A General Framework for Modeling Sub- and Ultraharmonics of Ultrasound Contrast Agent Signals with MISO Volterra Series

Abstract

Sub- and ultraharmonics generation by ultrasound contrast agents makes possible sub- and ultraharmonics imaging to enhance the contrast of ultrasound images and overcome the limitations of harmonic imaging. In order to separate different frequency components of ultrasound contrast agents signals, nonlinear models like single-input single-output (SISO) Volterra model are used. One important limitation of this model is its incapacity to model sub- and ultraharmonic components. Many attempts are made to model sub- and ultraharmonics using Volterra model. It led to the design of mutiple-input singe-output (MISO) Volterra model instead of SISO Volterra model. The key idea of MISO modeling was to decompose the input signal of the nonlinear system into periodic subsignals at the subharmonic frequency. In this paper, sub- and ultraharmonics modeling with MISO Volterra model is presented in a general framework that details and explains the required conditions to optimally model sub- and ultraharmonics. A new decomposition of the input signal in periodic orthogonal basis functions is presented. Results of application of different MISO Volterra methods to model simulated ultrasound contrast agents signals show its efficiency in sub- and ultraharmonics imaging.

1. Introduction

Medical diagnostic using ultrasound imaging was greatly improved with the introduction of ultrasound contrast agents. In ultrasound imaging, contrast agents are microbubbles [1]. Historically, the important difference between the acoustic impedance of the tissue and the gas encapsulated within the microbubbles was the first step to improve the contrast of echographic images. However, the contrast was still improved by taking into account the nonlinear behavior of microbubbles. In fact, when microbubbles were insonified by a sinusoidal excitation, they respond by generating harmonic components [2]. For example, second harmonic imaging (SHI) [3] consists in transmitting a signal at frequency 𝑓₀ and receiving echoes at twice the transmitted frequency 2𝑓₀. However, harmonic generation during the propagation of ultrasound in the nonperfused tissue limits the contrast [4].

Many years ago, experimental studies have shown the existence of subharmonics at 𝑓₀/2 [5] and ultraharmonics at ((3/2)𝑓₀,(5/2)𝑓₀,…) [6] in the microbubble response under specific conditions of frequency and pressure. The absence of these components in the backscattered signal by the tissue has enabled the introduction of sub- and ultraharmonics as an alternative of the harmonic imaging in order to enhance the contrast. Sub- and ultraharmonic imaging consists of transmitting a signal of frequency 𝑓₀ and extracting components at 𝑓₀/2,(3/2)𝑓₀,(5/2)𝑓₀,….

Many models have been developed to understand the dynamics of the microbubble [2]. Microbubble oscillation can be accurately described using models such as Rayleigh-Plesset modified equation [7–9]. However, to enable optimal separation of harmonic components, other nonlinear models like single-input single-output (SISO) Volterra model have been preferred [10]. A well known limitation of SISO Volterra model is its capacity to model exclusively harmonic components sub- and ultraharmonics are not modeled [11].

To overcome this difficulty, Boaghe and Billings [12] have proposed a multiple-input single-output (MISO) Volterra-based method. Input signals are specified by having sub-harmonic component at frequency 𝑓₀/𝑁. This approach has been applied in ultrasound medical imaging [13].

However, neither Boaghe and Billings [12] nor Samakee and Phukpattaranont [13] have clearly justified the required conditions to design a MISO Volterra decomposition able to model sub- and ultraharmonics.

To answer this untreated point, we propose a more general framework which firstly gives a clear justification regarding the choice of the model and secondly can offer interesting alternatives.

This paper is organized as follows: after recalling Volterra model and presenting the general framework of MISO Volterra methods, simulations of contrast ultrasound medical imaging followed by results are presented. Finally, a discussion completed by a conclusion closes the paper.

2. SISO Volterra Model

Volterra series were introduced like Taylor series with memory [10]. Let 𝑥(𝑛) and 𝑦(𝑛) be, respectively, the input and the output signals in the discrete time domain 𝑛 of the nonlinear system (see Figure 1). The output 𝑦(𝑛) of Volterra model of order 𝑃 and memory 𝑀 is given in [14]. Note that, in our study focused on ultrasound imaging, a third-order Volterra model 𝑃=3 is sufficient for the available transducers bandwidths. The output 𝑦(𝑛) of SISO Volterra model of order 𝑃=3 and memory 𝑀 is given by 𝑦(𝑛)=ℎ₀+^𝑀−1_𝑘1=0ℎ₁𝑘₁𝑥𝑛−𝑘₁+^𝑀−1_𝑘1=0^𝑀−1_𝑘2=0ℎ₂𝑘₁,𝑘₂𝑥𝑛−𝑘₁𝑥𝑛−𝑘₂+^𝑀−1_𝑘1=0^𝑀−1_𝑘2=0^𝑀−1_𝑘3=0ℎ₃𝑘₁,𝑘₂,𝑘₃×𝑥𝑛−𝑘₁𝑥𝑛−𝑘₂𝑥𝑛−𝑘₃, (1) where ℎ_𝑝(𝑘₁,𝑘₂,…,𝑘_𝑝) is the kernel of order 𝑝 of the filter, with 𝑝∈{1,2,3}.

Figure 1 

Block diagram of SISO Volterra model.

Equation (1) could be rewritten as follows 𝐲=𝐗⋅𝐡, (2) where the output signal is: 𝐲=𝑦(𝑀−1),𝑦(𝑀),…,𝑦(𝐿)^𝑇, (3) where 𝐿 is the length of the signal 𝑦(𝑛), and the vector of kernels is 𝐡=ℎ₁(0),ℎ₁(1),…,ℎ₁(𝑀−1),ℎ₂(),ℎ0,0₂(0,1),…,ℎ₂()ℎ𝑀−1,𝑀−1,…,_𝑝(0,0,0),…,ℎ₃(𝑀−1,𝑀−1,𝑀−1^𝑇, (4) where the input matrix is 𝐗=𝐱_𝑀−1,𝐱_𝑀,…,𝐱_𝐿^𝑇, (5) with vector 𝐱_𝑛=[𝑥(𝑛)()(),𝑥,𝑥𝑛−1,…,𝑥𝑛−𝑀+1₂(𝑛),𝑥(𝑛)𝑥(𝑛−1),…,𝑥²(),𝑥𝑛−𝑀+1₃(𝑛)(𝑛)𝑥(𝑛)𝑥()𝑥,𝑥𝑛−1,…,₃(𝑛−𝑀+1)^𝑇, (6) with 𝑛∈{𝑀−1,𝑀,…,𝐿}.

The vector of kernels 𝐡 is calculated to minimize the mean square error (MSE) between 𝑦(𝑛) and 𝑦(𝑛) according to the equation argmin_𝐡𝔼𝑦(𝑛)−𝑦(𝑛)², (7) where 𝔼 is the symbol of the mathematical expectation.

Vector 𝐡 is calculated using the least squares method 𝐡=𝐗^𝑇𝐗⁻¹𝐗^𝑇𝐲, (8) if (𝐗^𝑇𝐗) is invertible. Otherwise, regularization techniques can be used.

Nevertheless, as reported in [12], it is not possible to model sub- and ultraharmonics with SISO Volterra model under this formulation. This is due to the fact mentioned in [12] that SISO Volterra model can only model frequencies at integer multiples of the input frequency.

To overcome this limitation, Boaghe and Billings [12] proposed a MISO Volterra-based solution and not any more a SISO Volterra. This point is discussed in Section 3.

3. General Framework of MISO Volterra Model

According to Boaghe and Billings’ claims [12], it is possible to model sub- and ultraharmonic components of the signal 𝑦(𝑛) if the excitation signal to Volterra model has the sub-harmonic component at 𝑓₀/𝑁. The solution proposed by Boaghe and Billings [12] to show up the sub-harmonic component at frequency 𝑓₀/𝑁 is to decompose the input signal 𝑥(𝑛) into multiple input signals 𝑥_𝑖(𝑛), each signal having frequency components at 𝑓₀ and 𝑓₀/𝑁. From our point of view, Boaghe and Billings’ approach [12] claimed two conditions that are intrinsically coupled by the choice of the decomposition method as follows:(i)the input signal to Volterra model has sub-harmonic frequency component at 𝑓₀/𝑁;(ii)Volterra system is a MISO system described by 𝑥(𝑛)=^𝑁_𝑖=1𝑥_𝑖(𝑛). (9)

The block diagram of MISO Volterra model is presented in Figure 2.

Figure 2 

Block diagram of MISO Volterra model.

A third condition that is not really explained in [12], however, it is a crucial condition to carry out this modeling procedure. It is the orthogonality condition between each multiple input of MISO Volterra model. Taking into account this third condition makes it possible to generalize Boaghe and Billings’ approach presented in [12] as follows: 𝑥𝑛,𝑓₀=^𝑁_𝑖=1𝑥_𝑖𝑛,𝑓₀,𝑁=^𝑁_𝑖=1𝛼_𝑖Ψ_𝑖𝑛,𝑓₀,𝑁, (10) where 𝛼_𝑖 are coefficients to be adjust and Ψ_𝑖(𝑛,𝑓₀,𝑁) is the periodic orthogonal basis functions having a spectral component at 𝑓₀/𝑁. Different bases could be proposed. In this study, two bases are presented as follows.(1)In [12] a first periodic basis of orthogonal functions is proposed as follows: Ψ_𝑖𝑛,𝑓₀,𝑁=𝑥𝑛,𝑓₀∗^+∞_𝑘=−∞Rect_1/𝑓0𝑛𝑇_𝑠−𝑘𝑁+𝑖−1𝑓₀, (11) where 𝑇_𝑠 is the sampling period, ∗ represents the convolution product, and Rect_1/𝑓0(𝑛) is the rectangular function equal to 1 when −1/2𝑓₀<𝑛<−1/2𝑓₀ and equal to zero otherwise. Note that this approach is MISO1. (2)In the present work, a new periodic basis of orthogonal functions is presented as follows: Ψ_𝑖𝑛,𝑓₀,𝑁=𝑥𝑛,𝑓₀+(−1)^(𝑖−1)×𝑥𝑛,𝑓₀cos𝑛𝑇_𝑠𝑤₀𝑁−1𝑁+𝑥𝑛,𝑓₀sin𝑛𝑇_𝑠𝑤₀𝑁−1𝑁, (12) where 𝑥(𝑛)=ℋ(𝑥(𝑛)) is the Hilbert transform of 𝑥(𝑛) and 𝑤₀=2𝜋𝑓₀. Note that this second is MISO2 approach.

For our application in contrast medical imaging, the sub-harmonic frequency is 𝑓₀/2 [5–7], so 𝑁=2.

As an illustration, when 𝑥(𝑛)=𝐴cos(𝑤₀𝑛𝑇_𝑠) and 𝑁=2, the decomposition is written: (1)for the first basis, as follows: 𝑥(𝑛)=𝑥₁(𝑛)+𝑥₂(𝑛)=𝛼₁Ψ₁𝑛,𝑓₀,2+𝛼₂Ψ₂𝑛,𝑓₀,2, (13) with 𝛼₁=𝛼₂=1, and Ψ₁𝑛,𝑓₀,2=𝐴cos𝑤₀𝑛𝑇_𝑠∗^+∞_𝑘=−∞Rect_1/𝑓0𝑛𝑇_𝑠−2𝑘𝑓₀,Ψ₂𝑛,𝑓₀,2=𝐴cos𝑤₀𝑛𝑇_𝑠∗^+∞_𝑘=−∞Rect_1/𝑓0𝑛𝑇_𝑠−2𝑘+1𝑓₀, (14)(2)and for the second basis, as follows: 𝑥(𝑛)=𝑥₁(𝑛)+𝑥₂(𝑛)=𝛼₁Ψ₁𝑛,𝑓₀,2+𝛼₂Ψ₂𝑛,𝑓₀,2, (15) with 𝛼₁=𝛼₂=1/2, and Ψ₁𝑓𝑛,⁰2=𝐴cos𝑤₀𝑛𝑇_𝑠∗²_𝑞=1𝛿𝑛𝑇_𝑠𝑞,Ψ₂𝑓𝑛,⁰2=𝐴cos𝑤₀𝑛𝑇_𝑠∗²_𝑞=1(−1)^(𝑞−1)𝛿𝑛𝑇_𝑠𝑞, (16) where 𝛿(𝑛) is the Dirac function. Finally, Ψ₁(𝑛,𝑓₀/2) and Ψ₂(𝑛,𝑓₀/2) can be simply rewritten as follows: Ψ₁𝑛,𝑓₀,2=𝐴cos𝑤₀𝑛𝑇_𝑠𝑤+𝐴cos⁰2𝑛𝑇_𝑠,Ψ₂𝑛,𝑓₀,2=𝐴cos𝑤₀𝑛𝑇_𝑠𝑤−𝐴cos⁰2𝑛𝑇_𝑠. (17)

The two signals 𝑥₁(𝑛) and 𝑥₂(𝑛) for the two previous bases are represented in Figure 3.

(a)

(b)

Figure 3 

From top to bottom: input signal 𝑥, modified inputs 𝑥1, and 𝑥2, and the product 𝑥1𝑥2, (a) for the rectangular basis (basis 1) and (b) for the new basis (basis 2).

It is obvious to show that for the two bases, the signals 𝑥₁(𝑛) and 𝑥₂(𝑛) are orthogonal because ∑𝑥₁(𝑛)𝑥₂(𝑛)=0 (From a statistical point of view, the two signals 𝑥₁(𝑛) and 𝑥₂(𝑛) are orthogonal if and only if 𝔼[𝑥₁(𝑛)𝑥₂(𝑛)]=0. If 𝑥₁(𝑛) and 𝑥₂(𝑛) are stationary and ergodic, then 𝔼[𝑥₁(𝑛)𝑥₂(𝑛)]=∑(𝑥₁(𝑛)𝑥₂(𝑛)).). The algebraic area of the signal 𝑧(𝑛)=𝑥₁(𝑛)𝑥₂(𝑛), shown in Figure 3, is equal to zero.

Finally, if the components 𝑥_𝑖(𝑛) are orthogonal to each other, then this also means that the output of Volterra model 𝑦(𝑛) can be decomposed as follows: 𝑦(𝑛)=^𝑁_𝑖=1𝑦_𝑖(𝑛), (18) where the components 𝑦_𝑖(𝑛) are also orthogonal to each other. A proof of this propriety is given in Appendix A.

The consequence of this statement is that MISO Volterra model can be considered as 𝑁 parallel SISO Volterra models as depicted in Figure 4.

Figure 4 

Block diagram of orthogonal MISO Volterra model.

4. Simulations

To validate the different proposed bases and to quantify its performances for application in contrast ultrasound medical imaging, realistic simulations are proposed. To carry out the simulations, the free simulation program BUBBLESIM developed by Hoff [7] was used to calculate the oscillations and scattered echoes for a specified contrast agent and excitation pulse. A modified version of Rayleigh-Plesset equation was chosen. The model presented by Church [15] and then modified by Hoff [7] is based on the theoretical description of microbubbles as air-filled particles with surface layers of elastic solids. In order to simulate the mean behavior of a microbubble cloud, we hypothesized that the response of a cloud of 𝑁_𝑏 microbubbles was 𝑁_𝑏 times the response of a single microbubble with the mean properties.

The incident burst to the microbubble is a sinusoidal wave of frequency 𝑓₀=4 MHz (The resonance frequency of a microbubble of 1.5 μm is about 2.25 MHz. Therefore, the emission frequency at 4 MHz is nearly the double of the resonance frequency.) To ensure the presence of sub- and ultraharmonics with moderate destruction of microbubbles, Forsberg et al. have proposed in [16] a pressure range from 1.2 MPa to 1.8MPa. To limit the destruction of microbubbles, we set the pressure level to the lowest value at 1.2MPa. The burst consists of 18 cycles. The sampling frequency is 𝑓_𝑠=60MHz. The parameters of the microbubble are given in the Table 1 [13].

Table 1 

The parameters of microbubbles [13].

5. Results

In this research, the performances of different modeling methods are evaluated qualitatively and quantitatively.

5.1. Qualitative Evaluation

To evaluate qualitatively the two MISO methods, MISO1 (with the basis proposed in [12]) and MISO2 (with the new basis proposed in the present work) with respect to SISO Volterra method, temporal representations of 𝑦(𝑛) and 𝑦(𝑛), and spectral representations |𝑌(𝑘)|² and |𝑌(𝑘)|² of the nonlinear system backscattered by the contrast agent in nonlinear mode are presented in Figure 5.

(a)

(b)

Figure 5 

(a) Comparison between the backscattered signal by the microbubble 𝑦(𝑛) (black) and its estimation 𝑦(𝑛) (green): (top) the modeled signal with SISO Volterra model, (middle) MISO1 method, and (bottom) MISO2 method. (b) Spectra of different signals are presented in (1). Here SNR=∞ dB, 𝑃=3, and 𝑀=19.

Results presented in Figure 5 are obtained for a signal to tissue ratio SNR=∞ and using Volterra model of order 𝑃=3 and memory 𝑀=19.

To better distinguish the different harmonic components of ultrasound signal, six cycles of 0.05 μs are presented in Figure 5(a), and a bandwidth of 13MHz covering the 3 harmonics potentially accessible in ultrasound imaging is presented in Figure 5(b). For both types of representations, the fundamental frequency, harmonics, sub- and ultraharmonics are well apparent. In Figure 5(a) (top), only harmonic components at 𝑓₀, 2𝑓₀, and 3𝑓₀ are modeled by SISO Volterra. This result confirms that SISO Volterra system is unable to correctly model sub- and ultraharmonics at frequencies 𝑓₀/2,(3/2)𝑓₀, and (5/2)𝑓₀. In Figure 5 (middle, bottom), all the spectral components are correctly modeled validating the two MISO approaches.

5.2. Quantitative Evaluation

To determine accurately the performances of the two methods and to know which Volterra approach provides the best performances a quantitative study is necessary. The relative mean square error (RMSE) defined as follows: 𝔼𝑦(𝑡)(𝑡)RMSE=−𝑦²𝔼𝑦(𝑡)² (19) is evaluated for different noise levels at the system output. The noise level, adjusted as a function of SNR, is Gaussian and white. Ten realizations are made to evaluate the fluctuations of RMSE. RMSE for SNR=∞,20,15, and 10dB is reported in Figure 6. A zoom in Figure 6(d) shows the fluctuations of the EQMR around a mean value.

(a)

(b)

(c)

(d)

Figure 6 

Variation of the RMSE in dB between the modeled signal with SISO Volterra (blue), MISO1 method (green), and MISO2 method (black) and the backscattered signal by the microbubble as a function of the memory of Volterra model in the presence of noisy output: (a) SNR=∞ dB, (b) SNR=20dB, (c) SNR=15dB, and (d) SNR=10dB.

The main result of these simulations shows that regardless the SNR values, MISO Volterra methods provide a much better RMSE than SISO Volterra method. In fact, a gap between SISO Volterra method and the two methods MISO1 and MISO2 going from 5 to 16dB can be obtained depending on the SNR conditions. These results confirm that SISO Volterra method is not suitable for sub- and ultraharmonic modeling. A zoom on Figure 6(d) emphasizes the small fluctuations of the RMSE. This result shows the robustness of the two MISO Volterra approaches towards noise.

Note that the RMSE obtained with the two MISO Volterra approaches are similar and follows the same trend. However, a small advantage in favor of MISO2 method with respect to MISO1 method for memory values 𝑚 smaller than 6 is noted.

Finally, the more the memory increases, the more the RMSE decreases, indicating that the different methods tend asymptotically toward the optimal solution.

6. Discussions and Conclusions

In the present research, we proposed a general framework that describes harmonic, sub-, and ultraharmonics modeling using Volterra decomposition. This framework allowed us to highlight three essential criteria instead of two, to accurately model sub- and ultraharmonics:(i)as suggested in [12], the basis should be periodic of period 𝑓₀/𝑁; (ii)as suggested in [12], Volterra system should be a MISO system; (iii)as suggested in this work, the decomposition of the input signal to Volterra model 𝑥(𝑛) must be done with an orthogonal basis.

This general framework has also justified the different steps of the decomposition thus allowing to propose new periodic orthogonal bases more efficient. It is the same for the choice of the order of Volterra model, which was limited to three. In fact, for more or less severe constraints on the ultrasound transducers bandwidth, the order can be reduced or increased.

This more general formulation provides a methodological basis for optimal sub- and ultraharmonics contrast imaging and opens a new research axis for more efficient periodic orthogonal bases of MISO Volterra systems and also for new MISO systems based on Hammerstein models or Wiener models.

Appendix

A. Decomposition of MISO Volterra Model of 2 Inputs to 2 SISO Volterra Models

A MISO Volterra model with 𝑁 inputs is equivalent to 𝑁 SISO Volterra models if and only if the mean square error to be minimized between 𝑦(𝑛) and 𝑦(𝑛) is the same in both cases. We will determine the conditions that must be satisfied by the inputs 𝑥₁(𝑛) and 𝑥₂(𝑛) of MISO Volterra when 𝑁=2, to have this equivalence.

Volterra kernels are calculated using the least squares method by minimizing the mean square error (MSE) between 𝑦(𝑛) and the modeled signal 𝑦(𝑛): 𝔼𝑦(𝑛)−𝑦(𝑛)². (A.1)

For MISO Volterra model, the decomposition of 𝑥(𝑛) into 𝑥₁(𝑛) and 𝑥₂(𝑛) such that 𝑥(𝑛)=𝑥₁(𝑛)+𝑥₂(𝑛) requires that 𝑦(𝑛)=𝑦₁(𝑛)+𝑦₂(𝑛). It follows that 𝑦(𝑛)=𝑦₁(𝑛)+𝑦₂(𝑛). The error to be minimized is 𝔼𝑦(𝑛)−𝑦(𝑛)²=𝔼𝑦(𝑛)²+𝔼𝑦(𝑛)²𝑦(𝑛)𝑦(𝑛𝑦−2𝔼)=𝔼₁(𝑛)+𝑦₂(𝑛)²+𝔼𝑦₁(𝑛)+𝑦₂(𝑛)²−2𝔼𝑦₁(𝑛)+𝑦₂(𝑛)𝑦₁(𝑛)+𝑦₂(𝑛)=𝔼𝑦₁(𝑛)²+𝔼𝑦₂(𝑛)²+2𝔼𝑦₁(𝑛)𝑦₂(𝑛𝑦)+𝔼₁(𝑛)²+𝔼𝑦₂(𝑛)²+2𝔼𝑦₁(𝑛)𝑦₂(𝑛𝑦)−2𝔼₁(𝑛)𝑦₁(𝑛)−2𝔼𝑦₁(𝑛)𝑦₂(𝑛𝑦)−2𝔼₂(𝑛)𝑦₁(𝑛)−2𝔼𝑦₂(𝑛)𝑦₂(𝑛). (A.2)

For the 2 SISO Volterra models of inputs 𝑥₁(𝑛) and 𝑥₂(𝑛) and outputs 𝑦₁(𝑛) and 𝑦₂(𝑛), respectively, the error to be minimized is 𝔼𝑦₁(𝑛)−𝑦₁(𝑛)²+𝔼𝑦₂(𝑛)−𝑦₂(𝑛)²=𝔼𝑦₁(𝑛)²+𝔼𝑦₁(𝑛)²−2𝔼𝑦₁(𝑛)𝑦₁(𝑛𝑦)+𝔼₂(𝑛)²+𝔼𝑦₂(𝑛)²−2𝔼𝑦₂(𝑛)𝑦₂(𝑛). (A.3)

A MISO Volterra model could be seen as 2 SISO Volterra models if (A.2) and (A.3) are equal. This equality gives 𝔼𝑦₁(𝑛)𝑦₂(𝑛)+𝔼𝑦₁(𝑛)𝑦₂(𝑛𝑦)−𝔼₁(𝑛)𝑦₂(𝑛)−𝔼𝑦₂(𝑛)𝑦₁(𝑛)=0. (A.4)

One possible solution is that each term of the equation is equal to zero: 𝔼𝑦₁(𝑛)𝑦₂(𝑛𝔼𝑦)=0,₁(𝑛)𝑦₂(𝑛𝔼𝑦)=0,₁(𝑛)𝑦₂(𝑛𝔼𝑦)=0,₂(𝑛)𝑦₁(𝑛)=0. (A.5)

Therefore 𝑦₁(𝑛)⟂𝑦₂(𝑛),𝑦₁(𝑛)⟂𝑦₂(𝑛),𝑦₁(𝑛)⟂𝑦₂(𝑛),𝑦₂(𝑛)⟂𝑦₁(𝑛), (A.6) where ⟂ means orthogonal. Elsewhere, 𝑦₁(𝑛) and 𝑦₂(𝑛) are calculated according to (1). That implies that 𝔼𝑦₁(𝑛)𝑦₂(𝑛)=𝔼^𝑀−1_𝑘1=0ℎ₁𝑘₁𝑥₁𝑛−𝑘₁^𝑀−1_𝑘1=0ℎ^′₁𝑘₁𝑥₂𝑛−𝑘₁+^𝑀−1_𝑘1=0^𝑀−1_𝑘2=0ℎ₂𝑘₁,𝑘₂𝑥₁𝑛−𝑘₁𝑥₁𝑛−𝑘₂×^𝑀−1_𝑘1=0^𝑀−1_𝑘2=0ℎ^′₂𝑘₁,𝑘₂𝑥₂𝑛−𝑘₁×𝑥₂𝑛−𝑘₂+⋯=0. (A.7)

One possible solution is that each term of the equation is equal to zero. For the first term, we get 𝔼^𝑀−1_𝑘1=0ℎ₁𝑘₁𝑥₁𝑛−𝑘₁^𝑀−1_𝑘1=0ℎ^′₁𝑘₁𝑥₂𝑛−𝑘₁=^𝑀−1_𝑘1=0ℎ₁𝑘₁ℎ^′₁𝑘₁𝔼𝑥₁𝑛−𝑘₁𝑥₂𝑛−𝑘₁=0. (A.8)

The last equation implies that 𝑥₁(𝑛)⟂𝑥₂(𝑛). For the other terms in (A.7), we obtain the same conclusion 𝑥₁(𝑛)⟂𝑥₂(𝑛). Therefore, 𝑦₁(𝑛) and 𝑦₂(𝑛) are orthogonal if 𝑥₁(𝑛) and 𝑥₂(𝑛) are also orthogonal.

Elsewhere, if 𝑦₁(𝑛) and 𝑦₂(𝑛) are the estimations of 𝑦₁(𝑛) and 𝑦₂(𝑛), then 𝑦(𝑛)≈𝑦₁(𝑛) and 𝑦₂(𝑛)≈𝑦₂(𝑛). This means that the orthogonality of 𝑦₁(𝑛) and 𝑦₂(𝑛) implies the orthogonality of each couple formed by the four signals presented in (A.5). This is true if and only if 𝑥₁(𝑛)⟂𝑥₂(𝑛).

Therefore, a MISO Volterra model with two inputs could be treated as two SISO Volterra models if the two inputs are orthogonal. This result could be generalized for MISO Volterra model with 𝑁 inputs.

Acknowledgment

The authors would like to thank the Lebanese council of scientific research (CNRSL) for financing this work.