Volterra Frequency Response Functions Analysis of ......C. Samakee and P. Phukpattaranont 17...

C. Samakee and P. Phukpattaranont 17

Volterra Frequency Response FunctionsAnalysis of Subharmonic Oscillations from

Bubble

Chinda Samakee 1 and Pornchai Phukpattaranont 2 ,

ABSTRACT

Subharmonic images are becoming interestingly inmedical ultrasound, as the improvement of contrast-to-tissue ratio (CTR). However, an important partfor image enhancement is processing techniques.In this study, frequency response functions (FRFs)of multiple-input-multiple-output (MIMO) Volterrasystems were used to investigate potentially possiblein separating subharmonic frequency for the qualityof the images. The motivation for this research isthat Volterra filter application in separating subhar-monic component need analysis which is FRFs (e.g.linear and quadratic Volterra kernels). The theme inthis study is the analysis of FRFs to investigate pos-sible subharmonic filtering of bubble frequency. Ma-terial of an experimental subharmonic represented byChurchs model was the excitation amplitude pressure0.6 MPa at twice of resonance frequency. Simulationresults of Volterra kernel are analyzed and displayedin graphs of FRFs. In conclusion, this study demon-strated the cause of multi-frequency response of FRFsand the direction in filtering the subharmonic withVolterra filter.

Keywords: subharmonic oscillation, bubble, non-linear system identification, MIMO Volterra series

1. INTRODUCTION

To increase the effectiveness of medical ultrasoundimaging, ultrasound contrast agents (UCAs) are usedfor improving diagnostic capabilities. UCAs basedsuspensions or gas bubbles have been developed forenhancing acoustic properties by the injection ofsmall bubbles with diameters less than 10 µm intothe blood flow in the body [1]. When bubble is ex-cited by an ultrasound at one frequency fs, echo sig-nals include the frequency components of harmonic(2fs, 3fs, . . .), subharmonic (fs/2, fs/3, . . .) and ul-traharmonic (3fs/2, 5fs/2, . . .) [2]. This nonlinearoscillation can be applied for ultrasound imaging byimaging techniques [3].

Manuscript received on May 14, 2013 ; revised on November7, 2013.1 The author is with Telecommunication Engineering Pro-

gram, the Department of Electrical Engineering, RajamangalaUniversity of Technology Srivijaya, Songkhla, Thailand, email:[email protected] The author is with the Department of Electrical Engineer-

ing, Prince of Songkla University, Songkhla, Thailand, email:[email protected]

Imaging techniques have designed in formattingultrasound images with the second harmonic fre-quency (2fs) in past years, such as bandpass filter(BPF), pulse inversion (PI) and Volterra filter [4], [5].From three methods, the Volterra filter provides highcontrast-to-tissue ratio (CTR) comparable to BPFand PI in separating 2fs echoes of UCA from tissueregion of fundamental components (fs). It has beendemonstrated in [5]. Many research groups have beeninvestigated the subharmonic (fs/2) generations frombubble in order to investigate specific responses andpotential use in imaging [6]-[9]. A main advantageof the use of fs/2 for subharmonic imaging, due tothe subharmonic generation in only contrast agentsregion, but the deficiency of generation in tissues re-gion, thus the CTR at fs/2 is better than at 2fs [10].

In this work, we are interested in potential towardsof the fs/2 and Volterra filter to increased CTR forimaging applications. The Volterra filter is parallelfilter on operators of linear, quadratic, cubic, etc.,which its output is independent of each operators [11].The use of Volterra filter can be categorized into twoprocedures: modeling and separating signals. Gener-ally, for a single-input-single-output (SISO) Volterraseries cannot be used for modeling the behavior ofsubharmonic oscillation case. However, previous re-search by Boaghe and Billings has presented a newidea for modeling subharmonic systems, which over-comes these problems by using multiple-input-single-output (MISO) Volterra series [12].

Recently, subharmonic modeling of bubble oscilla-tion using MIMO Volterra models has been investi-gated. In [13], it demonstrated the impulse responsefunctions and the frequency response in linear ker-nel. We show here the frequency response functions(FRFs) of each operators to analyze possible separa-tion of subharmonic frequency. Additionally, we ex-pand this study of motive of multi-frequency responseof the FRFs.

The remainder of this paper presents analytical re-sults of FRFs from MIMO Volterra models to findsolutions for separating subharmonic. Simulation re-sults for backscatter from a bubble and kernel proce-dures are also presented.

2. VOLTERRA SERIES

A mathematical model for the description of dy-namic systems is the Volterra series, which can repre-sent nonlinear dynamic systems. The Volterra series

18 INTERNATIONNAL JOURNAL OF APPLIED BIOMEDICAL ENGINEERING VOL.6, NO.1 2013

model can be written by an explicit input-output re-lation of the form of convolution summation [11]

y(n) = h0 +N∑i=1

M−1∑m1=0

· · ·M−1∑mi=0

hi(m1,m2, · · · ,mi)

×x(n−m1) · · ·x(n−mi), (1)

where y(n) the output from the model; x(n − mi)the input to the system; hi(m1,m2, . . . ,mi) i

th-orderVolterra kernels; N the maximum order of system; Ma memory length.

With the kernels or the ith-order impulse responsefunctions (IRFs) of the system, which are useful infinding system properties as sets of input and out-put data. Usually it is studied in frequency domainfor applying independent operations by each trans-fer functions. The Fourier transform of the kernelshi(m1,m2, . . . ,mi) in (1) can be written as

Hi(f1, . . . , fi) =M−1∑m1=0

. . .M−1∑mi=0

hi(m1,m2, . . . ,mi)

×e−(j 2πM m1p+...+j 2π

M mip), (2)

p ∈ {0, 1, . . . , N − 1} ,

where is called the ith-order FRFs of the system. TheFRF is used in studies on only filtering subharmonicwith Volterra kernel to analyze the system frequencyresponse, and looks new methodologies.

For the kernel estimation in Equ. (1), the Volterramodel system is written in linear algebra equationform, let y(n), x(n−mi), and hi(m1,m2, · · · ,mi) beoutput vector y, input matrix X and Volterra kernelvector h respectively, and L is length of the inputsequence.

y = Xh, (3)

where

y =

y(0)

y(1)

y(2)

...

y(n)

X =

1 x(0) x(−1) . . . xi(n−M + 1)

1 x(1) x(0) . . . xi(n−M + 2)

1 x(2) x(1) . . . xi(n−M + 3)

......

.... . .

...

1 x(n) x(t− 1) . . . xi(n−M + L)

h =

h0

h1(0)

h1(1)

...

hi(M − 1, . . . ,M − 1)

The Volterra kernel of system in Equ. (3) can besolved a the set of the ith-order Volterra series,

h = X†h. (4)

The notation on matrix X† is generalized inversewhich defined by performing of the singular value de-composition (SVD) method [14], which is given by

X† = (USVT) =r∑

i=1

σiuivTi . (5)

Then, the solution of the Volterra kernel is given by

h = (USVT)†y =r∑

i=1

vi1

σiuTi y, (6)

where S is a M × N diagonal matrix with sin-gular values sorted in descending order σ1 ≥σ2 ≥ σ3 . . . ≥ σr ≥ 0 (r = min{M,N}).U(M×N) and V(N×N) are unitary matrices,which U(u1, . . . ,uM) and V(v1, . . . ,vM) are or-thonormal columns.

3. MATERIALS AND METHODS

In this section, we explain materials process andsimulation methods in studying. First, materials con-tain bubble representation of generating subharmonicand receiving echoes from single bubble. Last, meth-ods contain the procedures for finding kernel andmodeling of subharmonic oscillation with Volterra se-ries are detailed.

3.1 Materials

3.1...1 Bubble

A theoretical representation of an encapsulatedbubble motion can be explained using Church model[15] to describe the dynamic behaviour of non-linearoscillation in a liquid. The equation of the model iswritten as

ρL

(R

′′R+

3

2(R

′)2)

= p0

((R0

R

)3κ

− 1

)(7)

−Pin(t)− 4µLR

′

R

−12µSdSR

20

R3

R′

R

−12GSdSR

20

R

(1− R0

R

),

where R, R′, and R

′′are, respectively, the radius of

bubble, the first and second time derivatives of thebubble radius; R0 is equilibrium radius of the bubble,dS is the shell thickness, GS is the shear modulus, ρLis the density of liquid, p0 is the atmospheric, µL isthe shear viscosity in surrounding medium, µs is the


Church model for

bubble motion

Conversion to

scatter pressure

Pin Psp R

Fig.1:: The block diagram of subharmonic simula-tion from bubble.

Fig.2:: MIMO Volterra series structure for findingthe kernel of subharmonic system.

shear viscosity of shell and κ is the polytropic expo-nent of the gas. The resonant frequency of bubbleis

f0 =ω0

2π=

1

2πR0

√1

ρL

(3κp0 + 12GS

dSR0

).(8)

The acoustic excitation of oscillation amplitude pres-sure Pin(t) is given by sinusoidal waveform:

Pin(t) = pAcos(2πfSt) (9)

where pA is peak pressure amplitudes, fS is the trans-mit frequency and t is the duration time of the inci-dent pressure.

The solution of Equ. 7 for the radial oscillations bythe fourth-order Runge-Kutta method at a samplingfrequency 60 MHz and the initial conditions used areR = R0 and R

′= 0 at time t = 0. In this study,

all the parameters used of bubble for simulation arelisted in Table I [15]. For the driving pressure wave-form, Pin(t), the parameters used in the numericalsimulation are as follows: pA = 0.6 MPa, fS = 2MHz, f0 = 4.5 MHz and t = 40µs.

3.1...2 Formulation of Predicted Echoes

Signals from bubble representation of the Churchequation, which has been modeled for the bubble ra-dius (R, R

′, and R

′′), are used for calculating the

received echo. The scattered pressure PSP (t) of bub-ble echoes can be expressed as [16]

PSP (t) = ρLR

r(2R

′2 +RR′′) (10)

where r is a distance from the scattered pressure tothe center of the bubble at 6 cm. For the acousti-cal simulation system, including simulation of subhar-monic generation of bubble and formulation of pre-dicting echo is shown in Fig. 1.

Fig.3:: MISO Volterra series structure for the mod-eling subharmonic oscillation.

Fig.4:: Inspection of the input-output waveforms toset the time interval at the same period as subhar-monic oscillation.

Fig.5:: Decomposition of the original input signalPin(t) into xnew

1 and xnew2 signals.


Fig.8:: Quadratic filter kernels: (top) Hxnew1

1Q in Equ. (17); Hxnew3

2Q (bottom) in Equ. (19)

3.2 Methods

3.2...1 MIMO Volterra System Kernel

The subharmonic system of bubble has been iden-tified to the frequency-domain analysis by the MIMOVolterra models [12], [13]. The identification proce-dures of kernels are as follows:

1) Generate subharmonic of behaviour of oscillatingbubble by numerical solution of the Church model inEqu. 7 and transformed into acoustic pressure of echothrough Equ. 10.2) Compare the period of waveform signals betweenPin(t) and PSP (t) to segment time interval by nTi >TS , where Ti is time interval, Ts is sampling periodand n = 2, 3, . . .3) Create the signals Pin(t) and PSP (t) to be periodic

in nTi into an n dimensional signal as rules: Equ. (11)

Pin(mod)(t), PSP (mod)(t) =

{x(t), y(t) t∈[(i−1)TS ;iTS

0 otherwise(11)

where i = 1, 2, . . . , n, x(t) and y(t) are the sequencescomposed of Pin(t) and PSP (t), respectively by x(t),y(t) = 0 ≤ TS ≤ nTi. From rule in Equ. (11), thePin(t) and PSP (t) signals are divided into the nth

order, that are,

Pin(mod)(t) = xnew1 (t)+xnew

2 (t)+ . . .+xnewi (t) (12)

and

PSP (mod)(t) = ynew1 (t)+ynew2 (t)+. . .+ynewi (t). (13)


Fig.6:: Decomposition of the original scattered pres-sure signal PSP (t) into ynew1 and ynew2 signals.

Fig.7:: Linear filter kernels Hx1new1L (solid line) in

Equ. (17) and Hx2new2L (dotted) in Equ. (19). (top)

Gain; (bottom) Phase.

4) Implement the system of MIMO Volterra modelson each individual decomposed inputs xnew

i and out-puts ynewi to determine the Volterra kernels.

The procedural setup of solving MIMO Volterrakernels from bubble simulated is illustrated in Fig. 2.

3.2...2 MISO Volterra Modeling

After obtaining the kernels from the MIMOVolterra models, the modeling of subharmonic oscilla-tion can be modeled using a MISO Volterra model asshown in Fig. 3. This model is composed of multipli-cation of a input multiplexer M with the obtainedkernels, and combination of each output Volterramodels by the single system S.

4. SIMULATION RESULTS AND ANALY-SIS

4.1 Generating and Modeling of Subharmonic

Fig. 4 (bottom) shows the variation of drivingacoustic pressure input Pin(t) with time into bubblerepresentation. Fig. 4 (top) shows the result of thescattered pressure PSP (t) from the bubble. Fig. 4(top) is compared with Fig. 4 (bottom), in order toknow the number of the time interval nTi as the sub-harmonic oscillation. The same period for the datauses of Pin(t) and PSP (t) are also plotted with marklines. It is seen that the double use of nTi is 2Ti.Figs. 5 and 6 show samples of produced the input[xnew

1 ; xnew2 ] and the output [ynew1 ; ynew2 ], compo-

nents xnewi and y2i for modified into two-dimension as

the rule in Equ. (11) for decompositions of Pin(t) andPsp(t), respectively.

Then the estimation of filter kernel for sub-harmonic systems can be expressed in an inputxnew1 /output ynew1 relationship corresponds to the fol-

lowing MIMO second-order Volterra series (N = 2)equation:

ynew1 (n) = h1b +

M−1∑m1=0

h1L(m1)xnew1 (n−m1)

+M−1∑m1=0

M−1∑m2=0

h1Q(m1,m2)xnew1 (n−m1)x

new1 (n−m2)

(14)

where h1b, h1L and h1Q are, respectively, bias, linearquadratic and cubic Volterra kernels for the compo-nents xnew

1 and ynew1 . The calculation of the firstkernel was shown to be

h1 = X†1y1 (15)

By the output ynew1 (n) after h1 solutions, it can berewritten in form the sum of the output of two differ-ent multivariate cross-term nonlinear subsystems.

ynew1 (n) = h1b + y1L(n) + y1Q(n) (16)


where the terms y1L, and y1Q denote the output oflinear Similarly, MIMO second-order Volterra seriesfor the components xnew

2 and ynew2 can be modeledas

ynew2 (n) = h2b +M−1∑m1=0

h2L(m1)xnew2 (n−m1)

+M−1∑m1=0

M−1∑m2=0

h2Q(m1,m2)xnew2 (n−m1)x

new2 (n−m2)

(17)

where h2b, h2L and h2Q are bias, linear and quadraticVolterra kernels, respectively. We obtain the kernelsby

h2 = X†2y2 (18)

And the output ynew2 (n) in form the sum of transferfunctions is

ynew2 (n) = h2b + h2L(n) + h2L(n) (19)

The terms y2L and y2Q denote the output of linearand quadratic transfer functions, respectively. Fi-nally we get the expression of MISO Volterra modelas:

y(n) = ynew1 (n) + ynew2 (n) (20)

where y(n) is the response of MISO Volterra outputof the modeling of subharmonic oscillation.

4.2 Frequency Response of Kernels Obtained

After the decomposition of each individual input-output (terms of the types ynew1 , xnew

1 and ynew2 ,xnew2 ) is successful, the resulting estimated linear

and quadratic kernels can be obtained by the MIMOVolterra models. They are obtained from solving theEqu. (15) and Equ. (18). Impulse response identifi-cation represent an implicit and in variant propertyof the underlying system state. They are used as fre-quency domain analysis to aid in filter applicationsfor characterizing subharmonic oscillation.

The FRFs are obtained using the Fourier trans-forms of kernels. The frequency response characteris-tics of linear and quadratic filter kernels with a serieslength M = 22 is plotted in Figs. 7-8. Fig. 7 shows

the linear functions Hxnew1

1L and Hxnew2

2L for the mod-els in Equ. (14) and Equ. (17) respectively. FromFig. 9, it can be seen that the difference of both

is in the gain and phase lines between Hxnew1

1L and

Hxnew2

2L . These gains are brandish in the entire fre-quency range, which, for the various frequencies isa follow-up from individual input decomposed. Thispoint will be adjudicated in section 4.4.

Fig. 8 shows the quadratic functions of Hxnew1

1Q and

Hxnew2

2Q for the models in Equ. (14) and Equ. (17)respectively. It can be seen that the gain of fre-

quency components for Hxnew1

1Q is mainly about the

Fig.9:: Responses of Church model (solid line) andMISO second-order Volterra model (dotted line):(top) time domain; (bottom) frequency domain.

range above 0 Hz, but it is gradually decreased forin the range of high frequency components. For the

gain response Hxnew2

2Q , it can be seen multi frequencycomponents, but the apparent frequency is at 9 MHz.

The estimated Volterra filter kernels were used theconvolution with the input data to predict and to an-alyze that oscillate subharmonics, which will be pre-sented in detail in section 4.3 and 4.4.

4.3 Response of MISO Volterra Model

In order to demonstrate the appropriate functionsof the system kernels approach for bubble oscilla-tions, data obtained from identification procedureswas overlaid to the system MISO Volterra model,which is shown in Fig. 9. The prediction for this sys-tem as illustrated in Fig. 9, which gives accuracy forthe output responses by comparing between Churchmodel and Volterra model in time domain. For the


Fig.10:: Driving acoustic pressure input used in sim-ulation: (top) Pin(t), (middle) xnew

1 , (bottom) xnew2 .

amplitude response frequency components, it can beaccurately investigated in the frequency domain.

The frequency responses of backscattering signalfrom bubble is exciting the input amplitude pressure,where the driving frequency is fs = 2f0 = 4.5 MHz.The frequency components obtained consists of sub-harmonic ( fs2 = 2.25 MHz), harmonic (fs = 4.5 MHz,2fs = 9 MHz, 3fs = 13.5 MHz, 4fs = 18 MHz, 5fs =22.5 MHz, 6fs = 27 MHz,) and ultraharmonic (3 fs

2

= 6.75 MHz, 5 fs2 = 11.25 MHz, 7 fs

2 = 15.75 MHz,

9 fs2 = 20.25 MHz, 11 fs

2 = 24.75 MHz, 13 fs2 = 29.25

MHz) as seen in Fig. 9.

We compare the prediction result to confirm theaccuracy estimation of MIMO Volterra system proce-dure and MISO Volterra system response from FRFsderived. This provides for the analysis in a separationof subharmonic frequency components.

Fig.11:: Linear transfer functions frequency re-sponse: (top) y1L in Equ. (21), (bottom) y2L inEqu. (23).

Fig.12:: Quadratic transfer functions frequency re-sponse: (top) y1Q in Equ. (22), (bottom) y2Q inEqu. (24).


4.4 Analysis of Results

In this section, we want to demonstrate the outputfrequency components from the linear and quadratictransfer functions MIMO in response to these inputs.The linear and quadratic transfer functions are usedfor analyzing a subharmonic Volterra filter driven bythe kernels, which is considered in this study.

We consider initially that the excitation value ofinput frequency is based on assumption of modifyingthe input in Equ. (12). The value of frequency inputPin(t) before signal partition is single-tone sinusoidalat fs = 4.5 MHz. But after signal partition to themulti-tone input xnew

1 and xnew2 signals as shown in

Fig. 5, they are multi value for the response of fre-quency components as shown in Fig. 10 (middle andbottom). From these pictures, it is observed that theresponse of is almost same. They give the frequencycomponents of xnew

1 and xnew2 which contains subhar-

monic, harmonic and ultraharmonic as well as the fre-quency response obtained from church model. But forat fs = 4.5 MHz and its around the frequencies, thepeak amplitude in gain is clearly strong over anotherregion. For ynew1 and ynew2 of individual multi-toneoutput signals, they do not change time interval anddo not effect in resulting the response of frequency.However, the peak level in gain is changed slightly.

The output responses of Volterra transfer func-tion consist of linear and quadratic transfer functions.The response of these operators is relation to cross-kernel and input transforms. Consider of the fre-quency response of the MIMO system is similar tothe SISO system, the response of linear term is atfundamental frequency and the N th-order (quadratic,cubic,) terms is at higher multiple fundamental fre-quency. The response of the linear and quadratictransfer functions on cross-kernel and input trans-forms in this study is given below.

From Equ. (14) and Equ. (17), each of the multi-tone inputs and outputs xnew

1 ynew1 and xnew2 ynew2

quadratic transfer functions in second-order Volterramodel in Equ. (14) are

y1L =

M−1∑m1=0

h1L(m1)xnew1 (n−m1) (21)

and

y1Q =M−1∑m1=0

M−1∑m2=0

h1Q(m1,m2)xnew1 (n−m1)

xnew1 (n−m2). (22)

The responses of the linear and quadratic transferfunctions in Equ. (17) are

y2L =M−1∑m1=0

h2L(m1)xnew2 (n−m1) (23)

and

y2Q =M−1∑m1=0

M−1∑m2=0

h2Q(m1,m2)xnew12 (n−m1)

xnew2 (n−m2). (24)

The output frequency components from the lineartransfer functions and quadratic transfer functionsare shown in Fig. 11 and Fig. 12, respectively.

Fig. 11 shows the frequency response componentsof linear transfer functions obtained for the systemin Equ. (21) and Equ. (24). The gain levels of lineartransfer functions are high in around the neighbour-hood of frequency 4.5 MHz. However, the peak ampli-tude of gain for every linear transfer function is thehighest at 4.5 MHz (fundamental frequency). Theresponse of linear transfer functions gives that themulti-frequency components like the input frequencyexcited from each of multi-tone signals are comparedin Fig. 10. This means that there is the responseof frequency in linear transfer functions ( y1L, y2L)depending on input signals (xnew

1 , xnew2 ) properties,

which the input frequency indicates the relation toset of the fundamental frequency.

Quadratic transfer functions of the system inEqu. (22) and Equ. (24) are given in Fig. 12. Thefrequency response of quadratic transfer functionsis twice the frequency components of the linear re-sponse while the linear response exists power in thefrequency range of the quadratic. In this figure, wesee that the gain levels of frequency generation aregrowth. One can see in this figure, the gain levelsof quadratic transfer functions are high in around theneighbourhood of frequency 9 MHz. (twice the funda-mental frequency). It shows significance the responsethat around the neighbourhood of twice the funda-mental frequency is similar to the response of a SISOVolterra system for exciting by a single-tone inputsignal.

From the above analysis, the main reason why theproblem of the kernels of the MIMO Volterra modelcannot be applied to separate the subharmonic fre-quency is that the exciting frequency multi-tone in-put resulting in generations of multi-input frequency.Therefore the operator of functions will be obtainedmulti-frequency as well. The key idea to overcomethis problem is to use a single-tone signal frequencyof input signal with the frequency like subharmonicoscillation.

5. CONCLUSION

To analyze the separation of the subharmonic fre-quency components of echo signal from bubble withVolterra filter, we used MIMO Volterra system ap-proach to model the subharmonic system with MISOVolterra system. We studied the output frequency oflinear and quadratic transfer functions to analyze thefrequency response for employing subharmonic sepa-ration.


The FRFs obtained from MIMO Volterra systemhas been successfully illustrated for driving MISOVolterra system in modeling subharmonic oscilla-tion. However, the output response of the linear andquadratic obtained from the convolution of the FRFswith the data input cannot be applied in filteringsubharmonic components. However, we are obtainedknowledge of application the Volterra filter for sepa-rating subharmonic frequency.

Future studies are modeling of subharmonic oscil-lation by SISO Volterra filter using a single-tone half-frequency input signal and to separate subharmonicfor improving CTR.

6. ACKNOWLEDGEMENT

This research project was funded by a grant fromthe Faculty of Engineering, Prince of Songkla Univer-sity through Contract No. Eng-51-2-7-02-0004-S.

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Chinda Samakee He received B.Eng.degree from Siam University, Bangkok,Thailand in 1996, and M. Eng. degreeand also Ph.D. degree in Electrical En-gineering at the Prince of Songkla Uni-versity in 2004 and 2013, respectively.His research interests include ultrasoundsignal processing, ultrasound contrastimaging, and signal processing

Pornchai Phukpattaranont was bornin Songkla, Thailand. He received the B.Eng. and M. Eng. degrees in electricalengineering from Prince of Songkla Uni-versity in 1993 and 1997, respectively,the Ph.D. degree in Electrical and Com-puter Engineering from the Universityof Minnesota, in 2004. He is currentlyan assistant professor of Electrical Engi-neering at Prince of Songkla University.His research interests include ultrasoundcontrast imaging, signal processing, im-

age processing, and biomedical instrumentation. Dr. Phuk-pattaranont is member of the IEEE

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