Semiblind Channel Estimation for MIMO Spatial Multiplexing Systems

Abdelkader Medles Dirk T.M. Slock Institut EurCcom*, 2229 route des Cretes, B.P. 193,

06904 Sophia Antipolis Cedex, FRANCE Tel: +33 4 9300 2910/2606; Fax: +33 4 9300 2627 e-mail:medles,slock)@eurecom.fr

Abstract In this paper; we analyze the semiblind mutual informa-

tion (MI) between the input and output of a MIMO channel. To that end we consider the popular block fading model. We assume that some trainingipilot symbols get inserted at the beginning of each burst. We show that the average MI over a transmission burst can be decomposed into symbol position dependent contributions. The MI component at a certain symbol position optimally combines semiblind in- formation up to that symbol position (with peflect input re- covery up to that position) with blind information from the rest of the burst. We also analyze the asymptotic regime

.for which we can formulate optimal channel estimates and evaluate the capacity loss with respect to the known chan- nel case. Asymptotically, the decrease in MI involves Fisher Information matrices for certain channel estimation prob- lems. We also suggest to exploit correlations in the channel model to improve estimation performance and minimize ca- pacity loss.

1: Introduction We consider single user spatial multiplexing systems

over flat fading MIMO channels. We furthermore consider the usual block fading model in which data gets transmitted over a number of bursts such that the channel is constant over a burst but fading independently between bursts. The transmitter is assumed to have no channel knowledge. The formidable capacity increase realizable with such dual an- tenna array systems has been shown [5],[2] to be propor- tional to the minimum of the antenna array dimensions for channel with i.i.d. fading entries. At least, this is the case when the receiver has perfect channel knowledge. This con- dition is fairly straightforward to approach in SISO systems by inserting pilot/training data in the transmission, with ac- ceptable capacity decrease [l]. For MIMO systems of large dimensions though, the training overhead for good channel

*Eur6com's research is partially supported by its industrial partners: Ascom, Swisscom, Thales Communications, ST Microelectronics, CEGE- "EL, Motorola, France T616com, Bouygues Telecom, Hitachi Europe Ltd. and Texas Instruments. The work reported herein was also partially sup- ported by the French RNRT project ERMITAGES.

estimation quality becomes far from negligible though, es- pecially for higher Doppler speeds such as in mobile com- munications. The effect of channel estimation errors on the MI has been analyzed in [4], whereas optimal design of training based channel estimation has been addressed in [6]. The true capacity of the system is higher though than the MI obtained by training based channel estimates [2],[3]. In this paper, we attempt to approach the true capacity by optimal semiblind channel estimation that is suggested by a decomposition of the MI. Related background material ap- peared in [7] where a variety of semiblind MIMO channel estimation techniques were introduced.

2: Capacity Decomposition We shall consider here the usual block fading model, except that we shall refer to a block as burst; consider then trans- mission over a MIMO flat fading channel y = H z + 'U, for a particular burst of T symbol periods. The accumulated received signal over the burst is then:

(1) Y = (IT @ H ) x + v where Y and V are N,.T x 1 and X is NtT x 1. H is a N,. x Nt channel matrix. Nt (resp. N,) is the number of transmit (resp. receive) antennas. We assume the use of a pilot training sequence of length NtTp, the length of the transmitted data is then NtTd so that Tp.+ Td = T. We can decompose the burst signal into training and data parts X = (XF,X:)' ,Y = (Y:lYz)T andV = (%',c)'. As stated in [6], the mutual information between the input and the output of this system verifies: I(Yp,Yd;XdlXp) = I (yd ;xdIxp ,yp ) + I (yp ;xpIxd)

= I (yd; x d l x p , y p ) (2)

I(Y,, X p ; Xd) = 0 due to the independence between xd and (Yp, X p ) . Consider now a partition of Xd in Q blocks X j , i = 1,. . . , &, Xd = ( X T , . . , , X z ) T , of different lengths T i , i = 1,. . .,&, x j T = Td. Xj and are assumed independent from block to block (block-wise coding across bursts). Let's define for, j 3' i , X! = ( X T , X s l , . . ., Xj')' and xj = (qT, X s l , . . . , yj')'. Then,

0-7803-7 147-X/01/$10.0002001 IEEE 999

=O (3)

h(K IX,, U,, X I , Xf), and considering the fact that Xf is independent of Y1 conditioned on (X, , Yp , XI), this leads to h ( Y l X p , Y p , X I , X?) = h( f i IX,, 5, XI) and finally I(Y1; X f I X , , Yp, XI) = 0. Iterating the equation for i = 2, . . . , Q, this leads to the following:

where ~(Yl;xfIxP,Y,,xl) = h(YlIXp,Yp,Xl) -

I (Yd;Xd lXprYp) = EL' I(Yt,Q;x~JXp,Yp,Xf-l,Y,-~) 9

= qy i ; xj IX,, Yp, xi-', Y p ) i=l

I 2 This clearly shows the way of processing to achieve capac- ity: for every block we use the already detected blocks as a (data-aided (DA)) training sequence (in addition to the ac- tual training) and use the not yet detected bIocks as blind information. Also, the mutual information can be seen as the sum of two parts: I(Yd; XdlXpl Yp) = 11 + 12 where: 11: is the rate that we can achieve by using only the already treated blocks for side information (DA training). 12: is the additional amount of rate that can be achieved by exploiting the blind information contained in the not yet de- tected blocks. The average mutual information is defined as Iawg = +I(&; XdlXpr Yp). We want to show below that this quan- tity goes to the coherent MI I(y; t l H ) as T and Q grow for finite Tp, where I(y; z / H ) = lim +I(Y; X I H ) = lim &I(X ; Xi 1 H) . An upper bound is given by:

i+oo

T-+m

I w g = +(h(Xd) - h(XdlXp7 5 , Y d ) ) $(h(Xd) - h(XdlXp,Yp,Yd, H ) )

= + ( h ( X d ) - h(XdlYd, H ) ) (4) = +.(I(Y; X ( H ) - I (Yp; X,IH))

that verifies limj+oo g(i) = H as. Given that, liw+, ~ ; I ( ~ ~ " x X ~ ~ X , , Y , , X ~ - ~ , Y ; : - ~ ) 2 ~ ( y ; z l ~ ) . This allows us to conclude that limT,, Iawg 2 I(y; %IN). Combining this result with (4) we conclude that: limT,, ~ I ( Y d ; x d [ x p , ~ ) = I(y;zlH) . This means that when T grows, adopting the detection method per block, and the associated channel estimation allows to reach the capacity of the system assuming perfect channel knowledge (see 131 for related results at high SNR). Remarkl: When T grows, the use of detected blocks only to estimate the channel allows to achieve asymptotically the MI IOwg of the system. But for finite T , it is necessary to also use the blind information to reach it. Remark2: For a fixed T , and when all the entries of X (training and data) are iid, it's easy to see that the average MI Iavs of the system is maximized when the number of the training symbols Tp is as small as possible (i.e. allows semiblind identifiability of the channel). Remark3: In the case where H is no longer constant, and varies from block to block H = H(i), the (coherent) ca- pacity of the system assuming perfect channel knowledge is no longer achievable €or large T and the average MI is bounded by:

I Q I Q i = l i = l

- x I ( X ; X i l g i ) 5 Iaug 5 - x I ( X ; X i \ H i ) T (6)

for any channel estimate Gi = & ( X P , Xi-' , Ti). 3: Channel estimation 3.1 Bayesian case (random channel with prior)

The capacity C of the system is the maximum MI Inug over all input distributions, under a given power constraint. We have

This is valid for all choices of the partitioning of Xd into Xi I i = I,. . . , Q, and in particular for Q = Td and Tj E 1. For an AWGN V with power U: and in the absence of side information on the channel at the transmitter (see [SI), the max in the upper bound of the capacity (coherent capacity) is attained for a centered white Gaussian input with covari-: ance Rx, = vz IN, T,, .

. , where g(i) = g ( X p , X ; - ' , y j ) . %e received signal 2 yt: = H xi + M = it(;) xi -( H ( i ) X i + V;: = H ( ' ) X j + V;: + Zj, where we as- sume H(') to satisfy the Pythagorean Theorem (PT) (i.e.

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i? is decorrelated with E). Now I ( ~ j X j l i ? ( i ) ) =

Zj IXj , e(')). Under the above conditions and for uncor- related and centered Gaussian Vi and X i = Xi" (vari- able with same 1st and 2nd order moments, but Gaus- sian), it _was shown in [6] that a lower bound for I ( K , X i (H( ' ) ) is given by considering Zi as an indepen- dent and white Gaussian noise, with covariance 6," I , where

and U&,, = I. The new lower bound is now:

h ( l q m ) - h(xIxi,N) = h(Yilii(')) - h(V;. +

U," = &tr E(Zj2:) = U, 2 tr E ( fi(*)fi(') N , = Nt&;(,) tr E(i;icoi;(:) H

Nr N c

c 2 + ~ T z 1 I ( K ; X j l 2 " ) ) 2 E l n d e t ( I + u z + N ~ u ~ g l f i ( i )G( i )H)

H ( a )

(9) - , then due to

the fact that the channel estimator satisfies the Pythagorean Theorem, we have that U&,, + a&,, = U;. Now:

tr E(ii(*)B(a) H) and z(i) fi(:) Nr Nt U i i ( 0

Let O&,) =

O ~ ( O L - U ~ ( , ) ) -(i)-(i) H 2 + o~+N~o: :L " 1

H ( * )

= CLB (10)

The expectation is over the distribution of $'I, which re- mains close to that of H('). Then the given capacity lower bound CLB depends primarily 05 the Mean Square Error (MSE) of the channel estimator Id'). Since CLB is a de- creasing function of the MSE, the optimum estimator is the Minimum Mean Square Error(MMSE) estimator:

which is an unbiased estimator of H . The performance of any unbiased estimator is bounded by the Cramer-Rao lower bound:

where h = [Re(vect(H))T I r n ( ~ e c t ( H ) ) ~ ] ~ and J ( j ) is the Bayesian Fischer Information Matrix (FIM) for the a posteriori distribution of H , and is in this case:

( 4 J D A t raining

We have U& 2 =. t r J - ( ' ) This is an absolute lower bound on the channel estimation MSE. The MMSE estima- tor achieves this bound asymptotically (Td -+ 00). The above result is also valid for the case when the channel varies from block to block, since channel re-estimation for every block is assumed. 3.2 Asymptotic Behavior

We focus here on the asymptotic behavior of the ca- pacity loss for small channel estimation error. We sup- pose that the channel is constant over every block, the mu- tual information for the particular block (i) at the-receiver assuming channel estimation is then I ( K ; XjlH( ' ) ) = E(IHc,) (K; XilZ('))) , where IH(;) (x; Xi$?( ' ) ) is the ca- pacity for a particular realization of the channel. It is as- sumed here that the channel H ( i ) may vary from block to block (while allowing an asymptotic regime). The MI as- suming perfect channel knowledge and for a particular re- alization is IH(;) (x; Xj lH( i ) ) . Let's temporarily drop the block index (i) . In the following, we derive a weighting matrix that appears in the asymptotic MI decrease and optimal semiblin? chan- nel estimate. The first order derivative of IH (Y; X IH) with respect to E = H - G evaluated at E = o is zero:

J =1

(13) The second order derivative w.r.t. E evaluated at 0 is:

(14) J y and J y l x are FIMs describing the MI decrease due to the channel estimation error and evaluated at a = 0. To compute J y and J y lx, we consider a receiver point of view i: which, given a certain realization H and? certain i?, H is an unknown constant. Then PH(Y IX, H ) = p(V + (e + @ X l X , @ = p(V + E X l X ) = pi;(VlX) where V = V + E X and the variable f i lX follows the same distribution as V but with an offset (mean) i? X (notation assumes Ni = 1). Now:

I

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For 2 in a small neighborhood of H , the MI decrease is approximated by a quadratic function:

-T IH(Y; XI@ - IH(Y; XIH) = -h W ( H ) FI = -(G - h)T(Jylx(H) - Jy(H))($ - h)

(15)

The weightingmatrix W ( H ) = Jylx(H) - J y ( H ) isnon- negative definite since IH (Y; XI E ) 5 IH (Y ; XI H ) . ExamJe: For white Gaussian and_decorrelated V;: and Xi, KIXj is Gaussian with mean H X and covariance

Then Jyi(H) = 0 and Jyilx,(H) = Nj$I . As a re-

sult W ( H ) = m = Ni$I is constant. More gen- erally, for any Gaussian noise and zero mean input Xi; Jyi(H) = 0, Jyi1xi(H) = &;!xi is constant and

We now want to find the best channel estimator, that minimizes the MI decrease. Then we needhto mini- mize the cost function I (Y;XIH) - I(Y;XIH(S)) = EIH(Y;XIH) - 1 ~ ( Y ; X l f i ( S ) ) whereforevery block i, r? is based on S(') = (X,, Xf-', Ti). Asymptotically

IPi - (IN; si i ) xi l a u : ~ : pg(%IXj) = ( ~ T U , ~ ) - N ~ N ~ Z exp -

0:

Wj(H) = Wj = Jy;lx;.

mi9i(s) (I(Y; XIH) - w; X l W ) ) ) j E[min- E(G(S) - h)TW(H)(%(S) - h)lS] h w * Goppt(S) = ( E W ( ~ ) l S ) - ' ( E ~ ( ~ > h I S ) )

(16) So Gopt is a weighted MMSE estimate. For centered in- put X and Gaussian noise V , W ( H ) = W and the opti- mal channel estimator is the MMSE estimator g0,t (S) = W-' EWhlS = EhlS. The minimum mean MI de- crease w.r.t. the perfectly known channel case is:

I (Y;XIH) - I(Y;Xlf?opt(S)) = E[hTW(H)h

-4 EW(H)hlsl)T( EW(H)lS)-l (EW(H)hlSl)I. and all this so far for block (i). The minimum mean MI de- crease for the complete burst becomes, using the recursive MI decomposition of section 2:

xF=i [ I (x ;x j IH( i ) ) - I ( K ; x ~ I ~ $ \ ( s ( ~ ) ) ) I = xzl [E hT W(H('))h

- ( E Wi hl S('))T ( E Wi IS('))-' ( E Wihl S( i ) ) ] .

where W' = Wi(H(')) = Jyilxi(H(')) - JY~(H(~)) . In the case of a constant channel over the different blocks (i.e H(' ) = H, i = 1,. . . , Q), CY- I ( X ; x~IH(')) gets modified to I&; X@) and Wj(Haf) to Wj(H).

3.3 Deterministic channel I

In this case we have no prior information on the channel. The channel is constant during the burst with an unknown value H . The coherent capacity is then I(Yd; XdIG, X,, H ) . For a given realization Yo of Y andia (variable) channel g, let's define G(Yo, X,, E ) = h(X) E(-lnq(XIYO,@) = h(Xd) - E(-lnq(XIYi,Yi,@)) where the expectation here is with respect to x d (X, is known), and q(XIy,O, Y i , 2) = p(Xly,O, ydo, H ) I H = ~ . Let's introduce a partition of Yd in which the different blocks have the same lengths IVi = NI, i = 1, . . . , Qand are i.i.d. Then in +G(Y",X,, H ) = $[lnq(X,IYpO, H ) + xzl (h(Xj) + E In q(Xj lyt:., g) ) ] , the averaging over the blocks tends asymptotically to an expectation w.r.t. Yi. We conclude that :

limT+, $G(Y", X,, H ) = limT+, $lnp(X,IYpO, H ) - x,S=,(h(Xi) - Elnp(Xilyt:., H ) = I(Y; zIH)

(17) so asymptotically G(Y", X,, H ) approaches the mutual infoi- mation. Let's define G,(H) = limT+, +G(Y", X,, E] . h

GW(H) - GW(@ h

= & EX,,Y,lHPnp(XlIYl,H) -lnq(XIIYl,H)l '

= ~ J P ( Y l I H ) ( S p ( X l l ~ , H ) I n ~ ~ ~ ~ l ) d ~ ~

= ~ E Y ~ I H [ D ( P ( X ~ I Y ~ , H ) I ( P ( X ~ I ~ , H ) ) ] 2 0 (Kullback-Leibner distance). This means that G, (r?) is maximized when q(X1 IY1,E) = p(X1 IK, H ) . This fact, combined with the hypothesis that the training part is suf: ficiently informative to allow, together with the blind in- formation, complete channel identifiability, ensures $at asymptotically (for T d and Tp big enough) G(Y", X, , H ) is maximized for H =_H. We can hence use the maxi- mization of G(Yo, X,, H ) as optimizationapproach to find a consistent channel estimator. This method is related to the first iteration of an itera; tive MAP&& estimation approach for input signallchannel with EM applied to the ML estimation part for the chan- nel. The maximum a posteriori (MAP) estimate of Xd iy the MAP/ML approach is: I

max Inq(XIY",H') . (18)

h

XdMAP = argXd xd,"

Various solution techniques exist for this type of problem. Consider an alternating maximization (between xd and H ) approach in which expectation over x d is introduced when maximizing over H (EM-like approach). The iterations comprise two steps. The first step for the first iteration gives:

h

H = argmaxxl E(lnq(XIYO,H')) (19) = argmaxHIG(Yo,Xp,H') .

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This is a semiblind cost function for the channel estimation. The second step consists of th_e MAP estimation of the input assuming the channel H‘ = H . Remarkl: The recursive decomposition of section 2 re- mains valid in the deterministic channel case, and we can process by successive detection of the symbols (blocks). To have an acceptable algorithm complexity, one can choose from a variety of channel updating techniques. Along the lines of section 2, this approach allows to maximize the MI for large T. Remark2: Similarly to the Bayesian case, we can evaluate the asymptotic MI decrease as h W ( H ) with W ( H ) = J y p ( H ) - J y ( H ) . In the deterministic case however, the direct minimization of the MI decrease does not lead to a meaningful channel estimator. 4 Correlated MIMO channel model

In order to improve channel estimation and reduce ca- pacity loss, it is advantageous to exploit correlations in the channel, if present. So consider the frequency-flat MIMO channel: H (Nrs x Ntm) , h = vect(H). The correlated channel model we suggest is: -

where the elements of g are taken to be i.i.d. Gaussian for a stochastic model. The correlations are captured by S. Special case 1: Bell-LabdSaturn model: H = G Rf” j S = Rf12 8 & I 2 , g = vect(G) Special case 2: Cioffi-Raleigh model (multipath model): H = C g j a i b F =$ S = [ b ; @ a i b ;@a2 a . . ] .

-T

h = S g ( + go h for direct path, 1901 = 1)

i The model h = S g is straightforwardly extendible to the non-zero delay-spread case. 5 Observations

Capacity approaching channel estimation should exploit prior info + datddecision aided info + (Gaussian) blind info. The symbol-wise decomposition of the block fading chan- nel capacity involves for each symbol position in a burst a channel estimate that is based on: prior channel distri- bution info, training and detected inputs up to that symbol position, (Gaussian) blind info in remaining channel out- puts. Hence, symbol-wise Gaussian semiblind (Bayesian) channel estimation is required (blind parts: detected data + Gaussian undetected data). To have asymptotically (in S N R and burst length) negligible capacity loss, enough training symbols are required to have (deterministic) identifiability of the parameters that cannot be identified blindly (from the Gaussian undetected symbols), hence blind (Gaussian) info reduces training data requirements. Exploiting channel correlations, the (effective) number of degrees of freedom in the channel and hence the training requirements get re- duced. The channel with i.i.d. entries, while optimal from a capacity point of view, is the worst case from the channel estimation point of view.

The recursive mutual info decomposition may suggest a practical approach for channel estimation. However, sim- pler practical approaches would pass through the bursts it- eratively, with semiblind (blind info = Gaussian undetected symbols) channel estimation in the first pass, and semiblind (blind info = detected data) channel estimation in the next iterations. Prior channel info (and S in the channel model) gets estimated (sufficiently well) by considering the data in multiple bursts jointly (assuming these parameters are in- variant across a (large) set of bursts).

Whereas we have considered block fading so far in this paper, we conjecture that these results extend to the continu- ous transmission (CT) case: in steady-state, channel estima- tion should be based on the semi-infinite detected past sym- bols, and semi-infinite future blind channel information. A Gauss-Markov model for the channel variations with a cer- tain Doppler bandwidth will prevent perfect channel extrac- tion from this infinite data though. Finally, the proposed channel model is useful for the introduction of partial chan- nel knowledge at the transmitter. Indeed, if the transmit- ter can know the channel correlations summarized in S in h = S g and only lacks knowledge of the fast fading pa- rameters g, the channel capacity may be close to that of the known channel case.

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L. Zheng and D. Tse “Communication on the Grass- mann Manifold: A Geometric Approach to the Non- coherent Multiple Antenna Channel”. To appear in IEEE Trans. Info. Theory. M. Medard ‘The effect upon channel capacity in wire- less communications of perfect and imperfect knowl- edge of the channel”. ZEEE Trans. Info. Theory, vol. 46, no. 3, pp. 933-946, May 2000. I.E. Telatar, “Capacity of Multi-antenna Gaussian Channels”. Eu,: Trans. Telecom., vol. 10, no. 6,

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