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discrete-time Wiener system with a pure cubic nonlinearity will be discussed in Chapter 6.

3.4

Discrete-time Volterra systems

So far the theory has been developed for a Wh structure. The theory can be gen- eralised to discrete-time Volterra systems that encompass a much wider range of time-invariant nonlinear systems (Volterra,1930/2005). If a nonlinear system con- tains feedback, the theory in this section should provide an adequate approximation if the input-output relationship of the said system can be represented sufficiently well by a Volterra series of a finite degree.

Following on from the Stone-Weierstrass theorem (Stone, 1948), any time- invariant, finite memory, causal, discrete-time nonlinear system that is a continuous functional1 _{of the input can (over a uniformly bounded set of input signals) be} uniformly approximated to an arbitrary accuracy by a discrete-time, finite-memory Volterra series of sufficient order (Korenberg,1991). Similar statements can also be made for continuous-time systems through the pioneering work of Fréchet (1910). Many time-invariant nonlinear dynamical systems including many block structured systems (for example,Whof Section3.3and parallel Wiener) can be modelled using Volterra series. The converse is also true—a parallel Wiener structure, for example, may represent any Volterra systems by incorporating more and more branches within any arbitrary level of accuracy (Westwick & Kearney, 2003). Hence, Bla theory applicable to Wiener systems can be, by principle of superposition, applied to a Volterra System.

Here, the theoreticalBlais derived for white zero-mean Gaussian and binary sequences for a generic discrete-time Volterra system (see Appendix A.2 for the assumptions made). The zero-mean requirement is not a big deal in practice, as a dc term simply changes the operating point at which the system is linearised. Discrepancies between the Gaussian and binaryBlahowever, depend on the power spectra and the higher order moments of both types of inputs. Hence, both the Gaussian and binary sequences are assumed to be white and have power normalised to unity.

A Siso Volterra system is represented by Fig. 3.5. Its output is given by a Volterra series, which contains a sum of multi-dimensional convolution integrals of 1_{The term ‘functional’ in mathematics refers to a function which maps a vector space to a scalar} field; in this case, from the vector of the system input in different past time points to the output at a given time.

3.4. Discrete-time Volterra systems

ℎ[𝑥]

𝑢[𝑡] 𝑦[𝑡]

Figure 3.5: Generic Volterra system structure

the system input with the Volterra kernels, from degree of one to infinity. A Volterra kernel may be considered a multi-dimensional analogue of anIrf. The zeroth kernel contribution𝑦 is a constant independent of system input. The first degree of the Volterra kernel ℎ [𝑘] with a dimension of one (containing a single argument) is known as the linear kernel and this is equivalent to the Irf𝑔[𝑘] for a linear system. A purely linear system has the linear kernel equal to its Irf and all higher order kernels are exactly zero; as such, a linear dynamical system is a degenerate case of a Volterra system. For nonlinear systems, one or more higher order kernels are non- trivial, i.e. having non-zero output contributions. These multi-dimensional kernels capture the nonlinear effects. Given a time-invariant nonlinear system depicted in Fig.3.5, the output 𝑦is an aggregate sum of contributions from an infinite number of Volterra kernels such that 𝑦 = ∑ 𝑦 , where 𝑦 is the output contribution from a single kernel of order 𝑞. 𝑦 for𝑞 ≥ 1 is given by:

𝑦 [𝑡] = ⋯ ℎ 𝑘 ,..., 𝑘 𝑢[𝑡 − 𝑘 ] (3.35)

where ℎ is the 𝑞th Volterra kernel. Without loss of generality, normalising the power of the input to unity and using (3.9), the Blafor the𝑞th _{kernel is given by:}

𝑔_Bla(𝑟) = Eq𝑦 [𝑡]𝑢[𝑡 − 𝑟]y = ⋯ ℎ 𝑘 ,..., 𝑘 E t 𝑢[𝑡 − 𝑟] 𝑢[𝑡 − 𝑘 ] | . (3.36)

The law of superposition applies since if 𝑦 = ∑ 𝑦 ,𝑔_Bla= ∑ 𝑔_Bla.

The mathematical work for deriving the theoretical contributions from an arbitrary Volterra kernel degree to theBlais shown in AppendixA.2. Based upon this, the theoretical contributions from the 3rd _{and 5}th _{degree Volterra kernels are} given in the next two sections.

3.4.1 Third degree Volterra contributions

The following results apply to time-invariant discrete-time Volterra systems excited by zero-mean inputs with normalised power of unity.

3.4. Discrete-time Volterra systems

Gaussian input case

AppendixA.2 has shown that for zero-mean white Gaussian inputs, (3.36) may be simplified to (A.2.10), reproduced as follows:

𝑔_Bla[𝑟] = 𝑞!! ⋯ ℎ 𝑘 , 𝑘 , 𝑘 , 𝑘 ,..., 𝑘 , 𝑘 , 𝑟 (3.37)

where 𝑞 is odd and 𝑝 = (𝑞 − 1)/2. From this equation, the Bla of a nonlinear system described by a 3rd degree Volterra kernel, excited by zero-mean Gaussian inputs with normalised power, is given by:

𝑔 _Bla[𝑟] = 3!! ℎ 𝑘, 𝑘, 𝑟

= 3 ℎ 𝑘, 𝑘, 𝑟 . (3.38)

Binary input case

With respect to zero-mean binary inputs, using (A.2.14), one can deduce that a 3rd degree Volterra series contains the following forms of non-trivial (non-zero) contri- butions: ⎧ ⎨ ⎩ ( ) ℎ [𝑟, 𝑟, 𝑟] for𝑘 = 𝑟 ( ) ℎ [𝑘, 𝑘, 𝑟] for𝑘 ≠ 𝑟 =⎧ ⎨ ⎩ ℎ [𝑟, 𝑟, 𝑟] for𝑘 = 𝑟 3ℎ [𝑘, 𝑘, 𝑟] for𝑘 ≠ 𝑟. (3.39) The binary BLA is then given by:

𝑔 _BLA[𝑟] = ℎ [𝑟, 𝑟, 𝑟] + 3 ℎ [𝑘, 𝑘, 𝑟].

Incorporating the first term into the second term to remove the restriction of𝑘 ≠ 𝑟 in the summation yields:

𝑔 _BLA[𝑟] = −2ℎ [𝑟, 𝑟, 𝑟] + 3 ℎ [𝑘, 𝑘, 𝑟]

= 𝑔 _Bla[𝑟] − 2ℎ [𝑟, 𝑟, 𝑟] . (3.40)

Note that if the nonlinear system is a Wiener system constructed from a linearity with Irf of 𝑔[𝑘] followed by a pure cubic nonlinearity, its Volterra kernel

3.4. Discrete-time Volterra systems

ℎ[𝑎, 𝑏, 𝑐]collapses into the product𝑔[𝑎]𝑔[𝑏]𝑔[𝑐]. In such a case, (3.40) is simplified to the previous result of (3.25), as required.

Arbitrary input case

Similar to the steps performed in deriving (3.39), for any zero-mean arbitrary inputs, there are non-trivial contributions of the Volterra series in the form of:

⎧ ⎨ ⎩ ( ) ℎ [𝑟, 𝑟, 𝑟] 𝔐 for𝑘 = 𝑟 ( ) ℎ [𝑘, 𝑘, 𝑟] (𝔐 ) for𝑘 ≠ 𝑟 (3.41)

where 𝔐 ≜ EJ𝑢 Kis the moment notation defined in Section3.2.

Recall from (3.27) that the definition of the non-Gaussian moment correction terms is 𝛿 ≜ Er𝑢 [𝑘]z− (𝑛 − 1)!! 𝜎 for even 𝑛, where 𝑢 [𝑘] is some zero-mean arbitrary sequence. Normalising the power of the sequences such that 𝜎 = 𝔐 = Er𝑢 [𝑘]z = 1, one can write the contributions to Bla of this arbitrary sequence in terms of Gaussian moment by applying (A.3.9) for both Gaussian and arbitrary signals as:

𝑔 _BLA = 𝑔 _BLA+ 𝑔 _BLA− 𝑔 _BLA = 𝑔 _BLA+ 𝛿 ℎ [𝑟, 𝑟, 𝑟] +

(3 − 3) ℎ [𝑘, 𝑘, 𝑟]

= 𝑔 _BLA+ 𝛿 ℎ [𝑟, 𝑟, 𝑟]. (3.42)

If the nonlinear system is a Wiener system with Irf of 𝑔_L[𝑘] followed by a pure cubic nonlinearity, (3.42) becomes:

𝑔 _BLA[𝑟] = 𝑔 _BLA[𝑟] + 𝛿 𝑔_L[𝑘]. (3.43) If the input is binary, using the definition of 𝛿 , 𝛿 = 1 − (4 − 1)!! = 1 − 3 = −2. Substituting these into (3.43) one obtains again (3.25) as expected.

3.4.2 Fifth degree Volterra contributions

The results for the 5th _{Volterra kernel are reproduced from similar workings shown} in Appendix A.3, up until where the expressions are adapted for Wiener systems. These results apply to time-invariant discrete-time Volterra systems excited by zero- mean white inputs with normalised power of unity.