In a broadcast channel in which one transmitter serves receivers, the capacity region highly depends on the amount of channel state information (CSI) at the transmitter. Assuming that the transmitter knows the SNR of all the receivers, opportunistic strategy maximizes the throughput (sum-rate) of the system. It is usually assumed that CSI is accurate, however, evaluating the SNR is basically an estimation problem in the receiver which cannot be done without error. In this paper, we analyze the effect of the noisy estimation of SNR on the throughput of a broadcast channel. We propose a generalization of the opportunistic transmission in which the transmitter still sends to the user with the highest estimated SNR, but backs off on the transmit rate based on the variance of the estimation error. We obtain the optimum amount of back off and compute the throughput for our scheduling scheme. Clearly, the estimation can be improved by using a longer training phase; however, longer training would deteriorate the throughput. In the final part of the paper, we address this trade off and obtain the optimum training strategy that maximizes the throughput of the system. 1

Ali Vakili

Babak Hassibi

Masoud Sharif

In a broadcast channel in which one transmitter serves n receivers, the capacity region highly depends on the amount of channel state information (CSI) at the transmitter. Assuming that the transmitter knows the SNR of all the receivers, opportunistic strategy maximizes the throughput (sum-rate) of the system. It is usually assumed that CSI is accurate, however, evaluating the SNR is basically an estimation problem in the receiver which cannot be done without error. In this paper, we analyze the effect of the noisy estimation of SNR on the throughput of a broadcast channel. We propose a generalization of the opportunistic transmission in which the transmitter still sends to the user with the highest estimated SNR, but backs off on the transmit rate based on the variance of the estimation error. We obtain the optimum amount of back off and compute the throughput for our scheduling scheme. Clearly, the estimation can be improved by using a longer training phase; however, longer training would deteriorate the throughput. In the final part of the paper, we address this trade off and obtain the optimum training strategy that maximizes the throughput of the system

Vakili, Ali

Sharif, Masoud

Hassibi, Babak

Caltech Authors - Main

THE EFFECT OF CHANNEL ESTIMATION ERROR ON THE THROUGHPUT OF
BROADCAST CHANNELS
Ali Vakili, Masoud Sharif, and Babak Hassibi
California Institute of Technology
Department of Electrical Engineering
Pasadena, CA 91125, USA
ABSTRACT
In a broadcast channel in which one transmitter serves   re-
ceivers, the capacity region highly depends on the amount of
channel state information (CSI) at the transmitter. Assum-
ing that the transmitter knows the SNR of all the receivers,
opportunistic strategy maximizes the throughput (sum-rate)
of the system. It is usually assumed that CSI is accurate,
however, evaluating the SNR is basically an estimation prob-
lem in the receiver which cannot be done without error. In
this paper, we analyze the effect of the noisy estimation of
SNR on the throughput of a broadcast channel. We pro-
pose a generalization of the opportunistic transmission in
which the transmitter still sends to the user with the high-
est estimated SNR, but backs off on the transmit rate based
on the variance of the estimation error. We obtain the op-
timum amount of back off and compute the throughput for
our scheduling scheme. Clearly, the estimation can be im-
proved by using a longer training phase; however, longer
training would deteriorate the throughput. In the final part
of the paper, we address this trade off and obtain the opti-
mum training strategy that maximizes the throughput of the
system.
1. INTRODUCTION
A multiuser channel in which a transmitter sends indepen-
dent streams of information to a number of receivers is com-
monly referred to as a Broadcast Channel (BC). In a time-
varying environment such as a wireless channel, it is known
that exploiting the multiuser nature of the channel by op-
portunistic transmission maximizes the throughput of such
a system [1]. This however requires the full knowledge of
the channel state information (CSI) at the transmitter and
the receivers. From a practical perspective, obtaining CSI
at the receivers is basically an estimation problem which in
general cannot be error free. Therefore, it would be worth-
This work was supported in part by the National Science Foundation
under grant no. CCR-0133818 and CCR-0326554, by the David and Lu-
cille Packard Foundation, and by Caltech’s Lee Center for Advanced Net-
working.
while to understand the effect of channel estimation error
on the capacity region and scheduling schemes in BC.
The effect of imperfect CSI on the capacity of point-to-
point SISO andMIMO channels has been studied in [2], [8],
and [3]. The robustness of the capacity achieving scheme
in a Gaussian flat-fading channel is studied in [4]. As for
the broadcast channel, [5] considered the effect of non-ideal
feedback only in the transmitter due to doppler effect.
In this paper, we first take into account the effect of
estimation noise and assume that the transmitter and re-
ceivers have an “estimate” of the channel and also know
the variance of the estimation error. We further assume that
the feedback link is error free. We consider a scheduling
scheme that transmits to the user with the best estimated
SNR at a rate which is backed off from the rate that would
have been given by opportunistic scheduling. In our model
we assume a packet is dropped if a capacity outage occurs.
Backing off on the rate would therefore decrease the prob-
ability of occurrence of an outage. We obtain the optimum
rate back off that maximizes the throughput. This optimum
value can be found by solving a non-linear equation. In
order to get more insight into the amount of back off as a
function of the estimation noise, we obtain the back off ex-
plicitly for two regimes: (1) small variance of the estimation
error and (2) large number of users. We then investigate the
throughput loss due to this type of imperfect CSI. We obtain
an approximation for the throughput in small error variance
regime. Simulation results are presented to show the effect
of channel estimation error on the throughput and also to
verify our asymptotic analysis.
In our model in the first part, we assume that the chan-
nel estimations are given by a “genie” without any cost. In
the second part of this paper, we consider a training phase
in time at the beginning of each coherence interval of the
channel [8]. Clearly, the variance of the estimation error is
a function of the quality of the training which depends on its
length and the amount of power spent during training. In-
creasing the throughput requires a better estimation for the
SNR or a longer training period which itself results in send-
ing fewer data symbols or less throughput. We analyze this
IV ­ 291­4244­0469­X/06/$20.00 ©2006 IEEE ICASSP 2006
trade off and show that the optimum training strategy which
maximizes the throughput is to dedicate only one channel
use for training in each coherence interval of the channel.
2. THE CHANNEL MODEL
We consider a single-antenna broadcast channel with   users.
We assume a block fading model for the channel with co-
herence interval of length
 
. Links to the users are assumed
to be uncorrelated flat-fading Gaussian channels. We de-
note the signal sent to the  ’th user in the  ’th channel use
by      . Therefore the signal received by the  ’th user can
be written as,      	 
                             (1)
in which      is the channel coefficient with complexGaus-
sian distribution        . We have normalized the signals
so that            and 
 is the average signal to noise
ratio of the received signal due to the channel noise alone.
In (1),      is the additive white noise with i.i.d.       
entries. Since we average the throughput over all channel
uses, we drop the time index  in (1).
We denote our estimation of the channel’s coefficient
and the estimation error by   and   , respectively. Thus,        (2)
where   and   have zero-mean complex Gaussian distri-
butions. In fact 
      is what the receiver  would feedback
to the transmitter as the estimated SNR. We further assume
the feedback link is error free.
If we assume that the channel is estimated by an MMSE
estimator, the orthogonality principle implies                          [6]. We further assume that the vari-
ance of the estimation error is equal to  , i.e.,           ,
and is known to the transmitter. The actual (instantaneous)
SNR of the  ’th user (denoted by   ) can be written as,
              (3)
For mathematical convenience, we denote the normalized
estimated SNR and the normalized estimation error for the ’th user by   and   . Therefore, both   and   have expo-
nential distribution of variance one and can be written as
                   (4)
We can now rewrite   in (3) as
      ! !  "   (5)
where "  # $ # and 
 % & &       
 denote “normalizederror variance” and ”effective channel SNR”, respectively.
3. THE OPPORTUNISTIC SCHEME WITH RATE
BACK OFF
In this paper, we assume that the transmitter knows " and
 % & & and the receivers feed back the estimated SNRs to
the transmitter. We consider an opportunistic scheduling in
which all the power is assigned to the user with the high-
est estimated SNR (referred to as the “best” user). Since
the received SNR is just an estimate of the actual SNR, the
transmitter backs off on the estimated SNR (or rate) in order
to reduce the probability of outage. The main goal of this
section is to compute the optimum amount of back off that
maximizes the throughput (or sum-rate) as a function of the
variance of the estimation error.
In order to find the throughput of this scheme we need
to obtain the distribution of the actual SNR defined in (5)
given the estimated SNR or   for the best user. We drop
the index  whenever we denote the quantities related to the
best user. The distribution of the maximum estimated SNR
can be written as' (       ) $ *+    ) $ *+  , $  -     (6)
where
-  .  denotes the unit step function. We can further
show that the distribution of the actual SNR of the best user
given its estimation can be written as [7],
' (       ) $ /0 1 23 3 $ /4  ! ! 5 "   -     
 % & &  (7)
If there was no error in our estimation, the actual SNRwould
have been 
  . In fact, the actual SNR is always less than 
 
in the presence of estimation error. In our scheduling, the
transmitter backs off from 
  to  6    which is called the
“assumed” SNR. Clearly, if the actual SNR (i.e.,  ) is below
the assumed SNR (i.e.,  6    ), a capacity outage occurs.
Therefore, the instantaneous rate will be,
7  8 9 : ;     6   6 <   6 =  (8)
Thus we can write the average throughput as,7 > ?  @ AB      ) $ *+  , $  ) $ *+ 9 : ;     6    C 8 @   ! ! *++ D 1 *+ 5 ) $ /0 1 23 3 $ /4  ! ! 5 "   E  F E  (9)
The following theorem obtains the optimum value for
the back off to maximize the throughput of the system.
Theorem 1. Consider a broadcast channel with   users
where each user can estimate its channel with an error vari-
ance of  . The optimum backed off SNR (denoted by  G H I6 )
IV ­ 30
that maximizes the throughput of the opportunistic schedul-
ing is the non-trivial solution to the equation,
             	 
           

                       
(10)
Proof: It is easy to see that the maximization can be in-
terchanged with the integration. Maximization of the inner
integral in (9) over    leads to (10). 
When    , Theorem 1 implies that          which
is consistent with the opportunistic scheduling in the ab-
sence of the estimation error.
Remark: It is worth mentioning that the actual SNR
(defined in (3)) is always less than the estimated one, i.e.,   !  " . Therefore, for any nonzero # , if the transmitter
sends at a rate of
	 
         , the packet will be dropped and
the average throughput will become zero. This implies that
a back off system is necessary in the presence of channel
estimation error.
Although numerical evaluation of    is quite straight-
forward, finding an analytical solution for      does not
seem to be tractable. Therefore we consider two important
asymptotic regimes, namely, when the estimation noise # is
small and when the number of users   is large.
3.1.      for small-  regime
From a practical point of view, we are mostly interested in
the case where the estimation error (i.e., # or equivalently ) is small. Therefore in this subsection we fix   and  and
let  go to zero. We can state the following theorem for this
regime which is proved in [7],
Theorem 2. Consider the setting of Theorem 1. If 
tends to zero, then      is equal to,
       $ % %      $ % % 	 
   &        (11)
Therefore we may write an approximation for      to the
first order as,
     '     	     
  $ % %      $ % % 	 
   &  (12)
This can be used to find an approximation for the throughput
of the system when  is small. In order to calculate  ' ( ,
one should plug in the solution of      into the integral of
(9) and calculate the integral; this is however analytically
intractable. Using Theorem 2 we can find an approximation
for
) ' (
when  is small by plugging (11) into (9) as
   '    ' ( 
      * +,        -.  /  &   -.
0 	 
      $ % %      $ % % 	 
   &   1   
0 0.05 0.1 0.15 0.2 0.25 0.3
1
1.5
2
2.5
3
β
R
av
R
av
 with exact λ
c
R
av
 with λ
c
 for small α regime
R
av
 with λ
c
 for large n regime
Fig. 1. Comparison of
) ' (
calculated by using the exact
value of      and small-  or large-   approximations for
      and   
This essentially implies that   '    ' ( 
       /   $  ' (                	 
   &  
where  /   $  ' (    is the sum-rate capacity in a broadcast
channel with no estimation error and with average SNR of . Therefore when # is small, the estimation error has two
effects on the throughput namely, a pre-log factor of     
and also the SNR hit of
&&   2          .
3.2.      for large-   regime
In a typical cellular system, we may have a large number of
users   . This implies that, almost surely   is about 	 
    	 
  	 
 
  . In this regime, we can obtain      and prove that
[7],
       $ % %      $ % % 	 
  	 
  -.    	 
  	 
  	 
      (13)
Fig. 1 shows the numerical results for average through-
put when      is computed numerically by solving (10), us-
ing small-  approximation, and using large-   approxima-
tion. Here we assume that there are       users in the
system and    . Fig. 2 shows the throughput as a func-
tion of the number of users for different # ’s. It is worth
noting that by only little estimation error of #      , the
throughput is decreased by about %   .
4. ESTIMATION VIA TRAINING IN TIME
In the previous section, we assumed that the channel esti-
mation is given to the system with the aid of a “genie” at no
cost. In this section, we take into account the benefits and
IV ­ 31
costs of a training phase in the system. Following [8], we
assume that at the beginning of each coherence interval of
length
  1, there is a training phase of duration     and data
will be transmitted in the remaining part, i.e.,
           .
We also assume that we have an energy budget equal to 
for each coherence interval. If we define    and   as the
power levels of the training and data phases respectively,
this implies that,
                (14)
It is intuitively clear that as we increase the training budget
(i.e.,
   
or    ), the estimation error will decrease and there-
fore the transmission rate during the data phase
  
will be
increased. This howeverwould decrease the overall through-
put as the duration of data transmission (or its power) will
diminish. Therefore there is a trade off on the amount of
training needed to maximize throughput.
Considering an MMSE estimation, we can show that the
variance of the estimation error is [6],
           

      (15)
The next theorem states the optimum training power and
duration that maximize the throughput.
Theorem 3. Consider the settings of Theorem 1 and
assume that the channel estimation is obtained by MMSE
training phase of length
   
with power level of    . In or-
der to maximize the throughput, we need only one symbol,
i.e.
      , for training. The optimum power level    is
the solution to a maximization problem that can be found
numerically.
Proof: The proof follows the method used in [8]. We
find the derivative of the expression for
	 
 
with respect to  
and show that it is positive. Since
   
can not be less
than  , we should fix it at this value. See [7] for the com-
plete proof and the formulation of the maximization prob-
lem.
 
The result of Theorem 3 implies that using more than
one channel use for training would deteriorate the through-
put although it does improve the channel estimation. This is
mainly due to the fact that using a longer training decreases
the pre-log factor while improves the signal to noise ratio in
the argument of the logarithm.
5. CONCLUSION
We considered a BC with receivers having channel estima-
tion error. We generalized the opportunistic scheme to a
transmission strategy where we send to the user with the
highest estimated SNR with a rate back off. We obtained
the optimum rate back off and the resulting throughput. We
1  denotes the number of channel uses and therefore is a discrete num-
ber.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
1
1.5
2
2.5
3
3.5
4
n
R
av
β = 0
β = 0.05
β = 0.1
β = 0.2
Fig. 2. The effect of the estimation error on the throughput
for   
further analyzed the behavior of the back off when the vari-
ance of the error is small or when the number of users is
large. We also looked into the trade offs when we have a
training phase on the throughput. We showed that in or-
der to maximize the throughput, we need to spend only one
channel use per coherence interval for training.
6. REFERENCES
[1] R. Knopp and P. Humblet, “Information capacity and power
control in single cell multiuser communications,” in Proc.
IEEE Inter. Conf. Comm., pp. 331– 335, 1995.
[2] M. Medard, “The effect upon channel capacity in wireless
communications of perfect and imperfect knowledge of the
channel,” IEEE Trans. on Info. Theory, vol. 46, no. 3, pp.
933–946, May 2000.
[3] T. Yoo and A. Goldsmith, “Capacity and optimal power al-
location for fading MIMO channels with channel estimation
error,” submitted to IEEE Trans. on Info. Theory, Aug. 2005.
[4] A. Lapidoth and S. Shamai, “Fading channels: How perfect
need ”perfect side information” be?,” IEEE Trans. on Info.
Theory, vol. 49, no. 10, pp. 951–964, April 2002.
[5] M. Kobayashi, G. Caire, and D. Gesbert, “Antenna diversity
vs. multiuser diversity: Quantifying the tradeoffs,” in Proc.
ISITA, October, 2004.
[6] T. Kailath, A.H. Sayed, and B. Hassibi, Linear Estimation,
Prentice-Hall, 2000.
[7] A. Vakili, M. Sharif, and B. Hassibi, “The effect of SNR
estimation error on the sum-rate of broadcast channels,” in
preparation.
[8] B. Hassibi and B. M. Hochwald, “How much training is
needed in a multiple-antenna wireless link?,” IEEE Trans.
on Info. Theory, vol. 48, no. 5, pp. 1118–1134, May 2003.
IV ­ 32


English

The Effect of Channel Estimation Error on the Throughput of Broadcast Channels

THE EFFECT OF CHANNEL ESTIMATION ERROR ON THE THROUGHPUT OF
BROADCAST CHANNELS
Ali Vakili, Masoud Sharif, and Babak Hassibi
California Institute of Technology
Department of Electrical Engineering
Pasadena, CA 91125, USA
ABSTRACT
In a broadcast channel in which one transmitter serves   re-
ceivers, the capacity region highly depends on the amount of
channel state information (CSI) at the transmitter. Assum-
ing that the transmitter knows the SNR of all the receivers,
opportunistic strategy maximizes the throughput (sum-rate)
of the system. It is usually assumed that CSI is accurate,
however, evaluating the SNR is basically an estimation prob-
lem in the receiver which cannot be done without error. In
this paper, we analyze the effect of the noisy estimation of
SNR on the throughput of a broadcast channel. We pro-
pose a generalization of the opportunistic transmission in
which the transmitter still sends to the user with the high-
est estimated SNR, but backs off on the transmit rate based
on the variance of the estimation error. We obtain the op-
timum amount of back off and compute the throughput for
our scheduling scheme. Clearly, the estimation can be im-
proved by using a longer training phase; however, longer
training would deteriorate the throughput. In the ﬁnal part
of the paper, we address this trade off and obtain the opti-
mum training strategy that maximizes the throughput of the
system.
1. INTRODUCTION
A multiuser channel in which a transmitter sends indepen-
dent streams of information to a number of receivers is com-
monly referred to as a Broadcast Channel (BC). In a time-
varying environment such as a wireless channel, it is known
that exploiting the multiuser nature of the channel by op-
portunistic transmission maximizes the throughput of such
a system [1]. This however requires the full knowledge of
the channel state information (CSI) at the transmitter and
the receivers. From a practical perspective, obtaining CSI
at the receivers is basically an estimation problem which in
general cannot be error free. Therefore, it would be worth-
This work was supported in part by the National Science Foundation
under grant no. CCR-0133818 and CCR-0326554, by the David and Lu-
cille Packard Foundation, and by Caltech’s Lee Center for Advanced Net-
working.
while to understand the effect of channel estimation error
on the capacity region and scheduling schemes in BC.
The effect of imperfect CSI on the capacity of point-to-
point SISO andMIMO channels has been studied in [2], [8],
and [3]. The robustness of the capacity achieving scheme
in a Gaussian ﬂat-fading channel is studied in [4]. As for
the broadcast channel, [5] considered the effect of non-ideal
feedback only in the transmitter due to doppler effect.
In this paper, we ﬁrst take into account the effect of
estimation noise and assume that the transmitter and re-
ceivers have an “estimate” of the channel and also know
the variance of the estimation error. We further assume that
the feedback link is error free. We consider a scheduling
scheme that transmits to the user with the best estimated
SNR at a rate which is backed off from the rate that would
have been given by opportunistic scheduling. In our model
we assume a packet is dropped if a capacity outage occurs.
Backing off on the rate would therefore decrease the prob-
ability of occurrence of an outage. We obtain the optimum
rate back off that maximizes the throughput. This optimum
value can be found by solving a non-linear equation. In
order to get more insight into the amount of back off as a
function of the estimation noise, we obtain the back off ex-
plicitly for two regimes: (1) small variance of the estimation
error and (2) large number of users. We then investigate the
throughput loss due to this type of imperfect CSI. We obtain
an approximation for the throughput in small error variance
regime. Simulation results are presented to show the effect
of channel estimation error on the throughput and also to
verify our asymptotic analysis.
In our model in the ﬁrst part, we assume that the chan-
nel estimations are given by a “genie” without any cost. In
the second part of this paper, we consider a training phase
in time at the beginning of each coherence interval of the
channel [8]. Clearly, the variance of the estimation error is
a function of the quality of the training which depends on its
length and the amount of power spent during training. In-
creasing the throughput requires a better estimation for the
SNR or a longer training period which itself results in send-
ing fewer data symbols or less throughput. We analyze this
IV ­ 291­4244­0469­X/06/$20.00 ©2006 IEEE ICASSP 2006
trade off and show that the optimum training strategy which
maximizes the throughput is to dedicate only one channel
use for training in each coherence interval of the channel.
2. THE CHANNELMODEL
We consider a single-antenna broadcast channel with   users.
We assume a block fading model for the channel with co-
herence interval of length
 
. Links to the users are assumed
to be uncorrelated ﬂat-fading Gaussian channels. We de-
note the signal sent to the  ’th user in the  ’th channel use
by      . Therefore the signal received by the  ’th user can
be written as,

     	 
                 ﬀ   ﬃ ﬀ    ﬀ
 
ﬀ (1)
in which      is the channel coefﬁcient with complexGaus-
sian distribution     ﬀ ﬃ  . We have normalized the signals
so that           ﬃ and 
 is the average signal to noise
ratio of the received signal due to the channel noise alone.
In (1),      is the additive white noise with i.i.d.     ﬀ ﬃ 
entries. Since we average the throughput over all channel
uses, we drop the time index  in (1).
We denote our estimation of the channel’s coefﬁcient
and the estimation error by   and   , respectively. Thus,
  

  

  (2)
where   and   have zero-mean complex Gaussian distri-
butions. In fact 
      is what the receiver  would feedback
to the transmitter as the estimated SNR. We further assume
the feedback link is error free.
If we assume that the channel is estimated by an MMSE
estimator, the orthogonality principle implies         
   

       

   [6]. We further assume that the vari-
ance of the estimation error is equal to  , i.e.,           ,
and is known to the transmitter. The actual (instantaneous)
SNR of the  ’th user (denoted by   ) can be written as,
  


   


 

  

(3)
For mathematical convenience, we denote the normalized
estimated SNR and the normalized estimation error for the
 ’th user by   and   . Therefore, both   and   have expo-
nential distribution of variance one and can be written as
  


   
ﬃ  
ﬀ   


   

(4)
We can now rewrite   in (3) as






  ! !
 "  
(5)
where "  # $
#
and 
 % & &   ﬃ    
 denote “normalized
error variance” and ”effective channel SNR”, respectively.
3. THE OPPORTUNISTIC SCHEME WITH RATE
BACK OFF
In this paper, we assume that the transmitter knows " and

 % & & and the receivers feed back the estimated SNRs to
the transmitter. We consider an opportunistic scheduling in
which all the power is assigned to the user with the high-
est estimated SNR (referred to as the “best” user). Since
the received SNR is just an estimate of the actual SNR, the
transmitter backs off on the estimated SNR (or rate) in order
to reduce the probability of outage. The main goal of this
section is to compute the optimum amount of back off that
maximizes the throughput (or sum-rate) as a function of the
variance of the estimation error.
In order to ﬁnd the throughput of this scheme we need
to obtain the distribution of the actual SNR deﬁned in (5)
given the estimated SNR or   for the best user. We drop
the index  whenever we denote the quantities related to the
best user. The distribution of the maximum estimated SNR
can be written as
' (
   
  )
$ *
+
 ﬃ 
)
$ *
+
 ,
$  -
    (6)
where
-
 .  denotes the unit step function. We can further
show that the distribution of the actual SNR of the best user
given its estimation can be written as [7],
' (
     
)
$ /
0 1 2
3
3
$ /
4
 ! ! 5


" 

-





ﬃ

 % & &
 (7)
If there was no error in our estimation, the actual SNRwould
have been 
  . In fact, the actual SNR is always less than 
 
in the presence of estimation error. In our scheduling, the
transmitter backs off from 
  to  6    which is called the
“assumed” SNR. Clearly, if the actual SNR (i.e.,  ) is below
the assumed SNR (i.e.,  6    ), a capacity outage occurs.
Therefore, the instantaneous rate will be,
7

8 9 : ;
 ﬃ   6   6 < 
  6 = 
(8)
Thus we can write the average throughput as,
7 > ?
 @ A
B
 
 ﬃ 
)
$
*
+

,
$ 
)
$
*
+
9 : ;
 ﬃ   6    
C
8
@
  ! !
*
+
+ D
1
*
+
5
)
$ /
0 1
2
3
3
$ /
4
 ! ! 5

" 
 E
 F
E
 (9)
The following theorem obtains the optimum value for
the back off to maximize the throughput of the system.
Theorem 1. Consider a broadcast channel with   users
where each user can estimate its channel with an error vari-
ance of  . The optimum backed off SNR (denoted by  G H I
6
)
IV ­ 30
that maximizes the throughput of the opportunistic schedul-
ing is the non-trivial solution to the equation,
    

 ﬃ   
  


	 
 
 ﬃ   
  



ﬃ


  


   

 

   
 ﬃ
 
  
  


(10)
Proof: It is easy to see that the maximization can be in-
terchanged with the integration. Maximization of the inner
integral in (9) over    leads to (10). 
When    , Theorem 1 implies that     

    which
is consistent with the opportunistic scheduling in the ab-
sence of the estimation error.
Remark: It is worth mentioning that the actual SNR
(deﬁned in (3)) is always less than the estimated one, i.e.,
   !  " . Therefore, for any nonzero # , if the transmitter
sends at a rate of
	 
 
 ﬃ      , the packet will be dropped and
the average throughput will become zero. This implies that
a back off system is necessary in the presence of channel
estimation error.
Although numerical evaluation of    is quite straight-
forward, ﬁnding an analytical solution for  
  

does not
seem to be tractable. Therefore we consider two important
asymptotic regimes, namely, when the estimation noise # is
small and when the number of users   is large.
3.1.     

for small-  regime
From a practical point of view, we are mostly interested in
the case where the estimation error (i.e., # or equivalently
 ) is small. Therefore in this subsection we ﬁx   and  and
let  go to zero. We can state the following theorem for this
regime which is proved in [7],
Theorem 2. Consider the setting of Theorem 1. If 
tends to zero, then     

is equal to,
 
  


 $ % %  
ﬃ    $ % %
	 
 

&

     
 (11)
Therefore we may write an approximation for     

to the
ﬁrst order as,
 
  
'  

 	
  

 

 $ % %  
ﬃ   
$ % %
	 
 

&


(12)
This can be used to ﬁnd an approximation for the throughput
of the system when  is small. In order to calculate  ' ( ,
one should plug in the solution of     

into the integral of
(9) and calculate the integral; this is however analytically
intractable. Using Theorem 2 we can ﬁnd an approximation
for
) ' (
when  is small by plugging (11) into (9) as

 
'  


' (


 ﬃ   
*
+
,
 
 ﬃ 


-
.
 /
 &


-
.
0
	 
  
ﬃ 

$ % %
 
ﬃ    $ % %
	 
 

&

 
1
  
0 0.05 0.1 0.15 0.2 0.25 0.3
1
1.5
2
2.5
3
β
R
av
R
av
 with exact λ
c
R
av
 with λ
c
 for small?α regime
R
av
 with λ
c
 for large?n regime
Fig. 1. Comparison of ) ' ( calculated by using the exact
value of     

and small-  or large-   approximations for
 
 ﬃ   and   
This essentially implies that

 
'  


' (


 ﬃ     /


$ ﬂ ﬂ
' ( ﬃ  
 
    
ﬃ   
 
	 
 

&

 
where 
/


$ ﬂ ﬂ
' (
   is the sum-rate capacity in a broadcast
channel with no estimation error and with average SNR of
 . Therefore when # is small, the estimation error has two
effects on the throughput namely, a pre-log factor of  ﬃ   
and also the SNR hit of
&
& 
 2      




.
3.2.     

for large-   regime
In a typical cellular system, we may have a large number of
users   . This implies that, almost surely   is about
	 
 
  
	 
  	 
 
  . In this regime, we can obtain     

and prove that
[7],
 
  



$ % %
 
ﬃ   
$ % %
	 
  	 
 
-
.

  
	 
  	 
  	 
 
 

 (13)
Fig. 1 shows the numerical results for average through-
put when     

is computed numerically by solving (10), us-
ing small-  approximation, and using large-   approxima-
tion. Here we assume that there are    ﬃ   users in the
system and    . Fig. 2 shows the throughput as a func-
tion of the number of users for different # ’s. It is worth
noting that by only little estimation error of #      , the
throughput is decreased by about %   .
4. ESTIMATION VIA TRAINING IN TIME
In the previous section, we assumed that the channel esti-
mation is given to the system with the aid of a “genie” at no
cost. In this section, we take into account the beneﬁts and
IV ­ 31
costs of a training phase in the system. Following [8], we
assume that at the beginning of each coherence interval of
length
  1, there is a training phase of duration
   
and data
will be transmitted in the remaining part, i.e.,
  

 

   
.
We also assume that we have an energy budget equal to 
for each coherence interval. If we deﬁne 
 
and 

as the
power levels of the training and data phases respectively,
this implies that,
  
     
 
   
 (14)
It is intuitively clear that as we increase the training budget
(i.e.,
   
or 
 
), the estimation error will decrease and there-
fore the transmission rate during the data phase
  
will be
increased. This howeverwould decrease the overall through-
put as the duration of data transmission (or its power) will
diminish. Therefore there is a trade off on the amount of
training needed to maximize throughput.
Considering an MMSE estimation, we can show that the
variance of the estimation error is [6],
 
ﬃ
ﬃ  
 
 
 

ﬃ
ﬃ  
 
 (15)
The next theorem states the optimum training power and
duration that maximize the throughput.
Theorem 3. Consider the settings of Theorem 1 and
assume that the channel estimation is obtained by MMSE
training phase of length     with power level of    . In or-
der to maximize the throughput, we need only one symbol,
i.e.
   
 ﬃ , for training. The optimum power level    is
the solution to a maximization problem that can be found
numerically.
Proof: The proof follows the method used in [8]. We
ﬁnd the derivative of the expression for
	 
 
with respect to
  
and show that it is positive. Since
   
can not be less
than ﬃ , we should ﬁx it at this value. See [7] for the com-
plete proof and the formulation of the maximization prob-
lem.  
The result of Theorem 3 implies that using more than
one channel use for training would deteriorate the through-
put although it does improve the channel estimation. This is
mainly due to the fact that using a longer training decreases
the pre-log factor while improves the signal to noise ratio in
the argument of the logarithm.
5. CONCLUSION
We considered a BC with receivers having channel estima-
tion error. We generalized the opportunistic scheme to a
transmission strategy where we send to the user with the
highest estimated SNR with a rate back off. We obtained
the optimum rate back off and the resulting throughput. We
1  denotes the number of channel uses and therefore is a discrete num-
ber.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
1
1.5
2
2.5
3
3.5
4
n
R
av
β = 0
β = 0.05
β = 0.1
β = 0.2
Fig. 2. The effect of the estimation error on the throughput
for   
further analyzed the behavior of the back off when the vari-
ance of the error is small or when the number of users is
large. We also looked into the trade offs when we have a
training phase on the throughput. We showed that in or-
der to maximize the throughput, we need to spend only one
channel use per coherence interval for training.
6. REFERENCES
[1] R. Knopp and P. Humblet, “Information capacity and power
control in single cell multiuser communications,” in Proc.
IEEE Inter. Conf. Comm., pp. 331– 335, 1995.
[2] M. Medard, “The effect upon channel capacity in wireless
communications of perfect and imperfect knowledge of the
channel,” IEEE Trans. on Info. Theory, vol. 46, no. 3, pp.
933–946, May 2000.
[3] T. Yoo and A. Goldsmith, “Capacity and optimal power al-
location for fading MIMO channels with channel estimation
error,” submitted to IEEE Trans. on Info. Theory, Aug. 2005.
[4] A. Lapidoth and S. Shamai, “Fading channels: How perfect
need ”perfect side information” be?,” IEEE Trans. on Info.
Theory, vol. 49, no. 10, pp. 951–964, April 2002.
[5] M. Kobayashi, G. Caire, and D. Gesbert, “Antenna diversity
vs. multiuser diversity: Quantifying the tradeoffs,” in Proc.
ISITA, October, 2004.
[6] T. Kailath, A.H. Sayed, and B. Hassibi, Linear Estimation,
Prentice-Hall, 2000.
[7] A. Vakili, M. Sharif, and B. Hassibi, “The effect of SNR
estimation error on the sum-rate of broadcast channels,” in
preparation.
[8] B. Hassibi and B. M. Hochwald, “How much training is
needed in a multiple-antenna wireless link?,” IEEE Trans.
on Info. Theory, vol. 48, no. 5, pp. 1118–1134, May 2003.
IV ­ 32


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The effect of channel estimation error on the throughput of broadcast channels

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