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Hindawi Publishing CorporationEURASIP Journal on Advances in
Signal ProcessingVolume 2008, Article ID 158037, 14
pagesdoi:10.1155/2008/158037
Research ArticleA Forward-Backward Kalman Filter-based STBC
MIMOOFDM Receiver
T. Y. Al-Naffouri and A. A. Quadeer
Electrical Engineering Department, King Fahd University of
Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Correspondence should be addressed to A. A. Quadeer,
[email protected]
Received 5 May 2008; Accepted 11 December 2008
Recommended by Kostas Berberidis
Orthogonal frequency division multiplexing (OFDM) has emerged as
a modulation scheme that can achieve high data ratesover frequency
selective fading channel by efficiently handling multipath effects.
This paper proposes receiver design for space-time block coded MIMO
OFDM transmission over frequency selective time-variant channels.
Joint channel and data recovery areperformed at the receiver by
utilizing the expectation-maximization (EM) algorithm. It makes
collective use of the data constraints(pilots, cyclic prefix, the
finite alphabet constraint, and space-time block coding) and
channel constraints (finite delay spread,frequency and time
correlation, and transmit and receive correlation) to implement an
effective receiver. The channel estimationpart of the receiver
boils down to an EM-based forward-backward Kalman filter. A
forward-only Kalman filter is also proposed toavoid the latency
involved in estimation. Simulation results show that the proposed
receiver outperforms other least-squares-basediterative
receivers.
Copyright © 2008 T. Y. Al-Naffouri and A. A. Quadeer. This is an
open access article distributed under the Creative
CommonsAttribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original
work isproperly cited.
1. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is atechnique
that enables high speed transmission over fre-quency selective
channels with simple equalizers. It doesthis by creating a set of
parallel frequency-flat channelsover which large constellation
signals can be transmitted.For frequency flat fading channels,
space-time codes providediversity and coding gain benefits compared
with single-input single-output (SISO) systems improving the BER
per-formance of the system [1]. (We concentrate in this work
onspace-frequency codes. We note however that a very
similarapproach can be used for space-time block codes (STBC).Given
the similarity between the two approaches and the factthat the
abbreviation STBC is more familiar, we continue touse this
abbreviation to refer to our frequency-space codes.)When multiple
antennas (MIMO) are combined with OFDM,space-time codes can be used
per tone, providing the benefitof multiple antennas with simple
channel equalization.
An OFDM receiver needs channel state information (CSI)to detect
the data. With no CSI at the receiver, boththe channel and the data
are unknowns that have to berecovered. The estimation process can
be carried out jointly
or separately. Techniques for channel estimation fall into
3distinct classes: (1) training/pilot-based, (2) semiblind, and(3)
blind methods. Training/pilot-based methods estimatethe channel
from a known preamble/pilot sequence sent atthe transmitter and use
the estimated channel to decode thedata [2–7]. Blind methods do not
use any preambles/pilotsbut rely instead on a priori constraints to
recover the channeland data. The data constraints include
redundancy due tothe cyclic prefix [8, 9], nonredundant, and
nonconstant-modulus precoding [10], constant-modulus
modulation[11], subspace-based constraints [12], and the finite
alphabetconstraint [13]. Semiblind methods make use of both
pilotsand additional channel/input data constraints to
performchannel identification and equalization. These methods
usepilots to obtain an initial channel estimate and improve
theestimate by using a variety of a priori information. Thus,in
addition to the aforementioned constraints, semiblindmethods use
the time and frequency correlation [14] and thesubspace of the
channel [15]. The channel estimate can alsobe improved iteratively
in a data-aided fashion [12], or morerigourously by the expectation
maximization (EM) approach[16–21].
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2 EURASIP Journal on Advances in Signal Processing
1.1. Paper contributions
This paper considers receiver design for OSTBC-OFDM
trans-mission over a frequency selective, time-variant channel.
Wepropose a semiblind iterative receiver using the EM algorithmfor
joint channel and data recovery. The main contributionsof the work
are summarized below.
(1) We make a collective use of the structure of
thecommunication problem (i.e., the constraints on thedata and on
the channel) in a transparent manner.The data constraints include
the finite alphabetconstraint [13], the cyclic prefix [8], pilots
[6, 7], andthe OSTBC. In addition, the receiver uses the
followingconstraints on the channel: the finite delay
spread,frequency and time correlation [22–24], and
spatialcorrelation (it is also straightforward to
incorporatechannel sparsity [15, 25, 26]). This is in contrastto
the work done by other researchers as none hasused all the above
mentioned constraints in such acollective and transparent
manner.
(2) The channel estimation and data detection as well asthe
exploitation of the system constraints is done in asemioptimal
manner through the EM algorithm. Thisguarantees a simple receiver
structure.
(3) In spite of the complexity of the problem that weaddress
here and the many constraints we incorpo-rate, our algorithm
maintains its transparency:
(a) the maximization step is used for channelestimation and
makes use of the channelconstraints by employing a
forward-backwardKalman filter.
(b) the expectation step is used for data detectionand makes use
of the data constraints.
(4) In contrast to prior approaches, we take a channelestimation
centric viewpoint and reverse the rolesof the channel and the
transmitted signal in ourimplementation of the EM algorithm [19]
using theestimation step for data detection and maximizationstep
for channel estimation. We do so because it isdifficult to employ
the expectation step for channelestimation while simultaneously
incorporating thevarious constraints on the channel (most notably,
thetime-correlation constraint).
A similar algorithm was proposed by the first authorin [27] for
the SISO case. Extending this algorithm to theMIMO case is
nontrivial. For in addition to the scale up inthe number of
transmit and receive antennas, this paper (1)makes use of the
space-time code and (2) makes full use ofthe frequency and time
correlation as well as transmit andreceive spatial correlation (see
Section 2.2 and Appendix Awhere we derive the channel model).
Moreover, and in spiteof the many dimensions we deal with, we
maintain thetransparency of the presentation.
1.2. Paper organization
The paper is organized as follows. After introducing
ournotation, we give an overview of the transceiver in Section
2.Section 3 then derives the input/output (I/O) equationsfor
MIMO-OFDM with ST coding (the equations are neededfor channel and
data recovery). Channel estimation usingthe forward-backward (FB)
Kalman filter is derived inSection 4 at the end of which we
summarize the transceiveralgorithm with an extension/modification
of the algorithm.Section 5 presents our simulations and Section 6
presentsour conclusions.
1.3. Notation
Proper choice of notation is essential for clarity and
con-sistency. One challenge in choosing notation is the
manydimensions we deal with in this paper including sample
time,(frequency) tone, (channel) tap, (OFDM) symbol index,
andspace.
In this paper, we denote scalars with small-case letters(e.g.,
x), vectors with small-case boldface letters (e.g., x),and matrices
with uppercase boldface letters (e.g., X).Calligraphic notation
(e.g., X) is reserved for vectors in thefrequency domain. A hat
over a variable indicates an estimate
of the variable (e.g., ̂h is an estimate of h). We use∗ to
denoteconjugate transpose, ⊗ to denote Kronecker product, IN
todenote the size N × N identity matrix, and 0M×N to denotethe all
zero M × N matrix. Given a sequence of vectors htxrxfor rx = 1 · ·
·Rx and tx = 1 · · ·Tx, we define the followingstack variables:
hrx =
⎡
⎢
⎢
⎢
⎣
h1rx...
hTxrx
⎤
⎥
⎥
⎥
⎦
, h =
⎡
⎢
⎢
⎣
h1...
hRx
⎤
⎥
⎥
⎦
. (1)
The notation vec(X) denotes a column vector consistingof the
concatenation of all column vectors of X while theoperation diag(X)
transforms the vector X into a matrixwith diagonal X .
2. AN OVERVIEWOF THE COMMUNICATION SYSTEM
In this section, we give an overview of the
communicationssystem: transmitter, channel, and receiver.
2.1. Transmitter
A block diagram of the transmitter is shown in Figure 1.The bit
sequence to be transmitted passes through aconvolutional encoder
that serves as an outer code for thesystem. The coded output is
punctured to increase the coderate. The punctured sequence then
passes through a randominterleaver which rearranges the order of
the bits accordingto a random permutation. The interleaved bit
sequence ismapped to QAM symbols (or any modulation scheme forthat
matter) using Gray coding and the QAM symbols are inturn mapped to
the OFDM symbols with some tones reservedfor the pilot symbols. The
STBC encoder uses the OFDM
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T. Y. Al-Naffouri and A. A. Quadeer 3
Pilot insert.
STBC encoder
IFFT
IFFT
IFFT
Cyclic prefix
Cyclic prefix
Cyclic prefix
PuncturerEncoder Interleaver Modulator
Figure 1: OSTBC OFDM transmitter.
symbols to construct the ST block by mapping the variousOFDM
symbols to a specific antenna and specific time slotdepending on
the ST code used. Each antenna performs anIFFT operation on the
OFDM symbols to produce the time-domain OFDM symbols and adds a
cyclic prefix to each priorto transmission.
2.2. Channel model
We consider a time-variant and frequency selective MIMOchannel.
For a general MIMO system, the I/O time-domainrelationship is
described by
y(m) =P∑
p=0H(p)x(m− p), (2)
where H(p) is the Rx × Tx MIMO impulse response at tapp and
where m represents the sample time index. The tapsH(p) usually
incorporate the effect of the transmit filter andthe effects of the
transmit and receive correlation makingH(p) correlated across space
and tap. We will assume forsimplicity that H(p) is iid for all p
(In Appendix A weconsider the general case where the channel
exhibits transmitand receive correlation). We will also assume that
the tapH(p) remains constant over a single ST block (and henceover
the constituent OFDM symbols) and changes from thecurrent block
(Ht(p)) to the next (Ht+1(p)) according tothe dynamical equation
(We will at times suppress the timedependence for notational
convenience.)
Ht+1(p) = α(p)Ht(p) +√
(1− α2(p))e−βpUt(p). (3)
Here, Ut(p) is an iid matrix with entries that are N(0, 1),and
α(p) is related to the Doppler frequency fD(p) byα(p) = J0(2π
fD(p)T), whereT is the time duration of one STblock. The variable β
in (3) corresponds to the exponent of
the channel decay profile while the factor√
(1− α2(p))e−βpensures that each link maintains the exponential
decayprofile (e−βp) for all time.
This channel model allows the channel to be time variant(as the
channel can change arbitrarily from one ST blockto the next) while
avoiding intercarrier interference andensuring the proper operation
of the space-time code (asthe channel remains fixed over any one
OFDM symbol or
ST block). This model was adopted in [23, 28, 29] in anSISO
context. (This model is based on approximating thenonrational Jakes
model by a first-order AR model. It is alsopossible to have a
better approximation by employing higher-order AR models but this
would increase the latency of thereceiver.)
In this paper, we scale up the model to the MIMO caseand also
show how to incorporate transmit and receivecorrelation in Appendix
A (see [27, 30] where we alsoincorporate the effect of the receive
filter).
Using this dynamical model, we can obtain the state-space model
for the impulse response htxrx between transmitantenna tx and
receive antenna rx. From (3), we can write
htxrx ,t+1(p) = α(p)htxrx ,t(p) +
√
(
1− α2(p))e−βputxrx ,t(p). (4)
By stacking (4) over the taps p = 0, 1, . . . ,P, we obtain
thedynamical model
htxrx ,t+1 = Fhtxrx ,t + Gu
txrx ,t, (5)
where
F =
⎡
⎢
⎢
⎣
α(0). . .
α(P)
⎤
⎥
⎥
⎦
,
G =
⎡
⎢
⎢
⎢
⎣
√
1− α2(0). . .
√
(
1− α2(P))e−βP
⎤
⎥
⎥
⎥
⎦
.
(6)
By further stacking (5) over all transmit and receiveantennas
(refer to our stacking notation in (1)), we obtain
ht+1 =(
ITxRx ⊗ F)
ht +(
ITxRx ⊗G)
ut, (7)
where ht+1,ht, and ut , are vectors of size TxRx(P+1)×1.
Thedynamical equation (7) shows explicit dependence on
thespace-time index t (so t = 1 for the first space-time
symbolwhich consists of two OFDM symbols in the Alamouti case,t = 2
for the second space-time symbol, etc.).
For complete characterization of the dynamical model,we need to
specify the covariance of ut, and also the
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4 EURASIP Journal on Advances in Signal Processing
covariance of the channel at the first time instant. It is
easyto show that
E[
utu∗t] = IRx ⊗ E
[
urxu∗rx
]
= IRx ⊗(
ITx ⊗ E[
utxrxutx∗rx
])
= IRx ⊗ ITx ⊗ IP+1 = ITxRx(P+1).(8)
We can similarly show that the channel covariance at the
firsttime instant is given by
E[
h0h∗0] = ITxRx ⊗GG∗. (9)
The covariance information is important for employing theKalman
filter which is used for channel estimation. We finallynote that
while (3) and (7) are equivalent, the latter modelis in vector form
and hence lends itself more to the Kalmanfilter operations, which
are used for channel estimation.
A note about time variation
One drawback of the approach in this paper is the blockfading
model that we adopt. For it is more realistic to assumethat the
channel continuously varies with time. There are 3justifications
for using the block fading model.
(1) While it is more realistic to assume that the
channelcontinuously varies over the OFDM symbol, the modelwe assume
is more valid than the block-fading modelthat is widely used in
literature. For in the block-fading model, it is usually assumed
that the channelremains constant over any one symbol and
variesindependently from one symbol to another. Here, weaccount for
the time correlation across symbols.
(2) The purpose of this paper is to design an algorithmthat
makes a collective use of the underlying structureof the
communication problem to lower the trainingoverhead required in the
time-variant case. Solvingthe general time-variant case is a future
researchproblem that builds upon the findings in this
paper[31].
(3) One could still envision applications where thechannel is
constant over an ST block, but variessubstantially from one symbol
to the next. Consider,for example, a multiuser application in which
thewireless channel is time shared. Imagine also thatthe channel is
very slowly time variant but the dutycycle is very large. In that
case, the channel that eachuser experiences during his transmission
burst is veryslow, but from one burst to another, the channelwould
change substantially due to the long duty cycle.This situation
would also make sense in randomaccess scenarios.
2.3. Receiver
This paper is concerned with designing a receiver for thesystem
described above. For completeness and as an allusionto the
developments further ahead, Figure 2 shows a blockdiagram of the
proposed receiver. As we will show, thereceiver’s core operation is
based on the EM algorithm whichperforms joint channel and data
recovery.
2.3.1. STBC decoder/data detector (estimation step)
The STBC decoder/data detector calculates the conditionalfirst
and second moments of the transmitted data (softestimate) to be
used by the channel estimator.
2.3.2. channel estimator (maximization step)
Pilots are used to initialize channel estimation. The
channelestimator then uses the soft data estimates together withthe
data and channel constraints to improve the channelestimate. These
two processes (channel estimation and datadetection) go on
iteratively until a stopping criterion issatisfied.
3. INPUT/OUTPUT EQUATIONS FOR MIMO-OFDM
As pointed out in Section 2.3, the receiver performs
twooperations, channel estimation and data detection. As such,we
need to derive two forms of the I/O equations: one thatlends itself
to channel estimation (i.e., treats the channelimpulse response as
the unknown) and a dual version thatlends itself to data detection
(i.e., treats the input in itsuncoded form as the unknown). To this
end, let Xtx bethe OFDM symbol transmitted through antenna tx which
firstundergoes an IDFT xtx = 1/NQXtx where Q is the N ×N IDFT
matrix. The system then appends a cyclic prefixbefore transmission.
At the receiver end, the receiver stripsthe cyclic prefix to obtain
the time domain symbol ytxrx . TheI/O equation of the OFDM system
between transmit antennatx and receive antenna rx is best described
in the frequencydomain
Ytxrx = diag(Xtx )Q∗P+1htxrx + Nrx , (10)where Ytxrx ,Xtx ,H
txrx , and N
txrx are the (length-N) DFTs of
ytxrx , xtx ,htxrx ,nrx , respectively, and where (10) follows
from the
fact that
H txrx = Q∗[
htxrxO(N−P−1)×1
]
= Q∗P+1htxrx . (11)
Here, QP+1 represents the first P + 1 rows of Q. Bysuperposition
and using the stacking notation (1), we canexpress the I/O equation
at receive antenna rx as
Yrx =[
diag(
X1) · · ·diag(XTx
)](
ITx ⊗Q∗P+1)
hrx + Nrx .(12)
3.1. I/O equations: channel estimation version
Consider a set of Nu uncoded OFDM symbols {S(1), . . . ,S(Nu)}
which we would like to transmit over Tx anten-nas and Nc time
slots. Following [32], we can per-form ST coding using the set of
Tx × Nc matrices{A(1),B(1), . . . ,A(Nu),B(Nu)} which characterizes
the STcode. We can now show that the OFDM symbol transmittedfrom
antenna tx at time nc is given by [32]:
Xtx(
nc) =
Nu∑
nu=1atx ,nc
(
nu)
ReS(
nu)
+ jbtx ,nc(
nu)
ImS(
nu)
,
(13)
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T. Y. Al-Naffouri and A. A. Quadeer 5
Cyclic prefix
removal
Cyclic prefix
removal
Cyclic prefix
removal
STBC decoder
Pilot removal
De-modulator
De-interleaver
De-puncturer Decoder
Initial channel
estimatorFFT
FFT
FFT Channel estimator
E[X]E[X2]
hest
hest, final
hest, initial
Figure 2: OSTBC OFDM receiver.
where atx ,nc(nu) is the (tx,nc) element of A(nu) and btx
,nc(nu)is the (tx,nc) element of B(nu). Thus, in the presence of
STcoding, (12) reads
Yrx(
nc) = [diag(X1
(
nc)) · · ·diag(XTx
(
nc))]
× (ITx ⊗Q∗P+1)
hrx + Nrx(
nc)
.(14)
(It should be clear that by using an ST code, a diversity
ofTx×Rx is achieved by spreading each symbol in time and spaceat
the transmitter antenna. The symbols are not spread infrequency and
thus, (multipath) diversity is lost at the priceof decoding
simplicity (i.e., bin by bin decoupling).) Thisrepresents the I/O
equation at antenna rx at OFDM symbol ncof an ST block. Collecting
this equation for all such symbolsyields
Yrx = Xhrx + Nrx , (15)where
Yrx =
⎡
⎢
⎢
⎣
Yrx (1)...
Yrx (Nc)
⎤
⎥
⎥
⎦
,
X =
⎡
⎢
⎢
⎢
⎢
⎣
{
diag(
X1(1)) · · ·diag(XTx (1)
)}(
ITx ⊗Q∗P+1)
{
diag(
X1(2)) · · ·diag(XTx (2)
)}(
ITx ⊗Q∗P+1)
...{
diag(
X1(
Nc)) · · ·diag(XTx
(
Nc))}(
ITx ⊗Q∗P+1)
⎤
⎥
⎥
⎥
⎥
⎦
.
(16)
Now, by further collecting this relationship over all
receiveantennas, we obtain
Yt =(
IRx ⊗Xt)
ht + Nt . (17)
This equation captures the I/O relationship at all
frequencybins, for all input and output antennas, and for all
OFDMsymbols of the tth ST block.
To perform initial channel estimation, we select thoseequations
where the pilots are present. Let Ip denote theindex set of the
pilots bins. Then, the pilot/output equationtakes the form
YtIp =(
IRx ⊗XtIp)
ht + NtIp . (18)
As can be seen, (17) and (18) are quite similar and using themin
a Kalman filter context will be similar as well.
3.2. I/O equations: data detection version
Signal detection in ST-coded OFDM is done on a tone-by-tonebasis
(as in SISO OFDM), except that the tones are collectedfor the whole
ST block (i.e., for Rx receive antennas and overNc time slots).
From (10), we can construct the following I/Oequation at any tone n
belonging to the OFDM symbol nc:
Yrx(
nc) =
[
H1rx · · · HTxrx
]
⎡
⎢
⎢
⎣
X1(
nc)
...XTx
(
nc)
⎤
⎥
⎥
⎦
+ Nrx(
nc)
.
(19)
We suppress the dependence on n for notational conve-nience.
Collecting this relationship for all receive antennasyields
⎡
⎢
⎢
⎣
Y1(
nc)
...YRx
(
nc)
⎤
⎥
⎥
⎦
=
⎡
⎢
⎢
⎣
H11 · · · HTx1... · · ·
...H1Rx · · · H
TxRx
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
X1(
nc)
...XTx
(
nc)
⎤
⎥
⎥
⎦
+
⎡
⎢
⎢
⎣
N1(
nc)
...NRx
(
nc)
⎤
⎥
⎥
⎦
,
(20)
or, more succinctly,
Y(nc) = HX(nc) + N (nc). (21)
By further concatenating this relationship for nc = 1, . . .
,Nc,we can show that the following relationship holds (see
[32]):
Y = C[
ReSImS
]
+ N , (22)
where
Y =
⎡
⎢
⎢
⎣
Y(1)...
Y(Nc)
⎤
⎥
⎥
⎦
, S =
⎡
⎢
⎢
⎣
S(1)...
S(Nu)
⎤
⎥
⎥
⎦
, C =[
Ca Cb]
,
(23)
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6 EURASIP Journal on Advances in Signal Processing
with Ca = [ vec(HA(1)) · · · vec(HA(Nu)) ] and Cb =[ vec(HB(1))
· · · vec(HB(Nu)) ]. We finally note that theSTBC code is
orthogonal if and only if the matrix C satisfies[32]
Re[C∗C] = ‖H‖2I2Nu ∀H. (24)
This property is essential to perform data detection. Westress
that the relationships (19) through (24) apply at aparticular tone
n and that this dependence has been omittedfor notational
convenience.
4. JOINT CHANNEL ANDDATA ESTIMATION:AN EM APPROACH
4.1. The EM algorithm
Consider the I/O equation (17) reproduced here for
conve-nience
Yt =(
IRx ⊗Xt)
ht + Nt . (25)
Ideally, we estimate ht by maximizing some
log-likelihoodfunction, for example,
̂hMAPt = maxht
ln p(
Yt | Xt,ht)
+ ln p(
ht)
. (26)
In our case, however, the input Xt is not available. Thus, weuse
the EM algorithm and maximize instead an averaged formof the
log-likelihood function. Specifically, starting from an
initial estimate ̂h(0)t , the estimate ̂ht is calculated
iteratively,with the estimate at the jth iteration given by
̂h( j)t = arg max
htEXt|Yt ,̂h( j−1)t ln p
(
Yt | Xt,ht)
+ ln p(
ht)
.
(27)
For example, when the system obeys the I/O relationship (25)and
ht isN (0,Π), the EM-based estimate (at the jth iteration)is given
by
̂h( j)t = arg min
ht
∥
∥Yt −(
IRx ⊗ E[
Xt])
ht∥
∥
21/σ2n
+∥
∥ht∥
∥
2I⊗Cov[X∗t ] +
∥
∥ht∥
∥
2Π−1,
(28)
where the two moments of Xt are taken given the output Ytand the
most recent channel estimate h
( j−1)t . (The weighted
norm notation ‖h‖2A stands for h∗Ah.) We now derive the
EMalgorithm for the time-variant case.
4.2. Channel estimation: an EM-basedforward-backward Kalman
Consider the OFDM system of this paper, essentially describedby
the state-space model
ht+1 =(
ITxRx ⊗ F)
ht +(
ITxRx ⊗G)
ut , (29)
Yt =(
IRx ⊗Xt)
ht + Nt, (30)
with h0 ∼ N (0,Π) and ut ∼ N (0,Ru). Given a sequenceof T + 1
input and output ST symbols XT0 and Y
T0 , (we use
XT0 to denote the sequence X0,X1, . . . ,XT) we obtain the
MAPestimate of the channel sequence hT0 by maximizing the
log-likelihood
L = ln p(YT0 | XT0 ,hT0)
+ ln p(
hT0)
. (31)
Now, using (30), we can express the first term of the
log-likelihood (up to some additive constant) as
ln p(
YT0 | XT0 ,hT0) =
T∑
t=0ln p
(
Yt |Xt,ht)
= −T∑
t=0
∥
∥Yt −(
IRx ⊗Xt)
ht∥
∥
21/σ2n
.
(32)
Similarly, using (29), we can express the second term of
(31)(again up to some additive constant) as
ln p(
hT0) =
T∑
t=1ln p
(
ht | ht−1)
+ ln p(
h0)
= −T∑
t=1
∥
∥ht −(
ITxRx ⊗ F)
ht−1∥
∥
2(GRuG∗)
−1 − ∥∥h0∥
∥
2Π−10
.
(33)
Combining these two expressions yields
L = −T∑
t=1
∥
∥Yt −(
IRx ⊗Xt)
ht∥
∥
21/σ2n
−T∑
t=1
∥
∥ht −(
ITxRx ⊗ F)
ht−1∥
∥
2(GRuG∗)
−1 − ∥∥h0∥
∥
2Π−10
.
(34)
Since the channel sequence hT0 is jointly Gaussian, the
MAPestimate of the channel sequence given the input and
outputsequences XT0 and Y
T0 is the same as the MMSE estimate given
the same sequences. The MMSE estimate itself is obtained bythe
forward-backward (FB) Kalman filter. This allows us tostate the
following theorem. (For a proof, see [33, problem10.9].)
Theorem 1 (channel estimation-known input case). Con-sider the
state-space model (29)-(30). Given the input andoutput sequences
XT0 and Y
T0 , the MAP (or equivalently MMSE)
estimate of hT0 is obtained by applying the following
(forward-backward Kalman) filter to the state-space model
(29)-(30).
Forward run
For i = 1, . . . ,T , calculateRe,t = σ2nITxRxN +
(
IRx ⊗Xt)
Pt|t−1(
IRx ⊗X∗t)
P0|−1 = Π0,
(35)
Kt = Pt|t−1(
IRx ⊗X∗t)
R−1e,t , (36)
̂ht|t =(
ITxRx(P+1) −Kt(
IRx ⊗Xt))
̂ht|t−1 + KtYt , (37)
̂ht+1|t =(
ITxRx ⊗ F)
̂ht|t, h0|−1 = 0, (38)Pt+1|t = (ITxRx ⊗ F)(Pt|t−1 −KtRe,tK∗t
)(ITxRx ⊗ F∗)
+ GRuG∗.(39)
-
T. Y. Al-Naffouri and A. A. Quadeer 7
Backward run
Starting from λT+1|T = 0 and for t = T ,T − 1, . . . , 0,
calculate
λt|T =(
IP+N −(
IRx ⊗X∗t)
K∗t)(
I⊗ F∗)λt+1|T+(
I⊗Xt)
R−1e,t(
Yt −(
I⊗Xt)
̂ht|t−1)
,(40)
̂ht|T = ̂ht|t−1 + Pt|t−1λt|T . (41)
The desired estimate is ̂ht|T .
(The Kalman filter has been initialized with zero in (38)as we
are dealing with Rayleigh channel. For the case ofRicean channel,
we only need to initialize the Kalman filterwith nonzero mean
corresponding to the direct line-of-sight(LOS) signal present in
Ricean channel.) This theorem allowsus to obtain the estimate of
hT0 when the input sequenceXT0 is not available. For in this case,
we maximize the log-likelihood (34) averaged over the sequence XT0
. Thus, the jthiteration of the EM algorithm is now obtained by
maximizingthe averaged log-likelihood
L = EXT0 |YT0 ,h
T( j−1)0
[L]. (42)
By inspecting (34), we note that the only term that ismodified
under expectation is the first summand, and itsexpectation is given
by
E∥
∥Yt −(
IRx ⊗Xt)
ht∥
∥
21/2σ2n
= ∥∥Yt −(
IRx ⊗ E[
Xt])
ht∥
∥
21/2σ2n
+∥
∥ht∥
∥
2(1/2σ2n )IRx⊗Cov[X∗t ]
=∥
∥
∥
∥
∥
[
Yt0TxRx(P+1)×1
]
−[
IRx ⊗ E[
Xt]
IRx ⊗ Cov[
X∗t]1/2
]
ht
∥
∥
∥
∥
∥
2
1/2σ2n
,
(43)
where the expectations are taken given the previous
estimatêh
( j−1)0 and the output symbols Y
T0 . We thus have
L = −T∑
t=0
∥
∥
∥
∥
∥
[
Yt0TxRx(P+1)×1
]
−[
IRx ⊗ E[
Xt]
IRx ⊗ Cov[
X∗t]1/2
]
ht
∥
∥
∥
∥
∥
2
1/2σ2n
−T∑
t=1
∥
∥ht − Fht−1∥
∥
2(GRuG∗)
−1 − ∥∥h0∥
∥
2Π−10
.
(44)
Note that we can obtain the averaged likelihood (44) fromthe
original likelihood (34) by performing the substitution
IRx ⊗Xt −→[
IRx ⊗ E[
Xt]
IRx ⊗ Cov[
X∗t]1/2
]
,
Yt −→[
Yt0TxRx(P+1)×1
]
,
(45)
we can thus state the following theorem.
Theorem 2 (channel estimation-unknown input case). Con-sider the
state-space model (29)-(30) and assume that thereceiver does not
have access to the transmitted data XT0 . The
channel estimate at the jth iteration hT( j)0 of the EM
algorithm is
obtained by applying the forward-backward Kalman (35)–(41)to the
following state-space model
ht+1 =(
ITxRx ⊗ F)
ht +(
ITxRx ⊗G)
ut, (46)[
Yt0TxRx(P+1)×1
]
=[
IRx ⊗ E[
Xt]
IRx ⊗ Cov[
X∗t]1/2
]
ht +
[
Ntnt
]
, (47)
where nt is virtual noise that is not physically present and
thatis independent of the physical noiseNt .
The virtual noise results from having a norm of thefollowing
form:
∥
∥
∥
∥
∥
[
Yt0TxRx(P+1)×1
]
−[
IRx ⊗ E[
Xt]
IRx ⊗ Cov[
X∗t]1/2
]
ht
∥
∥
∥
∥
∥
2
(1/σ2)I
. (48)
Note that the weighting matrixI is now of size N + P. Toaccount
for this, we assume the presence of virtual noise ntwhich is
independent of the actual physical noise Nt in theI/O equation
(47).
To fully implement the EM algorithm, we need to initializethe
algorithm and calculate the first and second moments ofthe input,
which we do next.
4.3. Initial channel estimation
We obtain the initial channel estimate from the
pilot/outputequation (18) together with the dynamical channel
model(7). Specifically, we do this by applying the FB Kalman to
thefollowing state-space model
ht+1 =(
ITxRx ⊗ F)
ht +(
ITxRx ⊗G)
ut, (49)
YtIp =(
IRx ⊗XtIp)
ht + NtIp , (50)
that is, by applying the FB Kalman filter (35)–(41) with
thefollowing substitution:
Yt −→ YtIp , Xt −→ XtIp , ITxRxN −→ ITxRx|Ip|. (51)
4.4. Data detection
To detect the data, we use the data detection version of theI/O
equation (22). Upon multiplying both sides by C∗ andtaking the real
part, we obtain
˜Y = ‖H‖2[
ReSImS
]
+ ˜N , (52)
where ˜Y and ˜N are 2Nu × 1 vectors defined by ˜Y =ReC∗Y and ˜N
= ReC∗N . Since C is orthogonal, thenoise ˜N remains white, and the
input can be detected onan element-by-element basis. (Equation (13)
holds for everySTBC but only the orthogonal case is considered in
thiswork.) We will now demonstrate how to detect the elementsof ReS
(the imaginary part can be treated similarly). Solet R = {r1, . . .
, r|R|} denote the alphabet set from whichthe elements of ReS take
their values. We can evaluate the
-
8 EURASIP Journal on Advances in Signal Processing
conditional pdf f (ReS(nu)| ˜Y(nu),H) by applying Bayesrule on
it which yields
f(
ri | ˜Y(
nu)
,H) = f (ri,
˜Y(nu) |H)f ( ˜Y(nu) |H)
= f (ri,˜Y(nu) |H)
∑|R|i=1 f ( ˜Y(nu), ri |H)
= f (˜Y(nu) | ri,H) f (ri |H)
∑|R|i=1 f ( ˜Y(nu) | ri,H) f (ri |H)
,
(53)
f(
ri | ˜Y(
nu)
,H) = e
−(| ˜Y(nu)−‖H‖2ri|2)/2σ2n∑|R|
i=1 e−(|˜Y(nu)−‖H‖2ri|2)/2σ2n
. (54)
We can use this pdf to calculate conditional expectation
ofReS(nu) and its second moment given the output ˜Y(nu):
E[
ReS(
nu) | ˜Y(nu
)
,H] =
∑|R|i=1 rie
−(| ˜Y(nu)−‖H‖2ri|2)/2σ2n∑|R|
i=1 e−(|˜Y(nu)−‖H‖2ri|2)/2σ2n
,
E[|ReS(nu
)|2 | ˜Y(nu)
,H] =
∑|R|i=1 r
2i e−(| ˜Y(nu)−‖H‖2ri|2)/2σ2n
∑|R|i=1 e−(|
˜Y(nu)−‖H‖2ri|2)/2σ2n.
(55)
We can similarly calculate the two moments of theimaginary part.
Now (55), just like (19)–(24), apply at acertain frequency tone n.
So collecting (55) for all tones(n = 1, . . . ,N) produces the two
moments of the uncodedOFDM symbols. Specifically, we can
calculate
E[ReS(nu)], E[ImS(nu)],
E[diag(ReS(nu))2], E[diag(ImS(nu))
2].(56)
We show in Appendix B that these moments are enough
tocharacterize the first and second moments E[X] and E[X∗X],which
are needed for channel estimation.
4.5. Summary of the EM-based receiver
We now have all the elements for the iterative receiver
forchannel and data recovery, and for ease of reference,
wesummarize the receiver algorithm in the following. Given
asequence of input and output symbols XT0 and Y
T0 perform
the following operations.
(1) Calculate the initial channel estimate hT0 (0) by apply-ing
the FB Kalman filter to the state-space model(49)-(50), that is, by
applying (35)–(41) with thefollowing substitutions:
Yt −→ YtIp , Xt −→ XtIp , ITxRxN −→ ITxRx|Ip|.
(57)
(2) Iterate between the expectation and maximizationsteps for j
= 1, . . . ,Niter:
(a) expectation:
(i) use (55) to compute the first two momentsof the uncoded OFDM
symbols S(1), . . . ,S(nu),given the output YT0 and the most
recentestimate of the channel, hT0 ( j − 1);
(ii) use these moments to calculate the moments ofX through the
relationships (13) and (55).
(b) maximization:
obtain the channel estimate hT0 ( j) by employingthe FB Kalman
to the state-space model (46)-(47),that is, by applying (35)–(41)
with the followingsubstitutions:
IRx ⊗Xt −→[
IRx ⊗ E[
Xt]
IRx ⊗ Cov[
X∗t]1/2
]
,
Yt −→[
Yt0TxRx(P+1)×1
]
,
ITxRxN −→ ITxRx(N+P+1).
(58)
The algorithm can be stopped when the maximum numberof
iterations Niter is reached or when the difference betweentwo
consecutive estimates ‖hT( j)0 −hT( j−1)0 ‖2 is below a
certainthreshold.
4.6. Modification: Kalman- (forward-only)based estimation
One disadvantage of the FB Kalman (summarized inSection 4.5
above) is the storage and latency involved. Thealgorithm needs to
wait for all T + 1 ST symbols before itcan execute the backward run
and hence obtain the channelestimate. One way around this is to
reduce the window sizeT. Alternatively, we can run the filter in
the forward directiononly (i.e., run (35)–(39)) for both the
initial estimation andthe EM iteration. The algorithm then
collapses to the Kalman-based filter proposed in [34] where the
data and channel arerecovered within one ST symbol.
5. SIMULATION RESULTS
In the following simulations, we test the performance of
theforward and forward-backward Kalman filters.
5.1. The forward-only Kalman
The transmitter and receiver illustrated in Figures 1 and2 were
implemented. The outer encoder is a rate 3/4convolutional encoder
and the coded bits are mapped to16-QAM symbols using Gray coding.
We use the OSTBCcommonly known as the Alamouti code with number of
timeslots Ns = 2 and number of transmitters Tx = 2 [35].
-
T. Y. Al-Naffouri and A. A. Quadeer 9
Our MIMO channel model is simulated using the state-space model
(7) with parameters, α = 0.985, β = 0.2, andP = 7. The number of
receive antennas is set to Rx = 2.
Three thousand packets were simulated per SNR-value.Each packet
is comprised of 12 OFDM symbols transmittedover six ST blocks. Each
OFDM symbol consists of 64frequency tones and a cyclic prefix of
length 16. The lengthof cyclic prefix is always kept greater than
or equal to thechannel length P + 1 to avoid any ISI. We employ 16
pilotsin the OFDM symbols making up the first ST block, while
thenumber of pilots we use in subsequent symbols vary betweentwo,
six, and ten.
In the following, we discuss the effect of various parame-ters
on the BER performance of the receiver design.
5.1.1. Benchmarking
We compare our algorithm with an EM-based iterative MMSEreceiver
such as the one proposed in [17, 21]. In contrastto our work, the
authors in [17, 21] take a data-centricapproach, treating the
transmitted signal as the desiredparameter and the channel as the
unobserved data. Thisalgorithm further confines its pilots to the
first ST block.The pilots are used to produce an initial channel
estimate forthe first ST block. This estimate is in turn used to
predictthe initial channel estimate for the subsequent ST blocksby
employing a time correlation filter [17]. These initialestimates
are used to kick-start the EM algorithm.
In this algorithm, the E-step is calculated by a
conditionalexpectation of the channel given the received symbol
andthe current estimate of the transmitted data (i.e., throughMMSE
estimation). The maximization step is simply the harddecision, that
is, the ML estimate of the transmitted data.
In Figure 3, we compare both schemes with 16 pilots inthe
initial ST block and zero pilots in the subsequent blocks.EMA
refers to the iterative MMSE scheme while EMB refersto the Kalman
filter-based scheme proposed in this paper.We also implement both
schemes with a total of 26 pilotsas shown in Figure 3. The EMA
confines the pilots to the firstST block while in EMB, we place 16
pilots in the first ST blockand 2 pilots each in subsequent blocks.
This ensures that bothschemes incur the same pilot overhead.
Our algorithm (EMB) outperforms EMA of [17] in bothpilot
scenarios. One reason for this performance improve-ment is that our
algorithm incorporates the time correlationinformation and the most
recent channel estimate in everyiteration of the EM algorithm,
while that of [17] does not.
5.1.2. Effect of number of iterations
In this section, we test the sensitivity of our algorithm tothe
number of EM iterations used. Here, we employ sixpilots per OFDM
symbol (in addition to the 16 pilots persymbol employed in the
first ST block). From Figure 4, wesee that the first iteration
yields substantial improvementover the pilot-based estimation. The
enhanced performancewith increased number of EM iterations is also
evident fromFigure 4. This improved performance comes at the cost
ofincreased computational complexity.
0 2 4 6 8 10 12 14 16 18
SNR (dB)
10−3
10−2
10−1
100
BE
R
EMA, pilots = [16 0 0 0 0 0]EMB , pilots = [16 0 0 0 0 0]EMA,
pilots = [26 0 0 0 0 0]EMB , pilots = [16 2 2 2 2 2]Perf. ch.
Figure 3: Comparison of EM-MMSE(EMA) and EM-FB-Kalman(EMB)
algorithms.
0 5 10 15
SNR (dB)
10−5
10−4
10−3
10−2
10−1
100
BE
R
Pilots onlyIter = 1
Iter = 4Perf. ch.
Figure 4: BER performance with different number of EM
iterations.
5.1.3. Effect of incorporating frequency and timecorrelation in
the channel estimation
The impact of using both frequency and time correlationsin
channel estimation is shown in Figure 5 for the six-pilotscenario.
In this figure, Ie = 1 (where Ie stands for “infor-mation used in
estimation”) refers to channel estimationusing either frequency or
time correlation information onlywhile Ie = 2 implies the use of
both frequency and timecorrelation in channel estimation. We
observe an error floorwhen either the frequency or time correlation
information is
-
10 EURASIP Journal on Advances in Signal Processing
0 5 10 15 20
SNR (dB)
10−4
10−3
10−2
10−1
100
BE
R
Pilots only, Ie = 1 (time)Iter = 1, Ie = 1 (time)Pilots only, Ie
= 1 (frequency)Iter = 1, Ie = 1 (frequency)
Pilots only, Ie = 2Iter = 1, Ie = 2Perf. ch.
Figure 5: Effect of a priori correlation knowledge on the
perfor-mance of the receiver.
used in channel estimation. (By using frequency correlationonly,
we mean a receiver that estimates the channel usingleast squares,
i.e., by performing the minimization in (28).This could
equivalently be performed by implementingthe FB-Kalman (35)–(41)
with the matrix F set to zero.Similarly, the time correlation only
case can be performedby implementing the FB-Kalman (35)–(41) with
the matrix
G set to√
1− α2(p)I. It should be clear that it is not possibleto ignore
the frequency correlation completely but by usingthis setting, its
effect is only decreased as the channel taps donot follow the
exponential channel decay profile and becomeidentically
distributed.) This error floor remains regardlessof the number of
iterations. However, when we incorporateboth frequency and time
correlation information, we observea significant improvement in
BER. We also note that asingle EM iteration provides substantial
improvement whencompared to the pilot-based estimation case. We
concludethat including both time and frequency correlations in
thechannel estimation process (especially for channels with
hightime correlation) increases the amount of information thatcan
be harnessed by iterating.
5.1.4. Effect of time variation
In this section, we test the performance of our receiveragainst
different degrees of time variation. This is parameter-ized by α (0
≤ α ≤ 1) with lower values of α indicating a moretime-variant
channel. (According to IEEE 802.16 standards,the typical range for
α is 0.677 to 0.9398 for a vehicle speeddecreasing from 120 km/h to
50 km/h, resp.) In Figure 6,we show the BER curves for a system
that employs six pilotsper OFDM symbol. We observe an error floor
as the channelvariation increases. So, we are unable to capture the
time
0 5 10 15
SNR (dB)
10−6
10−5
10−4
10−3
10−2
10−1
100
BE
R
Iter = 1, α = 0.7Iter = 1, α = 0.8Iter = 1, α = 0.985
Perf. ch., α = 0.985Perf. ch., α = 0.8Perf. ch., α = 0.7
Figure 6: BER performance with varying time correlation with
sixpilots.
0 2 4 6 8 10 12
SNR (dB)
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
BE
R
Iter = 1, α = 0.985Iter = 1, α = 0.7Iter = 1, α = 0.8Iter = 1, α
= 0.3
Perf. ch., α = 0.985Perf. ch., α = 0.8Perf. ch., α = 0.7Perf.
ch., α = 0.3
Figure 7: BER performance with varying time correlation with
tenpilots.
diversity. More pilots are thus needed to capture diversity
andimprove performance.
For comparison, in Figure 7, we show the BER curvesfor a system
with ten pilots per OFDM symbol for α =0.3, 0.7, 0.8 and 0.985.
From this figure, we observe that asα decreases (indicating more
channel variation), the BERimproves. This comes from increased time
diversity in thechannel. Therefore, with enough number of pilots,
we areable to track the channel and capture time diversity.
-
T. Y. Al-Naffouri and A. A. Quadeer 11
0 2 4 6 8 10 12 14
SNR (dB)
10−7
10−6
10−5
10−4
10−3
10−2
10−1
BE
R
Kalman with hard dataFB Kalman with hard dataKalman with soft
data
FB Kalman with soft dataPerf. ch.
Figure 8: BER performance of Kalman and FB-Kalman using hardand
soft estimate of data.
5.2. The FB Kalman
To test the FB Kalman filter, we use an input similar to theone
employed in the forward-only Kalman case. The outerencoder in this
case is a rate 1/2 convolutional encoder. Thenumber of transmitters
is two as Alamouti code is used inthis case also and the number of
receivers is also fixed at two.We use two MIMO channel models in
this case. One is spatiallywhite while the other one is spatially
correlated with transmitand receive correlation matrices
T(p) =[
1 ζζ 1
]
, R(p) = I , (59)
where ζ = 0.2. All other parameters are the same in bothchannel
models, that is, α = 0.8, β = 0.2, and P = 16.
Packets are transmitted at each SNR-value until a mini-mum
number of errors (five in this case) occur. Similar tothe
forward-only Kalman case, each packet consists of six STblocks. The
first ST block contains 16 pilots while the numberof pilots in
subsequent blocks remains fixed at 12.
Figure 8 compares the Kalman and the FB-Kalman overspatially
white channel. The results are shown for twoscenarios. In one, hard
estimate of data (the estimated data)is used while in the other
soft estimate (expected value of thedata) is used. The figure
clearly illustrates that the FB-Kalmanis better than the Kalman for
both scenarios. It can alsobe observed from this figure that the
FB-Kalman using softestimate of data outperforms the one using hard
estimate.
The performance of Kalman and the FB-Kalman iscompared in Figure
9 over the two channel models, thatis, spatially correlated and
spatially white channel model.In both cases, soft estimate of data
is used. In the perfectchannel scenario, the increased diversity in
uncorrelated(white) case gives a better BER compared to the
correlatedcase. However, in the estimated channel case, diversity
is
0 2 4 6 8 10 12
SNR (dB)
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
BE
R
Kalman over spatially correlated ch.FB Kalman over spatially
correlated ch.Kalman over spatially white ch.FB Kalman over
spatially white ch.Perf. spatially correlated ch.Perf. spatially
white ch.
Figure 9: BER performance of Kalman and FB-Kalman using softdata
over spatially white and correlated channel models.
a two-edged sword. On the one hand, increased diversityshould
improve performance if we manage to have a goodestimate of the
channel. On the other hand, increaseddiversity also increases the
channel’s degrees of freedomgiving an inferior estimate to the
uncorrelated case forthe same number of pilots used. This explains
why in theestimated case, the correlated channel performs better
thanthe uncorrelated channel for the channel estimate quality inthe
former is better than the channel quality in the latter.
6. CONCLUSION
In this paper, we have proposed a receiver for
MIMO-OFDMtransmission over time-variant channels. While the
paperassumed the channel to be constant within any ST block,the
channel was allowed to vary from one block to the next.This makes
the receiver suitable for operation in high-speedenvironments.
The receiver employs the EM algorithm to achieve channeland data
recovery. Specifically, the data recovery (or theexpectation step)
is as simple as decoding a space-timeblock code. Channel recovery
(or the maximization step) isperformed using a forward-backward
Kalman filter. We alsosuggested a relaxed (forward-only) version of
the algorithmthat is able to perform recovery with no latency and
henceavoid the delay and storage shortcomings of the FB-Kalman.
When compared with other MIMO receivers, our receivermakes the
most use of the underlying structure. Specifically,the algorithm
makes use of the finite alphabet constraints(55), the data in its
soft form (46)-(47), pilots (49)-(50),finite-delay spread (in that
channel estimation is done in thetime domain), frequency and time
correlation (7), spatial
-
12 EURASIP Journal on Advances in Signal Processing
correlation in (A.16), and space-time coding. It is
alsostraightforward to incorporate the effect of an outer
code,sparsity, and the cyclic prefix (see [27, 36]). Our
simulationsshow the favorable behavior of the two Kalman filters
ascompared to other receivers.
APPENDICES
A. CHANNELMODEL IN THE PRESENCE OFSPATIAL CORRELATION
In what follows, we derive the dynamical model for thechannel
impulse response in the transmit correlation case,and then
generalize our results of Section 2.2 to deal withthe general
(transmit and receive) correlation case. In thetransmit correlation
case, H(p), the MIMO impulse responseat tap p, is given by
H(p) =W(p)T1/2(p), (A.1)
where T1/2(p) is the transmit correlation matrix (of size Tx)at
tap p and where W(p) consists of iid elements. The matrixW(p)
remains constant over a single ST block and variesfrom one ST block
to the next according to
Wt+1(p) = α(p)Wt(p) +√
(
1− α2(p))e−βpUt(p), (A.2)
where α(p),β, and Ut(p) are as defined in Section 2.2.(We
suppress the time dependence at times for
notationalconvenience.)
Just as we did in Section 2.2, we would like to constructa
recursion for the tap htxrx (p) and subsequently scale it upfor the
SISO and MIMO cases. Now since htxrx (p) is the (rx, tx)element of
H(p), we deduce from (A.1) that it is the innerproduct of the rx
row of W(p) and the tx column of T1/2, thatis,
htxrx (p) = wrx (p)ttx (p). (A.3)
Moreover, from (A.2), we have the following recursion forwrx
(p):
wrx ,t+1(p) = α(p)wrx ,t(p) +√
(
1− α2(p))e−βpurx ,t(p).(A.4)
Postmultiplying both sides by ttx (p) yields
wrx ,t+1(p)ttx (p) = α(p)wrx ,t(p)ttx(p)
+√
(
1− α2(p))e−βpurx ,t(p)ttx (p).(A.5)
This means that htxrx (p) satisfies the dynamical equation
htxrx ,t+1(p) = α(p)htxrx ,t(p) +
√
(
1− α2(p))e−βputtxrx ,t(p),(A.6)
where uttxrx is defined by
uttxrx (p) = urx (p)ttx (p). (A.7)
Concatenating (A.6) for p = 1, 2, . . . ,P yields a
dynamicequation for the impulse response
htxrx =
⎡
⎢
⎢
⎢
⎣
htxrx (0)...
htxrx (P)
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎣
wrx (0)ttx (0)
...
wrx (P)ttx (P)
⎤
⎥
⎥
⎥
⎦
, (A.8)
which is the same as the dynamic equation (see (5)) for
thespatially uncorrelated case
htxrx ,t+1 = Fhtxrx ,t + Gut
txrx ,t . (A.9)
The only difference from the uncorrelated case is that uttxrx
isno more white. Rather, we have
E[
uttxrxuttx∗rx
]
Δ= E
⎡
⎢
⎢
⎢
⎢
⎢
⎣
urx (0)ttx (0)
urx (1)ttx (1)
...
urx (P)ttx (P)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
×[
ttx∗(0)u∗rx (0) ttx∗(1)u∗rx (1) · · · ttx∗(P)u∗rx (P)
]
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
tttxtx (0)tttxtx (1)
. . .
tttxtx (P)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Δ= diag(tttxtx)
,
(A.10)
where
tttxrx =
⎡
⎢
⎢
⎢
⎢
⎣
trx∗(0)ttx (0)trx∗(1)ttx (1)
...trx∗(P)ttx (P)
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
trx (0)ttx (0)
trx (1)ttx (1)
...trx (P)t
tx (P)
⎤
⎥
⎥
⎥
⎥
⎦
, (A.11)
and where the second line follows from the fact that trx∗(p)
=ttx (p) since T
1/2(p) is conjugate symmetric. In general, wecan show that
E[
utrxut∗r′x
] =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
diag(
tt11)
diag(
tt21) · · · diag(ttTx1
)
diag(
tt12)
diag(
tt22) · · · diag(ttTx2
)
...... · · ·
...diag
(
tt1Tx)
diag(
tt2Tx) · · · diag(ttTxTx
)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
(A.12)
for rx = r′x and is zero otherwise. Alternatively, we can
writethis as
E[
utrxut∗r′x
] =
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
P∑
p=0T(p)⊗ (IpBIp) for rx = r′x,
O otherwise,
(A.13)
-
T. Y. Al-Naffouri and A. A. Quadeer 13
where
B =
⎡
⎢
⎢
⎢
⎢
⎣
1 0 · · · 00 0 · · · 0...
... · · ·...
0 0 · · · 0
⎤
⎥
⎥
⎥
⎥
⎦
,
I =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
01 0
1. . .. . . 0
1 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
I =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 1
0. . .. . . 1
0 10
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(A.14)
Collecting (A.9) for all transmit and receive antennas
yields
ht+1 =(
ITxRx ⊗ F)
ht +(
ITxRx ⊗G)
utt , (A.15)
where
E[
uu∗] = IRx ⊗ E
[
utrxut∗rx
]
=P∑
p=0IRx ⊗ T(p)⊗
(
IpBIp).
(A.16)
When the channel exhibits both transmit and receive
corre-lations, the IRh continues to satisfy the dynamical
equation(A.15) except that the correlation of the innovation u is
nowgiven by
E[
uu∗] =
P∑
p=0R(p)⊗ T(p)⊗ (IpBIp). (A.17)
B. CALCULATING THEMOMENTS OFX
In this appendix, we demonstrate that the four moments (56)of
the uncoded OFDM symbol S(nu) are enough to calculatethe first two
moments of X,E[X], and E[X∗X]. Since Xdepends linearly on ReS(nu)
and ImS(nu) (see (13) and(16)), it is straight forward to calculate
the mean of X startingfrom the means of ReS(nu) and ImS(nu). Now
from (16),we note that evaluating E[X∗X] boils down to evaluating
thecross correlation E[diag(X∗i (nc))diag(X j(n
′c))], recall also
that
Xtx(
nc) =
Nu∑
nu=1atx ,nc
(
nu)
ReS(
nu)
+ jbtx ,nc(
nu)
ImS(
nu)
.
(B.1)
This means that calculating the cross expectation boils downto
calculating the cross correlation of ReS(nu), ImS(nu),ReS(n′u), and
ImS(n
′u) for nu,n
′u = 1, . . . ,Nu. It is easy
to see that these variables are independent for nu
/=n′u.Moreover, since the noise in (52) is white, one can alsosee
that ReS(nu) and ImS(nu) are independent. As aresult, we can
completely characterize the cross correlationE[diag(X∗i (nc))diag(X
j(n
′c))] and hence the expectations
E[X] and E[X∗X] starting from the first and secondmoments of
(56).
ACKNOWLEDGMENTS
This work was supported by King Abdul Aziz City forScience and
Technology (KACST), Saudi Arabia, Projectno. AR 27-98. Earlier
versions of this work appeared atSignal Processing Advances for
Wireless CommunicationsWorkshop (SPAWC), Helsinki, Finland, June
2007 [37]and IEEE Global Telecommunications Conference (GLOBE-COM),
Texas, USA, December 2004 [38].
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1. INTRODUCTION1.1. Paper contributions1.2. Paper
organization1.3. Notation
2. AN OVERVIEWO F THE COMMUNICATION SYSTEM2.1. Transmitter2.2.
Channel modelA note about time variation
2.3. Receiver2.3.1. STBC decoder/data detector (estimation
step)2.3.2. channel estimator (maximization step)
3. INPUT/OUTPUT EQUATIONS FOR MIMO-OFDM3.1. I/O equations:
channel estimation version3.2. I/O equations: data detection
version
4. JOINT CHANNEL AND DATA ESTIMATION: AN EM APPROACH4.1. The EM
algorithm4.2. Channel estimation: an EM-basedforward-backward
Kalman4.3. Initial channel estimation4.4. Data detection4.5.
Summary of the EM-based receiver4.6. Modification:
Kalman(forward-only) based estimation
5. SIMULATION RESULTS5.1. The forward-only Kalman5.1.1. Bench
marking5.1.2. Effect of number of iterations5.1.3. Effect of
incorporating frequency and time correlation in the channel
estimation5.1.4. Effect of time variation
5.2. The FB Kalman
6. CONCLUSIONAPPENDICESA. CHANNEL MODEL IN THE PRESENCE OF
SPATIAL CORRELATIONB. CALCULATING THE MOMENTS OF X
ACKNOWLEDGMENTSREFERENCES
