Pablo Piantanida and Pierre Duhamel

                                                                    e              e
                                     Laboratoire des Signaux et Syst` mes, CNRS/Sup´ lec,
                                               F-91192 Gif-sur-Yvette, France

                             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-
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
Thus, this matrix is a complex´normal distributed matrix, denoted
                  `                                                      the Fading MIMO-BC with ML channel estimation and channel es-
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)
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 =
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
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
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
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-
               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-
                                                                         ditional noise k−1 si . Then, the kth codeword sk is obtained by
In this paper, we use the notion of reliable communication based on      letting sk = uk − Fke, where uk is an auxiliary random vector cho-
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
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                +
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.
 Pk =
  Σ,j       Pj ,                                                                   4. SIMULATION RESULTS AND DISCUSSIONS
               b           b
 Pπ(k) = δπ(k) Hπ(k) Pπ(k) H† ,
                                                                            In this section, we illustrate our results via a realistic downlink wire-
          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† .
                                                           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
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
where Hk ∼ CN 0, InT ⊗ σh,k InR with σh,k = σE,k + σh,k . By
                                                          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
                                                                            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
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
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
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
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
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

                                                                                                                                                1.5               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
mates at the transmitter. If it is not the case, TDMA provides similar                                                                           0
                                                                                                                                                            1           2   3    4           5             6    7       8        9       10
performances to MIMO Broadcast channels.                                                                                                                                        R2 [bits/channel use]

                                                                                                                      Fig. 2. Similar plots of achievable rates, with four transmitter anten-
                            0.8                N=20                                                                   nas (nT = 4) and two receiver antennas (nR = 2).
                                                                                          TDMA with ML channel
                                                                                          estimation at the Rx
                                           N=5                                                                                                                              6. REFERENCES
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    R1 [bits/channel use]

                                                                                      −0.1 bit
                            0.5                                                                                           multi-antenna gaussian broadcast channel,” IEEE Trans. Infor-
                                                   −0.04 bits
                                                                                     +0.12 bits
                                                                                                                          mation Theory, vol. IT-49, pp. 1691–1706, july 2003.
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                                      SNR =10dB
                                           2                                                                              region of the gaussian MIMO broadcast channel,” in To appear
                                      n =n =1
                                       T       R                                                                          in IEEE Trans. on Information Theory, presented in ISIT 2004.
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                                           ML estimation at both Tx and Rx (N=20)
                                           ML estimation at both Tx and Rx (N=5)
                                                                                                                      [3] M. M´ dard, “The effect upon channel capacity in wireless com-
                            0.1            ML estimation only at the Rx (N=20)
                                           ML estimation only at the Rx (N=5)
                                                                                                                          munication of perfect and imperfect knownledge of the chan-
                                                                                                                          nel,” IEEE Trans. Information Theory, vol. IT-46, no. 3, pp.
                                  0                   0.5          1                1.5           2         2.5   3
                                                                                                                          933–946, May 2000.
                                                                        R [bits/channel use]
                                                                                                                      [4] Taesang Yoo and Andrea Goldsmith, “Capacity of fading
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and channel estimates known at the transmitter (dashed lines) vs the                                                      of International Conf. on Comunications (ICC), June 2004.
SNR, for training sequence lengths N = {5, 20}. Dashed-dot lines
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                                                                                                                          discrete memoryless channels under channel estimation errors,”
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scenario, we characterized an achievable rate region with ML chan-                                                        channel with partial side information,” IEEE Trans. Information
nel estimation. We derive the optimal Dirty-paper coding (DPC)                                                            Theory, vol. 51, no. 2, pp. 506– 522, Feb. 2005.
scheme under the assumption of Gaussian inputs. Our results, for                                                      [8] A. Lapidoth, S. Shamai, and M. Wigger, “On the capacity of
downlink communications, are useful to assess the amount of train-                                                        a MIMO Fading Broadcast Channel with imperfect transmitter
ing data to achieve target rates. These show that a BC, with a sin-                                                       side-information,” in Proceedings of Allerton Conf. on Com-
gle transmitter and receiver antenna and no channel information at                                                        mun., Control, and Comput., Sep. 2005.
the transmitter, can still achieve significant gains using the proposed                                                [9] P. Piantanida and P. Duhamel, “Achievable rates for the Fad-
DPC scheme. Further numerical results show that, under the as-                                                            ing MIMO Broadcast Channel with imperfect channel estima-
sumption of imperfect channel information at the receiver, the ben-                                                       tion,” in Proc. of the Forty-Fourth Annual Allerton Conference
efit of channel estimates known at the transmitter does not lead to                                                        on Communication, Control, and Computing, Sep. 27-29 2006.
large rate increases. However, we also showed that, for multiple
antenna BCs, in order to achieve large gain rates using multiuser

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