Fourier nonlinear filters

Signal Processing 94 (2014) 183–194

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Fourier nonlinear filters Alberto Carini a,n, Giovanni L. Sicuranza b a b

DiSBeF, University of Urbino “Carlo Bo”, 61029 Urbino, Italy DIA, University of Trieste, 34127 Trieste, Italy

a r t i c l e i n f o

abstract

Article history: Received 29 December 2012 Received in revised form 10 June 2013 Accepted 13 June 2013 Available online 28 June 2013

In this paper, two new sub-classes of linear-in-the-parameters nonlinear discrete-time filters, derived from the truncation of multidimensional generalized Fourier series, are presented. The filters, called Fourier nonlinear filters and even mirror Fourier nonlinear filters, are universal approximators for causal, time-invariant, finite-memory, continuous nonlinear systems, according to the Stone–Weierstrass approximation theorem. Their properties and limitations are discussed in detail. In particular, we show, by means of appropriate simulation examples, that an orthogonality property they satisfy for white uniform input signals is useful for improving the identification of nonlinear systems. & 2013 Elsevier B.V. All rights reserved.

Keywords: Nonlinear filters Volterra filters Fourier nonlinear filters Even mirror Fourier nonlinear filters

1. Introduction Truncated Volterra filters [1] are one of the most popular sub-classes of linear-in-the-parameters (LIP) nonlinear discrete-time filters used in common practice (see [2–13] for selected recent contributions on this topic). Their success derives from two key factors: (i) the ability to be universal approximators, i.e., to arbitrarily well approximate any causal, time-invariant, finite-memory, continuous nonlinear system according to the Stone–Weierstrass approximation theorem [14] and (ii) the simple LIP input–output relationship. Their main drawbacks are: (i) the large number of coefficients involved, which increases exponentially with the filter order and (ii) the slow convergence of the gradient descent adaptation algorithms, caused mainly by the poor conditioning of the input autocorrelation matrix. Many other nonlinear LIP models have been successively introduced to simplify the nonlinear filter structures, and thus model nonlinearities using less coefficients than Volterra filters (e.g., Hammerstein filters1 [1,16–18], memory polynomial filters [19], generalized memory polynomial

n

Corresponding author. Tel.: +39 0722 304412; fax: +39 0722 4475. E-mail address: [email protected] (A. Carini). 1 Hammerstein filters formed by a parameterized static nonlinearity followed by a parameterized linear system are not LIP nonlinear models [15]. Nevertheless, if the nonlinearity is assumed to be fixed a priori, as it often happens in the literature, the filter is a LIP model. 0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.06.018

filters [20], Functional Link Artificial Neural Networks (FLANN) [21] based on trigonometric [22,23] or other expansions [24], Radial Basis Function Networks [25,26]). A review under a unified framework of finite-memory LIP nonlinear filters is given in [27]. However, some of LIP nonlinear models do not possess the ability of Volterra filters of being universal approximators. This is the case, for example, of the well-known FLANN filters. These filters have been frequently used in applications in the fields of nonlinear active noise control [23], channel equalization [22] and nonlinear acoustic echo cancellation [28]. However, as pointed out in [29], the performance of FLANN filters may be negatively affected by the lack of cross-terms, i.e., products of samples with different time shifts. To overcome this difficulty, the structure of the conventional FLANN filter has been modified, including appropriate cross-terms, in [30], where a generalized FLANN (GFLANN) filter has been proposed. It should be noted that FLANN and GFLANN filters are not universal approximators since, differently from truncated Volterra filters, their basis functions do not fulfill the conditions of the Stone–Weierstrass theorem. In this paper we introduce two sub-classes of LIP nonlinear filters relying on trigonometric basis functions: the Fourier nonlinear (FN) filters,2 and the even mirror

2 FN filters have been first introduced in [31] with a different name and without a detailed discussion of their properties.

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Fourier nonlinear (EMFN) filters.3 Their basis functions satisfy all the requirements of the Stone–Weierstrass theorem, and thus FN and EMFN filters are universal approximators, as the truncated Volterra filters. On the other hand, in contrast to truncated Volterra filters, the basis functions of FN and EMFN filters are orthogonal for white uniform input signals. This property guarantees fast convergence of gradient descent adaptation algorithms and has been exploited to derive simple identification methods for nonlinear systems, allowing performance improvements with respect to the Volterra filters. FN filters are derived here by completing the set of trigonometric basis functions of the conventional FLANN filters and resorting to a multidimensional representation. It is also shown in the paper that FN filters can be considered as a truncation of a multidimensional generalized Fourier series [33]. We discuss their properties and point out their drawbacks, which affect also all FLANN filters [22,23,28]. To overcome these drawbacks, we introduce a second sub-class of LIP nonlinear filters based on trigonometric basis functions, the EMFN filters. We show that EMFN filters derive from the truncation of a multidimensional even symmetric generalized Fourier series. The models obtained for nonlinear systems using universal approximators as Volterra, FN, and EMFN filters, are equivalent when their order tends to infinity. Therefore, we provide in the paper the conversion formulas among them. Finally, significant examples showing the capabilities of FN and EMFN filters in comparison with truncated Volterra and FLANN filters are presented. Typical nonlinear situations are illustrated and discussed, while results involving the identification of real nonlinear systems will be deferred to another paper due to space limitations. The paper is organized as follows. In Section 2 a methodology for introducing novel classes of nonlinear filters on the basis of the Stone–Weierstrass theorem is outlined. In Sections 3 and 4, FN and EMFN filters, respectively, are introduced and their properties are described. The conversion among FN, EMFN, and Volterra filters is considered in Section 5. The simulation results are presented in Section 6. Conclusions follow in Section 7. Throughout the paper the following notation is used. Sets are represented with curly brackets, intervals with square brackets, while arguments of functions can be included within round, square, or curly brackets. N is the set of natural numbers, Nþ the set of positive natural numbers, R the set of real numbers, and Rλ is the interval ½λ; þλ⊂R. With xðnÞ∈Rλ , it is meant that all the elements of the sequence x(n) belong to the interval Rλ .

expressed in the following form: yðnÞ ¼ f ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ;

ð1Þ

where xðnÞ∈R1 is the input signal, yðnÞ∈R is the output signal and N is the memory length of the system. Eq. (1) involves present and past samples of the input signal and this representation is useful to efficiently implement the nonlinear filter, suitably exploiting its shifting property [27]. From the analysis point of view, Eq. (1) can be interpreted as a multidimensional function over the RN 1 space, where each dimension corresponds to a delayed input sample. This representation has been already exploited, for example, for truncated Volterra filters [1]. With this notation in mind, it is possible to represent the expansion of the function f in (1) with a series of basis functions fi, as in the following equation: f ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ 1

¼ ∑ ci f i ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ;

ð2Þ

i¼1

where ci ∈R and each fi is a continuous function from RN 1 to R. Every choice of the set of basis functions fi defines a different family of nonlinear filters, which can be used to identify or approximate the nonlinear systems in (1). In particular, we are interested in nonlinear filters that are able to arbitrarily well approximate any time-invariant, finite-memory, continuous nonlinear system. To this purpose, the Stone–Weierstrass theorem is exploited: “Let A be an algebra of real continuous functions on a compact set K. If A separates points on K and if A vanishes at no point of K, then the uniform closure B of A consists of all real continuous functions on K” [14]. According to the Stone–Weierstrass Theorem, every algebra of real continuous functions on the compact RN 1 which separates points and vanishes at no point is able to arbitrarily well approximate the continuous function f in (1). A family A of real functions is said to be an algebra if A is closed under addition, multiplication, and scalar multiplication, i.e., if (i) f þ g∈A, (ii) f  g∈A, and (iii) cf ∈A, for all f ∈A, g∈A and for any real constant c. For example, the following set of N-dimensional polynomial basis functions forms an algebra: f1; xðnÞ; xðn1Þ; …; xðnN þ 1Þ; x2 ðnÞ; …; x2 ðnN þ 1Þ; xðnÞxðn1Þ; …;xðnN þ 2ÞxðnN þ 1Þ; …; xðnÞxðnN þ 1Þ; x3 ðnÞ; …g:

ð3Þ

The linear combination of these functions defines the truncated Volterra filters. 3. Fourier nonlinear filters

2. Nonlinear filters and the Stone–Weierstrass theorem In this section, we consider the problem of the approximation with discrete-time filters of the input–output relationship of a time-invariant, finite-memory, continuous nonlinear system. Assuming for simplicity the system to be causal, its input–output relationship can be

A FLANN filter of order P approximates the nonlinear system in (1) with the following input–output relationship: yðnÞ ¼ a0 xðnÞ þ ⋯ þ aN1 xðnN þ 1Þ þc1;0 cos ½πxðnÞ þ ⋯ þ c1;N1 cos ½πxðnN þ 1Þ þs1;0 sin ½πxðnÞ þ ⋯ þ s1;N1 sin ½πxðnN þ 1Þ þ ⋯ þcP;0 cos ½PπxðnÞ þ ⋯ þ cP;N1 cos ½PπxðnN þ 1Þ þsP;0 sin ½PπxðnÞ þ ⋯ þ sP;N1 sin ½PπxðnN þ 1Þ:

3

A short introduction to EMFN filters can also be found in [32].

ð4Þ

A. Carini, G.L. Sicuranza / Signal Processing 94 (2014) 183–194

A FLANN filter can be considered as a hybrid structure that combines a linear filter and a nonlinear part, composed by sine and cosine functions of the input signal. According to (4), FLANN filters approximate the function f in (1) using the following set of basis functions: fxðnÞ; …; xðnN þ 1Þ; cos ½πxðnÞ; …; cos ½πxðnN þ 1Þ; sin ½πxðnÞ; …; sin ½πxðnN þ 1Þ; …; cos ½PπxðnÞ; …; cos ½PπxðnN þ 1Þ; sin ½PπxðnÞ; …; sin ½PπxðnN þ 1Þg: ð5Þ GFLANN filters add to the basis functions of the FLANN filters the cross products xðn1Þ cos ½πxðnÞ; …; xðnN þ 1Þ cos ½πxðnÞ; xðn1Þ sin ½πxðnÞ; …; xðnN þ 1Þ sin ½πxðnÞ; and suitably selected delayed versions, as shown in [30]. However, FLANN and GFLANN filters cannot approximate every nonlinear system in (1) with arbitrary accuracy. In fact, the basis functions of FLANN and GFLANN filters do not constitute an algebra, because they are not closed under multiplication: the product of sine and cosine functions cannot be expressed as a linear combination of their basis functions. Nevertheless, by using the prosthaphaeresis formulas, such products can be conveniently expressed as cosines and sines of sums and differences of the variables, as for example sin ½πxðnÞ  cos ½πxðn1Þ ¼ 12 sin fπ½xðnÞxðn1Þg þ12 sin fπ½xðnÞ þ xðn1Þg: Therefore, it is easy to find a set of trigonometric functions that form an algebra. Such a set is composed by all cosine and sine functions of every order P having the form cos fπ½xðni1 Þ 7 ⋯ 7 xðniP Þg; sin fπ½xðni1 Þ 7 ⋯ 7 xðniP Þg;

ð6Þ

with i1 ¼ 0; …; N1; i2 ¼ 0; …N1; …; iP ¼ 0; …; N1 [31]. For P¼0, the constant function equal to 1 is considered. The set of functions in (6) is redundant because it includes repeated arguments and arguments which may cancel one to each other when the minus sign is used. This problem has been preliminarily considered in [31] and a more complete solution will be presented in this paper. The resulting set of functions on the compact RN 1 is closed under addition, multiplication (for the prosthaphaeresis formulas), and scalar multiplication thus forming an algebra. The algebra vanishes at no point (since the constant function is also considered),4 but does not separates points on the entire compact RN 1 . For example, for any positive integer k it is sin ½kπð1Þ ¼ sin ½kπð þ 1Þ ¼ 0; cos ½kπð1Þ ¼ cos ½kπð þ 1Þ ¼ ð1Þk : Nevertheless, the algebra separates points in all the compacts RN 1ϵ for any arbitrarily small positive ϵ. Indeed, two separate 4

Alternatively, it can be observed that sin ½πxðnÞ and cos ½πxðnÞ cannot be simultaneously zero for any value of x(n) in R1 .

185

points must have at least one different coordinate xðniÞ and are separated by the following function:5 f ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ ¼ xðniÞ ¼

2 1 ð1Þk ∑ sin ½ðk þ 1ÞπxðniÞ: πk¼0kþ1

Thus, the algebra satisfies all the requirements of the Stone– Weierstrass theorem in RN 1ϵ for any ϵ 4 0. Therefore, a linear combination of basis functions from (6) can arbitrarily well approximate any continuous function f ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ from RN 1 to R at all points, apart possibly from the boundaries of RN 1 . Such filters are called here Fourier nonlinear (FN) filters. Their basis functions are specified as follows: The basis function of order 0 is the constant 1. The basis functions of order 1 are cos ½πxðnÞ; cos ½πxðn1Þ; …; cos ½πxðnN þ 1Þ; sin ½πxðnÞ; sin ½πxðn1Þ; …; sin ½πxðnN þ 1Þ: The basis functions of order P are cos fπ½xðni1 Þ 7 ⋯ 7 xðniP Þg; sin fπ½xðni1 Þ 7 ⋯ 7 xðniP Þg;

ð7Þ

with i1 ¼ 0; …; N1; i2 ¼ 0; …N1; …; iP ¼ 0; …; N1, avoiding repetitions and cancellations of arguments. It is worth noting that the linear term is not explicitly given, since it is indirectly obtained from the combination of the other trigonometric terms. A FN filter of uniform order P is defined by the linear combination of the Pth order basis functions in (7), while a FN filter of non-uniform order P is defined by the linear combination of all the terms in (7), with order ranging from 0 to P. It is worth noting that any linear combination of the constant and cosine terms leads to an even nonlinear system, since f ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ ¼ f ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ:

ð8Þ

Any linear combination of the sine terms leads to an odd nonlinear system, since f ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ ¼ f ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ:

ð9Þ

Thus, if we need to model an odd nonlinear system, as it often happens in practice, we can restrict our search to the sine basis functions. A simple method to generate all the nonlinear basis functions of any order P, avoiding repetitions and cancellations between terms, is provided in Appendix A. It is also shown that the number of terms of uniform order P and memory of N samples, is

 P1 P1 Nþi N ðP; NÞ ¼ 2 ∑ : ð10Þ i P i¼0 It is easy to see that the number of terms of uniform order tends to increase exponentially with the order P, well beyond that of a homogeneous Volterra filter with equal 5 The right-hand side of this equation is the Fourier series expansion of xðniÞ, which converges to xðniÞ in the entire interval ½1 þ ϵ; 1ϵ for ϵ 40.

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order P and memory N. This fact constitutes a drawback that is in part balanced by the orthogonality property of the basis functions of FN filters. Indeed, this property is not shared by Volterra filters, unless appropriate orthogonalization procedures are applied at the expense of additional computations [34,1]. Property. The set of basis functions of FN filters is orthogonal in RN 1 for a white uniform input signal. Considering two different basis functions fi and fj, it results Z þ1 Z þ1 ⋯ f i ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ 1

1

f j ½xðnÞ; xðn1Þ; …; xðnN þ 1Þ dxðnÞ⋯dxðnN þ 1Þ ¼ 0:

ð11Þ

This property can be easily proved since the product of two basis functions is a linear combination of basis functions. Each basis function can be expanded in the linear combination of products of factors having the form sin ½απxðnkÞ or cos ½απxðnkÞ, with α integer. Since Z þ1 Z þ1 sin ½απxðnkÞ dxðnkÞ ¼ cos ½απxðnkÞ dxðnkÞ ¼ 0; 1

1

ð12Þ (11) immediately follows. As a direct consequence of this orthogonality property, the following condition holds for a stochastic white uniform input signal: Z þ1 Z þ1 ⋯ f i ½xðnÞ; …; xðnN þ 1Þf j ½xðnÞ; …; xðnN þ 1Þ 1

1

p½xðnÞ; …; xðnN þ 1Þ dxðnÞ⋯dxðnN þ 1Þ ¼ 0;

ð13Þ

where p½xðnÞ; …; xðnN þ 1Þ is the probability density of the N-tuple ½xðnÞ; …; xðnN þ 1Þ, equal to the constant 1=2N for a white uniform distribution in R1 . This orthogonality property allows the derivation of simple identification algorithms using white uniform input signals. Moreover, it guarantees fast convergence of the gradient descent adaptation algorithms applied to FN filters. Another advantage of employing an orthogonal basis is the fact that the inclusion of more terms to a truncated approximation does not, in principle, require a recalculation of the previously estimated terms. According to the orthogonality property, the basis functions form a complete orthogonal system. As a consequence, the FN filters can also be considered as derived from the truncation of the N-dimensional generalized Fourier series [33] originated by this complete orthogonal system. For P-1, the FN filter of order P and memory N that approximates the nonlinear system in (1) converges to this N-dimensional Fourier expansion of f on RN 1. The basis functions of the FN filters are periodic, with period 2 in all dimensions, outside the unit hypercube RN 1 and the N-dimensional Fourier expansion of the function f in (1) converges to its periodic repetition at every point where the function is continuous. If there is a discontinuity in f, the Ndimensional Fourier series converges, as in the 1-dimensional case, to the average of the left hand and right hand limits. Even though f is assumed to be continuous, almost surely it is affected by discontinuities at the borders of the unit

hypercube. As a consequence, the FN filter is not able to approximate f at the unit hypercube borders. Moreover, as it is known from the Fourier theory, the discontinuity points originate high frequency components in the Fourier domain. This fact, transposed in our theory, means that the discontinuities on the borders of the unit hypercube originate relevant oscillations in the Fourier expansion of f. These limitations affect both the FN filters and the FLANN filters currently used in the literature. To overcome these limitations, in the next section we introduce the novel sub-class of EMFN filters, characterized by basis functions with even symmetry. 4. Even mirror Fourier nonlinear filters In this section, a novel sub-class of LIP nonlinear filters, characterized by basis functions with even symmetry, is introduced. These filters, called even mirror Fourier nonlinear (EMFN) filters are universal approximators with the same complexity of Volterra filters, but, in contrast to Volterra filters, satisfy the orthogonality property. Therefore, they can be a valid alternative to Volterra filters. Moreover, EMFN filters allow us to overcome the limitation of the FN filters due to the discontinuities on the borders of the unit hypercube R1 , caused by the periodic repetition of the function f in (1). These discontinuities require high order basis functions to accurately model f. It is well known from the theory of the Discrete Cosine Transform that a simple expedient to avoid this drawback is that of considering a mirror image periodic repetition of f [35]. Accordingly, as shown in Appendix B, the following set of basis functions with memory of N samples can be used to expand f:

 The basis function of order 0 is the constant 1.  Thehbasis functions i hπ of order i 1 arehπ i π sin

2

xðnÞ ; sin

2

xðn1Þ ; …; sin

2

xðnN þ 1Þ : ð14Þ

 The basis functions of order 2 are cos h½πxðnÞ;i cos h½πxðn1Þ;i…; cos h½πxðnN þ 1Þ;i π π π xðnÞ sin xðn1Þ ; … sin xðnN þ 2Þ 2 2 2 hπ i sin xðnN þ 1Þ ; 2 hπ i hπ i hπ i sin xðnÞ sin xðn2Þ ; … sin xðnN þ 3Þ 2 2 2 hπ i hπ i hπ i sin xðnN þ 1Þ ;…; sin xðnÞ sin xðnN þ 1Þ : 2 2 2 ð15Þ

sin

 The basis functions of any order P are similarly defined as the N possible basis functions     Pπ Pπ xðnÞ ; … cos xðnN þ 1Þ for P even; cos 2 2     Pπ Pπ xðnÞ ; … sin xðnN þ 1Þ for P odd; sin 2 2 and every possible product of functions with order lower than P without repetitions, such that the sum of these orders equals P. In Appendix B we provide an algorithm for deriving all the P-th order basis functions avoiding repetitions and ensuring

A. Carini, G.L. Sicuranza / Signal Processing 94 (2014) 183–194

completeness under product. It is easy to prove that these basis functions form an algebra that satisfies the requirements of the Stone–Weierstrass theorem on RN 1 . In contrast to FN filters, this algebra separates points on the entire RN 1 , because two different points must have at least one different coordinate xðniÞ and, for example, the function sin ½ðπ=2ÞxðniÞ separates these points. The linear combinations of the abovegiven basis functions of order P define the filters of uniform order P. As shown in Appendix B, the number of terms in an EMFN filter of uniform order P and memory N is equal to that of the triangular representation of a truncated Volterra filter of the same order and memory
 N þ P1 : ð16Þ P The number of terms in an EMFN filter of non-uniform order P and memory N, including the terms of order 0; …; P, is equal to
 NþP : ð17Þ N It should be noted that all even basis functions are even, all odd basis functions are odd. It is easy to verify that the basis functions are orthogonal in RN 1 , since for every i∈N, it results
 Z þ1 2i þ 1 πx dx ¼ 0 ð18Þ sin 2 1 and, for every i∈Nþ , it results Z þ1 cos ðiπxÞ dx ¼ 0

ð19Þ

1

Therefore, for a white uniform input signal in R1 , the EMFN filters satisfy the orthogonality property (13), and thus offer all the benefits already mentioned for FN filters. Finally, it is worth noting that their basis functions form a complete orthogonal system. Therefore, EMFN filters can be considered as derived from the truncation of the N-dimensional even mirror generalized Fourier series generated by this complete orthogonal system.

5. Representation conversion among Volterra, FN, and EMFN filters Since Volterra, FN and EMFN filters share the property of being universal approximators for time-invariant, finite-memory, continuous, nonlinear systems with input signal xðnÞ∈R1 , it is clear that all these representations are equivalent for order P-1. As a consequence, it is possible to pass from one representation to another. In this section, we provide simple conversion formulas between different representations. While, in principle, it is possible to pass from one infiniteorder representation to another infinite-order representation, it has to be emphasized that, given a FN or an EMFN filter of finite order, there exists no exact corresponding Volterra filter of finite order, and vice versa. Therefore, although the formulas we provide are valid for theoretical purposes or when an analytical expression is known for the nonlinear filter, in many practical situations the proposed formulas have to be truncated.

187

Table 1 Taylor series expansion of sin ðxÞ and cos ðxÞ. x3 x5 x7 þ  þ⋯ sin ðxÞ ¼ x 3! 5! 7! x2 x4 x6 cos ðxÞ ¼ 1 þ  þ⋯ 2! 4! 6!

Table 2 Expansion of x, x3, x5 as a function of sin ðkπxÞ.   2 1 1 1 sin ðπxÞ  sin ð2πxÞ þ sin ð3πxÞ  sin ð4πxÞ þ … π 2 3 4 " 2 22 π 2 6 32 π 2 6 sin ð2πxÞ þ sin ð3πxÞ x3 ¼ 3 ðπ 2  6Þ sin ðπxÞ  π 23 33 # 2 2 4 π 6  sin ð4πxÞ þ ⋯ 43 " 2 24 π 4  20  22 π 2 þ 5! x5 ¼ 5 ðπ 4  20π 2 þ 5!Þ sin ðπxÞ sin ð2πxÞ π 25 x¼

34 π 4  20  32 π 2 þ 5! 44 π 4 20  42 π 2 þ 5! þ sin ð3πxÞ  sin ð4πxÞ þ … 35 45

#

Table 3 Expansion of x2, x4, x6 as a function of cos ðkπxÞ.   1 4 1 1 1  cos ðπxÞ 2 cos ð2πxÞ þ 2 cos ð3πxÞ 2 cos ð4πxÞ þ … 3 π2 2 3 4 " 1 8 22 π 2 6 32 π 2 6 cos ð2πxÞ þ cos ð3πxÞ x4 ¼  4 ðπ 2 6Þ cos ðπxÞ 5 π 24 34 # 2 2 4 π 6  cos ð4πxÞ þ … 44 " 1 12 24 π 4 20  22 π 2 þ 5! x6 ¼  6 ðπ 4 20  π 2 þ 5!Þ cos ðπxÞ cos ð2πxÞþ 7 π 26 # 4 4 2 2 4 4 2 2 3 π 20  3 π þ 5! 4 π 20  4 π þ 5! cos ð3πxÞ cos ð4πxÞ þ … 36 46

x2 ¼

To pass from a FN filter or an EMFN filter to a Volterra filter it suffices to replace each sine and cosine term with its Taylor series expansion, according to the formulas in Table 1. To pass from a Volterra series representation to a FN or an EMFN representation we can exploit the orthogonality of the basis functions fi and compute the coefficients as R þ1 R þ1 1 ⋯ 1 f i ½xðnÞ; …; xðnN þ 1Þf ½xðnÞ; …; xðnN þ 1Þ  dxðnÞ…dxðnN þ 1Þ ci ¼ R þ1 R þ1 2 1 ⋯ 1 f i ½xðnÞ; …; xðnN þ 1Þ dxðnÞ…dxðnN þ 1Þ ð20Þ More conveniently, to pass from Volterra filters to FN filters, we can exploit the expansion formulas of Tables 2 and 3. To pass from Volterra filters to EMFN filters, we can exploit the expansion formulas of Tables 3 and 4. 6. Simulation results In this section we provide some simulation results to show capabilities and limitations of FN and EMFN filters in comparison to Volterra and FLANN filters. When dealing with

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Table 4 Expansion of x, x3, x5 as a function of sin ðð2k þ 1Þ=2πxÞ. 



  8 1 1 3 1 5 1 7 πx  2 sin πx þ 2 sin πx  2 sin πx þ … sin 2 2 2 2 π2 5 7 3 "


 24 1 32 π 2  8 3 52 π 2  8 5 πx  πx þ πx x3 ¼ 4 ðπ 2  8Þ sin sin sin 4 4 2 2 2 π 5 3 #
 72 π 2  8 7  πx þ … sin 2 74 "

 40 1 34 π 4  48  32 π 2 þ 24  4! 3 x5 ¼ 6 ðπ 4  48π 2 þ 24  4!Þ sin πx  πx þ sin 6 2 2 π 3 #

 54 π 4  48  52 π 2 þ 24  4! 5 74 π 4  48  72 π 2 þ 24  4! 7 πx  πx þ … sin sin 2 2 56 76

x¼

Table 5 MSEs for Volterra, FLANN, FN and EMFN filters with 1–5 coefficients. No. coeff. Volterra FLANN FN EMFN

1

2 2

0.48  10 0.48  10  2 0.22  10  1 0.11  10  2

3 3

0.35  10 0.29  10  2 0.15  10  1 0.21  10  4

ð21Þ

with K¼0.435. This system is modeled using Volterra, FLANN, FN, and EMFN filters of different orders. Since the system has no memory, none of the nonlinear filters has cross terms. To identify (21), a dense grid of points uniformly distributed in the interval ½1; þ1 is used and the filter that minimizes the mean-square error (MSE) is obtained by solving an overdetermined system of linear equations. Since the sigmoid is an odd function, we consider only the odd terms in the four filters (i.e., the odd order terms in Volterra and EMFN filters, and the sin terms in FLANN and FN filters). Tables 5 and 6 provide the MSEs and the condition numbers6 of the system of linear equations, respectively, for the four filters. The 6 Given an overdetermined system, Ax ¼ y, the condition number is defined as the ratio of the largest to smallest eigenvalue of ðAT AÞ.

5 5

0.27  10 0.23  10  2 0.11  10  1 0.18  10  6

different nonlinear filter structures, we must take into account that each structure is better fitted to given nonlinear conditions, and may work worse in other conditions. When modeling a nonlinear system with small or mild nonlinearities, a Volterra or FLANN filter can be a better model than FN and EMFN filters, because the two latter do not use explicitly the linear term. It is worth noting that a linear term can be added to FN and EMFN filters, too, thus obtaining a hybrid structure similar to FLANN filters. The hybrid structure may offer good performance in some cases, even though the orthogonality of the basis functions is lost. In contrast, in the presence of a strong nonlinearity as, for example, a saturation effect, EMFN filters can offer better performance than Volterra and FLANN filters. In the following experiments, we refer to the typical problem of modeling a saturation nonlinearity with or without memory. In the first set of simulations, we consider a system without memory, i.e., with N ¼1, specified by a sigmoidal input–output relationship yðnÞ ¼ K  tanh½xðnÞ=K;

4 4

0.17  10  6 0.16  10  2 0.68  10  2 0.86  10  7

0.21  10 0.19  10  2 0.83  10  2 0.17  10  6

Table 6 Condition number for Volterra, FLANN, FN and EMFN filters with 1–5 coefficients. No. coeff.

1

2

3

4

5

Volterra FLANN FN EMFN

1.0 1.0 1.0 1.0

2.7  101 2.5 1.0 1.0

8.3  102 3.0 1.0 1.0

2.6  104 3.3 1.0 1.0

8.3  105 3.4 1.0 1.0

number of coefficients ranges from 1–5, which means that Volterra and EMFN filters assume orders 1, 3, 5, 7, 9. FN and FLANN filters assume orders 1–5 and 0–4, respectively. From Table 5, we see that FN and FLANN filters do not provide good results for modeling the sigmoid in (21), because the filter output is affected by the discontinuity at x ¼ 7 1. In contrast, EMFN filters provide very good modeling, with performance better than Volterra filters. This fact is also evidentiated by the condition number in Table 6. While for Volterra filters the condition number increases exponentially with the filter order, for FN and EMFN filters there is a perfect conditioning because of the orthogonality of the basis functions. In FLANN filters, the linear term leads to an imperfect, even though sufficiently good, conditioning. Fig. 1 shows the sigmoid and its approximations obtained using Volterra, FLANN, FN, and EMFN filters, all with five coefficients. While the Volterra and the EMFN filters practically overlap with the sigmoid, the errors originated by the discontinuity at 7 1 in FLANN and FN filters are particularly evident. The aim of the second set of simulations is to point out the advantages, due to the orthogonality property in (13), offered by FN and EMFN filters in comparison to Volterra and FLANN filters. To this purpose, we consider a saturated dynamic system represented by a Wiener model obtained by cascading the linear filter with impulse response ½0:25; 0:5; 2:0; 0:5; 0:25 with the nonlinearity in (21). This system is modeled with Volterra, FLANN, FN

A. Carini, G.L. Sicuranza / Signal Processing 94 (2014) 183–194

0.4

0.4

0.2

0.2

0

0

y

0.6

y

0.6

0.2

0.2 0.4

0.4

Sigmoid Volterra FLANN

0.6 0.8

189

1

0.5

0

0.5

Sigmoid EMFN FN

0.6 1

0.8

1

0.5

0

x

0.5

1

x

Fig. 1. A sigmoid and its approximation with (a) Volterra and FLANN filters, (b) EMFN and FN filters with five coefficients.

Table 7 Order of the filter, number of coefficients, value of the step size, and steady-state MSE for the filters of the second example.

Volterra FLANN FN EMFN Linear+FN Linear+EMFN

Order 3 7 3 3 3 3

# Coeff. 40 40 115 40 120 45

Step size 4

2.0  10 1.3  10  4 2.0  10  4 2.0  10  4 1.2  10  4 1.2  10  4

MSE 6.4  10  3 1.3  10  2 1.7  10  2 4.0  10  3 5.0  10  3 4.0  10  3

MSE

Filter

101

(a) (b) (c) (d) (e) (f)

102

0

0.5

1

1.5

Samples and EMFN filters. For sake of completeness, even hybrid filters obtained by connecting in parallel a linear and a FN or EMFN filter are considered. All filters have memory N ¼5 samples. The standard LMS algorithm with a white uniform input signal in ½1; þ1 has been used for the identification. Since the nonlinear system is odd, we consider only the odd terms of Volterra and EMFN filters, and the sine terms of FN and FLANN filters. The order of the different filters and the corresponding number of coefficients are shown in Table 7. The order of all filters has been set to 3, apart from the FLANN filter for which an order 7 has been chosen to have the same number of coefficients of Volterra and EMFN filters. Since different nonlinear structures converge to different MSEs, we compare the performance of the six filters by tuning the step size in order to have the same initial convergence speed for all filters. As a matter of fact, Fig. 2 diagrams the ensemble averages, over 500 independent realizations, of the learning curves for the six filters in the first 2  104 time instants. As shown in Table 7, the step sizes used for each filter have all the same order of magnitude, since they range between 1  104 and 2  104 . Fig. 3 shows the ensemble averages, over 500 independent realizations, of the learning curves for the six filters on a length of 106 time instants. The mean values of the MSEs on the last 100,000 samples of 2  106 sample simulations are reported in the last column of Table 7. As a consequence of the orthogonality property in (13), the input data autocorrelation matrices of the FN and EMFN filters are

2

2.5 x 104

Fig. 2. Learning curves of (a) Volterra, (b) FN, (c) FLANN, (d) hybrid linear-FN, (e) hybrid linear-EMFN and (f) EMFN filter in the first 20,000 samples.

diagonal. This fact explains the very fast convergence of FN and EMFN filters in comparison to Volterra filters. With FLANN filters and the hybrid (linear-FN and linear-EMFN) structures, the orthogonality of the autocorrelation matrix is lost. However, since there is still a good conditioning, the convergence speed is only slightly worse than for FN and EMFN filters. As far as the steady-state is concerned, in this example the FN filter provides the worst performance with a large MSE, because of the discontinuities at the unit hypercube borders. The same discontinuities affect the FLANN filter, as it was shown in Fig. 1. Thanks to the linear term, the FLANN filter provides a slightly better performance than the FN filter, but still worse than the Volterra filter. By adding a linear term to the FN filter, we can improve the steady-state modeling performance and thus obtain results better than those of Volterra filters, even tough the improvement is achieved at the cost of a larger number of coefficients. The best performance in terms of both convergence speed and steady-state modeling capabilities is here offered by the EMFN filter. Adding a linear term to the EMFN structure does not give, in the given experimental conditions, any significant improvement in terms of the average accuracy. In conclusion, these experimental results show that EMFN filters are excellent candidates for dealing with strong saturation nonlinearities.

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10-1

MSE

MSE

10-1

10-2

10-3

0

2

4

6

8

Samples

10-3

10 x 105

MSE

MSE 0

2

4

6

8

Samples

4

6

8

10 x 105

8

10 x 105

8

10 x 105

10-2

10-3

10 x 105

0

2

4

6

Samples

10-1

10-1

MSE

MSE

2

10-1

10-2

10-2

10-3

0

Samples

10-1

10-3

10-2

0

2

4

6

Samples

8

10 x 105

10-2

10-3

0

2

4

6

Samples

Fig. 3. Learning curves of (a) Volterra, (b) FLANN, (c) FN, (d) EMFN, (e) hybrid linear-FN and (f) hybrid linear-EMFN filter.

7. Conclusions In this paper, two novel sub-classes of LIP nonlinear filters that can arbitrarily well approximate a causal, timeinvariant, finite-memory, continuous nonlinear system are presented. The methodology we use to reach this goal is based on the well-known Stone–Weierstrass approximation theorem. To specifically define the elements of the filters called here Fourier nonlinear filters and even mirror Fourier nonlinear filters, we resort to multidimensional generalized Fourier series expansions. The properties of these filters are analyzed and discussed. It is shown that

FN filters can be affected by discontinuities at the unit hypercube borders. In contrast, EMFN filters are not affected by these discontinuities and thus should be preferred for modeling nonlinear systems. The significance for the identification of nonlinear systems of the orthogonality property, in the presence of white uniform input signals, is pointed out. The reported simulation results show that EMFN filters offer better performance than Volterra filters when modeling systems affected by strong nonlinearities. Presently, work is in progress to derive efficient identification methods for modeling real-world nonlinear systems using EMFN filters.

A. Carini, G.L. Sicuranza / Signal Processing 94 (2014) 183–194

Appendix A. Basis functions of FN filters In this appendix we show how it is possible to generate the basis functions of FN filters, avoiding repetitions and cancellation of the arguments. Moreover, we compute their number, N ðN; PÞ, for a filter of uniform order P and memory N. As already mentioned, the basis function of order 0 is the constant 1. The 2N basis functions of order 1 are cos ½πxðnÞ; cos ½πxðn1Þ; …; cos ½πxðnN þ 1Þ; sin ½πxðnÞ; sin ½πxðn1Þ; …; sin ½πxðnN þ 1Þ: To generate all the other basis functions a recursive rule is applied. The basis functions of order P and memory N, can be found by applying the sin or cos operator to the summations of terms xðn þ i1 Þ 7xðn þ i2 Þ 7…xðn þ iP Þ;

ðA:1Þ

with i1 ¼ 0; …; N1, i2 ¼ i1 ; …; N1, …, and iP ¼ iP1 ; …; N1, to avoid repetitions. To avoid cancellations in (A.1), whenever ijþ1 ¼ ij the term 7 xðnijþ1 Þ must repeat the sign of xðnij Þ. It should be noted that the first term has always a positive sign. Thus, we can form all order P, memory N, summations by:

 Adding to x(n) all summations of order P1, memory N, 

 

starting at time n, and subtracting to x(n) all summations of order P1, memory N1, starting at time n1. Adding to xðn1Þ all summations of order P1, memory N1, starting at time n1, and subtracting to xðn1Þ all summations of order P1, memory N2, starting at time n2. … Adding to xðnN þ 1Þ the only addition of order P1, memory 1, starting at time nN þ 1, which is ðP1ÞxðnN þ 1Þ. Thus, by applying this rule we obtain

N ðP; NÞ ¼ N ðP1; NÞ þ N ðP1; N1Þ þ N ðP1; N1Þ þN ðP1; N2Þ þ ⋯ þ N ðP1; 1Þ N

N1

m¼1

m¼1

¼ ∑ N ðP1; mÞ þ ∑ N ðP1; mÞ

ðA:2Þ

Since N ð1; NÞ ¼ 2N, by recursively applying (A.2), we can compute N ðP; NÞ. In what follows we prove by induction that:

 P1 P1 Nþi N ðP; NÞ ¼ 2 ∑ : ðA:3Þ i P i¼0 For P ¼1, it results

 0 N N ðP; NÞ ¼ 2  ¼ 2N; 0 1

ðA:4Þ

N ðP1; NÞ ¼ 2 ∑

From (A.2), we have

 N P2 P2 mþi N ðP; NÞ ¼ 2 ∑ ∑ i P1 m¼1i¼0

 N1 P2 P2 mþi þ2 ∑ ∑ i P1 m¼1i¼0
 N

 P2 P2 P2 mþi P2 ∑ þ2 ∑ ¼2 ∑ i i P1 m¼1 i¼0 i¼0
 N1 mþi : ðA:6Þ ∑ P1 m¼1 Since it is [36] ! M mþj ¼ ∑ j m¼0


mþi P1




¼

P2

i¼0

and then prove (A.3).

i




Nþi P1

jþ1

! ;

ðm þ iP þ 1Þ þ P1 P1

ðA:7Þ  ;

ðA:8Þ

and ðaþP1 P1 ¼ 0Þ for a o0, we have

 P2 P2 Nþiþ1 N ðP; NÞ ¼ 2 ∑ i P i¼0

 P2 P2 Nþi þ2 ∑ i P i¼0



 P2 P2 Nþi N þ P1 N þ2 ¼2 ∑ þ2 i1 P P P i¼1

 P2 P2 Nþi þ2 ∑ i P i¼1 



 P2 P2 P2 N Nþi þ ¼2 þ2 ∑ i i1 P P i¼1
 N þ P1 þ2 P



 P2 P1 Nþi N þ P1 N þ2 ¼2 þ2 ∑ i P P P i¼1

 P1 P1 Nþi : ðA:9Þ ¼2 ∑ i P i¼0 In Table A1, the pseudocode for generating the basis functions up to order 3 of a FN filter with N sample memory is given. In the table the basis functions are indicated with gðℓ; nÞ, where ℓ is an index varying from 1 till the number of basis functions, and n is a time index. Appendix B. Basis functions of EMFN filters We first consider the 1-dimensional case and a continuous function f(x) from the unit interval ½1; 1 to R. We extend f(x) on the entire real axis R by considering its periodic even mirror repetition, such that f ð1 þ xÞ ¼ f ð1xÞ

f ðx þ 4Þ ¼ f ðxÞ:


Mþjþ1

ðB:1Þ

and

which is true. Let us assume P2

191

 ;

ðA:5Þ

ðB:2Þ

Since f(x) is periodic of period 4, we can consider its Fourier series expansion using the basis functions 
 π  π  3π 1; cos x ; sin x ; cos ðπxÞ; sin ðπxÞ; cos x ; 2 2 2

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Table A1 Pseudocode for generating the basis functions up to order 3 of a FN filter with N sample memory. %% Basis function of order 0: ℓ¼1 gðℓ; nÞ ¼ 1 %% Basis function of order 1: for i¼ 0:N-1 ℓ¼ℓþ1 gðℓ; nÞ ¼ cos ½πxðniÞ ℓ¼ℓþ1 gðℓ; nÞ ¼ sin ½πxðniÞ end %% Basis function of order 2: for i¼ 0:N-1 for j¼ i:N-1 ℓ¼ℓþ1 gðℓ; nÞ ¼ cos fπ½xðniÞ þ xðnjÞg ℓ¼ℓþ1 gðℓ; nÞ ¼ sin fπ½xðniÞ þ xðnjÞg if j4 i ℓ¼ℓþ1 gðℓ; nÞ ¼ cos fπ½xðniÞxðnjÞg ℓ¼ℓþ1 gðℓ; nÞ ¼ sin fπ½xðniÞxðnjÞg end end end %% Basis function of order 3: for i ¼ 0 : N1 for j ¼ i : N1 for k ¼ j : N1 ℓ¼ℓþ1 gðℓ; nÞ ¼ cos fπ½xðniÞ þ xðnjÞ þ xðnkÞg ℓ¼ℓþ1 gðℓ; nÞ ¼ sin fπ½xðniÞ þ xðnjÞ þ xðnkÞg if k 4 j ℓ¼ℓþ1 gðℓ; nÞ ¼ cos fπ½xðniÞ þ xðnjÞxðnkÞg ℓ¼ℓþ1 ðℓ; nÞ ¼ sin fπ½xðniÞ þ xðnjÞxðnkÞg end if j4 i ℓ¼ℓþ1 gðℓ; nÞ ¼ cos fπ½xðniÞxðnjÞxðnkÞg ℓ¼ℓþ1 gðℓ; nÞ ¼ sin fπ½xðniÞxðnjÞxðnkÞg end if k 4 j4 i ℓ¼ℓþ1 gðℓ; nÞ ¼ cos fπ½xðniÞxðnjÞ þ xðnkÞg ℓ¼ℓþ1 gðℓ; nÞ ¼ sin fπ½xðniÞxðnjÞ þ xðnkÞg end end end end

sin




 3π 5π 5π x ; cos ð2πxÞ; sin ð2πxÞ; cos x ; sin x ;… : 2 2 2 ðB:3Þ

The basis functions 

 π  3π 5π cos x ; sin ðπxÞ; cos x ; sin ð2πxÞ; cos x ;… 2 2 2 are not even mirror, i.e., they do not satisfy the condition in (B.1), and consequently they do not contribute to the expansion of the even mirror periodic function f(x). It can

be easily verified that the remaining basis function 
 π  3π 1; sin x ; cos ðπxÞ; sin x ; cos ð2πxÞ; 2 2
 5π sin x ;… ; 2

ðB:4Þ

satisfy all the requirements of the Stone–Weierstrass theorem and can be used to arbitrarily well approximate any 1-dimensional continuous function f(x) from the unit interval ½1; 1 to R. We define the order of the basis functions as follows: 1 is a basis function of order 0, sin ½ðπ=2xÞ is a basis function of order 1, cos ðπxÞ is a basis function of order 2, sin ½ð2k þ 1Þðπ=2xÞ is a basis function of order 2k þ 1, cos ðkπxÞ is a basis function of order 2k. To develop the even mirror nonlinear basis functions for the N-dimensional case, i.e., for a continuous function f ½xðnÞ; xðn1Þ; …; ðnN þ 1Þ with memory of N samples from RN 1 to R, we first consider the 1-dimensional basis functions in (B.4) for x ¼ xðnÞ; xðn1Þ; …; xðnN þ 1Þ   hπ i 3π xðnÞ ; … 1; sin xðnÞ ; cos ½πxðnÞ; sin 2 2   hπ i 3π 1; sin xðn1Þ ; cos ½πxðn1Þ; sin xðn1Þ ; … 2 2 ⋮ hπ i 1; sin xðnN þ 1Þ ; cos ½πxðnN þ 1Þ; 2   3π xðnN þ 1Þ ; … sin 2 Then, following the requirements of the Stone–Weierstrass theorem, we multiply the terms having different variables in any possible manner to guarantee completeness of the algebra under multiplication. We define the order of an Ndimensional basis function as the sum of the orders of the constituent 1-dimensional basis functions. For example, cos ½2πxðnÞ  cos ½πxðn1Þ  sin ½ðπ=2Þxðn2Þ has order 4 þ 2þ 1 ¼ 7. Avoiding repetitions, we thus obtain the following basis functions: The basis function of order 0 is the constant 1. The basis functions of order 1 are the N possible 1dimensional basis functions of the same order hπ i hπ i hπ i sin xðnÞ ; sin xðn1Þ ; …; sin xðnN þ 1Þ : 2 2 2 The basis functions of order 2 are the N possible 1-dimensional basis functions of the same order and the basis functions originated by the product of two 1-dimensional basis functions of order 1. Avoiding repetitions, we have the basis functions cos ½πxðnÞ; cos ½πxðn1Þ; … cos ½πxðnN þ 1Þ; hπ i hπ i hπ i sin xðnÞ  sin xðn1Þ ; … sin xðnN þ 2Þ  2 h2 i2 π sin xðnN þ 1Þ ; h2 i hπ i hπ i π sin xðnÞ  sin xðn2Þ ; … sin xðnN þ 3Þ  2 2 2 hπ i sin xðnN þ 1Þ ; 2 ⋮ hπ i hπ i sin xðnÞ  sin xðnN þ 1Þ : 2 2

A. Carini, G.L. Sicuranza / Signal Processing 94 (2014) 183–194

Table B1 Pseudocode for generating the basis functions up to order 3 of an EMFN filter with N sample memory. %% Basis function of order 0: ℓ¼1 gðℓ; nÞ ¼ 1 %% Basis function of order 1: for i ¼ 0 : N1 ℓ¼ℓþ1 hπ i gðℓ; nÞ ¼ sin xðniÞ 2 end %% Basis function of order 2: for i ¼ 0 : N1 for j ¼ i : N1 ℓ¼ℓþ1 if i ¼ ¼ j gðℓ; nÞ ¼ cos ½πxðniÞ else hπ i π xðniÞ sin ½ xðnjÞ gðℓ; nÞ ¼ sin 2 2 end end end %% Basis function of order 3: for i ¼ 0 : N1 for j ¼ i : N1 for k ¼ j : N1 ℓ¼ℓþ1 if i ¼ ¼ j ¼ ¼ k   3π gðℓ; nÞ ¼ sin xðniÞ 2 elseif i ¼ ¼ j hπ i gðℓ; nÞ ¼ cos ½πxðniÞ sin xðnkÞ 2 elseif j ¼ ¼ k hπ i gðℓ; nÞ ¼ sin xðniÞ cos ½πxðnjÞ 2 else hπ i hπ i hπ i gðℓ; nÞ ¼ sin xðniÞ sin xðnjÞ sin xðnkÞ 2 2 2 end end end end

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