n)\prod _{j=1}^{\overline {p}}x(n-\tau _{j})\right\}=E\left\{G_{\overline {p}}x(n)\prod _{j=1}^{\overline {p}}x(n-\tau _{j})\right\}={\overline {p}}!A^{\overline {p}}k_{\overline {p}}(\tau _{1},\dots ,\tau _{\overline {p}}).}
3844:
3699:
4678:
4960:
837:
1326:
3154:
1570:
This method, developed by Lee and
Schetzen, orthogonalizes with respect to the actual mathematical description of the signal, i.e. the projection onto the new basis functionals is based on the knowledge of the moments of the random signal.
3836:
824:. In many physical situations, this assumption about the input set is a reasonable one. The theorem, however, gives no indication as to how many terms are needed for a good approximation, which is an essential question in applications.
4092:
6437:-th-order kernel, all lower kernels must be identified again with the higher variance. However, an outstanding improvement in the output MSE will be obtained if the Wiener and Volterra kernels are obtained with the new formulas.
2350:
182:
1557:
An important aspect, with respect to which the following methods differ, is whether the orthogonalization of the basis functionals is to be performed over the idealized specification of the input signal (e.g. gaussian,
1681:
105:
for the integration of
Volterra analytic functionals. The use of the Volterra series for system analysis originated from a restricted 1942 wartime report of Wiener's, who was then a professor of mathematics at
2590:
1987:
1262:
as symmetrical. In fact, for the commutativity of the multiplication it is always possible to symmetrize it by forming a new kernel taken as the average of the kernels for all permutations of the variables
6155:{\displaystyle h_{1}=k_{1}^{(1)}-3A_{1}\sum _{\tau _{2}}k_{3}^{(3)}(\tau _{1},\tau _{2},\tau _{2})+15A_{1}^{2}\sum _{\tau 2}\sum _{\tau _{3}}k_{5}^{(5)}(\tau _{1},\tau _{2},\tau _{2},\tau _{3},\tau _{3}),}
2361:
6744:
inputs can be used for the orthogonalization and that fewer data points suffice to reach a desired level of accuracy. Also, estimation can be performed incrementally until some criterion is fulfilled.
2184:
2970:
2089:
4947:{\displaystyle k_{2}^{(2)}(\tau _{1},\tau _{2})={\frac {1}{2!A_{2}^{2}}}\left\{E\left\{y^{(2)}(n)\prod _{i=1}^{2}x^{(2)}(n-\tau _{i})\right\}-A_{2}k_{0}^{(2)}\delta _{\tau _{1}\tau _{2}}\right\},}
6466:
2978:
1670:
1110:
3454:
1260:
496:
5530:
682:
6756:
basis-functionals. For estimation, the order of the original should be known, since the
Volterra basis functionals are not orthogonal, and thus estimation cannot be performed incrementally.
4522:
3456:, conceived as the solution for the estimation of the diagonal elements themselves. Efficient formulas to avoid this drawback and references for diagonal kernel element estimation exist
6423:{\displaystyle h_{0}=k_{0}^{(0)}-A_{0}\sum _{\tau _{1}}k_{2}^{(2)}(\tau _{1},\tau _{1})+3A_{0}^{2}\sum _{\tau _{1}}\sum _{\tau _{2}}k_{4}^{(4)}(\tau _{1},\tau _{1},\tau _{2},\tau _{2}).}
1307:
741:
5709:
4079:{\displaystyle h_{1}=k_{1}-3A\sum _{\tau _{2}}k_{3}(\tau _{1},\tau _{2},\tau _{2})+15A^{2}\sum _{\tau 2}\sum _{\tau _{3}}k_{5}(\tau _{1},\tau _{2},\tau _{2},\tau _{3},\tau _{3}),}
6878:
Vito
Volterra. Theory of Functionals and of Integrals and Integro-Differential Equations. Madrid 1927 (Spanish), translated version reprinted New York: Dover Publications, 1959.
5522:
5465:
5399:{\displaystyle k_{3}^{(3)}(\tau _{1},\tau _{2},\tau _{3})={\frac {1}{3!A_{3}^{3}}}\left\{E\left\{y^{(3)}(n)\prod _{i=1}^{3}x^{(3)}(n-\tau _{i})\right\}-A_{3}^{2}\left\right\}.}
2628:
7025:
Korenberg, M. J.; Bruder, S. B.; McIlroy, P. J. (1988). "Exact orthogonal kernel estimation from finite data records: extending Wiener's identification of nonlinear systems".
3557:
4530:
1534:{\displaystyle \sum _{\tau _{1}=0}^{M}\sum _{\tau _{2}=\tau _{1}}^{M}\cdots \sum _{\tau _{p}=\tau _{p-1}}^{M}h_{p}(\tau _{1},\dots ,\tau _{p})\prod _{j=1}^{p}x(n-\tau _{j}).}
3459:
Once the Wiener kernels were identified, Volterra kernels can be obtained by using Wiener-to-Volterra formulas, in the following reported for a fifth-order
Volterra series:
3376:{\displaystyle k_{p}(\tau _{1},\dots ,\tau _{p})={\frac {E\left\{\left(y(n)-\sum \limits _{m=0}^{p-1}G_{m}x(n)\right)x(n-\tau _{1})\cdots x(n-\tau _{p})\right\}}{p!A^{p}}}.}
6732:
3549:
3503:
4437:
4399:
4369:
4342:
6772:). Franz and Schölkopf proposed that the kernel method could essentially replace the Volterra series representation, although noting that the latter is more intuitive.
2226:
1191:
767:
6702:
6672:
3707:
1137:
563:
535:
414:
806:
6645:
4302:{\displaystyle h_{0}=k_{0}-A\sum _{\tau _{1}}k_{2}(\tau _{1},\tau _{1})+3A^{2}\sum _{\tau _{1}}\sum _{\tau _{2}}k_{4}(\tau _{1},\tau _{1},\tau _{2},\tau _{2}).}
434:
1046:{\displaystyle y(n)=h_{0}+\sum _{p=1}^{P}\sum _{\tau _{1}=a}^{b}\cdots \sum _{\tau _{p}=a}^{b}h_{p}(\tau _{1},\dots ,\tau _{p})\prod _{j=1}^{p}x(n-\tau _{j}),}
3386:
The main drawback of this technique is that the estimation errors, made on all elements of lower-order kernels, will affect each diagonal element of order
2249:
4344:
has the advantage of stimulating high-order nonlinearity, so as to achieve more accurate high-order kernel identification. As a drawback, the use of high
379:{\displaystyle y(t)=h_{0}+\sum _{n=1}^{N}\int _{a}^{b}\cdots \int _{a}^{b}h_{n}(\tau _{1},\dots ,\tau _{n})\prod _{j=1}^{n}x(t-\tau _{j})\,d\tau _{j}.}
6891:
Radiation Lab MIT 1942, restricted. report V-16, no 129 (112 pp). Declassified Jul 1946, Published as rep. no. PB-1-58087, U.S. Dept. Commerce. URL:
2243:
Recalling that every
Volterra functional is orthogonal to all Wiener functional of greater order, and considering the following Volterra functional:
4401:
in the identification process can lead to a better estimation of lower-order kernel, but can be insufficient to stimulate high-order nonlinearity.
1866:{\displaystyle H_{p}x(n)=\sum _{\tau _{1}=a}^{b}\cdots \sum _{\tau _{p}=a}^{b}h_{p}(\tau _{1},\dots ,\tau _{p})\prod _{j=1}^{p}x(n-\tau _{j}).}
107:
2536:
1890:
775:: The use of the Volterra series to represent a time-invariant functional relation is often justified by appealing to a theorem due to
2519:{\displaystyle E\left\{y(n)H_{\overline {p}}^{*}x(n)\right\}=E\left\{\sum _{p=0}^{\infty }G_{p}x(n)H_{\overline {p}}^{*}x(n)\right\}.}
4440:
formulas for taking into account the use of different variances. Furthermore, new Wiener to Volterra conversion formulas are needed.
81:. The Volterra series, which is used to prove the Volterra theorem, is an infinite sum of multidimensional convolutional integrals.
7068:
Franz, Matthias O.; Bernhard Schölkopf (2006). "A unifying view of Wiener and Volterra theory and polynomial kernel regression".
58:. Its main advantage lies in its generalizability: it can represent a wide range of systems. Thus, it is sometimes considered a
7174:. Dept. Electr. Engrg, Univ.Tech. Eindhoven, NL 1977, T-H report 77-E-71. (Chronological listing of early papers to 1977) URL:
4371:
values causes high identification error in lower-order kernels, mainly due to nonideality of the input and truncation errors.
2097:
6617:{\displaystyle h_{n}(\tau _{1},\dots ,\tau _{n})=\sum _{i=1}^{M}(c_{i}a_{ni}\omega _{\tau _{1}i}\dots \omega _{\tau _{n}i}),}
3138:{\displaystyle k_{p}(\tau _{1},\dots ,\tau _{p})={\frac {E\left\{y(n)x(n-\tau _{1})\cdots x(n-\tau _{p})\right\}}{p!A^{p}}}.}
2915:
2002:
1584:
1056:
5696:{\displaystyle h_{3}=k_{3}^{(3)}-10A_{3}\sum _{\tau _{4}}k_{5}^{(5)}(\tau _{1},\tau _{2},\tau _{3},\tau _{4},\tau _{4}),}
821:
743:
are negative. The integrals may then be written over the half range from zero to infinity. So if the operator is causal,
603:!, a convention which is convenient when taking the output of one Volterra system as the input of another ("cascading").
7243:
7238:
6798:
3393:
98:
1203:
439:
6769:
6765:
612:
1876:
To allow identification orthogonalization, Volterra series must be rearranged in terms of orthogonal non-homogeneous
832:
The discrete-time case is similar to the continuous-time case, except that the integrals are replaced by summations:
609:: Since in any physically realizable system the output can only depend on previous values of the input, the kernels
148:
The latter functional mapping perspective is more frequently used due to the assumed time-invariance of the system.
776:
6952:
S. Orcioni; M. Pirani; C. Turchetti (2005). "Advances in Lee–Schetzen method for Volterra filter identification".
4449:
89:
The Volterra series is a modernized version of the theory of analytic functionals from the Italian mathematician
6445:
This method was developed by Wray and Green (1994) and utilizes the fact that a simple 2-fully connected layer
1266:
7127:
687:
74:
5862:{\displaystyle h_{2}=k_{2}^{(2)}-6A_{2}\sum _{\tau _{3}}k_{4}^{(4)}(\tau _{1},\tau _{2},\tau _{3},\tau _{3}),}
43:
3694:{\displaystyle h_{3}=k_{3}-10A\sum _{\tau _{4}}k_{5}(\tau _{1},\tau _{2},\tau _{3},\tau _{4},\tau _{4}),}
7187:
6450:
6433:
As can be seen, the drawback with respect to the previous formula is that for the identification of the
5473:
5416:
4673:{\displaystyle k_{1}^{(1)}(\tau _{1})={\frac {1}{A_{1}}}E\left\{y^{(1)}(n)x^{(1)}(n-\tau _{1})\right\},}
157:
78:
2595:
7216:
7152:
6819:
537:. If it is not symmetrical, it can be replaced by a symmetrized kernel, which is the average over the
6781:
1575:
6994:"Improving the approximation ability of Volterra series identified with a cross-correlation method"
6707:
3511:
3465:
7093:
7050:
6969:
4415:
4377:
4347:
4320:
2192:
7194:
Signal Processing, 81 2001 533–580. (Alphabetic listing to 2001) www.elsevier.nl/locate/sigpro
7175:
7085:
7042:
6839:
6752:
3831:{\displaystyle h_{2}=k_{2}-6A\sum _{\tau _{3}}k_{4}(\tau _{1},\tau _{2},\tau _{3},\tau _{3}),}
817:
6916:
Early MIT reports by Brilliant, Zames, George, Hause, Chesler can be found on dspace.mit.edu.
6704:
the coefficients of the polynomial expansion of the output function of the hidden nodes, and
1170:
746:
7147:
7139:
7077:
7034:
7005:
6961:
6831:
6768:. Consequently, this approach is also based on minimizing the empirical error (often called
6677:
510:
6650:
3148:
If we want to consider the diagonal elements, the solution proposed by Lee and Schetzen is
1115:
548:
520:
513:
of the system. For the representation to be unique, the kernels must be symmetrical in the
392:
782:
505:
128:
102:
55:
51:
97:
became interested in this theory in the 1920s due to his contact with Volterra's student
7208:, IEEE Trans. Circuits & Systems, vol.CS-11(4) Aug 1977; vol.CS-11(5) Oct 1977 2–6.
6630:
6446:
813:
419:
134:
94:
59:
6892:
6835:
31:
other times. This provides the ability to capture the "memory" effect of devices like
7232:
6926:
M. Pirani; S. Orcioni; C. Turchetti (Sep 2004). "Diagonal kernel point estimation of
6793:
2345:{\displaystyle H_{\overline {p}}^{*}x(n)=\prod _{j=1}^{\overline {p}}x(n-\tau _{j}),}
1881:
1551:
1313:
809:
90:
24:
7172:
Bibliography of Volterra series, Hermite functional expansions, and related subjects
7054:
6973:
7128:"Calculation of Volterra Kernels for Solutions of Nonlinear Differential Equations"
7097:
47:
7081:
416:
on the right side is usually taken to be zero by suitable choice of output level
122:
The theory of the Volterra series can be viewed from two different perspectives:
1559:
66:
7199:
The Identification of Nonlinear Biological Systems: Volterra Kernel Approaches
7143:
7010:
6993:
6965:
70:
32:
7089:
6843:
4443:
The traditional Wiener kernel identification should be changed as follows:
7046:
36:
7038:
16:
Model for approximating non-linear effects, similar to a Taylor series
1550:
by estimating the coefficients of an orthogonalized series, e.g. the
7217:
http://rfic.eecs.berkeley.edu/~niknejad/ee242/pdf/volterra_book.pdf
6780:
This method was developed by van Hemmen and coworkers and utilizes
2236:) is some stationary white noise (SWN) with zero mean and variance
1200:
We can always consider, without loss of the generality, the kernel
6868:. Vol. III. Italy: R. Accademia dei Lincei. pp. 97–105.
2585:{\displaystyle \tau _{1}\neq \tau _{2}\neq \ldots \neq \tau _{P}}
1982:{\displaystyle y(n)=\sum _{p}H_{p}x(n)\equiv \sum _{p}G_{p}x(n).}
69:, a Volterra series denotes a functional expansion of a dynamic,
6764:
This method was invented by Franz and Schölkopf and is based on
4317:
In the traditional orthogonal algorithm, using inputs with high
7113:
Determination of Volterra kernels for nonlinear RF amplifiers
7201:, Annals Biomedical Engineering (1996), Volume 24, Number 2.
7176:
http://alexandria.tue.nl/extra1/erap/publichtml/7704263.pdf
54:
distortion in many devices, including power amplifiers and
7126:
J. L. van Hemmen; W. M. Kistler; E. G. F. Thomas (2000).
2179:{\displaystyle E\{G_{i}x(n)G_{j}x(n)\}=0;\quad i\neq j,}
808:
for which the approximation will hold is required to be
7184:
Proc. IEEE, vol.62, no.8, pp. 1088–1119, Aug. 1974
6907:
MIT Dec 10 1951, tech. rep. no 217, Res. Lab. Electron.
6818:
Friston, K.J.; Harrison, L.; Penny, W. (2 April 2003).
2965:{\displaystyle {\tau _{i}\neq \tau _{j},\ \forall i,j}}
2084:{\displaystyle E\{H_{i}x(n)G_{j}x(n)\}=0;\quad i<j,}
7213:
Nonlinear System Theory: The Volterra–Wiener Approach.
7206:
Frequency-domain analysis of weakly nonlinear networks
176:) as output can be expanded in the Volterra series as
7223:
The Volterra and Wiener Theories of Nonlinear Systems
6710:
6680:
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2005:
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442:
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185:
50:. It is also used in electrical engineering to model
1665:{\displaystyle y(n)=h_{0}+\sum _{p=1}^{P}H_{p}x(n),}
1105:{\displaystyle P\in \{1,2,\dots \}\cup \{\infty \}.}
7182:
Analysis of nonlinear systems with multiple inputs,
6893:
http://www.dtic.mil/dtic/tr/fulltext/u2/a800212.pdf
7192:A bibliography on nonlinear system identification.
6726:
6696:
6666:
6639:
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3449:{\displaystyle \sum \limits _{m=0}^{p-1}G_{m}x(n)}
3448:
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3137:
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2901:
2622:
2584:
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1981:
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378:
144:from a function space into real or complex numbers
23:is a model for non-linear behavior similar to the
6866:Sopra le funzioni che dipendono da altre funzioni
1255:{\displaystyle h_{p}(\tau _{1},\dots ,\tau _{p})}
491:{\displaystyle h_{n}(\tau _{1},\dots ,\tau _{n})}
677:{\displaystyle h_{n}(t_{1},t_{2},\ldots ,t_{n})}
6735:represents the effective memory of the system.
42:It has been applied in the fields of medicine (
6954:Multidimensional Systems and Signal Processing
6930:-th order discrete Volterra-Wiener systems".
1574:We can write the Volterra series in terms of
1147:is finite, the series operator is said to be
77:. The Volterra series are frequently used in
8:
6932:EURASIP Journal on Applied Signal Processing
4517:{\displaystyle k_{0}^{(0)}=E\{y^{(0)}(n)\},}
4508:
4480:
2151:
2104:
2056:
2009:
1316:with symmetrical kernels we can rewrite the
1163:are finite, the series operator is called a
1096:
1090:
1084:
1066:
6987:
6985:
6983:
6947:
6945:
1996:operators can be defined by the following:
1545:Methods to estimate the kernel coefficients
7215:Baltimore 1981 (Johns Hopkins Univ Press)
7153:11370/eda737ae-40d1-4ff3-93d7-6b2434d23d52
6905:A method of Wiener in a nonlinear circuit.
1320:-th term approximately in triangular form
1302:{\displaystyle \tau _{1},\dots ,\tau _{p}}
820:set of functions, which is compact by the
7180:Bussgang, J.J.; Ehrman, L.; Graham, J.W:
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3123:
3097:
3069:
3030:
3018:
2999:
2986:
2980:
2937:
2924:
2919:
2917:
2882:
2863:
2845:
2830:
2813:
2796:
2769:
2758:
2731:
2702:
2675:
2664:
2638:
2614:
2609:
2597:
2576:
2557:
2544:
2538:
2490:
2480:
2458:
2448:
2437:
2399:
2389:
2363:
2330:
2303:
2292:
2267:
2257:
2251:
2200:
2194:
2133:
2111:
2099:
2038:
2016:
2004:
1958:
1948:
1923:
1913:
1892:
1851:
1829:
1818:
1805:
1786:
1773:
1763:
1750:
1745:
1732:
1719:
1714:
1689:
1683:
1641:
1631:
1620:
1607:
1586:
1519:
1497:
1486:
1473:
1454:
1441:
1431:
1418:
1405:
1400:
1387:
1380:
1367:
1362:
1352:
1339:
1334:
1328:
1293:
1274:
1268:
1243:
1224:
1211:
1205:
1172:
1123:
1117:
1058:
1031:
1009:
998:
985:
966:
953:
943:
930:
925:
912:
899:
894:
884:
873:
860:
839:
784:
748:
736:{\displaystyle t_{1},t_{2},\ldots ,t_{n}}
727:
708:
695:
689:
665:
646:
633:
620:
614:
550:
522:
479:
460:
447:
441:
421:
400:
394:
367:
359:
350:
328:
317:
304:
285:
272:
262:
257:
244:
239:
229:
218:
205:
184:
6889:Response of a nonlinear device to noise.
2912:So if we exclude the diagonal elements,
6810:
6674:the weights to the linear output node,
509:. It can be regarded as a higher-order
6460:-th-order Volterra kernel is given by
6784:to sample the Volterra coefficients.
4404:This phenomenon, which can be called
684:will be zero if any of the variables
7:
572:is finite, the series is said to be
7132:SIAM Journal on Applied Mathematics
4412:To overcome this limitation, a low
3398:
3238:
2228:is arbitrary homogeneous Volterra,
114:, came into use a few years later.
5517:{\displaystyle h_{4}=k_{4}^{(4)},}
5460:{\displaystyle h_{5}=k_{5}^{(5)},}
4374:On the contrary, the use of lower
2949:
2449:
1093:
14:
2623:{\displaystyle A=\sigma _{x}^{2}}
588:are finite, the series is called
93:, in his work dating from 1887.
2163:
2068:
812:. It is usually taken to be an
773:Fréchet's approximation theorem
101:. Wiener applied his theory of
7111:Siamack Ghadimi (2019-09-12),
6608:
6539:
6512:
6480:
6414:
6362:
6357:
6351:
6283:
6257:
6252:
6246:
6201:
6195:
6146:
6081:
6076:
6070:
6006:
5967:
5962:
5956:
5908:
5902:
5853:
5801:
5796:
5790:
5742:
5736:
5687:
5622:
5617:
5611:
5563:
5557:
5506:
5500:
5449:
5443:
5353:
5340:
5335:
5329:
5286:
5273:
5268:
5262:
5219:
5206:
5201:
5195:
5154:
5135:
5130:
5124:
5095:
5089:
5084:
5078:
5024:
4985:
4980:
4974:
4904:
4898:
4867:
4848:
4843:
4837:
4808:
4802:
4797:
4791:
4737:
4711:
4706:
4700:
4659:
4640:
4635:
4629:
4621:
4615:
4610:
4604:
4568:
4555:
4550:
4544:
4505:
4499:
4494:
4488:
4469:
4463:
4293:
4241:
4178:
4152:
4070:
4005:
3946:
3907:
3822:
3770:
3685:
3620:
3443:
3437:
3341:
3322:
3313:
3294:
3283:
3277:
3231:
3225:
3200:
3168:
3103:
3084:
3075:
3056:
3050:
3044:
3024:
2992:
2893:
2856:
2802:
2783:
2751:
2745:
2708:
2689:
2657:
2651:
2505:
2499:
2473:
2467:
2414:
2408:
2382:
2376:
2336:
2317:
2282:
2276:
2215:
2209:
2148:
2142:
2126:
2120:
2053:
2047:
2031:
2025:
1973:
1967:
1938:
1932:
1903:
1897:
1857:
1838:
1811:
1779:
1704:
1698:
1656:
1650:
1597:
1591:
1525:
1506:
1479:
1447:
1249:
1217:
1141:discrete-time Volterra kernels
1037:
1018:
991:
959:
850:
844:
795:
789:
671:
626:
485:
453:
356:
337:
310:
278:
195:
189:
1:
6836:10.1016/S1053-8119(03)00202-7
6456:The general notation for the
1193:, the operator is said to be
1165:doubly finite Volterra series
599:-th-order term is divided by
7082:10.1162/neco.2006.18.12.3097
6799:Polynomial signal processing
6727:{\displaystyle \omega _{ji}}
4409:used in the identification.
3544:{\displaystyle h_{4}=k_{4},}
3498:{\displaystyle h_{5}=k_{5},}
2887:
2850:
2835:
2818:
2774:
2736:
2680:
2485:
2394:
2308:
2262:
7197:Korenberg M.J. Hunter I.W:
6770:empirical risk minimization
6766:statistical learning theory
4432:{\displaystyle \sigma _{x}}
4394:{\displaystyle \sigma _{x}}
4364:{\displaystyle \sigma _{x}}
4337:{\displaystyle \sigma _{x}}
7260:
6820:"Dynamic causal modelling"
6739:Exact orthogonal algorithm
3390:by means of the summation
46:) and biology, especially
7144:10.1137/S0036139999336037
7011:10.1007/s11071-014-1631-7
6966:10.1007/s11045-004-1677-7
2221:{\displaystyle H_{i}x(n)}
7225:, New York: Wiley, 1980.
6992:Orcioni, Simone (2014).
4313:Multiple-variance method
541:! permutations of these
6864:Volterra, Vito (1887).
1566:Crosscorrelation method
1186:{\displaystyle a\geq 0}
762:{\displaystyle a\geq 0}
607:The causality condition
389:Here the constant term
6728:
6698:
6697:{\displaystyle a_{ji}}
6668:
6641:
6618:
6538:
6424:
6156:
5863:
5697:
5518:
5461:
5400:
5118:
4948:
4831:
4674:
4518:
4433:
4395:
4365:
4338:
4303:
4080:
3832:
3695:
3545:
3499:
3450:
3423:
3377:
3263:
3139:
2966:
2903:
2779:
2685:
2624:
2586:
2520:
2453:
2346:
2313:
2222:
2180:
2085:
1983:
1867:
1834:
1768:
1737:
1666:
1636:
1535:
1502:
1436:
1392:
1357:
1303:
1256:
1187:
1133:
1106:
1047:
1014:
948:
917:
889:
802:
763:
737:
678:
559:
531:
492:
430:
410:
380:
333:
234:
44:biomedical engineering
6782:Dirac delta functions
6776:Differential sampling
6729:
6699:
6669:
6667:{\displaystyle c_{i}}
6642:
6619:
6518:
6451:multilayer perceptron
6425:
6157:
5864:
5698:
5519:
5462:
5401:
5098:
4949:
4811:
4675:
4519:
4434:
4396:
4366:
4339:
4304:
4081:
3833:
3696:
3546:
3500:
3451:
3397:
3378:
3237:
3140:
2967:
2904:
2754:
2660:
2625:
2587:
2521:
2433:
2347:
2288:
2223:
2181:
2086:
1984:
1868:
1814:
1741:
1710:
1667:
1616:
1536:
1482:
1396:
1358:
1330:
1304:
1257:
1188:
1134:
1132:{\displaystyle h_{p}}
1107:
1048:
994:
921:
890:
869:
822:Arzelà –Ascoli theorem
803:
764:
738:
679:
560:
558:{\displaystyle \tau }
532:
530:{\displaystyle \tau }
493:
431:
411:
409:{\displaystyle h_{0}}
381:
313:
214:
158:time-invariant system
79:system identification
6708:
6678:
6651:
6631:
6467:
6169:
5876:
5710:
5531:
5474:
5417:
4961:
4687:
4531:
4450:
4416:
4378:
4348:
4321:
4093:
3845:
3708:
3558:
3512:
3466:
3394:
3155:
2979:
2916:
2637:
2596:
2537:
2362:
2250:
2193:
2098:
2003:
1891:
1682:
1585:
1327:
1267:
1204:
1171:
1116:
1057:
838:
801:{\displaystyle x(t)}
783:
747:
688:
613:
549:
521:
440:
420:
393:
183:
7244:Functional analysis
7239:Mathematical series
7190:& Serpendin E:
7115:, Microwaves&RF
6441:Feedforward network
6361:
6306:
6256:
6205:
6080:
6029:
5966:
5912:
5800:
5746:
5621:
5567:
5510:
5453:
5339:
5272:
5205:
5179:
5056:
4984:
4908:
4769:
4710:
4554:
4473:
2619:
2495:
2404:
2272:
267:
249:
118:Mathematical theory
7070:Neural Computation
7039:10.1007/BF02364581
6998:Nonlinear Dynamics
6724:
6694:
6664:
6637:
6614:
6420:
6341:
6340:
6323:
6292:
6236:
6235:
6185:
6152:
6060:
6059:
6042:
6015:
5946:
5945:
5892:
5859:
5780:
5779:
5726:
5693:
5601:
5600:
5547:
5514:
5490:
5457:
5433:
5396:
5319:
5252:
5185:
5165:
5042:
4964:
4944:
4888:
4755:
4690:
4670:
4534:
4514:
4453:
4429:
4391:
4361:
4334:
4299:
4230:
4213:
4141:
4076:
3994:
3977:
3896:
3828:
3759:
3691:
3609:
3541:
3495:
3446:
3373:
3135:
2962:
2899:
2620:
2605:
2582:
2516:
2476:
2385:
2342:
2253:
2218:
2176:
2081:
1979:
1953:
1918:
1863:
1662:
1531:
1299:
1252:
1183:
1129:
1102:
1043:
798:
759:
733:
674:
555:
527:
488:
426:
406:
376:
253:
235:
142:functional mapping
140:A real or complex
7076:(12): 3097–3118.
6753:Linear regression
6748:Linear regression
6640:{\displaystyle n}
6324:
6307:
6219:
6043:
6030:
5929:
5763:
5584:
5058:
4771:
4589:
4214:
4197:
4125:
3978:
3965:
3880:
3743:
3593:
3368:
3130:
2948:
2890:
2853:
2838:
2821:
2777:
2739:
2683:
2488:
2397:
2311:
2265:
1944:
1909:
818:uniformly bounded
429:{\displaystyle y}
137:(real or complex)
73:, time-invariant
7251:
7158:
7157:
7155:
7123:
7117:
7116:
7108:
7102:
7101:
7065:
7059:
7058:
7027:Ann. Biomed. Eng
7022:
7016:
7015:
7013:
7004:(4): 2861–2869.
6989:
6978:
6977:
6949:
6940:
6939:
6938:(12): 1807–1816.
6923:
6917:
6914:
6908:
6901:
6895:
6885:
6879:
6876:
6870:
6869:
6861:
6855:
6854:
6852:
6850:
6830:(4): 1273–1302.
6815:
6733:
6731:
6730:
6725:
6723:
6722:
6703:
6701:
6700:
6695:
6693:
6692:
6673:
6671:
6670:
6665:
6663:
6662:
6646:
6644:
6643:
6638:
6623:
6621:
6620:
6615:
6607:
6606:
6602:
6601:
6584:
6583:
6579:
6578:
6564:
6563:
6551:
6550:
6537:
6532:
6511:
6510:
6492:
6491:
6479:
6478:
6429:
6427:
6426:
6421:
6413:
6412:
6400:
6399:
6387:
6386:
6374:
6373:
6360:
6349:
6339:
6338:
6337:
6322:
6321:
6320:
6305:
6300:
6282:
6281:
6269:
6268:
6255:
6244:
6234:
6233:
6232:
6218:
6217:
6204:
6193:
6181:
6180:
6161:
6159:
6158:
6153:
6145:
6144:
6132:
6131:
6119:
6118:
6106:
6105:
6093:
6092:
6079:
6068:
6058:
6057:
6056:
6041:
6028:
6023:
6005:
6004:
5992:
5991:
5979:
5978:
5965:
5954:
5944:
5943:
5942:
5928:
5927:
5911:
5900:
5888:
5887:
5868:
5866:
5865:
5860:
5852:
5851:
5839:
5838:
5826:
5825:
5813:
5812:
5799:
5788:
5778:
5777:
5776:
5762:
5761:
5745:
5734:
5722:
5721:
5702:
5700:
5699:
5694:
5686:
5685:
5673:
5672:
5660:
5659:
5647:
5646:
5634:
5633:
5620:
5609:
5599:
5598:
5597:
5583:
5582:
5566:
5555:
5543:
5542:
5523:
5521:
5520:
5515:
5509:
5498:
5486:
5485:
5466:
5464:
5463:
5458:
5452:
5441:
5429:
5428:
5405:
5403:
5402:
5397:
5392:
5388:
5387:
5383:
5382:
5381:
5380:
5379:
5370:
5369:
5352:
5351:
5338:
5327:
5315:
5314:
5313:
5312:
5303:
5302:
5285:
5284:
5271:
5260:
5248:
5247:
5246:
5245:
5236:
5235:
5218:
5217:
5204:
5193:
5178:
5173:
5161:
5157:
5153:
5152:
5134:
5133:
5117:
5112:
5088:
5087:
5059:
5057:
5055:
5050:
5031:
5023:
5022:
5010:
5009:
4997:
4996:
4983:
4972:
4953:
4951:
4950:
4945:
4940:
4936:
4935:
4934:
4933:
4932:
4923:
4922:
4907:
4896:
4887:
4886:
4874:
4870:
4866:
4865:
4847:
4846:
4830:
4825:
4801:
4800:
4772:
4770:
4768:
4763:
4744:
4736:
4735:
4723:
4722:
4709:
4698:
4679:
4677:
4676:
4671:
4666:
4662:
4658:
4657:
4639:
4638:
4614:
4613:
4590:
4588:
4587:
4575:
4567:
4566:
4553:
4542:
4523:
4521:
4520:
4515:
4498:
4497:
4472:
4461:
4438:
4436:
4435:
4430:
4428:
4427:
4400:
4398:
4397:
4392:
4390:
4389:
4370:
4368:
4367:
4362:
4360:
4359:
4343:
4341:
4340:
4335:
4333:
4332:
4308:
4306:
4305:
4300:
4292:
4291:
4279:
4278:
4266:
4265:
4253:
4252:
4240:
4239:
4229:
4228:
4227:
4212:
4211:
4210:
4196:
4195:
4177:
4176:
4164:
4163:
4151:
4150:
4140:
4139:
4138:
4118:
4117:
4105:
4104:
4085:
4083:
4082:
4077:
4069:
4068:
4056:
4055:
4043:
4042:
4030:
4029:
4017:
4016:
4004:
4003:
3993:
3992:
3991:
3976:
3964:
3963:
3945:
3944:
3932:
3931:
3919:
3918:
3906:
3905:
3895:
3894:
3893:
3870:
3869:
3857:
3856:
3837:
3835:
3834:
3829:
3821:
3820:
3808:
3807:
3795:
3794:
3782:
3781:
3769:
3768:
3758:
3757:
3756:
3733:
3732:
3720:
3719:
3700:
3698:
3697:
3692:
3684:
3683:
3671:
3670:
3658:
3657:
3645:
3644:
3632:
3631:
3619:
3618:
3608:
3607:
3606:
3583:
3582:
3570:
3569:
3550:
3548:
3547:
3542:
3537:
3536:
3524:
3523:
3504:
3502:
3501:
3496:
3491:
3490:
3478:
3477:
3455:
3453:
3452:
3447:
3433:
3432:
3422:
3411:
3382:
3380:
3379:
3374:
3369:
3367:
3366:
3365:
3349:
3348:
3344:
3340:
3339:
3312:
3311:
3290:
3286:
3273:
3272:
3262:
3251:
3207:
3199:
3198:
3180:
3179:
3167:
3166:
3144:
3142:
3141:
3136:
3131:
3129:
3128:
3127:
3111:
3110:
3106:
3102:
3101:
3074:
3073:
3031:
3023:
3022:
3004:
3003:
2991:
2990:
2971:
2969:
2968:
2963:
2961:
2946:
2942:
2941:
2929:
2928:
2908:
2906:
2905:
2900:
2892:
2891:
2883:
2868:
2867:
2855:
2854:
2846:
2840:
2839:
2831:
2822:
2814:
2809:
2805:
2801:
2800:
2778:
2770:
2768:
2741:
2740:
2732:
2715:
2711:
2707:
2706:
2684:
2676:
2674:
2629:
2627:
2626:
2621:
2618:
2613:
2591:
2589:
2588:
2583:
2581:
2580:
2562:
2561:
2549:
2548:
2525:
2523:
2522:
2517:
2512:
2508:
2494:
2489:
2481:
2463:
2462:
2452:
2447:
2421:
2417:
2403:
2398:
2390:
2351:
2349:
2348:
2343:
2335:
2334:
2312:
2304:
2302:
2271:
2266:
2258:
2227:
2225:
2224:
2219:
2205:
2204:
2185:
2183:
2182:
2177:
2138:
2137:
2116:
2115:
2090:
2088:
2087:
2082:
2043:
2042:
2021:
2020:
1988:
1986:
1985:
1980:
1963:
1962:
1952:
1928:
1927:
1917:
1872:
1870:
1869:
1864:
1856:
1855:
1833:
1828:
1810:
1809:
1791:
1790:
1778:
1777:
1767:
1762:
1755:
1754:
1736:
1731:
1724:
1723:
1694:
1693:
1671:
1669:
1668:
1663:
1646:
1645:
1635:
1630:
1612:
1611:
1540:
1538:
1537:
1532:
1524:
1523:
1501:
1496:
1478:
1477:
1459:
1458:
1446:
1445:
1435:
1430:
1429:
1428:
1410:
1409:
1391:
1386:
1385:
1384:
1372:
1371:
1356:
1351:
1344:
1343:
1308:
1306:
1305:
1300:
1298:
1297:
1279:
1278:
1261:
1259:
1258:
1253:
1248:
1247:
1229:
1228:
1216:
1215:
1192:
1190:
1189:
1184:
1138:
1136:
1135:
1130:
1128:
1127:
1111:
1109:
1108:
1103:
1052:
1050:
1049:
1044:
1036:
1035:
1013:
1008:
990:
989:
971:
970:
958:
957:
947:
942:
935:
934:
916:
911:
904:
903:
888:
883:
865:
864:
807:
805:
804:
799:
768:
766:
765:
760:
742:
740:
739:
734:
732:
731:
713:
712:
700:
699:
683:
681:
680:
675:
670:
669:
651:
650:
638:
637:
625:
624:
564:
562:
561:
556:
536:
534:
533:
528:
511:impulse response
497:
495:
494:
489:
484:
483:
465:
464:
452:
451:
435:
433:
432:
427:
415:
413:
412:
407:
405:
404:
385:
383:
382:
377:
372:
371:
355:
354:
332:
327:
309:
308:
290:
289:
277:
276:
266:
261:
248:
243:
233:
228:
210:
209:
56:frequency mixers
7259:
7258:
7254:
7253:
7252:
7250:
7249:
7248:
7229:
7228:
7167:
7165:Further reading
7162:
7161:
7125:
7124:
7120:
7110:
7109:
7105:
7067:
7066:
7062:
7024:
7023:
7019:
6991:
6990:
6981:
6951:
6950:
6943:
6925:
6924:
6920:
6915:
6911:
6902:
6898:
6886:
6882:
6877:
6873:
6863:
6862:
6858:
6848:
6846:
6817:
6816:
6812:
6807:
6790:
6778:
6762:
6750:
6741:
6711:
6706:
6705:
6681:
6676:
6675:
6654:
6649:
6648:
6629:
6628:
6593:
6588:
6570:
6565:
6552:
6542:
6502:
6483:
6470:
6465:
6464:
6443:
6404:
6391:
6378:
6365:
6329:
6312:
6273:
6260:
6224:
6209:
6172:
6167:
6166:
6136:
6123:
6110:
6097:
6084:
6048:
5996:
5983:
5970:
5934:
5919:
5879:
5874:
5873:
5843:
5830:
5817:
5804:
5768:
5753:
5713:
5708:
5707:
5677:
5664:
5651:
5638:
5625:
5589:
5574:
5534:
5529:
5528:
5477:
5472:
5471:
5420:
5415:
5414:
5371:
5361:
5356:
5343:
5304:
5294:
5289:
5276:
5237:
5227:
5222:
5209:
5184:
5180:
5144:
5119:
5073:
5072:
5068:
5064:
5060:
5035:
5014:
5001:
4988:
4959:
4958:
4924:
4914:
4909:
4878:
4857:
4832:
4786:
4785:
4781:
4777:
4773:
4748:
4727:
4714:
4685:
4684:
4649:
4624:
4599:
4598:
4594:
4579:
4558:
4529:
4528:
4483:
4448:
4447:
4419:
4414:
4413:
4381:
4376:
4375:
4351:
4346:
4345:
4324:
4319:
4318:
4315:
4283:
4270:
4257:
4244:
4231:
4219:
4202:
4187:
4168:
4155:
4142:
4130:
4109:
4096:
4091:
4090:
4060:
4047:
4034:
4021:
4008:
3995:
3983:
3955:
3936:
3923:
3910:
3897:
3885:
3861:
3848:
3843:
3842:
3812:
3799:
3786:
3773:
3760:
3748:
3724:
3711:
3706:
3705:
3675:
3662:
3649:
3636:
3623:
3610:
3598:
3574:
3561:
3556:
3555:
3528:
3515:
3510:
3509:
3482:
3469:
3464:
3463:
3424:
3392:
3391:
3357:
3350:
3331:
3303:
3264:
3221:
3217:
3216:
3212:
3208:
3190:
3171:
3158:
3153:
3152:
3119:
3112:
3093:
3065:
3040:
3036:
3032:
3014:
2995:
2982:
2977:
2976:
2933:
2920:
2914:
2913:
2878:
2859:
2841:
2826:
2792:
2727:
2726:
2722:
2698:
2647:
2643:
2635:
2634:
2594:
2593:
2592:and by letting
2572:
2553:
2540:
2535:
2534:
2454:
2432:
2428:
2372:
2368:
2360:
2359:
2326:
2248:
2247:
2196:
2191:
2190:
2129:
2107:
2096:
2095:
2034:
2012:
2001:
2000:
1954:
1919:
1889:
1888:
1847:
1801:
1782:
1769:
1746:
1715:
1685:
1680:
1679:
1637:
1603:
1583:
1582:
1568:
1547:
1515:
1469:
1450:
1437:
1414:
1401:
1376:
1363:
1335:
1325:
1324:
1289:
1270:
1265:
1264:
1239:
1220:
1207:
1202:
1201:
1169:
1168:
1119:
1114:
1113:
1055:
1054:
1027:
981:
962:
949:
926:
895:
856:
836:
835:
830:
781:
780:
745:
744:
723:
704:
691:
686:
685:
661:
642:
629:
616:
611:
610:
547:
546:
519:
518:
475:
456:
443:
438:
437:
436:. The function
418:
417:
396:
391:
390:
363:
346:
300:
281:
268:
201:
181:
180:
168:) as input and
154:
152:Continuous time
135:function spaces
120:
112:Volterra series
103:Brownian motion
87:
52:intermodulation
21:Volterra series
17:
12:
11:
5:
7257:
7255:
7247:
7246:
7241:
7231:
7230:
7227:
7226:
7219:
7209:
7202:
7195:
7185:
7178:
7166:
7163:
7160:
7159:
7118:
7103:
7060:
7033:(2): 201–214.
7017:
6979:
6960:(3): 265–284.
6941:
6918:
6909:
6896:
6880:
6871:
6856:
6809:
6808:
6806:
6803:
6802:
6801:
6796:
6789:
6786:
6777:
6774:
6761:
6758:
6749:
6746:
6740:
6737:
6721:
6718:
6714:
6691:
6688:
6684:
6661:
6657:
6647:is the order,
6636:
6625:
6624:
6613:
6610:
6605:
6600:
6596:
6591:
6587:
6582:
6577:
6573:
6568:
6562:
6559:
6555:
6549:
6545:
6541:
6536:
6531:
6528:
6525:
6521:
6517:
6514:
6509:
6505:
6501:
6498:
6495:
6490:
6486:
6482:
6477:
6473:
6447:neural network
6442:
6439:
6431:
6430:
6419:
6416:
6411:
6407:
6403:
6398:
6394:
6390:
6385:
6381:
6377:
6372:
6368:
6364:
6359:
6356:
6353:
6348:
6344:
6336:
6332:
6327:
6319:
6315:
6310:
6304:
6299:
6295:
6291:
6288:
6285:
6280:
6276:
6272:
6267:
6263:
6259:
6254:
6251:
6248:
6243:
6239:
6231:
6227:
6222:
6216:
6212:
6208:
6203:
6200:
6197:
6192:
6188:
6184:
6179:
6175:
6163:
6162:
6151:
6148:
6143:
6139:
6135:
6130:
6126:
6122:
6117:
6113:
6109:
6104:
6100:
6096:
6091:
6087:
6083:
6078:
6075:
6072:
6067:
6063:
6055:
6051:
6046:
6040:
6037:
6033:
6027:
6022:
6018:
6014:
6011:
6008:
6003:
5999:
5995:
5990:
5986:
5982:
5977:
5973:
5969:
5964:
5961:
5958:
5953:
5949:
5941:
5937:
5932:
5926:
5922:
5918:
5915:
5910:
5907:
5904:
5899:
5895:
5891:
5886:
5882:
5870:
5869:
5858:
5855:
5850:
5846:
5842:
5837:
5833:
5829:
5824:
5820:
5816:
5811:
5807:
5803:
5798:
5795:
5792:
5787:
5783:
5775:
5771:
5766:
5760:
5756:
5752:
5749:
5744:
5741:
5738:
5733:
5729:
5725:
5720:
5716:
5704:
5703:
5692:
5689:
5684:
5680:
5676:
5671:
5667:
5663:
5658:
5654:
5650:
5645:
5641:
5637:
5632:
5628:
5624:
5619:
5616:
5613:
5608:
5604:
5596:
5592:
5587:
5581:
5577:
5573:
5570:
5565:
5562:
5559:
5554:
5550:
5546:
5541:
5537:
5525:
5524:
5513:
5508:
5505:
5502:
5497:
5493:
5489:
5484:
5480:
5468:
5467:
5456:
5451:
5448:
5445:
5440:
5436:
5432:
5427:
5423:
5407:
5406:
5395:
5391:
5386:
5378:
5374:
5368:
5364:
5359:
5355:
5350:
5346:
5342:
5337:
5334:
5331:
5326:
5322:
5318:
5311:
5307:
5301:
5297:
5292:
5288:
5283:
5279:
5275:
5270:
5267:
5264:
5259:
5255:
5251:
5244:
5240:
5234:
5230:
5225:
5221:
5216:
5212:
5208:
5203:
5200:
5197:
5192:
5188:
5183:
5177:
5172:
5168:
5164:
5160:
5156:
5151:
5147:
5143:
5140:
5137:
5132:
5129:
5126:
5122:
5116:
5111:
5108:
5105:
5101:
5097:
5094:
5091:
5086:
5083:
5080:
5076:
5071:
5067:
5063:
5054:
5049:
5045:
5041:
5038:
5034:
5029:
5026:
5021:
5017:
5013:
5008:
5004:
5000:
4995:
4991:
4987:
4982:
4979:
4976:
4971:
4967:
4955:
4954:
4943:
4939:
4931:
4927:
4921:
4917:
4912:
4906:
4903:
4900:
4895:
4891:
4885:
4881:
4877:
4873:
4869:
4864:
4860:
4856:
4853:
4850:
4845:
4842:
4839:
4835:
4829:
4824:
4821:
4818:
4814:
4810:
4807:
4804:
4799:
4796:
4793:
4789:
4784:
4780:
4776:
4767:
4762:
4758:
4754:
4751:
4747:
4742:
4739:
4734:
4730:
4726:
4721:
4717:
4713:
4708:
4705:
4702:
4697:
4693:
4681:
4680:
4669:
4665:
4661:
4656:
4652:
4648:
4645:
4642:
4637:
4634:
4631:
4627:
4623:
4620:
4617:
4612:
4609:
4606:
4602:
4597:
4593:
4586:
4582:
4578:
4573:
4570:
4565:
4561:
4557:
4552:
4549:
4546:
4541:
4537:
4525:
4524:
4513:
4510:
4507:
4504:
4501:
4496:
4493:
4490:
4486:
4482:
4479:
4476:
4471:
4468:
4465:
4460:
4456:
4426:
4422:
4388:
4384:
4358:
4354:
4331:
4327:
4314:
4311:
4310:
4309:
4298:
4295:
4290:
4286:
4282:
4277:
4273:
4269:
4264:
4260:
4256:
4251:
4247:
4243:
4238:
4234:
4226:
4222:
4217:
4209:
4205:
4200:
4194:
4190:
4186:
4183:
4180:
4175:
4171:
4167:
4162:
4158:
4154:
4149:
4145:
4137:
4133:
4128:
4124:
4121:
4116:
4112:
4108:
4103:
4099:
4087:
4086:
4075:
4072:
4067:
4063:
4059:
4054:
4050:
4046:
4041:
4037:
4033:
4028:
4024:
4020:
4015:
4011:
4007:
4002:
3998:
3990:
3986:
3981:
3975:
3972:
3968:
3962:
3958:
3954:
3951:
3948:
3943:
3939:
3935:
3930:
3926:
3922:
3917:
3913:
3909:
3904:
3900:
3892:
3888:
3883:
3879:
3876:
3873:
3868:
3864:
3860:
3855:
3851:
3839:
3838:
3827:
3824:
3819:
3815:
3811:
3806:
3802:
3798:
3793:
3789:
3785:
3780:
3776:
3772:
3767:
3763:
3755:
3751:
3746:
3742:
3739:
3736:
3731:
3727:
3723:
3718:
3714:
3702:
3701:
3690:
3687:
3682:
3678:
3674:
3669:
3665:
3661:
3656:
3652:
3648:
3643:
3639:
3635:
3630:
3626:
3622:
3617:
3613:
3605:
3601:
3596:
3592:
3589:
3586:
3581:
3577:
3573:
3568:
3564:
3552:
3551:
3540:
3535:
3531:
3527:
3522:
3518:
3506:
3505:
3494:
3489:
3485:
3481:
3476:
3472:
3445:
3442:
3439:
3436:
3431:
3427:
3421:
3418:
3415:
3410:
3407:
3404:
3400:
3384:
3383:
3372:
3364:
3360:
3356:
3353:
3347:
3343:
3338:
3334:
3330:
3327:
3324:
3321:
3318:
3315:
3310:
3306:
3302:
3299:
3296:
3293:
3289:
3285:
3282:
3279:
3276:
3271:
3267:
3261:
3258:
3255:
3250:
3247:
3244:
3240:
3236:
3233:
3230:
3227:
3224:
3220:
3215:
3211:
3205:
3202:
3197:
3193:
3189:
3186:
3183:
3178:
3174:
3170:
3165:
3161:
3146:
3145:
3134:
3126:
3122:
3118:
3115:
3109:
3105:
3100:
3096:
3092:
3089:
3086:
3083:
3080:
3077:
3072:
3068:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3039:
3035:
3029:
3026:
3021:
3017:
3013:
3010:
3007:
3002:
2998:
2994:
2989:
2985:
2960:
2957:
2954:
2951:
2945:
2940:
2936:
2932:
2927:
2923:
2910:
2909:
2898:
2895:
2889:
2886:
2881:
2877:
2874:
2871:
2866:
2862:
2858:
2852:
2849:
2844:
2837:
2834:
2829:
2825:
2820:
2817:
2812:
2808:
2804:
2799:
2795:
2791:
2788:
2785:
2782:
2776:
2773:
2767:
2764:
2761:
2757:
2753:
2750:
2747:
2744:
2738:
2735:
2730:
2725:
2721:
2718:
2714:
2710:
2705:
2701:
2697:
2694:
2691:
2688:
2682:
2679:
2673:
2670:
2667:
2663:
2659:
2656:
2653:
2650:
2646:
2642:
2617:
2612:
2608:
2604:
2601:
2579:
2575:
2571:
2568:
2565:
2560:
2556:
2552:
2547:
2543:
2527:
2526:
2515:
2511:
2507:
2504:
2501:
2498:
2493:
2487:
2484:
2479:
2475:
2472:
2469:
2466:
2461:
2457:
2451:
2446:
2443:
2440:
2436:
2431:
2427:
2424:
2420:
2416:
2413:
2410:
2407:
2402:
2396:
2393:
2388:
2384:
2381:
2378:
2375:
2371:
2367:
2353:
2352:
2341:
2338:
2333:
2329:
2325:
2322:
2319:
2316:
2310:
2307:
2301:
2298:
2295:
2291:
2287:
2284:
2281:
2278:
2275:
2270:
2264:
2261:
2256:
2217:
2214:
2211:
2208:
2203:
2199:
2187:
2186:
2175:
2172:
2169:
2166:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2136:
2132:
2128:
2125:
2122:
2119:
2114:
2110:
2106:
2103:
2092:
2091:
2080:
2077:
2074:
2071:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2041:
2037:
2033:
2030:
2027:
2024:
2019:
2015:
2011:
2008:
1990:
1989:
1978:
1975:
1972:
1969:
1966:
1961:
1957:
1951:
1947:
1943:
1940:
1937:
1934:
1931:
1926:
1922:
1916:
1912:
1908:
1905:
1902:
1899:
1896:
1874:
1873:
1862:
1859:
1854:
1850:
1846:
1843:
1840:
1837:
1832:
1827:
1824:
1821:
1817:
1813:
1808:
1804:
1800:
1797:
1794:
1789:
1785:
1781:
1776:
1772:
1766:
1761:
1758:
1753:
1749:
1744:
1740:
1735:
1730:
1727:
1722:
1718:
1713:
1709:
1706:
1703:
1700:
1697:
1692:
1688:
1673:
1672:
1661:
1658:
1655:
1652:
1649:
1644:
1640:
1634:
1629:
1626:
1623:
1619:
1615:
1610:
1606:
1602:
1599:
1596:
1593:
1590:
1578:operators, as
1567:
1564:
1546:
1543:
1542:
1541:
1530:
1527:
1522:
1518:
1514:
1511:
1508:
1505:
1500:
1495:
1492:
1489:
1485:
1481:
1476:
1472:
1468:
1465:
1462:
1457:
1453:
1449:
1444:
1440:
1434:
1427:
1424:
1421:
1417:
1413:
1408:
1404:
1399:
1395:
1390:
1383:
1379:
1375:
1370:
1366:
1361:
1355:
1350:
1347:
1342:
1338:
1333:
1296:
1292:
1288:
1285:
1282:
1277:
1273:
1251:
1246:
1242:
1238:
1235:
1232:
1227:
1223:
1219:
1214:
1210:
1182:
1179:
1176:
1126:
1122:
1112:Each function
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1042:
1039:
1034:
1030:
1026:
1023:
1020:
1017:
1012:
1007:
1004:
1001:
997:
993:
988:
984:
980:
977:
974:
969:
965:
961:
956:
952:
946:
941:
938:
933:
929:
924:
920:
915:
910:
907:
902:
898:
893:
887:
882:
879:
876:
872:
868:
863:
859:
855:
852:
849:
846:
843:
829:
826:
814:equicontinuous
797:
794:
791:
788:
758:
755:
752:
730:
726:
722:
719:
716:
711:
707:
703:
698:
694:
673:
668:
664:
660:
657:
654:
649:
645:
641:
636:
632:
628:
623:
619:
595:Sometimes the
554:
526:
498:is called the
487:
482:
478:
474:
471:
468:
463:
459:
455:
450:
446:
425:
403:
399:
387:
386:
375:
370:
366:
362:
358:
353:
349:
345:
342:
339:
336:
331:
326:
323:
320:
316:
312:
307:
303:
299:
296:
293:
288:
284:
280:
275:
271:
265:
260:
256:
252:
247:
242:
238:
232:
227:
224:
221:
217:
213:
208:
204:
200:
197:
194:
191:
188:
153:
150:
146:
145:
138:
119:
116:
95:Norbert Wiener
86:
83:
60:non-parametric
15:
13:
10:
9:
6:
4:
3:
2:
7256:
7245:
7242:
7240:
7237:
7236:
7234:
7224:
7220:
7218:
7214:
7210:
7207:
7203:
7200:
7196:
7193:
7189:
7188:Giannakis G.B
7186:
7183:
7179:
7177:
7173:
7170:Barrett J.F:
7169:
7168:
7164:
7154:
7149:
7145:
7141:
7137:
7133:
7129:
7122:
7119:
7114:
7107:
7104:
7099:
7095:
7091:
7087:
7083:
7079:
7075:
7071:
7064:
7061:
7056:
7052:
7048:
7044:
7040:
7036:
7032:
7028:
7021:
7018:
7012:
7007:
7003:
6999:
6995:
6988:
6986:
6984:
6980:
6975:
6971:
6967:
6963:
6959:
6955:
6948:
6946:
6942:
6937:
6933:
6929:
6922:
6919:
6913:
6910:
6906:
6900:
6897:
6894:
6890:
6884:
6881:
6875:
6872:
6867:
6860:
6857:
6845:
6841:
6837:
6833:
6829:
6825:
6821:
6814:
6811:
6804:
6800:
6797:
6795:
6794:Wiener series
6792:
6791:
6787:
6785:
6783:
6775:
6773:
6771:
6767:
6760:Kernel method
6759:
6757:
6754:
6747:
6745:
6738:
6736:
6719:
6716:
6712:
6689:
6686:
6682:
6659:
6655:
6634:
6611:
6603:
6598:
6594:
6589:
6585:
6580:
6575:
6571:
6566:
6560:
6557:
6553:
6547:
6543:
6534:
6529:
6526:
6523:
6519:
6515:
6507:
6503:
6499:
6496:
6493:
6488:
6484:
6475:
6471:
6463:
6462:
6461:
6459:
6454:
6452:
6448:
6440:
6438:
6436:
6417:
6409:
6405:
6401:
6396:
6392:
6388:
6383:
6379:
6375:
6370:
6366:
6354:
6346:
6342:
6334:
6330:
6325:
6317:
6313:
6308:
6302:
6297:
6293:
6289:
6286:
6278:
6274:
6270:
6265:
6261:
6249:
6241:
6237:
6229:
6225:
6220:
6214:
6210:
6206:
6198:
6190:
6186:
6182:
6177:
6173:
6165:
6164:
6149:
6141:
6137:
6133:
6128:
6124:
6120:
6115:
6111:
6107:
6102:
6098:
6094:
6089:
6085:
6073:
6065:
6061:
6053:
6049:
6044:
6038:
6035:
6031:
6025:
6020:
6016:
6012:
6009:
6001:
5997:
5993:
5988:
5984:
5980:
5975:
5971:
5959:
5951:
5947:
5939:
5935:
5930:
5924:
5920:
5916:
5913:
5905:
5897:
5893:
5889:
5884:
5880:
5872:
5871:
5856:
5848:
5844:
5840:
5835:
5831:
5827:
5822:
5818:
5814:
5809:
5805:
5793:
5785:
5781:
5773:
5769:
5764:
5758:
5754:
5750:
5747:
5739:
5731:
5727:
5723:
5718:
5714:
5706:
5705:
5690:
5682:
5678:
5674:
5669:
5665:
5661:
5656:
5652:
5648:
5643:
5639:
5635:
5630:
5626:
5614:
5606:
5602:
5594:
5590:
5585:
5579:
5575:
5571:
5568:
5560:
5552:
5548:
5544:
5539:
5535:
5527:
5526:
5511:
5503:
5495:
5491:
5487:
5482:
5478:
5470:
5469:
5454:
5446:
5438:
5434:
5430:
5425:
5421:
5413:
5412:
5411:
5393:
5389:
5384:
5376:
5372:
5366:
5362:
5357:
5348:
5344:
5332:
5324:
5320:
5316:
5309:
5305:
5299:
5295:
5290:
5281:
5277:
5265:
5257:
5253:
5249:
5242:
5238:
5232:
5228:
5223:
5214:
5210:
5198:
5190:
5186:
5181:
5175:
5170:
5166:
5162:
5158:
5149:
5145:
5141:
5138:
5127:
5120:
5114:
5109:
5106:
5103:
5099:
5092:
5081:
5074:
5069:
5065:
5061:
5052:
5047:
5043:
5039:
5036:
5032:
5027:
5019:
5015:
5011:
5006:
5002:
4998:
4993:
4989:
4977:
4969:
4965:
4957:
4956:
4941:
4937:
4929:
4925:
4919:
4915:
4910:
4901:
4893:
4889:
4883:
4879:
4875:
4871:
4862:
4858:
4854:
4851:
4840:
4833:
4827:
4822:
4819:
4816:
4812:
4805:
4794:
4787:
4782:
4778:
4774:
4765:
4760:
4756:
4752:
4749:
4745:
4740:
4732:
4728:
4724:
4719:
4715:
4703:
4695:
4691:
4683:
4682:
4667:
4663:
4654:
4650:
4646:
4643:
4632:
4625:
4618:
4607:
4600:
4595:
4591:
4584:
4580:
4576:
4571:
4563:
4559:
4547:
4539:
4535:
4527:
4526:
4511:
4502:
4491:
4484:
4477:
4474:
4466:
4458:
4454:
4446:
4445:
4444:
4441:
4424:
4420:
4410:
4407:
4402:
4386:
4382:
4372:
4356:
4352:
4329:
4325:
4312:
4296:
4288:
4284:
4280:
4275:
4271:
4267:
4262:
4258:
4254:
4249:
4245:
4236:
4232:
4224:
4220:
4215:
4207:
4203:
4198:
4192:
4188:
4184:
4181:
4173:
4169:
4165:
4160:
4156:
4147:
4143:
4135:
4131:
4126:
4122:
4119:
4114:
4110:
4106:
4101:
4097:
4089:
4088:
4073:
4065:
4061:
4057:
4052:
4048:
4044:
4039:
4035:
4031:
4026:
4022:
4018:
4013:
4009:
4000:
3996:
3988:
3984:
3979:
3973:
3970:
3966:
3960:
3956:
3952:
3949:
3941:
3937:
3933:
3928:
3924:
3920:
3915:
3911:
3902:
3898:
3890:
3886:
3881:
3877:
3874:
3871:
3866:
3862:
3858:
3853:
3849:
3841:
3840:
3825:
3817:
3813:
3809:
3804:
3800:
3796:
3791:
3787:
3783:
3778:
3774:
3765:
3761:
3753:
3749:
3744:
3740:
3737:
3734:
3729:
3725:
3721:
3716:
3712:
3704:
3703:
3688:
3680:
3676:
3672:
3667:
3663:
3659:
3654:
3650:
3646:
3641:
3637:
3633:
3628:
3624:
3615:
3611:
3603:
3599:
3594:
3590:
3587:
3584:
3579:
3575:
3571:
3566:
3562:
3554:
3553:
3538:
3533:
3529:
3525:
3520:
3516:
3508:
3507:
3492:
3487:
3483:
3479:
3474:
3470:
3462:
3461:
3460:
3457:
3440:
3434:
3429:
3425:
3419:
3416:
3413:
3408:
3405:
3402:
3389:
3370:
3362:
3358:
3354:
3351:
3345:
3336:
3332:
3328:
3325:
3319:
3316:
3308:
3304:
3300:
3297:
3291:
3287:
3280:
3274:
3269:
3265:
3259:
3256:
3253:
3248:
3245:
3242:
3234:
3228:
3222:
3218:
3213:
3209:
3203:
3195:
3191:
3187:
3184:
3181:
3176:
3172:
3163:
3159:
3151:
3150:
3149:
3132:
3124:
3120:
3116:
3113:
3107:
3098:
3094:
3090:
3087:
3081:
3078:
3070:
3066:
3062:
3059:
3053:
3047:
3041:
3037:
3033:
3027:
3019:
3015:
3011:
3008:
3005:
3000:
2996:
2987:
2983:
2975:
2974:
2973:
2958:
2955:
2952:
2943:
2938:
2934:
2930:
2925:
2921:
2896:
2884:
2879:
2875:
2872:
2869:
2864:
2860:
2847:
2842:
2832:
2827:
2823:
2815:
2810:
2806:
2797:
2793:
2789:
2786:
2780:
2771:
2765:
2762:
2759:
2755:
2748:
2742:
2733:
2728:
2723:
2719:
2716:
2712:
2703:
2699:
2695:
2692:
2686:
2677:
2671:
2668:
2665:
2661:
2654:
2648:
2644:
2640:
2633:
2632:
2631:
2615:
2610:
2606:
2602:
2599:
2577:
2573:
2569:
2566:
2563:
2558:
2554:
2550:
2545:
2541:
2532:
2513:
2509:
2502:
2496:
2491:
2482:
2477:
2470:
2464:
2459:
2455:
2444:
2441:
2438:
2434:
2429:
2425:
2422:
2418:
2411:
2405:
2400:
2391:
2386:
2379:
2373:
2369:
2365:
2358:
2357:
2356:
2355:we can write
2339:
2331:
2327:
2323:
2320:
2314:
2305:
2299:
2296:
2293:
2289:
2285:
2279:
2273:
2268:
2259:
2254:
2246:
2245:
2244:
2241:
2239:
2235:
2231:
2212:
2206:
2201:
2197:
2173:
2170:
2167:
2164:
2160:
2157:
2154:
2145:
2139:
2134:
2130:
2123:
2117:
2112:
2108:
2101:
2094:
2093:
2078:
2075:
2072:
2069:
2065:
2062:
2059:
2050:
2044:
2039:
2035:
2028:
2022:
2017:
2013:
2006:
1999:
1998:
1997:
1995:
1976:
1970:
1964:
1959:
1955:
1949:
1945:
1941:
1935:
1929:
1924:
1920:
1914:
1910:
1906:
1900:
1894:
1887:
1886:
1885:
1883:
1882:Wiener series
1879:
1860:
1852:
1848:
1844:
1841:
1835:
1830:
1825:
1822:
1819:
1815:
1806:
1802:
1798:
1795:
1792:
1787:
1783:
1774:
1770:
1764:
1759:
1756:
1751:
1747:
1742:
1738:
1733:
1728:
1725:
1720:
1716:
1711:
1707:
1701:
1695:
1690:
1686:
1678:
1677:
1676:
1659:
1653:
1647:
1642:
1638:
1632:
1627:
1624:
1621:
1617:
1613:
1608:
1604:
1600:
1594:
1588:
1581:
1580:
1579:
1577:
1572:
1565:
1563:
1561:
1555:
1553:
1552:Wiener series
1544:
1528:
1520:
1516:
1512:
1509:
1503:
1498:
1493:
1490:
1487:
1483:
1474:
1470:
1466:
1463:
1460:
1455:
1451:
1442:
1438:
1432:
1425:
1422:
1419:
1415:
1411:
1406:
1402:
1397:
1393:
1388:
1381:
1377:
1373:
1368:
1364:
1359:
1353:
1348:
1345:
1340:
1336:
1331:
1323:
1322:
1321:
1319:
1315:
1314:causal system
1310:
1294:
1290:
1286:
1283:
1280:
1275:
1271:
1244:
1240:
1236:
1233:
1230:
1225:
1221:
1212:
1208:
1198:
1196:
1180:
1177:
1174:
1166:
1162:
1158:
1154:
1150:
1146:
1142:
1124:
1120:
1099:
1087:
1081:
1078:
1075:
1072:
1069:
1063:
1060:
1040:
1032:
1028:
1024:
1021:
1015:
1010:
1005:
1002:
999:
995:
986:
982:
978:
975:
972:
967:
963:
954:
950:
944:
939:
936:
931:
927:
922:
918:
913:
908:
905:
900:
896:
891:
885:
880:
877:
874:
870:
866:
861:
857:
853:
847:
841:
833:
828:Discrete time
827:
825:
823:
819:
815:
811:
792:
786:
778:
774:
770:
756:
753:
750:
728:
724:
720:
717:
714:
709:
705:
701:
696:
692:
666:
662:
658:
655:
652:
647:
643:
639:
634:
630:
621:
617:
608:
604:
602:
598:
593:
591:
590:doubly finite
587:
583:
579:
575:
571:
566:
552:
544:
540:
524:
516:
512:
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6903:Ikehara S:
1880:operators (
1576:homogeneous
1560:white noise
67:mathematics
7233:Categories
7211:Rugh W J:
6887:Wiener N:
6824:NeuroImage
6805:References
2630:, we have
545:variables
517:variables
502:-th-order
75:functional
33:capacitors
7204:Kuo Y L:
6713:ω
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99:Paul LĂ©vy
71:nonlinear
37:inductors
7090:17052160
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6974:57663554
6849:24 April
6844:12948688
6788:See also
4406:locality
2972:, it is
2533:is SWN,
129:operator
62:model.
7098:9268156
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810:compact
777:Fréchet
131:mapping
85:History
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1195:causal
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160:with
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1648:x
1643:p
1639:H
1633:P
1628:1
1625:=
1622:p
1614:+
1609:0
1605:h
1601:=
1598:)
1595:n
1592:(
1589:y
1529:.
1526:)
1521:j
1510:n
1507:(
1504:x
1499:p
1494:1
1491:=
1488:j
1480:)
1475:p
1467:,
1461:,
1456:1
1448:(
1443:p
1439:h
1433:M
1426:1
1420:p
1412:=
1407:p
1389:M
1382:1
1374:=
1369:2
1354:M
1349:0
1346:=
1341:1
1318:n
1295:p
1287:,
1281:,
1276:1
1250:)
1245:p
1237:,
1231:,
1226:1
1218:(
1213:p
1209:h
1181:0
1175:a
1161:P
1157:b
1153:a
1145:P
1125:p
1121:h
1100:.
1097:}
1091:{
1085:}
1079:,
1076:2
1073:,
1070:1
1067:{
1061:P
1041:,
1038:)
1033:j
1022:n
1019:(
1016:x
1011:p
1006:1
1003:=
1000:j
992:)
987:p
979:,
973:,
968:1
960:(
955:p
951:h
945:b
940:a
937:=
932:p
914:b
909:a
906:=
901:1
886:P
881:1
878:=
875:p
867:+
862:0
858:h
854:=
851:)
848:n
845:(
842:y
796:)
793:t
790:(
787:x
757:0
751:a
729:n
725:t
721:,
715:,
710:2
706:t
702:,
697:1
693:t
672:)
667:n
663:t
659:,
653:,
648:2
644:t
640:,
635:1
631:t
627:(
622:n
618:h
601:n
597:n
586:N
582:b
578:a
570:N
543:n
539:n
515:n
500:n
486:)
481:n
473:,
467:,
462:1
454:(
449:n
445:h
424:y
402:0
398:h
374:.
369:j
361:d
357:)
352:j
341:t
338:(
335:x
330:n
325:1
322:=
319:j
311:)
306:n
298:,
292:,
287:1
279:(
274:n
270:h
264:b
259:a
246:b
241:a
231:N
226:1
223:=
220:n
212:+
207:0
203:h
199:=
196:)
193:t
190:(
187:y
174:t
172:(
170:y
166:t
164:(
162:x
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