MIMO Frequency-Selective Channel Modelingbased on Pathwise Dynamics

Maxime GuillaudEurecom Institute

2229 route des Cretes, B.P. 19306904 Sophia Antipolis Cedex, FRANCE

Tel: +33 4 9300 2647; Fax: +33 4 9300 2627maxime.guillaud@eurecom.fr

Dirk T.M. SlockEurecom Institute

2229 route des Cretes, B.P. 19306904 Sophia Antipolis Cedex, FRANCE

Tel: +33 4 9300 2606; Fax: +33 4 9300 2627dirk.slock@eurecom.fr

Abstract— A specular approach to model MIMO frequency-selective channel variations, that yields a parsimonious channelrepresentation, is proposed. The specular structure is shownto simplify linear estimation and prediction of the channel.Identifiability of specular channels is studied, and an algorithmachieving identification is proposed. The performance of theproposed method is evaluated through computer simulations.

I. INTRODUCTION

The use of specular models for channel analysis andtracking has been proposed by various authors seeking toimprove the ability to accurately estimate [1], represent andtransmit [2], or predict [3], [4] Channel State Information(CSI). Specular methods constitue viable candidates forchannel tracking and prediction, since the insight theyprovide into the actual channel structure – namely, separationof the channel variation into its space and time components –can improve the performance and decrease the complexity ofchannel tracking and prediction. Various methods have beenproposed to estimate the underlying parameters, includingMUSIC in [3], ESPRIT in [4] and SAGE in [1].

In the present contribution, we propose to use a specular(pathwise) approach in order to gain access to a reducedparameter set representing the channel state, in order toimprove channel estimation (smoothing) and prediction.After recalling the specular channel model in section II, andoutlining in section III how it makes channel estimationand prediction easier, we provide sufficient conditionsfor identifiability of a specular channel in section IV,and an algorithm, based on simultaneous diagonalizationof the covariance matrices, that achieves identification isproposed in section V. Section VII presents simulation results.

II. SPECULAR CHANNEL MODEL

Let us consider a Multiple-Input Multiple-Output (MIMO)frequency-selective channel, with

� �transmit (Tx) and

� �receive (Rx) antennas. The impulse response of the channelbetween the � � �

Tx antenna and the � � �Rx antenna is denoted

by � � � � � � , where � is the time and is the lag.

We will henceforth work under the assumption that thechannel state evolves according to a specular model. In sucha model, each impulse response � � � � � � � is the superpositionof a finite number � of discrete paths at lag � � � �� � � � � � �� � � ,� � � � � � � , resulting from either line-of-sight propagation,or one or several reflections. This model relies upon the factthat the paths between all the Tx-Rx antenna pairs have mostof their characteristics in common, except for what happensnear the antenna arrays. Hence, they share some properties,namely their speed w.r.t. the reflectors, and the reflectioncharacteristics (hence their Doppler and gain are the samewhatever antenna pair is considered). Each path coefficientcan be decomposed into a product of two components:� a space component � � � � �� , which depends on the physical

properties of path � between Tx antenna � and Rx antenna� , including antennas and reflectors position, path loss,etc.� a time component � � � � which includes the Dopplerdue to reflectors motion and the relative speed of thetransmitter w.r.t. the receiver.

The time components � � � � are assumed to be independentbetween paths, hence � �� � � ! " # � � � � � � $ � � � % �& ' � � � � � . Note that we consider a time scale where theyevolve significantly during time, e.g. due to the Dopplereffect, and hence can be considered random processes,whereas the physical properties of the problem, comprised ofthe � � � � �� , do not vary.

In discrete time, the specular channel model yields

� � � � �" � ( )� � � � * � � � � � � � �+ , - . / 01

� 2 3 � � � � �� � " � � 4 ( , - . / 05 � � � (1)

where we used the discretized version of the time component� " � � )� � � * � � � where � � is the sampling interval at thereceiver. Let us assume that the impulse response has finitesupport, and consider its discretized version

6 � � � �" )� 7 � � � � �" � 8 � � � � � � � � �" � 9 : 3 ; < � (2)

11680-7803-8622-1/04/$20.00 ©2004 IEEE

with � chosen such that all the channel coefficients outsidethe lag interval � � � � � � � � � � � are zero. Let us further stackthese into a row vector with

� � � � coefficients� � �� � � � � � � �� � � � � � � � � � � �� � � � � � � � �� � � � � � � � � � � � �� � � � � (3)

We emphasize the fact that� � consitutes a snapshot of all the

channel impulse response coefficients at time � � . With thisnotation, (1) can be rewritten in more compact form as� � � � � (4)

where ! � " � # �$ �� % � " � # �$ & ' ( ) * + , -. � � � � � � ' ( ) * + , -. � � � � � / , ! $ ��& ! � � � � �$ � � � ! � � � � � �$ � ! � � � � �$ � � � ! � � � � � � �$ / � , � �� � ! � � � � � � ! 0 ,and � �� � 1 � � � � � � � � 1 � � 0 � � .

III. SPECTRAL FACTORIZATION AND LINEAR ESTIMATION

In this section, we outline the possible improvements inchannel tracking that can be achieved by deconstructing aspecular channel, i.e. by separating the time and space proper-ties as enounced in the previous section before doing any kindof smoothing or prediction. We seek to model the discrete-time random process 2 � � 3 from its noisy measurements4� � � � � 5 6 � where the noise 2 6 � 3 is white Gaussian, iid,independent from 2 � � 3 . Assuming that both 2 � � 3 and 2 6 � 3are wide-sense stationary (WSS), let us define the (matrix)covariances7 89 89 � : � �� ; � � 4� � < = 4� >� � and

7 9 89 � : � �� ; � � � � < = 4� >� �where ; � � ? is the expectation operator taken over � , and the@ -transformsA 89 89 � @ � �� < BC= D E B 7 89 89 � : � @ E =

andA9 89 � @ � �� < BC= D E B 7 9 89 � : � @ E = � (5)

The best linear estimator (in terms of mean square error) of� � < F � G H �given 2 4� I 3 � I D E B isJ� � < F �� < BC

" D E B K � E " � " (6)

where the matrix filter coefficients K = are determined in the@ -transform domain

K � @ � �� < BC= D E B K = @ E =(7)

by [5]

K � @ � � L @ F A9 89 � @ � M E N � @ E N � O < 7 P E � M E � � @ � � (8)

where M � @ � and7 P

come from the spectral factorizationA 89 89 � @ � � M � @ � 7 P M N � @ E N � � (9)

In general, spectral factorization is hard to compute in the caseof vector-valued processes, and (8) is not feasible. Note thatdue to the independence between 2 6 � 3 and 2 � � 3 ,

A9 89 � @ � �

A9 9 � @ � and

A 89 89 � @ � �A

9 9 � @ � 5A

Q Q � @ � � (10)

A. Specular model and spectral factorization

Under the assumption that the channel variations follow thespecular model (4), the covariances become7 89 89 � : � � � ; � � � < = >� � � > 5 ; � R 6 � < = 6 >� S (11)� � 7 T T � : � � > 5 7 Q Q � : � � (12)7 9 89 � : � � � ; � � � < = >� � � > � � 7 T T � : � � > � (13)

Note that in eqs. (11) and (13) the factor � is independent ofthe lag : . Therefore, the @ -transforms can be factored as

A 89 89 � @ � � �A T T � @ � � > 5

AQ Q � @ � and (14)

A9 89 � @ � � �

A T T � @ � � > � (15)

Let us define

U � $ �V V � @ � �� < BC= D E B ; � R 1 � < = � $ 1 N� � $ S @ E =for W � � � � � X �

(16)Note that

A T T � @ � is diagonal, since we assume that the 1 $ � Y �are independent:A T T � @ � � diag & U � � �V V � @ � � � � � � U � 0 �V V � @ � / � (17)

Note that this structure allows to obtain the spectralfactorization of

A 89 89 � @ � by performing X independent, scalar

spectral factorizations of the U � $ �V V � @ � (equivalently, this meansthat the random process 2 � � 3 can be acurately tracked bytracking X scalar processes).

In the following sections, we will address the followingidentifiability problem: assuming the knowledge of thespectrum

A 89 89 � @ � , we show that if the U � $ �V V � @ � are linearlyindependent polynomials, it is possible to identify them up toa permutation and a complex scalar coefficient. Fortunately,this is sufficient for our needs, since all possible solutionsyield the same predictor K � @ � . Then, we discuss an algorithmthat achieves this identification.

IV. IDENTIFIABILITY

In this section, we show that if the spectrums U � $ �V V � @ � arelinearly independent polynomials, the spectral factorization(15) is unique up to a permutation and a scalar coefficientapplied to the columns of � .

Let us assume that � has full column rank, and let� Z � � � � � � Z 0 � be an orthonormal base of the column subspaceof � . Let [ �� � Z � � � � � � Z 0 . Let \ denote the representationof � in this base, such that � � [ \ . Let us assume that

A9 89 � @ � has an alternative factorizationA

9 89 � @ � � ]A T ^ T ^ � @ � ] > � (18)

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and show that � and � are identical up to a permutation and alinear scaling of their columns. Using the fact that � � � � � � ,the decomposition in (15) yields

� � � � � � � � ��

� �� � � � � � (19)

Hence, using (18) and defining the � � � matrix � ��� � � � � ,� � � � �

� � � � � � ��

� � (20)

Therefore we need to prove that

�is the product of a

permutation matrix and a diagonal matrix. Let � � � � � � � � � �denote the columns of � . The diagonal structure of

� � � � � �lets us rewrite � � � � � ��

� � � � � � �� � � � � � � � � � (21)

This implies that each � � � �� is diagonal, otherwise theoff-diagonal terms would yield an identically zero linearcombination of � � � � � � � � ’s, which contradicts the linearindependence assumption. This implies that each � � has atmost one non-zero coefficient, which is equivalent to sayingthat

�represents the product of a permutation matrix and a

diagonal matrix.

V. PRACTICAL IDENTIFICATION METHOD

The previous discussion has shown that any factorizationof the form of (18) is an equally good way of decomposing! " # $ into scalar, independent processes. In this section, wepresent an algorithm to find one of these decompositions.

Let us assume that the noise level is known, or has beenestimated, hence

�� � is known, or equivalently,

% � � & �is known for & ' ( . The algorithm that we present hereprovides a matrix ) that decomposes

�� � into independent

processes, i.e. )�

� � ) � is diagonal. We restrict the problemto the signal subspace, and consider � � % � � * � � . Since it ispositive semi-definite, it can be decomposed according to itseigenstructure: � � % � � * � � � + , + � � (22)

where + is a � � � unitary matrix, and , is diago-nal, and contains the (non-negative) eigenvalues. Notice that + - , � + - , � � constitutes a Cholesky decomposition of� � % � � * � � . Since . � / % � � * � 0 . � / % � � * � 0 �

is also aCholesky decomposition of the same matrix, they are unitarilysimilar [6], i.e. there exist an unitary matrix 1 s.t.

2 � 3 % � � * � 4 � � 1 2 + - , 4 � � (23)

Obviously, finding 1 would let us identify � up to the scalaruncertainties contained in / % � � * � . In order to find it, weuse the fact that2 + - , 4 � � � % � � & � � 2 + - , 4 �

� 1 � % � � & � % � � * � � 1 5 & ' ( � (24)

Under our assumptions, there is a unique way (up to apermutation 6 ) of diagonalizing the spectrum matrix, asdemonstrated in section IV, hence if 7 is a unitary matrixthat diagonalizes 7 - , � + � � � % � � & � � + - , � 7 � forall & ' ( , then 1 � 6 8 7 . Finding 7 is the well-knownsimultaneous diagonalization problem [7], and can be solvedefficiently using an algorithm based on Jacobi angles [8].It follows that ! " # $ is transformed into an arbitrary vectorprocess ! 9 # $ , with

9 # �� 7 - , � + � � � " # � 6 3 % � � * � � : # � (25)

and

�; � is diagonal: 7 - , � + � � � is a possible ) . The

uncertainties outlined in section IV appear clearly in (25):each component of ! 9 # $ is normalized to unitary variance,and the permutation 6 is unknown.

Obviously, the theoretical identifiability outlined insection IV, is not realistic in practice, since implementationconstraints would restrict the knowledge of

% � � & � toa limited range of & . Also, the requirement of linearindependence of the columns of � , as well as the linearindependence of the -spectrums of the time coefficients,are not guaranteed to be fulfilled in real life. However, oursimulations show that these limitations do not seem to incursignificant problems in practice.

VI. APPLICATIONS

One of the interest of a specular channel model is its long-term validity. Since it closely follows the physical channelstructure, and hence separates the spatial and temporal prop-erties of the channel, estimation or prediction of the timecoefficients only (the < = > � ) together with the knowledge of� provides knowledge of the channel state, following from" # � � � : # (26)� � + - , 1 � 3 % � � * � � : # (27)� � + - , 1 � 6 � 9 # (28)� � + - , 7 � 9 # (29)

where we used successively (4), (23), (25), and the definitionof 1 . Note that these matrices need to be estimated first,and in particular the choice of the number of independentcomponents to consider. In practice, this parameter can bechosen as the number of non-noise eigenvalues of

% �� �� * � .Then, � can be obtained by orthogonalizing the correspondingset of eigenvectors. Subsequently, the channel estimate can beobtained through ?" # � � + - , 7 � @9 # �where @9 # is obtained by any estimation method (for instancesmoothing, linear prediction...) from the

?9 # . Since thecoefficients in 9 # are independent processes, the burden ofthe estimation method is greatly decreased. Several kinds

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−15 −10 −5 0 5 10 15 20 25 300

5

10

15

20

25

30

Channel estimate SNR

Out

put S

IR

N=1000 samples, P=P’=4N=100 samples, P=P’=4N=100 samples, P=4, P’=6N=100 samples, P=4, P’=3N=20 samples, P=P’=4

Fig. 1. SIR of the process separation

of processings can be applied at this point, for instancesmoothing if the goal is to increase channel estimationacuracy. Prediction can be useful in systems relying onChannel State Information (CSI) at the Transmitter (CSIT) toenhance the link quality: since duplex systems mainly rely ona feedback scheme to transmit the channel state informationfrom the receiver, and since the channel state informationcan not be fed back in a negligible amount of time, theability for the transmitter to extrapolate CSI from past valuescan therefore be an important asset in the actual use of aCSIT-exploiting transmission scheme [9]. The nature of theunderlying processes must also be considered. For instance,pure Doppler effect would yield an autoregressive process oforder 1.

VII. SIMULATION RESULTS

The algorithm proposed in section V has been proven insection IV to identify perfectly the system in the noiselesscase. We present here some results obtained in a more practicalsimulation setting. We simulated a setting with � � � antennas,and a delay spread limited to � � � samples. The channel isan actual specular channel with � � � paths, where each pathis determined by randomly generated integer lags, uniformDoD and DoAs, a Gaussian gain and a random AR3 process.The algorithm works with approximate covariance matricesestimated from a finite-length measurement interval of

�successive realizations of � � . The number � � of estimatedindependent random processes to estimate is set artificially,and several cases ( � � � , � � � � , � � � � ) are presentedhere.

The figure of merit used in these simulations is derivedfrom the fact that in the absence of noise, with the notationsof section V, � � � � � � � � � � � � � � � � � � � � � �

, i.e. itis a diagonal matrix. When noise is present, this matrix iscomputed from the true � and the specular model as estimated

by the proposed algorithm, hence it is not perfectly diagonal.Hence, denoting ! " # $ % " # $ &� � � � � � � � � � � � � , ! ' # ' isthe amount of energy from � ( # ' � that is correctly attributedto the ) th estimated process, and the ! " # ' * + ,� ) representsthe crosstalk from other processes. Thus, we define a globalsignal-to-interference ratio (SIR) as

- . / &� 01' 2 � ! 3' # '

01' 2 � 0 " 42 ' ! 3' # "

5 (30)

This value is plotted on Figure 1 for various configurations,with respect to the SNR of the raw channel estimates

6 .These results clearly show the influcence of the the quality

of estimation of the covariances: increasing the number ofrealizations used for estimating the channel statistics from� � � �

to� � 7 � �

, then 7 � � �, has a relatively bigger

influence than the SNR variations over the range picturedhere. The influence of overestimating the number of pathscan be estimated by comparing the � � � � � � to the

� � � 8 case. Overestimation of the number of paths yieldsan almost negligible decrease in the SIR of the � correctlyidentified paths. The case where � � � � is interesting in thatit represents the resilience of the identification algorithm tomodel mismatch, i.e. when the signal does not conform to theassumptions that support our method. In this case, the SIRcriterion is only computed on the � � separated processes,which are statistically the strongest. This explains the fact thatwe observe a slight SIR increase in this case, for low inputSNR values, and goes to prove that the � � strongest paths arecorrectly identified. However, this metric is hiding the fact thatthe channel is not fully analyzed, and hence not as predictable.Evidencing this would require a more involved simulationsetup, where the modeling error variance would be considered.

VIII. CONCLUSION

We presented a channel modeling method based on theassumption that the channel follows a specular structure.We showed how this structured model, by separating spaceand time-components, lends itself to simplified tracking,including smoothing and prediction, once the underlyingspace and time characteristics are separated. We showed thatunder mild assumptions on the channel characteristics, thesecomponents are identifiable, and proposed a method based onsimultaneous diagonalization of the covariance matrices thatachieves the identification. We evaluated the performance ofthe proposed method through simulations.

ACKNOWLEDGMENT

Eurecom’s research is partially supported by its industrialpartners: Ascom, Swisscom, Thales Communications,ST Microelectronics, CEGETEL, Motorola, FranceTelecom, Bouygues Telecom, Hitachi Europe Ltd. andTexas Instruments.

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[5] T. Kailath, A.H. Sayed, and B. Hassibi, Linear Estimation, Prentice Hall,2000.

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