Adaptive channel aided decision feedback
equalisation for SISO and MIMO systems
Y. Lee and W.-R. Wu
Abstract: Error propagation can seriously affect the performance of an adaptive decision feedback equaliser (DFE), especially when operated in time-varying channel environments. Error propagation not only affects DFE decisions, but also disturbs the DFE adaptation. The paper focuses on improving the robustness against error propagation for the least-mean-square (LMS) based minimum mean-squared-error DFE (MMSE-DFE). A specifically designed channel estimator is introduced to help the DFE adaptation in the decision-directed (DD) mode. Unlike the conventional DFE, the proposed adaptive channel-aided DFE (ACA-DFE) only adapts the feedforward filter with the LMS algorithm. The feedback filter, however, is obtained from the postcursors of the estimated channel convolved with the feedforward filter. As a result, the proposed ACA-DFE can reduce the error propagation effect and perform better than the conventional adaptive DFE. We also demonstrate that the ACA-DFE can be extended to multiple-input multiple-out (MIMO) systems improving the performance of the conventional MIMO DFE.
1 Introduction
Decision feedback equaliser (DFE) is a well-known channel equaliser in single-input single-output (SISO) systems[1–3]. It has been widely used in digital communications to suppress inter-symbol interference (ISI) for over several decades. When the channel spectrum exhibits spectral nulls due to multipath propagation, the DFE performs signifi-cantly better than the linear equaliser (LE). Though the maximum likelihood sequence estimator (MLSE) [4] can have better performance than the DFE, the computational complexity is much more higher.
A DFE incorporates a feedforward filter (FFF) operating on the received signal to suppress precursor ISI, and a feedback filter (FBF) operating on previously detected symbols to suppress postcursor ISI. A DFE uses a nonlinear decision device at the output, and the output represents a noise-free replica of the transmitted symbol assuming that the probability of decision error is small. However, if a symbol is detected incorrectly, the next input to the FBF will be in error. As this error advances through the feedback loop, the probability of making an error in the detection of subsequent symbols will be increased. It can result in error propagation (EP) that causes bursts of incorrect decisions and a corresponding increase in the decision-error rate[5]. A number of schemes were proposed to reduce EP for DFE. A technique combining DFE with partial response precoding and detection was presented in [6]. In [7–9], soft decisions and specifically designed constraints were suggested to prevent questionable decisions
from being used in the feedback loop and thereby the probability of error burst was reduced. Besides, a periodic transmission of a short resetting sequence calculated based on a certain steady-state error probability was used to control the error behaviour of DFE[10].
Recently, much attention is paid in the development of multiple-input multiple-out (MIMO) systems. With the use of multiple antennas at both transmitter and receiver, the spectral efficiency of a communication system can be increased dramatically[11]. For high data-rate transmission, frequency selective fading is present between pairs of transmit and receive antennas. This brings a great design challenge at MIMO receivers. One solution for this problem is to use an MIMO DFE, where both the FFF and the FBF are extended to have multiple inputs and multiple outputs, i.e., multi-dimensional FFF and multi-dimensional FBF performing multi-dimensional channel equalisation[12–14]. For the MIMO DFE, the problem of EP is even more severe than that of its SISO companion owing to the complicated channel configuration and the need to detect signal buried in ISI in addition to co-channel interference (CCI), in addition to noise.
Since the communication environment may be time-varying, tap weights in the DFE should be updated dynamically for better performance [15]. The least-mean-square (LMS) adaptive algorithm[16]is well-known for its simplicity and robustness, and is often utilised to adapt both the FFF and the FBF in SISO DFE systems. It can be shown that the LMS algorithm is also attractive to the adaptive MIMO DFE for dispersive MIMO channels [17, 18]. As described, the EP effect will have a greater impact in the adaptive implementation of the DFE. A decision error not only affects the DFE future outputs, but also disturbs the reference signal of the adaptive algorithm. As a result, the DFE will be adapted toward an incorrect direction. In the worst case, EP can diverge the DFE adaptation.
The most popular design strategy for channel equalisa-tion by far is the use of the minimum mean-squared-error (MMSE) criterion. Its well-accepted theoretical framework E-mail: ymlee@ncnu.edu.tw
Y. Lee is with the Graduate Institute of Communication Engineering, National Chi Nan University, Nantou Hsien 545, Taiwan, Republic of China W.-R. Wu is with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, Republic of China
rThe Institution of Engineering and Technology 2006 IEE Proceedings online no. 20050569
doi:10.1049/ip-com:20050569
and amenability to adaptive implementation make it very attractive for practical usage. Another strategy for equaliser design is to use the minimum bit-error rate/minimum symbol-error rate (MBER/MSER) criterion [19]. Various adaptive realisations for the MBER/MSER equalisation were proposed in [20–23]. Though better results can be obtained in terms of this criterion, there is no guarantee that the global minimum can be reached. In addition, the convergence rate may be slower and the computational complexity may be higher. All these may make the MBER/ MSER equaliser less effective in time-varying channel environments. In the following, we only consider the DFE optimised by the MMSE criterion.
In this paper, an LMS-based MMSE-DFE is proposed to reduce the EP effect. A particularly designed channel estimator is introduced to the conventional DFE structure. The resultant adaptive channel-aided (ACA) DFE can perform better than the conventional adaptive DFE and the EP effect can be effectively reduced. This approach is different from those channel-estimation-based DFEs pro-posed in [24, 25], where both the FFF and the FBF are calculated based on the estimated channel response. Since matrix multiplications and inversions are involved, its computational complexity will be high for time-varying channels. In the proposed ACA-DFE, however, the adaptive structure is remained. Only the FFF is adapted with the LMS algorithm, and the FBF is obtained from the postcursors of the up-to-date estimated channel convolved with the FFF. Generally, this will result in lower com-putational complexity. We will also show that our SISO ACA-DFE can be extended to an MIMO ACA-DFE.
This paper is organised as follows. In Section 2, the background materials for the MMSE-DFE for both SISO and MIMO systems are described. In Section 3, we propose the new ACA-DFE and explain its operation mechanisms. This result can be extended to use in MIMO channels resulting an MIMO ACA-DFE. Finally, simulation results and conclusions are presented in Section 4 and 5, respectively. Throughout the paper, we utilise the super-scripts ( )*
, ( )T
, and ( )H
to denote conjugation, transposition, and Hermitian transposition, respectively, and the operator E{ } to denote mathematical expectation.
2 Background
2.1
Conventional DFE for SISO systems
Let the tap weights of the FFF and the FBF of a DFE be denoted by the column vectors f with length a and b with length b, respectively. The complex dispersive channel is modelled by discrete path hlwith 0 l L 1, in which L is the channel order. We assume that the transmitted symbol aðkÞ is randomly generated and the noise sample sequence nðkÞ is zero mean, white, and Gaussian distrib-uted. The received discrete-time equivalent baseband signal at the kth time instant can then be modelled as
xðkÞ ¼X
L1
l¼0
hlaðk lÞ þ nðkÞ ¼ hTaðkÞ þ nðkÞ ð1Þ with h¼ ½h0 h1 hL1T
and aðkÞ ¼ ½aðkÞ aðk 1Þ
aðk L þ 1ÞT. Let x(k) be the input vector of the FFF with length a, i.e. xðkÞ ¼ ½xðkÞ xðk 1Þ xðk a þ 1ÞT, and ^aðkÞ be the input vector of the FBF with length b, i.e. ^aðkÞ ¼ ½^aðk k 1Þ ^aðk k 2Þ ^aðk k bÞT, where k is a suitably chosen decision delay. For the training based MMSE-DFE, the error signal can then be written as
eðkÞ ¼ aðk kÞ ðfHxðkÞ bH^aðkÞÞ ð2Þ
Assuming decisions are correct, i.e., ^aðkÞ ¼ aðkÞ, we can write the MSE as
EfjeðkÞj2g ¼ fH Rxxf fHRxab fHpxa b H RHxaf þ bH Raabþ bHpaa p H xaf þ p H aabþ s2a ð3Þ with Rxx¼ EfxðkÞxHðkÞg, Raa¼ EfaðkÞaHðkÞg, Rxa¼
EfxðkÞaHðkÞg, p
xa¼ EfxðkÞaðk kÞg, paa¼ EfaðkÞ
aðk kÞg, and s2
a¼ Efaðk kÞa
ðk kÞg. To obtain
the optimum solution, we set the gradient of EfjeðkÞj2g with respect to f*and b*to zero. This results in
fopt¼ Rxx 1 s2 a RxaRHxa 1 pxa ð4Þ bopt¼ 1 s2 a RHxafopt ð5Þ
As we can see, the optimum solution relies on the correlation matrices which cannot be known in advance, and the matrix inverse operation in (4) requires extensive computation. A simple alternative to find the optimum tap weights is to use an adaptive training method. The LMS algorithm is known to be a simple yet effective choice. The LMS update equations for f and b are expressed as[16]
fðk þ 1Þ ¼ f ðkÞ þ mfxðkÞeðkÞ ð6Þ
bðk þ 1Þ ¼ bðkÞ mb^aðkÞeðkÞ ð7Þ
where fðkÞ and bðkÞ are the estimates of foptand boptat the
kth time instant, mfand mbare the step sizes controlling the
convergence rate, and eðkÞ is the error signal given in (2). A typical adaptation process consists of a training mode and a decision-directed (DD) mode. Initially, the training mode is launched and sufficient training symbols are transmitted to let both fðkÞ and bðkÞ converge around the optimum. Then, the DFE switches to the DD mode in which DFE decisions are used as the reference signal and the DFE is continuously adapted. However, DFE decisions may not be always reliable, especially in time-varying channels. Deci-sion errors not only affect the DFE future output, but also disturb the DFE adaptation. In the worst case, the adaptive DFE can diverge, and another training period needs to be re-initiated.
2.2
Conventional DFE for MIMO systems
The SISO DFE can be extended to an MIMO DFE for the equalisation of MIMO channels. Here, we use M N to signify the configuration with M transmit and N receive antennas, and L to indicate the maximum order for the multi-dimensional channel. Generally, Mr N is assumed. A sequence of data symbols amðkÞ (1 m M) is
transmitted from the mth antenna. We define amðkÞ ¼
½amðkÞ amðk 1Þ amðk L þ 1Þ T
as a collection of L successive data symbols from the mth antenna. These data symbols are randomly generated (both in time and space domain) and drawn from the same signal constellation with a variance of s2
a. All M data sequences are transmitted over
the MIMO channel. The sampled channel response from the mth transmit antenna to the nth receive antenna is given by hnm ¼ ½h0nm h1nm h L1 nm T ð8Þ for m¼ 1; 2; . . . ; M and n ¼ 1; 2; . . . ; N . We can assemble the vectors hnm into a matrix of size L N as
for m¼ 1; 2; . . . ; M. We also let nðkÞ ¼ ½n1ðkÞ n2ðkÞ
nNðkÞ T
be an N-dimensional noise vector with zero mean, white, and Gaussian distributed elements. With the formulation, M different symbols are simultaneously transmitted through M antennas and received by N antennas to yield the N-dimensional signal vector xðkÞ ¼ ½x1ðkÞ x2ðkÞ xNðkÞ
T
. With this premise, the received discrete-time equivalent baseband signal vector can be written as
xðkÞ ¼X
M
m¼1
HTmamðkÞ þ nðkÞ ð10Þ
The formulation of the MIMO DFE is similar to that of the SISO DFE. Nevertheless, M decision devices are employed for M different data sequences. For simplicity, we only consider the most basic form of the MIMO DFE which does not include any successive interference cancellation (SIC) action [2]. To be consistent with the previous derivation for the SISO DFE, we first arrange the structure of the FFF into M matrices Fm, for m¼ 1; 2; . . . ; M, with
dimension a N, and the FBF into M matrices Bm, for m¼ 1; 2; . . . ; M, with dimension b M. Both a and b are selected to be long enough to cover the ISI effect in the multi-dimensional channel. The matrices Fm and Bm
have the forms as
Fm¼ fm01 fm02 fmN0 .. . .. . . . . .. .
fma11 fma12 fmNa1
2 6 6 4 3 7 7 5 ¼ ½fm1 fm2 . . . fmN ð11Þ Bm¼ b0 m1 b0m2 b0mM .. . .. . . . . .. . bb1m1 bb1m2 bb1mM 2 6 6 4 3 7 7 5 ¼ ½bm1 bm2 . . . bmM ð12Þ To be more compact, we stack the components in the above matrices to form the following vectors
fm¼ ½ f T m1 f T m2 . . . f T mN T ð13Þ bm¼ ½bTm1 b T m2 . . . b T mM T ð14Þ for the mth FFF and the mth FBF, respectively. Similarly, the successive received signal of (10) for the nth antenna can be first grouped as xnðkÞ ¼ ½xnðkÞ xnðk 1Þ xnðk a þ 1ÞT, for n¼ 1; 2; . . . ; N , and then the total received signal vector is described as xðkÞ ¼ ½xT
1ðkÞ
xT
2ðkÞ xTNðkÞ T
, which serves as the input to the FFF. The most recent b decisions from the output of the mth decision device after delay km are labelled as ^amðkÞ ¼
½^amðk km 1Þ ^amðk km 2Þ ^amðk km bÞT, for m¼ 1; 2; . . . ; M. Here, we assume that all decision delays kmare known at the receiver. Then, we can write the overall
decision vector as aðkÞ ¼ ½aT
1ðkÞ aT2ðkÞ aTMðkÞ T
, which constitutes the input to the FBF. After that, we may express the estimate in the output of the MIMO DFE prior to the mth decision device at the kth time instant as
ymðkÞ ¼ f H
mxðkÞ b H
maðkÞ ð15Þ
and the estimation error for it as
emðkÞ ¼ amðk kmÞ ymðkÞ ð16Þ
for m¼ 1; 2; . . . ; M. We see that the error signal given in (16) is similar to that of the SISO case except dimension
expansion. Architecture-wise, the MIMO DFE can be treated as a generalisation of the SISO DFE, where the scalar delay line, the taps, and the decision are replaced by the vector delay line, the matrix taps, and the decision vector, respectively. With reference to (4) and (5), for each data sequence and the corresponding decision device, we may have the optimum FFF and FBF expressed as
fm;opt¼ Rxx 1 s2 a RxaRHxa 1 pxam ð17Þ bm;opt¼ 1 s2 a RHxafm;opt ð18Þ
with the matrix and vector elements defined similar to those for the SISO DFE. Again, to avoid the matrix inverse operation in the FFF calculation, we may adopt the LMS algorithm to find the optimum tap weights recursively. It is not difficult to obtain the update equations for the MIMO DFE as
fmðk þ 1Þ ¼ fmðkÞ þ mfxðkÞemðkÞ ð19Þ
bmðk þ 1Þ ¼ bmðkÞ mbaðkÞemðkÞ ð20Þ
for m¼ 1; 2; . . . ; M. Since the received signal is also corrupted by CCI in MIMO channel environments, it tends to make the error signal in the above update equations more noisy. For similar ISI conditions, the performance of the adaptive MIMO DFE is worse than that of the adaptive SISO DFE.
3 Proposed adaptive channel-aided DFE (ACA-DFE)
3.1
ACA-DFE for SISO systems
Figure 1 is the block diagram of the proposed ACA-DFE for SISO systems. To obtain the channel response, we first introduce a channel estimator in the DFE structure. Let the coefficients of the included channel estimator be denoted as q and its dimension is g 1. The value of g is chosen to be larger than or equal to that of the channel order L. For convenience, we choose g¼ L. According to Fig. 1, the channel estimator q is tuned by a new error signal eqðkÞ,
and the cost function for the optimisation of q can be written as min q EfjeqðkÞj 2g ¼ min q EfjxðkÞ q H~aðkÞj2g ð21Þ
where xðkÞ is the received signal and ~aðkÞ ¼ ½^aðkÞ ^
aðk 1Þ ^aðk g þ 1ÞT is the input vector to the channel estimator. Assume that decisions are correct and input data symbols are white. We can then calculate the input correlation matrix for q as R~a~a¼ Ef~aðkÞ~aHðkÞg ¼
s2
aIg, where Ig is a g g identity matrix, and the
cross-correlation vector for aðkÞ and xðkÞ as p~ax¼
Ef~aðkÞxðkÞg ¼ s2
ah
. From (21), the optimum q solved
by the classical Wiener solution is qopt¼ R1
~
a~apax¼ h
ð22Þ
We see that the optimum q equals the complex conjugate of the channel response. As previous, we can use the LMS algorithm to approach qoptrecursively. The update equation
is stated as
qðk þ 1Þ ¼ qðkÞ þ mq~aðkÞeqðkÞ ð23Þ
where qðkÞ is the estimate of qoptat the kth time instant and
mqis the step size for the adaptation. We observe that the
channel estimation problem is essentially a system identifi-cation problem. For uncorrelated input data symbols, the eigenvalues of the input correlation matrix are all identical, and thus the eigenvalue spread equals unity, which is the minimum possible value. It is well-known that the convergence rate of the LMS algorithm is inversely proportional to the eigenvalue spread [16]. Thus, the convergence of qðkÞ is expected to be fast and stable.
Here, we make use of this channel estimator and propose a new DFE structure, i.e. the ACA-DFE. Our approach uses a basic property of the DFE, i.e., the postcursors of the channel response convolved with the FFF is cancelled by the FBF. For completeness, we now show the property formally. It is simple to see that the convolution of the channel and the FFF results in a response of length aþ g 1. Thus, for perfect postcursor cancellation, we must have b a þ g 2 k. Without loss of generality, we let b¼ a þ g 2 k. Represent the convolution of qopt
and foptas Pfopt, where P is anða þ g 1Þ a matrix as
P¼ h0 0 0 h1 h0 0 0 .. . . . . .. . hg1 hg2 h0 0 0 0 hg1 hg2 h0 0 0 .. . . . . . . . .. . 0 0 hg1 hg2 h0 0 0 0 hg1 hg2 h0 .. . . . . .. . 0 0 hg1 hg2 0 0 hg1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð24Þ We can further partition P as P¼ ½PT
r P
T
p
T
, where Pris
of dimension ðk þ 1Þ a and Pp is of dimension
ða þ g 2 kÞ a. It is not difficult to observe that
Prfopt corresponds to the precursor response of Pfopt
while Ppfopt the postcursor response. Recall that the
optimum FFF and FBF for SISO systems is calculated using (4) and (5), respectively. With some manipulations, we can derive
1 s2
a
RHxa¼ Pp ð25Þ
From (5), we then obtain bopt¼ Ppfopt. This result can be
re-stated as
bopt¼ postfqopt foptg ð26Þ
where # denotes the convolution operation and post{ } denotes the postcursor-taking operation. This result
suggests an adaptation approach for the training-based MMSE-DFE. Let fðkÞ and bðkÞ be the FFF and FBF at the kth time instant. With reference to (26), we can let
bðkÞ ¼ postfqðkÞ f ðkÞg ð27Þ
in which qðkÞ is the channel estimate at the kth time instant. If qðkÞ converges to qopt, bðkÞ will converge to bopttoo. The
difference between this approach and the conventional method lies in that only fðkÞ is adapted (not both f ðkÞ and bðkÞ). For the conventional adaptive DFE in the DD mode, the scenario is that both fðkÞ and bðkÞ are adapted based on the LMS update equations as given in (6) and (7). We observe that if there is a decision error, the error will immediately reflect to ^aðkÞ and then eðkÞ. Note that the adaptation of fðkÞ involves erroneous eðkÞ only while that of bðkÞ involves both erroneous ^aðkÞ and erroneous eðkÞ. The two error sources in (7) will make bðkÞ quite sensitive to decision errors. Alternatively, in the proposed method, only fðkÞ is adapted as given in (6). By using (27) to calculate bðkÞ, the overall DFE will perform much better. Although the effect of decision error will also pass to eqðkÞ which will
perturb the adaptation of qðkÞ, the influence is smaller. This is because the convergence of qðkÞ for channel estimate is much faster and more stable than that of the DFE. In one word, with the proposed operation, the resultant ACA-DFE will be less sensitive to decision error and the EP effect will be reduced.
3.2
ACA-DFE for MIMO systems
In Section 2.2, we have already shown that for the mth decision device in MIMO DFE, the optimum formulation is similar to that for SISO DFE except dimension expansion. This motivates us to extend the idea of the SISO ACA-DFE to MIMO ACA-DFE. The block diagram of the MIMO ACA-DFE is described in Fig. 2. First, we define the channel estimators qnm with dimension g 1, for m ¼ 1; 2; . . . ; M and n ¼ 1; 2; . . . ; N , to estimate hnm given in (8). Following the development presented
previously, we can express this task as a system identifica-tion problem as min qnm Efjeq;nmðkÞj2g ¼ min qnm EfjxnðkÞ qHnm~amðkÞj2g ð28Þ
where xnðkÞ is the received signal from the nth antenna and
~
amðkÞ ¼ ½^amðkÞ ^amðk 1Þ ^amðk g þ 1Þ T
is the deci-sion vector from the mth decideci-sion device as the input to the corresponding channel estimators. Similar to (22), the solution is in the form as
qnm;opt¼ h
nm ð29Þ
for m¼ 1; 2; . . . ; M and n ¼ 1; 2; ; N . Referring to (23), we can then use the LMS algorithm to approach qnm;opt
too. Likewise, we define matrices Pmn with
dimen-sion ða þ g 1Þ a, for m ¼ 1; 2; . . . ; M and n¼ 1; 2; . . . ; N , as Pmn¼ h0nm 0 0 h1 nm h0nm 0 0 .. . . . . .. . hg1nm hg2nm h0 nm 0 0 0 hg1 nm hg2nm h0nm 0 0 .. . . . . . . . .. . 0 0 hg1 nm hg2nm h0nm 0 0 0 hg1nm hg2nm h0nm .. . . . . .. . 0 0 hg1nm hg2nm 0 0 hg1nm 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð30Þ We then partition Pmn as Pmn ¼ ½PTr;mn PTp;mnT, where Pr;mn is of dimension ðkmþ 1Þ a and Pp;mn is of
dimension ða þ g 2 kmÞ a. By the definition that
Rxa¼ EfxðkÞaHðkÞg, we can have
1 s2 a RH xa¼ Pp;11 Pp;12 Pp;1N .. . .. . . . . .. . Pp;M 1 Pp;M 2 Pp;MN 2 6 4 3 7 5 ð31Þ
and the mth optimum FBF given in (18) can be specified as
bm;opt¼
PN n¼1
postfqn1;opt fmn;optg
PN n¼1
postfqn2;opt fmn;optg
.. . PN
n¼1
postfqnM ;opt fmn;optg
2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð32Þ
in which fmn;optis the nth sub-vector in fm;opt, as represented
in (13). While both qnm;optand fm;optare estimated using the
LMS algorithm, the estimation of bm;opt at the kth time
instant can then be calculated as
bmðkÞ ¼ PN n¼1 postfqn1ðkÞ fmnðkÞg PN n¼1 postfqn2ðkÞ fmnðkÞg .. . PN n¼1 postfqnMðkÞ fmnðkÞg 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð33Þ
Since qnm can estimate the corresponding channel response
hnm, for the same reason described in the SISO case, the
proposed operation in (33) can enhance the adaptation of the FBF. The resultant MIMO ACA-DFE can then improve the robustness against EP for MIMO channel equalisation.
4 Simulations
Computer simulations are conducted to demonstrate the effectiveness of the proposed DFE and MIMO ACA-DFE. In the first part, we consider SISO channels. In the second part, we consider MIMO channels. All transmitted symbols are randomly generated and then modulated by quadrature phase-shift keying (QPSK). All decision delays are chosen to optimise the performance. In all figures, at least 500 simulation runs are averaged to obtain each simulated result.
4.1
Experiment 1: ACA-DFE
In this set of simulations, we demonstrate that the pro-posed ACA-DFE can provide more robust and stable performance than the conventional adaptive DFE against EP under severe ISI environments. We first consider a static channel chosen from [4, p. 616], which is ½0:227 0:460 0:688 0:460 0:227T ( Proakis C channel). The parameters a, b and g for those filters are set to be 9, 9 and 5, respectively. For comparison, we also show the case of ACA-DFE with perfect channel state information (CSI). The signal-to-noise ratio (SNR) is set as 25 dB. Here, mf ¼ mb¼ 0:005, mq¼ 0:002, and the number of training
symbols Tt¼ 2000. The DD mode follows immediately
after the training mode. Figure 3 gives the learning curves for various equalisation schemes. We see that there is no big difference in performance between the proposed ACA-DFE with and without perfect CSI. It implies that the channel estimator works fairly well. The ACA-DFE performs better than the conventional adaptive DFE in the DD mode in this severe ISI scenario. To demonstrate the merits of the proposed ACA-DFE further, we give the relation between the average SER and the step size used in the FFF (the same step size is used in the FBF of the conventional adaptive DFE as well) in Fig. 4. This figure reveals that the ACA-DFE always has lower SER than the conventional adaptive DFE with the same step size. There are a couple of things that we can observe from the figure. First, there is an optimum step size for a DFE. As known, for the LMS algorithm, the smaller the step size, the smaller the output MSE in the steady state ( possibly the lower the average SER). However, a smaller step size will also make the convergence slower. As a result, there exists an optimum step size balancing these two effects. The optimum mf giving
the lowest SER is around 0.005 for both schemes, and the
0 5000 10000 15000
0 2
number of iterations
decision MSE, dB
conventional adaptive DFE
) MMSE
SER improvement with the ACA-DFE is almost an order of magnitude. Second, the ACA-DFE outperforms the conventional adaptive DFE for any step size. Last, given a target SER, the applicable step-size range of the ACA-DFE is wider than that of the conventional adaptive DFE. We then consider a time-varying channel constructed from Proakis C channel used previously. The first, second, forth and fifth paths in the channel now undergo fading according to Jake’s model[26], and are upper bounded by their corresponding path magnitudes. The normalised Doppler frequency fdTs equals 5 10 4, in which fd is
the Doppler frequency and Tsis the symbol duration. This
time-varying channel is normalised to keep the SNR at a constant level. Figure 5 shows the average SER against the step size used in the DFE. We observe that the SER improvement for this time-varying case is quite significant, and is more than an order of magnitude most of the time. In practice, it is difficult to know the exact channel variation pattern and the optimum step size. For the ACA-DFE, since the applicable step-size range is wider and the resultant SER is always lower, making the choice of the step size becomes much easier. This enables the ACA-DFE to work adequately in general time-varying environments.
4.2
Experiment 2: MIMO ACA-DFE
In this part, we consider the MIMO ACA-DFE for dispersive MIMO channels. First, we use the static 2 2 MIMO channel given in [2]: h11¼ ½0:781 0:625T, h12¼
½0:781 0:625T, h21¼ ½0:895 0:447T, and h22¼ ½0:958
0:287T. The parameters used are Tt ¼ 200, SNR ¼ 20 dB,
mf ¼ mb¼ 0:005, and mq¼ 0:002. The learning curves for
the MIMO ACA-DFE and the conventional adaptive MIMO DFE are shown in Fig. 6. We see that the MIMO ACA-DFE can achieve an MSE lower than that of the conventional adaptive MIMO DFE in the DD mode. Similarly, Fig. 7 presents the relationship between the average SER and the step size used in the MIMO DFEs. The MIMO ACA-DFE generally achieves lower SER than the conventional adaptive MIMO DFE. Next, we conduct the experiment under time-varying channel environments, in which we let the second channel tap in hnm (n¼ 1, 2 and
m¼ 1, 2) be varied with Jake’s model and upper bounded by its corresponding path magnitude. The normalised Doppler frequency is now changed to 2 10 4. Figure 8 shows the average SER versus the step size used for the time-varying channel. We can see that the MIMO ACA-DFE still provides better performance. In this scenario, the
10 10 10 10 10 10 10 100 µf SER
conventional adaptive DFE
Fig. 4 Average SER for static Proakis C channel with different step sizes 10 10 10 10 10 10 10 10 10 10 100 µf SER
conventional adaptive DFE
Fig. 5 Average SER for time-varying Proakis C channel
( fdTs¼ 5 10 4) with different step sizes 0 1000 2000 3000 4000 5000
0
number of iterations
decision MSE, dB
conventional adaptive MIMO DFE
) MMSE
− −
Fig. 6 MSE learning curves for static MIMO channel
10 10 10 10 10 10 10 100 µf SER
conventional adaptive MIMO DFE MIMO −
Fig. 7 Average SER for static MIMO channel with different step sizes
optimum step size for both DFE schemes is around 0.006, and the SER improvement with the MIMO ACA-DFE is more than an order of magnitude. However, note that the performance obtained with the proposed method is not as good as that in the SISO scenario. It is because the MIMO environment induces CCI for each transmitted sequence and this lowers the input SNR. Also, the channel becomes multi-dimensional and is more difficult to estimate. 5 Conclusions
In this paper, we have developed the ACA-DFE for SISO systems and MIMO systems. With the additional channel estimator(s) and the special operation for the FBF, the stability and robustness against EP are improved. Simula-tion results confirm the usefulness of these proposed schemes. Note that the DFE considered here for MIMO systems is generally referred to as the parallel interference cancellation ( PIC) scheme in which decisions are made for all recovered bit streams simultaneously. To further enhance the performance in MIMO systems, we can apply the SIC technique where the decision for each bit stream is made sequentially. The decision sequence is determined according to the decision-error probabilities in the recovered bit streams. Combining SIC and the method proposed in this paper, we can reduce the EP effect even more effectively. Research on this subject is now underway.
6 References
1 Belfiore, C., and Park, Jr., J.: ‘Decision feedback equalization’, Proc. IEEE, 1979, 67, (8), pp. 1143–1156
2 Al-Dhahir, N., and Cioffi, J.M.: ‘MMSE decision-feedback equalizers: Finite-length results’, IEEE Trans. Inf. Theory, 1995, 41, (7), pp. 961–975
3 Cioffi, J.M., Dudevoir, G.P., Eyuboglu, M.V., and Forney, G.D.: ‘MMSE decision-feedback equalizers and coding. I. Equalization results’, IEEE Trans. Commun., 1995, 43, (10), pp. 2582–2594 4 Proakis, J.G.: ‘Digital communications’ (McGraw-Hill, New York,
1995, 3rd edn.)
5 Smee, J.E., and Beaulieu, N.C.: ‘Error-rate evaluation of linear equalization and decision feedback equalization with error propaga-tion’, IEEE Trans. Commun., 1998, 46, (5), pp. 656–665
6 Russell, M., and Bergmans, W.M.: ‘A technique to reduce error propagation in M-ary decision feedback equalization’, IEEE Trans. Commun., 1995, 43, (12), pp. 2878–2881
7 Chiani, M.: ‘Introducing erasures in decision-feedback equalization to reduce error propagation’, IEEE Trans. Commun., 1997, 45, (7), pp. 757–760
8 Fertner, A.: ‘Improvement of bit-error-rate in decision feedback equalizer by preventing decision error propagation’, IEEE Trans. Signal Process., 1998, 46, (7), pp. 1872–1877
9 Reuter, M., Allen, J.C., Zeidler, J.R., and North, R.C.: ‘Mitigating error propagation effects in a decision feedback equalizer’, IEEE Trans. Commun., 2001, 49, (11), pp. 2028–2041
10 Wulich, D., and Geva, A.: ‘On a periodic training sequence in DFE to reduce the steady-state error probability’, IEEE Trans. Commun., 1999, 47, (9), pp. 1288–1292
11 Foschini, G.J., and Gans, M.J.: ‘On the limits of wireless commu-nications in a fading environment when using multiple antennas’, Wirel. Pers. Commun., 1998, 6, (3), pp. 315–335
12 Al-Dhahir, N., and Sayed, A.H.: ‘The finite-length input multi-output MMSE-DFE’, IEEE Trans. Signal Process., 2000, 48, (10), pp. 2921–2936
13 Tidestav, C., Ahl!en, A., and Sternad, M.: ‘Realizable MIMO decision feedback equalizers: Structure and design’, IEEE Trans. Signal Process., 2001, 49, (1), pp. 121–133
14 Fischer, R., Huber, J., and Windpassinger, C.: ‘Signal processing in decision-feedback equalization of intersymbol-interference and multi-ple-input/multiple-output channels: A unified view’, Signal Process., 2003, 83, (8), pp. 1633–1642
15 Qureshi, S.U.H.: ‘Adaptive equalization’, Proc. IEEE, 1985, 73, (9), pp. 1349–1387
16 Haykin, S.: ‘Adaptive filter theory’ ( Prentice-Hall Inc, 1996, 3rd edn.) 17 Frigon, J.F., and Daneshrad, B.: ‘A multiple input-multiple output (MIMO) adaptive decision feedback equalizer (DFE) with cancella-tion for wideband space-time communicacancella-tions’, Int. J. Wirel. Inf. Netw., 2002, 9, (1), pp. 13–23
18 Komninakis, C., Fragouli, C., Sayed, A.H., and Wesel, R.D.: ‘Multi-input multiple-output fading channel tracking and equalization using Kalman estimation’, IEEE Trans. Signal Process., 2002, 50, (5), pp. 1065–1076
19 Aaron, M.R., and Tufts, D.W.: ‘Intersymbol interference and error probability’, IEEE Trans. Inf. Theory, 1966, 12, (1), pp. 26–34 20 Chen, S., Mulgrew, B., Chng, E.S., and Gibson, G.J.: ‘Space
translation properties and the minimum-BER linear-combiner DFE’, IEE Proc. Commun., 1998, 145, (10), pp. 316–322
21 Chen, S., Hanzo, L., and Mulgrew, B.: ‘Adaptive minimum symbol-error-rate decision feedback equalization for multilevel pulse-amplitude modulation’, IEEE Trans. Signal Process., 2004, 52, (7), pp. 2092–2101
22 Yeh, C.-C., and Barry, J.R.: ‘Adaptive minimum bit-error rate equalization for binary signaling’, IEEE Trans. Commun., 2000, 48, (7), pp. 1226–1235
23 Yeh, C.-C., and Barry, J.R.: ‘Adaptive minimum symbol-error rate equalization for quadrature-amplitude modulation’, IEEE Trans. Signal Process., 2003, 51, (12), pp. 3263–3269
24 Shukla, P.K., and Turner, L.F.: ‘Channel-estimation-based adaptive DFE for fading multipath radio channels’, IEE Proc. I, Commun. Speech Vis., 1991, 138, (12), pp. 525–543
25 Rontogiannis, A.A., and Berberidis, K.: ‘Bandwidth efficient transmis-sion through sparse channels using a parametric channel-estimation-based DFE’, IEE Proc. Commun., 2005, 152, (4), pp. 251–256 26 Dent, P., Bottomley, G.E., and Croft, T.: ‘Jakes fading model
revisited’, Electron. Lett., 1993, 29, (6), pp. 1162–1163
10 10 10 10 10 10 10 10 10 100 µf SER
conventional adaptive MIMO DFE MIMO −
Fig. 8 Average SER for time-varying MIMO channel
