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ON THE CAPACITY OF THE FADING MIMO BROADCAST CHANNEL WITHOUT CHANNEL INFORMATION AT THE TRANSMITTER AND IMPERFECT ESTIMATION AT THE RECEIVERS Pablo Piantanida and Pierre Duhamel e e Laboratoire des Signaux et Syst` mes, CNRS/Sup´ lec, F-91192 Gif-sur-Yvette, France Email:{piantanida,pierre.duhamel}@lss.supelec.fr ABSTRACT In most practical situations however, only a channel estimate is available at the receivers that differs from the true channel. Here Consider a base station transmitting information over a downlink we concentrate on the case where no channel information is avail- wireless communication channel, where the mobiles (the receivers) able at the transmitter, i.e., there is no feedback from the receivers only dispose of a noisy estimate of the channel parameters, and these back to the transmitter conveying the channel estimates. In this sce- estimates are not available at the base station (the transmitter). In this nario, it is well-known that the performances of single user channels paper, we examine the effects of imperfect channel estimation at the are severely affected by the imperfect channel information at the re- receivers and no channel knowledge at the transmitter on the capac- ceiver (cf. [3], [4] and [5]). Nevertheless, for multiuser channels, of ity of the multiuser Fading MIMO Broadcast Channel. We derive particular interest is the joint effect of the imperfect knowledge at the the optimal Dirty-paper coding (DPC) scheme and its correspond- receivers without channel information at the transmitter. In fact, the ing achievable rates with the assumption of Gaussian inputs. Our channel estimation error of one user affects the achievable rates of results, for uncorrelated Rayleigh fading, are particularly useful for many other users if interference cancellation, or DPC, is used. Con- a system designer to assess the amount of training data and the chan- sequently, when the channel is not available at the transmitter and nel characteristics (e.g. SNR, fading process, number of antennas) imperfectly estimated at the receivers, it is not immediately clear to achieve target rates. We provide numerical results for a two-users whether it is more efﬁcient to send information only to one user at MIMO Broadcast Channel with maximum-likehood (ML) channel a time (i.e. time-division multiple-access TDMA) rather than to use estimation. These illustrate a practical trade-off between the amount multiuser interference cancellation (cf. [6]). In this matter, the study of training and its impact to the multiuser interference cancellation of limits of reliable information rates of the Fading MIMO-BC under performance. In particular, we observe the surprising result that a these assumptions is particularly relevant. Indeed, intensive recent Broadcast Channel with a single transmitter and receiver antenna, research has been conducted, e.g. [7] proposed an opportunistic cod- and imperfect channel estimation at the receivers, does not need the ing scheme that employs only partial channel information. Whereas knowledge of estimates at the transmitter to achieve large rates. in [8], the authors show that when the transmitter only knows chan- Index Terms— Broadcast channels, Maximum likelihood esti- nel estimates, the limiting ratio between the sum-rate capacity and mation, Channel capacity, Fading channels. the capacity of a single-user channel with cooperating receivers is upper bounded by 2/3. Also note that achievable rates with channel estimates available at both transmitter and all receivers have been 1. INTRODUCTION derived in [9]. Throughout this paper we assume that the channel, which nei- The increasing applications of multiuser wireless communications ther the transmitter nor the receivers know exactly, is estimated at have spawned much research about the best manner to send informa- each receiver by using maximum-likelihood (ML) estimation, dur- tion to multiple users at the same time (downlink channel). In the re- ing an independent training phase (Section 2). In this work, based cent years, the Gaussian Multiple-Input-Multiple-Output Broadcast on previous results [9], we derive the optimal Dirty-paper coding Channel (MIMO-BC) has been extensively studied. Most of the lit- (DPC) scheme and its corresponding achievable rates assuming that erature focuses on the information-theoretic performances under the all channel estimates are fully unknown at the transmitter (Section assumption that perfect channel information is available at both sides 3). We address this problem through the notion of reliable commu- (transmitter and all receivers). In [1], the authors have established an nication based on the average of the error probability over all chan- achievable rate region, referred to as Dirty-paper coding (DPC) re- nel estimation errors. This notion allows to change the problem to gion. They conjectured that this achievable region is the capacity. that of evaluating the achievable DPC region of a composite (more Recently in [2], this conjecture was proved by showing that the DPC noisy) MIMO-BC. Finally, in Section 4, we use a two-users uncor- region is equal to the capacity region. The most notable characteris- related Rayleigh-fading MIMO-BC to illustrate average rates over tic of the fading MIMO-BC is that under the assumption of perfect all channel estimates, for different amount of training and antenna channel knowledge, as the signal-to-noise ratio (SNR) tends to inﬁn- conﬁgurations. ity, the limiting ratio between the sum-rate capacity and the capacity of a single-user channel that results when the receivers are not al- 2. CHANNEL MODEL lowed to cooperate is one. Thus, for a BC where the receivers do not cooperate, the interference cancellation implemented by DPC results Consider a memoryless Fading MIMO-BC with m-users. Assume in no asymptotic loss. that the transmitter has nT antennas and each receiver has nR an- tennas, with nT ≥ nR . The discrete-time channel at the time t is i.e., emax,k ≤ . This reliability notion allows us to consider the ca- ¯ modeled by pacity region of a composite (more noisy) BC model. A more robust yk (t) = Hk (t)s(t) + zk (t), k=1,. . . ,m (1) notion of reliability over estimated channels is proposed in [5]. where s(t) ∈ CnT ×1 is the vector of transmitter symbols and yk (t) ∈ 3.1. Channel estimates known at the transmitter and all receivers CnR ×1 is the vector of received symbols at Terminal k. Throughout this paper upper case boldface letters denote matrices, while lower In [9], by combining DPC scheme (as well as that proposed in [1]) case boldface denote vectors. Here, Hk (t) ∈ CnR ×nT is the com- and the above notion of reliable communication, the authors, assum- plex random matrix of the Terminal k whose entries are indepen- ing the channel estimates available at both transmitter and all re- dent identically distributed (i.i.d.) zero-mean circularly symmet- ceivers, derived the following achievable rate region. ric complex Gaussian (ZMCSCG) random variables CN(0, σh,k ). 2 ¯ e ¯ b Theorem 3.1 An achievable region RTxRx = EH {RTxRx (P , H)} for b Thus, this matrix is a complex´normal distributed matrix, denoted ` the Fading MIMO-BC with ML channel estimation and channel es- 2 Hk (t) ∼ CN 0, InT ⊗ ΣH,k , with ΣH,k = σh,k InR the Her- timates known at both transmitter and all receivers, is given by mitian covariance matrix of the columns of Hk . The noise vector ˘ [ ` ´¯ zk (t) ∈ CnR ×1 at Terminal k consists in ZMCSCG random vec- e ¯ b RTxRx (P , H) = co f b A π, {Pk }, {Wk }, H , (3) 2 tor with covariance matrix Σ0,k = σz,k InR , where InR denotes the `P ´ π,{Pk }: Pk 0 ∀ k, tr ¯ Pk ≤ P nR × nR identity matrix. We assume both Hk (t) and zk (t) are er- k godic and stationary random process, and the channel matrix Hk (t) ` ´ ˘ f b where A π, {Pk }, {Wk }, H = R ∈ Rm : Rk ≤ Rπ(k) , k = + eDPC is independent of s(t) and zk (t). This leads to a stationary and ¯ discreet-time memoryless BC with marginal pdfs Wk (yk |s, Hk ) = 1, . . . , m , and ˛ ˛ ` ´ ˛ 2 “ k ” ˛ CN Hk s, Σ0,k , k = 1, . . . , m. The input symbols are constrained ˛δπ(k) Hπ(k) P Pπ(i) H† b b e + Σ0,π(k) ˛ ` ´ ˛ π(k) ˛ ¯ to satisfy tr Es (s(t)s(t)† ) ≤ P , where tr(·) denotes the matrix eDPC i=1 ˛ , (4) Rπ(k) = log2 ˛ ˛ ˛ † “ k−1 P ” trace and (·) denotes Hermitian transposition. ˛ 2 b b† e 0,π(k) ˛ ˛δπ(k) Hπ(k) Pπ(j) Hπ(k) + Σ ˛ A standard technique to allow the receivers to estimate the chan- ˛ j=1 ˛ nel matrix consists in transmitting training sequences among the data, i.e., a set of symbols whose location and values are known to e ¯ with Σ0,π(k) = Σ0,π(k) + δπ(k) P ΣE,π(k) and δπ(k) deﬁned by the receivers. We assume that the channel matrix is constant during 2 SNRT ,π(k) σh,π(k) the transmission of an entire codeword so that the transmitter, before δπ(k) = 2 SNRT ,π(k) σh,π(k) +1 . sending the data s, can teach the channel to the receivers by sending a training sequence of N vectors ST = (sT,1 , . . . , sT,N ). This se- Let π be a permutation on the set of indexes {1, . . . , m}, such that π quence is affected by the channel matrix Hk , allowing the receiver determines the encoding order for the DPC scheme, i.e. the message at the Terminal k to observe separately YT,k = Hk ST + ZT,k , of user π(m) is encoded ﬁrst while the message of user π(m − where ZT,k is the noise matrix affecting the transmission of train- 1) is encoded second and so on. The DPC region RBC e(DPC) is the ` ´ ing symbols. The average energy of the training symbols is PT = ` ´ f convex hull co{·} of the union of all sets A π, {Pk }, {Wk }, H ofb 1 N nT tr ST S† . We focus on ML estimation of Hk , for each user T achievable rates over all permutations π and admissible covariance k = 1, . . . , m, from the observed signals YT,k and ST . This yields matrices {Pk }. Here, admissibility Pk 0 means that Pk is a b to Hk = Hk + Ek [9], where Ek denotes the estimation error ma- positive semi-deﬁnite matrix and | · | stands for determinant. 2 2 trix yielding to a white error matrix ΣE,k = σE,k InR and σE,k = −1 N PT SNRT,k with SNRT,k = σ2 , when the training sequences are or- 3.2. Channel estimates known only at the receivers Z,k thogonal. Then, by using the fading pdf, the expression of the ML We now focus on the optimal design of successive interference can- estimator and some algebra. We obtain a composite BC cellation, assuming that the channel estimates are only available at ` ´ the receivers, as well as in DPC scheme. A successive encoding f b b Wk (yk |s, Hk ) = CN δk Hk s, Σ0,k + δk ΣE,k s 2 , (2) strategy corresponds to the following approach : (i) the users are or- 2 SNRT ,k σh,k dered (ii) each user is encoded by considering the previous users as where δk = 2 SNRT ,k σh,k +1 . This composite channel model is used non-causally known interference. The situation here is signiﬁcantly in the next section (further details are provided in [9]). different of that with perfect channel knowledge (cf. [1]) or when the In the following section, assuming that the channel estimates channel estimates are also availables at the transmitter (cf. [9]). The b b b are not availables at the transmitter H = (H1 , . . . , Hm ), we de- reason is that the transmitter cannot use the channel estimates to ﬁnd rive the optimal DPC scheme and its achievable rates for the Fading optimal precoding matrices for DPC scheme. In the DPC scheme, MIMO-BC (1). For comparison, we ﬁrst review the corresponding users codeword {sk } with corresponding covariance matrices {Pk } rate region with channel estimates known at the transmitter. are independent and added up to form the transmittedP P codeword s = k sk . The encoder considers the interference e = ms i=k+1 si due to users i > k to encode the user codeword sk . The remaining 3. OPTIMAL DPC SCHEME AND ACHIEVABLE RATES codewords (s1 , . . . , sk−1 ) are considered by the kth decoder as ad- P ditional noise k−1 si . Then, the kth codeword sk is obtained by i=1 In this paper, we use the notion of reliable communication based on letting sk = uk − Fke, where uk is an auxiliary random vector cho- s the average of the error˘ probability over all channel estimation er- ¯ sen according to the message for the kth user and Fk ∈ CnR × nR is b rors emax,k = EHk |Hk emax,k (ϕ, φk , Hk ; Hk ) , k = 1, . . . , m, ¯ b the corresponding precoding matrix. Finally, the best choice is taken where (ϕ, φk ) are the respective coding and decoding functions. among all permutations of the encoding order and assuming Gaus- This requires that the maximum of the averaged error probability oc- sian inputs {sk }. This DPC scheme has been shown to be optimal curs with probability less than some arbitrary small for each user, for the MIMO-BC with perfect channel information [1]. a0,k a−1 p Using the successive encoding strategy, we ﬁrst determine an and λ(α) = √ 1,k , β±,k (α) = bk ± b2 − 4α, bk = a3,k b2 −4α k achievable rate region for the composite BC (2), which results of “ k ” ` 2 ´2 a0,k 2a1,k a2,k imperfect channel estimation at the receivers. Then, we investigate a1,k a3,k a0,k − 1 , a0,k = δk Pk + δk PΣ,k+1 α , a1,k = 2 m optimal precoding matrices, inspired by the optimal solution when 2 ¯ ¯ 2 2 δk Pk +δk Pm Σ,k+1 α , a2,k = δk P and a3,k = σz,k +δk σE,k P . Un- 2 2 2 the estimates are availables at the transmitter. ∗ fortunately, (10) does not lead to an explicit solution for αk . How- ¯ e ¯ b Theorem 3.2 An achievable region RRx (F) = EH {RRx (P , H, F)} b ever, this maximization can be numerically solved for each k = for the Fading MIMO-BC with ML channel estimation and channel ¯ 1, . . . , m, to compute (6) and then RRx (F∗ ). Both solutions were estimates not available at the transmitter, is given by ¯ tested, and we observed that the achievable rates with F are very ˘ [ ` ´¯ close to those provided by the optimal solution F∗ . Curves are omit- e ¯ b RRx (P , H, F) = co f b B π, {Pk }, {Wk }, H, F , (5) ted due to lack of space. As a result, we have chosen in the sim- `P ´ π,{Pk }: Pk 0 ∀ k, tr k ¯ Pk ≤ P ulations below to use the mean parameter to design the ”close to optimal” DPC scheme. ` ´ ˘ f b + g eDPC B π, {Pk }, {Wk }, H, F = R ∈ Rm : Rk ≤ Rπ(k) (Fπ(k) ), Note that, under channel estimation errors, the channel noise of ¯ g eDPC the resulting composite BC (2) is correlated to the channel input. k = 1, . . . , m , and Rπ(k) (Fπ(k) ) = Thus, its probability distribution is actually depending on the prob- |Pπ(k) ||Pπ(k) + Qπ(k) + Nπ(k) | ability distribution of the input s. As a consequence, non Gaussian log2 ˛ ˛ P ˛, ˛ input distributions are expected to attain the boundary points of the ˛ π(k) + Fπ(k) Qπ(k) F†π(k) Pπ(k) + Fπ(k) Qπ(k) ˛ capacity region. Therefore, our assumption of Gaussian inputs only ˛ ˛ ˛ Pπ(k) + Qπ(k) F† π(k) Pπ(k) + Qπ(k) + Nπ(k) ˛ leads to an achievable rate region, which is not the capacity region. (6) P k Pk = Σ,j Pj , 4. SIMULATION RESULTS AND DISCUSSIONS j=i b b Pπ(k) = δπ(k) Hπ(k) Pπ(k) H† , 2 In this section, we illustrate our results via a realistic downlink wire- π(k) 2 b Qπ(k) = δπ(k) Hπ(k) Pm b† less communication scenario involving a two-users (m = 2) Fading Σ,π(k)+1 Hπ(k) , ¯ b π(k)−1 b MIMO Broadcast Channel (1). Consider ﬁrst that the base station Nπ(k) = Σ0,π(k) + δπ(k) P ΣE,π(k) + δπ(k) Hπ(k) PΣ,1 H† . 2 π(k) and the mobiles have a single antenna (nT = nR = 1). We show Actually, it remains to ﬁnd the optimal precoding matrices F = the average (over all channel estimates) of achievable rates with ML (F1 , . . . , Fm ) maximizing the rates in (6). We emphasize that this channel estimation (5) and channel estimates unknown at the trans- maximization must be taken over all matrices not depending on the mitter, for different amount of training N . For comparison, we also b channel estimates H (these are assumed to be unknown at the trans- show similar plots with channel estimates known at the transmitter, mitter). First consider an intuitive suboptimal choice for Fk , k = the time-division rate region where the transmitter sends information 1, . . . , m. This choice consists in taking the average over all channel to only a single user at a time and the ergodic capacity under perfect estimates of the optimal matrices with channel estimates availables channel information. We assume that the transmitter must satisfy ¯ the short-term power constraint P , for every fading state. Then, we at the transmitter. This amounts to the following computation ˘ ` ´−1 ¯ investigate the evolution of achievable rates by increasing the num- ¯ b b b F k = E H Pk ( Hk ) Pk ( Hk ) + N k ( Hk ) , (7) b ber of transmitter and receiver antennas. To this end, we consider ` ´ a transmitter with four antennas (nT = 4) and receivers with two b where Hk ∼ CN 0, InT ⊗ σh,k InR with σh,k = σE,k + σh,k . By 2 ˆ 2 ˆ 2 2 antennas (nR = 2). using some algebra, we can easily show the following result. Fig. 1 shows the average of the achievable rate region (in bits per Lemma 3.3 The average in (7) is given by channel use) with estimated channel information at both transmitter Pk and all receivers (Theorem (3.1)) and with channel estimates only ¯ Fk = k [1 − ρk exp(ρk )E1 (ρk )] InR , (8) availables at the receivers (Theorem (3.2)), for different amount of PΣ,1 training N = {5, 20}. We suppose very different signal-to-noise ra- 2 2 ¯ σz,k + δk σE,k P R ∞ tios SNR1 = 0dB and SNR2 = 10dB, and equal fading distributions where ρk = 2 2 k and E1 (z) = t−1 exp(−t)dt de- 2 2 σh1 = σh2 = 1. Here, the training assumes same channel SNR, i.e., δk σh,k PΣ,1 ˆ z ¯ the average energy of the training symbols is PT = P . Observe that notes the exponential integral function. the achievable rates with channel estimation are still quite large irre- The other (obviously optimal, but solvable numerically only) spective of the small training sequence length N = 5 (dashed and possibility is to ﬁnd directly the optimal matrix F∗ maximizing (6). danshed-dot lines), i.e. 0.2 bits less comparing to the capacity with k Observe that perfect channel information (solid line). Suppose now that user-2 n ˛ ˛o needs to send information at a rate R2 = 1.5 bits. We want to deter- ˛ P + FQk F† Pk + FQk ˛ mine, how large performance gains can be achieved for user-1, when F∗ = arg min EH log2 ˛ k ˛ Pk + Q k F ˛ . Pk + Q k + N k ˛ k b † F the channel estimates are not availables at the transmitter. We inves- (9) tigate this by observing the gain for the ﬁrst user when the second Therefore, by using some algebra and after factorizing the matrix user is transmitting at 1.5 bits. Note that this gain is −0.1 bits (with b Hk . The solution of (9) is shown to be given by F∗ = α∗ InR , k k N = 20) and −0.22 bits (with N = 5) less compared to the case of n h „ « „ « perfect channel information. On the other hand, only 0.04 bits more β−,k (α) β−,k (α) are expected when the transmitter knows the channel estimates. This α∗ = arg min λ(α) exp k E1 0≤α≤1 4α 4α rate gain is slightly smaller, and consequently we can conclude that „ « „ « β+,k (α) β+,k (α) io the knowledge of the channel estimates at the transmitter is not really − exp E1 , (10) necessary with the proposed DPC scheme. 4α 4α Fig. 2 shows similar plots with nT = 4 and nR = 2. In this interference cancellation instead of TDMA, the transmitter requires multiple antenna scenario, without channel information at the trans- the knowledge of all channel estimates, i.e., some feedback channel mitter, there can be no adaptive spatial power allocation. However, (perhaps rate-limited) must go from the receivers to the transmitter, at equal power, it is seen that a small increase in the number of trans- conveying these channel estimates. mitter antennas can cause signiﬁcant improvement, comparing with 4.5 the single antenna case. Note that our short-term power constraint Ergodic capacity with perfect channel information is averaged over all transmitter antennas, so that this power con- 4 ML channel estimation straint is independent of the number of transmitter antennas. Con- N=20 at both Tx and Rx 3.5 sider now that user-2 needs to send information at a rate R2 = 5 bits. −1.2 bits We observe that, with channel estimates availables at the transmit- 3 ML channel estimation N=5 ter, signiﬁcant gains can be achieved compared to the case where R [bits/channel use] only at the Rx +1.2 bits the estimates are unknown at the transmitter (approximately 2 bits). 2.5 −2 bits Whereas, a multiple antenna Broadcast Channel achieves rates close N=20 2 to those of the time-division multiple access (dot line). The gain, by using DPC instead of TDMA, is reduced to only 0.12 bits with 1 1.5 N=5 N = 20, while not signicative gain is observed for N = 5. Note that this gain is equal to that obtained with a single antenna. Therefore, 1 +0.12 bits nT=4, nR=2 for MIMO-BC, taking a real beneﬁt from a large number of transmit 0.5 antennas would require an instantaneous knowledge of channel esti- SNR1=0dB SNR2=10dB mates at the transmitter. 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However, we also showed that, for multiple antenna BCs, in order to achieve large gain rates using multiuser