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# A reduced-rank approach for implementing higher-order Volterra filters

- Eduardo L. O. Batista
^{1}Email authorView ORCID ID profile and - Rui Seara
^{1}

**2016**:118

https://doi.org/10.1186/s13634-016-0420-5

© The Author(s) 2016

**Received:**28 June 2016**Accepted:**2 November 2016**Published:**11 November 2016

## Abstract

The use of Volterra filters in practical applications is often limited by their high computational burden. To cope with this problem, many strategies for implementing Volterra filters with reduced complexity have been proposed in the open literature. Some of these strategies are based on reduced-rank approaches obtained by defining a matrix of filter coefficients and applying the singular value decomposition to such a matrix. Then, discarding the smaller singular values, effective reduced-complexity Volterra implementations can be obtained. The application of this type of approach to higher-order Volterra filters (considering orders greater than 2) is however not straightforward, which is especially due to some difficulties encountered in the definition of higher-order coefficient matrices. In this context, the present paper is devoted to the development of a novel reduced-rank approach for implementing higher-order Volterra filters. Such an approach is based on a new form of Volterra kernel implementation that allows decomposing higher-order kernels into structures composed only of second-order kernels. Then, applying the singular value decomposition to the coefficient matrices of these second-order kernels, effective implementations for higher-order Volterra filters can be obtained. Simulation results are presented aiming to assess the effectiveness of the proposed approach.

## Keywords

- Nonlinear filtering
- Reduced-rank implementation
- Volterra filter

## 1 Introduction

The first challenge in filtering applications involving nonlinear systems is to choose an adequate model of the nonlinear filter [1]. To meet this challenge, one important filter characteristic that needs to be considered is the trade-off between implementation complexity and approximation capability. The well-known Volterra filter [1] represents one extreme of this trade-off, since its universal approximation capability [2–4] comes at the cost of a high computational complexity (which is due to the large number of coefficients required for the implementation) [1, 5–9]. In this context, one topic that has drawn attention from researchers in the last decades is the development of Volterra implementations having an enhanced trade-off between computational complexity and approximation capability.

Several different approaches have been proposed in the open literature aiming to obtain reduced-complexity implementations of Volterra filters. Some of these approaches are based on sparse Volterra implementations that are obtained by zeroing the less significant filter coefficients [9]. Examples of these implementations are the Volterra delay filter [10], the power filter [11], and general diagonally-pruned implementations [12–14]. Other approaches combine interpolation and sparse implementations for the sake of performance [15–17]. Frequency-domain approaches also have been used for obtaining effective reduced-complexity Volterra implementations [18, 19]. All aforementioned approaches are, in some sense, based on the use of predefined forms of basis vectors for identifying and discarding the less significant coefficients of a Volterra filter. In contrast, the approaches from [20–25] involve the identification of particular basis vectors that can then be exploited aiming to reduce the complexity of a Volterra filter. These approaches are typically based on the definition of coefficient matrices, which are decomposed aiming to obtain the basis vectors. The singular value decomposition is often used for carrying out such a matrix decomposition and, as a result, the obtained basis vectors are singular vectors of the coefficient matrix considered. Thus, discarding the singular vectors related to the smaller singular values, effective reduced-complexity reduced-rank Volterra implementations are obtained.

The first reduced-rank approaches used for implementing Volterra filters are focused on second-order Volterra kernels [20–22]. This is due to the fact that the second-order Volterra coefficients have two indices, which makes the definition of a second-order coefficient matrix a straightforward task. For higher-order filters, matrix-based reduced-rank approaches are usually obtained by considering non-trivial definitions of (often rectangular) coefficient matrices [24], which occasionally lead to ineffective reduced-rank implementations. In this context, the present paper is focused on the development of a novel reduced-rank approach for implementing higher-order Volterra filters. This approach is based on a new form of Volterra kernel implementation that allows converting a higher-order Volterra kernel into a structure composed of second-order kernels. Then, applying second-order reduced-rank implementation strategies to such a structure, effective reduced-rank implementations for higher-order Volterra filters can be achieved.

The remainder of this paper is organized as described in the following. Section 2 revisits the Volterra filters, discussing the redundancy-removed and matrix-form representations of kernel input-output relationships. Also, in Section 2, the reduced-rank implementations of Volterra filters are briefly described. Section 3 is dedicated to the contributions of this paper, comprising a new form of kernel implementation and the description of the proposed approach for implementing reduced-rank Volterra filters. Finally, Sections 4 and 5 are dedicated to present the experimental results and the concluding remarks, respectively.

The mathematical notation considered in this paper is based on the standard practice of using lowercase boldface letters for vectors, uppercase boldface letters for matrices, and both italic Roman and Greek letters for scalar quantities. Moreover, superscript T stands for transpose, ⊗ represents the Kronecker product, and ||·||_{2} denotes a quadratic norm. Additionally, underbars specify variables related to the redundancy-removed Volterra representation and overbars indicate variables related to the proposed approach.

## 2 Volterra filters and reduced-rank implementations

*P*th-order Volterra filter is composed of

*P*kernels, each corresponding to a certain order of polynomial nonlinearity [1]. The output

*y*(

*n*) of such a filter is obtained from

*y*

_{ p }(

*n*) representing the output of the

*p*th-order kernel. In its standard form, the input-output relationship of the

*p*th-order kernel is given by

with *x*(*n*) denoting the input signal and \(h_{p(m_{1},m_{2},\dots,m_{p})}\), the *p*th-order coefficients. In this work, the standard kernels are assumed to be symmetric, which means that \(h_{p(m_{1},m_{2},\dots,m_{p})} = h_{p(m_{2},m_{1},\dots,m_{p})} = \cdots = h_{p(m_{p},m_{p-1},\dots,m_{1})}\) (i.e., all *p*th-order coefficients with permutated indices have identical values). This assumption is used without loss of generality, since any standard Volterra kernel can be represented in symmetric form [1, 25].

*N*coefficients. Thus, (3) can be rewritten in a vector form as

*p*≥2) are nonlinear kernels whose outputs depend on cross products of samples of the input signal. As described in [1], the input-output relationships of the nonlinear kernels can also be expressed in vector form. Thus, for the

*p*th-order kernel, one has

where **h**
_{
p
} is the *p*th-order coefficient vector (composed of coefficients \(h_{p(m_{1},m_{2},\dots,m_{p})}\) with *m*
_{1},…,*m*
_{
p
} ranging from 0 to *N*−1) and **x**
_{
p
}(*n*)=**x**
_{1}(*n*)⊗**x**
_{
p−1}(*n*) is the *p*th-order input vector.

*p*th-order Volterra kernel has one coefficient for each

*p*th-order cross product of input samples, resulting in a number of coefficients given by

This number increases exponentially with both the memory size and the order of the kernel. As a consequence, the computational cost for implementing a Volterra filter may become prohibitive even in applications involving kernels with relatively small memory size.

### 2.1 Redundancy-removed implementation

*h*

_{2(0,1)}and

*h*

_{2(1,0)}) are multiplied by the same cross product of the input signal (e.g.,

*x*(

*n*)

*x*(

*n*−1) in the case of

*h*

_{2(0,1)}and

*h*

_{2(1,0)}) when the kernel output is evaluated. Thus, merging redundant coefficients into a single coefficient, the input-output relationship of the

*p*th-order kernel, given by (2), can be rewritten as

*p*th-order coefficients of the redundancy-removed kernel. Such representation of the kernel input-output relationship, known as triangular [1] or even redundancy-removed [15] representation, results in a number of coefficients given by

Moreover, it is important to highlight that the reduction in the number of coefficients from (8) to (10) obtained by using the redundancy-removed implementation comes without loss of generality (i.e., a given kernel can be equivalently implemented by using either the standard or the redundancy-removed implementation).

*p*th-order redundancy-removed coefficient vector, which is composed of coefficients \(\underline {h}_{p(m_{1},m_{2},\dots,m_{p})}\), and \(\underline {\mathbf {x}}_{p}(n) = \mathbf {L}_{p} [\mathbf {x}_{p-1}(n) \otimes \mathbf {x}_{1}(n)] \) is the

*p*th-order redundancy-removed input vector with

**L**

_{ p }denoting the

*p*th-order elimination matrix [9]. For instance, considering the second-order kernel, (9) results in

### 2.2 Matrix-form kernel representation

*m*

_{1}and

*m*

_{2}as coordinates of a Cartesian space to define the following second-order coefficient matrix:

*p*th-order coefficient matrix \(\hat {\mathbf {H}}_{p,p_{1},p_{2}}\) with

*p*=

*p*

_{1}+

*p*

_{2}. This matrix contains the \(\underline {N}_{p}\) coefficients of the

*p*th-order redundancy-removed kernel arranged in such a way that the kernel output can be written as

where \(\underline {\mathbf {x}}_{p_{1}}(n)\) and \(\underline {\mathbf {x}}_{p_{2}}(n)\) are the redundancy-removed input vectors with orders *p*
_{1} and *p*
_{2}, respectively. It is important to mention that the number of coefficients \(\underline {N}_{p}\) of the *p*th-order kernel is smaller than the number of entries (\(\underline {N}_{p_{1}} \times \underline {N}_{p_{2}}\)) of \(\hat {\mathbf {H}}_{p,p_{1},p_{2}}\). As a result, several elements of \(\hat {\mathbf {H}}_{p,p_{1},p_{2}}\) are in fact equal to zero [24].

### 2.3 Reduced-rank implementations

**H**

_{2}given by (16). Since this matrix is symmetric, its left singular vectors are equal to its right singular vectors and, as a result, one obtains

*λ*

_{ k }and \(\tilde {\mathbf {h}}_{2,k}\) are, respectively, the

*k*th singular value and the

*k*th singular vector of

**H**

_{2}. Now, substituting (21) into (17), one gets

*N*FIR filters with their squared outputs multiplied by the singular values of

**H**

_{2}. Moreover, since the singular vectors \(\tilde {\mathbf {h}}_{2,k}\) (with

*k*=0,…,

*N*−1) are unit vectors, the branches of the structure from Fig. 1 involving small values of

*λ*

_{ k }can be disregarded, resulting in a reduction of computational complexity with low impact on the implementation precision.

with \(\tilde {\mathbf {h}}_{2\mathrm {l},k}\) representing the *k*th left singular vector of **H**
_{2} and \(\tilde {\mathbf {h}}_{2\mathrm {r},k}\) denoting the *k*th right singular vector. By comparing (22) and (23), one can notice that the latter (which is based on the redundancy-removed implementation) results in a structure with higher computational cost than the structure resulting from the former (based on the standard representation). Thus, one verifies that the standard representation is in fact more advantageous than the less costly redundancy-removed one for obtaining this type of reduced-rank implementations of second-order kernels.

*p*≥3), one appealing approach for obtaining reduced-rank implementations is the one used in [24] to obtain the so-called parallel-cascade Volterra structures. Such an approach is based on the application of the singular value decomposition to the coefficient matrix of the general matrix-form kernel representation given by (20). For instance, considering the case of a third-order kernel, (20) becomes

*σ*

_{ k }is the

*k*th singular value of \(\hat {\mathbf {H}}_{3,1,2}, \hat {\mathbf {h}}_{3,1,k}\) is the

*k*th left singular vector, and \(\hat {\mathbf {h}}_{3,2,k}\) is the

*k*th right singular vector. Then, substituting (25) into (24) and manipulating the resulting expression, one gets

*N*branches, with the output of the

*k*th branch given by the product of the outputs of two kernels (a first-order kernel and a second-order one), weighted by the

*k*th singular value

*σ*

_{ k }of \(\hat {\mathbf {H}}_{3,1,2}\). As in the case of the structure of Fig. 1, reduced-complexity reduced-rank Volterra implementations can be obtained from the parallel-cascade structure of Fig. 2, removing the branches related to the smallest singular values of \(\hat {\mathbf {H}}_{3,1,2}\). In addition, as mentioned in [24], the second-order blocks of the structure of Fig. 2 can be further decomposed by using reduced-rank approaches, which allows a more detailed singular-value-dependent kernel pruning. However, this pruning is not a straightforward task due to the hierarchical nature of the resulting structure (i.e., it involves reduced-rank decompositions with different levels of importance).

## 3 Novel reduced-rank approach for implementing Volterra filters

This section is devoted to the development of a novel reduced-rank approach for implementing Volterra filters. To this end, a new strategy for implementing higher-order kernels using a parallel structure composed of lower-order kernels is first discussed. Then, such a strategy is exploited along with the singular value decomposition to obtain the proposed reduced-rank implementation approach.

### 3.1 Kernel implementation redesigned

*N*−

*m*

_{1}, coefficient vector

Note that, in (30), the output of the second-order kernel is evaluated by summing the outputs of *N* first-order kernels multiplied by delayed samples of the input signal. Therefore, (30) can be seen as a decomposition of the second-order kernel into a parallel structure composed of first-order kernels.

which corresponds to the decomposition of the third-order kernel into a structure composed of second-order kernels.

*p*th-order kernel, the following input-output relationship can be obtained:

*p*−1)th-order kernel with memory size

*N*−

*m*

_{1}. Thus, from (35), we can infer that any

*p*th-order kernel can be decomposed into

*N*kernels with order

*p*−1, as illustrated in Fig. 3.

### 3.2 Proposed approach

Aiming to develop the proposed reduced-rank approach for implementing Volterra filters, we first consider that the kernel implementation strategy introduced in Section 3.1 can be used to decompose any higher-order kernel (with *p*≥3) into a parallel structure composed exclusively of second-order kernels. This decomposition is straightforward for the third-order kernel, since, by using *p*=3 in (35), one obtains a structure composed of *N* second-order kernels, as described by (31)–(34). In the case of the fourth-order kernel, (35) can be used first for obtaining a structure composed of *N* third-order kernels and then for decomposing each of these third-order kernels into second-order kernels. As a result, a structure composed of *N*
^{2} second-order kernels is obtained for the implementation of the fourth-order kernel. Following this rationale, one can notice that, by using (35) to carry out successive kernel decompositions, an implementation composed of *N*
^{(p−2)} second-order kernels can always be obtained for a *p*th-order kernel.

Now, the idea behind the proposed reduced-rank approach for implementing Volterra filters is to exploit the fact that any *p*th-order kernel can be decomposed into a parallel structure of second-order kernels as previously described. Taking into account this fact, a reduced-rank implementation for the *p*th-order kernel can be obtained by applying, to each second-order kernel resulting from the kernel decomposition, the strategy used for obtaining the reduced-rank implementation of Fig. 1 (see Section 2.3). Thus, one obtains a structure composed of *N*
^{(p−2)} blocks, each having the form of the structure of Fig. 1. Then, disregarding the branches of these blocks related to the smaller singular values, a reduced-complexity reduced-rank kernel implementation is obtained. In this context, the proposed approach can be summarized as follows:

- i)
Exploit the strategy described in Section 3.1 (see Fig. 3) to obtain an implementation of the

*p*th-order kernel of interest in the form of a parallel structure composed of second-order kernels. - ii)
Use the standard matrix-form representation (see Section 2.2) to represent all second-order kernels that compose the kernel of interest.

- iii)
Obtain reduced-rank implementations of such second-order kernels by using the singular value decomposition as described in Section 2.3.

- iv)
Remove (prune) the branches of the resulting structure related to the smaller singular values of the involved second-order coefficient matrices.

The proposed reduced-rank approach for implementing Volterra filters consists of the application of these four steps to all kernels, which allows obtaining effective reduced-complexity Volterra filter implementations.

## 4 Simulation results

with \(\underline {\mathbf {h}}_{\mathrm {k}}\) denoting the coefficient vector of the kernel to be implemented, and \(\underline {\mathbf {h}}_{\text {rr}}\), the coefficient vector obtained by using the reduced-rank approach. A hierarchical branch-level pruning is applied to the parallel structures obtained by using the approaches considered, which means that one branch is removed at a time, with the branches related to the smallest singular values removed first. After the removal of each branch, both the normalized misalignment and the number of required arithmetic operations are evaluated, resulting in the curves used here for comparing the different approaches. Markers are presented along these curves, indicating each evaluated pair of normalized misalignment and number of operations per sample.

### 4.1 Example 1: modeling of a third-order kernel

*N*

_{a}=10, sampling rate of

*f*

_{s}=44.1 kHz, and white input signal). The PC and the proposed reduced-rank approaches are used here for implementing the third-order kernel of the Volterra model obtained in such an experiment. As a result, the curves of normalized misalignment as a function of the number of operations per sample shown in Fig. 4 are obtained. A vertical dotted line pointing out the number of operations per sample required by the corresponding redundancy-removed Volterra implementation (around 495 operations) is also included in this figure aiming to establish a limit after which it is no longer interesting to use reduced-rank implementations. One can notice from Fig. 4 that the proposed approach outperforms the PC one for the case considered here; since, a smaller normalized misalignment is obtained for any given number of operations per sample. For instance, if a misalignment below −15 dB is desired, a PC-based implementation would require at least 280 operations per sample (i.e., a reduction of about 43% with respect to the number of coefficients of the redundancy-removed Volterra kernel), whereas an implementation based on the proposed approach would require only 139 operations per sample (a reduction of almost 72%).

### 4.2 Example 2: modeling of a fourth-order kernel

## 5 Conclusions

In this paper, a novel reduced-rank approach for implementing higher-order Volterra filters was introduced. Such an approach is based on a new form of kernel implementation that allows converting any higher-order kernel into a structure composed exclusively of second-order kernels. Then, exploiting the singular value decomposition along with the coefficient matrices of these second-order kernels, a reduced-rank implementation for the higher-order Volterra filter can be achieved. Numerical simulation results were shown, confirming the effectiveness of the proposed approach in obtaining reduced-complexity implementations of Volterra filters.

## Declarations

### Acknowledgements

The authors are thankful to Gabriel Celso Kulevicz da Silva for providing the experimental data used in Example 1 of Section 4. Thanks are also due to the editors and the anonymous reviewers for their valuable and constructive comments and suggestions, from which the revision of this paper has benefited significantly.

### Competing interests

The authors declare that they have no competing interests.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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