On the Capacity of OFDM-Based Spatial

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002 225
On the Capacity of OFDM-Based Spatial
Multiplexing Systems
Helmut Bölcskei, Member, IEEE, David Gesbert, Member, IEEE, and Arogyaswami J. Paulraj, Fellow, IEEE
Abstract—This paper deals with the capacity behavior of wireless
orthogonal frequency-division multiplexing (OFDM)-based
spatial multiplexing systems in broad-band fading environments
for the case where the channel is unknown at the transmitter and
perfectly known at the receiver. Introducing a physically motivated
multiple-input multiple-output (MIMO) broad-band fading
channel model, we study the influence of physical parameters such
as the amount of delay spread, cluster angle spread, and total angle
spread, and system parameters such as the number of antennas
and antenna spacing on ergodic capacity and outage capacity. We
find that, in the MIMO case, unlike the single-input single-output
(SISO) case, delay spread channels may provide advantages over
flat fading channels not only in terms of outage capacity but
also in terms of ergodic capacity. Therefore, MIMO delay spread
channels will in general provide both higher diversity gain and
higher multiplexing gain than MIMO flat-fading channels.
Index Terms—Broad-band fading channels, diversity gain, ergodic
capacity, MIMO, multiplexing gain, OFDM, outage capacity.
I. INTRODUCTION AND OUTLINE
THEUSEOFmultiple antennasatbothendsofawireless link
has recently been shown to have the potential of achieving
extraordinary bit rates [1]–[4]. The corresponding technology is
known as spatial multiplexing [1] or BLAST [2], [5] and allows
an increase in bit rate in a wireless radio link without additional
power or bandwidth consumption. So far, most of the research in
this context has focused on the narrow-band flat-fading case. Extensive
investigations on the capacity of narrow-band flat-fading
(deterministic and stochastic) multiple-input multiple-output
(MIMO) channels (assuming different levels of channel state
information at the transmitter and the receiver) can be found in
[2], [3], and [5]–[8].
1) Contributions: For a broad-band MIMO fading channel
model, which is based on previous work reported in [9], [10], we
provide expressions for the ergodic capacity and the outage capacity
of orthogonal frequency-division multiplexing (OFDM)-
Paper approved by C. Tellambura, the Editor for Modulation and Signal Design
of the IEEE Communications Society. Manuscript received October 12,
1999; revised September 18, 2000, and April 7, 2001. This work was supported
in part by FWF under Grants J1629-TEC and J1868-TEC. This paper was presented
in part at the IEEE ICASSP-00, Istanbul, Turkey, June 2000.
H. Bölcskei was with the Coordinated Science Laboratory and the Department
of Electrical Engineering, University of Illinois at Urbana-Champaign,
Urbana, IL 61801 USA. He is now with the Communication Technology Laboratory,
ETH Zurich, ETH Zentrum, ETF EIZZ, CH-8092 Zurich, Switzerland
(e-mail: bolcskei@nari.ee.ethz.ch)).
D. Gesbert was with Iospan Wireless Inc., San Jose, CA 95134 USA. He
is now with the Department of Informatics, University of Oslo, N-0316 Oslo
Norway (e-mail: gesbert@ifi.uio.no).
A. J. Paulraj is with the Information Systems Laboratory, Department of Electrical
Engineering, Stanford University, Stanford, CA 94305-9510 USA (e-mail:
apaulraj@stanford.edu).
Publisher Item Identifier S 0090-6778(02)01368-5.
based spatial multiplexing systems [4], [11] considering the case
where the channel is unknown at the transmitter and perfectly
known at the receiver. These expressions are then used to study
(analytically and numerically) the influence of propagation parameterssuchasdelayspread,
clusteranglespread,andtotalangle
spread, and system parameters such as the number of antennas
and antenna spacing on capacity.We find that, in theMIMOcase,
unlike the single-input single-output (SISO) case, delay spread
channels may provide an advantage over flat-fading channels not
only in terms of outage capacity but also in terms of ergodic capacity.
Consequently, MIMO delay spread channels provide not
only higher diversity gain than MIMO flat-fading channels but
also higher multiplexing gain.
2) Relation to Previous Work: Our channel model builds on
research reported in [9] and [10]. In particular, it is an extension
of the space–time channel model proposed in [9] to the case of
multiple antennas at both the transmitter and the receiver. The capacity
of deterministic MIMO channels with memory and full
channel knowledge at the transmitter and the receiver was derived
in [12]. In [13], the capacity of deterministic two-user multiaccess
channels with memory is computed. Using a parametric
MIMOchannelmodel in which each path is described by an angle
of departure, an angle of arrival, a (complex) path gain, and a
pathdelay,thecapacityofthecorrespondingdeterministicMIMO
delay spread channel (full channel knowledge at the transmitter
and the receiver) has been provided in [4]. Using the same parametric
channel model and defining the underlying parameters as
random variables, a parametric MIMO fading channel model is
established in [11], and an expression for the ergodic capacity is
provided for the cases where the channel is either knownat the receiver
only or known at both the transmitter and the receiver. The
channel model used in [4] and [11] does not capture the effects
of spatial fading correlation, diffuse scattering, and scattering radius
on capacity and is therefore fundamentally different from the
MIMO fading channel model used in this paper. Furthermore, in
the channel model used in [4] and [11], each path can only be a
rank-1 contributor to capacity,1 whereas in ourmodel the rank depends
on physically meaningful parameters such as cluster angle
spread and antenna spacing. OurMIMOfading channel model is
therefore more flexible than the one used in [4] and [11]. An interesting
asymptotic (in thenumberof antenna elements) analysis
for both the flat-fading and the frequency-selective fading cases
appears in [14].
The use of OFDM in the context of spatial multiplexing has
been proposed previously in [4], [11], [15], and [16]. However,
it appears that no capacity studies of OFDM-based spatial multiplexing
systems using the physically motivated MIMO fading
1This statement will be made more precise in Section II-B.
0090–6778/02$17.00 © 2002 IEEE
226 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002
channel model provided in this paper have been performed so
far. For the single-carrier narrow-band flat-fading case, the impact
of spatial fading correlation and antenna array geometry
on capacity has been studied in [7] and [8]. To the best of our
knowledge, the impact of physical parameters (delay spread,
cluster angle spread, and total angle spread) and system parameters
(number of antennas and antenna spacing) on ergodic capacity
and outage capacity in the broad-band OFDM case has
not been studied in the literature so far.
3) Organization of the Paper: The rest of this paper is organized
as follows. In Section II, we introduce our broad-band
MIMO fading channel model. In Section III, we derive expressions
for the ergodic capacity and the outage capacity of OFDMbased
spatial multiplexing systems taking into account the new
channel model. In Section IV, we study the influence of propagation
parameters and system parameters on ergodic capacity
and outage capacity. We furthermore demonstrate that, in the
MIMO case, delay spread channels may provide advantage over
flat-fading channels not only in terms of outage capacity but also
in terms of ergodic capacity. In Section V, we provide numerical
results complementing the analytical results in Section IV.
Finally, Section VI provides our conclusions and some future
research directions.
II. BROAD-BAND MIMO FADING CHANNEL MODEL
In this section, we shall introduce a newmodel for broad-band
MIMO fading channels based on a physical description of the
propagation environment. Our channel model builds on previous
work reported in [9] and [10].
A. General Assumptions
1) Propagation Scenario: We assume that the subscriber
unit (SU) is surrounded by local scatterers so that fading at
the SU antennas is spatially uncorrelated. The base transceiver
station (BTS), however, is sufficiently high so that it is unobstructed
and no local scattering occurs. Therefore, spatial
fading at the BTS will be correlated with the exact correlation
depending on the BTS antenna spacing and the angle spread
observed at the BTS array [17]. Our model incorporates the
power delay profile of the channel, but neglects shadowing.
These assumptions on the propagation scenario are typical for
cellular suburban deployments [17], where the BTS is on a
tower or on the roof of a building and the terminal is on the
street level and experiences local scattering. For the sake of
simplicity, throughout the paper, we restrict our attention to the
uplink case. The results for the downlink case are similar. In
the following, and denote the number of transmit (i.e.,
SU) and receive (i.e., BTS) antennas, respectively.
2) Channel: Following [9] and [10], we model the delay
spread by assuming that there are significant scatterer clusters
(see Fig. 1) and that each of the paths emanating from within the
same scatterer cluster experiences the same delay. In practice,
local scatterers in the cluster introduce micro delay variations,
which will be neglected in our model. With denoting the
discrete-time transmitted signal vector and the discrete-
time received signal vector, respectively, we can
Fig. 1. Schematic representation of theMIMOdelay spread channel composed
of multiple clustered paths. Each path cluster has a mean angle of arrival  and
an angle spread  . The absolute antenna spacing is denoted by d.
write
(1)
where the complex-valued random matrix represents
the th tap of the discrete-time MIMO fading channel impulse
response. Note that in general there will be a continuum
of delays. The channel model (1) is derived from the assumption
of having resolvable paths, where with and
denoting the signal bandwidth and delay spread, respectively.
The elements of the individual are (possibly correlated) circularly
symmetric complex Gaussian random variables.2 Different
scatterer clusters are uncorrelated, i.e.,3
for (2)
where
with denoting the th
column of the matrix , and denoting the all-zero matrix
of size . Each scatterer cluster has a mean
angle of arrival at the BTS denoted as , a cluster angle spread
(proportional to the scattering radius of the cluster), and a path
gain (derived from the power delay profile of the channel).
3) Array Geometry: For the sake of simplicity, we assume
a uniform linear array (ULA) at both the BTS and the SU with
identical antenna elements. Most of our results, can however, be
extended to nonuniform arrays. The relative antenna spacing is
denoted as , where is the absolute antenna spacing
and is the wavelength of a narrow-band signal with
center frequency .
4) Fading Statistics: We assume that the (
; ) have zero mean (i.e.,
2A circularly symmetric complex Gaussian random variable is a random variable
z = (x + jy)  CN(0;  ), where x and y are i.i.d. N(0;  =2):
3E denotes the expectation operator and the superscript stands for conjugate
transposition.
BOLCSKEI et al.: CAPACITY OF OFDM-BASED SPATIAL MULTIPLEXING SYSTEMS 227
pure Rayleigh fading) and that the correlation matrix
is independent of , or, equivalently, the
fading statistics are the same for all transmit antennas. Defining
for ,
to be the fading correlation between
two BTS antenna elements spaced wavelengths apart, the
correlation matrix can be written as
(3)
Note that we have absorbed the power delay profile of the
channel into the correlation matrices.
Factoring the correlation matrix according to
, where is of size , the
matrices can be written as
(4)
with the being uncorrelated matrices with i.i.d.
entries. We have therefore decomposed the th tap of
the stochastic MIMO channel impulse response into the product
of a deterministic matrix taking into account the spatial
fading correlation at the BTS and a stochastic matrix of i.i.d.
complex Gaussian random variables.
Let us next assume that the angle of arrival for the th
path cluster at the BTS is Gaussian distributed
around the mean angle of arrival , i.e., the actual angle of arrival
is given by with . The variance
is proportional to the angular spread and hence the scattering
radius of the th path cluster. It is shown in [9] that for
small angular spread the correlation function can be approximated
as
(5)
Although this approximation is accurate only for small angular
spread, it does provide the correct trend for large angular spread,
namely uncorrelated spatial fading. Note that in the case
, the correlation matrix collapses to a rank-1 matrix and
can be written as with the array response
vector of the ULA given by
(6)
B. Differences to the Parametric Fading Channel Model
In the parametric fading channel model proposed in [11], each
tap can be written as
where denotes the complex Gaussian distributed path gain,
and are the random angle-of-arrival and angle-of-departure,
respectively, of the th path, and and the
and receive and transmit array response vectors
(cf. (6)), respectively. Note that in this model every realization
of has rank 1. In our MIMO fading channel model, the rank
of the matrices is controled by the fading correlation at the
BTS. If the angular spread of the th path cluster is large, will
have high rank; for decreasing angular spread the rank of
will decrease. This follows from (4) and the fact that the correlation
matrix loses rank if the angular spread decreases. The
MIMO fading channel model proposed in this paper is therefore
more flexible than the parametric fading channel model [11] and
seems to be a more adequate description of a real-world scattering
environment.
III. MUTUAL INFORMATION AND CAPACITY OF OFDM-BASED
SPATIAL MULTIPLEXING SYSTEMS
In this section, we derive an expression for the mutual information
of OFDM-based spatial multiplexing systems. This expression
is then used to compute the ergodic capacity and study
the outage properties of the system.
A. OFDM-Based Spatial Multiplexing
Spatial multiplexing [1], also refered to as BLAST [2], [5],
has the potential to drastically increase the capacity of wireless
radio links with no additional power or bandwidth consumption.
The technology requires multiple antennas at both ends of
the wireless link. The gain in terms of ergodic capacity over
SISO systems resulting from the use of multiple antennas is
termed multiplexing gain. The main reason for using OFDM
in this context is the fact that OFDM modulation turns a frequency-
selectiveMIMO fading channel into a set of parallel frequency-
flat MIMO fading channels. This renders multichannel
equalization particularly simple, since for each OFDM tone a
narrow-band receiver can be employed [4], [11]. In OFDMbased
spatial multiplexing, the (possibly coded) data streams
are first passed through OFDM modulators and then launched
from the individual antennas. Note that this transmission takes
place simultaneously from all transmit antennas. In the receiver,
the individual signals are passed through OFDM demodulators,
separated, and then decoded. Fig. 2 shows a schematic
of an OFDM-based spatial multiplexing system. Throughout the
paper, we assume that the length of the cyclic prefix (CP) in the
OFDM system is greater than the length of the discrete-time
baseband channel impulse response. This assumption guarantees
that the frequency-selective fading channel indeed decouples
into a set of parallel frequency-flat fading channels [18].
Organizing the transmitted data symbols into frequency
vectors with
denoting the data symbol transmitted from the th antenna
on the th tone and defining
, it can be shown
that
(7)
where denotes the reconstructed data vector for the th tone,
and is additive white Gaussian noise (AWGN) satisfying
(8)
where is the identity matrix of size . From (7), it can
be seen that equalization requires application of a narrow-band
receiver for each tone .
228 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002
Fig. 2. OFDM-based spatial multiplexing system (OMOD and ODEMOD denote an OFDM-modulator and demodulator, respectively).
B. Mutual Information
We start by stacking the vectors , , and according to
where and are vectors and is an
vector. Note that (8) implies that the noise vector is white,
i.e.,
We furthermore define the block-diagonal matrix
With these definitions, (7) can be rewritten as
(9)
In the following we assume that for each channel use (corresponding
to at least one OFDM symbol) an independent realization
of the random channel impulse response matrices is
drawn and that the channel remains constant within one channel
use. Using (9), the mutual information (in b/s/Hz) of the OFDMbased
spatial multiplexing system under an average transmitter
power constraint is given by4 [19], [20]
(10)
where with5 is the covariance matrix of the
Gaussian input vector and is the maximum overall transmit
power. Note that mutual information is normalized by , since
data symbols are transmitted in one OFDM symbol and that
we ignored the loss in spectral efficiency due to the presence of
the CP. The matrix is a block-diagonal matrix
given by
4Throughout the paper, all logarithms are to the base 2.
5Tr(A) stands for the trace of the matrix A.
where the matrices are the covariance matrices
of the Gaussian vectors , and as such determine the power
allocation across the transmit antennas and across the OFDM
tones. If the channel is perfectly known at the transmitter, the
optimum power allocation is obtained by distributing the total
available power according to the water-filling solution [4].
In OFDM-based spatial multiplexing systems, statistically independent
data symbols are transmitted from different antennas
and different tones and the total available power is allocated uniformly
across all space–frequency subchannels [4], [11]. In the
following, we set
, which is easily verified to result in . Using (10),
we therefore obtain
(11)
where . The quantity is the mutual information
of the th MIMO OFDM subchannel. Note that, since
is random, is a random entity as well. We
shall next show that the distribution of is independent of
and hence all the have the same distribution.
In the following, the notation means that the
distribution of the random variable is equal to the distribution
of the random variable .
Proposition 1: The distribution of
is independent of and given by
for
(12)
where , is an i.i.d.
random matrix with entries, and . Finally,
denotes the th eigenvalue of .
Proof: Gaussianity of the implies Gaussianity of
for . Now, using
(13)
BOLCSKEI et al.: CAPACITY OF OFDM-BASED SPATIAL MULTIPLEXING SYSTEMS 229
it follows that the columns of
are uncorrelated and have the same statistics.
Denoting the first column of as
and the first column of as and using (2) it follows
that
Note that the correlation matrix is independent of . We have
thus shown that
with denoting an i.i.d. matrix with entries. Hence,
it follows that
which can be rewritten as
Using the fact that [21], we obtain
Now, without changing the distribution, we can right-multiply
by to obtain
Using and exploiting the unitarity
of , we finally get
which concludes the proof.
C. Ergodic Capacity and Outage Capacity
We shall next establish the information-theoretic value of the
results derived in Section III-B. Two scenarios are considered,
the ergodic and the nonergodic case. In both cases, we assume
that the channel remains fixed within one channel use (at least
one OFDM symbol) and then changes in an independent fashion
to a new realization.
1) Ergodic Case: The basic assumption here is that the
transmission time is long enough to reveal the long-term ergodic
properties of the fading channel. In this case, a Shannon
capacity exists and is given by with defined in
(11). At rates lower than , the error probability (for a good
code) decays exponentially with the transmission length. The
assumption here is that the fading process is ergodic, coding
and interleaving are performed across OFDM symbols, and
that the number of fading blocks spanned by a codeword goes
to infinity whereas the block size (which equals the number of
tones in the OFDM system multiplied by the number of OFDM
symbols spanning one channel use) remains constant (and
finite). Capacity can be achieved in principle by transmitting
a codeword over a very large number of independently fading
blocks. We furthermore note that the capacity obtained for an
OFDM-based spatial multiplexing system is a lower bound
for the capacity of the underlying broad-band MIMO fading
channel.
2) Nonergodic Case: In this case, we assume that a codeword
spans an arbitrary but fixed number of blocks while the
block size goes to infinity. This situation typically occurs when
stringent delay constraints are imposed, as is the case, for example,
in speech transmission over wireless channels. These assumptions
give rise to error probabilities which do not decay
with an increase of block length. A capacity in the Shannon
sense does not exist since, with nonzero probability, which is independent
of the code length, the mutual information in (11)
falls below any positive rate, as small as it may be. Thus, the
concept of capacity versus outage [22], [23] has to be invoked.
Assuming that codewords extend over a single block, the outage
(or failure) probability for a given rate is the probability that
falls below that rate. In this case, capacity is viewed as a random
entity [22], [23] since it depends on the instantaneous random
channel parameters.
IV. INFLUENCE OF CHANNEL AND SYSTEM PARAMETERS ON
CAPACITY
In this section, we study the influence of channel and system
parameters on ergodic capacity and outage capacity. In particular,
we demonstrate that in the MIMO case, unlike the SISO
case, delay spread channels may provide advantage over flatfading
channels in terms of ergodic capacity. While the ergodic
case is to some extent amenable to analytic studies, the nonergodic
case will mainly be discussed by means of simulation
results in Section V. Analytic results seem hard to obtain in the
nonergodic case. Some statements of qualitative nature on the
nonergodic case will be made in this section.
A. The Ergodic Case
The ergodic capacity is obtained from (11) as
Now, using Proposition 1, which says that the
all have the same distribution given by (12),
the ergodic capacity is obtained as
(14)
where expectation is taken with respect to . A semi-analytic
result for this expectation has been provided by Telatar in [3] for
the case where . In the general case, the evaluation of the
expectation in (14) requires the concept of zonal polynomials
[21] and is significantly more complicated. We shall therefore
resort to a simple asymptotic analysis by assuming that is
large. It follows from the law of large numbers that, for fixed
230 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002
as gets large, . Hence, in the
large limit, we get
(15)
where . In the low SNR regime, i.e.,
for small , it follows from (15) that in the large limit
where all the higher order terms in have been neglected. Thus,
in the low SNR regime, the ergodic capacity is driven by the
Trace of the sum correlation matrix . Here, comparing channels
on the basis of fixed energy, i.e., fixed leads to the
conclusion that delay spread has no impact on ergodic capacity.
In the high- SNR case, we obtain
(16)
The eigenvalue spread of the sum correlation matrix
therefore critically determines ergodic
capacity. In fact, we have
Lemma 2: For , the right-hand side (RHS) in (16)
is maximized for .
Proof: The proof follows easily by applying Jensen’s inequality
to the RHS of (16).
Using the developments in Section IV-A and [3, Theorem
1], it can even be shown that, for ,
maximizes the
exact (finite ) expression (14). A deviation of as
a function of from a constant function will therefore result
in a loss in terms of ergodic capacity or equivalently reduced
multiplexing gain. In the following, we restrict our attention to
the high SNR case. We shall next show how the propagation
and system parameters impact the eigenvalues of and hence
ergodic capacity. Since the individual correlation matrices
are Toeplitz, the sum correlation matrix is Toeplitz as well.
We can thus invoke Szegö’s theorem [24] to obtain the limiting
distribution6 of the eigenvalues of as
Using (5), we obtain
(17)
with the third-order theta function given by [25]
. Although this expression yields the exact
eigenvalue distribution only in the limiting case , in
6Note that forM !1the eigenvalues ofRare characterized by a periodic
continuous function [24]. Thus, whenever we use the term eigenvalue distribution,
we actually refer to this function.
(a)
(b)
Fig. 3. Limiting eigenvalue distribution of the correlation matrix R for the
cases of (a) high spatial fading correlation and (b) low spatial fading correlation.
the case of finite good approximations of the eigenvalues
can be obtained by sampling uniformly on the unit circle
[24], which allows us to assume that the eigenvalue distribution
in the finite case follows the distribution given by . We are
now able to study the impact of various propagation and system
parameters on the eigenvalue distribution of and hence the
ergodic capacity.
1) Impact of Cluster Angle Spread and Antenna
Spacing: Let us first investigate the influence of cluster angle
spread and antenna spacing on ergodic capacity. For the sake
of simplicity, take one path only and its associated correlation
matrix . The limiting eigenvalue distribution of is given
by .
Now, noting that the correlation function as a
function of is essentially a modulated Gaussian function with
its spread increasing for increasing antenna spacing and/or
increasing cluster angle spread and vice versa, it follows that
will be more flat in the case of large antenna spacing
and/or large cluster angle spread (i.e., low spatial fading correlation).
For small antenna spacing and/or small cluster angle
spread, will be peaky. Fig. 3(a) and (b) show the limiting
eigenvalue distribution of for high and low spatial fading
correlation, respectively. From our previous discussion, it thus
follows that the ergodic capacity will decrease for increasing
concentration of and vice versa.
2) Impact of Total Angle Spread: We shall next study the
impact of total angle spread on ergodic capacity. Assume that
either the individual scatterer cluster angle spreads are small
or that antenna spacing at the BTS is small or both. Hence,
the individual are peaky. Now, from (17), we can see
that the limiting distribution is obtained by adding the
individual limiting distributions . Note furthermore that
is essentially a Gaussian centered around .
Now, if the total angle spread, i.e., the spread of the , is large,
the sum-limiting distribution can still be flat even though
the individual are peaky. For a given small cluster angle
spread, Fig. 4(a) and (b) show example limiting distributions
BOLCSKEI et al.: CAPACITY OF OFDM-BASED SPATIAL MULTIPLEXING SYSTEMS 231
(a)
(b)
Fig. 4. Limiting eigenvalue distribution of the sum correlation matrix R =
R for fixed cluster angle spread and for the cases of (a) small total
angle spread and (b) large total angle spread.
for a three-path channel with a total angle spread of 22.5
degrees and a total angle spread of 90 degrees, respectively.
We can clearly see the impact of total angle spread on the
limiting eigenvalue distribution and hence on ergodic
capacity. Large total angle spread renders flat and therefore
increases ergodic capacity, whereas small total angle
spread makes peaky and hence reduces ergodic capacity.
This impact can further be illustrated by studying the extreme
case , i.e., small cluster angle spread (or equivalently
large distance between BTS and SU). In this case, the sum
correlation matrix is given by
with defined in (6). Take the simple example ,
, and . In this case, for and , we
get and for and we have
. For , the ergodic capacities obtained by
Monte Carlo evaluation of (14) are b/s/Hz in the case
of small total angle spread and b/s/Hz in the case of
large total angle spread.
3) Ergodic Capacity in the SISO and in the MIMO Case: It
is well known that in the SISO case delay spread channels do not
offer advantage over flat-fading channels in terms of ergodic capacity[
22],[23].Thiscaneasilybeseenfrom(14)bynotingthatin
theSISOcase andhenceergodiccapacity
is only a function of the total energy in the channel. In theMIMO
case, the situation is in general different. Fix , and take a
flat-fading scenario with small cluster angle spread where
has rank 1. In this case, the matrix has rank 1 with
probabilityoneandhenceonlyonespatialdatapipecanbeopened
up, or equivalently there is no multiplexing gain. Now, compare
this scenario to a delay-spread scenario where and each
of the has rank 1 but the sum-correlation
matrix has full rank. For this to happen, a sufficiently
large total angle spread is necessary. Clearly, in this case, spa-
(a)
(b)
Fig. 5. Example histograms of the mutual information I in b/s/Hz in the (a)
flat-fading case and (b) the high delay-spread case.
tial data pipes can be opened up and we will get a higher ergodic
capacity because the rank of is higher than in the flat-fading
case.We note that in the case where all the correlation matrices
satisfy this effect does not
occur.However, since this scenario corresponds to fully uncorrelated
spatial fading it is very unlikely.We can therefore conclude
that in practiceMIMOdelay spreadchannelsofferadvantageover
MIMOflat-fading channels in terms of ergodic capacity.We caution
the reader that this conclusion is a result of the assumption
thatdelayedpathsincreasethetotalanglespread.Thisassumption
hasbeenverifiedbymeasurementforoutdoorMIMObroad-band
channels in the 2.5-GHz band [26].
B. The Nonergodic Case
In [22], it has been demonstrated that SISO delay-spread
channels offer significant advantage over flat-fading channels
in terms of outage probabilities or outage capacity. The outage
properties are determined by the number of diversity degrees
of freedom in the channel. In our case, we have both spatial
diversity and frequency diversity available. We can therefore
expect that both diversity sources will contribute to the outage
characteristics of the system. Assuming that a codeword spans
one block, we recall that the outage probability for a given rate
is the probability that falls below that
rate. The distribution of is hard to compute analytically7. We
7In this context, we would like to point out an error in [27] and simplifications
of some of the results reported in [27], in which the D defined in [27, eq.
(9)] should read C  log det(I + DV HH V D )
and [27, eq. (11)] should be replaced by EfCg = E (1=N)
log det I +  H H  . Furthermore, it follows from
Proposition 1 in this paper that [27eq. (11)] can be simplified to yield
EfCg = E log det I + H H with  defined in
Proposition 1 in this paper. It can also be shown that the j j in [27]
are independent of k and equal the  (R) defined in Proposition 1 in this
paper, and that the r in [27] are independent of k and equal rank(R)
with R = R . With this, [27, eq. (12)] can be simplified to
yield EfCg  log (1 + M  (R)). Furthermore, [27, eq. (13)]
should be replaced by EfCg  E log(1+  (R)jr j )
and the last equation in [27] has to be replaced by EfCg 
E log 1+  (R)jr j +  (R)jr j .
232 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002
therefore resort to numerical studies presented in Section V and
make a few qualitative statements below.
The individual all have the same distribution. The correlation
between the , however, strongly depends on the amount
of delay spread in the system. In order to establish the value of
space–frequency diversity in terms of outage properties, let us
consider the two limiting cases of flat-fading (i.e., no frequency
diversity) and high delay spread. In the high delay spread case,
we assume that the correlation between the is small, which
corresponds to the assumption of a high number of independently
fading taps in the channel. The mean of is independent
of the correlation between the and is given by (14).
The variance of , however, and hence the outage properties
depend significantly on the amount of space–frequency diversity.
Denote the variance of as (recall that the distribution
of is independent of ). In the flat-fading case, we have
, whereas in the high delay-spread case (under
the idealistic assumption of full decorrelation of the ) we obtain
. Fig. 5(a) and (b) illustrate example
histograms of for a 64-tone OFDM system in the flat-fading
case and in the high delay-spread case, respectively. It can be
seen that in the high delay-spread case the distribution is significantly
more concentrated around the mean. Take a rate of say
7.5 b/s/Hz. For this rate, clearly from Fig. 5 the outage probability
will be much lower for the high delay-spread case than for
the flat-fading case.
Since the rank of the individual correlation matrices
determines the number of spatial degrees of
freedom in each path of the MIMO channel, it is to be expected
that the rank of the individual correlation matrices and not the
rank of the sum correlation matrix determines the outage
properties. This can be illustrated by assuming a simple example
where all the have rank 1 but are such that the sum correlation
matrix has full rank. In this case, it readily follows from (4)
that the number of degrees of freedom in each path is and
hence the total number of degrees of freedom in the channel is
, irrespectively of the rank of the sum correlation matrix
. In the case where the individual correlation matrices are
full rank, the sum correlation matrix is also full rank,8 but the
number of degrees of freedom in the channel will be ,
and hence significantly better outage properties than in the fully
correlated case can be expected. These statements will be corroborated
by means of simulation results in Section V. We conclude
by noting that, while the multiplexing gain is determined
by the rank of the sum correlation matrix , the diversity gain
will be governed by the rank of the individual correlation matrices
.
V. SIMULATION RESULTS
In every simulation example, 1000 independent Monte Carlo
runs were performed. Unless specified otherwise, the power
delay profile was taken to be exponential, the number of tones
in the OFDM system was , the CP length was 64, and
the relative antenna spacing was set to . For the sake
of simplicity, we assume uniform tap spacing in all simulation
8This follows from application of [28, Lemma 4.1] to R = R and
noting that # (; q) is nonnegative.
(a)
(b)
Fig. 6. Ergodic capacity (in b/s/Hz) as a function of SNR for various values of
L and (a) small cluster angle spread and (b) large cluster angle spread.
examples. Finally, the SNR was defined as SNR
.
A. Simulation Results
1) Simulation Example 1: In the first simulation example,
we study the impact of delay spread on ergodic capacity corroborating
the statement that in the MIMO case delay spread
may provide advantage over the flat-fading case in terms of ergodic
capacity (provided that the total angle spread is large).
The number of antennas was . In order to make
the comparison fair, we normalize the energy in the channel by
setting for all cases. The cluster angle spread was
assumed to be . In the flat-fading
case, the mean angle of arrival was set to . In the
delay-spread case, we assumed a total angle spread of 90 degrees.
Fig. 6(a) shows the ergodic capacity (in b/s/Hz) as a function
of SNR for different values of . We can see that ergodic
capacity indeed increases for increasing delay spread. We can
furthermore observe that increasing the number of resolvable
BOLCSKEI et al.: CAPACITY OF OFDM-BASED SPATIAL MULTIPLEXING SYSTEMS 233
Fig. 7. Outage probability for L = 1, 5, and 16 at an SNR of 10 dB.
Fig. 8. Outage probability for various values of s =  at an SNR of 10 dB.
taps beyond 4 does not further increase ergodic capacity. The
reason for this is that the number of transmit and receive antennas
was set to 4 and hence the maximum rank of the sum
correlation matrix is 4. Fig. 6(b) shows the ergodic capacity
for the same parameters as above except for the cluster angle
spread whichwas increased to .
In this case, the rank of the individual correlation matrices is
higher than 1 and the improvement in terms of ergodic capacity
resulting from the presence of multiple taps is less pronounced.
We emphasize that, as already stated in Section IV-A, this result
is a consequence of the assumption that delayed paths tend
to increase the total angle spread.
2) Simulation Example 2: In the second simulation example,
we investigate the impact of delay spread on the outage
properties of the system. Again, for fixed , Fig. 7
shows the outage probability for , 5 and 16 and an
SNR of 10 dB. Here, we assumed that there is no spatial
fading correlation. It is clearly seen that the outage probability
decreases significantly with increasing delay spread.
3) Simulation Example 3: In the last simulation example,
we investigate the impact of spatial fading correlation on outage
probability. For , and ,
0.5, and 0.7, Fig. 8 shows the outage probability as a function
of rate for an SNR of 10 dB. In all three simulations, the mean
angles of arrival were chosen such that the sum correlation matrix
was full rank. Again, in all cases . It can be
seen that, even though the sum correlation matrix has full
rank, the outage probability depends critically on the individual
cluster angle spreads and hence the rank of the individual correlation
matrices .
VI. CONCLUSION
Based on a physically motivated model for broad-band
MIMO fading channels, we derived expressions for the ergodic
capacity and for outage capacity of OFDM-based spatial multiplexing
systems for the case where the channel is unknown at
the transmitter and perfectly known at the receiver. We studied
the influence of propagation parameters and system parameters
on ergodic capacity and outage probability and demonstrated
the beneficial impact of delay spread and angle spread on
capacity. Specifically, we showed that, in the MIMO case as
opposed to the SISO case, delay-spread channels may provide
advantage over flat-fading channels not only in terms of outage
capacity but also in terms of ergodic capacity (provided the
assumption that delayed paths tend to increase the total angle
spread is true). We furthermore found that, while the multiplexing
gain is governed by the rank of the sum correlation
matrix , the diversity gain seems to be governed by the rank
of the individual correlation matrices .
Directions for further work include the analysis of the case
where there is scattering at both the transmitter and the receiver.
A question of particular importance seems to be the analysis
of the influence of scattering radii and distance between BTS
and SU on capacity. Furthermore, a detailed study of the influence
of different antenna geometries on the capacity of OFDMbased
spatial multiplexing systems appears to be of interest.
This problem has been studied to some extent in [7] for the
narrow-band frequency-flat fading case.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers
for their constructive criticism which helped to significantly improve
the quality of the paper and the exposition. They would
furthermore like to thank G. Wunder and R. W. Heath, Jr., for
their detailed comments on the paper.
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Helmut Bölcskei (M’98) was born in Mödling, Austria,
on May 29, 1970. He received the Dipl.-Ing. and
Dr. techn. degrees in electrical engineering/communications
from Vienna University of Technology, Vienna,
Austria, in 1994 and 1997, respectively.
From 1994 to 2001, he was with the Institute of
Communications and Radio-Frequency Engineering,
Vienna University of Technology. From March 2001
to January 2002, he was an Assistant Professor of
Electrical Engineering at the University of Illinois
at Urbana-Champaign. Since February 2002, he
has been an Assistant Professor of Communication Theory at ETH Zürich,
Switzerland. From February to May 1996, he was a Visiting Researcher at
Philips Research Laboratories Eindhoven, The Netherlands. From February
to March 1998, he visited the Signal and Image Processing Department at
ENST Paris, France. From February 1999 to February 2001, he was an Erwin
Schrödinger Fellow of the Austrian National Science Foundation (FWF)
performing research in the Smart Antennas Research Group in the Information
Systems Laboratory, Department of Electrical Engineering, Stanford University,
Stanford, CA. From 1999 to 2001, he was a consultant for IospanWireless,
Inc. (formerly Gigabit Wireless Inc.), San Jose, CA. His research interests
include communication and information theory and statistical signal processing
with special emphasis on wireless communications, multi-input multi-output
(MIMO) antenna systems, space-time coding, orthogonal frequency division
multiplexing (OFDM), and wireless networks.
Dr. Bölcskei received a 2001 IEEE Signal Processing Society Young Author
paper awards and serves as an Associate Editor for the IEEE TRANSACTIONS ON
SIGNAL PROCESSING.
David Gesbert (S’96–M’99) was born in France in
1969. He received the M.Sc. degree in electrical engineering
from National Institute for Telecommunications
(INT), Evry, France, in 1993 and the Ph.D. degree
from Ecole Nationale Superieure des Telecommunications,
Paris, France, in 1997.
From 1993 to 1997, he was with France Telecom
Research, where he was involved in the development
and study of receiver algorithms for digital radio
communications systems, with emphasis on blind
signal detection. From April 1997 to October 1998,
he was a Post-Doctoral Fellow in the Smart Antennas Research Group, Information
Systems Laboratory, Stanford University, Stanford, CA. In October
1998, he took part in the founding engineering team of Iospan Wireless, Inc.,
formerly Gigabit Wireless Inc., San Jose, CA, a startup company promoting
high-speed wireless data networks using smart antennas. In January 2001,
he joined the Signal Processing Group, Department of Informatics at the
University of Oslo, Norway, as an Adjunct Associate Professor in parallel to his
activities at Iospan. His research interests are in the area of signal processing
for digital communications, blind array processing, multi-input multi-output
(MIMO) systems, multi-user communications, and adaptive wireless networks.
Arogyaswami J. Paulraj (SM’85–F’91) received
the Ph.D. degree from the Naval Engineering College
and Indian Institute of Technology, Bangalore, in
1973.
He has been a Professor at the Department
of Electrical Engineering, Stanford University,
Stanford, CA, since 1993, where he supervises
the Smart Antennas Research Group. This group
consists of approximately 12 researchers working
on applications of space–time signal processing for
wireless communications networks. His research
group has developed many key fundamentals of this new field and helped
shape a worldwide research and development focus onto this technology. His
nonacademic positions included Head, Sonar Division, Naval Oceanographic
Laboratory, Cochin, India; Director, Center for Artificial Intelligence and
Robotics, Bangalore; Director, Center for Development of Advanced Computing;
Chief Scientist, Bharat Electronics, Bangalore, and Chief Technical
Officer and Founder, Iospan Wireless, Inc., San Jose, CA. He has also held
visiting appointments at Indian Institute of Technology, Delhi, Loughborough
University of Technology, and Stanford University. He sits on several boards
of directors and advisory boards for U.S. and Indian companies/venture partnerships.
His research has spanned several disciplines, emphasizing estimation
theory, sensor signal processing, parallel computer architectures/algorithms
and space–time wireless communications. His engineering experience included
development of sonar systems, massively parallel computers, and more recently
broad-band wireless systems. He is the author of over 250 research papers and
holds eight patents.
Dr. Paulraj has won several awards for his engineering and research contributions.
These include two President of India Medals, the CNS Medal, the
Jain Medal, the Distinguished Service Medal, the Most Distinguished Service
Medal, the VASVIK Medal, and the IEEE Best Paper Award (Joint), amongst
others. He is a member of the Indian National Academy of Engineering.
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