Communication-Inspired SensingWenshu Zhang and Liuqing Yang

Dept. of ECE, Univ. of Florida, Gainesville, FL 32611, USA

Abstract—Information theory, and particularly the mutualinformation (MI) has provided fundamental guidance for com-munications research. In [1], the MI was first applied to radarwaveform design. However, the practical meaning of MI inthe sensing context remains unclear. Recently, [6] shows thatunder the white noise assumption, the water-filling schemesimultaneously maximizes the MI and minimizes the minimumMSE (MMSE). Such an equivalence disappears when the targetparameter statistics are not perfectly known [7]. To further theunderstanding of the practical meaning of MI and to establish aconnection between MI and commonly adopted MSE measuresfor sensing, this paper takes a fresh look at the target estimationproblem, by considering the general colored noise, incorporatingthe normalized MSE (NMSE), and establishing joint robustdesigns for both the transmitter (waveforms) and the receiver(estimator) under various target and noise uncertainty levels.Our results show that: i) the optimum waveform designs resultedfrom the MI, MMSE and NMSE criteria are all different; andii) compared to MMSE, the NMSE-based designs share moresimilarities with the MI-based ones, especially when the targetand noise statistics are not perfectly known.

I. INTRODUCTIONIn multi-input multi-output (MIMO) communication sys-

tems, multiple transmit and/or receive antennas can increasethe diversity for enhanced reliability and increase the degreesof freedom for improved data rate. Partly inspired by these,MIMO sensing is recently drawing great interests. In such sys-tems, multiple waveform optimization is particularly critical.Bell’s 1993 paper first used the MI to design radar waveformsfor the estimation of an extended target [1].In order to link the MI criterion with more direct per-

formance indicators in sensing applications, Yang and Blumstudied the extended target estimation problem in a widelyseparated MIMO scenario. In [6], it was shown that the MI andMMSE criteria lead to identical optimum water-filling strategyunder perfectly known target and noise power spectral density(PSD) and white noise assumptions. These waveform designswere then extended in [7] to account for a bounded uncertaintyof the target PSD. Different from [6], it was shown in [7] thatthe MI and MMSE criteria result in different robust waveformdesigns.The MI is an essential measure in the field of communica-

tions. However, its role in sensing is not yet clear. Although theresults in [6], [7] shed some light on the possible connectionbetween the MI and MSE measures, they are based on limitingassumptions such as white noise and perfectly known noisePSD, and lack generality. In this paper, we will further theseexisting work and reveal more intrinsic connections betweenthe MI and MSE measures in a sensing setup.Our contributions in this paper are three-fold. First, we

take into consideration the more general and practical coloredGaussian noise. Secondly, we introduce the normalized MSE

(NMSE) minimization criterion for radar waveform design.It is more meaningful for parameter estimation problemsand exhibits more similar behaviors with the MI criterion,especially in robust designs. Thirdly, we provide joint robustdesigns of both the transmitted waveforms and the estimator atthe receiver, under various uncertainty models. Results showthat the MI- and NMSE-based robust designs are built on anidentical least favorable set (LFS), which is different from thatof the MMSE-based designs.

II. SYSTEM MODELDespite the many similarities between MIMO radar and

MIMO communications, they have some fundamental differ-ences. Take the transmitted signal optimization as an example,where we consider a simple setup of M = 2 transmit andN =2 receive antennas. In communications, the objective isto optimize the transmitted signals for their better estimation.Assume that duration of the transmitted signals is L, lengthof the channel delay is K , and the white noise ξ is zero-meanGaussian distributed. Then the system representation is:[

y1

y2

]= H

[x1

x2

]+ ξ =

[H1,1 H1,2

H2,1 H2,2

] [x1

x2

]+ ξ , (1)

where xi is the ith L×1 transmitted signal vector, yj theL×1 received signal vector at the jth receiver, and the L×L Toeplitz matrix Hj,i represents the channel response fromthe ith transmitter to the jth receiver. Denote the covariancematrices as Σx for the transmitted signals, and Σξ=σ2

ξI forthe white noise. Then the MI between the transmitted andreceived signals is:

MI = log∣∣∣σ−2

ξ ΣxHH

H + I2L

∣∣∣ , (2)

and the resulting MMSE after the MMSE estimator is

MMSE = tr{(σ−2

ξ HH

H + Σ−1x )−1

}. (3)

In communications, the classical optimum transmitted signaldesign maximizing MI or minimizing MMSE is obtained whenthe matrix (σ−2

ξ ΣxHHH + I2L) or (σ−2

ξ HHH + Σ−1

x ) isa scaled identity matrix (see e.g., [3], [4]).For a radar system, on the other hand, we desire to opti-

mize the transmitted signal for better estimation of the targetresponse. The 2 × 2 system model for radar is [6]:[

y1

y2

]=

[X

X

] [g1

g2

]+ ξ , (4)

where X = [X1 X2] and Xi is the L×K Toeplitz signalmatrix emitted from the ith transmit antenna, yj is the L×1signal vector at the jth receive antenna, gj = [gT

j,1 gTj,2]

T

and gj,i is the K×1 target response vector from the ithtransmit antenna to the jth receive antenna. Accordingly, theMI between the target response and the received signal is:

MI = log∣∣∣σ−2

ξ Σg(I2 ⊗ XH

X) + I4K

∣∣∣ , (5)

193978-1-4244-2941-7/08/$25.00 ©2008 IEEE Asilomar 2008

where Σg is the covariance matrix of the target response, andthe resulting MMSE after the MMSE estimator is:

MMSE = tr{(

σ−2

ξ (I2 ⊗ XH

X) + Σ−1g

)−1}

(6)In order to maximize MI or minimize MMSE, the optimum

strategy again requires the matrix (σ−2

ξ Σg(I2⊗XHX)+I4K)

or (σ−2

ξ (I2⊗XHX)+Σ−1

g ) to be a scaled identity matrix,as detailed in [6]. However, unlike the communications case,the special structure of I2⊗XHX makes such a conditionimpossible. In other words, one cannot design the transmittedsignal such that the estimation of g1 and g2 is simultaneouslyoptimized – there is simply not sufficient degrees of freedom.This simple comparison suggests that the transmitted signal

optimization problem is ill-formulated for the widely sepa-rated MIMO radar setup in [6]. To avoid this problem, weconsider a mixed MIMO structure with a widely separatedM -element transmit array and a closely spaced N -element receivearray. For an extended target of interest, the widely separatedtransmit antennas can impinge distinct scatterers from differentangles. For each of theM reflected signals,N coherent returnsare acquired at the receiver, the only difference among whichis a phase shift. One can then combine them coherently toobtain a processing gain of N . Having M transmitted signalsto design, this mixed MIMO setup has sufficient degrees offreedom in the optimum signal design for the estimation ofthe M viewing aspects of the target.Bearing the goal of comparing MI- and MSE-based radar

waveform designs, we will adopt the “mode” space signalmodel in [6]. Though originally developed for widely sepa-rated MIMO radar, it can be readily modified for our mixedMIMO setup. Specifically, the response of the extended targetfrom allM different viewing aspects is captured by theMK×1“mode” vector h � [h1, · · · , hMK ]

T . As in [6], h is modeledas Gaussian distributed with zero-mean and covariance matrixΛ = diag{λ1, · · · , λMK}, where λi’s can be regarded as thesamples of target PSD [6]. The coherent combining of thearrival signals at the receive array is rather straightforward.Hence we set N = 1 without loss of generality. As a result,the mode space system representation is given by

y = D1

2 h + η , (7)

where y is an MK × 1 vector representing the observedsignal in mode space, D = diag{d1, · · · , dMK} is the powerallocation matrix with transmit power di allocated to thecorresponding mode space waveform, with a total transmitpower constraint

∑MK

i=1di = P0. η is the MK × 1 Gaussian

noise vector in mode space with zero mean and diagonalcovariance matrix Ση whose diagonal entries can also beregarded as its PSD samples.With this representation, the waveform design problem sim-

plifies to a power allocation problem, where the total power isoptimally assigned toMK orthogonal waveforms in the modespace. It is worth noting that, though we adopted the modespace representation in [6] for comparison convenience withthe results therein, this model is actually very general. Ourmodel and the results hereafter can be readily generalized tocover “MIMO” radar systems resulted not only from multiplespatial viewing aspects, but also by alternative means such asfrequency agility.

III. OPTIMUM POWER ALLOCATION IN COLORED NOISE

The optimum power allocation problem has been studiedin [6] under the white Gaussian noise assumption. In asensing scenario, however, unwanted interferences includingjammers and antenna effects are often inevitable, suggestingthe necessity of incorporating more general colored noise.Hence throughout our analysis, we will consider coloredGaussian noise with zero mean and covariance matrix Ση=diag

{σ2

η1,· · ·, σ2

ηMK

}, where the diagonal elements can be

different.In this section, we assume that both the target and noise

PSDs are known exactly at both the transmitter and thereceiver. This assumption will be relaxed in the next section,where uncertainty of such knowledge will be taken intoaccount.

A. MI and MMSE CriteriaThe MI between the observed signal y and the target mode

response h given power allocation matrix D is given by:

I(y; h|D) =

MK∑i=1

log(σ−2ηi

λidi + 1)

. (8)

The logarithm is base-2 and the unit for the MI is bit. Notethat, instead of the white noise with a flat PSD in [6], weconsider non-flat colored noise here.Proposition 1 (MI-based Optimum Power Allocation): Theoptimum power allocation maximizing MI in the presence ofcolored Gaussian noise has the following water-filling form:

di =

(γMI −

σ2ηi

λi

)+

, for i = 1, · · · , MK , (9)

where γMI is a constant satisfying the total power constraint.

Due to space limit, all proofs are omitted here. For thedetailed proofs, please refer to our journal paper [8].For the MMSE-based designs, one needs to first specify the

MMSE estimator, denoted by Φ, as follows:

h = Φy =(DΣ

−1η + Λ

−1)−1

D1

2 Σ−1η y . (10)

Accordingly, the MMSE is given by

MMSE= tr{E

[(h−h

)(h−h

)H]}=

MK∑i=1

λi

σ−2ηi

λidi+1. (11)

Proposition 2 (MMSE-based Optimum Power Allocation):The optimum power allocation minimizing the MMSE in thepresence of colored Gaussian noise has the following form:

di =

(γMMSE

√σ2

ηi−

σ2ηi

λi

)+

, for i = 1, · · · , MK , (12)

where γMMSE is a constant ensuring the total power constraint.

From Propositions 1 and 2, we see that in the presence ofcolored Gaussian noise, the MMSE criterion does not lead tothe water-filling solution as in [6]. Hence the MI and MMSEcriteria are not equivalent as observed in [6] under the whitenoise assumption.

194

B. NMSE CriterionThe MMSE design minimizes the sum of target mode

estimation errors, but there is no guarantee on the MSEs ofindividual modes. In a radar problem, these weak modes mayassume significant information useful in describing the target[1]. A natural amendment is to introduce the normalized MSEs(NMSE), which is a common exercise in various estimationproblems (see e.g., [2], [5]).Specifically, normalizing individual MSEs with respect to

their average strength, we obtain the following expression:

NMSE � tr{E

[Λ−

1

2

(h − h

) (h − h

)HΛ−

1

2

]}

=

MK∑i=1

1

σ−2ηi

λidi + 1. (13)

Proposition 3 (NMSE-based Optimum Power Allocation):The optimum power allocation minimizing NMSE in the pres-ence of colored Gaussian noise has the following form:

di =

⎛⎝γNMSE

√σ2

ηi

λi

−σ2

ηi

λi

⎞⎠

+

, for i = 1, · · · , MK , (14)

where γNMSE is a constant ensuring the total power constraint.Propositions 1–3 show that the optimum power allocation

obtained via maximizing MI differs from those obtained byminimizing MMSE or NMSE, when colored noise is takeninto account. These results are based on the exact knowledgeof the target and noise PSDs. In practice, this knowledge canonly be obtained with some uncertainty. In the next section,we will study the behavior of MI-, MMSE- and NMSE-basedrobust designs under different uncertainty levels.

IV. ROBUST DESIGN: JOINT ESTIMATOR AND POWERALLOCATION OPTIMIZATION

When the PSDs of the target and the colored noise arenot precisely available, some robust approaches need to beadopted to design the optimum probing waveform at thetransmitter, as well as the estimator at the receiver. In [7], theminimax approach is adopted to address the bounded targetPSD uncertainty, while assuming a white noise PSD that isperfectly known. Here, we not only consider a colored noisePSD, but also allow for various uncertainty levels in the noisePSD. Another significant improvement upon [7] is that: [7]focuses on the transmitted waveform optimization only whileassuming an MMSE receiver based on the exact target PSD;whereas we jointly design the transmitted waveforms and theMMSE estimator via robust optimization.A. Robust Minimax Design CriteriaIn our minimax problem formulation, the estimator and

the power allocation are jointly designed to: i) provide theoptimum performance for the least favorable set (LFS) of thetarget and noise PSDs; and ii) provide equivalent or betterperformance for all other possible sets within the uncertaintyregion. Mathematically, we jointly design the MMSE estimatormatrix Φ and the power allocation matrix D such that:

maxD

{ infΛ,Ση

MI(Λ,Ση; D)}, MI-based,

minΦ,D

{ supΛ,Ση

MMSE(Λ,Ση;Φ, D)}, MMSE-based, (15)

minΦ,D

{ supΛ,Ση

NMSE(Λ,Ση;Φ, D)}, NMSE-based.

To solve the above problems, we will look for a sad-dle point for each case by finding the LFS (ΛR =diag{λR

1,· · ·, λR

MK},ΣRη = diag{σ2R

η1,· · ·, σ2R

ηMK}), the robust

estimator ΦR and the robust power allocation matrix DR =

diag{dR1 , · · · , dR

MK} such that:

MI(Λ,Ση;DR) −MI(ΛR,Σ

Rη ;DR)≥0

MMSE(Λ,Ση;ΦR, D

R) −MMSE(ΛR,Σ

Rη ;ΦR

, DR)≤0 (16)

NMSE(Λ,Ση;ΦR, D

R) − NMSE(ΛR,Σ

Rη ;ΦR

, DR)≤0

Next, a brief derivation for the differences at the left-handsides of the above three inequalities will be reviewed for MI,MMSE and NMSE criteria, separately.MI-based: As defined in (8), the MI formula has nothing todo with the receiver design. To calculate the MI difference in(16), one can substitute with (8) as follows:

MI(Λ,Σn;DR)−MI(ΛR,Σ

Rn ;DR)=

MK∑i=1

logσ−2

niλid

Ri +1

σ−2Rni

λRi dR

i +1. (17)

where the robust power allocation DR is simply the optimumone for the LFS, which can be obtained using (9).MMSE-based: The calculation of MMSE difference is morecomplicated, because one needs to jointly consider the powerallocation DR and the estimator ΦR.As mentioned before, the robust power allocation matrix

DR is the optimum one for the LFS (see (12)). Likewise,the robust estimator Φ

R is also optimum (MSE-minimizing)for the LFS. Using (10) and (11), the MMSE for the LFS,MMSE(ΛR,ΣR

η ;ΦR, DR) can be obtained.In the lack of exact knowledge of the true target and

noise PSDs, the transmitter should always use the robustpower allocation design DR at the transmitter, and the robustestimator design Φ

R at the receiver. Accordingly, for targetresponse h with arbitrary PSD Λ and noise η with Ση , wecould calculate MMSE(Λ,Ση;ΦR, DR), and then obtain theMMSE difference:

MMSE(Λ,Σn;ΦR, D

R) −MMSE(ΛR, Σ

Rn ;ΦR

, DR)

=

MK∑i=1

{1(

σ−2Rni

λRi dR

i + 1)2

(λi − λ

Ri

)

+

(σ−2R

niλR

i

)2dR

i(σ−2R

niλR

i dRi + 1

)2

(σ

2ni

− σ2Rni

)}. (18)

NMSE-based: Using the robust MMSE estimator ΦR and

the robust power allocation DR for the LFS, we get theNMSE for the LFS as NMSE(ΛR,ΣR

η ;ΦR, DR) [c.f. (13)].For target response and noise with arbitrary PSDs, we haveNMSE(Λ,Ση;ΦR,DR), and the NMSE difference as:

NMSE(Λ,Σn;ΦR, D

R) − NMSE(ΛR,Σ

Rn ;ΦR

, DR)

=

MK∑i=1

(σ−2R

niλR

i

)2dR

i(σ−2R

niλR

i dRi + 1

)2

(σ2

ni

λi

−σ2R

ni

λRi

). (19)

Recall that our robust minimax designs based on MI, MMSEand NMSE criteria are described by the three inequalities in(16), respectively. Now with specific expressions available in(17), (18), and (19), we will next find the LFS (ΛR, Σ

Rη )

satisfying these inequalities.

195

P0 (dB)

NM

SE

MMSE-based

NMSE-based

MI-based

Fig. 1. NMSE performance of three criteria: MMSE, NMSE and MI.

B. Joint Robust DesignWe have seen in the preceding subsection that the robust

transmitter and receiver designs DR and ΦR will be uniquely

specified once the LFS is determined. However, the LFS is notonly determined by the inequalities in (16), but also heavilydependent on the uncertainty model. We will consider twomodels allowing for uncertainty in both target and noise PSDs.Uncertainty Model I: In [7], a banded uncertainty modelfor the target PSD is considered. The exact target PSD isunknown, but lies within a band whose upper and lowerbounds are known. Here, we adopt this model for both thetarget and noise; that is,

λLi ≤ λi ≤ λ

Ui , σ

2Lηi

≤ σ2ηi

≤ σ2Uηi

, for i = 1, · · · , MK . (20)

Proposition 4 (LFS for Uncertainty Model I): When theuncertain target and noise PSDs fall within banded regionswith known upper and lower limits and no other constraint,the LFS for robust minimax design consists of:• ({λL

i }MKi=1

, {σ2Uηi

}MKi=1

) for MI-based optimization;• ({λU

i }MKi=1

, {σ2Uηi

}MKi=1

) for MMSE-based optimization;• ({λL

i }MKi=1

, {σ2Uηi

}MKi=1

) for NMSE-based optimization.We notice that the LFS for MI- and NMSE-based designs

is identical, but differs from that of MMSE-based design.Uncertainty Model II: In this model, we add an average powerratio constraint to the noise PSD uncertainty. That is,

σ2Lηi

≤σ2ηi≤σ

2Uηi

, i = 1,· · ·, MK, and ρ=1

MK

∑i

σ2ηi

λi

. (21)

Proposition 5 (LFS for Uncertainty Model II): Under theaverage power ratio constraint, the LFS is given as follows:

σ2Rηi

=

⎧⎨⎩

σ2Lηi

, if σ2Lηi

> knλi

σ2Uηi

, if σ2Uηi

< knλi

knλi, otherwise(22)

for MI and NMSE criteria, and

σ2Rηi

=

⎧⎨⎩

σ2Lηi

, if σ2Lηi

> kmλ2i

σ2Uηi

, if σ2Uηi

< kmλ2i

kmλ2i , otherwise

(23)

for MMSE criterion, where kn and km, (typically, kn �= km),are constants ensuring the average power ratio constraint.

Under the average power ratio constraint, the MI and NMSEcriteria again give rise to an identical LFS. In fact, the LFSgiven by (22) is one where σ2R

ηi/λi is made as flat as possible.

This result is very intuitive since the worst interference PSDis the one that perfectly matches the target PSD. On the otherhand, MMSE-based design suggests a different LFS wherethe noise PSD is maximumly matched to the target PSDsquare. This result is consistent with the NMSE-based one,considering that the latter is obtained by normalizing the MSEagainst the target PSD.

V. NUMERICAL RESULTSIn this section, we provide simulation results to verify our

analytical results and provide further comparisons among theMI, MMSE and NMSE criteria.

A. Optimum Power Allocation in Colored NoiseIn the simulation, we consider an extended target which is

described by five modes, {λi}={0.5, 0.8, 1.8, 1.5, 0.35} and{σ2

ni} = {0.2, 1, 2.3, 0.1, 1.7}. The total power constraint is

P0 =10 dB.The performance curves are plotted in Fig. 1 for the NMSE

merit based on all three criteria. Evidently, the NMSE-baseddesign is optimum, while MI- and MMSE-based designs bothexhibit performance losses. Again, This result validates thefact that all the criteria are not equivalent in the presence ofcolored noise.

B. Joint Robust Estimator and Power Allocation OptimizationSince the LFS for robust design with uncertainty model I

is straightforward, we only verify the case with uncertaintymodel II here due to space limit. The target PSD is assumedto be available, {λi}={0.5, 0.8, 1.8, 1.5, 0.35}. Uncertainty incolored noise normalized by {λi} is modeled in Fig. 2(a), withupper bound {σ2U

ηi/λi} and lower bound {σ2L

ηi/λi} subject to

average power ratio constraint ρ = 1. This noise uncertaintymodel normalized by {λ2

i } is given in Fig. 2(b). A set ofnominal values is used to make performance comparisons.The normalized LFS for colored noise is identical for MI

and NMSE criteria, and is supposed to be as flat as possible.In Fig. 2(a). When the constant kn = 1, the straight line liesbetween the upper and lower bounds. According to MI andNMSE criteria, the normalized noise LFS which satisfies theaverage power ratio constraint should be

{σ2R

ni/λi

}= 1. For

MMSE criterion, on the other hand, the straight line km =5ρ (

∑i λi) = 1.0101 is living between its corresponding upper

i

Lower Bound

Upper Bound

Nominal

LFS

Lower Bound

Upper Bound

Nominal

LFS

i

σ2

ηi

λi

σ2

ηi

λ2

i

(a) (b)

Fig. 2. Noise PSD uncertainty band model: (a) normalized by {λi};(b) normalized by {λ2

i }

196

P0 (dB)

MM

SE

Nominal PSD, Nominal design

Nominal PSD, Robust design

LFS PSD, Robust design

LFS PSD, Nominal design

Fig. 3. MMSE performance of robust designs.

and lower bounds in Fig. 2(b). Therefore the noise LFS forthe MMSE criterion is

{σ2R

ni/λ2

i

}= 1.0101.

We plot the MMSE, NMSE and MI curves in Figs. 3, 4 and5 to show how the robust designs make the worst performancebest. There are four curves for each case: A) nominal PSD withnominal design, which is the best achievable performance ifthere is no uncertainty; B) LFS PSD with nominal design,which is the case when one assumes the nominal PSD indesign but encounters the LFS scenario; C) LFS PSD withrobust design, which indicates the “worst-case” performancefor the robust design; and D) nominal PSD with robust design,which corresponds to the actual performance when the truePSD is the nominal but the robust design is used. For MMSEand NMSE criteria, large gaps between LFS PSD with nominaldesign (B curves) and LFS PSD with robust design (C curves)illustrates the significant improvement provided by the robustdesign for the worst case. This “best worst case” performanceprovides a performance lower bound. This is evidenced bythe D curves, which are bounded by C curves. For MI-baseddesign, we see that the performance is mainly determined bythe actual PSD (nominal or LFS) but hardly affected by thepower allocation design (nominal or robust), especially at highSNR. However, it still provides performance lower bound asa reference.

VI. CONCLUSIONSIn this paper, we studied the optimum waveform design

problem for target parameter estimation. Different from ex-isting works, we considered a mixed MIMO radar setupfor which the waveform optimization problem is meaningful,took into account the colored noise, incorporated the NMSEas a design criterion in addition to the MI and MMSE,and derived robust designs for various uncertainty modelsby jointly optimizing both the transmitter (waveforms) andthe receiver (estimator). The analytical and numerical resultssuggest that: i) the equivalence between the MI and MMSEcriteria disappears when the noise is colored; and ii) the NMSEcriterion seems to share more similarities with the MI. Inparticular, they lead to identical LFS in the robust design under

P0 (dB)

NM

SE

Nominal PSD, Nominal design

Nominal PSD, Robust design

LFS PSD, Robust design

LFS PSD, Nominal design

Fig. 4. NMSE performance of robust designs.

P0 (dB)

MI

(bit

s)

Nominal PSD, Nominal design

Nominal PSD, Robust design

LFS PSD, Robust design

LFS PSD, Nominal design

Fig. 5. MI performance of robust designs.

various uncertainty models, while the MMSE criterion alwayssuggests otherwise.

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