TheVolterra seriesis a model for non-linear behavior similar to theTaylor series. 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 this 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 atallother times. This provides the ability to capture the 'memory' effect of devices like capacitors and inductors.It has been applied in the fields of medicine (biomedical engineering) and biology, especially neuroscience. It is also used in electrical engineering to modelintermodulationdistortion in many devices including power amplifiers andfrequency mixers. Its main advantage lies in its generality: it can represent a wide range of systems. Thus it is sometimes considered anon-parametricmodel.Inmathematics, a Volterra series denotes a functional expansion of a dynamic,nonlinear, time-invariantfunctional. Volterra series are frequently used insystem identification. The Volterra series, which is used to prove the Volterra theorem, is an infinite sum of multidimensional convolutional integrals.

Mathematical theoryThe theory of Volterra series can be viewed from two different perspectives: either one considers an operator mapping between two real (or complex) function spaces or a functional mapping from a real (or complex) function space into the real (or complex) numbers. The latter, functional perspective is in more frequent use, due to the assumed time-invariance of the system.Continuous timeA continuous time-invariant system withx(t) as input andy(t) as output can be expanded in Volterra series as:

whereand.The function,is called then-th orderVolterrakernel. It can be regarded as a higher-order impulse response of the system.IfNis finite, the series is said to betruncated. Ifa,b, andNare finite, the series is calleddoubly-finite.Sometimes then-th order term is divided by n!, a convention which is convenient when taking the output of one Volterra system as the input of another ('cascading').The causality condition: Since in any physically realizable system the output can only depend on previous values of the input, the kernelswill be zero if any of the variablesare negative. The integrals may then be written over the half range from zero to infinity. So if the operator is causal,.Frchet's approximation theorem: The use of the Volterra series to represent a time-invariant functional relation is often justified by appealing to a theorem due toFrchet. 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 functionsfor which the approximation will hold is required to becompact. It is usually taken to be anequicontinuous,uniformly boundedset of functions, which is compact by theArzelAscoli theorem. 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.

Discrete time

whereand.,are called Volterra kernels.IfPis finite, the series operator is said truncated.Ifa,bandPare finite the series operator is called doubly-finite Volterra series.Ifthe operator is causal.We can always consider, without loss of the generality, the kernelas symmetrical. In fact, for the commutativity of the multiplication it is always possible to symmetrize it without changing.So for a causal system with symmetrical kernels we can write

Block-structured systemsBecause of the problems of identifying Volterra models other model forms were investigated as a basis for system identification for nonlinear systems. Various forms of block structured nonlinear models have been introduced or re-introduced.[5][6]The Hammerstein model consists of a static single valued nonlinear element followed by a linear dynamic element. The Wiener model is the reverse of this combination so that the linear element occurs before the static nonlinear characteristic. The Wiener-Hammerstein model consists of a static linear element sandwiched between two dynamic systems, and several other model forms are available. All these models can be represented by a Volterra series but in this case the Volterra kernels take on a special from in each case. Identification consists of correlation based and parameter estimation methods. The correlation methods exploit certain properties of these systems, which means that if specific inputs are used, often white Gaussian noise, the individual elements can be identified one at a time. This results in manageable data requirements and the individual blocks can sometimes be related to components in the system under study.More recent results are based on parameter estimation and neural network based solutions. Many results have been introduced and these systems continue to be studied in depth. One problem is that these methods are only applicable to a very special form of model in each case and usually this model form has to be known prior to identification.