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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 10, OCTOBER 2004
Achievable Information Rates and Coding
for MIMO Systems Over ISI Channels and
Frequency-Selective Fading Channels
Zheng Zhang, Student Member, IEEE, Tolga M. Duman, Senior Member, IEEE, and Erozan M. Kurtas, Member, IEEE
Abstract—We propose a simulation-based method to compute
the achievable information rates for general multiple-input multiple-output (MIMO) intersymbol interference (ISI) channels with
inputs chosen from a finite alphabet. This method is applicable
to both deterministic and stochastic channels. As an example of
the stochastic MIMO ISI channels, we consider the multiantenna
systems over frequency-selective fading channels, and quantify the
improvement in the achievable information rates provided by the
additional frequency diversity (for both ergodic and nonergodic
cases). In addition, we consider the multiaccess multiantenna
system and present some results on the achievable information-rate region. As for the deterministic MIMO ISI channels, we
use the binary-input multitrack magnetic recording system as an
example, which employs multiple write and read heads for data
storage. Our results show that the multitrack recording channels have significant advantages over the single-track channels,
in terms of the achievable information rates when the intertrack interference is considered. We further consider practical
coding schemes over both stochastic and deterministic MIMO
ISI channels, and compare their performance with the information-theoretical limits. Specifically, we demonstrate that the
performance of the turbo coding/decoding scheme is only about
1.0 dB away from the information-theoretical limits at a bit-error
rate of 10 5 for large interleaver lengths.
Index Terms—Channel capacity, frequency-selective fading, information rates, intersymbol interference (ISI), iterative decoding,
multiantenna systems, multiple-input multiple-output (MIMO)
systems, multitrack recording, turbo coding.
I. INTRODUCTION
R
ECENTLY, the multiantenna systems have received
a lot of attention due to their capability in taking
advantage of the spatial diversity. In [1] and [2], the authors have independently proved that the use of the multiple-transmit/multiple-receive antennas significantly improves
the capacity for a Rayleigh flat-fading channel, compared
with the single-transmit/single-receive antenna systems. More
specifically, when the subchannels experience independent
Paper approved by R. A. Valenzuela, the Editor for Transmission Systems
of the IEEE Communications Society. Manuscript received August 15, 2003;
revised November 3, 2003. This work was supported in part by a grant from
Seagate Technology, and in part by the National Science Foundation under CAREER Award CCR-9984237. This paper was presented in part in the IEEE International Symposium on Information Theory (ISIT), Lausanne, Switzerland,
July 2002, and in part at ISIT, Yokohama, Japan, June–July 2003.
Z. Zhang and T. M. Duman are with the Electrical Engineering Department,
Arizona State University, Tempe, AZ 85287-5706 USA (e-mail: zheng.zhang@
asu.edu; duman@asu.edu).
E. M. Kurtas is with Seagate Technology, Pittsburgh, PA 15222-4215 USA
(e-mail: Erozan.M.Kurtas@seagate.com).
Digital Object Identifier 10.1109/TCOMM.2004.836449
Rayleigh fading, and the same number of antennas are used
at both the transmitter and the receiver, the capacity increases
linearly with the number of the transmit or receive antennas.
For high-data-rate wireless communications, since the
symbol duration is small, compared with the multipath spread
of the channel, the channel experiences intersymbol interference (ISI), resulting in a frequency-selective fading channel.
Although the frequency selectivity may complicate the signaling algorithms, it provides additional (multipath) diversity.
In this paper, we consider the multiantenna systems over a frequency-selective fading channel, and refer to these systems as
stochastic multiple-input/multiple-output (MIMO) ISI systems.
MIMO systems are also considered in many other areas. For
example, in magnetic recording channels, multitrack recording
is proposed, where data are written to a group of adjacent tracks
simultaneously and read back by multiple heads in parallel, in
order to reduce the effect of the intertrack interference (ITI) [3].
For narrowtrack systems of the future, the multitrack approach
is an efficient solution to achieving high recording densities in
the presence of severe ITI. Note that the magnetic recording
channel is a typical ISI channel, therefore, the multitrack
recording systems can be modeled as deterministic MIMO ISI
channels. With this motivation, we also consider deterministic
MIMO ISI channels throughout this paper.
The capacity of the MIMO Gaussian channel with both deterministic and stochastic channel-transfer functions is derived
in [1] and [2], and related work for MIMO frequency-selective
fading channels is reported in [4] and [5]. For the single-input
single-output (SISO) ISI channel with additive white Gaussian
noise (AWGN), the capacity is derived in [6]. It is shown that
the capacity-achieving signals are Gaussian random variables
for these cases. On the other hand, in practice, signals are
selected from a finite alphabet. For example, signals used
in the magnetic recording channels are binary, and in the
wireless communications, specific constellations, such as the
phase-shift keying (PSK) and the quadrature amplitude modulation (QAM), are employed. Therefore, in general, one cannot
achieve the “unconstrained” capacity, especially in the high
signal-to-noise ratio (SNR) region, and achievable information
rates under suitable input constraints should also be considered.
When the inputs are selected from specific signal constellations (instead of the Gaussian inputs), there are usually no
closed-form solutions to the achievable information rates, even
for the simple ISI channels with AWGN. Instead, upper and
lower bounds on the information rates are provided in [7] and
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ZHANG et al.: ACHIEVABLE INFORMATION RATES AND CODING FOR MIMO SYSTEMS
[8]. More recently, a simulation-based technique has been proposed to compute the information rates of SISO ISI channels
with inputs selected from a finite alphabet in [9] (see also [10]
and [11]). The main idea is to use a simulation of the channel
excited by inputs with a specific distribution and employ the
Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm [12] to estimate
the joint probability of the output sequence. Then this joint probability is used to estimate the differential entropy of the output
sequence and the mutual information between the input and the
output, thus the achievable information rate under the specific
input constraint.
In the first part of the paper, we extend the technique of
[9] to the deterministic MIMO ISI channels and the multiantenna systems over frequency-selective fading channels.
We take both the ISI and interchannel interference (ICI) into
account and compute the mutual information between multiple inputs and multiple outputs for a given input–output
simulation. In [13] and [14], some preliminary results on the
simulation-based algorithm for computing the achievable information rates of MIMO ISI channels were presented. Here,
we discuss the method in detail and extend it further. We consider both the ergodic and nonergodic multiantenna systems
over frequency-selective fading channels. Our results quantify the improvement provided by the frequency diversity and
show the difference between the achievable information rates
using specific constellations and the unconstrained capacity.
We further note that there are some important factors that affect
the achievable information rates of the multiantenna systems,
including the existence of the line-of-sight (LOS) signals and
the presence of the spatial correlation in the channel-transfer
function. With the proposed method, we can easily compute
the achievable information rates by taking these considerations into account as well. Moreover, in addition to the case
with independent identically distributed (i.i.d.) inputs, we also
consider the channels with Markov inputs and perform the
maximization of the information rates over such inputs.
The techniques developed are quite general, and they are applicable to other important scenarios. As an example, we consider the multiaccess multiantenna system with inputs from specific constellations. We apply the simulation-based algorithms
to find the corresponding information-rate regions and compare the results with the capacity region [15]. We will see that
the information-rate region for the two-user case is shown as a
pentagon, which asymptotically becomes a rectangle (or square)
when the SNR is increased.
In the second part of the paper, we adopt a joint turbo coding
and iterative decoding scheme to the MIMO ISI channels, where
a soft-input soft-output channel detector (or equalizer) and an
outer decoder, corresponding to the outer code used, work cooperatively in an iterative manner. We design the maximum a
posteriori (MAP) detector, as the channel equalizer, for both the
deterministic and stochastic channels, and use a turbo code or
a single convolutional code as the outer code. The results show
that, to achieve a bit-error rate (BER) of
, which can be
considered as reliable transmission, the required SNR for the
proposed tubo-coding scheme with a block size of around
bits is only about 1-dB away from the information-theoretical
limits for a variety of MIMO ISI channels.
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The paper is organized as follows. In the next section, we
present the general MIMO ISI channel model. As examples for
the stochastic and deterministic channels, we describe the multiantenna system over a frequency-selective fading channel and
the multitrack magnetic recording system. In Section III, we
give the details of the simulation-based methods used to compute the achievable information rates for the general MIMO ISI
channels when the inputs are chosen from a finite alphabet. We
also discuss some special cases in detail, including the multiantenna system over frequency-flat fading channels, the multiaccess multiantenna systems, and the single-track recording
systems with the ITI. In Section IV, we study several examples
of the information rates, compare them with the unconstrained
capacity (if available), and discuss the results. In Section V, we
describe the MAP detector and the joint turbo coding/decoding
scheme for MIMO ISI channels, and present several examples.
Finally, we conclude the paper in Section VI.
II. CHANNEL MODEL
We consider a general discrete-time MIMO ISI channel with
inputs and outputs (denoted by a
system), and memory
. The th received signal at the time instance is given by
(1)
is the channel gain of the th
where is the th input,
tap for the subchannel from the th input to the th output, and
is the AWGN in the th receiver output. When the channel
. Also, we assume that the
is time invariant,
noise terms are temporally and spatially independent. In vector
notation
(2)
, the input
where the output vector
, the noise vector
vector
(where denotes the transpose), and the
.
channel-transfer function
We define the average SNR at each receiver as
(3)
where
denotes expectation,
is the energy per symbol at
is the variance of the AWGN if the
each transmitter, and
noise is real, and it is the variance per dimension if it is complex. We also note that the channel is assumed to be stationary,
is independent of , and
therefore, the expectation of
if the channel is deterministic, the expectation operation can be
omitted altogether.
A. Multiantenna Systems Over Frequency-Selective Fading
Channels
For the frequency-selective fading case, the channel in (2)
is nothing but a tapped-delay-line model where the number of
. The channel coefficients
the taps or resolvable paths is
are complex Gaussian random variables with equal
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variance. Unless otherwise stated, we assume that the channel is
Rayleigh fading, i.e., the channel coefficients have zero mean,
are independent, except when
and all the elements in
we consider the effects of the spatial correlation on the achievable information rates. In this case, the noise term in (2) is also
zero-mean complex Gaussian distributed.
For the multiantenna wireless systems, we consider both
ergodic channels (e.g., independent fading from one symbol
to the next, or block fading), and nonergodic channels [16]
(e.g., quasi-static fading, where the fading coefficients are fixed
during the transmission of an entire block). We note that although both the i.i.d. fading and block fading can be considered
as ergodic frequency-selective fading channels, they do not
refer to the exact same channel model, unlike the flat-fading
scenario, and the capacities and information rates may be
different. We assume that the channel state information (CSI)
is available at the receiver, but not at the transmitter. Therefore,
no waterfilling is performed at the transmitter, that is, the signal
energy is evenly allocated among all the transmitters.
A. Information Rates of the Deterministic and Stochastic
MIMO ISI Channels
B. Multitrack Recording Systems
The multitrack recording channel is an example of deterministic MIMO ISI channels, where the channel-transfer function
is constant. For clarity, let us first consider the discrete-time
channel model of the single-track system without ITI. The th
received symbol can be expressed by
(4)
where
is the set of ISI coefficients of the channel due to
the channel pulse response [17]. For practical systems, we also
need to incorporate the ITI among the adjacent tracks into the
multitrack system, where there are
channel model. In a
heads reading tracks simultaneously, the th received symbol
for the th read head is then given by
Let us first consider the deterministic channel and ergodic
wireless channel, and compute the Shannon type capacity, or
more precisely, the achievable information rate with the input
constraint, which is given by
(6)
where we omit the conditioning on the channel-transfer function
for simplicity. In this paper, we mainly consider the achievable
information rates with independent and uniformly distributed
(i.u.d.) inputs, called symmetric information rates. Since the capacity is the achievable information rate maximized over the distribution of the constrained inputs, the symmetric information
rate can be viewed as a lower bound. This lower bound should
be very close to the capacity in the high SNR region, if there
is no ambiguity about the channel inputs, given the noiseless
channel outputs. This is justified by the fact that, in the limiting
case when the SNR approaches infinity, the achievable information rate can be expressed by the entropy of the input vectors,
since the entropy of the input vectors conditioned on the noiseless output vectors is zero [18], and this entropy is maximized by
i.u.d. inputs. Furthermore, for the ergodic stochastic channels,
without the CSI known to the transmitter, using i.u.d. inputs is
a reasonable assumption that will result in a meaningful lower
bound on the constrained capacity.
However, the method we will present is more general. For example, it can be used to estimate the information rates for any
given Markov input, and even maximization can be performed
over all possible Markov inputs. This would generalize the algorithms given in [9] and [19], which deal with the SISO systems
over ISI channels, to the case of MIMO systems. At the end of
this section, we will discuss these extensions.
Since the additive noise is independent of the input signals,
the information rate can be expressed as
(5)
where we assume that the pulse responses from different tracks
are identical, i.e., the set of ISI coefficients is the same, except for the amplitude varying with the distance between the
track and the read head, which is reflected by the coefficients
. Clearly, a more general case of different pulse responses
on different tracks can also be considered in a similar fashion.
.
Comparing with (2), we see that
In the examples shown later, we consider the ideal normalized
and
Class IV partial-response (PR4) channel [17], where
,
,
, even though the technique for information-rate computation is very general.
III. COMPUTATION OF THE ACHIEVABLE INFORMATION RATES
In this section, we describe the algorithm that is used for computing the achievable information rates of the MIMO ISI channels in detail, for both the deterministic and stochastic cases. We
also focus on several specific cases, including the multiantenna
systems over the frequency-flat fading, the multiaccess multiantenna systems, and the single-track systems with ITI.
(7)
where the entropy of the noise sequence can be easily computed
when the covariance matrix of the noise is known [18]. Therefore, the problem reduces to the computation or estimation of
the entropy of the output sequence. Clearly
(8)
that is, the entropy can be expressed as the expectation of the
logarithm of the joint probability of the output sequence. Since
the inputs are selected from a finite alphabet, we can generate
the channel output sequence by simulation, set up a trellis based
on the memory of the channel and the multiple inputs, then use
the forward recursion of the BCJR algorithm [12] to compute
the joint probability of the output sequences. We can estimate
the entropy by conducting many simulations of the channel and
averaging the logarithm of the joint probability estimates. However, for our model, the channel outputs are stationary ergodic
hidden-Markov processes, so the Shannon–McMillan–Breiman
ZHANG et al.: ACHIEVABLE INFORMATION RATES AND CODING FOR MIMO SYSTEMS
theorem holds [18], [20]. Thus, we can estimate the entropy by
conducting a single simulation with a very large block length,
which simplifies the computations considerably.
Let us explain in detail the estimation of the joint probability of the output sequence obtained from a channel simulation. Assume that the size of the finite input alphabet is .
Since there are inputs and the memory of the channel is ,
there are
states in the trellis describing the MIMO
ISI channel. We denote the trellis state at the time instance as
, and define
(9)
(10)
is short for
and
,
. Then we can compute
for every trellis
state by using the forward recursion of the BCJR algorithm,
as follows:
where
(11)
possible transitions or branches for every state,
There are
for these branches when i.u.d. inputs are
and
used. For other transitions,
. For a given state tranare indesition, the elements of
pendent Gaussian random variables, and the real and imaginary
parts of any output sample are also independent if they are complex. Therefore, for the complex outputs, we have
(12)
where the subscripts of and represent the real and imaginary
parts, respectively. Otherwise
(13)
and
.
Initially, we set
With this recursive computation, we update the metrics with the
forward processing of the trellis. At the final stage, the joint
, is given by the sum
probability of the outputs,
of the metrics over all trellis states
(14)
The procedure for computing the information rates of the
MIMO ISI channels is summarized as follows.
1) Generate the output sequences with a large length
through the simulated MIMO ISI channel, where the
inputs are chosen i.i.d. from a finite alphabet of size .
states and
valid
2) Set up the MIMO trellis with
state transitions.
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3) Set the initial values of the metrics
) and set the time instance
.
4) Compute the probabilities of the output samples conditioned on the trellis transition using (12) or (13).
5) Update the metrics using (11).
if
.
6) Repeat steps 4) and 5) with
and
7) Compute
by (14) and (8), respectively.
8) Finally, estimate the achievable information rate by (7).
We emphasize again that, for stochastic channels, although
we did not express the probabilities as conditional ones (conditioned on the channel-transfer function) for simplicity in the
above computations, it is important to note that they are known
to the receiver and we can treat them as constants at every stage
of the trellis processing.
For the nonergodic channels, the fading coefficients are
chosen randomly and independently, but, in contrast to the
ergodic case, they are fixed during the transmission of the entire
block. Thus, the Shannon type capacity is zero, as no matter
how small the transmission rate is, there is still a nonzero
probability that the instantaneous information rate supported
by the channel will be smaller. In this case, the capacity can
be considered as a random variable corresponding to the instantaneous channel fading coefficients, and the cumulative
distribution function (CDF) of the capacity can be obtained by
a large number of channel realizations. Then we can compute
the information rate versus outage probability, or the outage
information rate, as compared with the outage capacity defined
in [2] and [16] for the case without constrained signaling.
1) Information Rates With Markov Inputs: In the above discussion, we have assumed that i.i.d. inputs are employed at the
transmitter. But sometimes, the constrained inputs may be correlated. For example, the run-length limited (RLL) modulation
codes are generally used in recording channels in order to reduce the ISI and to provide easier synchronization [21]. With
these codes, the channel inputs are, in fact, Markov inputs with
the memory determined by the parameters of the RLL codes.
As stated before, the information rates for the MIMO ISI channels with any Markov input can also be computed by using the
simulation-based algorithm. To accomplish this, we need to use
a joint trellis that incorporates both the Markov inputs and the
MIMO ISI channel. Suppose that the maximum memory of the
Markov inputs is , then the memory of the new trellis becomes
, where
stands for the maximization. In addition, the transition probability
depends directly on the
distribution of the Markov inputs.
Furthermore, we can perform waterfilling (that is, the maximization of the mutual information over the distribution of the
Markov inputs) if the channel is known at the transmitter (as in
the multitrack recording systems). By taking the correlated inputs into consideration, we may achieve a higher information
rate than the case with i.i.d. inputs, especially in the low SNR
region. In [19], the information rates for the SISO ISI channels
are maximized over all Markov inputs with a certain memory
by optimizing the transition probabilities of the input source via
an iterative technique. This simulation-based algorithm is performed by using the forward and backward recursions of the
BCJR algorithm that operate on the trellis of the ISI channel.
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We can extend this method to the MIMO ISI channels as well by
considering the new trellis of the MIMO ISI channel describing
the channel and the Markov inputs jointly, with number of states
.
Similar to the SISO case [19], the information rate for the
MIMO ISI channel with constrained (Markov) inputs can be
equivalently expressed by
(
,
). Then, we compute
for
. In the second half of the iteration, we
the given set of
update the transition probabilities using (22). We repeat these
steps until the transition probabilities converge (or the information rate converges). By relating this with the Arimoto–Blahut
algorithm [22], [23], it is conjectured in [19] that this iterative
method will lead to a convergence to the maximized information rate, which is also supported by empirical evidence.
(15)
represents the steady-state probability satisfying
. Denoting
and
by
and
, respectively, the term
for
to can be written as
a valid transition from state
where
(16)
To compute
, in addition to the forward recursion of the
BCJR algorithm, we need the following backward recursion
[12]:
(17)
where we define
(18)
for
all trellis states
, and initialize
. Then, we obtain [12]
as
for
(19)
where
can be computed by (14). Similarly
(20)
or
(21)
can be computed by substituting these quantities
Then,
into (16).
in (15) for fixed
(
,
To maximize
), we define the noisy adjacency matrix with size of
, where
for all the valid transitions, and
for the others [19]. Suppose that the maximum real
and the corresponding eigenvector is
eigenvalue of is
, then the maximization of
in
(15) is achieved by
(22)
The iterative algorithm works as follows. In the initialization, we arbitrarily choose a valid set of transition probabilities
B. Special Case: Multiantenna Systems Over Frequency-Flat
Fading Channels
We now consider the information rates of the MIMO systems
over frequency-flat fading channels with inputs chosen from a
finite alphabet, which can be considered as a special case of
. The mutual information
the MIMO ISI channels with
between the input vector and output vector conditioned on the
channel-transfer function is expressed by
(23)
where we use
,
,
to
denote the input, output, and noise vectors, respectively
(instead of using the general denotations in (1) for simcan be easily obtained for a
plicity). Since
given noise covariance matrix, we only need to compute
. In [24] and
[25], numerical integration is used in the calculations of the
information rates over SISO flat-fading channels. We may
apply this approach to the MIMO channels, but the resulting
complexity is very high. For example, for a (2, 1) system,
where we ignore the subscripts for the receiver and express
, we need to compute
the output as
(the subscripts of and
represent real and imaginary parts, respectively). It is clear
that a six-fold integration is necessary for such a small-sized
multiantenna system. We can easily see that we need an
-fold integration for a general
system with this
, we can reapproach. Although in the special case when
duce this computation to a -fold integration by dephasing, or
even to a -fold one with the binary PSK (BPSK) modulation
by ignoring the imaginary parts, the computational burden to
obtain an accurate result is still very high.
Instead, we can also adopt the simulation-based method and
estimate the entropy of the output sequence conditioned on the
channel-transfer function in a much more efficient way. In this
case, we generate channel realizations, obtain output vectors, and estimate the expectation directly by
(24)
is the computed joint
with large , where
probability of the output vector for the th realization, given by
(25)
ZHANG et al.: ACHIEVABLE INFORMATION RATES AND CODING FOR MIMO SYSTEMS
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where is the value set of the input vector with size
. Given the inputs and fading coefficients, the probability of
,
the output vector can be computed by (12) and
assuming the i.u.d. inputs are used.
C. Multiaccess Multiantenna Systems
We can also apply the Monte Carlo simulation technique to
the case of multiantenna systems with multiple users, which is,
in fact, a more general setting including the single-user MIMO
system as a special case. We consider a multiple-access channel
consisting of users, each equipped with transmit antennas,
and a common receiver with antennas. We denote this setting
system. The output vector of the multiaccess system
as a
resolvable
over a frequency-selective fading channel with
paths is given by
(26)
Fig. 1.
where the subscript
can rewrite (26) as
Information rates of the (2, 2) systems over ergodic channels.
denotes the th user. Similar to (2), we
(27)
where
is a
vector, and
is an
matrix. Assume
that all the transmitted signals go through independent and idensystem takes the same channel
tical channels, then the
one. Therefore, we can apply
representation as the
the algorithm described above for the multiaccess multiantenna
systems in a straightforward manner.
We assume that the CSI is known only at the receivers.
In this case, the ergodic capacity region is obtained in [15],
where it is reported that the ergodic capacity region for a
two-user system is a pentagon, and all the boundary points are
achievable by i.i.d. Gaussian input signals with evenly allocated
transmit power. Here we consider the inputs chosen from a
finite alphabet and compute the information-rate region for such
channels with two users. The information-rate region under
the specific input constraint is the closure of the convex hull of
satisfying
,
all
and
, where
and
represent the
transmitted signals for users 1 and 2, respectively, and
represents the received signals. We assume that there is a
power constraint for each user (which can be identical for
and
can be
both). The boundaries
computed using the same method as in the case of single-user
multiantenna system, since the interference from the other
by using the
user is known. We can compute
achievable information rate for the
system when the
subchannels are i.i.d..
without loss of generality, we assume that the signal on the first
track is the desired signal. To compute the information rates
for this single-track system, we cannot use (7), since there is
interference in the received signal. However, the information
rate can be written as
(28)
where we drop the subscript for the output sequence since there
is only one read head, and
stands for the interference-plusnoise term given by
(29)
Therefore, we can consider an imaginary channel model with
and use the BCJR algorithm to estimate both terms
output
on the right-hand side (RHS) of (28) at the same time. There
is another approach obtained by applying the chain rule of
and
the mutual information [18]. Denoting
by
and , respectively, we rewrite (28)
by
(30)
where the two terms on the RHS can be computed by the inforand
systems, respectively,
mation rates for the
with the assumption that the signals transmitted through different tracks are independent.
IV. EXAMPLES OF ACHIEVABLE INFORMATION RATES
D. Single-Track Systems With ITI
A. Information Rates and Outage Information Rates of the
Ergodic and Nonergodic Multiantenna Systems
For comparison, we also consider the single-track channels,
which take the interference from adjacent tracks into account,
but have only one read head. The receiver detects the desired
signal from the corresponding track and treats the others as pure
interference, thus does not perform joint decoding. For this case,
We first consider the multiantenna wireless systems with
BPSK modulation. Fig. 1 shows the information rates of a (2,
2) system over the ergodic frequency-selective fading channels
and three
equal-energy taps,
with two
respectively. The channel fading is assumed to be independent
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Fig. 2. Outage capacity and information rates of the (2, 2) systems over
nonergodic channels (outage probability = 0:1).
Fig. 3.
from one symbol to the next. Also shown are the unconstrained capacity and information rates of the frequency-flat
. We observe that, as expected, the
fading channels
frequency-selective fading channel has a higher information
rate than the frequency-flat one. This is in contrast to the
deterministic ISI channels, where ISI typically degrades the
system performance. We can further increase the number of
independent taps to obtain higher frequency diversity. However, we know from the simulation results that the additional
improvement in terms of achievable information rates is only
marginal with increasing .
In Fig. 2, we present the outage capacities and information
rates for the (2, 2) system over both frequency-flat and frequency-selective fading channels with two equal-energy taps,
when the outage probability is 0.1. The outage-capacity result
is obtained by using the method in [5], where i.i.d. Gaussian inputs are used. Similarly, we observe that both the capacity and
information rates for the frequency-selective fading channel are
considerably higher than the ones for frequency-flat fading. In
addition, we observe that there is almost no difference between
the capacity and information rates in the low SNR region, while
there is a significant difference in the high SNR region.
show the linear increase of the information rates as well. These
results (which are specifically for the case with constrained signaling schemes) are new and analogous to the (Gaussian input)
capacity results known for MIMO flat-fading channels.
B. Effects of the Number of Antennas
With the simulation-based algorithm, we can easily compute
the information rates with any number of transmit/receive an, and show
tennas. In Fig. 3, we consider the case when
the achievable information rates as a function of the number
of antennas for the ergodic frequency-selective fading channel
with two balanced taps. The SNR shown is from 10 to 10 dB
in 4-dB increments. Again, we assume that the fading is independent from one symbol to the next. We observe that there is
a linear increase in the achievable information rates, just as in
the case of the unconstrained capacity over the frequency-flat
fading channels [2]. The major difference is that there is a rate
limit, i.e., bits per channel use, that cannot be exceeded by
increasing the SNR for the constrained case. We can easily obtain similar results for the frequency-flat fading channel, which
Information rates of the (t; t) system over frequency-selective fading.
C. Effects of the Spatial Correlation
In the above simulations, we assumed that the fading coefficients from different antennas are independent, which is a necessary condition to achieve full space diversity. But in reality, the
antennas may not be separated far enough, thus there may be significant correlations among different subchannels. Here we use
one simple example to illustrate the effect of the spatial correlation on the achievable information rates. Now we consider a (2,
1) system with BPSK over the ergodic frequency-flat Rayleigh
fading channels. For every symbol instance, there are two fading
coefficients, both of which are zero-mean complex Gaussian
random variables with independent real and imaginary parts. We
.
denote them by and , and assume
, with
We define the correlation coefficient by
denoting the complex conjugate. In Fig. 4, we give an example
of the information rates for different values of , together with
the information rates for the SISO system. We see that, even
, the SNR loss due to the spatial correlation is less
when
than 2 dB for all the transmission rates, and the highly correlated (2, 1) system can still offer a large capacity improvement
, the unconstrained
over a SISO system. Note that when
capacity for the fully correlated (2, 1) system is exactly the same
as the SISO system when the transmission power is normalized according to the number of transmit antennas. However,
the achievable information rates are different. In particular, the
asymptotical information rate with infinite SNR is 1.5 bits per
. Intuitively, this is because we can
channel use when
detect the two input bits when they are the same, 1 or 1,
otherwise, we can only tell that one is 1 and the other is 1.
Mathematically, the achievable information rate in this case is
just the entropy of the channel output, which takes three values
for a fixed fading coefficient and has the entropy of 1.5 bits if
i.u.d. binary inputs are used.
ZHANG et al.: ACHIEVABLE INFORMATION RATES AND CODING FOR MIMO SYSTEMS
Fig. 4. Information rates of the (2, 1) system with spatial correlation.
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Fig. 5.
Information rates of the (2, 2) Ricean fading channels.
Fig. 6.
Information-rate and capacity regions of the (2, 1, 2) system.
D. Information Rates Over Ricean Fading Channels
We now consider the channels with a dominant LOS component, thus we use the Ricean fading channel model instead.
is defined as
In this case, the channel-transfer function
, where is the random transfer function
for the MIMO Rayleigh fading channel as described before, and
is the deterministic matrix that represents the LOS com. That is, the elements
ponent, given by
are complex Gaussian random variables with
of the matrix
independent real and imaginary parts that, respectively, have the
and
, and have the common
mean
for all
and
variance, say . We assume
, and define the Ricean factor
as
with the normalization
. When
, there
is no LOS and the channel exhibits Rayleigh fading, and when
, there is no fading and we can consider the channel as
a Gaussian channel. It is well known that the LOS is helpful for
the system performance in a SISO system, but it is generally not
the case in MIMO systems. In Fig. 5, the information rates of
the (2, 2) system over ergodic Ricean fading are shown, where
for the all the elements. We observe that the
we set
LOS is not desirable in terms of achievable information rates,
i.e., rich scattering provides more diversity for MIMO systems,
in this case.
E. Information-Rate Region of the Multiaccess Multiantenna
Systems
In this subsection, we present an example for a two-user multiaccess channel, where each user is equipped with a single
transmit antenna and there are two antennas at the receiver. In
Fig. 6, the ergodic frequency-flat fading channel is considered,
and both the information-rate region and capacity region [15]
are shown with a total SNR of 0 and 10 dB, respectively, where
the two users have the same power constraint.
We observe that the information-rate region becomes almost a
rectangle (or even square, in the case of two symmetric users) for
high SNR values, which is not true for the capacity region. The
reason behind this result is that the information rate
almost equals
when the SNR is large, i.e., the
knowledge of the interference cannot help much to increase the
information about the desired signal. In the limiting case with
infinite SNR, the detection of the signals can be done with zero
error probability only if the signals from the two users experience different fading. But for the capacity with Gaussian inputs,
the knowledge of the interference always helps significantly,
even with a very large SNR. Similar observations can be made
for the frequency-selective fading channels.
F. Information Rates of the Multitrack and Single-Track
Systems
Now we turn our attention to the deterministic MIMO ISI
channels. We consider a (2, 2) multitrack PR4 system with
and
, i.e.,
(31)
In order to obtain a fair comparison with the single-track system
with ITI, we consider the information rates per track, that is, we
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 10, OCTOBER 2004
Fig. 7. Achievable information rates for the multitrack system (LHS: information rates and capacity bound. RHS: comparison with the single-track system).
compute
(in bits per track use), where
is
defined by (6). On the left-hand side (LHS) of Fig. 7, we show
the unconstrained capacity and the achievable information rates
0.2 and 0.5, respectively. The
for the (2, 2) system with
capacity is obtained by using i.i.d. Gaussian inputs, thus it is, in
fact, a lower bound. Following a similar approach as in [5], the
capacity bound is given by
(32)
where
is the identity matrix with size of 2
given by
2 and
is
(33)
This technique is an extension of the one introduced in [6]
for the SISO ISI channel. First, a hypothetical channel model
is set up, where the linear convolution of the ISI channel is
replaced by the circular convolution. Then discrete Fourier
transform (DFT) is applied to convert the channel into parallel
and independent memoryless subchannels. The capacity for
this hypothetical channel is thus obtained, which is shown
to be the same as the capacity for the original channel with
linear convolution.
In the same figure, we also show the maximized information
. We
rates over all the Markov inputs with a memory
observe that in the low-to-medium SNR region, the information
rate achieved by the optimized Markov inputs is significantly
larger than the one with i.i.d. inputs, and it is even larger than
the capacity lower bound. Particularly, to achieve a rate of 0.4
bits per track use, there is a gain of 2.6 dB by using optimized
. However,
Markov inputs over the i.i.d. inputs when
in the high SNR region, the i.i.d. inputs are almost optimum,
as we expect. In addition, there is a significant SNR loss in the
high-rate region when is increased from 0.2 to 0.5.
For comparison purposes, we consider the single-track
. The insystem with the transfer function
formation rates for the multitrack and single-track PR4 systems
are shown on the RHS of Fig. 7 with i.u.d. inputs and different
values of . We observe that the improvement in achievable
information rates by adopting the multitrack system is large,
especially for high ITI levels. For example, to achieve 16/17
, about 11.6 dB is needed for
bits per track use when
the single-track system, but only 6.0 dB is necessary for the
multitrack system.
V. TURBO CODING AND ITERATIVE DECODING
FOR THE MIMO ISI CHANNELS
In this section, we describe a turbo coding/decoding scheme
for the MIMO ISI channels, in order to assess how close the
practical coding schemes may be to the information-theoretical
limits. In the first part, we describe the setting in detail. Then we
derive the MAP detector for the MIMO ISI channels, using a (2,
2) system as an example. Finally, we present several examples
for the multiantenna systems over ergodic frequency-selective
fading channels, and for multitrack and single-track recording
systems.
A. Concatenated Coding for MIMO ISI Channels
Both serial and parallel concatenation schemes have been
used for the recording systems [26], [27]. In parallel concatenation, the information bits are first encoded using a turbo code,
i.e., two parallel concatenated convolutional codes (PCCC) separated by an interleaver, then the coded bits are interleaved and
transmitted over the ISI channel. In serial concatenation, instead
of a turbo code, a single convolutional code (SCC) is employed.
The decoder in both cases is an iterative decoder composed of
MAP decoders for the component codes and a MAP detector for
the ISI channel.
The turbo coding and decoding algorithms are also employed
in the multiantenna systems over frequency-flat fading channels
(e.g., [28]–[30]). In [30], the turbo-coded bits are transmitted
through a number of antennas, and the obtained signals from
several receive antennas are sent to the turbo decoder for joint
decoding. The log-likelihood ratios (LLR) of the coded bits are
ZHANG et al.: ACHIEVABLE INFORMATION RATES AND CODING FOR MIMO SYSTEMS
Fig. 8.
Block diagram of the turbo coding scheme for MIMO ISI systems.
first computed based on the channel observations, and then they
are used in the turbo decoder. Turbo equalization by using an
iterative demodulation-decoding algorithm is also performed.
We now extend these techniques to the case of MIMO
ISI channels by designing the corresponding channel MAP
detectors for several scenarios, including stochastic channels
(multiantenna wireless channels) and deterministic channels
(multitrack and single-track recording channels).
The block diagrams of the transmitter and receiver are shown
in Fig. 8. At the transmitter, we first encode the message bit sequence, denoted by , by using an outer encoder, such as the
SCC or turbo code. After being passed through a random interleaver, the coded bits, represented by , are divided evenly into
groups, which are sent through the transmitters or tracks. In the
simulations, we will use the (2, 2) system as an example, where
we choose the odd indexed bits for one group and the even ones
for the other. The output sequences of the (deterministic or
stochastic) MIMO ISI channels, corrupted by the AWGN, constitute the received signal . At the receiver, the turbo equalization [31] is used, where a modified channel MAP detector
takes the channel outputs and the extrinsic information fed back
from the outer decoder as its inputs, and generates the soft information about the coded bits. This soft information is deinterleaved and passed to the outer decoder. The outer decoder, i.e.,
a turbo decoder or a MAP decoder for a SCC, generates the extrinsic information, which is then fed back to the channel MAP
detector after appropriate processing for the next iteration step.
The LLRs of the message bits are used to make hard decisions
after a number of iterations.
B. MAP Detector for the MIMO ISI Channels
We use a binary-input (2, 2) MIMO system with ISI memory
as an example to illustrate the necessary modifications
of
in the MAP algorithm. First of all, the channel trellis is set
up according to the multiple inputs and multiple outputs. The
states, and, in fact, it is the same as the one
trellis has
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used in the computation of the information rates. Suppose that
and
are the coded bits with
block lengths of transmitted through the (2, 2) ISI channel,
and the output vector is
.
Thus, the LLRs of the two coded bits at the time instance
can be computed as shown in (34) and (35) at
is the trellis state at the time
the bottom of the page, where
instance , as defined in Section III,
is short for
, and
is the set of valid state
and
(
,
transitions corresponding to
). We observe that there are four common terms in (34)
and (35), which can be computed by the BCJR algorithm operating on the trellis of the MIMO ISI channel. Similar to the
algorithm for computing the information rates, we have
(36)
,
, and
are defined as in (9), (10),
where
and (18), respectively.
Particularly, for a specific trellis state transition correand
, we rewrite
as
sponding to
(37)
(38)
where (37) holds because the two observations are independent
for a given state transition, and the approximation (38) follows
due to the use of the interleaver. Then, the a priori information
fed back from the outer decoder can be used to update
and
in the iterative decoding.
We also use this MAP detector and the coding/decoding
scheme for the single-track system, where there is only one
received signal and one desired transmitted signal, and no
detection is performed for other transmitted signals, which
are considered as pure interferences. In this case, we set up a
multiple-input single-output (MISO) trellis and follow a similar
procedure as the one described above. Since we have only one
output and there is no a priori information about the other input
bit in any iteration, we modify (38) by
(39)
(34)
(35)
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 10, OCTOBER 2004
Fig. 9. Performance of the SCC scheme for the (2, 2) multiantenna system.
Fig. 10. Performance of the SCC scheme for the multitrack and single-track
PR4 channels.
where is a constant corresponding to the fixed a priori information for the interfering input. We usually assume
, therefore can be omitted in the
computation. We only compute the LLRs of the desired information bits by (34).
When the channel response is random and known to the receiver, we just use the channel coefficients in the decoding, as
they are constant during every symbol duration. That is, in the
, the probability of the output samcomputation of
ples for a given trellis transition is conditioned on the random
channel response, just as the case of the computation of the information rates.
C. Examples
We employ this coding/decoding scheme in the multiantenna
system over the frequency-selective fading channels, where we
use the SCC as the outer encoder with generators of (33/31) in
(the overall rate is
octal form and a code rate of
). The block length of the input sequence to the outer encoder is 10 016 and 15 iterations are used. In Fig. 9, we consider
the ergodic frequency-selective channels with two balanced taps
and three balanced taps
, respectively. We assume that the channel fades are i.i.d. from one symbol to the
, where
next. We employ a precoder, defined by
indicates the modulo-2 addition and
denotes the delay operator. The performance of the uncoded system is obtained by
passing the uncoded and nonprecoded information bits through
the channel and using the channel MAP detector to obtain the
decisions. The BER for the coded system is plotted in terms of
, with SNR given in
the SNR per bit, which is defined as SNR
(3). As expected, the performance over the channel with larger
frequency diversity is better. We also see that if we do not use
the precoder in this case, we cannot observe the waterfall property of the BER curves. For the precoded system, the required
for the case with
is
SNR per bit to achieve a BER of
about 6.8 dB, while the theoretical limit shown in Fig. 1 is about
dB
5.9 dB per bit (where the rate loss of
is considered). Therefore, the gap between the theoretical limit
and the performance achieved by this practical coding/decoding
as reliable
scheme is about 0.9 dB, if we consider BER of
transmission. Similarly, the gap between these two values for
is also about 0.9 dB. We observe that the
the case with
SNR gain of the three-tap system over the two-tap one is about
0.6 dB, which is consistent with the information-theoretical results shown in Fig. 1. Furthermore, compared with the uncoded
system, a coding gain of about 8–9 dB is achieved. Since we are
using an ergodic channel, the diversity order is mostly dependent on the minimum distance of the turbo code or the convolutional code, and that is the reason we observe a very sharp slope
for the BER curves. We conducted extensive simulations with
other codes of different rates, and observed that a performance
of about 1.0 dB away from the theoretical limit is typical.
In Fig. 10, we show the BERs of this coding/decoding
scheme over the multitrack and single-track systems with
different values of . We employ the SCC as the outer code
as well, and the same parameters are used as in the previous
is employed. Compared
example. A precoder of
with Fig. 7, we are only about 0.7–1.2 dB (the rate loss is
considered as well) away from the theoretical limits for all the
cases. We also observe a great improvement of the multitrack
systems over the single-track systems with the same level of
ITI, which is again in line with the information-theoretical
results. Although no results are shown here, we have also used
the PCCC as the outer encoder, and found that the SCC achieves
a better performance than the PCCC for both the stochastic and
deterministic MIMO ISI channels.
VI. CONCLUSIONS
In this paper, we have computed the achievable information
rates for the general MIMO ISI channels with the constraint that
the inputs are chosen from a finite alphabet. The methods developed are applicable to both deterministic and stochastic channels. As an example of the stochastic MIMO ISI channels, we
have considered the multiantenna system over frequency-selective fading channels and have computed the information rates
ZHANG et al.: ACHIEVABLE INFORMATION RATES AND CODING FOR MIMO SYSTEMS
when the CSI is known to the receiver only. We have quantified
the improvement in the achievable information rates provided
by the additional diversity over the flat-fading channels for both
ergodic and nonergodic cases. In addition, we have discussed
the effects of the number of antennas, the spatial correlation of
the fading coefficients, and the existence of the LOS signals on
the achievable information rates with the signaling constraints.
This setting also includes as special cases the single-antenna
systems over frequency-selective fading channels, and the multiantenna systems over flat-fading channels. We have also applied this technique to compute the information-rate region of
the multiaccess multiantenna systems with inputs from a specific constellation. As for the deterministic MIMO ISI channels,
we have used the multitrack magnetic recording systems, which
employ multiple write and read heads, as examples. Both information rates with i.i.d. inputs and optimized Markov inputs have
been considered. Our results show that the multitrack recording
channels have significant advantages over the single-track channels, in terms of the achievable information rates when the ITI
is considered.
The “constrained” capacity results are important for practical
communication systems, since they can be used to compare the
performance of specific coding schemes with the informationtheoretical limits. In this paper, we have also described a
turbo coding/decoding scheme for the MIMO ISI channels
with a MAP detector developed for such channels, and we
have demonstrated that a performance of about 1.0 dB away
from the information-theoretical limits can be achieved.
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Zheng Zhang (S’00) received the B.E. degree with
honors from Nanjing University of Aeronautics and
Astronautics, Nanjing, China, in 1997, and the M.S.
degree from Tsinghua University, Beijing, China, in
2000, both in electronic engineering. Currently, he is
working toward the Ph.D. degree in electrical engineering at Arizona State University, Tempe.
His current research interests are in digital communications, wireless and mobile communications, information theory, channel capacity, channel coding,
turbo codes, LDPC codes, MIMO systems, and relay
channels.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 10, OCTOBER 2004
Tolga M. Duman (S’96–M’98–SM’03) received the
B.S. degree from Bilkent University, Ankara, Turkey,
in 1993, and the M.S. and Ph.D. degrees from Northeastern University, Boston, MA, in 1995 and 1998,
respectively, all in electrical engineering.
Since August 1998, he has been with the Electrical
Engineering Department, Arizona State University,
Tempe, first as an Assistant Professor (1998–2004),
and currently as an Associate Professor. His current
research interests are in digital communications,
wireless and mobile communications, channel
coding, turbo codes, coding for recording channels, and coding for wireless
communications.
Dr. Duman is the recipient of the National Science Foundation CAREER
Award, the IEEE Third Millennium medal, and the IEEE Benelux Joint Chapter
best paper award (1999). He is an editor for the IEEE TRANSACTIONS ON
WIRELESS COMMUNICATIONS.
Erozan M. Kurtas (M’98) received the B.Sc. degree from Bilkent University,
Ankara, Turkey, in 1991, and the M.Sc. and Ph.D. degrees from Northeastern
University, Boston, MA, in 1993 and 1997, respectively.
He is currently the Research Director of the Channels Department in
the Research Division of Seagate Technology, Pittsburgh, PA. His research
interests cover the general field of digital communication and information
theory with special emphasis on coding and detection for intersymbol
interference channels. He has published over 75 book chapters, and journal
and conference papers on the general fields of information theory, digital
communications, and data storage. He is the co-editor of the book Coding
and Signal Processing for Magnetic Recording System (Boca Raton, FL:
CRC Press, 2004). He has 11 pending patent applications.