IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL...

IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007 67

Optimal Sequential Energy Allocationfor Inverse Problems

Raghuram Rangarajan, Student Member, IEEE, Raviv Raich, Member, IEEE, and Alfred O. Hero, III, Fellow, IEEE

Abstract—This paper investigates the advantages of adaptivewaveform amplitude design for estimating parameters of anunknown channel/medium under average energy constraints. Wepresent a statistical framework for sequential design (e.g., designof waveforms in adaptive sensing) of experiments that improvesparameter estimation (e.g., unknown channel parameters) perfor-mance in terms of reduction in mean-squared error (MSE). Wetreat an time step design problem for a linear Gaussian modelwhere the shape of the input design vectors (one per time step)remains constant and their amplitudes are chosen as a function ofpast measurements to minimize MSE. For = 2, we derive theoptimal energy allocation at the second step as a function of thefirst measurement. Our adaptive two-step strategy yields an MSEimprovement of at least 1.65 dB relative to the optimal nonadap-tive strategy, but is not implementable since it requires knowledgeof the noise amplitude. We then present an implementable designfor the two-step strategy which asymptotically achieves optimalperformance. Motivated by the optimal two-step strategy, wepropose a suboptimal adaptive -step energy allocation strategythat can achieve an MSE improvement of more than 5 dB for

= 50. We demonstrate our general approach in the context ofMIMO channel estimation and inverse scattering problems.

Index Terms—Channel estimation, energy management, inversescattering, maximum likelihood, parameter estimation, sequentialdesign.

I. INTRODUCTION

ADAPTIVE sensing has been an important topic of researchfor at least a decade. Many of the classical problems in

statistical signal processing such as channel estimation, radarimaging, target tracking, and detection can be presented in thecontext of adaptive sensing. One of the important components inthese adaptive sensing problems is the need for energy manage-ment. Most applications are limited by peak power or averagepower. For example, in sensor network applications, sensorshave limited battery life and replacing them is expensive. Safetylimits the peak transmit power in medical imaging problems.Energy is also a critical resource in communication systemswhere reliable communication is necessary at low signal-to-noise ratios. Hence, it is important to consider energy limitationsin waveform design problems. Most of the effort in previous re-search has focussed on waveform design under peak power con-straints, e.g., sensor management. There has been little effort

Manuscript received September 1, 2006; revised February 14, 2007. Thiswork was supported in part by AFOSR MURI under Grant FA9550-06-1-0324.The associate editor coordinating the review of this manuscript and approvingit for publication was Prof. Antonia Papandreou-Suppappola.

The authors are with the Department of Electrical Engineering and Com-puter Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/JSTSP.2007.897049

in developing adaptive waveform design strategies that allocatedifferent amounts of energy to the waveforms over time. Ourgoal in this paper is to perform waveform amplitude design foradaptive sensing in order to estimate the set of unknown channelparameters or scattering coefficients under an average energyconstraint. We formulate this problem as an experimental designproblem in the context of sequential parameter estimation. Weexplain the methodology of experimental design, derive optimaldesigns, and show performance gains over nonadaptive designtechniques. As a final step, we describe in detail how some appli-cations of adaptive sensing such as channel estimation and radarimaging can be cast into this experimental design setting therebyleading to attractive performance gains compared to current lit-erature. Next, we present a review of waveform design and se-quential estimation literature to provide a context for our work.

Note: The term “sequential” is used in different contexts inthe literature. In this paper, “sequential” means that at every timeinstant, the best signal to transmit is selected from a library thatdepends on past observations.

A. Related Work—Waveform Design

Early work in waveform design focussed on selecting amonga small number of measurement patterns [1]. Radar signal de-sign using a control theoretic approach subject to both averageand peak power constraints was addressed in [2] and [3]. Thedesign was nonadaptive and the optimal continuous waveformswere shown to be on-off measurement patterns alternating be-tween zero and peak power levels for a tracking example. Inour design, the energy allocation to the waveforms over timeare optimally chosen from a continuum of values. Parameter-ized waveform selection for dynamic state estimation was ex-plored in [4] and [5] where the shape of the waveforms wereallowed to vary under constant transmit power. Closed-formsolutions to the parameter selection problem were found for avery restrictive set of cases such as one-dimensional target mo-tions. More recently a dynamic waveform selection algorithmfor tracking using a class of generalized chirp signals was pre-sented in [6]. In contrast to these efforts, we focus our workin finding optimal waveform amplitudes under an average en-ergy constraint for static parameter estimation. Sensor sched-uling can be thought of as an adaptive waveform design problemunder a peak power constraint [7] where the goal is to choose thebest sensor at each time instant to provide the next measurement.The optimal sensor schedule can be determined a priori and in-dependent of measurements for the case of linear Gaussian sys-tems [8], [9]. The problem of optimal scheduling for the case ofhidden Markov model systems was addressed in [10]. In Table I,we compare our work with existing literature via different cat-egories.

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68 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007

TABLE IKEY TO THE TABLE: D-DETERMINISTIC, R-RANDOM, LSD-LINEAR STATE DYNAMICS, NLSD-NON LINEAR STATE DYNAMICS, SQ-SEQUENTIAL DESIGN,

NSQ-NON SEQUENTIAL DESIGN, EN-ENERGY, SN-SENSORS, WV-WAVEFORM PARAMETERS

B. Related Work—Sequential Design for Estimation

The concept of sequential design has been studied by statisti-cians for many decades [17]–[22] and has found applications instatistics, engineering, biomedicine, and economics. Sequentialanalysis has been used to solve important problems in statisticssuch as change-point detection [23], [24] point and interval esti-mation [25], multi-armed bandit problems [26], quality control[27], sequential testing [28], and stochastic approximation [29].Robbins pioneered the statistical theory of sequential allocationin his seminal paper [26]. Early research on the applicationof sequential design to problems of estimation was limitedto finding asymptotically risk-efficient point estimates andfixed-width confidence intervals [11], [12], [30], i.e., sequentialdesign was used to solve problems in which a conventionalestimate, based on a sample whose size is determined by asuitably chosen stopping rule, achieves certain properties suchas bounded risk. For the problem of estimating the mean underunknown variance, it was shown that a sequential two-stepmethod guaranteed specified precision [23], [31], [32], whichis not possible using a fixed sample. The statistical sequentialdesign framework assumes a fixed measurement setup whileacquiring the data and does not consider energy constraints. Inthis paper, we adaptively design input parameters to alter themeasurement patterns under an average energy constraint toobtain performance gains over nonadaptive strategies.

Another class of problems in sequential estimation is onlineestimation, where fast updating of parameter estimates are madein real time, called recursive identification in control theory, andadaptive estimation in signal processing. For example, considerthe problem of estimating parameter in the following model

where are the sequence of inputs to the system,are independent identically distributed (i.i.d) Gaussianrandom variables with zero mean, and are the setof received signals. The maximum likelihood (ML) es-timate of is given by the least squares (LS) solution,

. One way of computingthe LS estimate is the recursive least squares approach (RLS)[13] which can be written as

where . The recursive process avoids thecomputational complexity of inverting the matrix.

In the above formulation it was assumed that the inputsequence remains fixed. The problem of waveform designis relevant when input can be adaptively chosen based onthe past measurements . Measurement-adaptiveestimation has application to a wide variety of areas such ascommunications and control, medical imaging, radar systems,system identification, and inverse scattering. By measure-ment-adaptive estimation we mean that one has control overthe way measurements are made, e.g., through the selection ofwaveforms, projections, or transmitted energy. The standardsolution for estimating parameters from adaptive measurementsis the ML estimator. For the case of classic linear Gaussianmodel, i.e., a Gaussian observation with unknown mean andknown variance, it is well-known [16] that the ML estimatoris unbiased and achieves the unbiased Cramér Rao lowerbound (CRB). Many researchers have looked at improving theestimation of these parameters by adding a small bias to reducethe MSE. Stein showed that this leads to better estimators thatachieve lower MSE than the ML estimator for estimating themean in a multivariate Gaussian distribution with dimensiongreater than two [14], [15]. Other alternatives such as theshrinkage estimator [33], Tikhonov regularization [34], andcovariance shaping least squares (CSLS) estimator [35] havealso been proposed in the literature. While these pioneeringefforts present interesting approaches to improve static param-eter estimation performance by introducing bias, none of themincorporate the notion of sequential design of input parameters.Our adaptive design of inputs effectively adds bias to achievereduction in MSE.

In this paper, we formulate a problem of sequentially se-lecting waveform amplitudes for estimating parameters ofa linear Gaussian channel model under an average energyconstraint over the waveforms and over the number of trans-missions. In Section II, the problem of experimental design[36], [37] for sequential parameter estimation is outlined andthe analogy between this problem and the waveform designproblem is explained. In Section III, closed-form expressionsfor the optimal design parameters and the correspondingminimum MSE in the scalar parameter case are derived for atwo-step procedure (two time steps). Since the optimal solutionrequires the knowledge of the parameter to be estimated, it isshown in Section IV that the performance of this omniscient so-lution can be achieved with a parameter independent strategy.

RANGARAJAN et al.: OPTIMAL SEQUENTIAL ENERGY ALLOCATION 69

In Section IV-A, we describe an -step sequential energyallocation procedure, which yields more than 5 dB gain overnonadaptive methods. These results are extended to the vectorparameter case in Section V. Finally in Section VI, we showthe applicability of this framework for channel estimation andradar imaging problems.

II. PROBLEM STATEMENT

We begin by introducing nomenclature commonly usedthroughout the paper. We denote vectors in by boldfacelowercase letters and matrices in by boldface uppercaseletters. The identity matrix is denoted by . We use and

to denote the transpose and conjugate transpose opera-tors, respectively. We denote the -norm of a vector by ,i.e., . A circularly symmetric complex Gaussianrandom vector with mean and covariance matrix is denotedas . and denote the statistical expectationand trace operators, respectively. The terms MSE and SNR areabbreviations to mean-squared error and signal-to-noise ratio,respectively.

Let be the -element vector of unknownparameters. The problem of estimating in noise can then bewritten as

(1)

where is an i.i.d. random process corrupting the function ofthe parameters of interest and denotes the time index.The -element design parameter vectors, can dependon the past measurements: , where isthe th -element observation vector. In the context of adaptivesensing, represents the response of the medium, and

denote the number of transmit and receive antennas respec-tively, are the set of waveforms to be designed, andare the set of channel parameters or scattering coefficients tobe estimated using the set of received signals . For theclassic estimation problem in a linear Gaussian model, we have

isa known matrix and linear in and is arandom vector. When is linear in , we can write

. In this case is uniquelydetermined by the matrices . The linearGaussian model has been widely adopted in many studies[38], [39] including channel estimation [40] and radar imaging[41] problems. The set of observations for the case of a scalarparameter are

(2)

An -step design procedure specifies a sequence of functionscorresponding to the transmitted

signal waveforms after receiving the previous measurements.An optimal -step procedure selects the design vectors sothat the MSE of the ML estimator,is minimized subject to the average energy constraint,

, where is the total available en-ergy. The ML estimator of for the -step procedure is given

by

(3)

and the corresponding MSE is

(4)

Denote , whererepresents the energy allocated to each time

step . Define as the average energyin the design parameters for the -step procedure. The averageenergy constraint can be written as

(5)

Our goal is to find the best sequence of the design vectorsto minimize the in (4) under the

average energy constraint in (5).

A. Nonadaptive Strategy

As a benchmark for comparison, we consider the nonadap-tive case where is deterministic, independentof , and . Simpli-fying the expression for MSE in (4), we have

(6)

where equality is achieved iff , the normalizedeigenvector corresponding to , the maximumeigenvalue of the matrix . Note

. Furthermore,the performance of the ML estimator does not depend on theenergy allocation. Hence, without loss of generality we canassume all energy is allocated to the first transmission whichimplies that any -step nonadaptive strategy is no better thanthe optimal one-step strategy. We define as

(7)

Then the average energy constraint in (5) is equivalentto , where .We show in [42] that the problem of minimizingsubject to is equivalent to minimizing

. Thus, we use the two minimizationcriteria interchangeably in the remainder of this paper. Theproduct of MSE and SNR is

(8)


and the minimum MSE for the one-step (or nonadaptive -step)strategy satisfies

(9)

While our goal is to find optimal input design parameters,which achieve minimum MSE, any

suboptimal design that guarantees isalso of interest. We first look at a two-step sequential designprocedure. A word of caution: in Sections III and III-A wedevelop optimal and suboptimal strategies where the solutionsrequire the knowledge of the unknown parameter . However,in Section IV we present a -independent design which asymp-totically achieves the performance of the ‘omniscient’ strategies.

III. OMNISCIENT OPTIMAL TWO-STEP SEQUENTIAL STRATEGY

In the two-step sequential procedure, we have timesteps where in each time step , we can control inputdesign parameter to obtain observation . For a two-stepprocess, we have

(10)

(11)

The ML estimator of for a two-step procedure from (3) is

(12)

and its MSE from (4) is given by

(13)

We assume that the shape of the optimal designs, i.e.,is the one-step optimum given by defined below (6) andminimize the MSE over the energy of the design parameters.Denote and . Underthe sequential design framework, we select

(14)

(15)

where and are real-valued scalars. The average energyconstraint from (5) can then be written as

(16)

We use Lagrangian multipliers to minimize the MSE in (13)with respect to and under the energy constraint in (16).The objective function to be minimized can then be written as

Fig. 1. Plot of the optimal and suboptimal solution to the normalized energytransmitted at the second stage as functions of received signal at first stage.

In [42], we show that the optimal solution to dependson only through the function , where

(17)

. Hence, we denote the solution as .Let . Setting thederivative of the objective function with respect to to zeroyields

(18)

where . The function that minimizes MSE isthe root of the third-order polynomial in (18), real-valued, andgreater than or equal to 1. If more than one real-valued solutiongreater than 1 to the cubic equation exists, the optimal solutionto will be the root that achieves minimum MSE. The optimal

for every and is denoted by . Also. Therefore, finding that minimizes MSE is equiva-

lent to finding that minimizes MSE. We obtain forevery and use a brute force grid search to find the optimalthat minimizes the objective function. The MSE is minimizedat , or . The optimal is given by therelation . Sincethis solution depends on the unknown parameter , we call thisminimizer an “omniscient” energy allocation strategy. For theoptimal solution, the product of is

(19)

This corresponds to a 32% improvement in performance or a1.67 dB gain in terms of SNR for the two-step design when com-pared to the one-step procedure for which

.The optimal energy allocation at the second step,

as shown in Fig. 1 (solid) is a thresholdingfunction, i.e., is zero for . This solution implies


that when the actual realization of the normalized noise alongin the first step is small enough, then the second

measurement becomes unnecessary. On the other hand, whenthe normalized noise along exceeds a threshold, thenthere is some merit in incorporating the information from thesecond measurement. The solution also suggests that the higherthe noise magnitude at the first step, the more the energy thatneeds to be used. However, the probability of allocating energygreater than a particular value decreases exponentially withthat energy value. Nevertheless in applications with a peakenergy constraint, the transmission of the optimal energy atthe second stage may not always be possible. Hence, in thefollowing subsection we look at a suboptimal solution whichtakes into account this constraint and still achieves near optimalperformance.

A. Omniscient Suboptimal Two-Step Strategy

The optimal solution in Section III is a thresholding function,where energy allocated to the second stage is zero if the noisemagnitude at the first step is less than a threshold and increaseswith increasing noise magnitudes otherwise. For the suboptimalsolution, we use a binary energy allocation strategy at the secondstage based on the noise magnitude at the first step, i.e., weallocate a fixed nonzero energy if the noise magnitude is greaterthan a threshold else we allocate zero energy. The suboptimalsolution to the design vectors and is then of the form

(20)

(21)

where is defined in (17), are design parameters inde-pendent of and is the indicator function, i.e.,

The SNR of the suboptimal two-step procedure is

(22)

The MSE of the ML estimator under this suboptimal solutionusing (13) is

(23)

Denote . Substitutingfor in the expressions for and in (23)and (22), and using the fact that and

when , the expres-sion simplifies to

(24)

Minimizing with respect to and througha grid search for and yields

. It follows that and .Substituting for the optimal values of in (24) and(22), and simplifying yields

(25)

This translates to a 28.57% improvement in MSE performanceor a 1.5 dB savings in terms of SNR. The suboptimal solutionto the energy design is shown in Fig. 1 by a dashed dotted lineindicated as Suboptimal-I. Thus, while the suboptimal strategylimits the peak transmit power to , it is able toachieve near optimal performance.

In the previous section, we addressed the problem ofminimizing MSE subject to an average energy constraint,

. An average energy constraint impliesthat the total allocated energy averaged over repeated trials ofthe two-step experiment is constrained to be less than or equalto . This is less restrictive than the strict energy constraint

, as any solution satisfying this constraintsatisfies the average energy constraint but not vice versa. Theproblem of minimizing the MSE in (13) under this strict energyconstraint was addressed in the context of radar imaging in[43]. We show in [42] that the optimal two-step design underthe strict constraint is given by

where , and . Theminimum MSE is then given by .The optimal solution satisfies the strict energy constraint withequality but the average energy used is

. The solution to the two-step strategy under this strictenergy constraint can also be derived by imposing an additionalconstraint, to the suboptimal design problem de-scribed earlier in this section. In the following section, we de-sign a -independent design strategy that achieves the optimalperformance asymptotically and allows for any peak power con-straint in the design.

IV. PARAMETER INDEPENDENT TWO-STEP DESIGN STRATEGY

Consider the optimal design for the two-step procedure

(26)

We showed that by designing and optimally we cangain up to 32% improvement in estimator performance. Butthe “omniscient” solution (26) depends on the parameter to beestimated. Here, we prove that we can approach the optimaltwo-step gain by implementing a -independent energy allo-cation strategy when is bounded, i.e.,

. We describe the intuition behind the proposed


Fig. 2. Plot of reduction in MSE versus percentage error in the guess of param-eter � for various SNR.

solution in this section. The details of the proof can be found in[42]. Since we do not know the value of the actual parameter,we replace by a “guess” of , say , in the optimal solutionto the design at the second step given in (26). The resultingsuboptimal design is

(27)

(28)

where

(29)

and , which is defined in (17) is . Substi-tuting the above suboptimal solution in the expression for

in (8) and simplifying, we obtain

(30)

Fig. 2 shows in (30) as a function of the percentage errorin the guess of for varying . Theplot indicates that when , the optimal performance ofthe adaptive two-step strategy is achieved for all SNR. At highSNR, for certain values of , the two-step strategy de-fined by (27) and (28) performs worse than a single step strategywith signal-to-noise ratio . This is because the solutionpresented in (27) and (28) in terms of scalar and thresholdingfunction were optimized for , i.e.,

when . When , the following happens:, and the design parameters and ,

which were found optimally for areno longer optimal. When is large, in (29) is a largeconstant and hence is a negligible term compared to withhigh probability. In other words, can be made arbi-trarily close to with high probability as tends to infinity.This implies that the strategy becomes equivalent to a two-stepnonadaptive strategy with a specific nonadaptive energy distri-bution between the two steps whose performance is given by

from Section II-A. Thus, we observe thatthe performance of the two-step strategy tends to 1 for large

.The optimal solution to is achieved

when . There are two ways that drive . If ,then and we have

, the optimal two-step performance. Since is arbitrary,; the two-step design is not optimal and therefore

. The other way to achieve theoptimal solution is to make as small as possible. Note thatif is sufficiently small approaches itsminimal value. Since , driving the tozero, drives the to infinity. To overcome this problem,we propose an -step procedure to allow the to befixed while driving . The -step algorithm is outlinedin Fig. 3. Any peak power constraint can also be satisfied usingthe -step strategy by choosing a sufficiently large . Themost important information in Fig. 2 is the performance of thetwo-step strategy under the low SNR regime since each 2-stepprocedure in the -step strategy works at th of thetotal SNR. Hence, as becomes large, SNR in each experimentis very small and the lack of knowledge of plays a negligibleeffect on the performance as is made close to zero through theSNR factor.

A. Design of -Step Procedure

In Sections III and IV, we derived the omniscient optimaltwo-step design to minimize the MSE and proved that the op-timal performance can be achieved asymptotically using an

-step strategy. But the -step strategy is a specific case ofa -step design. In this subsection, we generalize the subop-timal solution from the 2-step case to the -step case as follows:we assume that the shape of the design vector is fixed and look atthe energy allocation among the various steps. The set of obser-vations are as defined in (2). Let the shape of the design vector

be and the energy at step , i.e.,. Then

(31)

where are design parameters. This approximate solu-tion is motivated from the suboptimal thresholding solution tothe two-step case derived in Section III-A. Note that the defi-nition of the amplitudes at each stage is recursive, i.e., the am-plitude design depends on past inputs which


Fig. 3. Description of the N� two-step procedure.

Fig. 4. Plot of gains obtained through suboptimalN -step procedure as a func-tion of N through theory and simulations.

in turn depends on . To simplify our analysis, wemake the assumption . Then

(32)

where are i.i.drandom variables. Following the same procedure, a general ex-pression for can be written as

(33)

where . This form statesthat the stopping criteria at time step is when the magnitude ofthe average noise, drops below the threshold . The goalis to minimize of this -step proce-dure with respect to and .

Since there is no closed-form solution to this dimensionaloptimization [42], we evaluate the performance of suboptimal

solutions to the design vectors and . For our simulations,we choose . Further-more, we choose as ,where are optimal values from the suboptimal solutionpresented in Section III-A and is chosen to satisfy the averageenergy constraint. The intuition behind this choice of andis motivated through an asymptotic result derived in [42]. Weevaluate the performance of the -step procedure with theseparameters through theory and verify the theory using simula-tions. Performance gains, (in dB) are presented in Fig. 4.By designing this -step procedure, we are essentially alteringthe Gaussian statistics of the measurement noise to obtain im-provements in performance. In Fig. 5, we illustrate how the dis-tribution of the estimation residuals changes with the numberof the steps. We see that in 50 steps, we are able to achievegains of more than 5 dB. In Section IV, we showed that thetwo-step gain can be achieved using an -step strategy,i.e., in steps. The basic motivating factor was to reduce theSNR in each experiment and achieve the diversity gain by in-creasing the number of steps. For the general -step strategy,progressive reduction in SNR of each experiment implies thatas the number of steps increases, the error of guessing has areduced effect on the overall performance. We demonstrate theachievability of performance for any -step design through thefollowing theorem.

Theorem 4.1: For an -step procedure, we need to designa sequence of input vectors optimally under an av-erage energy constraint to minimize the MSE in (4). Let

be any design of the input parame-ters satisfying the following conditions.

• Average energy constraint:.

• Continuity: the design vector is acontinuous function of or can assume the formof a thresholding function in (31).

Then there exists a -independent strategy whose performancecan come arbitrarily close to which assumes theknowledge of parameter .

Proof: The proof is similar to the -step strategy pre-sented in Section IV, where the actual value of in the optimalsolution is replaced with a guess of . Refer to [42] for details.


Fig. 5. Distribution of noise versus number of steps.

V. SEQUENTIAL DESIGN FOR VECTOR PARAMETERS

A general -step procedure for the case of unknown pa-rameters can be written as

(34)

where is an -element vector, , andis a matrix. For the multiple parameter case, MSE isno longer a scalar. Various criteria such as trace, minmax, de-terminant of the MSE matrix can be considered as measures ofperformance under the multiple unknown setting.

A. Trace Criteria

For the multiple parameter case, the MSE is a matrix andwe consider the trace as a measure of performance, i.e.,

. The problem of multiple param-eter estimation is more complicated than estimation of a singleparameter for the following reason. We showed in Section II-Athat independent of the shape of , any nonadaptive energyallocation strategy is to assign all energy to the first step, i.e.,a one-step strategy with energy . But this is not true forthe multiple parameter setting. Let us consider a simple ex-ample of estimating two parameters in the model

, where

(35)

,and . Then for a one-step process, we have

and . Mini-mizing over the

energy constraint , we obtainand . Now consider the following two-stepnonadaptive strategy

Minimizing theover the energy constraint, we obtain

and . This translates to a3 dB gain in SNR for the two-step nonadaptive strategy overthe one-step approach. We control the shape of the input

such that we have different energy allocation for eachcolumn of the matrix . By specifically designing the two-stepnonadaptive strategy given in steps 1 and 2, we have reduced theestimation of the vector parameter to two indepen-dent problems of estimating scalar parameters and respec-tively. For each of these scalar estimators, we design two -stepsequential procedures ( steps in total) as in Section IV-A forscalar controls and to obtain an improvement in perfor-mance of estimating . Applying the -step design to bothand , we have for the firststeps and for the next steps.Hence, , where is definedbelow (33). In other words, the MSE gains of the -step proce-dure carry over to the vector parameter case as well.

B. Worst Case Error—Min Max Approach

The component wise MSE for estimating specific pa-rameters is given by the diagonal elements of the matrix


. We seek to find the op-timal energy allocation between the two design vectors,

that mini-mizes the worst case mean-squared error of theunknown parameters, where is any unit norm vector inde-pendent of past measurements, e.g., is chosen to minimizethe one-step MSE. The ML estimate for a one-step process withenergy and its corresponding MSE are given by

(36)

where

(37)

Define . Then

(38)

where is an -element vector with all zeros except for 1 inthe th position. Then for a one-step process

(39)

where indicates the and

(40)

The set of observations for the two-step process are

(41)

(42)

For a two-step procedure, we need to design and tominimize - . We show in [42] that

(43)

where

(44)

is a random variable. The error in (43) when mini-mized under the constraint is exactly thesame minimization derived for the single parameter case in Sec-tion III. It follows that the optimal and suboptimal solutions to

and will hold for the multiple parameter case. In otherwords . It follows that

(45)

and this performance can be achieved using a -independentstrategy along similar lines to the derivation for the scalar pa-rameter case in Section IV [42]. The reduction in MSE in (45)holds for any , the number of unknown parameters, as , theindex of the worst case error, can always be computed from (39)and (40) for any . A similar result can be derived for the

-step procedure.

VI. APPLICATIONS OF SEQUENTIAL ESTIMATION

A. MIMO Channel Estimation

It has been shown that multiple-input and multiple-outputsystems (MIMO) greatly increase the capacity of wireless sys-tems [44]–[46] and hence MIMO has become an active area ofresearch over the last decade [47], [48]. One important compo-nent in a MIMO system is the need to accurately estimate thechannel state information (CSI) at the transmitter and receiver.This estimate has shown to play a crucial role in MIMO com-munications [49]. A recent and popular approach to channel es-timation has been through the use of training sequences, i.e.,known pilot signals are transmitted and channel is estimatedusing the received data and the pilot signals. A number of tech-niques for performing training based channel estimation havebeen proposed: maximum likelihood training method [50], leastsquares training [51], minimum mean squared estimation [52].Recently, [40] proposed four different training methods for theflat block-fading MIMO system including the least squares andbest linear unbiased estimator (BLUE) approach for the case ofmultiple LS channel estimates.

1) Problem Formulation: In order to estimate thechannel matrix for a MIMO system with transmit andreceive antennas, training vectorsare transmitted. The corresponding set of received signals canbe expressed as [40], [53]

(46)

where is an matrix,is the matrix of sensor noise, is

the complex vector of transmitted signals, and isthe complex zero mean white noise vector. Let bethe transmitted training power constraint, i.e.,

indicates Frobenius norm anddenote the variance of receiver noise. Though is random,

we estimate for a particular realization corresponding to theblock of received data. The task of channel estimation is torecover the channel matrix based on the knowledge ofand as accurately as possible under a transmit power con-straint on . The standard LS solution and the correspondingestimation error can then be written as

(47)

Assuming co-located transmitter and receiver arrays [54],[55] and multiple training periods available within thesame coherency time (quasi-static) to estimate the channel,the set of received signals at the time steps given by


, can be rewritten in thefollowing form:

(48)

wheredenotes the column-wise concatenation of the matrix, and

is a linear function of the input ,which is the same model described in (34). In [40], a methodof linearly combining the estimates from each of the stageswas proposed and the MSE of the stage estimator wasshown to be , where is the totalpower used in the steps, i.e., . If thereare enough training samples, we could completely control thematrix through the input and make orthog-onal. In this case (48) along with the average power constraint

can benefit from adaptive energy allocationdesigns in Sections IV-A and V-A, where the problem is thenseparable into independent estimation problems of scalarparameters. Having steps in the training sequence suggestsan -step energy allocation strategy. Hence, it follows thatusing our strategy we are guaranteed to achieve the optimalerror given by , which we have shownto be at least 5 dB (in 50 steps) better than any nonadaptivestrategy.

B. Inverse Scattering Problem

The problem of imaging a medium using an array of trans-ducers has been widely studied in many research areas such asmine detection, ultrasonic medical imaging [56], foliage pene-trating radar, nondestructive testing [57], and active audio. Thegoal in imaging is to detect and image small scatterers in aknown background medium. A recent approach [58] uses theconcept of time reversal, which works by exploiting the reci-procity of a physical channel, e.g., acoustic, optical, or radio-fre-quency. One implication of reciprocity is that a receiver can re-flect back a time reversed signal, thereby focusing the signalat the transmitter source [59]. Furthermore, with suitable pre-filtering and aperture, the signal energy can also be focused onan arbitrary spatial location. This analysis assumes the noiselessscenario. For the noisy case, ML estimation of point scattererswas performed for both the single scattering and the multiplescattering models in [41]. We apply our concept of designing asequence of measurements to image a medium of multiple scat-terers using an array of transducers under a near-field approxi-mation of the scatterers in the medium.

1) Problem Setting: We have transducers located at po-sitions , that transmit narrowband signals with centerfrequency rad/s. The imaging area (or volume) is divided into

voxels at positions . The channel, denoted , be-tween a candidate voxel and the transducers is given by thehomogeneous Green’s function as

(49)

where is the speed of light and . This channelmodel is a narrowband near-field approximation, which ig-

nores the effect of multiple scattering and has been widelyadopted in other scattering studies, e.g., [60]. Each voxel canbe characterized by its scatter coefficient, e.g., radar crosssection (RCS), , which indicates the proportion of thereceived field that is re-radiated. Thus, the channel between thetransmitted field and the measured backscattered field at thetransducer array is , where

, and denotes a diagonalmatrix with as its th diagonal element.

The probing mechanism for imaging of the scatter cross sec-tion follows a sequential process, generating the following se-quence of noise contaminated signals

(50)

where . The noises are i.i.drandom vectors. The goal is to find estimates for

the scattering coefficients under the average energy constraintto minimize the MSE. If is a square matrix, then we cancondition to have a single non zero component onany one of the diagonal elements, which translates to isolatingthe th column of for any . As in Section V-A, we canperform independent -step experiments to guarantee the

-step gains of at least 5 dB over the standard single step MLestimation for imaging [41]. If we are interested in optimallyestimating any linear combination of the scattering coefficients,then the sequential strategy proposed in Section V-B can beused to achieve improvement in performance.

VII. CONCLUSION

In this paper, we considered the -step adaptive waveformamplitude design problem for estimating parameters of an un-known channel under average energy constraints. For a two-stepproblem, we found the optimal energy allocation at the secondstep as a function of the first measurement for a scalar param-eter in the linear Gaussian model. We showed that this two-stepadaptive strategy resulted in an improvement of at least 1.65dB over the optimal nonadaptive strategy. We then designed asuboptimal -stage energy allocation procedure based on thetwo-step approach and demonstrated gains of more than 5 dBin steps. We extended our results to the case of vectorparameters and provided applications of our design to MIMOand inverse scattering channel models.

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Raghuram Rangarajan (S’04) was born in India in1981. He received the B.Tech. degree from the De-partment of Electrical Engineering, Indian Instituteof Technology, Madras, in 2002 and the M.S. degreein electrical engineering:systems from the Universityof Michigan, Ann Arbor, in 2004. He is currently pur-suing the Ph.D. degree in the Department of Elec-trical Engineering and Computer Science, Universityof Michigan.

His research interests include statistical signalprocessing and design of experiments for sensor

networks and agile sensing applications.


Raviv Raich (S’98–M’04) received the B.Sc.and M.Sc. degrees in electrical engineering fromTel-Aviv University, Tel-Aviv, Israel, in 1994 and1998, respectively, and the Ph.D. degree in electricalengineering from Georgia Institute of Technology,Atlanta, in 2004.

Between 1999 and 2000, he worked as a Re-searcher with the communications team, IndustrialResearch Ltd., Wellington, New Zealand. Currently,he is a Postdoctoral Fellow with the University ofMichigan, Ann Arbor. His main research interest

is in statistical signal processing with specific focus on manifold learning,sparse signal reconstruction, and adaptive sensing. Other research interests liein the area of statistical signal processing for communications, estimation, anddetection theory.

Alfred O. Hero, III (S’79–M’80–SM’96–F’98)received the B.S. degree (summa cum laude) fromBoston University, Boston, MA in 1980 and thePh.D. degree from Princeton University, Princeton,NJ, in 1984, both in electrical engineering.

Since 1984, he has been with the University ofMichigan, Ann Arbor, where he is a Professor in theDepartment of Electrical Engineering and ComputerScience and, by courtesy, in the Department ofBiomedical Engineering and the Department ofStatistics. His recent research interests have been in

areas including inference in sensor networks, adaptive sensing, bioinformatics,inverse problems, and statistical signal and image processing.

Dr. Hero has received a IEEE Signal Processing Society Meritorious ServiceAward (1998), a IEEE Signal Processing Society Best Paper Award (1998), andthe IEEE Third Millenium Medal (2000). He is currently President of the IEEESignal Processing Society (2006–2008).

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