Training System
Resume:
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On the Overhead of Interference Alignment:
Training, Feedback, and Cooperation
Omar El Ayach, Student Member, IEEE, Angel Lozano, Senior Member, IEEE,
and Robert W. Heath, Jr., Fellow, IEEE
achieved. Realizing the gains of IA is therefore contingent
Abstract Interference alignment (IA) is a cooperative trans-
mission strategy that, under some conditions, achieves the upon providing systems with suf ciently accurate CSI at a
interference channel s maximum number of degrees of free- manageable overhead cost.
dom. Realizing IA gains, however, is contingent upon providing
Several approaches have been proposed to ful ll IA s trans-
transmitters with suf ciently accurate channel knowledge. In
mit CSI requirement [7] [9], typically assuming perfect CSI
this paper, we study the performance of IA in multiple-input
at the receiver. The feedback strategy in [7] proposes to
multiple-output systems where channel knowledge is acquired
through training and analog feedback. We design the training use Grassmannian codebooks to compress and improve CSI
and feedback system to maximize IA s effective sum-rate: a feedback in single-antenna frequency extended IA systems.
non-asymptotic performance metric that accounts for estimation
The feedback strategy was then extended to multiantenna
error, training and feedback overhead, and channel selectivity.
frequency extended systems in [8]. Both [7] and [8] guarantee
We characterize effective sum-rate with overhead in relation to
that limited feedback preserves the number of DoF by scaling
various parameters such as signal-to-noise ratio, Doppler spread,
and feedback channel quality. A main insight from our analysis the number of feedback bits with SNR, thus making codebooks
is that, by properly designing the CSI acquisition process, IA prohibitively large [10]. To overcome the problem of scaling
can provide good sum-rate performance in a very wide range
codebook size, and relax the reliance on frequency selectivity
of fading scenarios. Another observation from our work is
for quantization, [9] proposed an analog feedback strategy for
that such overhead-aware analysis can help solve a number of
constant MIMO interference channels. Using analog feedback,
practical network design problems. To demonstrate the concept
of overhead-aware network design, we consider the example a constant data rate gap from perfect CSI performance was
problem of nding the optimal number of cooperative IA users shown, as long as the SNRs on the forward and feedback
based on signal power and mobility.
links are order-wise equal. A limitation of the analysis in [7]
[9], however, is that the number of DoF remains the primary
I. I NTRODUCTION performance metric considered. IA s sum-rate performance at
nite SNR, especially when accounting for the time spent on
Interference alignment (IA) for the multiple-input multiple-
overhead signaling, has yet to be considered.
output (MIMO) interference channel is a cooperative transmis-
Attempts to more directly analyze or reduce overhead are
sion strategy that attempts to structure interfering signals such
limited to [11] [14]. To analyze the effect of overhead, [11]
that they occupy a reduced dimensional space when observed
considers the effective number of spatial DoF of an IA system
at the receivers [1], [2]. Alignment often enables achieving
with training and feedback. By considering DoF, however, [11]
the maximum number of degrees of freedom (DoF) [1],
implicitly characterizes performance at in nitely high SNR.
[2]. Precoding transmitted signals to carefully align them at
Alternatively, [12] reduces codebook size to limit overhead in
the receivers, however, requires knowledge of the interfering
limited feedback IA systems by leveraging temporal correla-
channels in the system, collectively known as channel state in-
tion without providing any overhead-aware analysis. In another
formation (CSI). Perfect CSI is assumed to be available when
line of work, information about the network topology is used
designing most IA algorithms [1], [3] [5] or reporting genie-
to partition users into optimally sized alignment groups [13].
aided IA gains. Practical systems, however, acquire receiver
In [14], IA is applied to partially connected interference
CSI with the help of training sequences or pilots [6]. Such
channels. User grouping and partial connectivity, however,
CSI can then be shared with the transmitters via feedback. As
only reduces the number of channels that must be shared
a result, practical CSI is imperfect and comes with an overhead
without suggesting an ef cient training and feedback strategy.
signaling cost, both of which penalize the effective data rates
In this paper, we characterize the performance of a MIMO
Manuscript received April 26, 2012; revised July 8, 2012; accepted August IA system that is designed for perfect CSI operation yet
24, 2012. The associate editor coordinating the review of this paper and
only has access to imperfect CSI through training and analog
approving it for publication was Y. Li.
feedback [9], [15], [16]. Thus, the performance demonstrated
Omar El Ayach and Robert W. Heath, Jr. are with the Wireless Networking
and Communications Group, Department of Electrical and Computer En- in this paper constitutes a lower bound for systems that are
gineering, The University of Texas at Austin, Austin, TX 78712 USA (e-
designed to be more robust to imperfect CSI through improved
mail: ********@******.***, ******@***.******.***). Angel Lozano is with
precoding strategies such as [17] for example. We adopt a
Universitat Pompeu Fabra, Barcelona, Spain (e-mail: *****.******@***.***).
The authors at The University of Texas were supported by the Of ce of block-fading model wherein the channel remains constant over
Naval Research (ONR) under grant N000141010337 and the Army Research
the block length, and varies independently across blocks. In
Laboratory contract: W911NF-10-1-0420. The work of A. Lozano is supported
contrast with earlier work on IA with feedback, we pre-
by the FET FP7 Project 265578 HIATUS .
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 2
cisely model channel selectivity by leveraging the relationship
between block-fading and continuous-fading channels shown Hk-1,k-1
TX RX
in [18]. This relationship allows us to de ne the concept of Hk
k-1 k-1,k
H -1
Doppler spread in a block fading channel and explicitly relate k+
1
the size of the coherence block to that Doppler spread. Since,k-
1
both CSI acquisition and data transmission must now occur,k
Hk-1
TX RX
Hk,k
within the limits of a single coherence block, the IA system
k k
Hk
is faced with a non-trivial tradeoff: too much overhead leaves +1,k
little time for payload data transmission, whereas too little 1,k+
overhead results in large sum-rate losses due to poor CSI -1
Hk Hk,k+
1
quality [18] [22]. In this paper, we design the training and TX RX
Hk+1,k+1
analog feedback system to maximize IA s effective sum-rate, k+1 k+1
a non-asymptotic performance metric that accounts for both
CSI quality and CSI acquisition overhead. CSI acquisition
overhead is a fundamental concept that was largely neglected
K -User MIMO interference channel model
Fig. 1.
in earlier work on IA with imperfect CSI.
We begin by giving a tractable expression for the IA sum-
rate in genie-aided systems with perfect CSI, and extend
II. S YSTEM M ODEL
the analysis under a general model for imperfect CSI. We
Consider the K -user narrowband MIMO interference chan-
then specialize our results to a system with training and
nel shown in Fig. 1 in which transmitter i communicates with
analog feedback by characterizing CSI quality as a function
its paired receiver i and interferes with all other receivers,
of system parameters such as training overhead, feedback
= i. For simplicity of exposition, consider a homogeneous
overhead and transmit power on both forward and reverse
network where all transmitters are equipped with N T antennas
links. This results in a tractable expression for IA s effective
and all receivers with N R antennas, and each node pair
sum-rate, which we proceed to optimize. To give a closed-form
communicates via d min(N T, NR ) independent spatial
solution for the optimal effective sum-rate, we build on the
streams. The results can be generalized to a different number
method in [18] and optimize a series expansion of the objective
of streams or antennas at each node, provided that IA remains
function. Initial results were reported in our previous work
feasible [24].
[23]. In this paper, we complete IA s performance analysis by
Assuming perfect time and frequency synchronization, the
analytically characterizing its maximum achievable effective
sampled baseband signal at receiver i can be written as
sum-rate and the corresponding optimum overhead budget.
The main insights and conclusions that can be drawn from
P P
yi = Hi,i Fi si + Hi, F s + vi,
the effective sum-rate analysis can be summarized as follows: (1)
d d
=i
Practical IA performance is not only a function of basic
where yi is the NR 1 received signal vector, P is the
system parameters such as network size and SNR, but
transmit power, H i, is the NR NT discrete-time effective
is tightly related to quantities such as Doppler spread,
to receiver i,
baseband channel matrix from transmitter
and feedback channel quality. Moreover, the dependence
Fi = fi1, . . ., fid is transmitter i s NT d precoding matrix,
of both the maximum effective sum-rate, and the corre-
si is the d 1 transmitted symbol vector at node i such that
sponding optimal overhead budget, on the various system
E [si s ] = Id, and vi is a vector of i.i.d complex Gaussian
i
parameters can be characterized accurately.
noise samples with covariance matrix 2 INR . The channels
By properly designing the training and feedback stages,
Hi, are assumed to be independent across users and each with
IA can be made both feasible and bene cial in a wide
i.i.d CN (0, 1) entries. Large-scale fading can be included in
range of fading scenarios, even when its relatively high
the system model at the expense of a more involved exposition
overhead is considered.
in Section IV.
Overhead-aware analysis is essential to the design of IA
The received signal at transmitter i on the feedback channel
networks. As an example of this observation, we use the
is
overhead analysis to give simple results on the optimal
=
Gi,i i +
G,i + i,
PF PF
number of cooperative IA users for channels with varying
yi x x v (2)
levels of selectivity. NR NR
=i
Throughout this paper, we use the following notation: A where PF is the feedback power available such that P F /P =
is a matrix; a is a vector; a is a scalar; denotes the, G,i is the NT NR discrete time feedback channel between
receiver and transmitter i with i.i.d CN (0, 1) entries, i is
conjugate transpose; a denotes the 2-norm of a; a is x
the absolute value of a; I N is the N N identity matrix; the symbol vector with unit variance entries sent by receiver i,
and i is a complex vector of i.i.d circularly symmetric white
CN (a, A) is a complex Gaussian random vector with mean a v
Gaussian noise with covariance matrix 2 INT . The forward
and covariance matrix A; (a 1, . . ., ak ) is an ordered set; E [ ]
denotes expectation. and feedback channels are assumed to be independent in the
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Overhead signaling Payload data transmission channels are known perfectly, as well as practical systems
where CSI is imperfect.
1
Tframe=
2fD
A. Interference Alignment with Perfect CSI
Fig. 2. The overhead model adopted in which training and feedback consume
resources that would otherwise be used for data transmission. IA often achieves the full number of DoF supported by
MIMO interference channels. In cases where the full DoF
cannot be guaranteed, IA has been shown to provide signi cant
error analysis of Section IV, i.e., a frequency division duplexed
gains in high-SNR sum-rate [3], [4], [26]. While this paper
system or a general non-reciprocal system is assumed.
focuses on IA, even better performance could be achieved
We adopt a block-fading channel model in which channels
with other precoding algorithms that seek a balance between
remain xed for a period, T frame, but vary independently from
interference minimization and signal power maximization [3],
block to block. To model the effect of channel selectivity on
[27], [28]. The algorithms in [3], [27], [28], however, do not
1
IA performance, we set the block length to T frame = 2fD,
readily lend themselves to average sum-rate analysis.
where fD plays the role of the block fading channel s effective
To analyze IA sum-rates, we begin by examining the
Doppler spread. The de nition of f D is motivated by the
effective channels created after precoding and combining. For
results in [18] showing a relationship between continuous
tractability, we focus on IA with a simple per-stream zero-
fading and block fading systems. To enable IA over such a
forcing (ZF) receiver. Recall that in the high (but nite) SNR
channel, both CSI acquisition and payload data transmission
regime, where IA is most useful, gains from more involved
must occur within the coherence time T frame, or else the CSI
receiver designs are limited. In such a system, receiver i
acquired becomes obsolete. The IA system then encounters a
projects its signal onto the columns of the zero-forcing com-
well-known tension between CSI acquisition and data trans- 1
biner Wi = wi, . . ., wi, . . ., wi which gives
m d
mission [18] [22], and must allocate resources to each of the
processes to optimize overall performance. P m
(wi ) yi = (w ) Hi,i fim sm
m
To account for CSI acquisition overhead, and to accurately di i
characterize the effective data rate achieved by IA, we adopt
P
(wi ) Hi,k fk sk + (wi ) vi .
+ m m
the overhead model shown in Fig. 2. In this model, overhead
d
signaling consumes time resources that could otherwise be (k, )=(i,m)
(4)
used for data transmission, i.e., CSI acquisition penalizes
effective sum-rate. For such an overhead model, the effective wi,
m
At the output of these linear receivers the conditions for
sum-rate (in bits/s/Hz) can be written as [18] [20] perfect IA can be stated as [4]
Tframe TOHD
(wi ) Hi,k fk = 0,
Re (P, TOHD ) = Rsum (P, TOHD ), (3) (k, ) = (i, m)
m
(5)
Tframe
(wi ) Hi,i fim c > 0, i, m,
m
(6)
where TOHD is the total time spent on training and feeding
back channels, and Rsum (P, TOHD ) is the average sum-rate where alignment is guaranteed by (5), and (6) is satis ed
almost surely [1], [4].
in bits/s/Hz achieved by IA on the channel uses allocated for
As a result of conditions (5) and (6), the combination of IA
payload transmission. Using (3), and previous insights into
and ZF effectively creates Kd non-interfering scalar channels.
IA performance, we highlight the tradeoff between overhead
The maximum mutual information across these channels is
signaling and data transmission. Increasing overhead improves
CSI quality and in turn improves Rsum (P, TOHD ), but the achieved via Gaussian signaling which yields an instantaneous
relative period over which Rsum (P, TOHD ) can be achieved sum-rate given by
shrinks. A similar tension exists when lowering overhead;
(wi ) Hi,i fim 2
K d P m
less overhead allows more channel uses for data transmission Rsum = log2 1+ d
. (7)
2
but the sum-rate per channel use suffers due to poor CSI i=1 m=1
quality. The objective then becomes maximizing the effective
To derive an expression for the average sum-rate, i.e., Rsum =
sum rate given in (3) by optimally trading off overhead
E [Rsum ], we rst give the following lemma.
with data transmission [18] [22]. Throughout this paper, we
Lemma 1 ( [9, Appendix A]): The effective direct channels
treat Rsum (P, TOHD ) as an information-theoretic quantity, and
(wi ) Hi,i fim are independent and Gaussian distributed with
m
thus derive mutual information-based sum-rates achievable
unit variance if: (i) the precoders F i are unitary and are
without errors. IA performance can also be analyzed from the
generated by an IA solution that does not consider the direct
perspective of xed-rate transmission where metrics such as
channels Hi,i, and (ii) the combiners W i are calculated to
bit error rate may be of interest [25].
simply zero-force inter-user and inter-stream interference.
The conditions Lemma 1 places on precoder and combiner
III. I NTERFERENCE A LIGNMENT: A N AVERAGE calculation are satis ed by most IA solutions such as [1], [3]
S UM -R ATE A NALYSIS [5]. Hence, as a result of Lemma 1, the scalar point-to-point
This section derives the average sum-rate achieved by channels created by the combination of IA and ZF experience
interference alignment in both genie-aided networks where Rayleigh fading. As a result, the average sum-rate can be
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 4
written in exponential integral form as [29], [30] Analyzing the maximum sum-rates achievable on the chan-
nel in (11) is in general dif cult, as it requires optimizing the
m2
m
K d
d (wi ) Hi,i fi
P
distribution of the input symbols s i for the interference channel
Rsum = E log2 1 +
2 in (11). Recall, however, that our objective is to analyze a
i=1 m=1
system optimized for perfect CSI operation, i.e. one that does
1
= Kd log2 (e)e1/ E1, (8) not account for CSI imperfection. This enables making the
following assumptions that would be expected from a system
which is written as a function of the per-stream SNR, = P
d 2, optimized for perfect CSI operation.
and E1 = 1 t 1 e t dt is an exponential integral.
Assumption 1: Transmitters use a typical Gaussian code-
book made up of i.i.d. symbols to form the symbol vectors
si . Such a signaling codebook, which was optimal for the
B. Interference Alignment with CSI from Training and Feed-
back interference free channels created by IA with perfect CSI, may
no longer be optimal now that CSI is imperfect.
When the channels are not known perfectly, interference
cannot be aligned perfectly. Misalignment leads to leakage
Assumption 2: Receivers perform nearest neighbor decod-
interference, which reduces the signal-to-interference-plus-
ing using the estimates Hi,i . Nearest neighbor decoding would
noise ratio (SINR) in the desired signal space. Moreover, im-
again be optimal with perfect CSI. The nearest neighbor
perfect knowledge of the direct channel implies that receivers
decoder, the channel estimates and the signaling codebook
will perform mismatched decoding [31], again reducing effec-
together satisfy the conditions outlined in [31] for Corollary
tive SINR. In this section, we examine the effect of imperfect
3.0.1 of [31] to hold with equality, meaning that the estimation
CSI on the performance of an IA system that is optimized
error plays the role of an additional source of Gaussian noise
for perfect-CSI operation, i.e., a system that does not consider
irrespective of its actual distribution.
CSI imperfection in its design. Thus, the performance results
demonstrated in this paper can be improved upon by adopting Under Assumptions 1-2, and combining the results of [31]
precoding algorithms that are more robust to CSI errors such and [34], the average sum-rate achieved can be written as
as [17].
2
(wi ) Hi,i fim
Consider an IA system in which transmitters use a common m
P
d
Rsum = E log2 1 +,
set of channel estimates as input to an IA solution such as
2
[1], [3] [5], i.e., they calculate imperfect IA precoders Fi and E (wi ) Hi,k fk + 2
P m
i,m
d
combiners Wi . Denote the channel estimates as Hi, and the k,
(12)
corresponding error as Hi, = Hi, Hi, . In this system, the where we note that the outer expectation is now only over
IA solution satis es the fading on the direct channel and not the interference.
Therefore, the leakage interference terms ( wi ) Hi,k fk indeed
m
(wi ) Hi,k fk = 0, (k, ) = (i, m)
m
(9)
play the role of independent sources of additive Gaussian
(wi ) Hi,i fim c > 0, i, m.
m
(10) noise, regardless of their distribution.
We assume receivers obtain perfect knowledge of the When the entries of Hi,k are zero-mean and uncorrelated
combiners Wi and the imperfect effective direct channels
with a variance of H, it follows that E [ P (wi ) Hi,k fk 2 ] =
2 m
wi Hi,i fim for detection, an assumption similar to [7] [9],
m d
P2 2 2
d H, thus the denominator in (12) is simply KP H + .
[20], [32] 1 whose relaxation is a topic of future work. In
Moreover, if the estimates Hi,k are MMSE estimates of Hi,k,
general, receiver side information about the effective channels
2
the entries of Hi,k have a variance of 1 H . This results in
can be acquired blindly [33] or via additional training or silent
an effective average SINR that can be written as
phases [16]. For such an IA system, the received signal after
2
projection is (1 H )
e =, (13)
2
Kd H + 1
P m
(wi ) yi = (w ) Hi,i fim sm
m
di i
where is the per-stream SNR de ned in (8). If the esti-
(11)
P mated direct channels Hi,i is Gaussian, the average sum-
(wi ) Hi,k fk sk + (wi ) vi,
+ m m
d rate achieved by IA with imperfect CSI is again given in
k,
exponential integral form as
where we have used the fact that conditions (9) and (10)
1
are satis ed, thus (wi ) Hi,k fk = (wi ) (Hi,k + Hi,k )fk =
m m
Rsum ( e ) = Kd log2 (e)e1/ eff E1 . (14)
m
(wi ) Hi,k fk . e
To evaluate sum-rate achieved by interference alignment, one
2
1 In
must now characterize e or equivalently H . In Section IV
fact [7], [8], [32] place a stronger assumption summarized by the
receivers knowledge of the exact imperfect CSI known to the transmitters.
we specialize our result for a system with training and analog
The two assumptions are functionally equivalent from the perspective of the
CSI feedback and later optimize IA s effective data rate with
sum-rate analysis, i.e., all that is needed is the receivers knowledge of wi
m
and of the scalars wi Hi,i fi .
m m overhead in Section V.
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IV. T RAINING AND A NALOG FEEDBACK where the leading scalar is to ensure that the average transmit
power constraints are satis ed with equality, i.e., one can
We propose to split the acquisition of CSI at the transmitter
verify that E [trace( X i X )] = f PF . We write the concate-
into three main phases. First, the transmitters train the forward i
nated KNT f matrix of feedback symbols observed by all
channels via pilots. Second, the receivers train the feedback
transmitters as
channels via pilots, setting the stage for the forward transmit-
1
ters to estimate the feedback information in the next stage.
f PF t P/NT
Yf =
Finally, the receivers feedback information about the forward
2 + t P/NT
KNT NR
channels in an analog fashion, i.e., as unquantized complex
Gi,1 (18)
symbols. We can characterize the CSI error introduced in the K
.
. Hi,1 . . . Hi,K i + V,
r r
CSI acquisition phase by examining the three stages. .
i=1 Gi,K
A. Forward and Feedback Channel Training where V is the KNT f matrix of i.i.d Gaussian noise.
In the rst training phase, each transmitter k sends an To simplify the performance analysis, we make the same
orthogonal pilot sequence matrix k, i.e., i = ik INT, assumption as in [9]: at the end of the feedback phase, the
k
over a training period t [35]. Pilot orthogonality imposes transmitters cooperate by sharing their rows of the received
the constraint t KNT . Each receiver i then observes the feedback matrix Y f which enables them to form a uni ed
NR t matrix estimate of the forward channels H i,k . We refer the reader
K to [9] for a discussion of this cooperative assumption and
t P
Yi = Hi,k k + Vi, i, (15) for alternative non-cooperative approaches that are shown to
NT
k=1 perform close to this special case.
where Vi is an NR t matrix of noise terms. Using Y i, Under this cooperative assumption, the transmitters estimate
Hi,k k by rst isolating the feedback sent by receiver i. They
receiver i calculates an MMSE estimate of its incoming
channels Hi,k k given by post-multiply their received symbols by to compute
i
t P 1
f PF t P/NT
NT
Yi, Y f i =
Hr = k, (16) 2 + P/N
i,k k
2 + t P KNT NR
t T
NT
Gi,1
where the superscript r emphasizes that Hr are the channel (19)
.
. Hr 1 . . . Hr
i,k
i,K + V i .
.
estimates gathered at the receiver before they are relayed i,
Gi,K
back to the transmitters and further corrupted. At the output
of this rst training stage, the channel estimates Hr have Gi
i,k
i.i.d. CN (0, 2 t P/NT T ) entries with corresponding errors
+ t P/N The transmitters then compute a common linear MMSE esti-
2
Hr CN (0, 2 + P/NT ). mate of the forward channels H i,k i, k using their feedback
i,k t
The feedback channel training phase proceeds similarly. channel estimates Gi,k i, k, and assuming that KN T NR
Namely, the receivers transmit orthogonal pilot sequences over so that the estimation problem is well posed. After a lengthy
a training period p KNR . The transmitters independently yet standard application of the orthogonality principle and the
compute MMSE estimates of their incoming channels, result- matrix inversion lemma, the MMSE estimate is given by
p P /NR
ing in estimates Gk,i CN (0, 2 + pFPF /NR ) with correspond-
1
2
K NT NR t P/NT
ing error terms Gk,i CN (0, 2 + p PF /NR ).
Hi =
2 + t P/NT
f Pf (20)
1
G Gi 1 G Gi G Y f,
+ + 2 INR
B. Analog Feedback i i i i
After forward and feedback channel training, the receivers
where we have written (20) in terms of Hi =
feedback their channel estimates Hr in an analog fashion
i,k
[Hi,1, . . ., Hi,K ] i, the concatenated estimate of the chan-
during a feedback period f . This is accomplished by rst post-
nels Hi = [Hi,1, . . ., Hi,K ] i, for the sake of notational
multiplying each N R KNT feedback matrix [ Hr 1 . . . Hr ]
i, i,K
brevity. The constants 1 and 2 are the MMSE regularization
with a KNT f matrix i such that i = i,k IKNT [9],
k
factors. For completeness, 1 and 2 are given by
[15]. The spreading matrices i orthogonalize the feedback
NT 2
from different users and facilitate estimation. This orthogo-
1 =, (21)
nality constraint requires that f K 2 NT . The transmitted
P t
NR f feedback matrix X i from receiver i can be written NT 2 2 KNT NR NR 2
2 = 1 + +2 .
as [9], [15] + p PF /NR
t P f PF
(22)
1
f PF t P/NT
Xi = Hr 1 . . . Hr
i,K i,
2 + P/N i, In essence, 1 captures the effect of the noise in the transmitted
KNT NR t T
estimates Hr, while 2 captures the effect of the noise in the
(17) i,k
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estimates Gi,k as well as the noise observed during feedback. covariance matrix equal to a scaled identity [9], [15], [37].
Thus marginalizing (24) over Gi, we nd the columns of Hi
Having formalized the three training and analog feedback
2
stages, we now analyze the variance, 2 H, of the CSI error are independent with scaled identity covariance matrices with
Hi,k Hi,k, which automatically yields an estimated CSI diagonal entries given by
2 2
variance of 1 H . Unfortunately, writing H exactly yields NT 2 2 NR 2 KNT NR NR 2
2
H = + + 1+ .
rather cumbersome expressions. For this reason, we replace (KNT NR )PF p
ead.dvi
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