Output frequency response function-based analysis for nonlinear ...

Abstract 

Mechanical Systems and Signal Processing 22 (2008) 102–120 

Mechanical Systems 

and 

Signal Processing 

Output frequency response function-based analysis 

for nonlinear Volterra systems 

Xing Jian Jing , Zi Qiang Lang, Stephen A. Billings 

Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK 

Received 11 December 2006; received in revised form 21 June 2007; accepted 28 June 2007 

Available online 4 July 2007 

This paper deals with the frequency domain analysis of nonlinear systems based on some recently developed results. A 

new systematic approach to the analysis and design of system output frequency response in terms of system time domain 

model parameters is proposed for nonlinear Volterra systems. A general procedure to conduct an output frequency 

response function (OFRF)-based analysis is given, the potential applications of the OFRF-based analysis are discussed, 

and some useful results and techniques are established, especially for the analysis of nonlinear systems with respect to any 

specific model parameters. These results facilitate and illustrate the practical application of the new method. A case study 

for the analysis of a mechanical system is provided to illustrate this new method. 

r 2007 Elsevier Ltd. All rights reserved. 

Keywords: Output frequency response function; Nonlinear systems; Volterra series 

1. Introduction 

ARTICLE IN PRESS 

www.elsevier.com/locate/jnlabr/ymssp 

Frequency domain analysis of a linear system can usually provide some intuitional insights into the system 

of interest, and thus is extensively adopted in engineering analysis and design. However, when it comes to a 

nonlinear system, it is not as easy as that for a linear system to perform a frequency domain analysis. It is 

known that nonlinear systems are often observed to have harmonics and complex inter-modulations, which 

transfer energy between different frequencies and give outputs at some quite different frequencies to those of 

the input, and in many systems chaos and bifurcation are also frequently encountered. These phenomena 

complicate the study of nonlinear systems in the frequency domain, and the frequency domain theory for 

linear systems cannot directly be extended to the nonlinear case. A systematic approach to the analysis of 

nonlinear systems in the frequency domain is yet to be developed. 

In the past decades, some progress has been made in this topic in spite of these difficulties, such as the 

describing function method [1–6] and the harmonic balance method [7,8]. For a class of nonlinear systems, 

frequency domain analysis can also be conducted based on Volterra series theory [9,10]. The nonlinear systems 

which have a Volterra series expansion [11–15] are simply referred to as nonlinear Volterra systems in this paper. 

Corresponding author. 

E-mail address: X.J.Jing@sheffield.ac.uk (X.J. Jing). 

0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. 

doi:10.1016/j.ymssp.2007.06.010

Based on the Volterra series expansion, the generalized frequency response function (GFRF) is proposed in 

George [16]. Thereafter, many researches on the frequency domain analysis of nonlinear Volterra systems are 

carried out [17–21]. By using this kind of frequency domain analysis method, nonlinear behaviour of a 

nonlinear Volterra system driven by a general input can be studied through its GFRFs and output spectrum. 

Therefore, nonlinear characteristics in the frequency domain of a Volterra system can be revealed from a novel 

and effective perspective. A limitation of this method may lie in that estimation and computation of the 

GFRFs and Volterra kernels for a nonlinear Volterra system usually involve some complicated computation 

and symbolic operations [22]. It can be seen that a straightforward analytical expression for the relationship 

between system time domain model parameters and system output frequency response can considerably 

facilitate the analysis and design of nonlinear Volterra systems in the frequency domain. 

For this purpose, a concept of the output frequency response function (OFRF) for nonlinear Volterra 

systems was proposed in Lang et al. [23], which defines an analytical relationship between system output 

spectrum and model parameters. The detailed structure of this relationship was studied in Jing et al. [24]. By 

using this explicit relationship, system model parameters of interest can be directly related to an engineering 

analysis and design objective which is dependent on system output frequency response, and the system 

output frequency response can be analysed in terms of some model parameters of interest. Therefore, 

based on the theoretical results established recently by the authors, a novel systematic approach to the 

frequency domain analysis of nonlinear Volterra systems is proposed in this paper. The new approach, 

referred to as OFRF-based analysis, allows analysis and design of output frequency response of nonlinear 

Volterra systems to be conducted in terms of system time domain model parameters with respect to any a 

specific input signal. A general procedure is proposed for the new frequency domain analysis method, and 

some fundamental results and practical techniques are developed to support and facilitate the application of 

this new method. 

The paper is structured as follows. Section 2 introduces the background of this study and provides some 

theoretical results established recently. Based on these theoretical results, a general procedure and some 

algorithms for the determination of the OFRF for a given NDE or NARX model are proposed and discussed 

in Section 3. Some useful results and techniques are developed in Section 4.1, especially for the analysis of 

nonlinear Volterra systems with respect to any specific model parameters, and what the OFRF-based analysis 

can deal with are summarized in Section 4.2. Section 5 provides an example to illustrate the new method. A 

summary is given in Section 6. 

2. Fundamental concept of the OFRF-based analysis for nonlinear systems 

Nonlinear systems considered in this paper can be described by the following nonlinear differential equation 

(NDE) model: 

X M 

X m 

X K 

m¼1 p¼0 l1;lpþq¼0 

where d 1 xðtÞ=dt 1 

l¼0 

cp;qðl1; ...; lpþqÞ Yp 

i¼1 

d li yðtÞ 

dt li 

¼ xðtÞ, p+q ¼ m, PK 

Y pþq 

i¼pþ1 

¼ 

l1;lpþq¼0 

PK 

l1¼0 

d li uðtÞ 

dt li 

¼ 0, (1) 

P K 

lpþq¼0 

ð Þ, M is the maximum degree of nonlinearity in 

terms of y(t) and u(t), and K is the maximum order of the derivative. In this model, the parameters such as 

c0,1(.) and c1,0(.) are linear parameters, which correspond to the linear terms in the model, i.e., d l yðtÞ=dt l and 

d l uðtÞ=dt l for k ¼ 0,1, y, K, and cpqð Þ for p+q41 are nonlinear parameters corresponding to nonlinear terms 

in the model of the form: 

Y p 

i¼1 

d li yðtÞ 

dt li 

Y pþq 

i¼pþ1 

d li uðtÞ 

dtli , 


X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 103 

e.g., y(t) p u(t) q . p+q is called the nonlinear degree of the nonlinear parameter cpqð Þ. Moreover, cpqð Þ and cp0q0ð Þ 

are referred to as different type of nonlinearity if p6¼p0 or q6¼q0 . Similar to the NDE model (1), a discrete 

nonlinear model known as NARX model is often used for practical nonlinear system identification from

104 

experimental data, which is given by 

yðtÞ ¼ XM 

X m 

X K 

m¼1 p¼0 k1;kpþq¼1 

cp;qðk1; ...; kpþqÞ Yp 

i¼1 

yðt kiÞ Ypþq 

i¼pþ1 

uðt kiÞ. (2) 

In engineering practise, the NDE or NARX model can represent a wide class of nonlinear dynamical systems 

and include several well-known nonlinear input–output models as special cases [25]. 

The input–output relationship of model (1) or (2) can both be approximated in the neighbourhood of the 

equilibrium by a Volterra series of sufficient orders [15]. In order to conduct a frequency domain analysis for 

this class of nonlinear systems, the GFRF was proposed in [16] 

Z 1 Z 1 

Hnðjo1; ...; jonÞ ¼ ... hnðt1; ...; tnÞ exp ð jðo1t1 þ þontnÞÞdt1 dtn, (3) 

1 

1 

where hnðt1; ...; tnÞ is a real valued function of t1; ...; tn called the nth-order Volterra kernel of the system. 

By probing method [10], the recursive algorithms for the computation of the GFRFs for model (1) and (2) 

were developed in [26,27]. The system output frequency response was studied in Lang and Billings [28]. 

Although these results provide an important basis for frequency domain analysis of nonlinear systems, it can 

be seen that the direct analytical computation of system output spectrum involves very complicated integral 

and symbolic operations in multi-dimensional complex space when the involved nonlinearity order is high, 

and the analytical relationship between model parameters and output spectrum cannot be demonstrated 

clearly by the recursive algorithms [22,24]. This inhibits the application of these theoretical results to a certain 

extent. For these reasons, the concept of OFRF was proposed in Lang et al. [23], which was also studied in 

Jing et al. [24] through the parametric characteristic analysis. The result in [24] explicitly reveals the analytical 

polynomial relationship between system model parameters and system output spectrum, and allows the 

detailed structure of the OFRF to be determined up to any high orders without complicated symbolic 

computations in multi-dimensional complex space. Based on the results in Jing et al. [24], the nth-order GFRF 

in Eq. (3) can be written as 

Hnðjo1; ...; jonÞ ¼CEðHnðjo1; ...; jonÞÞfnðjo1; 

...; jonÞ (4) 

and the system output spectrum can therefore be written into a more explicit polynomial form as follows, 

which is referred to as the OFRF in Lang et al. [23]: 

YðjoÞ ¼ XN 

n¼1 

CEðHnð ÞÞj nðjoÞ T , (5a) 

where 

1 

jnðjoÞ ¼pffiffi 

n 

nð2pÞ 

Z 

1 

o1þ 

f nðjo1; ...; jonÞ 

þon¼o 

Yn 

UðjoiÞdso. 

i¼1 

CE(.) is a novel coefficient extraction operator which has two fundamental operations ‘‘ ‘‘and ‘‘ ‘‘. The 

detailed definitions can refer to Jing et al. [24], and CE(Hn( )) is the parametric characteristics of the nth-order 

GFRF Hnð Þ, which can be recursively determined by 

CEðHnðjo1; ...; jonÞÞ 

¼ C0;n 

n 1 n q 

q¼1 p¼1 

Cp;q CEðHn q pþ1ð ÞÞ 


X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 

n 

p¼2 

Cp;0 CE Hn pþ1ð Þ , ð5bÞ 

where Cp;q ¼½cp;qð0; ...; 0Þ; cp;qð0; ...; 1Þ; ...; cp;qðK; ...; KÞŠ 

for some p, q. The terminating condition is 

|fflfflfflfflfflffl{zfflfflfflfflfflffl} 

pþq 

CE(H 1(.)) ¼ 1. Obviously, the elements of CE(H n( )) are functions of the system time domain model 

parameters which define nonlinearities. For more clarity, Eq. (5a) can be rewritten as 

YðjoÞ ¼cFðjoÞ T , (6)

N 

where c ¼ CEðHnð ÞÞ; FðjoÞ ¼ f1ðjoÞ; f2ðjoÞ; ...; fNðjoÞ . Note that f1ðjoÞ ¼H1ðjoÞ is the first-order 

n¼1 

GFRF, which represents the linear part of model (1) or (2). 

Eq. (5a) or (6) provide a more straightforward analytical expression for the relationship between system 

time domain model parameters and system output frequency response. The system output frequency response 

can therefore be analyzed in terms of any interested model parameters by using this explicit relationship. 

Hence, it can considerably facilitate the analysis and design of the output frequency response characteristics of 

nonlinear Volterra systems in the frequency domain. The main idea for the OFRF-based analysis proposed in 

this paper is that, given the model of a nonlinear system in the form of model (1) or (2), CE(Hn( )) can be 

computed according to Eq. (5b) and jn(jo) can be obtained according to a numerical method which will be 

discussed later, thus the OFRF of the nonlinear system subject to any a specific input function can be 

achieved, which is an analytical function of nonlinear parameters of system model, and finally frequency 

domain analysis for the nonlinear system can be conducted in terms of the specific model parameters of 

interest. In order to conduct an OFRF-based analysis, a general procedure is provided in this paper and 

several basic algorithms and related results are discussed. In many practical applications, the problems may be 

how some specific model parameters affect system output spectrum and what the effect is. Therefore, some 

fundamental results are also developed to facilitate the application of the new method for this purpose. 

3. A general procedure for the determination of the OFRF 

In this section, a general procedure for the determination of the OFRF for a given model (1) or (2) is 

proposed, and some useful algorithms and techniques are provided. 

3.1. Computation of the parametric characteristics of OFRF 

This step is to derive c ¼ CEðHnð ÞÞ in Eq. (6). 

n¼1 

N 



3.1.1. Determination of the largest order N 

To derive the parametric characteristics of OFRF, the first task is to compute the largest order, i.e., N, of 

the Volterra series expansion for the nonlinear system, which is basically determined by the significance of the 

truncation error in the Volterra series expansion of finite order. This can alternatively be done by evaluating 

the magnitude of the nth-order output frequency response Yn(jo). For example, the magnitude bound of 

Yn(jo) for the NARX model (2) can be evaluated by [29] 

Y nðjoÞ pan bn _ T 

n , (7) 

where an; _n are complex valued functions, and bn is a function vector of the system model parameters. For the 

detailed definitions for an; bn; _n refer to Jing et al. [29]. If the magnitude bound of a certain order of Yn(jo) is 

less than a predefined value (for instance 10 8 ), then the largest order N is obtained. It should be noted that 

the magnitude bound is a function of the model nonlinear parameters, therefore, the largest ranges of interest 

for each nonlinear parameter should be considered in the evaluation of |Yn(jo)|. 3.1.2. Determination of the parametric characteristics 

Once the largest order N is determined, the next step is to derive the parametric characteristics of GFRFs 

for the nonlinear system, i.e., CE(Hn( )) from n ¼ 2 to N, which will be used in the computation of 

N 

c ¼ CEðHnð ÞÞ. Note that CE(Hn( )) is computed in terms of the parameter vectors Cp;q ¼ 

n¼1 

½cp;qð0; ...; 0Þ; cp;qð0; ...; 1Þ; ...; cp;qðK; ...; KÞŠ 

for some p,q in Eq. (5b). 

|fflfflfflfflfflffl{zfflfflfflfflfflffl} 

pþq 

Basically, for some specific parameters to be analysed for a system, CE(H n( )) can be recursively computed 

by Eq. (5b) with respect to these parameters of interest with the other non-zero nonlinear parameters being 1.

106 

Alternatively, CE(Hn( )) can also be determined directly without recursive computations by using the 

following result. 

Proposition 1. [24]. The elements of (CEðHnðjo1; ...; jonÞ) includes the nonlinear parameter in C0n and all the 

parameter monomials in Cpq Cp 1q 1 Cp 2q 2 Cp kq k for 0pkpn 2, whose subscripts satisfy 

p þ q þ Xk 

i¼1 

ðp i þ q iÞ¼n þ k; 2pp i þ q ipn k; 2pp þ qpn k and 1pppn k. (8) 

Based on Proposition 1, the computation of the parametric characteristic CE(Hn( )) can be conducted as 

follows, which is referred to as Process A. For 0pkpn 2, 

(A1) Generate all the combinations (r 0, r 1, r 2y, r k) satisfying r0 þ P k 

i¼1 ri ¼ n þ k and 2pripn k with 

respect to a specific value of k; 

(A2) Generate all the possible combinations (pi,qi) with respect to each ri satisfying pi+qi ¼ ri, and note that 

when it is for r0, 1pp 0pn k; 

(A3) All the possible combinations can now be generated based on Steps (A1) and (A2), then remove all the 

repetitive terms; 

(A4) CE(Hn( )) is obtained in terms of the parameter vectors Cp,q for some p,q, which can be stored for any 

future usage. For a specific nonlinear system, CE(H n( )) can be obtained only by replacing the 

corresponding interested parameter vector C p,q with respect to the specific nonlinear system, and the 

other parameters in CE(Hn( )) are set to be zero if it is zero or set to be 1 if it is not of interest to be 

analysed. 

(A5) Achieve the final result by manipulating CE(Hn( )) and removing the repetitive terms according to the 

operation rules of ‘‘ ‘‘and ‘‘ ‘‘(see [24]). 

By this method, the parametric characteristic CE(Hn( )) can be obtained without recursive computations. 

N 

For a summary, the parametric characteristic vector c ¼ CEðHnð ÞÞ can be computed by following the 

n¼1 

process below, which is referred to as Process B: 

(B1) Determine the set of the nonlinear parameters of interest, denoted by SC; 

(B2) Determine the largest possible ranges for the interested nonlinear parameters, denoted by qSC; 

(B3) Determine the largest order N of the Volterra series expansion according to Eq. (7) and the discussions in 

Section 3.1.1. 

(B4) Computation of CE(Hn( )) with respect to the interested parameters SC following Process A or Eq. (5b) 

from n ¼ 2toN. 

N 

(B5) Combine the final parametric characteristic vector c ¼ CEðHnð ÞÞ. 

n¼1 

Therefore, based on Processes A and B, the parametric characteristics of the output frequency response 

with respect to any specific model parameters of interest can be determined, which are the coefficients 

of the polynomial function (6). Thus the structure of the polynomial (6) is also explicitly determined 

at this stage. 

3.2. A numerical method 

This step is mainly to determine FðjoÞ ¼ f 1ðjoÞ; f 2ðjoÞ; ...; f NðjoÞ in Eq. (6), then the OFRF in Eq. (6) 

is obtained consequently. Since the system model is known and the parametric characteristic vector c ¼ 

N 

n¼1 



CEðHnð ÞÞ is achieved, and note that F(jo) is invariant with respect to c ¼ CEðHnð ÞÞ, thus F(jo) can be 

n¼1 

N

derived with respect to any a specific input by following a numerical method as follows, which is referred to as 

Process C: 

(C1) Choose a series of different values of the interested nonlinear parameters, which are properly distributed 

in qSC, to form a series of vectors c1 that 

crðNÞ, where rðNÞ ¼ c denotes the dimension of vector c, such 

c ¼½c T 

1 c T 

rðNÞŠT is non singular: (9) 

(C2) Given a frequency o where the output frequency response of the nonlinear system is to be analysed or 

designed. Actuate the system using the same input under the different values of the nonlinear parameters 

c1 crðNÞ, then collect the time domain output y(t) for each case, and consequently obtain a series of 

output frequency response YðjoÞ1 YðjoÞrðNÞ at the frequency o by FFT technique. 

(C3) Step 2 yields 

CFðjoÞ T 2 3 2 3 2 3 

c1 j1ðjoÞ YðjoÞ1 6 c 

7 6 

2 7 j 

7 6 

6 2ðjoÞ 7 YðjoÞ 

7 

6 2 7 

6 7 6 7 6 7 

¼ 6 . 7 6 

6 

4 . 7 . 7 ¼ 6 

6 

5 4 . 7 . 7 ¼: YYðjoÞ. (10) 

6 

5 4 . 7 

5 

crðNÞ jrðNÞðjoÞ YðjoÞrðNÞ Hence, 



FðjoÞ T ¼ f1ðjoÞ; f2ðjoÞ; ...; fNðjoÞ T 

h iT ¼ j1ðjoÞ; j2ðjoÞ; ...; jrðNÞðjoÞ ¼ c 1 YYðjoÞ. ð11Þ 

Remark 1. In Step C1, r(N) different values of the parameter vector c in the parameter space qSC, such that 

detðcÞa0 can be obtained by choosing a grid of parameter values of the interested nonlinear parameters 

properly spanned in qSC, or using a stochastic-based searching method or other optimization methods such as 

GA to generate a non-singular matrix c. In practices, it is not difficult to find such a matrix with a proper 

inverse, which will be illustrated in Section 5. In Step C2, given the largest order N of the system output 

spectrum, it can be seen that this algorithm needs r(N) simulations to obtain r(N) output frequency responses 

under different parameter values. Note from Step C1 that rðNÞ ¼ c ¼ CEðHnð ÞÞ , which is not only a 

n¼1 

function of the largest order N but also dependent on the number of interested parameters. This implies the 

simulation burden will become heavier with the number of the interested parameters and the largest order N. 

In Step C3, since detðcÞa0, the complex valued function vector F(jo) in Eq. (11) is unique, which implies the 

result in Eq. (11) can sufficiently approximate to their real values if the truncation error incurred by the largest 

order N of the Volterra series is sufficiently small. 

Therefore, by following Process C, the complex valued function vector F(jo) can accurately be obtained for 

the specific input function used in Step C2 and at the given frequency o. Consequently, the OFRF (6) subject 

to the specific input function is now explicitly determined by following the method in Sections 3.1 and 3.2 for 

the nonlinear system of interest. Although the function vector F(jo) is obtained by using the numerical 

method above and consequently the obtained OFRF is not an analytical function of the frequencies and the 

input, the achieved relationship between the output spectrum and model nonlinear parameters is analytical 

and explicit for the specific input function at the given frequency o. Moreover, note that since CE(Hn( )) is 

known, and FðjoÞ ¼ j1ðjoÞ; j2ðjoÞ; ...; jNðjoÞ is determined, then Y nðjoÞ ¼CEðHnð ÞÞ jnðjoÞ T is also 

determined, which represents the analytical function for the nth-order output frequency response of nonlinear 

systems. 

It shall also be noted that, compared with the analytical computation by using the recursive algorithm for 

the GFRFs in Billings and Peyton-Jones [26] and the output spectrum provided in Lang and Billings [28], the 

N

108 

proposed method above as demonstrated in Sections 3.1 and 3.2 enable the OFRF to be obtained directly in a 

concise polynomial form as Eq. (6) without the complex integration in the high-dimensional super-plane 

o ¼ o1 þ þon especially when the nonlinearity order n is high. By using the proposed method above, the 

OFRF can be determined up to very high orders with respect to any specific model parameters of interest and 

any specific input signal at any given frequency. The cost may lie in that the new method needs r(N) 

simulations. 

4. OFRF-based analysis for specific model parameters 

Once the OFRF is obtained, the analysis and design of nonlinear systems described by model (1) or (2) can 

be carried out in terms of the interested model parameters which define system nonlinearities and may 

represent some structural and controllable factors of a practical engineering system. For example, the 

sensitivity of system output frequency response with respect to a nonlinear parameter can be studied based on 

the analytical expression (6). By using the link between the nonlinear terms of interest and the components of a 

practical engineering system and structure, the OFRF may provide a useful insight into the design of nonlinear 

components in the system to achieve a desired output performance. Therefore, the OFRF-based analysis 

method provides a novel approach to the analysis and synthesis of a considerably wide class of nonlinear 

systems subject to any specific input signal in the frequency domain. 

Note that the parametric characteristic vector CE(Hn( )) for all the model nonlinear parameters can be 

obtained according to Eq. (5b) or Process A, and if there are only some parameters of interest, the 

computation can be conducted by only replacing the other non-zero parameters with 1 as mentioned above. In 

many cases, only several specific model parameters, for example parameters in Cp,q, are of interest for a 

specific nonlinear system. Then the computation of the parametric characteristic vector in Eq. (6) can be 

simplified greatly. This section provides some useful results for the OFRF-based analysis with respect to one 

or more specific parameters in C p,q which can effectively facilitate the determination of the OFRF and the 

analysis based on the OFRF. Moreover, some potential applications of the OFRF-based analysis are also 

discussed. 

4.1. Parametric characteristic of the OFRF with respect to C p,q 

Proposition 2. Consider only the nonlinear parameter Cp,q ¼ c. The parametric characteristic vector of the nthorder 

GFRF with respect to the parameter c is 

n 1 

n 1 

pþq 1 dðpÞ posðn qÞ d pþq 1 

n 1 

pþq 1 , (12) 

CEðHnðjo1; ...; jonÞÞ ¼ 1; c; c 2 ; ...c 

( 

1 

where bcis to get the integer part of (.),dðpÞ ¼ 

0 

Proof. See Appendix B. & 

( 

if p ¼ 0; 

1 

, and posðxÞ ¼ 

else; 

0 

if x40; 

. 

else; 

Note that here c may be one parameter or a vector of some parameters of the same nonlinear degree and 

type in Cpq. Also note that cn ¼ c ...c c and is the reduced Kronecker product defined in Jing et al. [24], 

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 

n 

when c is a vector. Proposition 2 establishes a very useful result to study the effects on the output frequency 

response from a specific nonlinear degree and type of nonlinear parameters. Note also that if some nonlinear 

parameters in model (1) or (2) are zero, only part terms in CEðHnðjo1; ...; jonÞÞ take an effective role. The 

detailed form of CEðHnðjo1; ...; jonÞÞ can be derived from Processes A and B. However, direct using Eq. (12) 

does not affect the final result and is more convenient. 

Corollary 1. If all the other degree and type of nonlinear parameters are zero except that Cp,q ¼ c is non-zero. 

Then the parametric characteristic vector of the nth-order GFRF with respect to the parameter c is: if (n4p+q 

and p40), or (n ¼ p+q), and if additionally ðn1Þ= ðpþq1Þ is an integer, then 

CEðHnðjo1; ...; jonÞÞ ¼ c 

ðn1Þ= pþq 1 ð Þ , 


X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120

else 

CEðHnðjo1; ...; jonÞÞ ¼ 0, 

which can be summarized as 

CEðHnðjo1; ...; jonÞÞ ¼ c 

ðn1Þ= pþq 1 ð Þ d 

n 1 

p þ q 1 

n 1 

p þ q 1 

Proof. The results are directly followed from Propositions 1 and 2. & 

ð1dðpÞposðn qÞÞ. 

(13) 

Corollary 1 provides a more special case of nonlinear system (1) or (2). There are only several nonlinear 

parameters of the same nonlinear type and degree in the considered system. This result will be used in the example 

of Section 5. The following results can be achieved for the output frequency response. 

Corollary 2. Consider only the nonlinear parameter Cp,q ¼ c. The parametric characteristic vector of the output 

spectrum in (6) with respect to the parameter c can be written as 

N 

N 1 

pþq 1 dðpÞ posðN qÞd 

N 1 

pþq 1 

N 1 

pþq 1 . 

c ¼ CEðYðjoÞÞ ¼ CEðHnð ÞÞ ¼ 

n¼1 

1; c; c 2 ; ...; c 

If all the other degree and type of nonlinear parameters are zero except that Cp,q ¼ c is non-zero (p+q41). 

Then the parametric characteristic vector of the output spectrum in (6) with respect to the parameter c is 

if p ¼ 0 

c ¼ CEðYðjoÞÞ ¼ 1 CEðHqð ÞÞð1 posðq NÞÞ ¼ ½1cð1posðq NÞÞŠ, 

else 

bðN c ¼ CEðYðjoÞÞ ¼ 

1Þ= 

ðpþq1Þc CEðHðpþq 1Þiþ1ð ÞÞ ¼ 1; c; c 2 h 

ðN ; ...; cb 1Þ= 

ðpþq i 

1Þc 

. 

i¼0 

Proof. The results are straightforward from Proposition 2 and Corollary 1. & 

The results above involve the computation of c n .Ifc is an I-dimension vector, there will be many repetitive 

terms involved in c n . To simplify the computation, the following lemma can be used. 

Lemma 1. Let be c ¼ [c1,c2,y,cI] which can also be denoted by c [1:I], and cn ¼ c 

Kronecker product defined in Jing et al. [24], nX1 and IX1. Then 

c c,‘‘ 

’’ is the reduced 


n 

c n ¼ c n 1 c1; ...; c n 1 ½sð1Þn sðiÞn þ 1 : sð1ÞnŠci; ...; c n 1 ½sð1ÞnŠcI , 

where sðiÞn ¼ PI 

sðjÞn 1, s(.) 1 ¼ 1, and 1pipI. Moreover, DIM(c 

j¼i 

n n n 

) ¼ s(1) n+1, and the location of ci in c is 

s(1)n+1–s(i)n+1+1. 

Proof. See Appendix C. & 

4.2. Analysis based on the OFRF with respect to Cp,q 



Based on the result in Corollary 2 and Eq. (6), with respect to a specific parameter c, the OFRF can be 

written as 

YðjoÞ ¼ ¯j 0ðjoÞþc ¯j 1ðjoÞþc 2 j2ðjoÞþ þc ‘ j ‘ðjoÞþ . (14a) 

Since Y(jo) is also a function of c, therefore, Eq. (14a) is rewritten more clearly as 

Yðjo; cÞ ¼ ¯j 0ðjoÞþc ¯j 1ðjoÞþc 2 j2ðjoÞþ þc ‘ j ‘ðjoÞþ . (14b) 

Y(jo;c) is in fact a series of an infinite order, ‘ is a positive integer which can be determined by Corollary 2, 

¯j iðjoÞ is the complex valued function corresponding to the coefficient c i in Eq. (6), which can be obtained by 

following the method in Process C. If all the other degree and type of nonlinear parameters are zero

110 

except that Cp,q ¼ c is non-zero (p+q41), then ¯j iþ1ðjoÞ ¼j iðjoÞ (ji(jo) is defined in Eqs. (5) and (6). Based 

on Eq. (14), the following analysis can be conducted. 

(1) Sensitivity of the output frequency response to nonlinear parameters 

Based on Eq. (14), this can be obtained easily as 

qYðjo; cÞ 

¼ j 

qc 

1ðjoÞþ2cj2ðjoÞþ þ‘c ‘ 1 j ‘ðjoÞþ . (15) 

Similarly, the sensitivity of the magnitude of the output frequency response with respect to the nonlinear 

parameters can also be derived. Note that 

Yðjo; cÞ 2 ¼ Yðjo; cÞYð jo; cÞ 

¼ð¯j 0ðjoÞþc ¯j 1ðjoÞþc 2 ¯j 2ðjoÞ þ Þð ¯j 0ð joÞþc ¯j 1ð joÞþc 2 ¯j 2ð joÞþ Þ, 

¼ ¯j 0 ¯j 0 þ X1 

‘¼1 

qYðjo; cÞ 

qc 

X‘ 

‘ 

c 

i¼0 

¯j i ¯j ‘ i 

Yð jo; cÞ 

Yðjo; cÞ 

! 

:¼ p0 þ cp1 þ c 2 p2 þ þc 2‘ p2‘ þ . ð16Þ 

It is obvious that the spectral density of the output frequency response is still a polynomial function of the 

parameter c. Eq. (16) can also be directly derived by following Process C. Thus, the sensitivity of the 

magnitude of the output spectrum to the parameter c can be obtained as 

q Yðjo; cÞ 1 q Yðjo; cÞ 

¼ 

qc 2 Yðjo; cÞ 

2 

1 X 

¼ 

qc 2 Yðjo; cÞ 

1 X‘ 

‘ 1 

‘c ¯j i ¯j ‘ 

‘¼1 i¼0 

Given Eq. (15), Eq. (17a) can also be computed as 

! 

iÞ . (17a) 

q Yðjo; cÞ 

qc 

1 q Yðjo; cÞ 

¼ 

2 Yðjo; cÞ 

2 

qc 

1 

¼ 

2 Yðjo; cÞ 

qYðjo; cÞ 

Yð 

qc 

! 

qYð jo; cÞ 

jo; cÞþYðjo; cÞ 

qc 

¼< . ð17bÞ 

The sensitivity function for system output spectrum with respect to a nonlinear parameter provides a useful 

insight into the effect on system output performance from the specific model parameters. This will be 

illustrated in Section 5. Another possible application of the sensitivity function is for the output oscillation 

suppression problem. In many engineering practices, the oscillation in system output should be suppressed as 

small as possible. Based on Eq. (17), it can be seen that it should be q Yðjo; cÞ =qco0 for some c in order to 

reduce the magnitude level of output frequency response by properly designing the value of c. Consider Eq. 

(16), the following conclusion is obvious: 

(a) q Yðjo; cÞ =qco0 for some c )9some n40; Pn 



i¼0 

sign ðc n 1 Þ ¯j i ¯j n i o0 

(b) p1 ¼

It is followed from Eq. (14) that the radius of convergence is given by 

R ¼ lim 

‘!1 

¯j ‘ 1ðjoÞ 

¯j ‘ðjoÞ 

. (18) 

Obviously, if |c|oR, then the series is convergent. Define a ratio function 

Rð‘; cÞ ¼ ¯j ‘ 1ðjoÞc , 

¯j ‘ðjoÞc (19) 

which is a function of ‘ and also varying with different nonlinear parameters. It can be seen that, if 

DRð‘; c1Þ DRð‘; c2Þ 

4 , (20) 

D‘ D‘ 

then the output spectrum has a larger radius of convergence with respect to c1 than that with respect to c2.Eq.(19) and inequality Eq. (20) can be used as an evaluation of the effect on the convergence of the Volterra series 

expansion for the nonlinear system under study from a model parameter and the comparative advantage between 

different parameters. Note that divergence of the Volterra series expansion can sometimes imply the instability of 

the system under study or the non-existence of a Volterra series expansion. Thus this analysis can provide some 

useful information for the design of system output frequency response in terms of different model parameters. 

(3) Optimization of the output frequency response in terms of nonlinear parameters 

Given a desired magnitude of the output frequency response Y * , an optimal c * in qSc can be found such that 

minð 

Yðjo; cÞ Y Þ. 

c2@Sc 

A systematic method for this purpose is yet to be developed, which will be discussed in a future study. 

5. A case study 



To demonstrate the application of the proposed OFRF-based analysis method, a nonlinear spring-damping 

system is studied as shown in Fig. 1. The system has two nonlinear passive components and one nonlinear 

active unit. The active unit is described by F ¼ c1 _x 2x þ c2x _x 2 , the output property of the spring satisfies 

F ¼ Kx+c3x 3 , and the damper F ¼ B _x þ c4 _x 3 . u(t) is the external input force. The system dynamics can be 

described by 

M €x ¼ Kx B _x c1 _x 2 x c2 _xx 2 

c3x 3 

c4 _x 3 þ uðtÞ. (21a) 

Let the output be the transmitted force measured by 

y ¼ Kx. (21b) 

This kind of SDOF systems can be found in many engineering practices. The task for this case study is to 

investigate how the nonlinear terms included both in passive and active unites affect the output and what the 

effect might be, and thus to provide a useful insight into the design of corresponding nonlinear parameters to 

achieve a desired output frequency response. 

u(t) 

M 

Active 

K unit B 

x(t) 

Fig. 1. A mechanical system.

112 

For clarity in discussion, let M ¼ 240, K ¼ 16000, and B ¼ 296, then Eq. (21) can be rewritten as 

240 €x ¼ 16000x 296 _x c1 _x 2 x c2 _xx 2 

c3x 3 

c4 _x 3 þ uðtÞ, (22a) 

y ¼ 16000 x, (22b) 

Eq. (22a) is a simple case of the NDE model (1) with M ¼ 3, K ¼ 2, c10(2) ¼ 240, c10(1) ¼ 296, c10(0) ¼ 16000, 

c30(111) ¼ c4, c30(110) ¼ c1, c30(100) ¼ c2, c30(000) ¼ c3, c01(0) ¼ 1, and all the other parameters are zero. 

Therefore, what is interested in for this study is to analyse the effect of the nonlinear terms with coefficients c1, c2, c3 and c4 on the system output frequency response. To achieve this objective, the procedure proposed in 

Section 3 are adopted to derive the OFRF of system (22), and the results in Section 4 will be used for the 

computation of the parametric characteristic of the OFRF with respect to the nonlinear parameters c1, c2, c3 

and c4. Moreover, though the method proposed in this paper is suitable for a general input function u(t), yet 

for convenience in discussion, the input of system (22) is considered to be a sinusoidal function 

u(t) ¼ 100 sin(8.lt). To illustrate the new results more clearly and conveniently, the parameter c2 under the 

case of c1 ¼ c3 ¼ c4 ¼ 0 is studied firstly in detail. 

(1) Determine the parametric characteristics of OFRF 

Note that all the parameters of interest belong to C30, and the other degrees of nonlinear parameters are all 

zero. Thus Corollary 1 and 2 can be utilised directly. Therefore, 

n 1 

CEðHnðjo1; ...; jonÞÞ ¼ c ð Þ=2 d 

n 1 

2 

n 1 

2 

ð1dð3ÞposðnÞÞ ¼ c 

bðN1Þ= ðpþq1Þc c ¼ CEðYðjoÞÞ ¼ 

CEðHðpþq 1Þiþ1ð ÞÞ 

i¼0 

¼ 1; c; c 2 h i 

ðN1Þ= ðpþq1Þ ; ...; cb c ¼ 1; c; c 2 h i 

ðN1Þ=2 ; ...; cb c , 

n 1 ð Þ=2 d 

where c ¼ c2. To derive the detailed form for c, the largest order N should be determined first according to 

Process B in Section 3.2. In order to have a larger range in which the parameters can vary, in this case let 

c2A(0,10 8 ). Then the magnitude bound of Yn(jo) can be evaluated as mentioned in Process B. However, for 

paper limitation, the detailed computation is omitted in this case. It can be verified that N ¼ 23 is enough for 

use in this case. Therefore, 

c ¼ 1; c; c 2 h i 

ð23 1Þ=2 

; ...; cb c 

(2) Determination of F(jo) for the OFRF 

Following Process C, the matrix c ¼½c T 

1 

¼½1; c2; c 2 2 ; c32 ; c42 ; c52 ; ...; c11 2 Š. 

n 1 

2 

n 1 

2 

c T 

r ŠT should be constructed first. In this case, for any 12 

different values of c 2, the matrix c is a Vandermonde matrix and thus non-singular. Note that in many cases, 

the parameters may be set to be some large values and cover a large range. This will make the element values in 

the matrix c extraordinarily large. Then when the inverse of matrix c is computed, there may be some 

computation error involved in Matlab. To overcome this problem, c can be written as 

bðN1Þ= ðpþq1Þc c ¼ 

kCEðHðpþq 1Þiþ1ð ÞÞ=k ¼ 1; kðc=kÞ; k 2 ðc=kÞ 2 ðN1Þ=2 ; ...; kb c h i 

ðN1Þ=2 ðc=kÞb c . 

i¼0 

Then Eq. (6) can be written as 



Yðjo; cÞ ¼cFðjoÞ T ¼ 1 ðc=kÞ ðc=kÞ 2 

ðc=kÞ ‘ j1ðjoÞ; kj2ðjoÞ; ...; k ‘ j ‘ðjoÞ T , (23) 

where ‘ ¼ ðN1Þ=2 . Moreover, the range for each parameter can be divided into several sub-range, and the 

final result is the combination of these results obtained for each sub-range. In this study, let k ¼ 10 5 , then 

¯c2 ¼ c2=k 2½0; 1000Š. Choose ¯c2 to be the following values to construct c ¼½c T 

1 c T 

r ŠT , i.e., 

0.1,1,50,65,80,100,150,200,250,300,350,400,450,500,550, 600,650,700,750,800,850,900,950,980,1000. The output 

frequency response 

h 

YYðjoÞ ¼ YðjoÞ1; YðjoÞ2; ...; YðjoÞr i 

,

of system (22) at o ¼ 8.1 rad/s corresponding to different values of c2 can be obtained through FFT of the 

time domain output response. Then using Eq. (23), it can be derived from Eq. (11) that 

FðjoÞ T ¼ j1ðjoÞ; kj2ðjoÞ; ...; k ‘ j ‘ðjoÞ T ¼ðc T cÞ 1 c T YYðjoÞ. Therefore, the OFRF of system (22) with 

respect to nonlinear parameter c2 in the case of c1 ¼ c3 ¼ c4 ¼ 0 is obtained as 

Yðjo; c2Þ ¼ð2:060893505718041e þ 002 2:402014548824790e þ 002iÞ 

þ k 1 ð 5:14248529981906 þ 5:35676372314361iÞc2 

þ k 2 ð0:08589533966805 0:08827649204263iÞc 2 2 

þ k 3 ð 8:068953639113292e 004 þ 8:248154776018186e 004iÞc 3 2 

þ k 4 ð4:598423724418538e 006 4:686570228695798e 006iÞc 4 2 

þ k 5 ð 1:679591261850433e 008 þ 1:708497491564935e 008iÞc 5 2 

þ k 6 ð4:056287337706451e 011 4:120496550333245e 011iÞc 6 2 

þ k 7 ð 6:544911009113156e 014 þ 6:641760366680977e 014iÞc 7 2 

þ k 8 ð6:976300614229155e 017 7:073928662624432e 017iÞc 8 2 

þ k 9 ð 4:713366512185836e 020 þ 4:776287453573993e 020iÞc 9 2 

þ k 10 ð1:827866445826756e 023 1:851299290299388e 023iÞc 10 

2 

þ k 11 ð 3:098310700824303e 027 þ 3:136656793561425e 027iÞc 11 

2 . 

Based on this function, Eq. (16) can be further computed as 

Yðjo; cÞ 2 ¼ p0 þ cp1 þ c 2 p2 þ þc 2‘ p2‘ þ 

¼ð1:001695593467675e þ 005Þþk 1 ð 4:693027791051078e þ 003Þc2 

þ k 2 ð1:329525858242289e þ 002Þc 2 2 þ k 3 ð 2:55801250200731Þc 3 2 

þ k 4 0:03645314106899c 4 2 þ k 5 ð 3:968756773045435e 004Þc 5 2 

þ k 6 0:01517275811829c 6 2 þ . 



Note that this is an alternating series and it holds that p i 4 p iþ1 and p i ! 0. Hence the series may keep 

decreasing when c is going larger and within its radius of convergence. By following the similar method 

demonstrated above, the OFRFs of system (22) with respect to nonlinear parameters c 1, c 2, c 3 and c 4 of 

different cases can all be obtained, for instance Y(jo;c1), Y(jo;c3), and Y(jo;c4) (The other nonlinear 

parameters are zero if not appearing in the function). The results are shown in Figs. 2–4. 

Fig. 2 shows that the variation of the magnitude of the OFRFs with respect to each nonlinear parameter. It 

can be seen that the larger the nonlinear terms, the larger the effect they have on the system output frequency 

response; there is a good matching between the theoretical computation results and the simulation results to 

which they have been fitted, and there is also a good match between the theoretical computation results and 

the simulation tests (for parameter c 3) which are independent of the fitted simulation results. From both 

Figs. 2 and 3 it can also be seen that the system output frequency response is much more sensitive to the 

variation of the nonlinear parameters when they are small. Once the value of a nonlinear parameter is 

sufficiently large, then the sensitivity will tend to be zero. From comparisons between these four nonlinear 

terms, it can be concluded that the system output frequency response is more sensitive to the variation 

of the nonlinear parameter c4 when the values are small; however, when the values of each nonlinear 

parameters are sufficiently large, the system output spectrum is more sensitive to the nonlinear parameter c2. 

From Fig. 4 it can be seen that the convergence of the OFRFs are all very fast. This can also be 

understood that the energy disperses quickly with the nonlinear order going larger. It is noted that the 

ratio functions of c 2 and c 3 go up much faster than that of c 1, especially c 2. This implies that the radius of

114 

Magnitude 

Magnitude of the sensitivity functions (10e-5) 

350 

300 

250 

200 

150 

100 

50 



Output frequency response fuctions 

c1 

Simulation data 

c4 


c2 


c3 


Simulation tests 

0 1 2 3 4 5 6 7 8 9 

x 107 0 

Nonlinear parameters c1,c2,c3 and c4 

Fig. 2. Output frequency response functions with respect to c 1 to c 4, respectively. 

10 

5 

0 

-5 

-10 

-15 

Sensitivity of OFRF to nonlinear parameters 

1 2 3 4 5 6 7 8 9 10 11 

x 106 Nonlinear parameters c1, c2 and c3 

Fig. 3. Sensitivity function of the OFRFs with respect to c1 to c3, respectively. 

convergence of the output spectrum corresponding to c 2 should be larger. Simulation tests verify that the 

system is still stable when c 2 ¼ 10 17 where the magnitude of the output spectrum is 0.0216, while the system 

tends to be unstable when c 1 tends to be larger than 10 8 . 

c1 

c2 

c3

Ratio 



6 

5 

4 

3 

2 

1 

x 10 8 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

3.5 

1.5 2 2.5 3 3.5 4 4.5 5 

3 

2.5 

2 

1.5 

Ratio functions 

10 10.2 10.4 10.6 10.8 11 11.2 11.4 

0 

0 2 4 6 8 10 12 

Fig. 4. Ratio functions with respect to c 1 to c 4, respectively. 

From the analysis above for the four nonlinear parameters of nonlinear degree 3, respectively, it can be seen that 

(1) The computed system output spectrum has a larger radius of convergence with respect to c2, c3 and c4. 

(2) The system output spectrum is more sensitive to c4 and less sensitive to c3; 

(3) If the output spectrum with respect to a nonlinear parameter is an alternating series satisfying p i 4 p iþ1 

and p i ! 0, then the system output spectrum may be reduced to zero if additionally the radius of 

convergence for this parameter is sufficiently large. 

(4) The magnitude of output spectrum decreases with the increase of the values of the nonlinear parameters; 

thus, introduction of some simple nonlinear terms into a linear system may greatly improve the 

performance of output frequency response, and the stability of a nonlinear system is not necessarily 

deteriorated with increasing of the values of nonlinear parameters; This also shows that a larger value of 

the parameter for a nonlinear term does not lead to a bad performance of a system. 

(5) For system (22), the nonlinear parameters c3 and c4 can be designed large enough to have an as small as 

possible transmitted force since they correspond to passive elements, and several nonlinear terms in the 

active part can work together to achieve a better performance. 

To demonstrate further the advantage of the OFRF-based analysis and to show more clearly the effect on the 

system output spectrum from several nonlinear parameters together, the OFRF with respect to c1, c2 and c3, i.e., 

Y(jo;c1,c2,c3) is derived. Let c1 2½0; 10 5 Š; c2 2½0; 6 10 5 Š; c3 2½0; 5 10 5 Š; c4 ¼ 500, and the largest order N 

of the output spectrum is determined to be 11, then the parametric characteristic can be obtained as (c ¼ [c1,c2,c3]) 

c ¼ 1; c; c 2 h 

ð11 ; ...; cb i 

1Þ=2c 

l 

¼½1; c1; c2; c3; c 2 1 ; c1c2; c1c3; c 2 2 ; c2c3; c 2 3 ; c3 1 ; c2 1c2; c 2 1c3; c1c 2 2 ; c1c2c3; c 3 2 ; c2 2c3; c2c 2 3 ; c3 3 ; c4 1 ; c3 1c2; c 3 1c3; c 2 1c2 2 , 

c 2 1c2c3; c 2 1c23 ; c1c 3 2 ; c1c 2 2c3; c1c2c 2 3 ; c1c 3 3 ; c42 ; c32 c3; c 2 2c23 ; c2c 3 3 ; c43 ; c51 ; c41 c2; c 4 1c3; c 3 1c22 ; c31 c2c3; c 3 1c23 ; c21 c32 , 

c 2 1c22 c3; c 2 1c2c 2 3 ; c21 c33 ; c1c 4 2 ; c1c 3 2c3; c1c 2 2c23 ; c1c2c 3 3 ; c1c 4 3 ; c52 ; c42 c3; c 3 2c23 ; c22 c33 ; c2c 4 3 ; c53 Š: 

c1 

c4 

c2 

c3

116 



In order to construct the non-singular matrix c, the series of r(N) ¼ 55 different points c ¼ [c1, c2, c3] inqSC ¼ 

fc ¼½c1; c2; c3Šjc1 2½0; 1Š; c2 2½0; 6Š; c3 2½0; 5Šg can be obtained by using a simple stochastic-based searching 

method. In simulations, it is noticed that is easy to find such a series of points that det(c)6¼0. For example, a series of 

points c ¼ [c1, c2, c3] are illustrated in Fig. 5, and it can be obtained in this case that det(c) ¼ 0.08608811188201. It 

can be seen from simulations that it is easy to find a non-singular matrix c with a proper inverse. 

Then following the same procedure as demonstrated above, the OFRF Y(jo; c1, c2, c3) in this case can be 

obtained. The results are shown in Figs. 6 and 7. It can be seen that 

(1) By using the OFRF, the output spectrum can be plotted and analyzed under different combinations of the 

nonlinear parameters c1, c2 and c3. This provides a straightforward understanding of the relationship 

between system output spectrum and model parameters that define nonlinearities. 

6 

5 

4 

3 

2 

1 

0 

0 10 20 30 40 50 60 

Different values of c1, c2 and c3 

Fig. 5. A series of 55 points c ¼ [c1,c2,c3]. 

Fig. 6. Output spectrums with respect to c 1, c 2 and c 3. 

c1 

c2 

c3

(2) The OFRF is varying with different values of c1, c2 and c3. The effect on the output spectrum from any two 

nonlinear terms is not necessarily the simple combination of the contributions from each term respectively. 

Thus the parameters can be analyzed in order to get the best output frequency response performance. The 

OFRF provides a useful basis for this kind of analysis and optimization. 

From the discussions above, it can be concluded that the OFRF-based analysis provides a novel, 

effective and useful approach to the analysis and design of nonlinear Volterra systems in the frequency 

domain. 

6. Conclusions and discussions 



Fig. 7. Output spectrums with respect to any two combinations of c1, c2 and c3. 

Based on some recently developed results, the OFRF-based analysis for nonlinear Volterra systems is 

proposed in this paper. The OFRF-based analysis provides a novel and effective approach to the analysis and 

design of nonlinear Volterra systems in the frequency domain by using the explicit relationship between the 

system output frequency response and model parameters. The OFRF is characterized by its parametric 

characteristic timing a complex valued frequency dependent function vector. Thus in stead of the direct 

analytical computation of the OFRF, the proposed method simplifies the computation of the OFRF by 

splitting the computation procedure into two parts—one is the computation of the parametric characteristics 

for the OFRF, which is analytical in the determination of the relationship between the output spectrum and 

model parameters, and simpler to be carried out, and the other is the determination of the complex valued 

frequency-dependent function vectors, which are obtained by using the least-square method. Some

118 

fundamental results, techniques, and a general procedure for the determination of the OFRF for a given NDE 

or NARX model subject to any specific input signal are provided. Although the proposed method needs r(N) 

simulation data for the numerical method of Process C, and the OFRF obtained by the proposed method is 

not analytical with respect to the input signal and frequency variants at present, the case study for a simple 

mechanical system shows that the OFRF-based analysis is a useful approach to the analysis and design of 

nonlinear Volterra systems in the frequency domain. 

Acknowledgements 

The authors take their gratitude to the anonymous reviewers for their useful and constructive comments 

and suggestions on the manuscript, and also gratefully acknowledge the support of the EPSRC-Hutchison 

Whampoa Dorothy Hodgkin Postgraduate Award, for this work. 

Appendix A. Proof of Proposition 1 

CEðHnðjo1; ...; jonÞÞ 

¼ C0;n 

n 

p¼2 

n 1 n q 

q¼1 p¼1 

0 

B 

@ 

Cp;0 

0 

B 

@ 

Cp;q 

0 

B 

@ 

n pþ1 

0 

B 

@ 

p 

r1 rp¼1 P i¼1 

ri ¼n 

n q pþ1 

p 

r1 rp¼1 P i¼1 

ri ¼n q 

CE Hri ðjorXþ1 ; ...; jorXþr i Þ 

CE H ri ðjorXþ1 ; ...; jorXþr i Þ 

11 

CC 

AA 

11 

CC 

AA. 

ðC1Þ 

C0,n is the first term in Eq. (C1). For clarity, consider a simpler case that there is only output nonlinearities in 

Eq. (C1), then Eq. (C1) is reduced to only the last term of Eq. (C1), i.e., 

n 

CEðHn;pð ÞÞ ¼ p¼2 

n 

Cp;0 

n pþ1 

p 

r1 rp¼1 P i¼1 

ri ¼n 

[24] in order to finish the proof. & 

Appendix B. Proof of Proposition 2 

Cp;0 

p¼2 

CEðHri ð ÞÞ. At this stage, the reader is recommended to refer to Jing et al. 

Regard all other nonlinear parameters as constants or 1. From Proposition 1, if p+q4n then the parameter 

has no contribution to CE(Hn(.)), in this case CE(Hn(.)) ¼ 1 with respect to this parameter. Similarly, if 

p+q ¼ n then the parameter is an independent contribution in CE(Hn(.)), thus CE(Hn(.)) ¼ [1c] with respect to 

this parameter in this case. If p+qon and p40, then the independent contribution in CE(Hn(.)) for this 

ðn1Þ= ðpþq1Þ parameter should be cb c , and the monomials c x for 0pxo n 1=p þ q 1 are all not 

independent contributions in CE(Hn(.)). Hence CEðHnð ÞÞ ¼ 1; c; c2 h i 

ðn1Þ= ðpþq1Þ ; ...; cb c for this case. The 

similar result is held for the case p+qon and p ¼ 0. However, since there should be at least one p40 ina 

complete monomial, thus in this latter case c x 

for any x are not complete, which follows 

CEðHnð ÞÞ ¼ 1; c; c2 h i 

ðn1Þ= ðpþq1Þ ; ...; cb c 1 . The parametric characteristic vector for the nonlinear 

parameter c for all the cases above can be summarized into 

CEðHnð ÞÞ ¼ 1; c; c 2 ; ...; c 

This completes the proof. & 



n 1 

n 1 

pþq 1 dðpÞ posðn qÞ d pþq 1 

n 1 

pþq 1 .

Appendix C. Proof of Lemma 1 

The lemma is summarized by the following observation. For clarity, let I ¼ 3. 

cn ¼ 


c c ... c 

n 

sðjÞn j¼i 

1 for i ¼ 1; 2; 3; 

n ¼ 1½c1 c2 c3Š 1 1 1; 

n ¼ 2½c1 c2 c3Š ½c1 c2 c3Š 3 2 1; 

¼½c1 2 c1 c2 c1 c3 c2 2 c2 c3 c2 3Š n ¼ 3½c2 1 c1 c2 c1 c3 c2 2 c2 c3 c2 3Š ½c1 c2 c3Š 6 3 1; 

¼½c3 1 c21 c2 c2 1 c3 c1 c2 2 c1 c2 c3 c1 c2 3 c3 2 c22 c3 c2 c2 3 c3 3Š n ¼ 4½c3 1 c21 c2 c2 1 c3 c1 c2 2 c1 c2 c3 c1 c2 3 c3 2 c22 c3 c2 c2 3 c3 3Š ½c1 c2 c3Š 10 4 1; 

¼½c4 1 c3 1 c2 c3 1 c3 c2 1 c22 c21 c2 c3 c2 1 c3 2 c1 c3 2c1 c2 2 c3 c1 c2 c2 3 c1 c3 3 

c4 2 c3 2 c3 c2 2 c23 c2 c3 3 c43 Š 

n ¼ 5½c4 1 c3 1 c2 c3 1 c3 c2 1 c22 c21 c2 c3 c2 1 c23 c1 c3 2 c1 c2 2 c3 c1 c2 c2 3 c1 c3 3 

c4 2 c3 2 c3 c2 2 c23 c2 c3 3 c43 Š ½c1 c2 c3Š 15 5 1: 

¼½c5 1 c41 c2 c4 1 c3 c3 1 c22 c3 1 c2 c3 c3 1 c23 c21 c3 2 c21 c22 c3 c2 1 c2 c2 3 c21 c3 3 

c1 c4 2 c1 c3 2 c3 c1 c2 2 c23 c1 c2 c3 3 c1 c4 3 c5 2 c42 c3 c3 2 c23 c22 c3 3 c2 c4 3 c5 3Š sðiÞ n ¼ PI 

To complete the proof, the complete mathematical induction can be adopted. An outline for this proof is 

given here. Note that 

c n ¼ c n 1 c1; ...; c n 1 ½sð1Þn sðiÞn þ 1 : sð1ÞnŠ ci; ...; c n 1 ½sð1ÞnŠ cI 

includes all the non-repetitive terms of form c k1 

1 ck2 

2 c kI 

I with k1 þ k2 þ þkI ¼ n and 0pk1; k2; ...; kIpn. 

These terms can be separated into I parts, the ith part of which, i.e., cn 1½sð1Þn sðiÞn þ 1 : sð1ÞnŠ ci, includes all 

the non-repetitive terms of degree n which are obtained by the parameter ci timing the components of degree 

n– 1inc n 1 from sð1Þn sðiÞn þ 1tosð1Þn. Assume that the lemma holds for step n. Then for the step n+1, the 

ith part of the components in c n+1 must be cn½sð1Þnþ1 ðsðiÞn þ þsðIÞnÞþ1 : sð1Þnþ1Š ci which is 

cn½sð1Þnþ1 sðiÞnþ1 þ 1 : sð1Þnþ1Š ci. This completes the proof of Lemma 1. & 

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