5296 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER X/$ IEEE

Transcription

1 5296 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008 [13] E. A. Jorswieck and E. G. Larsson, Linear precoding in ultiple antenna broadcast channels: Efficient coputation of the achievable rate region, in Proc. IEEE/ITG Int. Workshop on Sart Antennas, Darstadt, Grtsny, Feb , 2008, pp [14] D. Tse and P. Viswanath, Fundaentals of Wireless Counication. Cabridge, U.K.: Cabridge Univ. Press, Multiode Precoding for MIMO Systes: Perforance Bounds and Liited Feedback Codebook Design Xiaofei Song and Heung-No Lee, Meber, IEEE Fig. 2. Pareto boundary for a saple channel realization with N =2 two transit antennas at high SNR 30 db. VI. CONCLUDING REMARKS The otivation for this correspondence has been the recent, huge interest in IFCs as a odel for spectru resource conflicts (see, e.g., [5], [7], [9], [10], and [12], and the references therein). Our ain contribution has been a characterization of the MISO IFC for arbitrary SNR, and specifically a paraetrization of the Pareto boundary of the rate region. Our hope is that the results will be useful for future research on resource allocation and spectru sharing for situations that are well odeled via the MISO IFC. REFERENCES [1] R. Ahlswede, The capacity region of a channel with two senders and two receivers, Ann. Prob., vol. 2, pp , [2] A. B. Carleial, Interference channels, IEEE Trans. Inf. Theory, vol. 24, no. 1, pp , Jan [3] T. Han and K. Kobayashi, A new achievable rate region for the interference channel, IEEE Trans. Inf. Theory, vol. 27, no. 1, pp , Jan [4] M. H. M. Costa, On the Gaussian interference channel, IEEE Trans. Inf. Theory, vol. 31, no. 5, pp , Sep [5] X. Shang, B. Chen, and M. J. Gans, On the achievable su rate for MIMO interference channels, IEEE Trans. Inf. Theory, vol. 52, no. 9, pp , Sep [6] S. A. Jafar and M. Fakhereddin, Degrees of freedo for the MIMO interference channel, IEEE Trans. Inf. Theory, vol. 53, no. 7, pp , Jul [7] M. Charafeddine, A. Sezgin, and A. Paulraj, Rate region frontiers for n-user interference channel with interference as noise, in Proc. 45th Allerton Conf. Counication, Control, and Coputing, Monticello, IL, Sep , [8] X. Shang and B. Chen, Achievable rate region for downlink beaforing in the presence of interference, in Proc. IEEE Asiloar Conf. Signgals, Systes, and Coputers, Pacific Grove, CA, Nov. 4 7, 2007, pp [9] E. A. Jorswieck and E. G. Larsson, The MISO interference channel fro a gae-theoretic perspective: A cobination of selfishness and altruis achieves Pareto optiality, in Proc. Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), [10] S. Vishwanath and S. A. Jafar, On the capacity of vector Gaussian interference channels, in Proc. IEEE Intforation Theory Workshoop, San Antonio, TX, Oct , 2004, pp [11] E. G. Larsson and P. Stoica, Space-Tie Block Coding for Wireless Counications. Cabridge, U.K.: Cabridge Uni. Press, [12] E. G. Larsson and E. A. Jorswieck, Copetition versus collaboration on the MISO interference channel, IEEE J. Sel. Areas Coun., vol. 26, no. 7, pp , Sep Abstract This correspondence investigates the proble of designing the precoding codebook for liited feedback ultiple-input ultiple-output (MIMO) systes. We first analyze the asyptotic capacity loss of a suboptial ultiode precoding schee as copared to optial waterfilling and show that the suboptial schee is sufficient when negligible capacity loss is allowed. This knowledge is then applied to the design of the liited feedback codebook. In the design, the generalized Lloyd algorith is eployed, where the coputation of the centroid is forulated as an optiization proble and solved optially. Nuerical results show that the proposed codebook design outperfors the coparable algoriths reported in the literature. Index Ters Given s rotation, liited feedback codebook design, Lloyd algorith, waterfilling. I. INTRODUCTION A well-known result of inforation theory establishes that feedback does not iprove the capacity of a discrete eoryless channel [1]. Nonetheless, for the cases where the channel is selective in either tie, frequency, or space, feedback of the channel state to the transitter can bring substantial benefits to the forward counications syste in ters of either capacity, perforance, or coplexity. The theoretical study of capacity and coding with channel state inforation at the transitter (CSIT) can be traced back as early as to Shannon [2]. More recently, inforation-theoretic capacity on channels with both perfect [3] [5] and iperfect [6] CSIT and practical coding schees using CSIT [7], [8] have been studied. With the advent of ultiple-input and ultiple-output (MIMO) antenna systes, investigation on the potential benefits of CSIT for MIMO systes has been intensified and design of a practical schee to achieve the potential benefits as closely as possible becoes very iportant. The channel estiation done at the receiver needs to be sent back to the transitter to provide the potential CSIT benefit. Thus, the study of MIMO syste with liited feedback is of practical interests. In the past, various options in MIMO transit beaforing Manuscript received October 15, 2007; revised June 6, First published July 18, 2008; current version published Septeber 17, The associate editor coordinating the review of this paper and approving it for publication was Dr. Athanasios P. Liavas. This research was supported by the University of Pittsburgh CRDF award and in part by ADCUS, Inc. X. Song was with the University of Pittsburgh, Pittsburgh, PA USA. He is now with Sandbridge Technologies, Lowell, MA USA (e-ail: xsong@sandbridgetech.co). H.-N. Lee is with the University of Pittsburgh, Pittsburgh, PA USA (e-ail: hnlee@pitt.edu). Color versions of one or ore of the figures in this correspondence are available online at Digital Object Identifier /TSP X/$ IEEE

2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER with liited rate feedback have been considered in [9] [13]. In the beaforing setting, however, one notes that the capacity loss is usually large, as copared to the optial water-filling (WF) solution [14]. To reedy this shortcoing, in [15] [17], the proble is approached fro the perspective of designing a codebook to achieve the WF gain. The optial WF solution is a ixture of optial antenna phase rotation and power adaptation, which changes subject to a particular realization of the channel. The proposed codebook design ethodology in [15] includes both the phase rotation and power allocation atrix. Whereas in the ultiode precoding schee [16], [17], the codebook contains only the first coluns of the phase rotation atrix (first eigenodes) and the total transit power is equally allocated to the eigenodes. In this correspondence, we also address the codebook design proble in the fraework of ultiode precoding. We first bound the asyptotic capacity loss of ultiode precoding, and show the sufficiency of it when negligible capacity loss is allowed. We then use the generalized Lloyd algorith [18] to design the ultiode precoding codebook. One novelty of our design ethodology lies in the coputation of the centroid, which is forulated as an optiization proble by the Given s angle paraetrization of unitary atrix (orthonoral colun atrix). The proposed algorith, although it is based on a suboptial ultiode precoding schee rather than on the optial WF as in [15], outperfors the algorith in [15] as the nuerical results indicate. We clai the gain in perforance is largely owing to the optial coputation of the centroid. Our proposed design also brings perforance superior, we show, to that of the sae ultiode precoding schee in [16]. The gain over [16] coes ainly fro the fact that the cardinality distribution of the designed codebook eploying our proposed algorith is uch closer to optial (if not exactly optial) as copared to the prederived one in [16]. The reainder of the correspondence is organized as follows. In Section II, the MIMO syste with ultiode precoding is introduced. In Section III, the asyptotic capacity loss of ultiode precoding as copared to WF is developed. The codebook design algorith is presented in Section IV, and the nuerical results are shown in Section V. Finally, conclusions are drawn in Section VI. II. SYSTEM OVERVIEW We consider a MIMO syste with N t inputs and N r outputs. The [N r 2 1] output signal vector is odeled as r = Hs + n, where H is a [N r 2N t ] channel atrix with circularly syetric coplex Gaussian entries of zero ean and unit variance; s is a [N t 2 1] transitted signal with total power constraint P t, such as IE[s 3 s] P t ; n is the [N r 2 1] additive white Gaussian noise present at the receiver, with IE [nn 3 ]= 2 I N 2N. Without loss of generality, we assue 2 =1 throughout the correspondence. By the singular value decoposition, the channel atrix H can be written as H = U3V 3, where both U and V are the unitary atrices and 3 is a diagonal atrix with the singular values of H on its diagonal. When the channel is known to both the transitter and the receiver, the capacity achieving power allocation solution is the well-known WF, and the instantaneous WF capacity [14] is C w(h) = log( i): (1) 1=2 i is the ith largest singular value of H and log is the logarith to base 2. For convenience, we define the subchannel loss as g i := 1= i and the set of good sub-channels as E = fg i : 0 g i > 0;i = 1; 2;...;N tg. Then 3 denotes the cardinality of the good subchannels, i.e., 3 = jej. The water level relates to P t by satisfying the total power constraint, i.e., P t = ( 0 g i) or S 0 := P t 3 = g i = 0 g (2) where g := 1= 3 g i is the ean. While perfect channel knowledge at the receiver can be assued, perfect CSIT ay not be available, in particular for frequency division duplex (FDD) systes where no reciprocity exists between the forward and reverse channel. In such a case, the CSI estiated at the receiver needs to be fed back to the transitter through a feedback channel. While the optial WF requires knowing both V and 3 at the transitter, under the constraint on the feedback channel capacity, we consider a suboptial transission schee which is tered ultiode precoding in [16]. With liited feedback, only the quantized first coluns of the eigenode atrix V are used at the transitter under ultiode precoding, and an equal aount of power P t = is allocated to each eigenode. The N t 2 unitary atrix ^V serves as the precoding atrix at the transitter. This then yields a odified input output relationship r = H ^V s + n (3) where s is 2 1 input with IE [s s 3 ]=(P t =)I 2. The instaneous channel capacity then is C p(h; ^V ) = log det I N + P t H ^V ^V 3 H 3 : (4) III. ASYMPTOTIC CAPACITY LOSS OF MULTIMODE PRECODING As the ultiode precoding schee is based on a suboptial rather than optial WF, it is worthwhile to investigate its loss in capacity as copared to the WF based optial ethod. However, the capacity of the channel for either schee can not be easily obtained with a given feedback rate, thus we resort to the asyptotic capacity loss of the ultiode precoding schee when the feedback rate is assued infinite. The optial ultiode precoding schee will use the exact first 0 coluns of V and V as the precoeding atrix, where 0 = arg ax 1N C p (H;V ). The instaneous capacity then is C p (H; V )= log 1+P t i 0 : (5) The asyptotic instaneous capacity loss of ultiode precoding, using (5) and (1) can be then be upperbounded as 1 C := C w(h) 0 C p(h; V ) C w(h) 0 C p(h; V ) 1+P = 0 log i = 0 log 1+ g i 0 g where the inequality is by the fact C p (H; V ) C p (H; V ) and the last equality follows fro (2). The capacity loss (6) depends on the optial power level. Thus, it can only be evaluated nuerically as no analytical solution to exists. To offer an insight into the capacity loss, we expand the suand in (6) by the Taylor series and obtain the following two upper bounds. (6)

3 5298 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008 Theore 1: The asyptotic capacity loss of ultiode precoding schee is upper bounded by 1 C 1 ln (g) 2 (7) where 2 (g) denotes the saple-variance of the subchannel losses (inverse sub-channel gains) in the set of good subchannels E. Meanwhile, by using the absolute value of the first-order ter, we have 1 C 1 ln 2 jg i 0 gj : (8) Proof: See Appendix A. These bounds, though easy to obtain, provide useful tools to ake inference on the loss of a suboptial solution. It is noteworthy that the asyptotic ultiode precoding schee subsues the beaforing schee in low signal-to-noise ratio (SNR) regie and the equal power allocation schee to all transit antennas in high SNR regie (when N t N r). Therefore, aking use of these upper bounds, we can explain why beaforing and the equal power allocation are optial in the low and the high SNR regie, respectively. In the high SNR regie, i.e., P t 1 and 1, the upper bound (7) becoes zero, and thus the equal power allocation is optial; in the low SNR regie, i.e., P t 1, 3 =1and 2 (g) =0, the upper bound also goes to zero, thus beaforing is optial in this case. Based on the enseble of randoly generated channel atrix H, the average asyptotic capacity loss of ultiode precoding is plotted in Fig. 1. The capacity loss is also copared with the bounds derived. Selected syste paraeters are N t = N r =16.Yu Cioffi bound [19] is also provided for the purpose of coparison. Fro the siulation results, we can see that the capacity loss of ultiode precoding is indeed sall as reflected in this figure. This suggests the suboptial schee based ultiode precoding can be used in practice at the cost of negligible capacity loss. IV. FEEDBACK CODEBOOK DESIGN Motivated by the sall asyptotic capacity loss of ultiode precoding, we now consider the proble of designing the ultiode precoding codebook C under the cardinality constraint jcj = N tot. Assue the feedback channel is error free, this precoding codebook C at the transitter is also the feedback codebook used at the receiver. Under the ultiode precoding schee, each codeword ^V k (1 k N tot) is a N t 2 unitary atrix, and the codebook C is a finite subset of unitary atrices, i.e., CU= N U =1 where U is the set of coplex N t 2 unitary atrix. With a slight change of notation, we drop the diensionality signifying subscript of the kth codeword ^V k. The proble of codebook design is essentially a vector quantization proble and hence the conventional generalized Lloyd algorith is applied here to find a codebook that optiizes an overall distortion easure. According to the generalized Lloyd algorith, a set of channel atrix H will be generated randoly according to a given channel statistics as the training sequence. We also randoly generate N tot nuber of orthonoral colun atrices, f ^V k g N k=1, as the initial codebook C. Depending upon the interests of syste perforance, various criteria such as probability of error [16], [20], capacity [15], [16] and the error exponent [21] have been utilized to design the codebook. In this section, our design goal is to find the codebook C that axiizes the forward channel capacity. It is equivalently the codebook that iniizes capacity loss copared to the optial WF. The capacity loss is thus eployed as our distortion easure, i.e., d(h; ^V k )=C w(h) 0 C p(h; ^V k ) (9) where C p (H; ^V k ) is defined in (4). Given a codebook C,wefirst find the optial partition of the training sequence according to the distortion easure, i.e., ^V k =argind(h; ^V ^V k ) 2C = arg ax C p (H; ^V ^V k ):=T(H): (10) 2C Define the kth cluster as R k = H : T (H) = ^V k. Then, the kth partial distortion is a conditional expectation D( ^V k ) = IE d(h; ^V k )jh 2R k. To update the codebook, we then recopute the optial centroid (new codeword) within each cluster R k such that the distortion in the cluster is iniized, i.e., ^V k = arg in D( ^V ^V k ): (11) 2U Once the new codebook is generated, we can repeat the previous N process until the overall distortion D = k=1 D( ^V k )p k has changed little since the last iteration, where p k is the probability that a specific channel atrix H falls into the cluster R k. The signal design procedure is suarized in the following two-step algorith. S1) Specify P t the total available transit power fro a given SNR value. Randoly generate the enseble of channel atrices H according to the given channel distribution which serve as the training set. Randoly generate an initial codebook C with N tot codewords. S2) Repeat the following sub-steps until the change of distortion D becoes negligible. a) Given a codebook C, redistribute each channel atrix H into one of the clusters in C by selecting the one whose centroid is closer to H, i.e., H 2R k () d(h; ^V k ) d(h; ^V k ) for k 3 6= k. b) Recopute the centroid for each cluster R k created, i.e., ^V k = arg in ^V 2U D( ^V k ) to obtain a new codebook C. If an epty cluster is generated in (a), randoly generate another replaceent centroid. c) Copute the overall distortion D for the new generated codebook C. We next discuss a few issues with the design algorith. A. Coputation of the Centroid One difficult part of the Lloyd algorith is the coputation of the centroid (11). In conventional scalar or vector quantization probles with the Euclidian distance easure [22], an explicit expression of coputing the centroid can be obtained. However, this is atheatically intractable in our proble as the distortion easure (9) eployed is non-linear. In [15], the authors eployed a siilar Lloyd algorith with a difference such that their design cobines optial antenna phase rotation and power adaptation, as copared to phase rotation only in the ultiode precoding schee. They used a heuristic approxiation to copute the centroid rather than an exact derivation. Their approxiation enjoyed a closed for expression, but it inevitably caused a perforance degradation. In this correspondence, we setup an optiization proble to copute the centroid (11), where the optiization is taken over the space

4 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER U = N =1 U. The optiization is solved in two steps. First by confining the optiization space to be a subspace U for each 1 < N t, we can obtain N t 0 1 unitary atrices, with each one lying in a different subspace U.For = N t, fro the fact that C p (H; ^V k )=C p (H; I N ) we set ^V k = I N. We then choose fro those N t optiized unitary atrices the one that gives the iniu D( ^V k ) as the kth centroid. To facilitate the solution of the optiization proble for each such that 1 < N t, we paraeterize the unitary atrix using the Given s angle rotation. For paraetrization, we define a N t 2 Nt coplex rotation atrix U pq ( p;q ; p;q ) with p<qand p;q ; p;q 2 [0; ) as U pq jk (p;q;p;q) = 1 if j = k and j 6= p; q cos( p;q) if j = k and j = p; q 0 sin( p;q)exp(0ip;q) if j = p and k = q sin( p;q ) exp(i p;q ) if j = q and k = p 0 otherwise. Then, any N t 2 ( <Nt) unitary atrix U can be written as the product of the Given s rotation atrices and a diagonal atrix, i.e., Fig. 1. Bounds on asyptotic capacity loss of ultiode precoding. TABLE I CODEWORD ALLOCATION IN DIFFERENT MODES U = U 3 N 01 p+1 p=1 q=n U p;q ( p;q; p;q) 1 I 0 (N 0)2 (12) where U 3 =diag exp(i 1);...; exp(i N ). Let 2 be the collection of Nt 2 paraeters (, and ). By this paraetrization, the coputation of centroid proble (11) becoes an unconstraint optiization over the paraeter set 2. To obtain the optial solution, we randoly generate the initial values of the paraeters and then update the along the direction of the gradient. The gradient with respect to the paraeter set 2 is derived in Appendix B. B. SNR Adaptive Codebook Design In theory, there is an optial codebook for each SNR point and the designed codebook shall vary with the SNR value. However, codebook for each SNR is ipossible as it causes too uch overhead in design as well as in adaptive feedback practice. As the operating SNR of a syste ay drift over tie, we take the approach to design a codebook which works well for a range of SNR values. We call this SNR adaptive codebook design in this correspondence. In practice, we can partition the dynaic range of SNR of a certain syste into saller regions and apply the SNR adaptive codebook design for each region. As long as the operating SNR of the syste does not change very fast, we can designate a codebook to use with little overhead incurred. The goal of designing the codebook that works for a range of SNR can be achieved with slight odifications of the proposed algorith. First, Step 1 of the algorith needs to be odified. We assue the instaneous operating SNR is a rando variable taking values in a range according to a given distribution. Then, in Step 1 of the algorith, instead of specifying a single power P t for all randoly generated channel atrices, we randoly generate P t, and use, according to the given distribution for each channel atrix H. Second, to strike a balance aong different SNR values, the noralized capacity difference, i.e., d 0 (H; ^V k )=10C p(h; ^V k )=C w(h), is eployed as the distortion easure. V. NUMERICAL RESULTS In Fig. 1, the noralized (with respect to the optial WF capacity) average channel capacity with feedback is shown for MIMO syste. Our codebook is designed according to the proposed algorith given in Section IV. In the design, 1000 channel atrices H were randoly generated as the training sequence. The schee by Lau, Liu, and Chen (covariance feedback schee) in [15] and the ultiode precoding schee by Love and Heath in [16] have also been ipleented and their perforances are depicted for coparison purpose. In the ultiode precoding by Love and Heath, the applied codebook distribution is the one derived fro the capacity allocation criterion in [16] for fair coparison. For exaple, for 3 bit feedback case, the nuber of codewords lies in U 1, U 2, U 3, and U 4 is respectively 4, 3, 0, and 1. The codebook by Love and Heath is generated so as to iniize the Fubini-study distance of each ode as in [16]. The codebook in covariance feedback schee is designed according to [15]. Fro the siulation results, we see our designed codebook outperfors the other two schees throughout all SNR regions. The gain over the codebook by Love and Heath coes fro the fact that the codebook distribution (shown in Table I) according to the proposed algorith is closer to optial as copared to the derived one in [16]. Meanwhile, our designed codebook based on a suboptial ultiode precoding schee surprisingly outperfors the optial WF based covariance feedback schee. We clai that the gain coes fro the ore exact coputation of the centroid. In Fig. 3, we depict the noralized forward channel capacity of our codebook design in the case when the operating SNR of the syste is set to vary over a range. We also copare our ultiode precoding schee with the one by Love and Heath for fair coparison. In our SNR adaptive codebook design, the SNR of the syste is assued to be uniforly distributed fro 03 to 3 db. Different fro the previous setting, the training sequence used to design the codebook has channel atrices. Seen fro Fig. 3, our adaptive codebook design again outperfors the ultiode precoding schee by Love and Heath for

5 5300 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008 APPENDIX A PROOF OF THEOREM 1 For all g i 2 E we have g i <, i.e., for all i 3, jg i 0 g=j < 1. Applying the Taylor series expansion to (6), we have 0 ln(1 + g i 0 g=) =0 1 (gi 0 g)i =i 1 i. In this series, the absolute value of ith-order ter is greater than the suation fro the (i +1)th-order ter up. By this property, both the second-order and first-order upperbound in Theore 1 can be obtained trivially. APPENDIX B COMPUTATION OF THE GRADIENT As the partial derivative is a linear operator, we can exchange the order of expectation and partial derivative to copute the gradient of the partial distortion D( ^V k ) with respect to 2, i.e., Fig. 2. Capacity coparison of our ultiode precoding with covariance feedback [15] and the ultiode precoding by Love and Heath [16] for MIMO syste. r 2D( ^V k )=r 2 IE H2R C w 0 C p H; U (2) = 0 IE H2R r 2 C p (H; U (2)) : If 2 = k with 1 k N t, to copute r 2C p(h; U(2)), we rewrite the Given s paraetrization (12) as U = U3U +1, where N 01 p+1 U +1 = p=1 q=n U p;q ( p;q ; p;q ) I 0 (N 0)2 : As C p H; U (2) depends on k only through U 3, using the results (Theore 2 and Lea 3) in [23], we have r 2 C p (H; U (2)) = 2ReTr H 3 HU EU r 2U 3 3 where E = (=N ti + UH 3 3 HU ) 01 is the MMSE atrix defined in [23] and r 2U 3 3j 2= = 0i exp(0i k )e k e 3 k; Fig. 3. Capacity coparison of our SNR adaptive ultiode precoding codebook with the ultiode precoding by Love and Heath [16] for MIMO syste. the sae feedback rate, although the difference is saller as copared to that in Fig. 2. In soe SNR region, even the perforance of a lower rate feedback adaptive codebook surpasses that of a higher rate ultiode schee. For exaple, our 2-bit feedback outperfors 3-bit feedback ultiode precoding schee by Love and Heath at SNR fro 01 to 3 db. VI. CONCLUSION We have considered the proble of designing the feedback codebook under the feedback channel rate constraint under ultiode precoding schee. A nuber of bounds on the asyptotic capacity loss of ultiode precoding have been obtained in this correspondence. These bounds explain why ultiode precoding is close to optial WF in throughput. We have then utilized the generalized Lloyd algorith to design the optial liited feedback codebook with respect to a distortion easure the loss in forward channel capacity as copared to the optial WF. In each iteration of the Lloyd algorith, the codebook is constructed by a gradient search ethod applied on Given s angle paraeterization of the unitary atrix (orthonoral colun atrix). The proposed algorith can also be adaptive to the SNR value with slight odifications. Nuerical results showed that the proposed algorith outperfors coparable algoriths reported in the literature. with e k being a unit nor colun vector of length N t with the kth eleent being 1. To copute the gradient r 2 C p (H; U (2)) when 2 = p;q or 2= p;q, we rewrite U as U = U 3 N 01 p+1 p=1 q=n U p;q ( p;q ; p;q ) I 0 (N 0)2 = U 3 U N 01;N ( N 01;N ; N 01;N ) 111 U p;q ( p;q; p;q) U U 111U 1;N ( 1;N ; 1;N ) U I 0 (N 0)2 The input output relationship with this precoding atrix U is then r = HU 01 U 0 U +1 s + n. Use the results in [23] again, we obtain r 2 C p (H; U (2))= 2ReTr U01H 3 3 HU EU r 2 U 3 0, where r 2 U 3 0j 2= = 0 sin( p;q ); if j = k and j = p; q cos( p;q) exp(0ip;q); if j = p and k = q 0 cos( p;q ) exp(i p;q ); if j = q and k = p 0; otherwise :

6 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER and r 2 U 3 0j 2= 0i sin( p;q)exp(0i p;q); if j = p and k = q = 0i sin( p;q) exp(i p;q); if j = q and k = p 0; otherwise. An Efficient Water-Filling Solution for Linear Coherent Joint Estiation Wenbin Guo, Meber, IEEE, Jin-Jun Xiao, Meber, IEEE, and Shuguang Cui, Meber, IEEE REFERENCES [1] T. M. Cover and J. A. Thoas, Eleents of Inforation Theory. New York: Wiley, [2] C. E. Shannon, Channels with side inforation at the transitter, IBM J. Res. Develop., vol. 2, pp , Oct [3] A. J. Goldsith and P. Varaiya, Capacity, utual inforation, and coding for finite-state arkov channels, IEEE Trans. Inf. Theory, vol. 42, no. 3, pp , May [4] A. J. Goldsith and P. Varaiya, Capacity of fading channels with channel side inforation, IEEE Trans. Inf. Theory, vol. 43, pp , Nov [5] G. Caire and S. S. Shaai, On the capacity of soe channels with channel state inforation, IEEE Trans. Inf. Theory, vol. 45, no. 6, pp , Sep [6] H. Viswanathan, Capacity of arkov channels with receiver CSI and delayed feedback, IEEE Trans. Inf. Theory, vol. 45, no. 2, pp , Mar [7] A. J. Goldsith and S.-G. Chua, Variable-rate variable-power MQAM for fading channels, IEEE Trans. Coun., vol. 46, pp , May [8] D. L. Goeckel, Adaptive coding for tie-varying channels using outdated fading estiates, IEEE Trans. Coun., vol. 47, pp , Jun [9] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, Efficient use of side inforation in ultiple-antenna data transission over fading channels, IEEE J. Sel. Areas Coun., vol. 16, no. 8, pp , Oct [10] K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, On beaforing with finite rate feedback in ultiple-antenna systes, IEEE Trans. Inf. Theory, vol. 49, no. 10, pp , Oct [11] D. J. Love, R. W. Heath, and T. Stroher, Grassannian beaforing for ultiple-input ultiple-output wireless systes, IEEE Trans. Inf. Theory, vol. 49, no. 10, pp , Oct [12] P. Xia and G. Giannakis, Design and analysis of transit-beaforing based on liited-rate feedback, IEEE Trans. Signal Process, vol. 54, no. 5, pp , May [13] J. C. Roh and B. D. Rao, Transit beaforing in ultiple-antenna systes with finite rate feedback: A vq-based approach, IEEE Trans. Inf. Theory, vol. 52, no. 3, pp , Mar [14] I. E. Telatar, Capacity of ulti-antenna Gaussian channels, Eur. Trans. Telecoun., vol. 10, no. 6, pp , Nov./Dec [15] V. K. N. Lau, Y. Liu, and T.-A. Chen, On the design of MIMO blockfading channels with feedback-link capacity constraint, IEEE Trans. Coun., vol. 52, pp , Jan [16] D. J. Love and J. R. W. Heath, Multiode precoding for MIMO wireless systes, IEEE Trans. Signal Process, vol. 53, no. 10, pp , Oct [17] J. Roh and B. Rao, Design and analysis of MIMO spatial ultiplexing systes with quantized feedback, IEEE Trans. Signal Process., vol. 54, no. 8, pp , Aug [18] S. Lloyd, Least squares quantization in pc, IEEE Trans. Inf. Theory, vol. 28, pp , Mar [19] W. Yu and J. Cioffi, Constant-power waterfilling: Perforance bound and low-coplexity ipleentation, IEEE Trans. Coun., vol. 54, no. 1, pp , Jan [20] S. Zhou and B. Li, BER criterion and codebook construction for finiterate precoded spatial ultiplexing with linear receivers, IEEE Trans. Signal Process, vol. 54, no. 5, pp , May [21] V. K. N. Lau, Y. Liu, and T.-A. Chen, Capacity of eoryless channels and block-fading channels with designable cardinality-constrained channel state feedback, IEEE Trans. Inf. Theory, vol. 50, no. 9, pp , Sep [22] A. Gersho and R. Gray, Vector Quantization and Signal Copression. Norwell, MA: Kluwer Acadeic, [23] D. Paloar and S. Verdu, Gradient of utual inforation in linear vector Gaussian channels, IEEE Trans. Inf. Theory, vol. 52, no. 1, pp , Jan Abstract We consider a sensor network with distributed sensors observing ultiple sources and transitting local observations over a Gaussian ultiple access channel to a fusion center where the signals are coherently cobined. We develop a joint estiation ethod based on the linear analog forwarding schee at each local node, which can be cast as a nonlinear optiization proble. By iniizing the gap to the perforance benchark, we obtain a closed-for solution that follows the water-filling strategy. Nuerical results are given to deonstrate the average estiation perforance in a fading counication environent. Index Ters Linear coherent estiation, water-filling, wireless sensor networks. I. INTRODUCTION Consider a sensor network where sensors observe ultiple sources and transit the local observations to a fusion center. The fusion center aggregates the received data to generate a final estiate. We assue that the sensors encode their observations in a distributed way, since in any applications cooperative encoding is infeasible. For exaple, in a battleground scenario, there ay be ultiple sensors tracking the oveent of the eney tank and sending their observations to a coander. The cooperative counications between sensors would cost local energy and increase the syste coplexity, which ay not be desirable. Therefore, distributed (noncooperative locally) schees that achieve acceptable perforance are of ore practical iportance. In wireless sensor networks, the liited energy and counication bandwidth are two ain constraints. Specifically, due to the difficulty of changing sensor batteries, low power consuption is iportant to guarantee a long lifetie for a sensor network. Counication bandwidth becoes a precious resource when a large nuber of sensors need to counicate with the fusion center under a strict delay requireent. There are two ways to odel the counication bandwidth constraint. One is to directly liit the transitted bits of each sensor node, which usually applies to a digital counication schee, for which soe of the recent work in the literature can be found in [1] [7]. The second one is to liit the nuber of real essages sent fro each sensor, which applies to analog forwarding schees [8]. The analog forwarding schees with linear processing at both local sensors and the Manuscript received October 5, 2007; revised June 12, First published July 18, 2008; current version published Septeber 17, The associate editor coordinating the review of this anuscript and approving it for publication was Prof. Praod K. Varshney. This research was supported in part by the National Science Foundation under Grants CNS and CCF , and by the Departent of Defense under Grant HDTRA W. Guo was with the Departent of Electrical and Coputer Engineering, University of Arizona, Tucson, AZ USA. He is now with the Wireless Counication Center, School of Telecounication Engineering, Beijing University of Posts and Telecounications, Beijing, , China (e-ail: gwb@bupt.edu.cn). J.-J. Xiao is with the Departent of Electrical and Systes Engineering, Washington University, St. Louis, MO USA (e-ail: xiao@ese.wustl. edu). S. Cui is with Departent of Electrical and Coputer Engineering, Texas A&M University, College Station, TX USA (e-ail: cui@ece.tau.edu). Color versions of one or ore of the figures in this paper are available online at Digital Object Identifier /TSP X/$ IEEE