1

BLISS: A Blind Spectrum Separation Approach forJamming-Resistant Communications

Srikanth Pagadarai

Department of Electrical and Computer EngineeringWorcester Polytechnic Institute, Worcester, MA, USA

Email:srikanthp@wpi.edu

Research Advisor: Professor Alexander M. Wyglinski

Abstract—In this presentation, we discuss some priliminaryresults pertaining to an ongoing project on the developmentofan adaptive signal processing solution combining antenna subsetselection, spectral subtraction, and blind source separation inorder to mitigate the impact of both wideband jamming andco-site interference by extracting individual transmissions frommultiple intercepted mixtures of wireless signals.

I. I NTRODUCTION

With the increasing volume of wireless traffic that theatreoperations require, the probability of transmissions interferingwith each other is steadily growing to the point that new tech-niques need to be employed. Furthermore, to combat remotelyoperated improvised explosive devices, many ground convoystransmit high-power broadband jamming signals, which blocksboth hostile as well as friendly communications. We aim todevise, implement, and evaluate an adaptive signal processingsoftware solution for mitigating the effects of both intentionaland unintentional jamming (including wideband jamming) viathe combination of antenna subset selection, spectral subtrac-tion, and blind source separation (BSS) techniques in orderto extract specific transmissions from a mixture of interceptedwireless signals. The goal of our proposed solution, calledBLInd Spectrum Separation (BLISS), is to enable reliable,high throughput, and robust end-to-end wireless communi-cations in the support of all Department of Navy missions,especially high capacity multimedia (voice, data, imagery)transmissions.

II. T ECHNICAL APPROACH

The BLISS solution (see Figure 1) integrates three well-known adaptive signal processing algorithms found in theopen literature, namely: antenna subset selection, spectralsubtraction, and blind source separation (BSS). Each of thesealgorithms is employed within the BLISS framework in orderto enable the process of extracting individual transmissionsintercepted from several mixtures of wireless signals. Al-though BSS can readily extract transmissions under ideal con-ditions, the BLISS framework will be deployed in challengingoperational scenarios which could significantly degrade theperformance of the communication system, or even cause itto fail. Hence, the other two algorithms, i.e., antenna subsetselection and spectral subtraction, are employed to make the

Fig. 1. Block diagram of the BLISS framework.

BSS process more robust in challenging operational condi-tions. Once the implementation of each of these techniques isperformed, the next step is their integration into a seamlessBLISS framework. Finally, validation of the novel frameworkis performed via computer simulations over a range of differentoperating scenarios.

Blind Signal Separation

Blind signal separation is the task of separating signals whenonly their mixtures are observed. The process is often termed“blind”, with the understanding that both source signals andmixing procedure are unknown. The mathematical model thatdescribes the observed mixtures as a function of independentcomponents is as follows:

x = Hs + n (1)

In reality, the specific mixing model is the paramount pieceof prior information required, and in many scenarios evenknowledge of certain source statistics is necessary. Underthissetting, the channel may generally be construed as a lineartime-invariant (LTI) system, though there is some activityoccurring with nonlinear mixing models. Algorithms whichrely on this concept, the separation-independence equivalence,may be classed as those performing Independent Compo-nent Analysis (ICA). The problem of BSS is then reducedto a mathematical optimization problem, of which severalapproaches exist, e.g., kurtosis, mutual information, crosspower-spectra, entropy, log-likelihood [1]. For instance, theoptimization problem based on kurtosis in two-dimensions can

2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−1

−0.5

0

0.5

1

t

s 1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−1

−0.5

0

0.5

1

t

s 2

(a) Original sources.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−2

−1

0

1

2

t

s1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

−2

−1

0

1

2

t

s2

(b) Estimated sources.

Fig. 2. An illustration of BSS.

be formulated as the minimization of

F(q) = q4

1+ q

4

2(2)

over q2

1+ q

2

2= 1 where the vector,q is a function of the

mixing matrix. The BSS problem for the case of an idealized2× 2 BPSK system is illustrated in Figure 2.

In this work, we will devise an unmixing system designedto separate wireless transmissions operating in the presence ofjamming fields, especially those with wideband characteristics.Furthermore, we will investigate technical challenges that willbe potentially encountered by the BLISS-enabled platform anddevise solutions for them.

Antenna Subset Selection

Multiple-input multiple-output (MIMO) wireless solutionsi.e, the use of multiple antennas at transmitter and receiver,has emerged as a cost-effective technology in making highdata rates a reality. This is due to the inefficiency associatedwith the implementation of a brute force approach of achievinghigh data rates using a single transmit-single receive antennasystem which is theoretically feasible provided that the productof the bandwidth (Hertz) and spectral efficiency (bits persecond per Hertz) matches the desired data rate. However,deploying transceivers with multiple antennas requires mul-tiple RF chains that are typically very expensive. Therefore,there is considerable incentive for low-cost, low-complexity

techniques with the benefits of multiple antennas. Optimalantenna subset selection is one such technique. A selectionof antenna elements, which are typically much cheaper thanRF chains, is made available at the transmitter and/or receiver.Transmission/reception is performed through the optimal sub-set. Several citerion have been proposed in order to arrive atthe optimal subset under a variety of operating conditions [2]–[5].

Spectral Subtraction

Spectral subtraction is a technique which has been tradition-ally employed for enhancing speech corrupted by broadbandnoise. In the context of speech enhancement, the techniqueinvolves subtracting an estimate of the noise power spectrumfrom the speech power spectrum, setting negative differencesto zero, recombining the new power spectrum with the originalphase, and then reconstructing the time waveform [6]–[8].In our implementation of the spectral subtraction, insteadofsubtracting noise, we subtract the power spectral densities ofknown signals from those that need further processing.

Integrated Framework

As for the overall framework that combines antenna subsetselection, spectral subtraction and blind signal separation, byapplying an optimal antenna subset selection algorithm, thesignal at the end of the transmitter passes through the RFchain corresponding to a subset of the available antennas andexperiences the distortion caused by the wireless channel.Thiswireless channel is modeled as a mixing matrix as shown inFigure 2. At the receiver end, a spectral subtraction techniqueis applied to subtract all of the known signals. After the knownsignals are subtracted, the problem is essentially a blind signalseparation problem. We would have a set of N signals that arereceived over M sensors. Nongaussianity of noise is assumedand by applying a BLISS technique especially tailored tothe case of jamming-resistant communications, the signalsarereceived.

REFERENCES

[1] A. Hyvarinen, J. Karhunen, and E. Oja”,Independent Component Anal-ysis. John Wiley & Sons Inc., 2001.

[2] R. U. Nabar, D. A. Gore, and A. J. Paulraj”, “Optimal selection anduse of transmit antennas in wireless systems,” inProc. ICT, (Acapulco,Mexico), 2000.

[3] D. A. Gore, R. U. Nabar, and A. J. Paulraj”, “Selecting an optimal set oftransmit antennas for a lowrank matrix channel,” inProc. IEEE ICASSP,(Istanbul, Turkey), pp. 2785–2788, 2000.

[4] A. Molisch, M. Win, and J. Winters”, “Capacity of MIMO systems withantenna selection,” inProc. IEEE Int. Contr. Conf., (Helsinki, Finland,2001), pp. 570–574, 2001.

[5] D. A. Gore and A. J. Paulraj”, “Space-time block coding with opti-mal antenna selection,” inProc. IEEE ICASSP, (Salt Lake City, UT),p. 24412444, 2001.

[6] S. A. M. Manzuri, R. Dianat, and J. Kabudian”, “An improved spectralsubtraction speech enhancement system by using an adaptivespectralestimator,” in Proc. IEEE Canadian Conf. on Elec. and Comp. Engg.,May 2005.

[7] R. Singh and P. Rao”, “Spectral subtraction speech enhancement withrasta filtering,” inProc. of National Conf. on Commns., (Kanpur, India),2007.

[8] Y. Nomura, J. Lu, H. Sekiya, and T. Yahagi”, “Spectral subtraction basedon speech/noise-dominant classification,” inProc. IEEE Int. Workshop onAcoustics, Echo and Noise Control, pp. 127–130, 2003.

1

System Modeling, Simulation, and Design forBeamforming Using Simulink HDL Coder

Yanjie PengDepartment of Electrical and Computer Engineering

Worcester Polytechnic Institute, Worcester, MA, USAEmail: yjpeng@WPI.EDU

Research Advisor: Professor Xinming Huang and Professor Andrew G. Klein

Abstract—Beamforming is a signal processing technique usedin sensor arrays for directional signal transmission or reception.It has found numerous applications in radar, sonar, seismology,wireless communications, speech, acoustics, and biomedicine.Adaptive beamforming is used to detect and estimate the signal-of-interest at the output of a sensor array by means of data-adaptive spatial filtering and interference rejection. We focus onthe study of using Matlab/Simulink tools for model simulationand hardware design of the traditional linear array beamformingusing QR-RLS (QR-decomposition-based recursive least-squares)algorithm. We present the details on the theoretical algorithm,corresponding simulation model, and the FPGA result of thehardware generated by Simulink HDL Coder.

I. INTRODUCTION

In the recent decade, several technologies have been pro-posed to achieve MIMO (multiple-input and multiple-output)from a conventional SISO (single-input and single-output)system. Among them beamforming [1] is a signal processingtechnique used in sensor arrays for directional signal trans-mission or reception. It has found numerous applications inradar, sonar, seismology, wireless communications, speech,acoustics, and biomedicine. Adaptive beamforming is used todetect and estimate the signal-of-interest at the output of asensor array by means of data-adaptive spatial filtering andinterference rejection.

We present the design of the traditional linear array beam-forming based on QR-RLS (QR-decomposition-based recur-sive least-squares) algorithm [1] using Simulink HDL Coder[5]. The theoretical algorithm is discussed in Section II.System modeling using Simulink HDL Coder is introducedin Section III, followed by hardware implementation result inSection IV. Conclusions and future work are given in SectionV.

II. LINEAR ARRAY ADAPTIVE BEAMFORMING

The block diagram of the traditional linear array beamform-ing (M sensors) is presented in Fig. 1. s(φ) is the steeringvector containing the information of the direction of the targetsignal. w∗1(n), w∗2(n), . . . w∗M (n) is the adjustable complexweight for sensor 1,2,...M , respectively.

The requirements for MVDR (minimum-variance distortion-less response) beamforming is: 1) protecting the target signal,

This work is supported by The Mathworks, Inc., Natick , Massachusetts01760, USA

...

Adaptive Weight Computation

Array of sensors

Output

Adjustable complex weights

Steering vector ( )φs

*

1( )w n

...

...

2

* ( )w n

* ( )M

w n

...

Figure 1. Beamforming

and 2) minimizing the interference. The QR-RLS algorithmis a numerically stable solution to MVDR problem. Thealgorithm is shown in Algorithm 1. The output of beamformeris e(n) = e′(n)

||a(n)||2 .

III. SIMULATION MODEL

The systolic array structure [3] is very suitable for im-plementing the QR-RLS algorithm. The simulation model ofsystolic array for 3-sensor beamforming build by SimulinkHDL Coder is shown in Figure 2. The array contains twotypes of processing cells: boundary cells (blue) and internalcells (red). The boundary cells perform the vectoring operationon received signal to form rotation angles used by internalcells. The internal cells perform Givens rotations of thereceived signal by the angles passed from the boundary cells.Simulation result given 50 snapshots is shown in Fig. 3.

IV. HARDWARE IMPLEMENTATION RESULTS

After fixed point simulation, we can obtain synthesizableverilog and/or hdl code directly by HDL Coder. The FPGAresult is shown in Table 2.

V. CONCLUSIONS AND FUTURE WORK

We present the design of the traditional linear array beam-forming based on QR-RLS (QR-decomposition-based recur-sive least-squares) algorithm using Simulink HDL Coder. The

2

Algorithm 1 QR-RLS Algorithm

• Initial condition Φ1/2(0) = I, and a(n) = s(φ)• For n = 1, 2, . . ., compute Φ1/2(n− 1) u(n)

aH(n− 1) 00T 1

Θ(n) =

Φ1/2(n) 0aH(n) −e′(n)γ−1/2(n)

uH(n)ΦH/2(n− 1) γ1/2(n)

where s(φ) = [1, e−jφ, ..., e−j(M−1)φ]T is the steering vector, u(n) is the received data at time n, and Θ(n) denotes Givens rotation

Output10

aH39

aH28

aH17

phiSqrt326

phiSqrt315

phiSqrt214

phiSqrt333

phiSqrt222

phiSqrt111

0

cell43

uIn

cIn

sIn

uOut

xOut

cell42

uIn

cIn

sIn

uOut

xOut

cell41

uIn

cIn

sIn

uOut

xOut

cell33

uIn

gIn

c s gOut

xOut

cell32uI

n

cIn

sIn

cOut

sOut

uOut

xOut

cell31

uIn

cIn

sIn

cOut

sOut

uOut

xOut

cell22

uIn

gIn

c s gOut

xOut

cell21

uIn

cIn

sIn

cOut

sOut

uOut

xOut

cell11

uIn

gIn

c s gOut

xOut

1/z1/z

1/z

1

uIn33

uIn22

uIn11

Figure 2. Systolic array for 3-sensor beamforming

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−60

−50

−40

−30

−20

−10

0

10

sinθ

Am

plitu

de r

espo

nse

(dB

)

MVDR Beamforming (3 sensors, 50 snapshots)

Targe Angle = sin−1(0.2)Interference Angle = sin−1(0)

Interference−to−noise ratio = 20 dbInterference−to−noise ratio = 30 dbInterference−to−noise ratio = 40 db

Figure 3. Simulation result of 3-sensor beamforming

Product Version Xilinx ISE V11.2

Target Device Xilinx Virtex5 XC5VLX330

Number of Slice Registers 1,824 out of 207,360 (1%)

Number of Slice LUTs 5,355 out of 207,360 (2%)

Number of DSP48Es 103 out of 192 (53%)

Maximum Speed 72.6 MHz

Table IFPGA RESULT FOR 3-SENSOR BEAMFORMING

theoretical algorithm, system modeling and simulation, andhardware implementation results are discussed.

In Section II boundary cells require implementations ofsquare root (sqrt) and reciprocal function. One well-knownand relatively simple method for computing sqrt and reciprocalis the CORDIC (Coordinate Rotation Digital Computer) [4]algorithm. The elemental operations required in the CORDICalgorithm are addition, subtraction, bit-shift, and table look-up.All of these functions are efficiently supported by FPGA archi-tectures, so the CORDIC algorithm is a good candidate for theboundary cells implementation. In [4], two pipelined CORDICengines are in use in the boundary cell. Since CORDICimplementation of sqrt and reciprocal is not supported so far inSimulink HDL coder, we used the Newton–Raphson methodin our design which would consume significant DSP resource,but also constrain the circuit speed. We will develop CORDICblock for sqrt and reciprocal implementation in the future.

REFERENCES

[1] S. Haykin, “Adaptive filter theory,” Prentice Hall, 4th edition, 2002.[2] J. G. McWhirter,“Recursive least-squares minimization using a systolic

array,” in Proc. SPIE, Real-Time Signal Processing VI, vol.431, pp.105-112, San Diego, CA, 1983.

[3] J. E. Volder, “The CORDIC trigonometric computing technique,” IRETrans. on Electronic Computers, pp. 330-334, Sep. 1959.

[4] C. Dick, F. Harris, M. Pajic, and D. Vuletic, “Real-time QRD-basedbeamforming on an FPGA platform,” in Proc. the 40th Asilomar Conf.on Signals, Systems and Computers, pp. 1200-1204, Oct, 2006.

[5] Simulink HDL Coder 1.6, The Mathworks, Inc.http://www.mathworks.com/products/slhdlcoder/

Wind Energy Conversion System Operation asPower Generator and Active Filter

Grazia Todeschini

Department of Electrical and Computer EngineeringWorcester Polytechnic Institute, Worcester, MA, USA

Email: grazia@wpi.edu

Research Advisor: Professor Alexander E. Emanuel

Abstract—This research deals with the performance of aWind Energy Conversion System (WECS) operating as powergenerator and Active Filter simultaneously. As a power generator,the WECS converts wind energy into electric energy; as an AF,it sinks the harmonic currents injected by Non-Linear Loads(NLLs) connected at the same feeder.

Three control systems are developed to ensure the describedoperation and a specific study regarding the compensation of thetriplen harmonics is carried out. WECS derating and voltagedistortion are expressed as function of the harmonic currentsinjected by the system.

The WECS performance as generator and AF has been studiedfor steady-state analysis, fast transients (voltage variations)andslow transients (wind speed variations). Simulation results ofa typical plant show that the proposed control systems allowoperating the WECS as power generator and AF both duringsteady state and transient operation; the described operationcauses power loss increase and voltage distortion that require aconservative choice of the WECS components and require systemderating.

I. I NTRODUCTION

Distributed generation represents an answer to the growingdemand of electric energy and to the increasing environmentalconcerns. Among alternative sources, wind energy is one of themost promising technologies. In the developed countries, windcapacity has been growing at a rate of 20% to 30% a year overthe last decade [1]. The rapid growth of the wind industry isdue to different factors, including [2]: supporting governmentpolicies, advances in wind power technology that causedcost reduction and performance improvement, environmentalconcerns and the deregulation process.

Given the advances in power electronics and control, ap-plications such as reactive power compensation, static transferswitches, energy storage, variable-speed generation, voltagecontrol and dynamic reactive power support are commonlyfound in modern wind power plants [3]: these applications areknown as ‘ancillary services’. The present research presentedin this paper investigate one of the ancillary services: activefilter operation.

II. T HE STUDIED SYSTEM

The studied system is shown in Fig. 1. The Doubly-FedInduction Generator (DFIG) stator terminals are connectedto

the Point of Common Coupling (PCC) through a transformerand a feeder, represented by the equivalent resistance andinductanceRc and Lc. The feeder that connects the Non-Linear Load (NLL) to the PCC is represented by the equivalentresistance and inductanceRh andLh. The DFIG rotor is sup-plied by two back-to-back connected converters: the rotor sideconverter (RSC) and the line side converter (LSC). The feederthat connects the LSC to the PCC has the equivalent resistanceand inductanceRL andLL. The RSC and LSC power switchesare driven by means of Pulse Width Modulation (PWM).The control system design is performed in an equivalentdq0

domain obtained by applying Park transformation to the three-phase variables [4].

III. C OMPENSATION BY MEANS OF COMBINED

MODULATION

As proved in [5], the most effective AF strategy uses bothpower converters and the DFIG by combining compensationby means of RSC and LSC modulation. This strategy is named‘combined modulation’ (CM). The aim of this method is todistribute the harmonic power flow and therefore the powerloss increase among the DFIG, the RSC and the LSC.

When compensation by means of CM is applied, the zero-sequence harmonics injection is obtained by modulating theLSC, and compensation of the symmetrical harmonic compo-nents is obtained by modulating the RSC. In terms ofdq0

components, the reference harmonic currents for the LSC andthe RSC control systems are as follows:

idL,ref = −i0h (1)

ids,ref = −idh (2)

iqs,ref = −iqh (3)

whereidh, iqh, i0h are thedq0 currents measured at the NLLterminals.

A. Steady-state analysis

Steady-state analysis allows verifying the validity of theproposed control system and quantifiyng its effects.

The current and voltage THD (Total Harmonic Distortion)are measured at the PCC for a case study, before and after

Fig. 1. WECS (Wind Energy Conversion System) configuration: the Doubly-Fed Induction Generator (DFIG) stator is connected to the PCC through a line;the DFIG rotor is supplied by two back-to-back connected converters; the dc-link central tap is tied to the neutral. A NLLis connected to the PCC.

TABLE ICURRENT AND VOLTAGE THD AT THE PCCFOR THE CASE STUDY: COMPARISON BETWEEN THE THREE COMPENSATION METHODS.

No Compensation With compensation Limit USRSC mod. LSC mod. Combined mod.

(a) Current THD 25% 16% 7.5% 7.0% 5-20%

(b) Voltage THD 115 4.7% 5.5% 4.0% 5%

harmonic compensation is implemented. Tab. I lists the re-sults and shows a significant THD reduction when harmoniccompensation is applied.

Two effects of the proposed application on the WECSoperation are also identified:

1) power loss increase and consequent temperature rise; un-der certain operating conditions, derating of the WECSis necessary to limit the power loss and the devicestemperature.

2) voltage distortion due to the harmonic current flowand harmonic voltage drop on the line impedances.It results that the the measured peak voltage at thestator terminals is above the rated value and requiresconservative insulation design.

B. Transient analysis

The system transient response is investigated for two typesof disturbances: voltage variation (fast transients) and windspeed variation (slow transients). An extensive number ofcases have been studied and the results are summarized asfollows:

• harmonic compensation is not affected by transient oper-ation, since the reference harmonic currents are measuredat the NLL terminals;

• derating helps reducing severity of current during thetransients because it causes a reduction of the fundamen-tal current amplitude;

• LVRT ability is verified: the WECS continues to provide

power and is not damaged according to the requirementspresented in [6];

• wind speed variation introduces low frequency currentdisturbances, due to the mechanical system response.

IV. CONCLUSION

The simulation results show that the WECS can be usedsimultaneously as Active Filter and Power Generator. Steady-state analysis shows that to safely operate the system athigh wind speeds, derating and a conservative choice of thesystem components are necessary. Transient phenomena haveno effects on the WECS ability to inject harmonic currents.

REFERENCES

[1] Z. Chen and F. Blaabjerg, “Wind energy: The world’s fastest growingenergy source,”IEEE Power Electronics Society Newsletter, vol. 18, no.3, pp. 15–19, 2006.

[2] T. Ackermann, Wind Power in Power Systems, John Wiley and Sons,Hoboken, NJ 07030-5774, US, 2005.

[3] Wachtel S., J. Marques, E. Quitman, and M. Schellschmidt, “Wind energyconverters with FACTS capabilities and the benefits for the integrationof wind power plants into power systems,” inProc. European WindEnergy Conference and Exhibition (EWEC), Milan, 4, May 7-10 2007,pp. 1761–65.

[4] B. K. Bose, Power Electronics and AC drives, Prentice-Hall, UpperSaddle River, NJ, 07458, US, 2001.

[5] G. Todeschini and A. E. Emanuel, “Wind energy conversion system asan active filter: Design and comparison of three control systems (part I),”Submitted to IET, Companion paper.

[6] FERC Order No. 661A,Interconnection for Wind Energy, 2005.

1

Stress Field Calculation for Quasi-static Ultrasound

Elastography via Force Sensor Integration and SLE Lili Yuan, Ultrasound Laboratory, ECE Department

Advisor: Professor Peder C. Pedersen

Abstract—Most current elastography methods remain qualitative

or display only strain information due to the inexact internal stress

distribution. A more accurate result may be obtained by employing

force sensors to gauge the applied vertical compression force on

skin surface through the ultrasound transducer. A computationally

efficient superposition algorithm based on Love’s closed-form

equation (SLE) is presented. It performs analytical calculation of

the 3D dynamic stress field inside a soft tissue phantom with non-

free boundary conditions under a non-uniformly stressed

rectangular linear array transducer, using the pressure on each

element of the contact surface. The validity of the SLE method was

tested by comparison to the stress field calculated with Finite

Element Analysis (FEA). For the region of interest (directly

underneath the ultrasound transducer), the SLE provides a

sufficiently precise solution and can be exploited for absolute

Young’s modulus reconstruction in real time.

I. INTRODUCTION

Ultrasound elastography is a recent technology for non-

invasively imaging diseased arteries, detecting prostate tumors

and categorizing breast lesions [1]. Well developed quasi-static

free-hand techniques mostly involve only tissue strain. However,

the stiffness may be dependent on the magnitude and direction

of the force applied, especially if the linear stress-strain region

(typically 1%-2%) is exceeded [2]. Inner stress as the response

of external force is also required in addition to the strain.

Inexpensive, ultra-thin, flexible piezoresistive force sensors

(Tekscan, MA) were utilized to quantitatively measure the

contact force. The force sensor output was then integrated into

the SonixRP (Ultrasonix, Vancouver, Canada) strain imaging

system. 3D stress field was achieved by means of the computed

pressure on each force plate element and SLE algorithm.

II. FORCE SENSOR INTEGRATION

A force plate with a slot for the ultrasound transducer was

designed, as shown in Fig. 1. Force sensors were mounted

beneath the force plate and connected to the signal amplification,

noise filtering, and data acquisition system. The force plate was

decomposed into small rectangular elements with each

individual constant pressure. Contact pressure on every sub-

region with a sensor was estimated by measured force divided

by sensing area. For the remaining sub-regions without sensors,

pressure was obtained by bilinear interpolation.

Force Sensors

Force Plate

Transducer Arrayx

y (0,0)

l

w

b

a

Force Plate Element

Fig. 1. Bottom view of linear array rectangular transducer with size of 60 mm ×

20mm, force plate geometry a × b, partitioned into small element with l × w.

A combined display of strain imaging and forcing function

was implemented. Figure 2 illustrates the outcome with agar-

based phantom. Force versus time was used to depict external

force magnitude and loading frequency.

Container filled with Water

PhantomSofter Cyst

SonixRPLinear Array Transducer

Force Sensors

Data Acquisition

System

Force Plate

Strain Image overlaid on B-mode

Forcing Function

Vertical Force

Fig. 2. Front view of sensor integration into strain image system, the upper

image is strain image superimposed on B-mode with blue and red color

respectively representing softer and harder material.

III. SUPERPOSITION ALGORITHM BASED ON LOVE’S FORMULA

SLE was introduced for analytical computation of 3D stress

field under the assumption of a homogeneous, isotropic, semi-

infinite, elastic solid material, stressed non-uniformly by

rectangular compressor. The stress at each field point is the

overall contribution from every force plate element under

individual constant pressure.

node A

z

yx

uniform compression

force plate th element

cross-section of 3D matrix

in the same plane of

transducer beam

1a

4d 3c

2b

0( )' ', ,x y

r

l

w

( )P ti

( , , 0)m ni i

( , , )x y z

( , , 0)2 2

l wm ni i ( , , 0)

2 2

l wm ni i

( , , 0)2 2

l wm ni i ( , , 0)

2 2

l wm ni i

i

field point

node Cnode D

node B

Fig. 3. Geometry of the 𝑖th force element with center and nodes coordinates

Expression for the distribution of vertical stress component

𝑆𝑧𝑧 at field point (𝑥, 𝑦, 𝑧) is

𝑆𝑧𝑧 𝑥, 𝑦, 𝑧 = 𝑆𝑧𝑧𝑖 𝑥, 𝑦, 𝑧

𝑥𝑛𝑜 ∗𝑦𝑛𝑜

𝑖=1 = 1

2𝜋[𝜕𝑉

𝜕𝑧− 𝑧

𝜕2𝑉

𝜕𝑧 2]

𝑥𝑛𝑜 ∗𝑦𝑛𝑜

𝑖=1 (1)

where, 𝑥𝑛𝑜 and 𝑦𝑛𝑜 denote the number of force elements in 𝑥 and

𝑦 direction, 𝑉 is the Newtonian potential, and 𝜕𝑉

𝜕𝑧,

𝜕2𝑉

𝜕𝑧 2 are

calculated by replacing contact area geometry in corresponding

equations as 𝑥𝑑 = 𝑚𝑖 −𝑙

2, 𝑥𝑢 = 𝑚𝑖 +

𝑙

2, 𝑦𝑑 = 𝑛𝑖 −

𝑤

2, 𝑦𝑢 = 𝑛𝑖 +

𝑤

2.

Elaborate description of these formulas omitted here for saving

space can be found in [3].

IV. RESULTS

A phantom (120mm × 60mm × 40mm , Young’s modulus

5 × 104 𝑃𝑎 , Poisson’s ratio 0.42 , density 1040 𝑘𝑔/𝑚3 ) and

2

coaxial force plate ( 108mm × 34mm ) were simulated for 3D

stress distribution calculation using both SLE and FEA

(COMSOL, Sweden).

Cross-sections coincident with transducer image plane from

the two methods were compared to evaluate the accuracy of SLE

for stress estimation in visco-elastic material with fixed

boundary.

A. Stress Field Calculation by SLE

Stress distribution, due to uniform pressure 1000 Pa, along the

axis of compressor decays with depth in a semi-infinite medium,

presented in Fig. 4. The counterpart by FEA with the same

tendency was described in [4].

Fig. 4. Stress field cross section due to uniform vertical compression by SLE.

As an initial experiment, two force sensors were allocated at

two ends of the force plate and the contact surface were divided

into three equal-size rectangle, where the pressure are 1000 𝑃𝑎,

1200 𝑃𝑎, and 1400 𝑃𝑎 in turn along 𝑥 axis. Uneven distributed

stress closer to surface and gradual reduction with depth is

shown in Fig. 5.

Fig. 5. Stress field cross-section for non-uniform normal compression by SLE

B. SLE Performance Evaluation

Stress results from FEA with fixed bottom and prescribed-

displacement surrounding boundary (only displacement in z

direction is allowed) is presented in Fig. 6, followed by the

normalized difference image (Fig. 7). SLE is far more efficient

than FEA (9.2 × 10−6 𝑠/𝑒𝑙𝑒𝑚𝑒𝑛𝑡 vs 1.07 × 10−2 𝑠/𝑒𝑙𝑒𝑚𝑒𝑛𝑡 ) and

less than 5.62% deviation from FEA from surface to depth at

22.3 mm except around the edges.

Fig. 6. 2D stress for non-uniform normal compression using FEA

Fig. 7. Normalized stress difference between SLE and FEA for variable pressure

The mismatch that exists because of gradient of pressure

variation, plate and phantom geometry, mesh fineness and

interpolation method for FEA simulation is not mentioned in

detail due to space limitation.

V. CONCLUSION

By virtue of force sensors and SLE, fast 3D dynamic stress

field calculations are achievable, which can be combined with

strain to obtain real time quantitative elastography for elastic

solid material, therefore improving the quality of tumor and

cancer diagnosis.

ACKNOWLEDGEMENT

The support from the Telemedicine and Advanced

Technology Research Center are greatly appreciated. The

authors also thank research IT support Siamak Najafi for

providing COMSOL software for free and thank lab manager

Frederick Huston for supplying sensor calibration equipments.

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on Part of the Boundary,” Phil. Trans. Roy. Soc. London, vol. A228, 1929,

pp. 377-420.

[3] Lili Yuan, Peder C. Pedersen, “Stress Field Simulation for Quantitive

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