Optimal Sequential Energy Allocation for Inverse Problems

Abstract

This paper investigates the advantages of adaptive waveform amplitude design for estimating parameters of an unknown channel/medium under average energy constraints. We present a statistical framework for sequential design (e.g., design of waveforms in adaptive sensing) of experiments that improves parameter estimation (e.g., unknown channel parameters) performance in terms of reduction in mean-squared error (MSE). We treat an N time step design problem for a linear Gaussian model where the shape of the N input design vectors (one per time step) remains constant and their amplitudes are chosen as a function of past measurements to minimize MSE. For N=2, we derive the optimal energy allocation at the second step as a function of the first measurement. Our adaptive two-step strategy yields an MSE improvement of at least 1.65 dB relative to the optimal nonadaptive strategy, but is not implementable since it requires knowledge of the noise amplitude. We then present an implementable design for the two-step strategy which asymptotically achieves optimal performance. Motivated by the optimal two-step strategy, we propose a suboptimal adaptive N-step energy allocation strategy that can achieve an MSE improvement of more than 5 dB for N=50. We demonstrate our general approach in the context of MIMO channel estimation and inverse scattering problems

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IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007
67
Optimal Sequential Energy Allocation
for 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 adaptive
waveform amplitude design for estimating parameters of an
unknown channel/medium under average energy constraints. We
present a statistical framework for sequential design (e.g., design
of waveforms in adaptive sensing) of experiments that improves
parameter estimation (e.g., unknown channel parameters) perfor-
mance in terms of reduction in mean-squared error (MSE). We
treat an
time step design problem for a linear Gaussian model
where the shape of the
input design vectors (one per time step)
remains constant and their amplitudes are chosen as a function of
past measurements to minimize MSE. For
optimal energy allocation at the second step as a function of the
first measurement. Our adaptive two-step strategy yields an MSE
improvement of at least 1.65 dB relative to the optimal nonadap-
tive strategy, but is not implementable since it requires knowledge
of the noise amplitude. We then present an implementable design
for the two-step strategy which asymptotically achieves optimal
performance. Motivated by the optimal two-step strategy, we
propose a suboptimal adaptive
that can achieve an MSE improvement of more than 5 dB for
? ??. We demonstrate our general approach in the context of
MIMO channel estimation and inverse scattering problems.
? ?, we derive the
-step energy allocation strategy
Index Terms—Channel estimation, energy management, inverse
scattering, maximum likelihood, parameter estimation, sequential
design.
I. INTRODUCTION
A
statistical signal processing such as channel estimation, radar
imaging, target tracking, and detection can be presented in the
contextofadaptivesensing.Oneoftheimportantcomponentsin
these adaptive sensing problems is the need for energy manage-
ment. Most applications are limited by peak power or average
power. For example, in sensor network applications, sensors
have limited batterylife and replacingthem is expensive.Safety
limits the peak transmit power in medical imaging problems.
Energy is also a critical resource in communication systems
where reliable communication is necessary at low signal-to-
noiseratios.Hence,itisimportanttoconsiderenergylimitations
in waveform design problems. Most of the effort in previous re-
searchhasfocussedonwaveformdesign underpeakpowercon-
straints, e.g., sensor management. There has been little effort
DAPTIVEsensinghasbeenanimportanttopicofresearch
for at least a decade. Many of the classical problems in
Manuscript received September 1, 2006; revised February 14, 2007. This
work was supported in part by AFOSR MURI under Grant FA9550-06-1-0324.
The associate editor coordinating the review of this manuscript and approving
it for publication was Prof. Antonia Papandreou-Suppappola.
The authors are with the Department of Electrical Engineering and Com-
puter Science,UniversityofMichigan, AnnArbor,MI 48109USA(e-mail:ran-
garaj@eecs.umich.edu; ravivr@eecs.umich.edu; hero@eecs.umich.edu).
Digital Object Identifier 10.1109/JSTSP.2007.897049
in developing adaptive waveform design strategies that allocate
different amounts of energy to the waveforms over time. Our
goal in this paper is to perform waveform amplitude design for
adaptivesensinginordertoestimatethesetofunknownchannel
parameters or scattering coefficients under an average energy
constraint.Weformulatethisproblemasanexperimentaldesign
problem in the context of sequential parameter estimation. We
explainthemethodologyofexperimentaldesign,deriveoptimal
designs, and show performance gains over nonadaptive design
techniques.Asafinalstep,wedescribeindetailhowsomeappli-
cationsofadaptivesensingsuchaschannelestimationandradar
imagingcanbecastintothisexperimentaldesignsettingthereby
leading 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 in
theliterature.Inthispaper,“sequential”meansthatateverytime
instant, the best signal to transmit is selected from a library that
depends on past observations.
A. Related Work—Waveform Design
Early work in waveform design focussed on selecting among
a small number of measurement patterns [1]. Radar signal de-
sign using a control theoretic approach subject to both average
and peak power constraints was addressed in [2] and [3]. The
design was nonadaptive and the optimal continuous waveforms
were shown to be on-off measurement patterns alternating be-
tween zero and peak power levels for a tracking example. In
our design, the energy allocation to the waveforms over time
are 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 were
allowed to vary under constant transmit power. Closed-form
solutions to the parameter selection problem were found for a
very restrictive set of cases such as one-dimensional target mo-
tions. More recently a dynamic waveform selection algorithm
for tracking using a class of generalized chirp signals was pre-
sented in [6]. In contrast to these efforts, we focus our work
in finding optimal waveform amplitudes under an average en-
ergy constraint for static parameter estimation. Sensor sched-
ulingcanbethoughtofasanadaptivewaveformdesignproblem
underapeakpowerconstraint[7]wherethegoalistochoosethe
bestsensorateachtimeinstanttoprovidethenextmeasurement.
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 of
hiddenMarkovmodelsystemswasaddressedin[10].InTableI,
we compare our work with existing literature via different cat-
egories.
1932-4553/$25.00 © 2007 IEEE

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68IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007
TABLE I
KEY 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 in
statistics, engineering, biomedicine, and economics. Sequential
analysis has been used to solve important problems in statistics
such as change-pointdetection [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 allocation
in his seminal paper [26]. Early research on the application
of sequential design to problems of estimation was limited
to finding asymptotically risk-efficient point estimates and
fixed-width confidence intervals [11], [12], [30], i.e., sequential
design was used to solve problems in which a conventional
estimate, based on a sample whose size is determined by a
suitably chosen stopping rule, achieves certain properties such
as bounded risk. For the problem of estimating the mean under
unknown variance, it was shown that a sequential two-step
method guaranteed specified precision [23], [31], [32], which
is not possible using a fixed sample. The statistical sequential
design framework assumes a fixed measurement setup while
acquiring the data and does not consider energy constraints. In
this paper, we adaptively design input parameters to alter the
measurement patterns under an average energy constraint to
obtain performance gains over nonadaptive strategies.
Another class of problems in sequential estimation is online
estimation,wherefastupdatingofparameterestimatesaremade
in realtime, called recursiveidentification in controltheory, and
adaptive estimation in signal processing. For example, consider
the problem of estimating parameter
in the following model
where
are
random variables with zero mean, and
of received signals. The maximum likelihood (ML) es-
timate of
is given by the least squares (LS) solution,
are the sequence of inputs to the system,
independent identically distributed(i.i.d)Gaussian
are the set
. One way of computing
the LS estimate is the recursive least squares approach (RLS)
[13] which can be written as
where
computational complexity of inverting the matrix.
In the above formulation it was assumed that the input
sequence
remains fixed. The problem of waveform design
is relevant when input
can be adaptively chosen based on
the past measurements
estimation has application to a wide variety of areas such as
communications and control, medical imaging, radar systems,
system identification, and inverse scattering. By measure-
ment-adaptive estimation we mean that one has control over
the way measurements are made, e.g., through the selection of
waveforms, projections, or transmitted energy. The standard
solution for estimating parameters from adaptive measurements
is the ML estimator. For the case of classic linear Gaussian
model, i.e., a Gaussian observation with unknown mean and
known variance, it is well-known [16] that the ML estimator
is unbiased and achieves the unbiased Cramér Rao lower
bound (CRB). Many researchers have looked at improving the
estimation of these parameters by adding a small bias to reduce
the MSE. Stein showed that this leads to better estimators that
achieve lower MSE than the ML estimator for estimating the
mean in a multivariate Gaussian distribution with dimension
greater than two [14], [15]. Other alternatives such as the
shrinkage estimator [33], Tikhonov regularization [34], and
covariance shaping least squares (CSLS) estimator [35] have
also been proposed in the literature. While these pioneering
efforts present interesting approaches to improve static param-
eter estimation performance by introducing bias, none of them
incorporate the notion of sequential design of input parameters.
Our adaptive design of inputs effectively adds bias to achieve
reduction in MSE.
In this paper, we formulate a problem of sequentially se-
lecting waveform amplitudes for estimating parameters of
a linear Gaussian channel model under an average energy
constraint 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 and
the analogy between this problem and the waveform design
problem is explained. In Section III, closed-form expressions
for the optimal design parameters and the corresponding
minimum MSE in the scalar parameter case are derived for a
two-step procedure (two time steps). Since the optimal solution
requires the knowledge of the parameter to be estimated, it is
shown in Section IV that the performance of this omniscient so-
lution can be achieved with a parameter independent strategy.
. The recursive process avoids the
. Measurement-adaptive

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RANGARAJAN et al.: OPTIMAL SEQUENTIAL ENERGY ALLOCATION 69
In Section IV-A, we describe an
allocation procedure, which yields more than 5 dB gain over
nonadaptive methods. These results are extended to the vector
parameter case in Section V. Finally in Section VI, we show
the applicability of this framework for channel estimation and
radar imaging problems.
-step sequential energy
II. PROBLEM STATEMENT
We begin by introducing nomenclature commonly used
throughout the paper. We denote vectors in
lowercase letters and matrices in
letters. The identity matrix is denoted by . We use
to denote the transpose and conjugate transpose opera-
tors, respectively. We denote the
i.e.,
. A circularly symmetric complex Gaussian
random vector with mean
and covariance matrix
as
.and denote the statistical expectation
and trace operators, respectively. The terms MSE and SNR are
abbreviations to mean-squared error and signal-to-noise ratio,
respectively.
Let
be the
parameters. The problem of estimating
written as
by boldface
by boldface uppercase
and
-norm of a vector by,
is denoted
-element vector of unknown
in noise can then be
(1)
where
the parameters of interest
The
-element design parameter vectors,
on the past measurements:
the th
-element observation vector. In the context of adaptive
sensing,
represents the response of the medium,
denote the number of transmit and receive antennas respec-
tively,
are the set of waveforms to be designed, and
are the set of channel parameters or scattering coefficients to
be estimated using the set of received signals
classic estimation problem in a linear Gaussian model, we have
isani.i.d.randomprocesscorruptingthefunctionof
and denotes the time index.
can depend
, whereis
and
. For the
is
a known
random vector. When
matrix and linear inandis a
, we can write
is uniquely
. The linear
is linear in
. In this case
determined by the matrices
Gaussian 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 scalar
parameter
are
(2)
An
-step design procedure specifies a sequence of functions
corresponding to the
signal waveforms after receiving the previous measurements.
An optimal
-step procedure selects the design vectors so
that the MSE of the ML estimator,
is minimized subject to the average energy constraint,
, where
ergy. The ML estimator of
for the
transmitted
is the total available en-
-step procedure is given
by
(3)
and the corresponding MSE
is
(4)
Denote
, where
represents the energy allocated to each time
step . Define
in the design parameters for the
energy constraint can be written as
as the average energy
-step procedure. The average
(5)
Our goal is to find the best sequence of the design vectors
to minimize the
average energy constraint in (5).
in (4) under the
A. Nonadaptive Strategy
As a benchmark for comparison, we consider the nonadap-
tive case where
of
fying the expression for MSE in (4), we have
is deterministic, independent
, and . Simpli-
(6)
where equality is achieved iff
eigenvector corresponding
eigenvalue of the matrix
, the normalized
the maximum
to
,
. Note
. Furthermore,
the performance of the ML estimator does not depend on the
energy allocation. Hence, without loss of generality we can
assume all energy is allocated to the first transmission which
implies that any
-step nonadaptive strategy is no better than
the optimal one-step strategy. We define
as
(7)
Then the average energy constraint in (5) is equivalent
to
, where
We show in [42] that the problem of minimizing
subject to
. Thus, we use the two minimization
criteria interchangeably in the remainder of this paper. The
product of MSE and SNR is
.
is equivalent to minimizing
(8)

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70IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 1, JUNE 2007
andtheminimumMSEfortheone-step(ornonadaptive
strategy satisfies
-step)
(9)
While our goal is to find optimal input design parameters,
which achieve minimum MSE, any
suboptimal design that guarantees
also of interest. We first look at a two-step sequential design
procedure. A word of caution: in Sections III and III-A we
develop optimal and suboptimal strategies where the solutions
require the knowledge of the unknown parameter
inSectionIVwepresenta
-independentdesignwhichasymp-
toticallyachievestheperformanceofthe‘omniscient’strategies.
is
. However,
III. OMNISCIENT OPTIMAL TWO-STEP SEQUENTIAL STRATEGY
In the two-step sequential procedure, we have
steps where in each time step
design parameter
to obtain observation
process, we have
time
, we can control input
. For a two-step
(10)
(11)
The ML estimator of
for a two-step procedure from (3) is
(12)
and its MSE from (4) is given by
(13)
Weassumethattheshapeoftheoptimaldesigns,i.e.,
is the one-step optimum given by
minimize the MSE over the energy of the design parameters.
Denote
and
the sequential design framework, we select
defined below (6) and
. Under
(14)
(15)
where
constraint from (5) can then be written as
andarereal-valuedscalars.Theaverageenergy
(16)
We use Lagrangian multipliers to minimize the MSE in (13)
withrespectto
andundertheenergyconstraintin(16).
The objective function to be minimized can then be written as
Fig. 1. Plot of the optimal and suboptimal solution to the normalized energy
transmitted at the second stage as functions of received signal at first stage.
In [42], we show that the optimal solution to
on
only through the function
depends
, where
(17)
. Hence, we denote the solution as.
Let
derivative of the objective function with respect to
yields
. Setting the
to zero
(18)
where
the root of the third-order polynomial in (18), real-valued, and
greater than or equal to 1. If more than one real-valued solution
greater than 1 to the cubic equation exists, the optimal solution
to
will be the root that achieves minimum MSE. The optimal
for everyandis denoted by
. Therefore, findingthat minimizes MSE is equiva-
lent to finding
that minimizes MSE. We obtain
every
and use a brute force grid search to find the optimal
that minimizes the objective function. The MSE is minimized
at
, or. The optimal
relation
this solution depends on the unknown parameter
minimizer an “omniscient” energy allocation strategy. For the
optimal solution, the product of
. The function that minimizes MSE is
. Also
for
is given by the
. Since
, we call this
is
(19)
This corresponds to a 32% improvement in performance or a
1.67dBgainintermsofSNRforthetwo-stepdesignwhencom-
pared to the one-step procedure for which
.
Theoptimalenergyallocation
as shown in Fig. 1 (solid) is a thresholding
function, i.e.,
is zero for
atthe second step,
. This solution implies

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RANGARAJAN et al.: OPTIMAL SEQUENTIAL ENERGY ALLOCATION71
that when the actual realization of the normalized noise along
in the first step is small enough, then the second
measurement becomes unnecessary. On the other hand, when
the normalized noise along
there is some merit in incorporating the information from the
second measurement. The solution also suggests that the higher
the noise magnitude at the first step, the more the energy that
needs to be used. However, the probability of allocating energy
greater than a particular value decreases exponentially with
that energy value. Nevertheless in applications with a peak
energy constraint, the transmission of the optimal energy at
the second stage may not always be possible. Hence, in the
following subsection we look at a suboptimal solution which
takes into account this constraint and still achieves near optimal
performance.
exceeds a threshold, then
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 noise
magnitude at the first step is less than a threshold and increases
with increasing noise magnitudes otherwise. Forthe suboptimal
solution,weuseabinaryenergyallocationstrategyatthesecond
stage based on the noise magnitude at the first step, i.e., we
allocate a fixed nonzero energy if the noise magnitude is greater
than a threshold else we allocate zero energy. The suboptimal
solution to the design vectors
andis then of the form
(20)
(21)
where
pendent of
is defined in (17),
and
are design parameters inde-
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 solution
using (13) is
(23)
Denote
for
and (22), and using the fact that
. Substituting
in (23)in the expressions forand
and
when, the expres-
sion
simplifies to
(24)
Minimizing
a grid search for
with respect to
and
and through
yields
and. It follows that.
Substituting for the optimal values of
(22), and simplifying yields
in (24) and
(25)
This translates to a 28.57% improvement in MSE performance
or a 1.5 dB savings in terms of SNR. The suboptimal solution
to the energy design is shown in Fig. 1 by a dashed dotted line
indicated as Suboptimal-I. Thus, while the suboptimal strategy
limits the peak transmit power to
achieve near optimal performance.
In the previous section, we addressed the problem of
minimizing MSE subject to an average energy constraint,
. An average energy constraint implies
that the total allocated energy averaged over repeated trials of
the two-step experiment is constrained to be less than or equal
to
. This is less restrictive than the strict energy constraint
, as any solution satisfying this constraint
satisfies the average energy constraint but not vice versa. The
problem of minimizing the MSE in (13) under this strict energy
constraint was addressed in the context of radar imaging in
[43]. We show in [42] that the optimal two-step design under
the strict constraint is given by
, it is able to
where
minimum MSE is then given by
The optimal solution satisfies the strict energy constraint with
equality but the average energy used is
. The solution to the two-step strategy under this strict
energy constraint can also be derived by imposing an additional
constraint,
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 optimal
performanceasymptoticallyandallowsforanypeakpowercon-
straint in the design.
, and. The
.
IV. PARAMETER INDEPENDENT TWO-STEP DESIGN STRATEGY
Consider the optimal design for the two-step procedure
(26)
We showed that by designing
gain up to 32% improvement in estimator performance. But
the “omniscient” solution (26) depends on the parameter to be
estimated. Here, we prove that we can approach the optimal
two-step gain by implementing a
cation strategy when
is bounded, i.e.,
. We describe the intuition behind the proposed
andoptimally we can
-independent energy allo-