Waveform Synthesis for Diversity-Based Transmit Beampattern Design1

Petre Stoica2 Jian Li3∗ Xumin Zhu4 Bin Guo4

AbstractTransmit beampattern design is a critically important task in many fields including defense and homeland

security as well as biomedical applications. Flexible transmit beampattern designs can be achieved byexploiting the waveform diversity offered by an array of sensors that transmit probing signals chosen atwill. Unlike a standard phased-array, which transmits scaled versions of a single waveform, a waveformdiversity-based system offers the flexibility of choosing how the different probing signals are correlatedwith one another. Recently proposed techniques for waveform diversity-based transmit beampattern de-sign have focused on the optimization of the covariance matrixR of the waveforms, as optimizing aperformance metric directly with respect to the waveform matrix is a more complicated operation. Givenan R, obtained in a previous optimization stage or simply pre-specified, the problem becomes that ofdetermining a signal waveform matrixX whose covariance matrix is equal or close toR, and whichalso satisfies some practically motivated constraints (such as constant-modulus or low peak-to-average-power ratio constraints). We propose a cyclic optimization algorithm for the synthesis of such anX,which (approximately) realizes a given optimal covariance matrixR under various practical constraints.A number of numerical examples and case studies are presented to demonstrate the effectiveness of theproposed algorithm.

1This work was supported in part by the Swedish Research Council (VR), by the National Science Foundation under Grant No. CCF-0634786, by the Office of Naval Research under Grant No. N000140710293, and by the Defense Advanced Research Projects Agency underGrant No. HR0011-06-1-0031. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarilyendorsed by the United States Government.

2Petre Stoica is with the Department of Information Technology, Uppsala University, Uppsala, Sweden. Phone: 46-18-471.7619; Fax:46-18-511925; Email: ps@it.uu.se.

3Jian Li is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611-6130, USA. Phone:(352) 392-2642; Fax: (352) 392-0044; Email: li@dsp.ufl.edu.∗Please address all correspondence to Dr. Jian Li.

4Xumin Zhu and Bin Guo are with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL32611-6130, USA. Phone: (352) 392-5241; Fax: (352) 392-0044.

1

I. I NTRODUCTION

Waveform diversity has been utilized both in multiple-input multiple-output (MIMO) communications

and in MIMO radar. In the past decade, communications systems using multiple transmit and receive

antennas have attracted significant attention from government agencies, academic institutions and research

laboratories, because of their potential for dramatically enhanced throughput and significantly reduced error

rate without spectrum expansion. Similarly, MIMO radar systems have recently received the attention of

researchers and practitioners alike due to their improved capabilities compared with a standard phased-

array radar. A MIMO radar system, unlike a standard phased-array radar, can transmit multiple probing

signals that may be chosen at will. This waveform diversity offered by MIMO radar is the main reason

for its superiority over standard phased-array radar; see, e.g., [1] - [29]. For colocated transmit and

receive antennas, for example, MIMO radar has been shown to have the following appealing features:

higher resolution (see, e.g., [1], [3]), superior moving target detection capability [6], better parameter

identifiability [13], [24], and direct applicability of adaptive array techniques [13], [15], [26]; in addition,

the covariance matrix of the probing signal vector transmitted by a MIMO radar system can be designed

to approximate a desired transmit beampattern – an operation that, once again, would be hardly possible

for conventional phased-array radar [13], [18], [22].

Transmit beampattern design is critically important not only in defense applications, but also in many

other fields including homeland security and biomedical applications. In all these applications, flexible

transmit beampattern designs can be achieved by exploiting the waveform diversity offered by the possi-

bility of choosing how the different probing signals are correlated with one another.

An interesting current research topic is the optimal synthesis of the transmitted waveforms. For MIMO

radar with widely separated antennas, waveform designs without any practical constraint (such as the

constant-modulus constraint) have been considered in [19]. For MIMO systems with colocated antennas,

on the other hand, the recently proposed techniques for transmit beampattern design or for enhanced

target parameter estimation and imaging have focused on the optimization of the covariance matrixR

2

of the waveforms [7], [11], [13], [18], [21], [22], [25], [27]. For example, in a waveform diversity-based

ultrasound system,R can be designed to achieve a beampattern that is suitable for the hyperthermia

treatment of breast cancer [30]. Now, instead of designingR, as in the cited references, we might think

of designing directly the probing signals by optimizing a given performance measure with respect to the

matrix X of the signal waveforms. However, compared with optimizing the same performance measure

with respect to the covariance matrixR of the transmitted waveforms, optimizing directly with respect to

X is a more complicated problem. This is so becauseX has more unknowns thanR and the dependence

of various performance measures onX is more intricate than the dependence onR (asR is a quadratic

function of X). In effect, there are several recent methods, as mentioned above, that can be used to

efficiently compute an optimal covariance matrixR, with respect to several performance metrics; yet

the same cannot be said about determining an optimal signal waveform matrixX, which is theultimate

goal of the designing exercise. Furthermore, in some cases, the desired covariance matrix is given (e.g.,

a scaled identity matrix), and therefore there is no optimization with respect toR involved (directly or

indirectly).

In this paper, we consider the synthesis of the signal waveform matrixX for diversity-based flexible

transmit beampattern design. WithR obtained in a previous (optimization) stage, our problem is to

determine a signal waveform matrixX whose covariance matrix is equal or close toR, and which also

satisfies some practically motivated constraints (such as constant-modulus or low peak-to-average-power

ratio (PAR) constraints). We present a cyclic optimization algorithm for the synthesis of such anX.

We also investigate how the synthesized waveforms and the corresponding transmit beampattern design

depend on the enforced practical constraints. Several numerical examples are provided to demonstrate the

effectiveness of the proposed methodology.

Notation. Vectors are denoted by boldface lowercase letters and matrices by boldface uppercase letters.

Thenth component of a vectorx is written asx(n). Thenth diagonal element of a matrixR is written as

Rnn. A Hermitian square root of a matrixR is denoted asR1/2. We use(·)T to denote the transpose, and

3

(·)∗ for the conjugate transpose. The Frobenius norm is denoted as‖·‖. The real part of a complex-valued

vector or matrix is denoted as Re(·).

II. FORMULATION OF THE SIGNAL SYNTHESIS PROBLEM

Let the columns ofX ∈ CL×N be the transmitted waveforms, whereN is the number of the transmitters,

andL denotes the number of samples in each waveform. Let

R , 1

LX∗X (1)

be the (sample) covariance matrix of the transmitted waveforms. We assume thatL > N (typically

L À N ). Note thatX has2NL real-valued unknowns, which is usually a much larger number than the

number of unknowns inR, viz. N2.

The class of (unconstrained) signal waveform matricesX that realize a given covariance matrixR is

given by

1√L

X∗ = R1/2U∗, (2)

whereU∗ is an arbitrary semi-unitaryN×L matrix (U∗U = I). Besides realizing (at least approximately)

R, the signal waveform matrix must also satisfy a number of practical constraints. LetC denote the set of

signal matricesX that satisfy these constraints. Then a possible mathematical formulation of the problem

of synthesizing the probing signal matrixX is as follows:

minX∈C;U

∥∥∥X−√

LUR1/2∥∥∥

2

. (3)

Depending on the constraint setC, the solutionX to (3) may realizeR exactly or only approximately.

Evidently asC is expanded (i.e., the constraints are relaxed), the matching error in (3) decreases. Whenever

the matching error is different from zero, we can useeither the solutionX to (3) as the signal waveform

matrix, in which case it will satisfy the constraints but it will only approximately realizeR, or√

LUR1/2,

whereU is theU-solution of (3), which realizesR exactly but satisfies only approximately the constraints

– the choice between these two signal waveform matrices may be dictated by the application at hand.

4

The minimization problem in (3) isnon-convexdue to the non-convexity of the constraintU∗U = I

and possibly of the setC, too. The constraintU∗U = I generates the so-called Stiefel manifold, and there

are algorithms that can be used to minimize a function over the said manifold (see, e.g., [31]). However,

these algorithms are somewhat intricate both conceptually and computationally, and their convergence

properties are not completely known; additionally, in (3) we also have the problem of minimizing with

respect toX ∈ C, which may also be non-convex.

With the above facts in mind, we prefer a cyclic (or alternating) minimization algorithm for solving

(3), as suggested in a related context in [32], [33]. We refer to the cited papers for more details on this

type of algorithm and its properties.

III. C YCLIC ALGORITHM FOR SIGNAL SYNTHESIS

We first summarize the steps of the cyclic minimization algorithm and then describe each step in detail.

Step 0: Set U to an initial value (e.g., the elements ofU can be independently drawn from a complex

Gaussian distribution with mean 0 and standard deviation 1); alternatively we can start with an initial

value for X, in which case the sequence of the next steps should be inverted (note that the initial

value of eitherU or X does not necessarily have to satisfy the constraints imposed on these variables

in the next steps of the cyclic algorithm).

Step 1: Obtain the matrixX ∈ C that minimizes (3) forU fixed at its most recent value.

Step 2: Determine the matrixU (U∗U = I) that minimizes (3) forX fixed at its most recent value.

Iteration: Iterate Steps 1 and 2 until a given stop criterion is satisfied. In the numerical examples presented

later, we terminate the iteration when the Frobenius norm of the difference between theU matrices

at two consecutive iterations is less than or equal to 10−4.

An important advantage of the above algorithm is that Step 2 has aclosed-form solution. This solution

can be derived in a number of ways (see, e.g., [34], [35]). A simple derivation of it runs as follows. For

5

given X, we have that

∥∥∥X−√

LUR1/2∥∥∥

2

= const− 2Re

tr[√

LR1/2X∗U]

.

Let

√LR1/2X∗ = UΣU∗ (4)

denote the singular value decomposition (SVD) of√

LR1/2X∗, whereU is N ×N , Σ is N ×N , andU

is L×N . Then

Re

tr[√

LR1/2X∗U]

= Re

tr[U∗UUΣ

](5)

=N∑

n=1

Re[

U∗UU]

nn

Σnn. (6)

Because

(U∗UU

)(U∗U∗U

)= U∗UU∗U

≤ U∗U

= I, (7)

it follows that

Re2[

U∗UU]

nn

≤

∣∣∣[U∗UU

]nn

∣∣∣2

≤[(

U∗UU)(

U∗U∗U)]

nn

≤ 1, (8)

and therefore that∥∥∥X−

√LUR1/2

∥∥∥2

≥ const− 2N∑

n=1

Σnn. (9)

The lower bound in (9) is achieved at

U = UU∗, (10)

which is thus the solution to the minimization problem in Step 2 of the cyclic algorithm.

6

The solution to the problem in Step 1 naturally depends on the constraint setC. For example, in radar

systems the need to avoid expensive amplifiers and A/D converters has led to the requirement that the

transmitted signals have constant modulus. Letxn(l)Ll=1 denote the elements in thenth columnxn of

the signal waveform matrixX. Then the constant-modulus requirement means that:

|xn(l)| = c, for some given constantc and for l = 1, · · · , L. (11)

(For example, we can choosec = R1/2nn ; we omit the dependence ofc on n for notational simplicity.)

Under the constraint in (11), Step 1 of the algorithm has also aclosed-formsolution. Indeed, the generic

problem to be solved in such a case is:

minψ

∣∣cejψ − z∣∣2 , (12)

wherec > 0 andz ∈ C are given numbers. Because

∣∣cejψ − z∣∣2 = const− 2c|z| cos [ψ − arg(z)] , (13)

the minimizingψ is evidently given by

ψ = arg(z). (14)

Therefore, under the constant-modulus constraint, both steps of the cyclic algorithm have solutions that

can be readily computed. However, (11) may be too hard a requirement on the signal matrix in the sense

that the corresponding minimum value of the matching criterion in (3) may not be as small as desired. In

particular, this means that1

LX∗X may not be a good approximation ofR (see, e.g., [25], where it was

shown that signals that have constant modulus and take on values in a finite alphabet may fail to realize

well a given covariance matrix).

With the above facts in mind, we may be willing to compromise and therefore relax the requirement

that the signals have constant modulus. In effect, in some modern radar systems this requirement can

be replaced by the condition that the transmitted signals have alow peak-to-average-power ratio(PAR).

7

Mathematically, the low PAR requirement can be formulated as follows:

PAR(xn)4=

maxl |xn(l)|21L

∑Ll=1 |xn(l)|2 ≤ ρ, for a givenρ ∈ [1, L], (15)

(where, once again, we omit the dependence ofρ on n for notational simplicity). If we add to (15) a

power constraint, viz.

1

L

L∑

l=1

|xn(l)|2 = γ, (e.g.,γ = Rnn), (16)

then the setC is described by the equations:

1L

∑Ll=1 |xn(l)|2 = γ,

|xn(l)|2 ≤ ργ, l = 1, · · · , L.

(17)

While the above constraint set is not convex, anefficient algorithmfor solving the corresponding problem

in Step 1 of the cyclic algorithm has been proposed in [32], [33]. Note that the constraints in (17) are

imposed onX in a column-wise manner. Consequently, the solution to Step 1 is obtained by dealing with

the columns ofX in a one by one fashion.

With the power constraint in (16) enforced, the diagonal elements ofR can be synthesized exactly. If

the exact matching ofRnn is not deemed necessary, we can relax the optimization by omitting (16). In

the Appendix we show how to modify the algorithm of [32], [33] in the case where only (15) is enforced.

(In all numerical examples presented in the following section, (16) will be enforced.)

IV. N UMERICAL CASE STUDIES

We present several numerical examples to demonstrate the effectiveness of CA for signal synthesis in

several diversity-based transmit beampattern design applications.

A. Beampattern Matching Design

We first review briefly the beampattern matching design (more details can be found in [18], [22]). We

then present a number of relevant numerical examples.

8

The power of the probing signal at a generic focal point with coordinatesθ can be shown to be (see,

e.g., [7], [15], [18]):

P (θ) = a∗(θ)Ra(θ), (18)

whereR is as defined before,

a(θ) =

[ej2πf0τ1(θ) ej2πf0τ2(θ) · · · ej2πf0τN (θ)

]T

, (19)

and wheref0 is the carrier frequency of the transmitted signal, andτn(θ) is the time needed by the signal

emitted via thenth transmit antenna to arrive at the focal point; unless otherwise stated,θ will be a

one-dimensional angle variable (expressed in degrees). The design problem under discussion consists of

choosingR, under a uniform elemental power constraint,

Rnn =C

N, n = 1, · · · , N, (20)

whereC is the given total transmitted power, to achieve the following goals:

(a) Control the spatial power at a number of given locations by matching (or approximating) a

(scaled version of a) desired transmit beampattern.

(b) Minimize the cross-correlation between the probing signals at a number of given locations (a

reason for this requirement is explained in [18], [22]); the cross-correlation between the probing

signals at locationsθ and θ is given bya∗(θ)Ra(θ).

Assume that we are given a desired transmit beampatternφ(θ) defined over a region of interestΩ. Let

µgGg=1 be a fine grid of points that coverΩ. As indicated above, our goal is to chooseR such that

the transmit beampattern,a∗(θ)Ra(θ), matches or rather approximates (in a least squares (LS) sense) the

desired transmit beampattern,φ(θ), over the region of interestΩ, and also such that the cross-correlation

(beam)pattern,a∗(θ)Ra(θ) (for θ 6= θ), is minimized (once again, in a LS sense) over a given setθkKk=1.

9

Mathematically, we therefore want to solve the following problem:

minα,R

1

G

G∑g=1

wg [αφ(µg)− a∗(µg)Ra(µg)]2 +

2wc

K2 −K

K−1∑

k=1

K∑

p=k+1

|a∗(θk)Ra(θp)|2

s.t. Rnn =C

N, n = 1, · · · , N

R ≥ 0, (21)

whereα is a scaling factor,wg ≥ 0, g = 1, · · · , G, is the weight for thegth grid point andwc ≥ 0 is the

weight for the cross-correlation term. Note that by choosingmaxg wg > wc we can give more weight to

the first term in the design criterion above, and vice versa formaxg wg < wc. We have shown in [13], [18],

[22] that this design problem is a semi-definite quadratic program (SQP) that can be efficiently solved in

polynomial time. Once the optimalR has been determined, we can use CA to synthesize the waveform

matrix X.

As mentioned in Section II, the CA solution to (3) may be chosen to realizeR exactly or only

approximately. When the signal waveforms are synthesized as√

LUR1/2, whereU is the solution to (3)

obtained via CA, then they realizeR exactly, but satisfy the PAR constraints only approximately. We

refer to the so-synthesized waveforms asthe CA synthesized waveforms with optimalR (abbreviated as

optimal R). When we use the solutionX to (3) obtained via CA as the transmitted signal waveform

matrix, thenX will satisfy the PAR constraints, but will realizeR only approximately. We refer to the

so-synthesized waveforms asthe CA synthesized waveforms with PAR≤ ρ (abbreviated asPAR≤ ρ).

In the following examples, the transmit array is assumed to be a uniform linear array (ULA) comprising

N = 10 sensors with half-wavelength inter-element spacing. The sample numberL is set equal to 256.

The uniform elemental power constraint withC = 1 is used for the design ofR. For Ω, we choose a

mesh grid size of 0.1. Finally, the CA algorithm is initialized using theU described in Step 0.

In the first example, the desired beampattern has one wide main-beam centered at 0 with a width of

60. The weighting factorwg in (21) is set to 1 andwc is set to 0. Figures 1(a), 1(b), and 1(c) show the

beampatterns using the CA synthesized waveforms under the constraints of PAR= 1 (constant-modulus),

10

PAR≤ 1.1, and PAR≤ 2, respectively. For comparison purposes, we also show the desired beampattern

φ(θ) scaled by the optimal value ofα. Note that the beampattern obtained using the CA synthesized

waveforms is close to the desired one even under the constant-modulus constraint.

We also note from Figure 1 that the beampatterns obtained usingthe CA synthesized waveforms with

optimal R are slightly different from those obtained usingthe CA synthesized waveforms with PAR≤ ρ.

Let

R =1

LX∗X (22)

be the sample covariance matrix corresponding tothe CA synthesized waveforms with PAR≤ ρ. Let

δ =∥∥∥R−R

∥∥∥ (23)

denote the norm of the difference betweenR andR. Then we haveδ = −29.7891 dB,−41.7237 dB, and

−119.5251 dB for Figures 1(a), 1(b), and 1(c), respectively. As expected, the difference decreases as the

PAR value increases. For the case of PAR = 2, the difference is essentially zero. The mean-squared error

(MSE) of R (i.e., the average value ofδ2), obtained under PAR = 1 and estimated via 100 Monte-Carlo

trials, is shown in Figure 4 as a function of the sample numberL. Note that, as also expected, the MSE

decreases asL increases. Figures 3(a) - 3(c) show the corresponding beampattern differences as a function

of θ, as an ensemble of realizations obtained from the 100 Monte-Carlo trials. In each Monte-Carlo trial,

the initial value forU in Step 0 of CA was chosen independently. Among other things, Figure 3 shows that

CA is not very sensitive to the initial value ofU used, and that this sensitivity decreases asρ increases.

Figure 2 shows the actual PAR values ofthe CA synthesized waveforms with optimalR corresponding

to Figure 1. These PAR values are also compared to those associated with the waveform matrix obtained

by pre-multiplying R1/2 with a 256 × 10 matrix whose columns contain orthogonal Hadamard code

sequences of length 256. The colored Hadamard sequences also have the optimalR as their sample

covariance matrix. Note thatthe CA synthesized waveforms with optimalR have much lower PAR values

than the colored Hadamard code sequences. Note also that the actual PAR values ofthe CA synthesized

11

waveforms with optimalR obtained under PAR≤ 1.1 are slightly lower than the PAR values obtained

under PAR= 1.

Next, we consider a scenario where the desired beampattern has three pulses centered atθ1 = −40,

θ2 = 0, andθ3 = 40, each with a width of 20. The same mesh grid is used as before, and we choose

the weighting factors aswg = 1 and wc = 1. Figure 5 shows the corresponding beampatterns. Remarks

similar to those on Figure 1 can be made for this example as well.

B. Minimum Sidelobe Beampattern Design

The minimum sidelobe beampattern design problem we consider here (see [18], [22] for more details)

is to chooseR, under the uniform elemental power constraint in (20) or rather a relaxed version of it (see

later on), to achieve the following goals:

(a) Minimize the sidelobe level in a prescribed region.

(b) Achieve a predetermined 3 dB main-beam width.

Assume that the main-beam is directed towardθ0 and the prescribed 3-dB angles areθ1 and θ2 (i.e.,

the 3-dB mainbeam width isθ2 − θ1, with θ1 < θ0 < θ2). Let Ωs denote the sidelobe region of interest

and µg a grid covering it. Then the design problem of interest in this section can be mathematically

formulated as follows:

mint,R

−t

s.t. a∗(θ0)Ra(θ0)− a∗(µg)Ra(µg) ≥ t, ∀µg ∈ Ωs

a∗(θ1)Ra(θ1) = 0.5a∗(θ0)Ra(θ0)

a∗(θ2)Ra(θ2) = 0.5a∗(θ0)Ra(θ0)

R ≥ 0

0.8

(C

N

)≤ Rnn ≤ 1.2

(C

N

), n = 1, · · · , N,

N∑n=1

Rnn = C. (24)

12

Note that the relaxed elemental power constraint in (24), while still quite practical, offers more flexibility

than the strict elemental power constraint in (20). Note also that the total transmit power is the same for

both (24) and (20), viz.C. In the examples below, we setC = 1.

As shown in [22], this minimum sidelobe beampattern design problem is a semi-definite program (SDP)

that can be efficiently solved in polynomial time. Once the optimalR has been determined, we can again

use CA to synthesize the waveform matrixX.

Consider first an example where the main-beam is directed towardθ0 = 0 with a 3-dB width equal to

20 (θ1 = −10 andθ2 = 10). The sidelobe region is chosen to beΩs = [−90,−20]∪ [20, 90], which

allows for some transition between the main-beam and sidelobe region. The same mesh grid size of 0.1

is used here. Figure 6 shows the synthesized beampatterns obtained using the CA synthesized waveforms

under the constraints of PAR= 1 and PAR≤ 1.1. Note that the minimum sidelobe beampatterns obtained

from the CA synthesized waveforms with optimalR are similar to those obtained fromthe CA synthesized

waveforms with PAR≤ ρ even for PAR = 1.

We next consider a case with the same design parameters as in the above example except that now we

also wish to place a−40 dB or deeper null atµn = −30. To do this, we add the following constraint to

the minimum sidelobe beampattern design problem in (24):

a∗(µn)Ra(µn) ≤ −40 dB, µn = −30, (25)

(the so-obtained problem is still a SDP). Figures 7(a) - 7(c) show the beampatterns obtained by using the

CA synthesized waveforms under the constraints of PAR= 1, PAR≤ 1.1, and PAR≤ 1.2, respectively.

For the CA synthesized waveforms with PAR≤ ρ, the null depths at−30 for the three different PAR values

shown in Figure 7 are−22.7769 dB,−33.9913 dB and−39.9657 dB, respectively. Hence a stringent

PAR constraint can have a significant impact on the null depth. For PAR≤ 1.2, the beampatterns obtained

with the CA synthesized waveforms with PAR≤ ρ and, respectively, withthe CA synthesized waveforms

with optimalR are almost identical.

13

Finally, consider an example where we wish to form a broad null over the regionΩn = [−55,−45],

where the power gain must be at least 30 dB lower than the power gain atθ0. To do this, we add the

following constraint to the minimum sidelobe beampattern design in (24):

a∗(µn)Ra(µn) ≤ 10−3a∗(θ0)Ra(θ0), ∀µn ∈ Ωn. (26)

Figures 8(a) and 8(b) show the beampatterns corresponding to the CA synthesized waveforms obtained

under PAR= 1 and PAR≤ 1.1, respectively. Similar remarks to those on Figure 7 can be made in this

case as well.

C. Waveform Diversity-Based Ultrasound Hyperthermia

In this final example, we consider an application of the waveform diversity-based transmit beampattern

design to the treatment of breast cancer via ultrasound hyperthermia. Of all women diagnosed with

breast cancer, 20% have locally advanced disease and even with aggressive treatments, the risk of distant

metastases remains high. Thermal therapy provides a good treatment option for this type of cancer: the

breast tumor is heated [36], and the resulting heat distribution sensitizes tumor tissues to the anti-cancer

effects of ionizing radiation or chemotherapy [37], [38], [39]. Thermal therapy can also help achieve

targeted drug delivery.

A challenge in the local hyperthermia treatment of breast cancer is heating the malignant tumors to a

temperature above43C for about thirty to sixty minutes, while maintaining a normal temperature level in

the surrounding healthy breast tissue region. Ultrasound arrays have been recently used for hyperthermia

treatment because they can provide satisfactory penetration depths in the human tissue. Note that the

elemental power of an ultrasound array must be limited to avoid burning healthy tissue. As a result,

a large aperture array is needed to deliver sufficient energy for heating the tumor without harming the

healthy tissue. However, due to the short wavelength of the ultrasound, the focal spots generated by a

large ultrasound array are relatively small and therefore hundreds of focal spots are required for complete

tumor coverage, which results in excessively long treatment times.

14

We have shown recently that flexible transmit beampattern design schemes can provide a sufficiently

large focal spot under a uniform elemental power constraint, which can lead to more effective breast

cancer therapies [30]. In the cited reference, the goal of the transmit beampattern design was to focus the

acoustic power onto the entire tumor region while minimizing the peak power level in the surrounding

healthy breast tissue region, under a uniform elemental power constraint. The beampattern design problem

is therefore to choose the covariance matrixR of the transmitted waveforms to achieve the following

goals:

(a) Realize a predetermined main-beam width that is matched to the entire tumor region; in the said

region the power should be within10% of the power deposited at the tumor center;

(b) Minimize the peak sidelobe level in a prescribed region (the surrounding healthy breast tissue

region).

This problem can be mathematically formulated as:

mint,R

−t

s.t. a∗(θ0)Ra(θ0)− a∗(µ)Ra(µ) ≥ t, ∀ µ ∈ ΩB

a∗(ν)Ra(ν) ≥ 0.9a∗(θ0)Ra(θ0), ∀ ν ∈ ΩT

a∗(ν)Ra(ν) ≤ 1.1a∗(θ0)Ra(θ0), ∀ ν ∈ ΩT

R ≥ 0

Rnn =C

N, n = 1, 2, · · · , N, (27)

whereθ0 is the tumor center location (θ0 is now a coordinate vector), andΩT andΩB denote the tumor

and the surrounding healthy breast tissue regions, respectively. Once the optimalR has been determined,

we use CA to synthesize the waveform matrixX under the constant-modulus constraint (PAR =1).

We simulated a 2D breast model, as shown in Figure 9. The breast model is a semicircle with a 10

cm diameter, which includes breast tissues, skin, chest wall, and a 16 mm diameter tumor whose center

is located atx = 0 mm, y = 50 mm. There are 51 acoustic transducers arranged in a uniform array, as

15

shown in the figure, with half wavelength (relative to the carrier frequency) inter-element spacing. The

dots in Figure 9 mark the locations of the acoustic transducers. The sample numberL is chosen to be

128. The finite-difference time-domain (FDTD) method [30] is used to simulate the power densities and

temperature distributions inside the breast model when the synthesized waveforms are transmitted via the

acoustic transducers.

Figure 10 shows the actual PAR values ofthe CA synthesized waveforms with optimalR. Figure 11

shows the temperature distributions within the breast model, with Figure 11(a) corresponding tothe CA

synthesized waveforms with PAR= 1 and Figure 11(b) tothe CA synthesized waveforms with optimalR.

As shown in Figures 11(a) and 11(b), by transmitting either of the synthesized diversity-based waveforms,

the entire tumor region is heated to a temperature equal to or greater than43C, while the temperature

of the surrounding normal tissues is below40C. In contrast with this, when a phased-array is used for

transmission and the delay-and-sum technique is employed to ensure that the energy is focused on the

tumor center, the temperature distribution is far from satisfactory (see [30]).

V. CONCLUDING REMARKS

We have considered the problem of waveform synthesis for diversity-based flexible transmit beampattern

designs. Optimization of a performance metricdirectly with respect to the signal matrix can lead to an

intractable problem even under a relatively simple low PAR constraint. For this reason, we proposed the

following strategy:first optimize the performance metric of interest with respect to the signal covariance

matrix R; and thensynthesize a signal waveform matrix that, under the low PAR constraint, realizes (at

least approximately) the optimal covariance matrix derived in the first step. We have presented a cyclic

optimization algorithm for the synthesis of a signal waveform matrix to (approximately) realize a given

covariance matrixR under the constant-modulus constraint or the low PAR constraint. The output of the

cyclic algorithm can be used to obtain either a waveform matrix whose covariance matrix is exactly equal

to R but whose PAR is slightly larger than the imposed value, or a waveform matrix with the imposed

PAR but whose covariance matrix may differ slightly fromR – the type of application will dictate which

16

one of these two kinds of waveforms will be more useful. A number of numerical examples have been

provided to demonstrate that the proposed algorithm for waveform synthesis is quite effective.

APPENDIX: ON ENFORCINGSOLELY THE PAR CONSTRAINT

Consider the following generic form of the problem:

mins‖s− z‖2 s.t. PAR(s) ≤ ρ, (28)

wherez is given and PAR(s) is as defined in (15). Hence we have omitted the power constraint (16),

which should lead to a smaller matching error. Because PAR(s) is insensitive to the scaling ofs, let us

parameterizes as

s = cx; ‖x‖2 = 1; wherec ≥ 0 is a variable. (29)

Using (29) in (28) yields:

‖s− z‖2 = ‖cx− z‖2 = c2 − 2cRe(x∗z) + const. (30)

If Re(x∗z) ≤ 0, then the minimum value of (30) with respect toc ≥ 0 occurs atc = 0. If Re(x∗z) > 0,

then the minimization of (30) with respect toc ≥ 0 gives:

c = Re(x∗z), (31)

and the value of (30) corresponding to (31) is smaller than the value associated withc = 0. Because

PAR(s) = PAR(x) does not depend on the phases of the elements ofx, we can always choosex such

that Re(x∗z) > 0 – so that we achieve a smaller value of (30). Consequently, the minimizing valuec ≥ 0

of (30) is always given by (31). The remaining problem is:

maxx

Re(x∗z) s.t. ‖x‖2 = 1 and PAR(x) ≤ ρ, (32)

or equivalently

minx‖x− z‖2 s.t. ‖x‖2 = 1 and PAR(x) ≤ ρ, (33)

17

which has the form required by the algorithm of [32], [33]. Therefore, we can solve (33) using the said

algorithm and then computes = cx with c given by (31).

The alternative discussed in Sec. III is to constrain‖s‖2 = ‖z‖2 (which is the case when we choose

γ = Rnn in (16)). The use of this constraint is logical if we want to matchRnn exactly (for strict

transmission power control, for example). However, if matchingRnn exactly is not a necessary condition,

then a smaller matching error betweens andz is obtained using (31) and (33).

REFERENCES

[1] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output (MIMO) radar and imaging: degrees of freedom and resolution,”37thAsilomar Conference on Signals, Systems and Computers,Pacific Grove, CA, vol. 1, pp. 54–59, November 2003.

[2] D. J. Rabideau and P. Parker, “Ubiquitous MIMO multifunction digital array radar,”37th Asilomar Conference on Signals, Systemsand Computers,Pacific Grove, CA, November 2003.

[3] I. Bekkerman and J. Tabrikian, “Spatially coded signal model for active arrays,”The 2004 IEEE International Conference on Acoustics,Speech, and Signal Processing,Montreal, Quebec, Canada, vol. 2, pp. ii/209–ii/212, March 2004.

[4] E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R. Valenzuela, “MIMO radar: an idea whose time has come,”Proceedingsof the IEEE Radar Conference, pp. 71–78, April 2004.

[5] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Performance of MIMO radar systems: advantages ofangular diversity,”38th Asilomar Conference on Signals, Systems and Computers,Pacific Grove, CA, vol. 1, pp. 305–309, November2004.

[6] K. Forsythe, D. Bliss, and G. Fawcett, “Multiple-input multiple-output (MIMO) radar: performance issues,”38th Asilomar Conferenceon Signals, Systems and Computers,Pacific Grove, CA, vol. 1, pp. 310–315, November 2004.

[7] D. R. Fuhrmann and G. San Antonio, “Transmit beamforming for MIMO radar systems using partial signal correlations,”38th AsilomarConference on Signals, Systems and Computers,Pacific Grove, CA, vol. 1, pp. 295–299, November 2004.

[8] L. B. White and P. S. Ray, “Signal design for MIMO diversity systems,”38th Asilomar Conference on Signals, Systems and Computers,Pacific Grove, CA, vol. 1, pp. 973 – 977, November 2004.

[9] F. C. Robey, S. Coutts, D. Weikle, J. C. McHarg, and K. Cuomo, “MIMO radar theory and exprimental results,”38th AsilomarConference on Signals, Systems and Computers,Pacific Grove, CA, vol. 1, pp. 300–304, November 2004.

[10] J. Tabrikian and I. Bekkerman, “Transmission diversity smoothing for multi-target localization,”The 2005 IEEE International Conferenceon Acoustics, Speech, and Signal Processing,Philadelphia, PA, vol. 4, pp. iv/1041–iv/1044, March 2005.

[11] K. W. Forsythe and D. W. Bliss, “Waveform correlation and optimization issues for MIMO radar,”39th Asilomar Conference onSignals, Systems and Computers,Pacific Grove, CA, pp. 1306–1310, November 2005.

[12] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Spatial diversity in radars - models and detectionperformance,”IEEE Transactions on Signal Processing, vol. 54, pp. 823–838, March 2006.

[13] J. Li and P. Stoica, “MIMO radar – diversity means superiority,”The Fourteenth Annual Workshop on Adaptive Sensor Array Processing(invited), MIT Lincoln Laboratory, Lexington, MA, June 2006.

[14] J. Tabrikian, “Barankin bounds for target localization by MIMO radars,”4th IEEE Workshop on Sensor Array and Multi-channelProcessing,Waltham, MA, July 2006.

[15] L. Xu, J. Li, and P. Stoica, “Adaptive techniques for MIMO radar,”4th IEEE Workshop on Sensor Array and Multi-channel Processing,Waltham, MA, July 2006.

[16] V. F. Mecca, D. Ramakrishnan, and J. L. Krolik, “MIMO radar space-time adaptive processing for multipath clutter mitigation,”4thIEEE Workshop on Sensor Array and Multi-channel Processing,Waltham, MA, July 2006.

[17] L. Xu, J. Li, and P. Stoica, “Radar imaging via adaptive MIMO techniques,”14th European Signal Processing Conference,(invited),Florence, Italy, September 2006.

[18] J. Li, P. Stoica, and Y. Xie, “On probing signal design for MIMO radar,”40th Asilomar Conference on Signals, Systems and Computers(invited), Pacific Grove, CA, October 2006.

[19] Y. Yang and R. S. Blum, “MIMO radar waveform design based on mutual information and minimum mean-square error estimation,”IEEE Transactions on Aerospace and Electronic Systems, vol. 43, pp. 330–343, January 2007.

[20] T. Aittomaki and V. Koivunen, “Low-complexity method for transmit beamforming in MIMO radars,”The 2007 IEEE InternationalConference on Acoustics, Speech, and Signal Processing,Honolulu, Hawaii, USA, April 2007.

[21] L. Xu, J. Li, P. Stoica, K. W. Forsythe, and D. W. Bliss, “Waveform optimization for MIMO radar: A Cramer-Rao bound based study,”2007 IEEE International Conference on Acoustics, Speech, and Signal Processing,Honolulu, Hawaii, April 2007.

[22] P. Stoica, J. Li, and Y. Xie, “On probing signal design for MIMO radar,”IEEE Transactions on Signal Processing, vol. 55, pp. 4151–4161, August 2007.

[23] I. Bekkerman and J. Tabrikian, “Space-time coding for active arrays,”IEEE Transactions on Signal Processing, to appear.[24] J. Li, P. Stoica, L. Xu, and W. Roberts, “On parameter identifiability of MIMO radar,”IEEE Signal Processing Letters,to appear,

available at http://www.sal.ufl.edu/sal/MIMOIdentifiability.pdf.

18

[25] D. R. Fuhrmann and G. San Antonio, “Transmit beamforming for MIMO radar systems using signal cross-correlation,”IEEETransactions on Aerospace and Electronic Systems, to appear.

[26] L. Xu, J. Li, and P. Stoica, “Adaptive MIMO radar,”IEEE Transactions on Aerospace and Electronic Systems,to appear, available athttp://www.sal.ufl.edu/sal/MIMORadarAES.pdf.

[27] J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: A Cramer-Raobound based study,”IEEE Transactions on Signal Processing,to appear, available at http://www.sal.ufl.edu/sal/CRBOPT.pdf.

[28] J. Li and P. Stoica, “MIMO radar with colocated antennas: Review of some recent work,”IEEE Signal Processing Magazine,to appear,available at http://www.sal.ufl.edu/sal/SPMMIMO.pdf.

[29] J. Li and P. Stoica, eds.,MIMO Radar Signal Processing. New York, NY: John Wiley & Sons, Inc., to appear in 2008.[30] B. Guo and J. Li, “Waveform diversity based ultrasound system for hyperthermia treatment of breast cancer,”IEEE Transactions on

Biomedical Engineering, to appear.[31] A. Edelman, T. Arias, and S. Smith, “The geometry of algorithms with orthogonality constraints,”SIAM Journal on Matrix Analysis

and Applications, vol. 20, no. 2, pp. 303–353, 1998.[32] J. A. Tropp, I. S. Dhillon, R. W. Heath, and T. Strohmer, “CDMA signature sequences with low peak-to-average-power ratio via

alternating projection,”37th Asilomar Conference on Signals, Systems and Computers,Pacific Grove, CA, vol. 1, pp. 475–479, November2003.

[33] J. A. Tropp, I. S. Dhillon, R. W. Heath, and T. Strohmer, “Designing structured tight frames via an alternating projection method,”IEEE Transactions on Information Theory, vol. 51, pp. 188–209, January 2005.

[34] C. R. Rao, “Matrix approximations and reduction of dimensionality in multivariate statistical analysis.” InMultivariate Analysis,P. R.Krishnaiah (ed.), vol. 5, pp. 3–22. Amsterdam: North-Holland Publishing Company, 1980.

[35] R. A. Horn and C. R. Johnson,Matrix Analysis. Cambridge, U.K.: Cambridge University Press, 1985.[36] National Cancer Institute, “Hyperthermia in cancer treatment: Questions and answers,” available at

http://www.cancer.gov/cancertopics/factsheet/Therapy/hyperthermia.[37] C. C. Vernon, J. W. Hand, S. B. Field, D. Machin, J. B. Whaley, J. Van Der Zee, W. L. J. Van Putten, G. C. Van Rhoon, J. D. P. Van

Dijk, D. G. Gonzalez, F. Liu, P. Goodman, and M. Sherar, “Radiotherapy with or without hyperthermia in the treatment of superficiallocalized breast cancer: Results from five randomized controlled trials,”International Journal of Radiation Oncology Biology Physics,vol. 35, pp. 731–744, July 1996.

[38] M. H. Falk and R. D. Issels, “Hyperthermia in oncology,”International Journal of Hyperthermia, vol. 17, pp. 1–18, January 2001.[39] E. L. Jones, J. R. Oleson, L. R. Prosnitz, T. V. Samulski, Z. Vujaskovic, D. Yu, L. L. Sanders, and M. W. Dewhirst, “Randomized trial

of hyperthermia and radiation for superficial tumors,”Journal of Clinical Oncology, vol. 23, pp. 3079–3085, 2005.

19

−50 0 500

0.5

1

1.5

2

2.5

3

Angle (degree)

Bea

mpa

ttern

DesiredCA: Optimal RCA: PAR = 1

(a)

−50 0 500

0.5

1

1.5

2

2.5

3

Angle (degree)

Bea

mpa

ttern

DesiredCA: Optimal RCA: PAR ≤ 1.1

(b)

20

−50 0 500

0.5

1

1.5

2

2.5

3

Angle (degree)

Bea

mpa

ttern

DesiredCA: Optimal RCA: PAR ≤ 2

(c)

Fig. 1. Beampattern matching design with the desired main-beam width of 60 and under the uniform elemental power constraint. Theprobing signals are synthesized forN = 10 andL = 256 by using CA under (a) PAR= 1 (resulting inδ = −29.7891 dB), (b) PAR≤ 1.1(resulting inδ = −41.7237 dB), and (c) PAR≤ 2 (resulting inδ = −119.5251 dB).

2 4 6 8 10−1

0

1

2

3

4

5

6

7

8

Index of Transmit Antenna

PA

R

CA (PAR = 1): Optimal RCA (PAR ≤ 1.1): Optimal RCA (PAR ≤ 2): Optimal RColored Hadamard

Fig. 2. PAR values for CA synthesized waveforms with optimalR and for colorized Hadamard code.

21

−50 0 50

−0.2

−0.1

0

0.1

0.2

0.3

Angle (degree)

Bea

mpa

ttern

Diff

eren

ce

(a)

−50 0 50

−0.2

−0.1

0

0.1

0.2

0.3

Angle (degree)

Bea

mpa

ttern

Diff

eren

ce

(b)

22

−50 0 50

−0.2

−0.1

0

0.1

0.2

0.3

Angle (degree)

Bea

mpa

ttern

Diff

eren

ce

(c)

Fig. 3. Differences between the beampatterns obtained from optimalR and the CA synthesized waveforms under (a) PAR= 1, (b)PAR≤ 1.1, and (c) PAR≤ 2.

101

102

103

−32

−31

−30

−29

−28

−27

−26

−25

−24

−23

Sample Number L

MS

E (

dB)

Fig. 4. MSE of the difference betweenR andR (CA synthesized constant modulus waveforms) as a function of sample numberL obtainedwith 100 Monte-Carlo trials.R is obtained from the CA synthesized constant modulus waveforms.

23

−50 0 500

0.5

1

1.5

2

2.5

3

3.5

Angle (degree)

Bea

mpa

ttern

DesiredCA: Optimal RCA: PAR = 1

(a)

−50 0 500

0.5

1

1.5

2

2.5

3

3.5

Angle (degree)

Bea

mpa

ttern

DesiredCA: Optimal RCA: PAR ≤ 1.1

(b)

Fig. 5. Beampattern matching design with each desired beam width of 20 and under the uniform elemental power constraint. The probingsignals are synthesized forN = 10 andL = 256 by using CA under (a) PAR= 1 and (b) PAR≤ 1.1.

24

−50 0 50−30

−25

−20

−15

−10

−5

0

5

10

15

Angle (degree)

Bea

mpa

ttern

(dB

)

CA: Optimal RCA: PAR = 1

(a)

−50 0 50−30

−25

−20

−15

−10

−5

0

5

10

15

Angle (degree)

Bea

mpa

ttern

(dB

)

CA: Optimal RCA: PAR ≤ 1.1

(b)

Fig. 6. Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 20 and under the relaxed elemental powerconstraint. The probing signals are synthesized forN = 10 andL = 256 by using CA under (a) PAR= 1 and (b) PAR≤ 1.1.

25

−50 0 50

−40

−30

−20

−10

0

10

Angle (degree)

Bea

mpa

ttern

(dB

)

CA: Optimal RCA: PAR = 1

(a)

−50 0 50

−40

−30

−20

−10

0

10

Angle (degree)

Bea

mpa

ttern

(dB

)

CA: Optimal RCA: PAR ≤ 1.1

(b)

26

−50 0 50

−40

−30

−20

−10

0

10

Angle (degree)

Bea

mpa

ttern

(dB

)

CA: Optimal RCA: PAR ≤ 1.2

(c)

Fig. 7. Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 20 and a−40 dB null at−30, under the relaxedelemental power constraint. The probing signals are synthesized forN = 10 andL = 256 by using CA under (a) PAR= 1, (b) PAR≤ 1.1,and (c) PAR≤ 1.2.

27

−50 0 50−40

−30

−20

−10

0

10

Angle (degree)

Bea

mpa

ttern

(dB

)

CA: Optimal RCA: PAR = 1

(a)

−50 0 50−40

−30

−20

−10

0

10

Angle (degree)

Bea

mpa

ttern

(dB

)

CA: Optimal RCA: PAR ≤ 1.1

(b)

Fig. 8. Minimum sidelobe beampattern design with the 3-dB main-beam width equal to 20 and a null from−55 to −45, under therelaxed elemental power constraint. The power gain difference between 0 and the null is constrained to be less than or equal to 30 dB. Theprobing signals are synthesized forN = 10 andL = 256 by using CA under (a) PAR= 1 and (b) PAR≤ 1.1.

28

x (mm)

y (m

m)

−60 −40 −20 0 20 40 60

10

20

30

40

50

60

70

80Acoustic transducer array

Breast

Tumor

Chest wall

Fig. 9. Breast model.

10 20 30 40 50−2

−1

0

1

2

3

4

5

Index of Acoustic Transducer

PA

R

Fig. 10. PAR values for CA synthesized waveforms with optimalR.

29

x (mm)

y (m

m)

36

36

36

36

36

3939

39

39

41

41

42

42

43

43

20 40 60 80 100 120 140

10

20

30

40

50

60

70

80

34

36

38

40

42

C

(a)

x (mm)

y (m

m)

36

36

36

36

36

39

39

39

39

41

41

42

42

43

43

20 40 60 80 100 120 140

10

20

30

40

50

60

70

80

34

36

38

40

42

C

(b)

Fig. 11. Temperature distribution forN = 50 andL = 128. (a): CA synthesized constant modulus signals, and (b): CA synthesized signalswith optimal R).