IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
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pp. 848–853.
, “A new bandwidth efficient transmit antenna modulation diversity
scheme for linear digital modulation,” in Proc. IEEE Int. Communications Conf. (ICC’93), 1993, pp. 1630–1634.
L. Zheng and D. N. C. Tse, “Communication on the Granssmann
manifold: A geometric approach to the noncoherent multiple-antenna
channel,” IEEE Trans. Inform. Theory, vol. 48, pp. 359–383, Feb. 2002.
Capacity of Multiple-Transmit Multiple-Receive Antenna
Architectures
Angel Lozano, Senior Member, IEEE, and
Antonia Maria Tulino, Member, IEEE
Abstract—The capacity of wireless communication architectures
equipped with multiple transmit and receive antennas and impaired by
both noise and cochannel interference is studied. We find a closed-form
solution for the capacity in the limit of a large number of antennas. This
asymptotic solution, which is a sole function of the relative number of
transmit and receive antennas and the signal-to-noise and signal-to-interference ratios (SNR and SIR), is then particularized to a number of cases
of interest. By verifying that antenna diversity can substitute for time
and/or frequency diversity at providing ergodicity, we show that these
asymptotic solutions approximate the ergodic capacity very closely even
when the number of antennas is very small.
Index Terms—Adaptive antennas, antenna arrays, asymptotic analysis,
channel capacity, diversity, fading channels, multiantenna communication,
multiuser detection.
I. INTRODUCTION
With the explosive growth of both the wireless industry and the Internet, the demand for mobile data access is expected to increase dramatically in the near future. As a result, the ability to support higher
capacities will be paramount. Capacity can be pushed by exploiting
the space dimension inherent to any wireless communication system.
Nonetheless, due to economical and environmental aspects, it is highly
desirable not to increase the density of base stations. Under such constraint, antenna arrays are the tools that enable spatial processing on a
per-base-station basis. Recognizing this potential, the use of arrays at
base-station sites is becoming universal. Array-equipped terminals, on
the other hand, had not been contemplated in the past because of size
and cost considerations. However, recent results in information theory
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have shown that, with the simultaneous use of multiple transmit and receive antennas, very large capacity increases can be unleashed [1]–[4].
At the same time, it is reasonable to expect that terminals supportive of
progressively higher data rates will tend to be naturally larger in size
and, consequently, they will be able to accommodate multiple closely
spaced antennas. Hence, the deployment of arrays at both base stations
and terminals appears as an attractive scenario for the evolution of mobile data access.
Great progress has been made toward understanding the information-theoretical capacity and the performance of multiple-antenna architectures with thermal noise as the only impairment (see [5]–[16]).
Within the context of a wireless system, however, the dominant impairment is typically not thermal noise, but rather cochannel interference.
Thus, the objective of the present work is to extend this understanding
to the realm of spatially colored interference. We invoke, as central tool,
recent results on the asymptotic distribution of the singular values of
random matrices and their application to randomly spread code-division multiple access (CDMA) [17]–[19]. Although these distributions
pertain asymptotically in the number of antennas, the results we derive
therefrom become virtually universal under ergodic conditions.
Since the focus is on mobile systems, we consider only “open-loop”
architectures wherein the transmitter does not have access to the instantaneous state of the channel. Only large-scale information—defined as
information that varies slowly with respect to the fading rate—is available to the transmitter.
This correspondence is organized as follows. In Section II, the metrics and models are introduced. In Section III, the noise-limited capacity is reviewed using the tools of asymptotic analysis. Such analysis is generalized, in Section IV, to environments containing spatially
colored interference. The main result therein is an expression of the
asymptotic capacity in the presence of both noise and interference. Finally, Section V concludes the correspondence.
II. DEFINITIONS AND MODELS
A. Propagation Model
M
N M
N
With
transmit and
receive antennas, the channel responses
from every transmit antenna to every receive antenna can be assembled
into an 2
random matrix
whose underlying random process
is presumed zero-mean and ergodic. The propagation scenario and the
spatial arrangement of the antennas determine the correlation among
the entries of . The scenario we consider, typical of a mobile system,
is based on the existence of an area of local scattering around each terminal. Accordingly, the power angular spread is expected to be very
large—possibly as large as 360 —at the terminals rendering the antennas therein basically uncorrelated. At the base station, the angular
spread tends to be small [20], [21] but the antennas can be also decorrelated by spacing them sufficiently apart [22]–[25]. Consequently, we
focus on channel matrices containing only independent entries.
Since the elements of are identically distributed, it is possible to
define a normalized channel matrix with unit-variance entries such
that
=p .
G
G
G
gH
G
H
B. “Open-Loop” Capacity
Manuscript received May 3, 2001; revised April 8, 2002.
A. Lozano is with the Wireless Communications Research Department, Bell
Laboratories, Lucent Technologies, Holmdel, NJ 07733 USA (e-mail: aloz@lucent.com).
A. M. Tulino is with the Department of Electrical Engineering, Princeton
University, Princeton, NJ 08540 USA (e-mail: atulino@princeton.edu).
Communicated by D. N. C. Tse, Associate Editor for Communications.
Digital Object Identifier 10.1109/TIT.2002.805084
Perfect channel estimation at the receiver [26]–[28] is presumed.1
The impairment comprises additive white Gaussian noise (AWGN) as
1The penalty associated with channel estimation is small as long as the coherence time of the channel—measured in symbols—is large enough with respect
to the number of transmit antennas [27].
0018-9448/02$17.00 © 2002 IEEE
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
well as interference, which—conditioned on its fading—is also presumed Gaussian.2 As a consequence, we are interested in the spatial
covariance of the impairment conditioned on the fading of the interference. Such spatial covariance, which we define as , is also presumed to be estimated perfectly at the receiver (but unknown to the
transmitter). Notice that, since depends on the random fading of the
interference, it can be viewed as a random variable itself. Furthermore,
given that is—in general—not proportional to the identity, it follows
that the interference is spatially colored.
are independent and unknown to the transWhen the entries of
mitter, the mutual information is maximized by transmitting a Gaussian
signal with spatial covariance
Q
TABLE I
ERGODIC (AND ASYMPTOTIC) CAPACITIES PER RECEIVE ANTENNA WITH
SNR 10 dB. THE ERGODIC VALUES CORRESPOND TO THE AVERAGE OF
10 000 INDEPENDENT RAYLEIGH CHANNEL REALIZATIONS
=
Q
Q
G
8 = MP IM
(1)
given a total radiated power P [2]. Since the channel is time varying in
nature, such mutual information fluctuates with it. With a sufficiently
long coding horizon, it is possible to code over the short-term channel
fluctuations and approach the ergodic capacity given by [11]
IN + MP gHHy Q01
(2)
with expectation over the distributions of H and Q. It is also possible to
C
=E
log2 det
code over the short-term channel randomness in the frequency domain,
whereby the ergodic capacity is approached as the signal bandwidth
increases [30].
If it is not possible to code over the short-term channel variations,
one must resort to the idea of outage capacity, wherein the capacity
itself is regarded as a random variable that fluctuates with the channel.
As we shall see, nonetheless, the outage capacity hardens around its
average as the number of antennas increases and thus the outage and
ergodic capacities coincide asymptotically.
III. NOISE-LIMITED CAPACITY
by the spreading factor, corresponds in our case to the number of receive antennas, while the number of users corresponds to the number
of transmit antennas. There are, nonetheless, significant differences.
• The spatial signatures for the different transmit antennas are not
chosen by the system designer, as in CDMA, but rather imposed
by nature. Moreover, their distribution depends on the type of
fading being experienced. Remarkably, though, the distribution
y converges to the same exact function
of the eigenvalues of
regardless of the distribution of its entries so long as those are
independent and identically distributed [17].
HH
• Because of the total power constraint imposed at the transmitter,
the SNR typically used in CDMA has to be normalized by the
dimensionality ratio .
• The transmit antennas are colocated and part of a single transceiver whereas, in CDMA, the multiple users are geographically
dispersed. Hence, joint coding of the transmit signals is feasible
in our problem, but not in CDMA.
With these considerations, the asymptotic capacity per dimension
derived in [17] and [32] for synchronous CDMA can be modified to
express the noise-limited asymptotic capacity per receive antenna as
C
A. Nonasymptotic Noise-Limited Capacity
(;
SNR) = log2 1 + SNR 0 F
;
SNR
SNR
0F
;
When the impairment consists exclusively of AWGN, we have
Q = 2 IN
+ log2 1 +
(3)
where 2 is the noise power per receive antenna. Defining
SNR
def
=
P
g
2
(4)
as the average signal-to-noise ratio (SNR), we can write
C
=
E
log2 det
IN + SNR
HHy
M
:
(5)
B. Asymptotic Noise-Limited Capacity
As the number of antennas increases, the empirical distribution of the
y converges to a deterministic
eigenvalues of the random matrix
function [31]. Defining the ratio of transmit and receive antennas as
HH
def
=
M
N
(6)
and the capacity per receive antenna as
C
C def
=
N
(7)
we now look into finding C as the number of antennas is driven to
infinity with a constant ratio .
Fundamentally, we are faced with a multiple-input multiple-output
problem that is akin to that of a synchronous CDMA channel with
random spreading. The dimensionality of such a channel, represented
2If
the interference is not Gaussian, the capacities we derive serve as lower
bounds [29].
2 (e) F
0 logSNR
with
1
F (x; y) def
= 4
1+y 1+
px 2 0
;
SNR
SNR
(8)
1+y 1
0 px
2
2
: (9)
The asymptotic capacity is solely a function of and the SNR. Furthermore, it yields an extremely accurate approximation to the ergodic
capacity even when the number of antennas is very small (see Table I).
Next, we particularize C
for a number of special cases and study
in detail its dependence on both and SNR.
1) Dependence on : C
is maximized by letting ! 1.
Hence, the capacity with an optimal receiver3 increases monotonically
with , as seen in Fig. 1. Although additional antennas thus increase
capacity regardless of whether they are added to the transmitter or to
the receiver, they add different value depending on where they are deployed. Such difference stems from the fact that the total radiated power
is bounded whereas the total captured power increases with additional
receive antennas. Therefore, the capacities corresponding to and to
1= are different.4 Additionally, Fig. 1 hints that there is little incentive in increasing beyond unity. This result is formalized in the next
section.
3This
is not necessarily true if a suboptimal receiver is utilized [6].
“closed-loop” architectures, wherein the channel realization is known at
the transmitter, these capacities coincide [2].
4In
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
Fig. 1.
Asymptotic noise-limited capacity per receive antenna as a function of for various levels of SNR.
Of particular interest is C
[14] as
C
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(1;
SNR) = 2 log2
with 1+
p
= 1, which can be expressed
1 + 4 SNR
2
2 (e)
0 log
4 SNR
p
!1 C
(;
lim
01
2
: (10)
SNR) = log2 (1 + SNR)
(11)
1
y
!1 M HH = IN :
(12)
Notice that, for N = 1, this configuration reverts to that of transmit
diversity, which has been actively researched in recent times [33]–[35].
Conversely, if M is kept fixed while N ! 1, the capacity per
receive antenna behaves as
(;
SNR) = log2
SNR
+ O ( )
(13)
and the channel again decouples
1
y
!1 N H H = IM :
lim
N
(14)
Interestingly, this decoupling also implies that the capacity would not
increase if the channel realization
were known at the transmitter
[11]. Hence, the “open-loop” and “closed-loop” capacities coincide for
! 0. Intuitively, as the number of channel dimensions becomes
much larger than the number of modes, none of these gets favored over
the others. Moreover, as can be inferred from [19] for the corresponding
multiuser problem, capacity can be approached for small with scalar
coding at every transmit antenna and a simple linear receiver.
H
(;
SNR) =
SNR
e
which, for = 1, particularize to
lim
C
C
! 1, the capacity per
which can be obtained by taking the limit on (8) or, alternatively, by
observing [2] that the channel decouples asymptotically
M
0 ( 0 1)
1
1 log2 1 0 1 + O SNR
;
SNR
log2 e 0 (1 0 )
1
;
1 log2 (1 0 ) + O SNR
log2
1 + 4 SNR
On the other hand, if N is kept fixed while M
receive antenna becomes
2) Dependence on SNR: Insightful expressions can be obtained by
at high SNR, wherein a series expansion of (8)
particularizing C
yields
C
(1;
SNR) = log2
SNR
+O
1
1
(15)
1
(16)
SNR
as derived in [1]. Therefore, the high-SNR capacity is proportional to
min(M; N ) and grows logarithmically with the SNR. As shown in
Fig. 2, this trend sets in very fast.
For an in-depth study of the low-SNR capacity, the reader is referred
to [36], [37].
e
C. How Many Antennas Should be Used?
The asymptotic expressions derived thus far can be used to quantify
the benefit of pushing beyond unity. The largest gain, which occurs
for SNR ! 1 is
lim
C (1; SNR) 0 C (1; SNR) = log2 (e) (17)
!1
SNR
or merely the equivalent of a 4.34-dB increase in SNR (Fig. 2), a vanishingly small improvement in capacity per receive antenna as the SNR
grows.
Given the cost associated with deploying multiple antennas with separate radio-frequency chains, it is important to ensure that those antennas are as effective as possible. Ineffective antennas raise the cost
and add unnecessary complexity.
In “open-loop” noise-limited conditions, the following applies.
• Given a number of receive antennas, at most as many transmit
antennas should be used.
• Given a number of transmit antennas, at least as many receive
antenna should be used. Additional receive antennas are always
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
Fig. 2.
Asymptotic noise-limited capacity per receive antenna as a function of SNR for various levels of .
advantageous. However, if we denote by R the highest rate that
can be supported with a realizable constellation by each transmit
antenna, there is little point in further increasing the number of
receive antennas once
C
> R:
(18)
As we will see next, these conclusions vary in the presence of
colored interference.
IV. CAPACITY IN THE PRESENCE OF INTERFERENCE
A. Nonasymptotic Capacity
Within the context of a mature system, the dominant impairment is
usually not thermal noise, but rather cochannel interference. Furthermore, most emerging data systems feature time-multiplexed downlink
channels, certainly those evolving from time-division multiple access
(TDMA) [38], but also those evolving from CDMA [39]–[41]. Hence,
same-cell users are mutually orthogonal and thus the downlink interference arises exclusively from other cells. This holds also for a code-multiplexed downlink in frequency-flat fading channels5 and, of course, for
a time-multiplexed uplink.
In order to simplify the analyses, it is not unusual that such outside
interference be regarded as additional AWGN [44]. In single-antenna
systems, this approximation assumes the interference to be Gaussian. In
multiple-antenna systems, it has a second—and more profound—implication: it assumes the interference to be spatially white and it thus
neglects the fact that interference does have a spatial structure, or color,
that can be exploited by a multiple-antenna receiver. This structure
tends to be particularly strong in the downlink, wherein the entire interference contribution of every cell emanates from the base station, a
single localized source.
In the presence of outside interference, the covariance of the impairment conditioned on the fading of the interference can be expressed as
Q=
K Pg
k k Hk Hy + 2 IN
k
k=1 Mk
with K the number of outside interferers (assumed mutually independent) and with Pk and Mk the total radiated power and number of
transmit antennas for the k th interferer, respectively. In the downlink,
each interferer would correspond to a neighboring base station; in the
uplink, it would correspond to a terminal in a neighboring cell. Each
N 2 Mk channel matrix k contains the transfer coefficients between
every transmit antenna of interferer k and every receive antenna at the
desired user. We denote by k the normalized counterpart of each
p
channel matrix so that k = gk k . Defining the signal-to-interference ratio (SIR) with respect to each interferer as
G
G
H
SIRk
H
= PPk ggk
(20)
we can rewrite the capacity as
C
= E log2 det IN + HHy
K
1
M
k=1 Mk SIRk
M
Hk Hky + SNR
IN
01
(21)
with expectation over the distribution of the various channels and with
the aggregate SIR and signal-to-interference-and-noise ratio (SINR)
being
1 = K
SIR
k=1
1
(22)
SIRk
1 = 1 + 1
5In frequency-selective channels, same-cell interference can be suppressed
by preceding the receiver with a chip-level equalizer [42]. Alternatively, wellestablished multiuser detection principles can be applied [43].
(19)
:
SINR
SNR
SIR
The performance of receive antenna arrays in interference-limited
conditions has been thoroughly studied, but mostly for single-antenna
transmitters [45]–[47]. With the addition of transmit arrays, the
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
problem becomes much more complex (see [11] as well as [48]–[51]
for related work). In general, the individual channel matrices k
cannot be estimated, but only the aggregate covariance , and thus
the receiver can only perform linear processing against the outside interferers while optimally detecting the desired signals [52]. Hence, we
find it convenient to distinguish between two classes of interference.
H
Q
• The mutual interference among the transmit antennas of the desired user, which we shall refer to as multiantenna interference.
• The outside interference that those antennas suffer from the K
interferers within .
Q
B. Asymptotic Capacity
The asymptotic capacity in the presence of both AWGN and outside
interference (derived in the Appendix) constitutes the main result of
the correspondence. Although we present it next for a homogeneous
system, wherein the number of transmit antennas is the same for the
user of interest as well as each of the outside interferers, the solution in
its most general form is included in the Appendix. In a homogeneous
system, the asymptotic capacity per receive antenna is given by
C
+K (; SNR; fSIRk g) = SIRk + SNR K
k=1
log2
+ log2
2
+ log2
1
+ (1
1
0 2 ) log2 (e)
(23)
with 1 and 2 the positive solutions to
1 +
SNR 1
+
SNR + 1
2 +
K
k=1
K
k=1
SNR 1
=1
SNR + SIRk
SNR 2
= 1:
SNR + SIRk
(24)
Because of the implicit nature of these equations, obtaining explicit
expressions therefrom requires solving for 1 and 2 in equations of
order K + 2 and K + 1, respectively. Hence, the complexity of the
solution is directly determined by the number of outside interferers.
Fortunately, the equations containing 1 and 2 are decoupled and thus,
as we shall see, it is possible to obtain meaningful expressions for a
large number of cases. If the number of outside interferers is set to
K = 0, it is rather straightforward to solve for 1 and 2 as
1 = 1 0
2 = 1
SNR
F
;
how the detection (by a linear MMSE receiver) of each of the
transmit signals is impaired by the presence of the other transmit
antennas plus the outside interference.
• 2 represents the asymptotic ratio between i) the SNR at the
output of a linear MMSE receiver detecting the signal transmitted
from any outside antenna in the presence of the rest of outside antennas (excluding the desired user), and ii) the SNR at the output
of a matched filter detecting that same antenna without any interference. Although it may seem intriguing that 2 plays a role
in the computation of the capacity given that the actual receiver
does not make any attempt to decode the outside interference, a
justification is given in the Appendix.
Clearly, both 1 and 2 are bound to lie within [0; 1].
1) Limiting Cases: Solving for the asymptotic efficiencies becomes trivial in some limiting cases. For growing !1 1 = 1 + SNR
lim
1+
1
SIR
SINR
!1 2 = SNR
lim
1
(26)
indicating that both efficiencies become independent of the composition of the outside interference; they become a function of only the
aggregate SIR and SINR. These efficiencies, in turn, yield
SIRk + SNR 1 + SNR
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SNR
(25)
and, substituting them into (23), obtain the asymptotic noise-limited
capacity per receive antenna of (8). Therefore, the above solution is a
generalization of the one presented in Section III.
The roles of 1 and 2 can be interpreted by relating them to the
multiuser efficiency, a quantity commonly used in multiuser detection
problems [19], [43]. Specifically, the following holds.
• 1 represents the asymptotic ratio between i) the SNR at the
output of a linear minimum mean-square error (MMSE) receiver
detecting the signal transmitted from any of the desired antennas
in the presence of multiple-antenna as well as outside interference, and ii) the SNR at the output of a matched filter detecting
that same antenna without any interference. Hence, it quantifies
!1 C
lim
+K (; SNR; fSIRk g) = log
2 (1 + SINR)
(27)
which has the exact same form of the asymptotic noise-limited capacity
obtained in (11) for ! 1. Hence, as the number of interfering antennas grows much larger than the number of receive antennas, the progressively fine color of the interference cannot be discerned and it thus
appears white to the receiver. The capacity depends only on the total
impairment power, irrespective of how it breaks down into noise and
outside interference.
On the other hand, for diminishing and finite K
1 = 1 + O( )
2 = 1 + O( )
(28)
confirming that the penalty due to a fixed number of interfering
antennas vanishes as the number of receive antennas grows without
+K behaves as
bound. With that, C
C
+K (; SNR; fSIRk g) = log SNR + O( )
2 (29)
and becomes determined only by the underlying AWGN, irrespective
of the SIR.
Finally, as the SNR grows
lim
= [1 0 (K + 1) ]+
SNR!1 1
lim
= [1 0 K ]+ :
SNR!1 2
(30)
In this regime, the receiver operates in zero-forcing mode against the
interference and thus the loss in 1 is given exactly by the total number
of transmit antennas (desired plus outside interference) per receive antenna. If that total number of transmit antennas exceeds the number of
receive antennas, then 1 = 0. The loss in 2 , on the other hand, is
exactly the total number of outside interfering antennas per receive antenna. Again, if the number of outside interfering antennas exceeds the
number of receive antennas, then 2 = 0.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
Fig. 3. Asymptotic capacity per receive antenna as a function of for K
Homogeneous system.
= 1 and various combinations of SIR and SNR that correspond to SINR = 10 dB.
2) Low-SNR Behavior: The behavior of the asymptotic efficiencies
at low SNR is given by
1 = 1 0 SNR
1+
1
SIR
2
1 =
SNR
2
+ O (SNR ):
(31)
SIR
Analogous to ! 1, a low SNR renders both efficiencies independent of the composition of the outside interference; only the aggregate
+K
SIR becomes relevant. With the above efficiencies, C
becomes
+K
(;
0 2 1 + + O(SNR ): (32)
SINR
3) Interference-Limited Behavior: Single Outside Interferer:
High-capacity wireless systems are typically designed to operate in interference-limited conditions [53]. In the remainder, we concentrate,
therefore, on studying the asymptotic capacity at high SNR. We begin
by evaluating the capacity in the presence of a single outside interferer.
With K = 1, the asymptotic capacity becomes
+1
=
1
2
1
(;
1
3
SIR + SNR SIR + SNR 2
1
+ (1
+ log2
0 ) log (e)
2
2
(33)
given
1 +
;
1
SNR
SIR
01)SNR + O
+O
;
1
SNR
1
2
<
1
2
(35)
=1
;
1
SNR
=
>1
;
1
SNR
0+O
;
1
2
(36)
<1
1
2
p
SIR
1
+O
:
SNR
SNR
In terms of asymptotic capacity, the above efficiencies yield
1 =
log2
1
+1
=
1
2
e
0
0
1 2
log2 1 log2
(1
02
1
SNR
2 log2
+O
+O
SNR
e
0 ) log
2
1+
SNR
p
e
(37)
0 (1 0 )
;
<
1
2
1
pSNR
;
=
1
2
1
SNR
+ O (1);
SIR
log2 (1 + SIR) + O
+O
1
SNR
1
2
<<1
1
pSNR
; =1
;
! 1.
(38)
SNR 1
SNR 1
=1
+
SNR + 1
SNR + SIR
SNR 2
=1
2 +
SNR + SIR
1
SNR
>
with the first-order coefficient of 1 for > a convoluted function
of and SIR. For = 1, this coefficient admits a simple form
C
1
1 + SNR
+O
0 2 + O
SIR
SNR
1
SNR; SIR)
log2
+ log2
1
1
1+SIR
SNR
(
2 =
;
1
SNR
and
SNR; fSIRk g) = log2 (e) SNR2
1
C
O
+ O (SNR )
2 = 1 0
C
At high SNR, the asymptotic efficiencies can be found to be
We observe the following.
(34)
which requires solving cubic and quadratic equations for 1 and 2 ,
+1
respectively. Fig. 3 depicts C
as function of for various levels
of SNR and SIR corresponding to SINR = 10 dB.
+1
• As the SNR grows, C
becomes independent of the SIR for
12 . Hence, while the combined number of desired plus outside transmit antennas does not exceed the number of receive antennas, the receiver approaches capacity by suppressing—purely
through linear processing—the outside interference while simultaneously detecting the desired signals. Once that interference
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
Fig. 4.
Asymptotic noise- and interference-limited capacity per receive antenna with is suppressed, capacity becomes limited only by the underlying
noise.
• For 12 < 1, the number of receive antennas exceeds the
number of desired antennas, but not the combined number of
desired plus outside interfering antennas. Therefore, the receiver
must compromise between assigning its degrees of freedom to
interference suppression and to signal detection. As the SNR
increases without bound, such compromise favors sacrificing a
fraction of the receive antennas for interference suppression.
For = 12 , the capacity becomes exactly half that of an
architecture with = 1 operating, free of outside interference,
at the same SNR. That is,
SNR!1
lim
C
+1
1
2
; SNR; SIR
=
1C
lim
SNR!1 2
(1;
SNR):
+1 at = 1 which, in interference• Particularly revealing is C
limited high-SIR conditions, becomes
+1 (1; SNR; SIR)
= log2 SIR + O
p1
SIR
+O
p1
SNR
= 1 and K = 1 as a function of SINR. Homogeneous system.
corresponding, to first order, to the equivalent of a 4.34-dB improvement in SINR. Hence, as illustrated in Fig. 4, operating over
colored impairment is noticeably beneficial even when the receiver has no spare degrees of freedom. Interestingly, the advantage appears to hold throughout the entire range of SINR levels.
• Unlike in the noise-limited case, the interference-limited
C +1 does not increase monotonically with . It grows with
+1
up to = 12 and it diminishes thereafter. Hence, C
is maximized—in these conditions—by having the number of
transmit antennas equal half the number of receive antennas just
so the receiver has enough spare antennas to suppress the outside
interference in its entirety while decoding the desired signals.
Notice that this holds only if the SNR is sufficiently large with
respect to the SIR (Fig. 3), specifically—from (16) and (40)—if
(39)
• For 1, the dependence on the SNR becomes weak. Once the
number of outside interfering antennas has exceeded the number
of receive antennas, the receiver can no longer suppress the totality of that interference and thus even the high-SNR performance is determined mostly by the SIR. Nonetheless, the colored interference renders the capacity higher than in equivalent
noise-limited conditions. Only as ! 1 does this advantage
dissipate (Fig. 3).
C
3123
(40)
SNR
e
+1 (1; SNR; SIR) 0 C
= log2 e + O
(1;
p1
SIR
+O
p1
SNR
(41)
p
SIR
2
:
(42)
In the presence of colored interference, the capacity clearly depends
not only on the number of transmit and receive antennas and the SINR,
but also on the degrees of freedom of the interference. Specifically, a
given amount of interference is more benign if it occupies fewer degrees of freedom.
4) Interference-Limited Behavior: Multiple Outside Interferers: In
the presence of K > 1 outside interferers, the high-SNR asymptotic
efficiencies of the previous section generalize to
1 ;
O SNR
1 =
SINR)
1+
If this condition is not met, the underlying noise influences the
capacity sufficiently to render it monotonic in . Hence, (42) can
be used as a criterion to assess whether the system is effectively
interference-limited when the interference arises from a single
outside transmitter.
indicating an advantage over its high-SNR noise-limited counterpart—see (16)—of
C
>
1
K +1
1
1+
> K1+1
SIR
SNR
0 (K + 1) + O
+O
1
SNR ;
1
SNR ;
= K1+1
< K1+1
(43)
3124
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
and
O
2 =
1
SNR
;
> K1
SIR
1
K
1
SNR
+O
1
SNR
1
SNR
;
0 K + O
;
(44)
= K1
< K1 .
Unfortunately, the first-order coefficients for both 1 at > K1+1 and
2 at > K1 do not amend themselves to simplification and thus we
obtain a more restricted set of explicit expressions for the asymptotic
capacity
log2
0(K +1)
1
SNR
e
0(1 0 K ) log 0 0KK 1
2
+O
K=
C
1
2
+
1
SNR
1+
1 log
SIR
1+
< K1+1
+ K1+1
Fig. 5. Simplified downlink scenario with two outside interferers at SIR
5 dB and SIR
8 dB, respectively.
SIR
log2 (1 + SIR) + O
=
1
pSNR
;
+O
e
2
+1)
;
SNR
log2
(
1
= K1+1
! 1.
;
1
SNR
For other values of , obtaining tractable expressions does not appear
feasible. However, it is possible to show that the multiple-interferer
capacity can always be bounded by
C
+K
C
+K
C
+K
(45)
with the bounds obtained by replacing the K outside interferer by a
single “equivalent” interferer generating the same exact aggregate SIR
with the highest and lowest possible levels of structure.
• The upper bound corresponds to concentrating the aggregate interference contribution of all K outside interferers into a single
one and thus
C
+K
(;
SNR; SIR) = C
+1
(;
SNR; SIR):
(46)
• The lower bound corresponds to forcing the K outside interferer
to be of equal strength while preserving the aggregate SIR. Since
K equal-power interferers are equivalent to a single interferer
+K
can be derived from
with K times as many antennas, C
the Appendix to be
C
+K
(;
SNR; SIR) = K log2
+ log2
+log2
2
1
SIR + SNR K
SIR + SNR K
1+ SNR
+ (1
2
2
(47)
with
1 +
SNR 1
SNR 1
+
+ SIR = 1
SNR + 1
SNR K
2 +
SNR 2
= 1:
SNR + SIR
(48)
In general, the tightness of both bounds depend on the specific set
of fSIRk g. The upper bound becomes tight in the presence of a dominating interferer whereas the lower bound, in turn, tightens in the presence of comparable-strength interferers. Furthermore
lim
K !1
C
+K
(;
SNR; SIR) = C
(;
SINR)
indicating that an increasing number of equal-strength outside interferers will progressively whiten the impairment. Hence, C
can also
+K
serve as a lower bound, but it is never as tight as C
for finite K .
An example corresponding to a simple downlink scenario with two
outside interferers is illustrated in Fig. 5. A multiantenna terminal is
illuminated by its serving base as well as two nearby interfering bases,
all of them equipped with the same number of transmit antennas. Accounting for different range and shadow fading to every base, a typical
set of SIR and SNR levels are chosen. The corresponding asymptotic
capacity appears in Fig. 6 along with its upper and lower bounds. Also
shown is the corresponding noise-limited capacity at the same aggregate SINR.
Note how neglecting the fact that the interference is colored can
lead to gross miscalculations of the actual capacity or, conversely, of
the SINR required to achieve a certain level of capacity. This converse look is provided in Fig. 7, which displays curves of constant
capacity on a SNR–SIR plane. Specifically, the curves shown corre+K
spond to C
= 2.5 b/s/Hz per receive antenna with = 0:5,
= 1, and = 2. For each configuration, the SNR–SIR combinations
that achieve such capacity are depicted as a function of the number of
equal-strength outside interferers.
C. How Many Antennas Should be Used?
1
0 ) log (e)
=
(49)
It was shown in Section III that, in noise-limited conditions, there is
little incentive in pushing beyond unity. From the results presented
throughout this section, it is clear that the presence of interference can
only lessen that incentive, at least in homogeneous systems wherein all
transmitters are equipped with the same number of active antennas. In
such conditions, the presence of a dominant outside interferer can actually render the capacity for > 1 smaller than for = 1 (Fig. 3).
On the other hand, the capacity with K outside interferers increases
monotonically for < K1+1 . The best choice for will thus usually
lie somewhere within [ K1+1 ; 1] depending on the structure of the interference and, therefore, on the geometry and propagation environment
of the corresponding system. The following applies.
• If each transmitter is interfered by a small number of dominant
outside interferers, the best will lie around K1+1 with no advantage—or even possibly a loss—in pushing it any further. A
similar conclusion is reached in [54].
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
3125
=
Fig. 6. Asymptotic capacity per receive antenna (solid line) as a function of with K
2, SIR = 5 dB, SIR = 8 dB, and SNR = 12 dB. Upper and lower
bounds (dashed lines) and the corresponding noise-limited capacity at the same aggregate SINR (circles) are also shown. Homogeneous system.
Fig. 7. Combination of SNR and SIR levels required to attain an asymptotic capacity of C
= 2 as a function of the number of equal-strength outside interferers. Homogeneous system.
• With a large number of comparable-strength outside interferers,
the best will lie close to = 1.
As in the noise-limited case, practical considerations preclude
making so small that
C
+K
>R
with R the highest realizable rate per transmit antenna.
(50)
= 2.5 b/s/Hz per receive antenna with = 0:5, = 1, and
V. SUMMARY
The main result of the correspondence is the asymptotic capacity of
multiple-transmit multiple-receive antenna architectures impaired by
AWGN as well as spatially colored interference. From the asymptotic
capacity, we gathered insight on how the capacity behaves. Although
the derivation of the asymptotic capacity requires driving the number
of transmit and/or receive antennas to infinity, the asymptotic solution
applies to virtually any number of antennas under ergodic conditions.
While in noise-limited conditions, the asymptotic capacity depends
only on the number of antennas and the SNR, the capacity in the pres-
3126
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
ence of interference depends also on the color thereof. At any given
SINR, the capacity increases with such color. The lowest capacity at
any SINR is attained in the presence of AWGN exclusively.
into our problem with a simple SNR normalization to yield the following asymptotic results:
C1 =
+
k
APPENDIX
The starting point for our derivation is (21), which can be manipulated into
C
=E
log2 det
K
+
i=k
SNR
Mk SIRk
0 log2 det IN +
C2 =
K
i=k
Hk Hky
SNR
Mk SIRk
Hk Hky
:
(51)
H2 = [ H1 H2 1 1 1 HK ]
H1 = [ H H2 ]
M
M
SIR
0
B2 =
B1 =
IM
O
M
..
.
M
SIR
IM
..
.
0
IM O
0 B2
0
M
0
0
0
SIR
M
B
log2 det
IN +
C2 = E
log2 det
IN + SNR
H B Hy
M 2 2 2
B
M
2 +
1 + SNR
2
B
2
0 1) log2 (e)
(54)
k
k
B
k E
SNR1 B1
SNR1 B1 + =1
k E
SNR2 B2
SNR2 B2 + = 1:
(55)
With (54) and (55), we can now calculate the asymptotic capacity per
+K = C1 –C2 .
receive antenna, in its most general form, as C
We note here that the above result holds even if the columns of as
well as of k (k = 1; . . . ; K ) are correlated, that is, if the transmit
arrays of the user of interest and the interferers exhibit antenna correlation. In this case, the matrices 1 and 2 would be given by
H
H
IM
B2 =
H1 B1H1
B B
O
111
SIR 21
M
0
M SIR 22 1 1 1
M
..
.
y
B1 = 20 BO
2
(53)
with 1 and 2 diagonal matrices. Notice that—provided the desired
user and the various outside interferers are mutually independent—the
block matrices H 1 and H 2 are composed by independent and identically distributed entries if only the individual and k matrices are.
Both C1 and C2 have meaningful interpretations.
H
1
B
1
0 1) log2 (e)
log2
+ (2
1 + SNR
B
the capacity can be further manipulated into C = C1 –C2 given
C1 = E
+ (1
log2
where the expectation is with respect to the nonnegative random variables B1 and B2 whose distribution is identical to the asymptotic empirical distribution of the diagonal elements of 1 and 2 , respectively. The efficiencies 1 and 2 are the solutions to
M
SNR
1
2
(52)
.
111
k
1 + +
111
111
..
1
1
k E
+ log2
Defining some block matrices
and
+ log2
IN + SNR
HHy
M
k E
H
• C1 corresponds to the capacity of the entire set of desired plus
outside transmit antennas and the receiver. Hence, it regards the
outside interfering antennas as additional desired signals that
could be decoded with proper knowledge of their corresponding
channel matrices.
• C2 corresponds to the capacity of the outside transmit antennas
and the receiver, excluding the desired user transmit antennas.
The difference between both terms yields the actual capacity for the
desired user in the presence of outside interferers of which only the
aggregate covariance is known.
In order to obtain the asymptotic value of C1 and C2 as the dimensions of H 1 and H 2 are driven to infinity, we apply [19, Theorem IV.1].
With this result, derived within the context of randomly spread CDMA
in fading channels, Shamai and Verdú obtained the asymptotic capacity
of an optimum receiver using the Tse–Hanly equation [18] whose solution is the asymptotic SNR at the output of an MMSE receiver. Defining
C1 = CN , C2 = CN , and k = MN , the above theorem can be mapped
..
.
0
..
0
.
111
2
M
M
0
0
0
SIR
2K
where the M 2 M matrix contains the transmit antenna correlation
at the desired user and the Mk 2 Mk matrices k contain the transmit
antenna correlations at each of the interferers. The expectation in (54)
and (55) would, in this case, be with respect to the asymptotic empirical
distribution of the eigenvalues of 1 and 2 If, on the other hand, we
wanted to take into account antenna correlation at the receiver as well,
we would have to resort to more sophisticated tools [55], but that is
beyond the scope of this correspondence.
In most instances, the number of transmit antennas is the same for the
user of interest as well as for every outside interferer (that is, k = 8 k), in which case C +K simplifies to
2
B
C
+K
=
K
k=1
log2
B
SIRk + SNR SIRk + SNR + log2
2
1
+ log2
+ (1
1+ SNR
0 2 ) log2 (e)
1
(56)
given
1 +
SNR 1
+
SNR + 1
2 +
K
k=1
K
k=1
SNR 1
=1
SNR + SIRk
SNR 2
= 1:
SNR + SIRk
(57)
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 12, DECEMBER 2002
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MMSE Detection in Asynchronous CDMA Systems: An
Equivalence Result
Ashok Mantravadi, Student Member, IEEE, and
Venugopal V. Veeravalli, Senior Member, IEEE
Abstract—The analysis of linear minimum mean-square error (MMSE)
detection in a band-limited code-division multiple-access (CDMA) system
that employs random spreading sequences is considered. The key features
of the analysis are that the users are allowed to be completely asynchronous,
and that the chip waveform is assumed to be the ideal Nyquist sinc function.
It is shown that the asymptotic signal-to-interference ratio (SIR) at the detector output is the same as that in an equivalent chip-synchronous system.
It is hence been established that synchronous analyses of linear MMSE detection can provide useful guidelines for the performance in asynchronous
band-limited systems.
Index Terms—Asymptotic analysis, asynchronous systems, band-limited
communication, code-division multiple access (CDMA), least mean squares
methods, matched filters (MFs), minimum mean-square error (MMSE) detection, sinc function.
I. INTRODUCTION
Multiuser detection in code-division multiple-access (CDMA) systems has been a topic of intense research for more than a decade [1].
Several criteria have been used for designing multiuser detectors, and
a particularly appealing one is to minimize the mean-squared error
(MSE) of the symbol estimates at the output of the detector. When the
detector is further constrained to be linear we obtain the linear minimum mean-squared error (LMMSE or simply, MMSE) detector [2].
Equivalently, the MMSE detector also maximizes the output signal-tointerference ratio (SIR) over the class of linear detectors. In addition, it
Manuscript received August 14, 2001; revised July 22, 2002. This
work was supported by the National Science Foundation under Grant
CCR-9980616, through a subcontract with Cornell University, and by the NSF
CAREER/PECASE Award CCR-0049089. The material in this correspondence
was presented in part at the IEEE International Symposium on Information
Theory, Washington, DC, June 2001.
A. Mantravadi was with the School of Electrical Engineering, Cornell University, Ithaca, NY 14853 USA. He is now with Qualcomm, Inc., San Diego,
CA (e-mail: am77@ee.cornell.edu).
V. V. Veeravalli is with the Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, 128 Computer and Systems Research Laboratory, Urbana, IL
61801 USA (e-mail: vvv@uiuc.edu).
Communicated by D. N. C. Tse, Associate Editor for Communications.
Digital Object Identifier 10.1109/TIT.2002.805078
allows for an adaptive implementation [3]. Hence, the MMSE detector
has been a subject of considerable study.
Detailed performance analysis for the MMSE detector was first considered in [4]. The spreading sequences were assumed to be arbitrary
but fixed, and the Gaussianity of the multiaccess interference at the
output of the detector was analyzed under various asymptotic scenarios.
A more promising approach for analysis was introduced in [5], [6].
Here, the spreading sequences were treated as independent random
vectors, and limits of the SIR and capacity were studied as the number
of users (K ) and the processing gain (N ) tend to infinity with the ratio
K=N approaching a constant. The limitation of the analysis in [5], [6]
is that it is restricted to the situation where the users are symbol-synchronous. In [7], the SIR analysis of [5] was extended to the case where
the users are symbol-asynchronous but chip-synchronous, i.e., the delays of all the users are aligned to the chip timing.
While it allows for accurate large-system analysis, the synchronous
or chip-synchronous assumption is not realistic for the received signal
on the reverse link of a cellular CDMA system, especially with user
mobility and the resulting variations in the delay. Thus, we would
like to allow the users to be completely asynchronous, i.e., symbolas well as chip-asynchronous. Analysis of the MMSE detector with
random spreading sequences and completely asynchronous users was
considered in [8]. However, the performance measure was the average
near–far resistance of the detector and bounds were obtained on this
quantity for finite K and N . Furthermore, the analysis relied on the
assumption that the chip waveform was limited to a chip interval.
In this correspondence, we allow the users to be completely asynchronous and consider SIR at the detector output as the performance
metric. We also assume that the system employs the ideal band-limited
(and hence, of infinite duration) sinc chip waveform. For single-user
narrow-band systems, the sinc waveform maximizes the signaling rate
when the symbol waveforms are constrained to have a given bandwidth
and to have no intersymbol interference [9]. In spread-spectrum systems, we have an additional degree of freedom, since the processing
gain of the system can be varied with the excess bandwidth of the
chip waveform to keep the symbol rate and occupied bandwidth fixed.
In such a framework, the sinc waveform maximizes the processing
gain since it has zero excess bandwidth. For the matched-filter (MF)
detector, the maximum processing gain also results in the maximum
output SIR across all waveforms [10], [11]. Hence, practical CDMA
systems (e.g., [12]) employ waveforms that have an approximately flat
spectrum over the band of operation. Similar observations hold for the
MMSE detector as well, although a formal proof of the optimality of the
sinc waveform appears to be open [13]. Based on the above remarks,
the sinc waveform can be considered to be a benchmark for band-limited systems. Hence, analysis of the MMSE detector when the users are
completely asynchronous and employ the sinc waveform is of much interest, from a theoretical as well as a practical viewpoint.
II. SYSTEM MODEL AND MF DETECTION
1
We consider a direct-sequence CDMA (DS/CDMA) model with K +
users, where the received complex baseband signal is given by
r(t) =
K
k=0
sk (t
0 k Tc )ei
+ w (t);
t
2 [01; 1]
(1)
where sk (t) is the signal transmitted by user k
0018-9448/02$17.00 © 2002 IEEE
sk (t) =
1
m=01
p
Ek bkm ckm (t):
(
) (
)
(2)