# Iterative Mean Removal Superimposed Training for  SISO and MIMO Channel Estimation

Authors: O. Longoria-Gandara; R. Parra-Michel; M. Bazdresch; A. G. Orozco-Lugo
Journal: International Journal of Digital Multimedia Broadcasting
 (2008)

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## Abstract

This contribution describes a novel iterative radio channel estimation algorithm based on superimposed training (ST) estimation technique. The proposed algorithm draws an analogy with the data dependent ST (DDST) algorithm, that is, extracts the cycling mean of the data, but in this case at the receiver_s end. We first demonstrate that this mean removal ST (MRST) applied to estimate a single-input single-output (SISO) wideband channel results in similar bit error rate (BER) performance in comparison with other iterative techniques, but with less complexity. Subsequently, we jointly use the MRST and Alamouti coding to obtain an estimate of the multiple-input multiple-output (MIMO) narrowband radio channel. The impact of imperfect channel on the BER performance is evidenced by a comparison between the MRST method and the best iterative techniques found in the literature. The proposed algorithm shows a good tradeoff performance
between complexity, channel estimation error, and noise immunity.


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## Body

## I. Introduction

One of the most widely used approaches to channel estimation
is to employ pilot assisted transmission (PAT), where a known training
sequence, also referred as pilot, is inserted at each block of transmitted data
[1]. Using the knowledge of the training symbols and the corresponding received
signal, the channel estimator block at the receiver is able to make an estimate
of the channel impulse response (CIR). However, these training pilots, using
time division multiplexing (TDM) scheme, consume valuable bandwidth resulting
in the reduction of the data rate.There are two well-known channel
estimation techniques to avoid the bandwidth waste of PAT schemes: superimposed
training (ST) [2, 3] and data dependent ST (DDST) [4]. These techniques are
based on the arithmetic addition (superimposed) of a training sequence to the
information data. Both schemes provide a simple (unsophisticated) channel
estimation processes; they differ only in that the cyclic mean of the
transmitted data of the DDST scheme is superimposed into the transmitted
sequence in similar way that the training signal.Although DDST outperforms ST [4] in
terms of channel estimation error, it is worth mentioning that the decoding of
data under DDST is of iterative nature because it needs to extract the data-dependent
distortion. Considering this, DDST with DDD removal (henceforth we
will refer to this scheme as DDST-DDD removal) gives a similar performance as
TDM-based channel estimation, but with fewer bandwidth losses. There are however
some drawbacks in trade; DDST technique introduces a delay in the transmitted
data in order to calculate the cyclic mean. It also assigns less power to the
data signal, and hence the use of high-order symbol constellations has
repercussions on the data decoding process [5].The last constraints lead to research on
iterative implementations of ST as in
[6–10], starting
from the SISO radio channel. These works are based on the use of the decoded
data to eliminate the distortion introduced by the received data in the channel
estimation process. The first approach uses ST in combination with a
traditional least squares channel estimate (LSST) and was developed in
[6].  In
[6, 7] it is clearly shown that
in terms of the channel estimation mean square error (MSE), LSST converges to a
fully trained system for high SNR in two iterations, thereby outperforming both
conventional ST and DDST. The great disadvantage of LSST scheme is the
computational burden of 𝒪(NM2) for one iteration, where N is the block length and M is the order of the CIR. In [11], it is
demonstrated that DDST-DDD achieves similar performance than LSST but with
considerably less complexity.Alternative iterative procedures of both
ST and DDST methods were introduced in [7]. The first one, IST, uses the
equalized symbols, obtained via ST, to improve the channel estimate in an
iterative way but with less complexity than LSST [6]. The second one, LSDDST employs
the LSST iterative approach but based on DDST instead of ST. LSDDST scheme, has
the same computational burden as LSST, that is, 𝒪(NM2),
but it converges faster to the fully trained system. In terms of bit error rate
(BER), LSDDST scheme yields almost the same BER than LSST.From the previous works, we are now able
to establish this contribution in the context of low-complexity iterative
algorithms using ST that shows good BER performance. We introduce a new
iterative mean removal ST (MRST) proposal and compare its performance with the
previous and most relevant works.This MRST yields similar performance to
DDST-DDD removal and IST but with less complexity when they are compared with
LSDDST and LSST. Because the iterative channel estimation methods depend on and
work jointly with the equalization stage, we present the results using two
equalizers widely used in communication systems: the minimum mean square error
(MMSE) equalizer and the maximum likelihood sequence estimation (MLSE)
equalizer. The inclusion of both techniques is helpful to accentuate some
particularities of the channel estimation methods used.Additionally, we extend the results of
SISO to MIMO case, and study the performance of training-based flat
block-fading MIMO channel estimation. Three training-based channel estimators
are considered (TDM, DDST, and MRST), which offer different tradeoffs in terms
of performance. We analyze the error performance of the MRST method based on
the traditional least squares (LSs) method and obtain the corresponding MSE.
The proposed MRST estimator for the MIMO case is illustrated using an
orthogonal space-time block coder
(OSTBC) with two transmit and two receive antennas, that is, Alamouti2×2 space-time coding.This paper is organized as follows.
Section2 deals with the new iterative ST approach in the framework for
ST/DDST schemes for SISO systems. The performance analysis of the MRST for MIMO
systems is obtained in Section 3. In
Sections 4 and 5, the simulation
results and performance comparison are given for SISO and MIMO, respectively. Finally,
the conclusions are set down in Section 6.


## 2. Mean Removal ST for SISO Systems

### 2.1. SISO System Model

Assuming a frequency-selective channel, Figure1 depicts the discrete-time baseband block diagram of a digital communication
system, where the channel input signal {s(k)} represents a succession of information blocks
of length N,
and is given by (1)s(k)=b(k)+c(k)+e(k),taking
into account that
the block of interest is k indexed
and k=0,1,…,N−1.
The sequence {b(k)} represents the data with zero mean and
variance E{|b(k)|2}=σb2,{c(k)} is a deterministic periodic training sequence
with period P and power σc2=(1/P)∑k=0P−1|c(k)|2,
and {e(k)} refers to
a data-dependent sequence, with period P,
obtained by periodically repeating NP times  the signal (2)e(j)=−(1Np)∑i=0NP−1b(iP+j),j=0,…,P−1,where Np=N/P.Figure 1
SISO digital communication system model.For the ST case,e(k)=0 and the sequences {c(k)},{e(k)}  are arithmetically added in a superimposed way
according to [4], before transmission. Then, the received signal, assuming that
exact synchronization and DC-offset are provided, can be observed at the
receiver as (3)r(k)=∑l=0M−1h(l)s(k−l)+n(k),where h(k) is the impulse response of order M−1  (i.e., h(0)≠0 and h(M−1)≠0), and {n(k)} is the complex Gaussian random noise with
zero-mean, white, uncorrelated to {b(k)} and independent real and imaginary parts with
variance σn2=N0/2 per dimension.Assuming that the channel is
quasistatic, that is, the channel is time-invariant during the information
block received, and the exact channel order(M−1) is known in advance, then the strong
constraint (P≥2M+1) showed in [11], with exact synchronization
and DC-offset provided, can be relaxed to P=M.
We used this assumption for the mathematical analysis of the system. Despite
this relaxation, all simulations were carried out considering the strong
constraint.


### 2.2. Performance Analysis

It is clear that DDST method must
compute the sequence{e(k)} from the data block that will be transmitted; consequently
this data processing has an impact on the total delay of every transmitted block.
On the other hand, the sequence {s(k)} has a wider dynamic range in comparison with
the sequence obtained with ST method. This results in a higher peak-to-average
power ratio (PAPR) of the preamplifier communication building block. Another
implication using DDST is the fact that the sequence {b(k)} will have less power in comparison again with
ST and then less noise immunity. For these reasons, iterative ST schemes are
very attractive.We developed the MRST scheme starting
from the hypothesis that if we could obtain an estimate of the signal{e(k)} (i.e., a cyclic mean of sequence {b(k)}) at the receiver side, then we would achieve
the performance of DDST in terms of channel error estimate MSE but with more
power assigned to the data sequence {b(k)},
having an impact on a better performance in terms of BER.Estimating the cycling mean of periodP as in [3], we can write (4)y˜(j)=1NP∑i=0NP−1r(iP+j),with j=0,1,…,P−1.Combining (3) and (4)
yields (5)y˜(j)=∑m=0M−1h(m){b˜(j−m)+e(j−m)+c(j−m)}+n˜(j), where (6)b˜(k)=1NP∑i=0NP−1b(iP+j),n˜(k)=1NP∑i=0NP−1n˜(iP+j).  From (5), it follows
that in matrix form (7)(C+B˜+E)h=y˜−n˜,where C and E are P×P circulant matrices with first columns [c(0)c(1)⋯c(P−1)]T and [e(0)e(1)⋯e(P−1)]T, respectively, and h=[h(0)h(1)⋯h(P−1)].
The P×1 column vectors y˜ and n˜ have similar expression to h.
Assuming a cyclic prefix of length P (as was done in [4, 7]), B˜ is circulant matrix with first column [b˜(0)b˜(1)⋯b˜(P−1)]T.For the ST case (i.e., whenE=0 in (7)) we have y˜=(C+B˜)h+n˜,
and using the channel estimate C−1y˜ from [3] then (8)h^ST=C−1y˜=h+C−1B˜h+C−1n˜.For the DDST caseE=−B˜ in (7), so we have the channel estimate [4] as (9)h^DDST=C−1y˜=h+C−1n˜.Clearly, the difference
between these two channel estimation schemes is the factor C−1B˜h.Now using the hypothesis early mentioned
(i.e.,E=−B˜^≃−B˜ with ST scheme), we proceed to make the
channel estimation for the MRST scheme, then (7) becomes (10)y˜=(C+B˜−B˜^)h+n˜,and multiplying
(10)
by C−1 we obtain (11)h^MRST=C−1y˜=C−1{(C+B˜−B˜^)h+n˜}, and finally (12)h^MRST=h+C−1(B˜−B˜^)h+C−1n˜.From
(12) it follows that if B˜^=B˜ then h^MRST=h^DDST.
In order to make a good estimate (B˜^) of B˜,
the following steps are made.(1)
Use (8) to have an initial channel estimate as plain ST and make y˜old=y˜.(2)
Use the channel estimated to obtain the equalized symbols and employ a
hard decision detector.(3)
Use the hard-decision symbols detected to calculateB˜^.(4)
RemoveB˜^ from
the received signal to obtain a new y˜ according to (13)y˜new=y˜old−B˜^h^,where y˜new is a column vector P×1.(5)
Use (11) with y˜=y˜new and update the channel estimate h^MRST.(6)
Go to step 2 and repeat as need it.Defining theMSE(h^):=E{∑k=0M−1|h^(k)−h(k)|2},
then from [3, 4], and having a better
estimation (B˜^) of B˜,
it follows that h^MRST≃h^DDST,
then (14)MSE(h^MRST)≃σn2Npσc2.From
(9) and (11), DDST and MRST,
respectively, we note the fact that C−1 is calculated only once beforehand. The IST
scheme, however, uses the estimate (B˜^) of B˜ to calculate (C+B˜^)−1
(see  [7, (12)]) for every iteration and
information block received. This action implies an additional complexity to the
computational burden.On the other hand, as we will see in
Section4, the estimate B˜^ used to estimate the channel in IST and MRST
methods depends on what type of equalizer is used. Conversely, the scheme
DDST-DDD removal cannot take advantage of MLSE equalizer, because this equalizer
delivers hard-decision symbols, and DDST-DDD removal uses the equalized symbols
previous to hard-decision procedure.Table1 summarizes the computational burden,
the computation of inverse matrix (C−1),
and the MSE performance approach. The complexity of an algorithm is a quite
important metric when the algorithm is to be implemented in HW, as this is the
case we are very concern of this value. As for computational burden metric, we
choose the number of iterations as function of the transmitted block length N and the equalizer length 𝒬,
to be precise, and the number of coefficients in the equalizer.Table 1
Synoptic Table of the Computational
burden,C−1 Computation and the Channel Estimate MSE.
ProcedureComputational
burdenC−1MSE(h^)
approachDDST-DDD removal [11]𝒪(𝒬N)  1 iterationPre-calculatedDDST ([4])IST scheme in [7]𝒪(3𝒬N)
2 iterations(C+B˜^)−1 every iterationDDST ([4])MRST𝒪(3𝒬N)
2 iterationsPrecalculatedDDST ([4])𝒪(·) notation expresses complexity, 𝒬,
and N denote the equalizer length and the block
length, respectively.Although DDST has the least
computational complexity at the receiver side, it needs to calculate the
data-dependent sequence{e(k)},
that implies additional complexity and time delay. Furthermore, it assigns less
transmission power to the data signal.


## 2.1. SISO System Model

Assuming a frequency-selective channel, Figure1 depicts the discrete-time baseband block diagram of a digital communication
system, where the channel input signal {s(k)} represents a succession of information blocks
of length N,
and is given by (1)s(k)=b(k)+c(k)+e(k),taking
into account that
the block of interest is k indexed
and k=0,1,…,N−1.
The sequence {b(k)} represents the data with zero mean and
variance E{|b(k)|2}=σb2,{c(k)} is a deterministic periodic training sequence
with period P and power σc2=(1/P)∑k=0P−1|c(k)|2,
and {e(k)} refers to
a data-dependent sequence, with period P,
obtained by periodically repeating NP times  the signal (2)e(j)=−(1Np)∑i=0NP−1b(iP+j),j=0,…,P−1,where Np=N/P.Figure 1
SISO digital communication system model.For the ST case,e(k)=0 and the sequences {c(k)},{e(k)}  are arithmetically added in a superimposed way
according to [4], before transmission. Then, the received signal, assuming that
exact synchronization and DC-offset are provided, can be observed at the
receiver as (3)r(k)=∑l=0M−1h(l)s(k−l)+n(k),where h(k) is the impulse response of order M−1  (i.e., h(0)≠0 and h(M−1)≠0), and {n(k)} is the complex Gaussian random noise with
zero-mean, white, uncorrelated to {b(k)} and independent real and imaginary parts with
variance σn2=N0/2 per dimension.Assuming that the channel is
quasistatic, that is, the channel is time-invariant during the information
block received, and the exact channel order(M−1) is known in advance, then the strong
constraint (P≥2M+1) showed in [11], with exact synchronization
and DC-offset provided, can be relaxed to P=M.
We used this assumption for the mathematical analysis of the system. Despite
this relaxation, all simulations were carried out considering the strong
constraint.


## 2.2. Performance Analysis

It is clear that DDST method must
compute the sequence{e(k)} from the data block that will be transmitted; consequently
this data processing has an impact on the total delay of every transmitted block.
On the other hand, the sequence {s(k)} has a wider dynamic range in comparison with
the sequence obtained with ST method. This results in a higher peak-to-average
power ratio (PAPR) of the preamplifier communication building block. Another
implication using DDST is the fact that the sequence {b(k)} will have less power in comparison again with
ST and then less noise immunity. For these reasons, iterative ST schemes are
very attractive.We developed the MRST scheme starting
from the hypothesis that if we could obtain an estimate of the signal{e(k)} (i.e., a cyclic mean of sequence {b(k)}) at the receiver side, then we would achieve
the performance of DDST in terms of channel error estimate MSE but with more
power assigned to the data sequence {b(k)},
having an impact on a better performance in terms of BER.Estimating the cycling mean of periodP as in [3], we can write (4)y˜(j)=1NP∑i=0NP−1r(iP+j),with j=0,1,…,P−1.Combining (3) and (4)
yields (5)y˜(j)=∑m=0M−1h(m){b˜(j−m)+e(j−m)+c(j−m)}+n˜(j), where (6)b˜(k)=1NP∑i=0NP−1b(iP+j),n˜(k)=1NP∑i=0NP−1n˜(iP+j).  From (5), it follows
that in matrix form (7)(C+B˜+E)h=y˜−n˜,where C and E are P×P circulant matrices with first columns [c(0)c(1)⋯c(P−1)]T and [e(0)e(1)⋯e(P−1)]T, respectively, and h=[h(0)h(1)⋯h(P−1)].
The P×1 column vectors y˜ and n˜ have similar expression to h.
Assuming a cyclic prefix of length P (as was done in [4, 7]), B˜ is circulant matrix with first column [b˜(0)b˜(1)⋯b˜(P−1)]T.For the ST case (i.e., whenE=0 in (7)) we have y˜=(C+B˜)h+n˜,
and using the channel estimate C−1y˜ from [3] then (8)h^ST=C−1y˜=h+C−1B˜h+C−1n˜.For the DDST caseE=−B˜ in (7), so we have the channel estimate [4] as (9)h^DDST=C−1y˜=h+C−1n˜.Clearly, the difference
between these two channel estimation schemes is the factor C−1B˜h.Now using the hypothesis early mentioned
(i.e.,E=−B˜^≃−B˜ with ST scheme), we proceed to make the
channel estimation for the MRST scheme, then (7) becomes (10)y˜=(C+B˜−B˜^)h+n˜,and multiplying
(10)
by C−1 we obtain (11)h^MRST=C−1y˜=C−1{(C+B˜−B˜^)h+n˜}, and finally (12)h^MRST=h+C−1(B˜−B˜^)h+C−1n˜.From
(12) it follows that if B˜^=B˜ then h^MRST=h^DDST.
In order to make a good estimate (B˜^) of B˜,
the following steps are made.(1)
Use (8) to have an initial channel estimate as plain ST and make y˜old=y˜.(2)
Use the channel estimated to obtain the equalized symbols and employ a
hard decision detector.(3)
Use the hard-decision symbols detected to calculateB˜^.(4)
RemoveB˜^ from
the received signal to obtain a new y˜ according to (13)y˜new=y˜old−B˜^h^,where y˜new is a column vector P×1.(5)
Use (11) with y˜=y˜new and update the channel estimate h^MRST.(6)
Go to step 2 and repeat as need it.Defining theMSE(h^):=E{∑k=0M−1|h^(k)−h(k)|2},
then from [3, 4], and having a better
estimation (B˜^) of B˜,
it follows that h^MRST≃h^DDST,
then (14)MSE(h^MRST)≃σn2Npσc2.From
(9) and (11), DDST and MRST,
respectively, we note the fact that C−1 is calculated only once beforehand. The IST
scheme, however, uses the estimate (B˜^) of B˜ to calculate (C+B˜^)−1
(see  [7, (12)]) for every iteration and
information block received. This action implies an additional complexity to the
computational burden.On the other hand, as we will see in
Section4, the estimate B˜^ used to estimate the channel in IST and MRST
methods depends on what type of equalizer is used. Conversely, the scheme
DDST-DDD removal cannot take advantage of MLSE equalizer, because this equalizer
delivers hard-decision symbols, and DDST-DDD removal uses the equalized symbols
previous to hard-decision procedure.Table1 summarizes the computational burden,
the computation of inverse matrix (C−1),
and the MSE performance approach. The complexity of an algorithm is a quite
important metric when the algorithm is to be implemented in HW, as this is the
case we are very concern of this value. As for computational burden metric, we
choose the number of iterations as function of the transmitted block length N and the equalizer length 𝒬,
to be precise, and the number of coefficients in the equalizer.Table 1
Synoptic Table of the Computational
burden,C−1 Computation and the Channel Estimate MSE.
ProcedureComputational
burdenC−1MSE(h^)
approachDDST-DDD removal [11]𝒪(𝒬N)  1 iterationPre-calculatedDDST ([4])IST scheme in [7]𝒪(3𝒬N)
2 iterations(C+B˜^)−1 every iterationDDST ([4])MRST𝒪(3𝒬N)
2 iterationsPrecalculatedDDST ([4])𝒪(·) notation expresses complexity, 𝒬,
and N denote the equalizer length and the block
length, respectively.Although DDST has the least
computational complexity at the receiver side, it needs to calculate the
data-dependent sequence{e(k)},
that implies additional complexity and time delay. Furthermore, it assigns less
transmission power to the data signal.


## 3. Mean Removal ST for MIMO Systems

### 3.1. MIMO System Model

We consider a wireless MIMO
communication link withnT transmit and nR receive antennas, operating in a Rayleigh
flat-fading environment. The fading coefficient hi,j is the complex path gain from transmit antenna i to receive antenna j.
We assume that the coefficients are independently complex circular symmetric
Gaussian with unit variance, and then H=[hi,j]∈ℂnT×nR. The expression for the received symbols can be
expressed as  (15)r=xH+n,  where r is the 1×nR received signal vector, x is the 1×nT transmitted signal vector, and n is an 1×nR vector of additive noise terms, assuming that
noise is spatially and temporally white Gaussian with zero mean and independent
real and imaginary parts with variance σn2=N0/2 per dimension.Let us assume the block transmission
scheme with the block lengthN at times k=1,…,N, and we also assume that the channel matrix H remains constant within a block of N symbols, that is, the block length is much
small than the channel coherence time. Under these assumptions, the channel,
within one block, can be written as (16)R=XH+N,where (17)R≜[r(1)r(2)⋮r(N)],X≜[x(1)x(2)⋮x(N)],N≜[n(1)n(2)⋮n(N)] are the matrices of the
received signal, transmitted signals, and noise, respectively [12].Let us denote the set of complex
information symbols prior to space-time encoding as{sλ∣sλ∈𝒰λ,λ=1,2,…,Λ},
where each 𝒰λ denotes a set of signal constellation points.
The symbols s1,s2,…,sΛ are zero-mean mutually uncorrelated random
variables. Let us introduce the vector (18)s≜[s1s2⋯sΛ]T, where (·)T denotes transpose. Note that s∈𝒮,
where 𝒮={s(1)s(2)⋯s(L)} is the set of all possible symbol vectors and L is the cardinality of this set. The N×nT complex matrix-valued function X(s) is called OSTBC [13] if it satisfies that(1)
all the entries ofX(s) are linear functions of the Λ complex variables s1,s2,…,sΛ and their complex conjugates;(2)
for any arbitrarys in ℂΛ×1, (19)XH(s)X(s)=∥s∥2InT,
where InT is the
identity nT×nT matrix, ∥·∥ is the Euclidean norm, and (·)H denotes Hermitian transpose.


### 3.2. Performance Analysis

In order to estimate the channel matrixH,
it should  be emphasized that in
any statistical expectation below, the matrix H is treated as random; at the same time, any
estimator of H is supposed to obtain an estimate of a
particular realization of this random matrix that corresponds to the current
block of the received data.In the conventional ST estimation
technique for MIMO systems [9], a known training N×nT matrix C,
is added arithmetically to the data N×nT matrix B during every block transmitted. In this way,
the transmitted signal matrix X can be expressed as X=B+C.
It is clear that the total transmitted power is distributed between the data
and the training signals, that is, Px=Pb+Pc.The ST system is depicted in Figure2,
where a signal matrix  X
is transmitted over the radio MIMO channel
with the channel matrix H,
and distorted with the noise matrix N.Figure 2
MIMO digital communication system model.Based on the received signalR, the MRST delivers an estimate of the
MIMO channel matrix, denoted as H^;
subsequently the decoder obtains an estimate of  X,
denoted as X^,
and an estimate of the mean of the data, b˜^,
which can be used by the channel estimator block, in an iterative way, to
provide the decoder with a better estimate of H.Now, the task of this channel estimation
algorithm is to recover the channel matrixH based on the knowledge of R and C.
Assuming flat-frequency channel conditions then all the row vectors of the
training matrix C can be equal. Hence, a time-domain estimator
based on the synchronized averaging of the received signal can be implemented.
The time-average of the signal R is  given by (20)v=E{R}=(1N)uR=[(1N)(uB+uC)]H+(1N)uN=[b˜+c]H+n˜, where v is a 1×nR column vector, u is the unit 1×N vector, c is the 1×nT training vector repeated N times and b˜,n˜  represent the time-average of the data matrix B and noise matrix N,
respectively, for every block transmitted.Using the LS approach [14, 15], an
estimate of H can be obtained as (21)H^LS=(cHc)−1cHv=c†v, where (cHc)−1cH is the pseudoinverse of c.We will use the following transmitted
training power constraint(22)cHc=PcI, where I is the nT×nT identity matrix; now the channel estimate is (23)H^LS=H+(1Pc)cH(b˜H+n˜). It is clear that the average b˜ over the data signal B represents an extra term for the channel
estimate. This average is exploited by the DDST method at the transmitter or by
the MRST method at the receiver.In order to obtain the MSE achieved by
MRST method, we use the performance analysis of the DDST method, explicitly we
useb˜=(1/N)uB and define the perturbation matrix E=−uTb˜ that will be arithmetically added to the data
signal B every transmitted block. Hence, the
transmitted signal is X=B+E+C and the corresponding received signal is R=(B+E+C)H+N,
then from (21) we obtain (24)H^LS=H+(1Pc)cHn˜.Considering
(22) under optimal training
and E{tr(NHN)}=σn2nR,
the MSE for DDST [9] is given by (25)MSE(H^DDST)=σn2nTnRNPc.Instead to take away the contribution ofb˜ at the transmitter, the MRST uses the plain ST
and removes b˜ in an iterative manner at the receiver. From
(21), the channel estimation error can be expressed as (26)H^MRST=c†([b˜+c]H−b˜^H^+n˜)=H+(1Pc)cH(b˜H−b˜^H^)+(1Pc)cHn˜. It follows that if H^b˜^≃Hb˜, then H^MRST≃H^DDST,
therefore the MSE(H^MRST)≃MSE(H^DDST) as it will be corroborated by the simulation
results in Section 5. With respect to the iterative procedure to obtain a
better estimate of b˜, we follow the steps explained for the SISO
case.


### 3.3. MRST with Alamouti Space-Time Coding

Let us use MRST estimation method with
the𝒢2-OSTBC nT=2,nR=2  system (Alamouti coding scheme). Let us assume N to be even, and (27)X=[s1s2−s2*s1*s3s4−s4*s3*⋮⋮sN−1sN−sN*sN−1*],B=[b1b2−b2*b1*b3b4−b4*b3*⋮⋮bN−1bN−bN*bN−1*],C=[c1c2−c2*c1*c1c2−c2*c1*⋮⋮c1c2−c2*c1*] are the matrices of the transmitted
signal, data signal, and training signal, respectively, with (·)∗ denoting complex conjugation. The matrix’s
rows and columns indicate the transmission time and transmit antenna,
respectively. Assuming flat-fading scenario, no cyclic prefix is required, and
the signal training selection can be done choosing two symbols c1,c2  and exploiting the property of orthogonally
achieved with the OSTBC system, that is, the two column vectors of the matrix C are orthogonal.The estimate of the cycling mean in
flat-fading scenario can be done obtaining the mean of the receive matrixR, denoted as R˜.
Each element of R˜ is given by (28)R˜i,j=1N/2∑k=1N/2Ri+2(k−1),j, where the indexes i,j correspond to Alamouti’s emission time (i=1,2) and transmit antenna (j=1,2) respectively for this particular block
coding transmission.Let us estimate the channel matrixH using (28) and assuming a noisy free scenario
(without loss of generality), hence, we have (29)Avec{H}=vec{R˜},withA=[(c1−b˜^1)(c2−b˜^2)0000(c1−b˜^1)(c2−b˜^2)−(c2−b˜^2)∗(c1−b˜^1)∗0000−(c2−b˜^2)∗(c1−b˜^1)∗],and vec{·} is the vectorization operator stacking all
rows of a matrix on top of each other. Therefore, the channel estimate is given
by (30)vec{H^}=A−1vec{R˜}.Because matrix A is unitary, then (30) can be rewritten as (31)vec{H^}=AHvec{R˜},and finally, (32)H^=[vec{InR}T⊗InT][InR⊗vec{H^}], where InT and InR are the nT×nT and nR×nR identity matrix, respectively, and ⊗ denotes the Kronecker product. The matrix H^ represents the new MIMO channel estimate used
in the iterative procedure to get a new estimate of the mean of the data-bearing
(b˜^).The key idea of this implementation is the way to remove the mean
at the receiver. Instead of subtracts the mean of the data, from the received
matrixR, we incorporate the mean estimate in the matrix A and use the Alamouti decoding procedure to
estimate the channel matrix.


## 3.1. MIMO System Model

We consider a wireless MIMO
communication link withnT transmit and nR receive antennas, operating in a Rayleigh
flat-fading environment. The fading coefficient hi,j is the complex path gain from transmit antenna i to receive antenna j.
We assume that the coefficients are independently complex circular symmetric
Gaussian with unit variance, and then H=[hi,j]∈ℂnT×nR. The expression for the received symbols can be
expressed as  (15)r=xH+n,  where r is the 1×nR received signal vector, x is the 1×nT transmitted signal vector, and n is an 1×nR vector of additive noise terms, assuming that
noise is spatially and temporally white Gaussian with zero mean and independent
real and imaginary parts with variance σn2=N0/2 per dimension.Let us assume the block transmission
scheme with the block lengthN at times k=1,…,N, and we also assume that the channel matrix H remains constant within a block of N symbols, that is, the block length is much
small than the channel coherence time. Under these assumptions, the channel,
within one block, can be written as (16)R=XH+N,where (17)R≜[r(1)r(2)⋮r(N)],X≜[x(1)x(2)⋮x(N)],N≜[n(1)n(2)⋮n(N)] are the matrices of the
received signal, transmitted signals, and noise, respectively [12].Let us denote the set of complex
information symbols prior to space-time encoding as{sλ∣sλ∈𝒰λ,λ=1,2,…,Λ},
where each 𝒰λ denotes a set of signal constellation points.
The symbols s1,s2,…,sΛ are zero-mean mutually uncorrelated random
variables. Let us introduce the vector (18)s≜[s1s2⋯sΛ]T, where (·)T denotes transpose. Note that s∈𝒮,
where 𝒮={s(1)s(2)⋯s(L)} is the set of all possible symbol vectors and L is the cardinality of this set. The N×nT complex matrix-valued function X(s) is called OSTBC [13] if it satisfies that(1)
all the entries ofX(s) are linear functions of the Λ complex variables s1,s2,…,sΛ and their complex conjugates;(2)
for any arbitrarys in ℂΛ×1, (19)XH(s)X(s)=∥s∥2InT,
where InT is the
identity nT×nT matrix, ∥·∥ is the Euclidean norm, and (·)H denotes Hermitian transpose.


## 3.2. Performance Analysis

In order to estimate the channel matrixH,
it should  be emphasized that in
any statistical expectation below, the matrix H is treated as random; at the same time, any
estimator of H is supposed to obtain an estimate of a
particular realization of this random matrix that corresponds to the current
block of the received data.In the conventional ST estimation
technique for MIMO systems [9], a known training N×nT matrix C,
is added arithmetically to the data N×nT matrix B during every block transmitted. In this way,
the transmitted signal matrix X can be expressed as X=B+C.
It is clear that the total transmitted power is distributed between the data
and the training signals, that is, Px=Pb+Pc.The ST system is depicted in Figure2,
where a signal matrix  X
is transmitted over the radio MIMO channel
with the channel matrix H,
and distorted with the noise matrix N.Figure 2
MIMO digital communication system model.Based on the received signalR, the MRST delivers an estimate of the
MIMO channel matrix, denoted as H^;
subsequently the decoder obtains an estimate of  X,
denoted as X^,
and an estimate of the mean of the data, b˜^,
which can be used by the channel estimator block, in an iterative way, to
provide the decoder with a better estimate of H.Now, the task of this channel estimation
algorithm is to recover the channel matrixH based on the knowledge of R and C.
Assuming flat-frequency channel conditions then all the row vectors of the
training matrix C can be equal. Hence, a time-domain estimator
based on the synchronized averaging of the received signal can be implemented.
The time-average of the signal R is  given by (20)v=E{R}=(1N)uR=[(1N)(uB+uC)]H+(1N)uN=[b˜+c]H+n˜, where v is a 1×nR column vector, u is the unit 1×N vector, c is the 1×nT training vector repeated N times and b˜,n˜  represent the time-average of the data matrix B and noise matrix N,
respectively, for every block transmitted.Using the LS approach [14, 15], an
estimate of H can be obtained as (21)H^LS=(cHc)−1cHv=c†v, where (cHc)−1cH is the pseudoinverse of c.We will use the following transmitted
training power constraint(22)cHc=PcI, where I is the nT×nT identity matrix; now the channel estimate is (23)H^LS=H+(1Pc)cH(b˜H+n˜). It is clear that the average b˜ over the data signal B represents an extra term for the channel
estimate. This average is exploited by the DDST method at the transmitter or by
the MRST method at the receiver.In order to obtain the MSE achieved by
MRST method, we use the performance analysis of the DDST method, explicitly we
useb˜=(1/N)uB and define the perturbation matrix E=−uTb˜ that will be arithmetically added to the data
signal B every transmitted block. Hence, the
transmitted signal is X=B+E+C and the corresponding received signal is R=(B+E+C)H+N,
then from (21) we obtain (24)H^LS=H+(1Pc)cHn˜.Considering
(22) under optimal training
and E{tr(NHN)}=σn2nR,
the MSE for DDST [9] is given by (25)MSE(H^DDST)=σn2nTnRNPc.Instead to take away the contribution ofb˜ at the transmitter, the MRST uses the plain ST
and removes b˜ in an iterative manner at the receiver. From
(21), the channel estimation error can be expressed as (26)H^MRST=c†([b˜+c]H−b˜^H^+n˜)=H+(1Pc)cH(b˜H−b˜^H^)+(1Pc)cHn˜. It follows that if H^b˜^≃Hb˜, then H^MRST≃H^DDST,
therefore the MSE(H^MRST)≃MSE(H^DDST) as it will be corroborated by the simulation
results in Section 5. With respect to the iterative procedure to obtain a
better estimate of b˜, we follow the steps explained for the SISO
case.


## 3.3. MRST with Alamouti Space-Time Coding

Let us use MRST estimation method with
the𝒢2-OSTBC nT=2,nR=2  system (Alamouti coding scheme). Let us assume N to be even, and (27)X=[s1s2−s2*s1*s3s4−s4*s3*⋮⋮sN−1sN−sN*sN−1*],B=[b1b2−b2*b1*b3b4−b4*b3*⋮⋮bN−1bN−bN*bN−1*],C=[c1c2−c2*c1*c1c2−c2*c1*⋮⋮c1c2−c2*c1*] are the matrices of the transmitted
signal, data signal, and training signal, respectively, with (·)∗ denoting complex conjugation. The matrix’s
rows and columns indicate the transmission time and transmit antenna,
respectively. Assuming flat-fading scenario, no cyclic prefix is required, and
the signal training selection can be done choosing two symbols c1,c2  and exploiting the property of orthogonally
achieved with the OSTBC system, that is, the two column vectors of the matrix C are orthogonal.The estimate of the cycling mean in
flat-fading scenario can be done obtaining the mean of the receive matrixR, denoted as R˜.
Each element of R˜ is given by (28)R˜i,j=1N/2∑k=1N/2Ri+2(k−1),j, where the indexes i,j correspond to Alamouti’s emission time (i=1,2) and transmit antenna (j=1,2) respectively for this particular block
coding transmission.Let us estimate the channel matrixH using (28) and assuming a noisy free scenario
(without loss of generality), hence, we have (29)Avec{H}=vec{R˜},withA=[(c1−b˜^1)(c2−b˜^2)0000(c1−b˜^1)(c2−b˜^2)−(c2−b˜^2)∗(c1−b˜^1)∗0000−(c2−b˜^2)∗(c1−b˜^1)∗],and vec{·} is the vectorization operator stacking all
rows of a matrix on top of each other. Therefore, the channel estimate is given
by (30)vec{H^}=A−1vec{R˜}.Because matrix A is unitary, then (30) can be rewritten as (31)vec{H^}=AHvec{R˜},and finally, (32)H^=[vec{InR}T⊗InT][InR⊗vec{H^}], where InT and InR are the nT×nT and nR×nR identity matrix, respectively, and ⊗ denotes the Kronecker product. The matrix H^ represents the new MIMO channel estimate used
in the iterative procedure to get a new estimate of the mean of the data-bearing
(b˜^).The key idea of this implementation is the way to remove the mean
at the receiver. Instead of subtracts the mean of the data, from the received
matrixR, we incorporate the mean estimate in the matrix A and use the Alamouti decoding procedure to
estimate the channel matrix.


## 4. Simulation and Results for SISO Systems

Equalization is a well-known technique
used to combat intersymbol interference (ISI) whereby the receiver attempts to
compensate for the effects of the channel of the transmitted symbols. An
equalizer attempts to determine the transmitted data from the received
distorted symbols using an estimate of the channel that caused the distortions.
In this contribution, we consider two types of equalizers widely used in
communications systems: the MMSE and MLSE equalizers. The MMSE has lesser
complexity and performance in terms of BER than the MLSE equalizer that is
optimal for ISI.


### 4.1. SISO System Using MMSE Equalizer

We considered a
time-invariant random three-tap frequency-selective Rayleigh fading channelh(k) with h(0)≠0 and h(2)≠0.
The channel coefficients were complex Gaussian, i.i.d. with unit variance, rescaled
to achieve unitary mean energy. The sequence {b(k)}∈{−1,+1} is an R.V with uniform p.d.f. and variance σb2.
The parameters σb2 and σc2 are chosen for ST and DDST independently such
that σb+e2+σc2=1.
The training to information power ratio σc2/σb+e2 was set to −6.9798dB with P=7.
The block length is fixed to N=420 and a cyclic prefix of length P added at the beginning of each block in both
ST and DDST methods. The channel estimated is used to design an MMSE equalizer of
length 11 and equalization delay d=7.
All simulations that were run until 1000 blocks with errors were found.Figure3 shows the channel estimate MSE
obtained from the three iterative procedures considered (a) DDST with DDD
removal exposed in [11], (b) IST scheme introduced in [7], and (c) MRST method
presented here. It can be observed that IST and MRST have a significant
approach to DDST (from 10 dB to 20 dB) just after two iterations.Figure 3
Channel estimate MSE with MMSE equalizer.Figure4 shows the BER
performance comparison where for 2 iterations, all schemes show similar
behavior for low and medium SNR levels; DDST exhibits a slight advantage for
high SNRs because it achieves a better channel estimate at these levels as is
shown in Figure 3.Figure 4
BER performance of BPSK signals with MMSE equalizer.


### 4.2. SISO System Using MLSE Equalizer

In order to get a better estimate(B˜^) of B˜ for both  IST and MRST schemes, we
use an MLSE equalizer with traceback length of 11. To be more specific, this
equalizer comprises a Viterbi algorithm, which finds the most likely data
sequence transmitted. Although, to perform close to ideal MLSE, the equalizer
requires traceback lengths of the order of 5-6 times the ISI span, we chose the
traceback length of 11, that is, the same number of taps of the MMSE equalizer
used in Section 4.1.Due to the fact that DDST-DDD removal
[11] works with the equalizer output before proceeding with a new hard-decision
process, that is, equalization and detection stages are carried out separately,
it cannot exploit the benefits of the MLSE equalizer. In order to have similar
conditions for the DDST method in the decoding procedure, firstly we used the
MMSE equalizer, secondly we removed the data distortion, and finally we used
the MLSE equalizer to obtain the data symbols. Figure 5 shows the BER
performance of these three methods and the conventional ST method.Figure 5
BER performance of BPSK signals with MLSE equalizer.Clearly, we observe the benefits of use IST or MRST coupling with an MLSE
equalizer, where both schemes  outperform DDST-DDD removal
when it only uses the MLSE equalizer. However, there is no noticeable
performance difference when DDST uses the concatenation of the MMSE and MLSE
equalizers.Figure6 shows the block error rate (BLER)
performance.  BLER is the statistical
measurement of the ratio of the number of blocks with error received to the
total number of blocks transmitted and it is part of the performance
requirements of 3GPP test. We observe that there is no noticeable performance
difference between the MRST and IST with one iteration, and DDST-DDD removal
using the equalizer concatenation and removing the data distortion.Figure 6
BLER performance of BPSK signals with MLSE equalizer
using the three iterative methods and ST.


## 4.1. SISO System Using MMSE Equalizer

We considered a
time-invariant random three-tap frequency-selective Rayleigh fading channelh(k) with h(0)≠0 and h(2)≠0.
The channel coefficients were complex Gaussian, i.i.d. with unit variance, rescaled
to achieve unitary mean energy. The sequence {b(k)}∈{−1,+1} is an R.V with uniform p.d.f. and variance σb2.
The parameters σb2 and σc2 are chosen for ST and DDST independently such
that σb+e2+σc2=1.
The training to information power ratio σc2/σb+e2 was set to −6.9798dB with P=7.
The block length is fixed to N=420 and a cyclic prefix of length P added at the beginning of each block in both
ST and DDST methods. The channel estimated is used to design an MMSE equalizer of
length 11 and equalization delay d=7.
All simulations that were run until 1000 blocks with errors were found.Figure3 shows the channel estimate MSE
obtained from the three iterative procedures considered (a) DDST with DDD
removal exposed in [11], (b) IST scheme introduced in [7], and (c) MRST method
presented here. It can be observed that IST and MRST have a significant
approach to DDST (from 10 dB to 20 dB) just after two iterations.Figure 3
Channel estimate MSE with MMSE equalizer.Figure4 shows the BER
performance comparison where for 2 iterations, all schemes show similar
behavior for low and medium SNR levels; DDST exhibits a slight advantage for
high SNRs because it achieves a better channel estimate at these levels as is
shown in Figure 3.Figure 4
BER performance of BPSK signals with MMSE equalizer.


## 4.2. SISO System Using MLSE Equalizer

In order to get a better estimate(B˜^) of B˜ for both  IST and MRST schemes, we
use an MLSE equalizer with traceback length of 11. To be more specific, this
equalizer comprises a Viterbi algorithm, which finds the most likely data
sequence transmitted. Although, to perform close to ideal MLSE, the equalizer
requires traceback lengths of the order of 5-6 times the ISI span, we chose the
traceback length of 11, that is, the same number of taps of the MMSE equalizer
used in Section 4.1.Due to the fact that DDST-DDD removal
[11] works with the equalizer output before proceeding with a new hard-decision
process, that is, equalization and detection stages are carried out separately,
it cannot exploit the benefits of the MLSE equalizer. In order to have similar
conditions for the DDST method in the decoding procedure, firstly we used the
MMSE equalizer, secondly we removed the data distortion, and finally we used
the MLSE equalizer to obtain the data symbols. Figure 5 shows the BER
performance of these three methods and the conventional ST method.Figure 5
BER performance of BPSK signals with MLSE equalizer.Clearly, we observe the benefits of use IST or MRST coupling with an MLSE
equalizer, where both schemes  outperform DDST-DDD removal
when it only uses the MLSE equalizer. However, there is no noticeable
performance difference when DDST uses the concatenation of the MMSE and MLSE
equalizers.Figure6 shows the block error rate (BLER)
performance.  BLER is the statistical
measurement of the ratio of the number of blocks with error received to the
total number of blocks transmitted and it is part of the performance
requirements of 3GPP test. We observe that there is no noticeable performance
difference between the MRST and IST with one iteration, and DDST-DDD removal
using the equalizer concatenation and removing the data distortion.Figure 6
BLER performance of BPSK signals with MLSE equalizer
using the three iterative methods and ST.


## 5. Simulation and Results for MIMO Systems

In what follows, we illustrate the
performance of the proposed scheme on MIMO systems working in a flat-fading
scenario. We use the𝒢2-OSTBC system, nT=nR=2 transmit and receive antennas, respectively, with
ideally uncorrelated elements. The block length is fixed to N=256 symbols, and all simulations were run until 1000
blocks were in error. The BER is represented as a function of the average SNR,
where N0=EsnT/SNR, and Es is the average energy per symbol.A QPSK symbol constellation
is used, and the power transmitted for each antenna is normalized to one, that
is,Px=1 [Watt].Figure7 depicts the channel estimate
MSE comparisons for DDST and the MRST techniques with one and two iterations;
both of them use Pc=17% of the power transmitted Px,
that correspond to an approximate upper bound of the region with the best BER
shown in Figure 10. Making an analysis of this performance, we can realize that
the proposed scheme starts to approach the DDST performance from 10 [dB]. The
MRST plot for the second iteration reaches the best approach. Note that this
benefit comes at the price of decoding complexity.Figure 7
Channel estimate MSE of𝒢2-OSTBC 2×2.Figure8 shows the BER performance
comparison between TDM with 17% bandwidth loss, DDST with DDD removal and the MRST
algorithm. We observe that DDST and MRST plots with one or two iterations
practically achieved the same performance. TDM-based channel estimation
attained the best BER performance at the expenses of 17% bandwidth loss.Figure 8
BER comparisons for various channel estimation techniques.Figure9 depicts the MSE of MRST versus Pc when SNR=15dB. For example, with Pc=0.125, we have that MSE(Pc)=4[0.0316/(0.125*256)]=0.004031.
A better MSE performance
is directly proportional to the training power used.Figure 9
MSE of channel estimates versus power fraction for training signal.Figure 10
BER versus power fraction for training signal.Figure10 plots the BER of MRST versus Pc from SNR=10dB to 15 dB. Note that the minimum
is achieved in the range where 0.1<Pc≤0.17 approximately. Because the BER achievement is
the key and critical factor in digital communication systems, we worked in the
region with the best BER in the selection of the training power assigned. This
selection is only a guide or suggestion for this type of radio propagation
conditions at these particular SNR levels. Similar training power range was
chosen for DDST in [9], but using a spatial multiplexing system. A more
detailed analysis of the training power allocation problem is shown in [16]. Particularly,
the simulations showed in Figures 9
and 10 were run until 10 000 blocks
were in error.


## 6. Conclusion

low-complexity iterative superimposed
training schemes that work jointly with equalization stage can offer similar or
better performance than the iterative symbol-by-symbol detection DDST with DDD
removal. The MRST introduced in this work effectively compensates the data
dependent distortion that ST is unable to deal with, but at the receiver side.
This leads to communication systems that perform similar to DDST but without
suffering its drawbacks. Furthermore, MRST can be successfully applied to both
SISO and MIMO systems, and in cooperation with the most widely used equalizers.
Particularly, for the SISO case, MRST shows similar performance like the previous
proposed method IST, but it avoids the inverse matrix computation(C+B˜^)−1 that every iteration IST does. Although the
results of both iterative show insignificant gaps in the performance of BER and
MSE, MRST is preferred by its less complexity hardware implementation.For the MIMO case, the performance
results are closely similar, that is, the performance of DDST is attained with
the proposed method. Additionally, iterative ST methods can still be applied to
time-varying channels, while DDST based system does not. This is true because
the concept of cyclic mean in DDST is meaningless due to the fact that every
symbol of the block transmitted is distorted in different way by the channel.



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*Source: 535269-2008-10-28.xml*