# Differential Evolution Algorithm Aided Turbo Channel Estimation and Multi-
User Detection for G.Fast Systems in the Presence of FEXT

Jiankang Zhang, , Sheng Chen, , Rong Zhang, ,
Anas F. Al Rawi, , Lajos Hanzo J. Zhang, S. Chen, R. Zhang and L. Hanzo are
with School of Electronics and Computer Science, University of Southampton,
Southampton SO17 1BJ, U.K. (Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>lh@ecs.soton.ac.uk). S. Chen is also
with King Abdulaziz University, Jeddah 21589, Saudi Arabia.A. F. Al Rawi is
with British Telecommunications, Martlesham, Ipswich, U.K. (Email:
anas.mohsin@bt.com).The financial support of the European Research Council’s
Advanced Fellow Grant, the Royal Society Wolfson Research Merit Award as well
as of the EPSRC project EP/N004558/1 are gratefully acknowledged. The research
data for this paper is available at https://doi.org/10.5258/SOTON/D0550.

###### Abstract

The ever-increasing demand for broadband Internet access has motivated the
further development of the digital subscriber line to the G.fast standard in
order to expand its operational band from 106 MHz to 212 MHz. Conventional
far-end crosstalk (FEXT) based cancellers falter in the upstream transmission
of this emerging G.fast system. In this paper, we propose a novel differential
evolution algorithm (DEA) aided turbo channel estimation (CE) and multi-user
detection (MUD) scheme for the G.fast upstream including the frequency band up
to 212 MHz, which is capable of approaching the optimal Cramer-Rao lower bound
of the channel estimate, whilst approaching the optimal maximum likelihood
(ML) MUD’s performance associated with perfect channel state information, and
yet only imposing about 5% of its computational complexity. Explicitly, the
turbo concept is exploited by iteratively exchanging information between the
continuous value-based DEA assisted channel estimator and the discrete value-
based DEA MUD. Our extensive simulations show that 18 dB normalized mean
square error gain is attained by the channel estimator and 10 dB signal-to-
noise ratio gain can be achieved by the MUD upon exploiting this iteration
gain. We also quantify the influence of the CE error, of the copper length and
of the impulse noise. Our study demonstrates that the proposed DEA aided turbo
CE and MUD scheme is capable of offering near-capacity performance at an
affordable complexity for the emerging G.fast systems.

###### Index Terms:

Digital subscriber line, far-end crosstalk, G.fast upstream, vectoring, turbo
channel estimation and multi-user detection, differential evolution algorithm

##  I Introduction

The demand for high-speed broadband Internet access has motivated the
construction of the hybrid digital subscriber line (DSL) and optical fiber
infrastructure, which has been standardized as G.fast by the International
Telecommunication Union Telecommunication Standardization Sector (ITU-T) [1].
G.fast still relies on the copper access network and in-premises wiring for
its last hundred meters, because it is prohibitively expensive to replace
copper by optical fiber. The G.fast standard has already exploited a broad
spectrum spanning up to 106 MHz, while the band up to 212 MHz has been planed
for future broadband access [2, 3]. However, exploiting the spectrum beyond 30
MHz inevitably imposes significant electromagnetic coupling between the
neighboring twisted-pairs, which is referred to as crosstalk. There are two
types of crosstalk, depending on the specific source of coupling. Explicitly,
coupling originating from transmitter co-located with a receiver is called
near-end crosstalk (NEXT), while the coupling arriving from the opposite side
of the duplex link is known as far-end crosstalk (FEXT) [4]. NEXT can be
avoided by employing frequency duplexing division and transmission
synchronization for separating the downstream and upstream transmissions [4],
or by vectoring the source signals for the sake of increasing the total
throughput of the cable [5]. By contrast, FEXT remains a significant
impairment that hampers achieving high data rates for G.fast systems [1].

(a) (b)

Figure 1: (a) Measured direct channel and crosstalk channel strengths, and (b)
the average noise power amplified by the ZF FEXT canceller.

###  I-A Existing solutions for mitigating FEXT in the downstream

FEXT is generally mitigated by using spectrum shaping and vectoring [6] in the
downstream transmission. The vectoring technique has also been standardized in
G.993.5 by the ITU-T [7] and it was further developed with the objective of
approaching Gigabit rates [1]. More specifically, in the downstream, the
transmitters are co-located at the central office (CO) or at the optical
network unit (ONU). Hence vectoring (precoding) can be applied to the
transmitted signals. As a powerful nonlinear precoding scheme, the Tomlinson-
Harashima precoder (THP) [8] was demonstrated to approach the single-user
bound for a bandwidth of up to 17.6 MHz. However, the THP is incompatible with
the previous version of G.fast. Hekrdla _et al._ [9] developed a dynamic
ordering-based THP by taking into account the specific G.fast channel
statistics. Zhang _et al._ [10] conceived the concept of expanded
constellation mapping for maximizing the received signal power, while
cancelling the FEXT. However, the THP imposes a high computational complexity
both on the transmitter and receiver. The investigations of [11] revealed that
the THP was also sensitive to the channel state information (CSI) estimation
error. By comparison, low-complexity linear precoding used in the context of
very-high-bit-rate DSL (VDSL) exhibits almost the same performance as the more
complex nonlinear ones [12]. The vectoring concept of [5] was proposed for
cancelling the FEXT by exploiting user coordination at the either CO or ONU.
Furthermore, for the second-generation VDSL (VDSL2) system [13] zero-forcing
(ZF) precoding was adopted to mitigate the FEXT based on the column-wise
diagonal dominant (CWDD) nature of the copper channel [5, 14], which was
demonstrated to be near-optimal below 17.6 MHz. However, ZF precoding relies
on inverting a matrix at each tone, which imposes an excessive computational
complexity, particularly for a large number of copper pairs in a cable. Leshem
_et al._ [15] simplified the ZF precoder by only exploiting the first-order
and second-order statistics of the CSI. As a further development, Zanko _et
al._ [6] simplified the ZF precoder by adopting the least mean square
algorithm. Adaptive precoders [16, 17, 18] were also designed for cancelling
the FEXT by only exploiting the polarity of the symbol errors observed.
Moreover, a potent combination of precoding and dynamic spectrum management
was also conceived for downstream vectored transmission. Specifically, Lanneer
_et al._ [19] developed both a linear and a nonlinear precoding-based dynamic
spectrum management, which maximizes the weighted sum-rate under realistic
per-line total power and per-tone spectral mask constraints. Note that the
FEXT-contaminated DSL system can be viewed as a multi-user multiple-input
multiple-output (MIMO) system [20]. Therefore, the powerful multi-user
transmission techniques originally developed for wireless MIMO communications,
such as the vectors perturbation technique [21, 22], can be readily be invoked
for downstream transmission in G.fast systems.

###  I-B Existing solutions for mitigating FEXT in the upstream

By contrast, in the upstream, the transmitters are those of independent users
at different locations. Since there are no physical lines connecting the
distributed users, no coordination is possible amongst their distributed
transmitters, and it is impossible to apply centralized transmit precoding
techniques. Therefore, the FEXT is typically mitigated either at the CO or at
the ONU by exploiting sophisticated FEXT cancellation techniques [23, 5].
Explicitly, Im _et al._ [24] proposed a joint FEXT canceller and equalizer,
while Hormis _et al._ [25] viewed the FEXT-impaired channel as a MIMO channel
and proposed a soft interference canceller for quad-wire loops based on the
sequential Monte Carlo technique. Ginis _et al._ [5] proposed to successively
decode the received signal based on the QR decomposed channels and on the
previous decision, which may be viewed as a special case of the ZF aided
generalized decision feedback equalizer (ZF-GDFE) [26]. The achievable
performance of the ZF-GDFE critically depends on the decoding order used [27],
and to achieve its full performance potential, exhaustive search is required,
which may impose an excessive complexity. Chen _et al._ [27] considerably
reduced the computational complexity either with the aid of an efficient
successive ordering search or by a modified greedy search. Their results show
that the successive ordering search aided ZF-GDFE is capable of approaching
the rate of the optimal ordering based ZF-GDFE. For VDSL2 systems, the linear
ZF equalizer of [12] is capable of closely approximating the performance of
the ZF-GDFE, despite its lower complexity. The family of ZF-type FEXT
cancellers treats the alien cross-talk as one of the self-FEXT contributions,
which results in poor performance in the presence of alien noise. Zafaruddin
_et al._ [14] proposed a constrained linear minimum-output energy receiver for
cancelling both the self-crosstalk and the alien crosstalk in the VDSL
upstream. Biyani _et al._ [28] proposed to whiten the alien noise
contaminating the VDSL systems with the aid of a co-operative alien noise
cancellation algorithm. Explicitly, the co-operative alien noise cancellation
algorithm succeeds in removing the alien noise that persists after the ZF-FEXT
canceller by invoking a sophisticated recursive scheme, which is capable of
meeting the Cramer-Rao lower bound (CRLB), provided that the symbol decision
errors are perfectly known, but naturally its performance will erode in the
face of imperfect symbol decision error knowledge.

Similar to the ZF precoder of the downstream, the ZF-based FEXT canceller of
the upstream also requires matrix inversion at each tone of the multi-user DSL
systems for attaining a near-optimal performance. However, unlike the ZF
precoder, a ZF-FEXT canceller, which basically relies on a ZF detection
algorithm, will significantly enhance the additive noise power.
Quantitatively, Fig. 1 (b) depicts the average noise power amplified by the
ZF-FEXT canceller, where we have
$\gamma=\frac{\tilde{\sigma}^{2}}{\sigma_{o}^{2}}$, with $\sigma_{o}^{2}$
being the power of the original noise term, while $\tilde{\sigma}^{2}$ is the
power of the noise output by the ZF-FEXT canceller. Observe that the noise
power is amplified quite dramatically at higher operational frequencies. At
the time of writing, the ZF-FEXT canceller is successfully deployed in G.fast
systems operating in a bandwidth spanning up to 100 MHz. However, in the near
future, the bandwidth will be increased up to around 200 MHz. Observe from
Fig. 1 (b) that the noise enhancement in an operating bandwidth of 200 MHz is
30 dB higher than that at 100 MHz. Clearly, the existing ZF-FEXT canceller
fails to perform well in these future high-bandwidth G.fast systems because of
this dramatically increased noise enhancement. Moreover, the direct channels
are overwhelmed by the crosstalk in the G.fast systems in the frequency range
spanning from 100 MHz to 212 MHz, as shown in Fig. 1 (a). Hence the ZF-FEXT
canceller is no longer near-optimal, since the channel matrix does not hold
the property of being column-wise diagonal dominant. Hence, more powerful
solutions for mitigating FEXT resisted in upstream G.fast systems are needed.

###  I-C Motivations and contributions

The G.fast upstream system is reminiscent of a multi-user MIMO system,
intuitively, the maximum likelihood (ML) multi-user detector (MUD) is expected
to provide the ultimate optimal solution, albeit its computational complexity
increases exponentially both with the number of users and with the modulation
order. However, the ML-MUD is impractical for the G.fast system, which may
support up to 24 users with the aid of a 4096-quadrature amplitude modulation
(QAM) constellation, because it would require $4096^{24}$ cost function
evaluations [1].

For wireless systems, evolutionary algorithms (EAs) have been extensively
applied both for downlink precoder designs [29, 30, 31] as well as for uplink
MUD designs [32, 33, 34, 35, 36, 37]. In particular, it has been demonstrated
that the EA-aided MUD solutions are capable of approaching the optimal ML-MUD
performance at a fraction of the computational complexity imposed by the ML-
MUD [32, 33, 34, 35, 36, 37, 38]. Among the various EAs, the differential
evolution algorithms (DEAs) [39, 40] have been shown to be particularly
powerful in joint iterative channel estimation (CE) and MUD. Explicitly, they
are capable of approaching the CRLB of CE and the optimal ML-MUD performance
associated with the perfect CSI at a fraction of the ML-MUD complexity [36,
37, 38]. Furthermore, turbo CE and MUD/decoder techniques [41, 42, 43, 44, 45,
46, 47, 48] have been widely developed for powerful wireless systems, which
are capable of achieving near-capacity performance at an affordable
complexity. The conceptual similarity between the G.fast and the wireless
multi-user uplink arises from the fact that the joint optimization of CE and
MUD in the both systems relies on a similar multi-objective, multidimensional
joint optimization problem associated with continuous CE parameters and
discrete MUD parameters. Hence, it is beneficial to appropriately adapt these
reduced-complexity state-of-the-art wireless techniques to the G.fast systems,
which is capable of jointly detecting multi-user upstream signals relying on
powerful central signal processing unit at the central office.

Against this background, our novel contributions are:

  * 1)

We conceive a new turbo CE and MUD scheme relying on the DEA for the emerging
family of G.fast systems having an operational bandwidth spanning up to 212
MHz, which is capable of approaching the CRLB of channel estimation as well as
the optimal ML-MUD’s performance associated with the perfect CSI, while only
imposing a fraction of the ML-MUD complexity.

  * 2)

The joint optimization problem of turbo CE and MUD proposed for the G.fast
upstream is converted to iteratively procedure of a continuous-parameter DEA
assisted channel estimator, which searches through the channel space to find
the optimal CE solution, and a discrete-parameter DEA assisted MUD, which is
capable of finding the optimal ML solution of the transmitted data.

  * 3)

Furthermore, the continuous DEA assisted channel estimator and discrete DEA
aided MUD iteratively exchange their extrinsic information to attain turbo
gains for both the CE and MUD. Specifically, the reliably detected symbols are
iteratively fed back to the channel estimator to be exploited together with
the pilot symbols for further improving the accuracy of the estimated CSI,
while the enhanced CSI estimates further improve the MUD’s detection
reliability.

  * 4)

We carry our extensive investigations for the convergence of DEA aided CE and
DEA aided MUD, the impacts of system bandwidth, DSL loop length, impulse noise
and channel estimation error as well as the analysis of computational
complexity. Our extensive investigations show that 18 dB normalized mean
square error (NMSE) channel estimation gain and 10 dB MUD signal-to-noise
ratio (SNR) gain can be achieved by exploiting this iteration gain.
Furthermore, we quantitatively investigate the influence of both the CE error
and of the copper length as well as the impact of impulse noise. Our results
confirm that the proposed DEA aided turbo CE and MUD scheme offers near-
capacity performance at an affordable complexity for the emerging family of
G.fast systems.

The rest of this paper is organized as follows. The upstream G.fast system
model is described in Section II. Section III is devoted to our DEA assisted
turbo CE and MUD. The CRLB of the channel estimate is derived in Section IV.
Our simulation results and discussions are presented in Section V, whilst our
concluding remarks are offered in Section VI.

##  II Upstream System Model

We consider the upstream of DSL within a bandwidth spanning up to 212 MHz,
which supports $L$ users simultaneously transmitting their signals to the CO.
Discrete multi-tone modulation is employed by each user, which occupies
$\Omega$ MHz bandwidth of $N_{c}$ subcarriers, each allocated $\Omega/N_{c}$
MHz. We assume that the cyclic prefix is sufficiently long and that the users
synchronously transmit their signals. Thus, there is no intersymbol
interference and no inter-carrier interference. Hence we can process the
signals on a per tone basis. By omitting the tone index, the $L$-user received
signal vector on the tone of interest can be written as [5]

$\displaystyle\boldsymbol{Y}=$
$\displaystyle\boldsymbol{H}\boldsymbol{X}+\boldsymbol{W},$ (1)

where $\boldsymbol{Y}\in\mathbb{C}^{L}$ is the received signal vector,
$\boldsymbol{X}\in\mathbb{C}^{L}$ is the transmitted signal vector of the $L$
users, and $\boldsymbol{W}\in\mathbb{C}^{L}$ is the zero-mean white Gaussian
noise vector with the covariance matrix $\sigma_{w}^{2}\boldsymbol{I}_{L}$ in
which $\boldsymbol{I}_{L}$ is the $L\times L$ identity matrix, while
$\boldsymbol{H}\in\mathbb{C}^{L\times L}$ is the frequency-domain channel
matrix whose diagonal element $H_{l,l}$ represents the $l$-th direct path and
the off-diagonal elements $H_{l,m}$ for $m\neq l$ represents the FEXT coupling
coefficients between lines $l$ and $m$.

Figure 2: DEA assisted turbo CE and MUD for upstream telecommunications over
DSL. The subscripts $m$ and $c$ of $\boldsymbol{L}$ are associated with the
MUD and channel decoder, respectively, while the subscripts $pr$, $po$ and $e$
are used for representing the a priori, a posteriori and extrinsic
information, respectively.

In traditional DSL systems, including VDSL2 [13], the receiver only utilizes
the $l$-th entry of $\boldsymbol{Y}$ for the direct-path CE and for the
detection of the $l$-th user’s data. This single-line based detection, i.e.,
single-user detection (SUD), works well for the low-frequency bands, since the
magnitudes of the FEXT coefficients are much smaller than that of the direct
path, which is also known as the ‘diagonal-dominant property’ [14]. At the BT
Lab at Ipswich, U.K., we have measured the frequency-domain channel responses
of 100 m and 200 m BT cables, consisting 10 twisted copper pairs and each wire
having a diameter of 0.5 mm. The results are depicted in Fig. 1. The VDSL
standard [12] uses the frequency band spanning from 25 kHz to 12 MHz. As seen
from Fig. 1 (a), in this frequency band, the FEXT effects are negligible,
therefore a SUD is adequate. For higher frequency bands of up to 100 MHz, the
‘diagonal-dominant property’ of the channel matrix remains valid, but the FEXT
effects become non-negligible, as shown in Fig. 1 (a). Thus, in the upstream
transmission, the linear ZF based MUD (ZF-MUD), i.e., the ZF-FEXT canceller,
can be invoked for removing the interference imposed by adjacent lines, which
is formulated as

$\displaystyle\hat{\boldsymbol{X}}=$
$\displaystyle\boldsymbol{H}^{-1}\boldsymbol{Y}=\boldsymbol{X}+\tilde{\boldsymbol{W}},$
(2)

where $\tilde{\boldsymbol{W}}=\boldsymbol{H}^{-1}\boldsymbol{W}$ denotes the
noise at the output of the ZF-FEXT canceller.

It can be observed from (2) that the efficiency of the ZF-MUD relies on the
diagonal-dominant property. When the ratio of the direct channel magnitude to
the FEXT interference channel magnitude is high, i.e., $\boldsymbol{H}$ is
well-conditioned, the inversion $\boldsymbol{H}^{-1}$ is well defined and,
therefore, the ZF-FEXT canceller efficiently mitigates the interference.
Observe from Fig. 1 (a) that the higher the operational frequency, the less
pronounced this diagonal-dominant property becomes, consequently the more
seriously the ZF-MUD suffers from noise enhancement, as seen from Fig. 1 (b).
Again, the emerging G.fast system will expand the bandwidth up to 212 MHz [2],
where we observe from Fig. 1 that the direct channels are overwhelmed by the
crosstalk. Hence $\boldsymbol{H}$ is extremely poorly conditioned and
consequently the ZF-FEXT canceller suffers from an extremely high noise
enhancement, such as 60 dB. This motivates our research of more powerful MUDs.

##  III DEA Assisted Turbo Channel Estimation and Multi-User Detection

###  III-A Turbo channel estimation and multi-user detection

At the $l$-th user’s transmitter, where $1\leq l\leq L$, the bit sequence is
first encoded by a forward error correction (FEC) code encoder. After passing
through an interleaver $\Pi$, the coded bit sequence is mapped by an $M$-ary
modulator relying on the modulation constellation $\mathbb{M}$ into the symbol
sequence, which is then transmitted upstream. At the CO receiver, the task is
to jointly estimate the channel $\boldsymbol{H}$ and to detect the data
$\boldsymbol{X}$ based on the noisy received signal $\boldsymbol{Y}$. Thus the
objective or cost function (CF) of this joint CE and MUD is the log likelihood
function of $\boldsymbol{Y}$ conditioned both on $\boldsymbol{H}$ and
$\boldsymbol{X}$. Since the noise $\boldsymbol{W}$ is white Gaussian, this CF
is given by

$\displaystyle\mathcal{J}\left(\boldsymbol{H},\boldsymbol{X}\right)=$
$\displaystyle\left\|\boldsymbol{Y}-\boldsymbol{H}\boldsymbol{X}\right\|^{2}.$
(3)

The joint ML CE and MUD solution in theory can be found by solving the
optimization problem:

$\displaystyle\left(\hat{\boldsymbol{H}},\hat{\boldsymbol{X}}\right)^{\star}=$
$\displaystyle\arg\min_{\left(\boldsymbol{H}\in\mathbb{C}^{L\times
L},\boldsymbol{X}\in\mathbb{M}^{L}\right)}\mathcal{J}\left(\boldsymbol{H},\boldsymbol{X}\right),$
(4)

which is unattainable owing to the need of jointly searching the high-
dimensional continuous channel space and the high-dimensional discrete data
space. Note that in the upstream of a DSL system typically dozens of users are
served and each user employs an $M$-ary modulator.

A straightforward suboptimal approach, which is widely adopted in practice, is
to first estimate the CSI given the training pilots and then to detect the
data using the estimated channel. To acquire an adequately accurate CE,
however, the number of training pilots must be sufficiently large. This
approach is therefore inherently suboptimal and significantly reduces the
achievable throughput. A much better approach is to decompose the
computationally prohibitive joint ML CE and MUD optimization problem into an
iterative CE and MUD optimization by using a powerful turbo technique for
attaining an iterative gain, which is capable of reducing the pilot overhead,
while still attaining the optimal performance [46, 47, 48]. Specifically, the
joint optimization problem (4) is solved by the iterative procedure formulated
as

$\displaystyle\left(\hat{\boldsymbol{H}},\hat{\boldsymbol{X}}\right)\\!=$
$\displaystyle\arg\min_{\boldsymbol{X}\in\mathbb{M}^{L}}\mathcal{J}_{\text{mud}}\\!\left(\\!\boldsymbol{X}\Big{|}\arg\min_{\boldsymbol{H}\in\mathbb{C}^{L\times
L}}\mathcal{J}_{\text{ce}}\left(\boldsymbol{H}\big{|}\breve{\boldsymbol{X}}\right)\\!\right)\\!\\!,\\!$
(5)

where the ‘inner’-optimization performs the CE conditioned on the available
data $\breve{\boldsymbol{X}}$, which has the CF

$\displaystyle\mathcal{J}_{\text{ce}}\left(\boldsymbol{H}\big{|}\breve{\boldsymbol{X}}\right)=$
$\displaystyle\left\|\boldsymbol{Y}-\boldsymbol{H}\breve{\boldsymbol{X}}\right\|^{2},$
(6)

while the ‘outer’-optimization carries out MUD conditioned on the available
CSI estimate $\breve{\boldsymbol{H}}$, which has the CF

$\displaystyle\mathcal{J}_{\text{mud}}\left(\boldsymbol{X}\big{|}\breve{\boldsymbol{H}}\right)=$
$\displaystyle\left\|\boldsymbol{Y}-\breve{\boldsymbol{H}}\boldsymbol{X}\right\|^{2}.$
(7)

The schematic of the proposed turbo CE and MUD procedure solving the iterative
optimization (5) is illustrated in Fig. 2.

Let us denote the iteration index by the superscript (i). In the first
iteration, the available data $\breve{\boldsymbol{X}}^{(0)}$ represents the
pilot symbols allocated by the system. The $i$-th iteration starts by
performing the CE:

$\displaystyle\breve{\boldsymbol{H}}^{(i)}=$
$\displaystyle\arg\min_{\boldsymbol{H}\in\mathbb{C}^{L\times
L}}\mathcal{J}_{\text{ce}}\left(\boldsymbol{H}|\breve{\boldsymbol{X}}^{(i-1)}\right),$
(8)

followed by the MUD:

$\displaystyle\tilde{\boldsymbol{X}}=$
$\displaystyle\arg\min_{\boldsymbol{X}\in\mathbb{M}^{L}}\mathcal{J}_{\text{mud}}\left(\boldsymbol{X}\Big{|}\breve{\boldsymbol{H}}^{(i)}\right).$
(9)

Then soft decoding takes place by iteratively exchanging soft extrinsic
information between the MUD and the soft channel decoder of Fig. 2.

Specifically, each detected data symbol $\tilde{X}$ of the $l$-th user, where
the user index $l$ is omitted for simplicity, is converted into log likelihood
ratios (LLRs) by a soft demapper [49], denoted by $\boldsymbol{L}_{m,po}$,
which represents the _a posteriori_ soft encoded bit information calculated by
the soft MUD. After subtracting the _a priori_ information
$\boldsymbol{L}_{m,pr}$ of the encoded bits, the extrinsic information
delivered by the soft MUD is formulated as

$\displaystyle\boldsymbol{L}_{m,e}=$
$\displaystyle\boldsymbol{L}_{m,po}-\boldsymbol{L}_{m,pr}.$ (10)

This is passed through the de-interleaver $\Pi^{-1}$, which becomes the _a
priori_ soft information $\boldsymbol{L}_{c,pr}$ entered into the soft
decoder. The decoder then decodes $\boldsymbol{L}_{c,pr}$ to provide the _a
posteriori_ soft information $\boldsymbol{L}_{c,po}$ for the decoded bits. The
resultant extrinsic information provided by the soft decoder

$\displaystyle\boldsymbol{L}_{c,e}=$
$\displaystyle\boldsymbol{L}_{c,po}-\boldsymbol{L}_{c,pr},$ (11)

is then passed through the interleaver $\Pi$, and becomes the new _a priori_
information of the encoded bits. The iterative soft de-mapping and decoding
continues until the process converges, typically after a few iterations. After
the convergence of the soft MUD/decoding process, the decoder outputs the hard
bits, and this iterative detection/decoding is denoted by
$\mathcal{C}\left(\tilde{\boldsymbol{X}}\right)$. The decoded hard bits are
re-encoded and re-modulated into the data symbols

$\displaystyle\breve{\boldsymbol{X}}^{(i)}=$
$\displaystyle\mathcal{M}\left(\mathcal{C}\left(\tilde{\boldsymbol{X}}\right)\right),$
(12)

which becomes the data available for the next iteration between the CE and
MUD. Since the soft channel decoder is capable of producing a reliable bit
steam after the convergence of the soft MUD/decoder,
$\breve{\boldsymbol{X}}^{(i)}$ represents ‘virtual pilot symbols’, and this
iteration gain of the soft channel decoder will be fully exploited by the CE
to deliver a more accurate channel estimate $\breve{\boldsymbol{H}}^{(i+1)}$,
which in turn generates an even more reliable
$\breve{\boldsymbol{X}}^{(i+1)}$. The iteration gain of this turbo CE and MUD
process allows us to gradually approach the optimal solution of (4).

Figure 3: Flowchart of the continuous DEA assisted channel estimation.

###  III-B Continuous DEA assisted channel estimation

The optimization (8) of searching the high-dimensional channel space to find
the optimal $\breve{\boldsymbol{H}}^{(i)}$ can be efficiently carried out by
the continuous DEA. We now elaborate on this continuous DEA aided CE, whose
flowchart is shown in Fig. 3. For notational convenience, we stack the columns
of $\boldsymbol{H}\in\mathbb{C}^{L\times L}$ and convert it into a vector
$\boldsymbol{h}\in\mathbb{C}^{L^{2}}$.

  1. 1.

Initialization. At the first generation $g=1$, the initial population of
$P_{s}$ members ${\hat{\boldsymbol{h}}}_{g,p_{s}}\in\mathbb{C}^{L^{2}}$ for
$1\leq p_{s}\leq P_{s}$ is randomly and uniformly generated. The mean value of
the crossover probability $C_{r}$ is initialized to $\mu_{C_{r}}=0.5$, while
the location parameter of the scaling factor $\lambda$ is initialized to
$\mu_{\lambda}=0.5$. The archive that preserves the $B$ best population
members is initialized to be empty, where $B=pP_{s}$ and $0<p<1$ is the greedy
factor. The archive is introduced for preserving the best ‘genes’ of the
population.

  2. 2.

Mutation. Each individual ${\hat{\boldsymbol{h}}}_{g,p_{s}}$, $1\leq p_{s}\leq
P_{s}$, has the CF value
$\mathcal{J}_{ce}\big{(}{\hat{\boldsymbol{H}}}_{g,p_{s}}\big{)}$ calculated
using (6), where ${\hat{\boldsymbol{H}}}_{g,p_{s}}$ is the channel matrix
corresponding to ${\hat{\boldsymbol{h}}}_{g,p_{s}}$. Each population member
${\hat{\boldsymbol{h}}}_{g,p_{s}}$ is mutated by adding two scaled difference-
vectors, namely
${\hat{\boldsymbol{h}}}_{g,best,r_{1}}-{\hat{\boldsymbol{h}}}_{g,p_{s}}$ and
${\hat{\boldsymbol{h}}}_{g,r_{2}}-{\hat{\boldsymbol{h}}}_{g,r_{3}}$, to it

$\displaystyle{\tilde{\boldsymbol{h}}}_{g,p_{s}}=$
$\displaystyle{\hat{\boldsymbol{h}}}_{g,p_{s}}+\lambda_{p_{s}}\big{(}{\hat{\boldsymbol{h}}}_{g,best,r_{1}}-{\hat{\boldsymbol{h}}}_{g,p_{s}}\big{)}$
$\displaystyle+\lambda_{p_{s}}\left({\hat{\boldsymbol{h}}}_{g,r_{2}}-{\hat{\boldsymbol{h}}}_{g,r_{3}}\right),$
(13)

where ${\hat{\boldsymbol{h}}}_{g,best,r_{1}}$ is randomly selected from the
archive, i.e. $r_{1}$ is randomly selected from $\\{1,2,\cdots,B\\}$, $r_{2}$
and $r_{3}$ are two values randomly selected from
$\\{1,2,\cdots,(p_{s}-1),(p_{s}+1),\cdots,P_{s}\\}$, while
$\lambda_{p_{s}}\in(0,~{}1]$ is a randomly generated scaling factor according
to the following procedure. Draw a random number $\gamma$ according to the
Cauchy distribution [50] with the location parameter $\mu_{\lambda}$ and the
scale parameter $\sigma_{\lambda}$: If $\gamma\leq 0$, re-draw $\gamma$; if
$\gamma\in(0,~{}1]$, then set $\lambda_{p_{s}}=\gamma$; if $\gamma>1$, then
use $\lambda_{p_{s}}=1$. The ‘mutated’ individual
${\tilde{\boldsymbol{h}}}_{g,p_{s}}$, $1\leq p_{s}\leq P_{s}$, has the CF
value $\mathcal{J}_{ce}\big{(}{\tilde{\boldsymbol{H}}}_{g,p_{s}}\big{)}$,
where ${\tilde{\boldsymbol{H}}}_{g,p_{s}}$ is the channel matrix corresponding
to ${\tilde{\boldsymbol{h}}}_{g,p_{s}}$.

  3. 3.

Crossover. The continuous DEA generates a ‘trial’ individual
$\breve{\boldsymbol{h}}_{g,p_{s}}$ by exchanging some elements of the ‘target’
individual ${\hat{\boldsymbol{h}}}_{g,p_{s}}$ with the corresponding elements
of the ‘donor’ individual ${\tilde{\boldsymbol{h}}}_{g,p_{s}}$. Explicitly,
the crossover operation on the $\alpha$-th element is given by

$\displaystyle\breve{h}_{g,p_{s},\alpha}=$
$\displaystyle\left\\{\begin{array}[]{ll}\tilde{h}_{g,p_{s},\alpha},&\text{rand}_{\alpha}(0,1)\leq
C_{r_{p_{s}}},\\\
\hat{h}_{g,p_{s},\alpha},&\text{otherwise},\end{array}\right.$ (16)

where $\text{rand}_{\alpha}(0,1)$ denotes the random number drawn from the
uniform distribution in $[0,~{}1]$ for the $\alpha$-th element, while
$C_{r_{p_{s}}}\in[0,~{}1]$ is the crossover probability, which is randomly
generated according to the following procedure. Draw a random number $\gamma$
according to the normal distribution with the mean $\mu_{C_{r}}$ and the
standard deviation $\sigma_{C_{r}}$: If $\gamma<0$, re-draw $\gamma$; if
$\gamma\in[0,~{}1]$, then set $C_{r_{p_{s}}}=\gamma$; if $\gamma>1$, then use
$C_{r_{p_{s}}}=1$. In Fig. 3, this crossover operation for $p_{s}=1$ is
illustrated. The trial individual $\breve{\boldsymbol{h}}_{g,p_{s}}$, $1\leq
p_{s}\leq P_{s}$, has the CF value
$\mathcal{J}_{\text{ce}}\big{(}\breve{\boldsymbol{H}}_{g,p_{s}}\big{)}$, where
again the matrix $\breve{\boldsymbol{H}}_{g,p_{s}}$ corresponds to
$\breve{\boldsymbol{h}}_{g,p_{s}}$.

  4. 4.

Selection. The selection operation decides whether the target vector
${\hat{\boldsymbol{h}}}_{g,p_{s}}$ or the trial vector
$\breve{\boldsymbol{h}}_{g,p_{s}}$ will survive to the next generation
according to their CF values

$\displaystyle{\hat{\boldsymbol{h}}}_{g+1,p_{s}}\\!=$
$\displaystyle\\!\left\\{\\!\begin{array}[]{cl}\breve{\boldsymbol{h}}_{g,p_{s}},\\!\\!&\\!\\!\mathcal{J}_{\text{ce}}\big{(}\breve{\boldsymbol{H}}_{g,p_{s}}\big{)}\leq\mathcal{J}_{\text{ce}}\big{(}{\hat{\boldsymbol{H}}}_{g,p_{s}}\big{)},\\\
{\hat{\boldsymbol{h}}}_{g,p_{s}},\\!\\!&\\!\\!\text{otherwise}.\end{array}\right.\\!\\!\\!$
(19)

The archive is replaced by the $100pP_{s}\%$ best individuals of the new
population
$\left\\{{\hat{\boldsymbol{h}}}_{g+1,p_{s}}\right\\}_{p_{s}=1}^{P_{s}}$.

  5. 5.

Adaptation. To keep up with the ‘evolution’, the mean of the crossover
probability $\mu_{C_{r}}$ and the location parameter of the scaling factor
$\mu_{\lambda}$ are adaptively updated according to

$\displaystyle\mu_{C_{r}}=$ $\displaystyle(1-c)\cdot\mu_{C_{r}}+c\cdot
mean_{A}(\mathcal{S}_{C_{r}}),$ (20) $\displaystyle\mu_{\lambda}=$
$\displaystyle(1-c)\cdot\mu_{\lambda}+c\cdot mean_{L}(\mathcal{S}_{\lambda}),$
(21)

where $c\in(0,~{}1]$ is the adaptive update factor controlling the rate of the
parameter adaptation, $mean_{A}(\cdot)$ and $mean_{L}(\cdot)$ denote the
arithmetic-mean and Lehmer-mean [51] operators, respectively, while
$\mathcal{S}_{C_{r}}$ and $\mathcal{S}_{\lambda}$ denote the sets of the
successful crossover probabilities $C_{r_{p_{s}}}$ and scaling factors
$\lambda_{p_{s}}$ of generation $g$, respectively.

  6. 6.

Termination. The ideal stopping criterion would be the convergence of the
population. In practice, we opt for halting the optimization procedure, when
any of the following two stopping criteria is met:

     * •

The pre-set maximum number of generations $G_{\text{max}}$ has been exhausted.

     * •

$\Delta_{g}$ generations have been explored without any reduction in the CF
value associated with the best individual in the population.

Otherwise, set $g=g+1$, and go to 2) Mutation.

Figure 4: Flowchart of the discrete DEA assisted multi-user detection.

The scale parameter $\sigma_{\lambda}$ of the scaling factor and the standard
deviation $\sigma_{C_{r}}$ of the crossover probability should be set to a
small value, e.g., $\sigma_{\lambda}=0.1$ and $\sigma_{C_{r}}=0.1$. The
remaining algorithmic parameters to be set are the population size $P_{s}$,
the greedy factor $p$, the adaptive update factor $c$, the maximum number of
generations $G_{\text{max}}$ and/or the value of $\Delta_{g}$ for terminating
the continuous DEA aided CE.

###  III-C Discrete DEA aided multi-user detection

Clearly, it is impractical to find the optimal ML solution
${\tilde{\boldsymbol{X}}}$ for the optimization (9) by exhaustive search, when
the number of upstream users served $L$ is large and/or the modulation order
$M$ is very high. However, the optimization (9) of searching the high-
dimensional data space to find the optimal ${\tilde{\boldsymbol{X}}}$ can be
carried out by a discrete DEA, at a fraction of the computational complexity
imposed by the exhaustive-search based optimal ML-MUD. This discrete DEA
assisted MUD is depicted in Fig. 4. We will denote the bit vector mapped to
the symbol vector $\boldsymbol{X}$ by
$\boldsymbol{b}=\big{[}b_{1}~{}b_{2}\cdots b_{A_{b}}\big{]}^{\rm T}$, where
each bit $b_{i}$ takes the value of 1 or 0, and $A_{b}=L\cdot\log_{2}M$.

  1. 1.

Initialization. At the first generation of $g=1$, the discrete DEA randomly
initializes its population of $P_{s}$ individuals

$\displaystyle\hat{\boldsymbol{b}}_{g,p_{s}}\\!=$
$\displaystyle\Big{[}\hat{b}_{g,p_{s},1}~{}\hat{b}_{g,p_{s},2}\cdots\hat{b}_{g,p_{s},A_{b}}\Big{]}^{\rm
T},1\leq p_{s}\leq P_{s},\\!$ (22)

i.e., every bit $\hat{b}_{g,p_{s},i}$ is randomly assigned 1 or 0. The mean of
the crossover probability $C_{r}$ and the location parameter of the scaling
factor $\lambda$ are initialized to $\mu_{C_{r}}=0.5$ and $\mu_{\lambda}=0.5$,
respectively. The archive that preserves the $B$ best individuals of the
previous generation is set to empty.

  2. 2.

Mutation. Each individual $\hat{\boldsymbol{b}}_{g,p_{s}}$, $1\leq p_{s}\leq
P_{s}$, corresponds to a modulated symbol vector
${\hat{\boldsymbol{X}}}_{g,p_{s}}^{b}$ associated with the CF value
$\mathcal{J}_{mud}\big{(}{\hat{\boldsymbol{X}}}_{g,p_{s}}^{b}\big{)}$ that is
calculated using (7). The discrete DEA mutates each base population vector
$\hat{\boldsymbol{b}}_{g,p_{s}}$ by adding two appropriately scaled and
randomly selected difference-vectors to it. Note that in the binary
arithmetic, the scaling or multiplying operation is represented by the bit-
wise exclusive-AND operator $\otimes$, while the addition or difference
operation is represented by the bit-wise exclusive-OR operator $\oplus$.
Explicitly, the ‘mutated’ individual is given by

$\displaystyle\tilde{\boldsymbol{b}}_{g,p_{s}}=$
$\displaystyle\hat{\boldsymbol{b}}_{g,p_{s}}\oplus\left(\boldsymbol{z}^{b}_{p_{s}}\otimes\left(\hat{\boldsymbol{b}}_{g,best,r_{1}}\oplus\hat{\boldsymbol{b}}_{g,p_{s}}\right)\right)$
$\displaystyle\oplus\left(\boldsymbol{z}^{b}_{p_{s}}\otimes\left(\hat{\boldsymbol{b}}_{g,r_{2}}\oplus\hat{\boldsymbol{b}}_{g,r_{3}}\right)\right),$
(23)

where $\hat{\boldsymbol{b}}_{g,best,r_{1}}$ is randomly selected from the
archive, $\hat{\boldsymbol{b}}_{g,r_{2}}$ and $\hat{\boldsymbol{b}}_{g,r_{3}}$
are randomly selected from the rest of the current population, while the bit-
scaling factor
$\boldsymbol{z}^{b}_{p_{s}}=\big{[}z^{b}_{p_{s},1}~{}z^{b}_{p_{s},2}\cdots
z^{b}_{p_{s},A_{b}}\big{]}^{\rm T}$ is the $A_{b}$-length binary-valued vector
generated randomly according to the following procedure. First, the real-
valued vector $\boldsymbol{z}_{p_{s}}=\big{[}z_{p_{s},1}~{}z_{p_{s},2}\cdots
z_{p_{s},A_{b}}\big{]}^{\rm T}\in\mathbb{R}^{A_{b}}$ is generated, whose
elements all obey the Gaussian distribution of zero mean and unity variance.
Then the scaling factor $\lambda_{p_{s}}\in(0,~{}1]$ is generated according to
the Cauchy distribution with the location parameter $\mu_{\lambda}$ and the
scaling parameter of $\sigma_{\lambda}$, similar to the generation of the
scaling factor in the mutation step of the continuous DEA. By comparing the
elements of $\boldsymbol{z}_{p_{s}}$ with $\lambda_{p_{s}}$, the elements of
$\boldsymbol{z}^{b}_{p_{s}}$ are determined according to

$\displaystyle z^{b}_{p_{s},i}=$
$\displaystyle\left\\{\begin{array}[]{cl}1,&\text{if
}z_{p_{s},i}<\lambda_{p_{s}},\\\ 0,&\text{otherwise},\end{array}\right.$ (26)

for $1\leq i\leq A_{b}$. Each ‘mutated’ individual
$\tilde{\boldsymbol{b}}_{g,p_{s}}$, $1\leq p_{s}\leq P_{s}$, has the
associated CF value
$\mathcal{J}_{mud}\big{(}{\tilde{\boldsymbol{X}}}_{g,p_{s}}^{b}\big{)}$, where
${\tilde{\boldsymbol{X}}}_{g,p_{s}}^{b}$ is the symbol vector corresponding to
the bit vector $\tilde{\boldsymbol{b}}_{g,p_{s}}$.

  3. 3.

Crossover. The discrete DEA generates a trial vector
$\breve{\boldsymbol{b}}_{g,p_{s}}$ by replacing certain elements of the target
vector $\hat{\boldsymbol{b}}_{g,p_{s}}$ with the corresponding elements of the
donor vector $\tilde{\boldsymbol{b}}_{g,p_{s}}$. There exist diverse variants
of this crossover mechanism [39, 40], and we adopt the uniform crossover
algorithm. Specifically, the $\alpha$-th element of
$\breve{\boldsymbol{b}}_{g,p_{s}}$ is determined according to

$\displaystyle\breve{{b}}_{g,p_{s},\alpha}=$
$\displaystyle\left\\{\begin{array}[]{ll}\tilde{{b}}_{g,p_{s},\alpha},&\text{rand}_{\alpha}(0,1)\leq
C_{r_{p_{s}}},\\\
\hat{{b}}_{g,p_{s},\alpha},&\text{otherwise},\end{array}\right.$ (29)

where $C_{r_{p_{s}}}\in[0,~{}1]$ is the crossover probability, randomly
generated according to the normal distribution of mean $\mu_{C_{r}}$ and
standard deviation $\sigma_{C_{r}}$, similar to the generation of the
crossover probability in the crossover step of the continuous DEA. This
crossover operation for $p_{s}=1$ is depicted in Fig. 4. The trial individual
$\breve{\boldsymbol{b}}_{g,p_{s}}$, $1\leq p_{s}\leq P_{s}$, has the CF value
$\mathcal{J}_{mud}\big{(}\breve{\boldsymbol{X}}_{g,p_{s}}^{b}\big{)}$, where
$\breve{\boldsymbol{X}}_{g,p_{s}}^{b}$ denotes the symbol vector corresponding
to $\breve{\boldsymbol{b}}_{g,p_{s}}$.

  4. 4.

Selection. Whether the target vector $\hat{\boldsymbol{b}}_{g,p_{s}}$ or the
trial vector $\breve{\boldsymbol{b}}_{g,p_{s}}$ survives into the next
generation is decided according to their associated CF values. Specifically,
for $1\leq p_{s}\leq P_{s}$,

$\displaystyle\hat{\boldsymbol{b}}_{g+1,p_{s}}=$
$\displaystyle\left\\{\begin{array}[]{ll}\breve{\boldsymbol{b}}_{g,p_{s}},&{\cal
J}_{\text{mud}}\big{(}\breve{\boldsymbol{X}}^{b}_{g,p_{s}}\big{)}\leq{\cal
J}_{\text{mud}}\big{(}{\hat{\boldsymbol{X}}}^{b}_{g,p_{s}}),\\\
\hat{\boldsymbol{b}}_{g,p_{s}},&\text{otherwise}.\end{array}\right.$ (32)

The archive is replaced by the $B=100pP_{s}\%$ best individuals of the new
population
$\left\\{\hat{\boldsymbol{b}}_{g+1,p_{s}}\right\\}_{p_{s}=1}^{P_{s}}$.

  5. 5.

Adaptation. Similar to the continuous DEA, the mean of the crossover
probability $\mu_{C_{r}}$ and the location parameter of the scaling factor
$\mu_{\lambda}$ are updated according to [40]

$\displaystyle\mu_{C_{r}}=$
$\displaystyle(1-c)\cdot\mu_{C_{r}}+c\cdot{mean}_{A}\big{(}\mathcal{S}_{C_{r}}\big{)},$
(33) $\displaystyle\mu_{\lambda}=$
$\displaystyle(1-c)\cdot\mu_{\lambda}+c\cdot{mean}_{L}\big{(}\mathcal{S}_{\lambda}\big{)},$
(34)

where again $c\in(0,~{}1]$ is the adaptive update factor, $mean_{A}(\cdot)$
and $mean_{L}(\cdot)$ denotes the arithmetic mean and Lehmer mean,
respectively, while $\mathcal{S}_{C_{r}}$ and $\mathcal{S}_{\lambda}$ denote
the sets of the successful crossover probabilities $C_{r_{p_{s}}}$ and scaling
factors $\lambda_{p_{s}}$ of generation $g$, respectively.

  6. 6.

Termination. The optimization procedure is halted when any of the following
two stopping criteria are met:

     * •

The pre-set maximum number of generations $G_{\text{max}}$ has been exhausted.

     * •

$\Delta_{g}$ generations have been explored without any reduction in the CF
value associated with the best individual in the population.

Otherwise, set $g=g+1$, and go to 2) Mutation.

TABLE I: Default algorithmic and system parameters | Initialization | Randomly |  | Initialization | Randomly |  | Channel code | Turbo
---|---|---|---|---|---|---|---|---
|  Population size $P_{s}$ | 100 |  |  Population size $P_{s}$ | 100 |  | Code rate | 1/2
DEA assisted |  Greedy factor $p$ | 0.1 | DE assisted |  Greedy factor $p$ | 0.1 | System | Memory length | 16
CE |  Adaptive factor $c$ | 0.1 | MUD |  Adaptive factor $c$ | 0.8 | parameters | Polynomial | $(3,[7,5])$
| $G_{\max}$ | 100 |  | $G_{\max}$ | 100 |  | Users | 4
| $\Delta g_{\max}$ | 20 |  | $\Delta g_{\max}$ | 20 |  | Modulation | 16-QAM

Similar to the continuous DEA, the scale parameter $\sigma_{\lambda}$ of the
scaling factor and the standard deviation $\sigma_{C_{r}}$ of the crossover
probability can both be set to 0.1. The user has to set the population size
$P_{s}$, the greedy factor $p$, the adaptive update factor $c$, the maximum
number of generations $G_{\text{max}}$ and/or the value of $\Delta_{g}$ for
terminating the discrete DEA aided MUD.

##  IV Cramer-Rao lower bound of channel estimation

The CRLB provides the lowest achievable mean square error (MSE) of any
unbiased estimator [52]. In the simulation section, we will demonstrate that
our DEA assisted CE is capable of approaching the CRLB. Therefore, below we
derive the CRLB of the channel estimator. Since the CRLB is related to the
available training symbols, we will introduce the symbol index $[s]$. Thus,
upon recalling (1), we rewrite the $l$-th received signal at the $s$-th symbol
$Y_{l}[s]$ as

$\displaystyle Y_{l}[s]=$
$\displaystyle\boldsymbol{H}_{[l:\,]}\boldsymbol{X}[s]+W_{l}[s],$ (35)

where $\boldsymbol{H}_{[l:\,]}$ is the $l$-th row of the channel matrix
$\boldsymbol{H}$, $\boldsymbol{X}[s]=\left[X_{1}[s]~{}X_{2}[s]\cdots
X_{L}[s]\right]^{\rm T}$ is the transmitted signal vector at the $s$-th
symbol, and $W_{l}[s]$ is the $l$-th element of $\boldsymbol{W}[s]$.

Since $W_{l}[s]\in\mathbb{C}$ represents the white Gaussian noise with
covariance $\sigma_{w}^{2}$, the conditional probability density function
(PDF) $f\left(Y_{l}[s]|\boldsymbol{H}_{[l:\,]}\right)$ is given by

$\displaystyle f\left(Y_{l}[s]|\boldsymbol{H}_{[l:\,]}\right)=$
$\displaystyle\frac{1}{2\pi\sigma_{w}^{2}}\exp\left(-\frac{\left|Y_{l}[s]-\boldsymbol{H}_{[l:\,]}\boldsymbol{X}[s]\right|^{2}}{2\sigma_{w}^{2}}\right).$
(36)

Thus, the joint conditional PDF over the $S$ consecutive OFDM symbols,
$f\left(Y_{l}[1],Y_{l}[2],\cdots,Y_{l}[S]|\boldsymbol{H}_{[l:\,]}\right)$, can
be formulated as

$\displaystyle
f\left(Y_{l}[1],Y_{l}[2],\cdots,Y_{l}[S]|\boldsymbol{H}_{[l:\,]}\right)$
$\displaystyle\qquad=\prod\limits_{s=1}^{S}\left(\frac{1}{2\pi\sigma_{w}^{2}}\exp\left(-\frac{\left|Y_{l}[s]-\boldsymbol{H}_{[l:\,]}\boldsymbol{X}[s]\right|^{2}}{2\sigma_{w}^{2}}\right)\right).$
(37)

The Fisher information matrix is defined as [52]

$\displaystyle\boldsymbol{I}\big{(}\boldsymbol{H}_{[l:\,]}\big{)}=$
$\displaystyle-\mathcal{E}\left\\{\frac{\partial^{2}\log
f\left(Y_{l}[1],Y_{l}[2],\cdots,Y_{l}[S]|\boldsymbol{H}_{[l:\,]}\right)}{\partial\boldsymbol{H}_{[l:\,]}\partial\boldsymbol{H}_{[l:\,]}^{\rm
H}}\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sigma_{w}^{2}}\sum\limits_{s=1}^{S}\mathcal{E}\left\\{\boldsymbol{X}[s]\left(\boldsymbol{X}[s]\right)^{\rm
H}\right\\}\\!\\!,\\!$ (38)

where $\mathcal{E}\left\\{~{}\right\\}$ denotes the expectation operator.

The CRLB for the estimate of $\boldsymbol{H}_{[l:\,]}$ is defined as

$\displaystyle\text{CRLB}\left(\boldsymbol{H}_{[l:\,]}\right)$
$\displaystyle=\text{Tr}\left(\boldsymbol{I}^{-1}\big{(}\boldsymbol{H}_{[l:\,]}\big{)}\right)$
$\displaystyle=2\sigma_{w}^{2}\text{Tr}\left(\left(\sum\limits_{s=1}^{S}\mathcal{E}\left\\{\boldsymbol{X}[s]\left(\boldsymbol{X}[s]\right)^{\rm
H}\right\\}\right)^{-1}\right),$ (39)

where $\text{Tr}(\cdot)$ denotes the matrix trace operation. Since the optimal
training symbol sequence satisfies
$\mathcal{E}\left\\{\boldsymbol{X}[s]\left(\boldsymbol{X}[s]\right)^{\rm
H}\right\\}=E_{\rm s}\boldsymbol{I}_{L}$, where $\boldsymbol{I}_{L}$ denotes
the $L\times L$ identity matrix and $E_{\rm s}$ is the average power of the
transmitted data symbol. Thus, the CRLB can be expressed as

$\displaystyle\text{CRLB}\left(\boldsymbol{H}_{[l:\,]}\right)=$
$\displaystyle\frac{2\sigma_{w}^{2}}{SE_{\rm s}}.$ (40)

Furthermore, the normalized CRLB (NCRLB), which represents the lower-bound of
the achievable NMSE, is given by

$\displaystyle\text{NCRLB}\left(\boldsymbol{H}_{[l:\,]}\right)=$
$\displaystyle\frac{2\sigma_{w}^{2}}{SE_{\rm
s}\left\|\boldsymbol{H}_{[l:\,]}\right\|^{2}}.$ (41)

##  V Simulation Results

In this section, we evaluate the performance of the proposed DEA assisted
turbo CE and MUD for upstream G.fast systems operated in the frequency range
of 2 MHz to 212 MHz, which are split into 4096 tones. The channel was
characterized by the measurements of BT’s twisted copper lines at the BT
Ultra-Fast lab. The number of simultaneous upstream users has the default
value of $L=4$ and each user employs the same 16-QAM scheme combined with a
half-rate turbo channel code of memory 16 using the octal generator
polynomials of $(3,[7,5])$. The total number of CF evaluations for the ML-MUD
is $16^{4}=65536$. The default loop length of the DSL lines is 100 m. The
iterative procedures of the inner turbo decoding is automatically terminated,
when they have converged. The number of iterations between the DEA assisted CE
and the DEA aided MUD is set to 6 in our investigation. Note that our proposed
scheme is readily applicable to systems supporting a number of users and
higher-order modulation. However, since we use the optimal ML solution as the
ultimate benchmark of our proposed scheme and we can only compute the optimal
ML solution for the system supporting a low number of users associated with
relatively low-order modulation, we restrict our simulation study to $L=4$ and
16-QAM modulation. The default algorithmic parameters used for the continuous
and discrete DEAs and the system parameters are summarized in Table I. Unless
otherwise specified, these default parameter values are used throughout.

We first investigate the per subcarrier performance and the convergence
performance of the individual continuous DEA aided CE and discrete DEA aided
MUD components, respectively, as well as the impact of the system bandwidth on
the achievable performance. Both the least square (LS) CE and the NCRLB are
used as the benchmarks for evaluating the continuous DEA aided CE (DEA-CE),
where the NCRLB indicates the best achievable performance of the channel
estimator. Furthermore, the SUD, the ZF-MUD and the ML-MUD are used as the
benchmarks of the discrete DEA aided MUD (DEA-MUD), where the ML-MUD provides
the best achievable detection performance. Then the performance of the
proposed turbo DEA-CE and DEA-MUD is investigated for quantifying the
achievable iterative gain of this turbo CE and MUD scheme. Furthermore, the
impact of the impulse noise as well as that of the CE error and the loop
length on the detection performance are also investigated.

(a) (b)

Figure 5: NMSE versus the subcarrier index: (a) $E_{b}/N_{0}=20$ dB, and (b)
$E_{b}/N_{0}=30$ dB.

(a) (b)

Figure 6: Idealized SER based on perfect CSI versus the subcarrier index: (a)
$E_{b}/N_{0}=20$ dB, and (b) $E_{b}/N_{0}=30$ dB.

(a) (b)

Figure 7: NMSE versus the number of iterations: (a) $E_{b}/N_{0}=20$ dB, and
(b) $E_{b}/N_{0}=30$ dB.

###  V-A Per subcarrier NMSE and SER performance of non-turbo CE and MUD

Let us now consider the non-turbo DEA-CE and DEA-MUD, where the channel
estimator relies purely on the pilot symbols only, while the turbo MUD and the
decoder perform iterative detection and decoding based on the perfect CSI.
This enables us to investigate both the NMSE of the DEA-CE and the ideal
symbol error ratio (SER) of the DEA-MUD, separately, on each subcarrier. Note
that the interleaving operation makes it impossible for us to investigate the
bit error rate (BER) of each individual subcarrier.

The NMSE performance of the DEA-CE at $E_{b}/N_{0}=20$ dB and 30 dB is shown
in Fig. 6 (a) and Fig. 6(b), respectively, where $E_{b}$ is the energy per bit
and $N_{0}=\sigma_{w}^{2}$ is the noise power. Observe that the DEA-CE
achieves an almost identical estimation performance to the LS-CE relying on
the same pilot symbols. The idealized SER performance based on the perfect CSI
achieved by the DEA-MUD at $E_{b}/N_{0}=20$ dB and 30 dB is depicted in Fig. 6
(a) and Fig. 6 (b), respectively, in comparison to the idealized SERs of the
SUD, ZF-MUD and ML-MUD. As expected, the DEA-MUD exhibits a much better
detection performance than both the SUD as well as the ZF-MUD, and it is
capable of approaching the performance of the ML-MUD. Observe that the
detection performance improvement of the DEA-MUD over both the SUD and over
the ZF-MUD is more significant at high frequencies. Interestingly, the SUD
outperforms the ZF-MUD at low frequencies, because the latter suffers from
serious noise enhancement.

(a) (b)

Figure 8: Idealized BER based on perfect CSI versus the number of iterations:
(a) $E_{b}/N_{0}=20$ dB, and (b) $E_{b}/N_{0}=30$ dB.

(a) (b)

Figure 9: (a) Training-based NMSE performance versus $E_{b}/N_{0}$, and (b)
idealized BER performance versus $E_{b}/N_{0}$ relying on perfect CSI, for the
system bandwidths of 52.5 MHz, 105.0 MHz, 157.5 MHz and 210.0 MHz.

###  V-B Convergence of DEA aided CE and DEA aided MUD

By operating the non-turbo DEA aided CE and DEA aided MUD, we can further
investigate the convergence of the DEA-CE and the convergence of the DEA-MUD
separately. The convergence performance of the DEA-CE and the DEA-MUD are
shown in Figs. 7 and 8, respectively, at $E_{b}/N_{0}=20$ dB and 30 dB.
Explicitly, we investigate three subcarriers, the 500-th, 2500-th and 4000-th
tones having the central frequencies of $f_{c}=26.85$ MHz, 130.35 MHz and
207.97 MHz, respectively, where $f_{c}=26.85$ MHz is in the frequency range of
the VDSL2 standard [13], while $f_{c}=130.35$ MHz and 207.97 MHz are in the
frequency range of G.fast.

It can be seen from Fig 7 that the DEA-CE converges to the LS-CE solution
within 30 iterations at $E_{b}/N_{0}=20$ dB and within 40 iterations at
$E_{b}/N_{0}=30$ dB, respectively. At $E_{b}/N_{0}=20$ dB, the BERs of the
DEA-MUD converge to those of the ML-MUD around 20 iterations, as seen from
Fig. 8 (a). Note that at $E_{b}/N_{0}=30$ dB, the BER of the SUD at
$f_{c}=26.85$ MHz, the BERs of the ZF-MUD at $f_{c}=26.85$ MHz and 130.35 MHz
as well as the BERs of the ML-MUD at all the three subcarriers are not
included in Fig. 8 (b), because they are infinitesimally low based on the
perfect CSI. Observe from Fig. 8 (b) that at $E_{b}/N_{0}=30$ dB, the DEA-MUD
converges to the ML-MUD after 18 iterations for $f_{c}=26.85$ MHz, 19
iterations for $f_{c}=130.35$ MHz and 45 iterations for $f_{c}=207.97$ MHz,
respectively.

(a) (b)

Figure 10: Achievable performance of the DEA aided turbo CE and MUD: (a) NMSE
versus $E_{b}/N_{0}$ parametrized by the number of turbo iterations, and (b)
BER versus $E_{b}/N_{0}$ parametrized by the number of versus turbo
iterations.

###  V-C Impact of system bandwidth on the achievable performance

As mentioned previously, the channel quality is critically dependent on the
system’s bandwidth (BW) or the frequency range. To investigate the influence
of the system’s frequency range, we consider four cases: the 1st case at the
BW of 52.5 MHz covers the first 1024 subcarriers of the lowest frequency
range, the 2nd case covers the first 2048 subcarriers with the BW of 105.0
MHz, the 3rd case has the BW of 157.5 MHz including the first 3072
subcarriers, and the 4th case includes all the 4096 subcarriers and covers the
total system BW of 210 MHz.

Fig. 9 (a) depicts the NMSE performance of the DEA-CE based on the pilot
symbols. As expected, the NMSE of the DEA-CE is identical to that of the LS-
CE, and the system BW has no impact on the training performance of an unbiased
channel estimator. Observe from Fig. 9 (a) that there is an approximately 18
dB gap between the training-based NMSE and the NCRLB, where the NCRLB is
calculated based on both the pilots and the data. Fig. 9 (b) shows the
idealized BER performance of the DEA-MUD based on the perfect CSI for
different system bandwidths. As expected, the BER performance is better for
the system having a lower frequency range, because the channel quality at a
lower frequency range is better. The results of Fig. 9 (b) also confirm that
the DEA-MUD and the ML-MUD exhibit almost identical detection performance, and
the DEA-MUD outperforms both the SUD and the ZF-MUD, particularly for the
systems including higher frequencies. For example, for the system including
all the 4096 subcarriers, the DEA-MUD attains an approximately 7 dB SNR gain
over the ZF-MUD at the BER level of $10^{-5}$. For this system, the SUD
exhibits a high error floor.

###  V-D Achievable performance of the DEA aided turbo CE and MUD

We now investigate the achievable performance of the proposed DEA aided turbo
CE and MUD. Again, we consider the system relying on all the 4096 subcarriers
and having the total system bandwidth of 210 MHz. The NMSE and BER versus
$E_{b}/N_{0}$ performance of this DE aided turbo CE and MUD scheme is
parametrized by the number of turbo iterations, as depicted in Fig. 10 (a) and
Fig. 10 (b), respectively.

The results of Fig. 10 need further explanations. Initially, given the
training pilots, the DEA-CE estimates the channel, and the NMSE of this
channel estimate is given by the $\text{Iter}=0$ curve of Fig. 10 (a), which
is identical to that of the training based LS-CE. Given this estimate, the
DEA-MUD and the channel decoder perform detection and decoding by iteratively
exchanging soft extrinsic information, and after convergence, the detected
data exhibits the BER represented by the $\text{Iter}=0$ curve of Fig. 10 (b).
The detected data are then fed back to the DEA-CE, which carries out the next
CE iteration based on both the detected data and pilots, leading to the
improved NMSE as seen from Fig. 10 (a). The enhanced estimated CSI is in turn
exploited by the DEA-MUD/channel decoder for producing the detected data at an
even lower BER. This ‘turbo’ procedure continues until at the 6th iteration
and around the SNR of 27 dB, the BER of the detected data becomes
infinitesimally low. Observe from Fig. 10 (a) that at this point, the NMSE of
the DEA aided turbo CE and MUD approaches the NCRLB. Explicitly, at the 6th
iteration and around the SNR of 27 dB, the achievable NMSE has approached the
NCRLB. This is because at this point, the detected data becomes the true data.
Not surprisingly, at this point, the BER of the DEA aided turbo CE and MUD
approaches that of the idealized ML-MUD associated with perfect CSI, which is
also identical to that of the idealized DEA-MUD relying on perfect CSI.

It is worth emphasizing the significance of the iterative gain obtained.
Specifically, by iteratively exchanging extrinsic information between the
continuous DEA aided CE and the discrete DEA aided MUD, approximately 18 dB of
NMSE gain as attained for the channel estimator and around 10 dB of SNR gain
is achieved for the MUD.

Figure 11: Impact of loop length on the achievable detection performance of
four idealized MUDs associated with perfect CSI.

###  V-E Impact of loop length

Although the lengths of the users’ DSL lines are different in practical
deployments, for the convenience of investigating the impact of loop length,
we assume that all the users have the same loop length. Fig. 11 shows the
influence of the loop length of DSL lines on the achievable BER performance of
the four idealized MUDs based on perfect CSI. It can be clearly seen from Fig.
11 that increasing the loop length significantly degrades the achievable BER
performance. Observe furthermore from Fig. 11 that the DEA-MUD attains the
optimal performance of the ML-MUD, and it considerably outperforms the ZF-MUD.
For these two systems, the SUD exhibits high BER floors.

###  V-F Impact of impulse noise

Next we investigate the impact of impulse noise on the achievable BER
performance. When the additive impulse noise is taken into account, the system
model (1) can be rewritten as

$\displaystyle\boldsymbol{Y}=$
$\displaystyle\boldsymbol{H}\boldsymbol{X}+\boldsymbol{W}+\boldsymbol{U},$
(42)

where $\boldsymbol{U}\in\mathbb{C}^{L}$ is the impulse noise vector. An OFDM
symbol is infected by the impulse noise with the probability of $\kappa$,
where typically we have $\kappa\in[0.01,~{}0.1]$ [53, 54]. Note that an OFDM
symbol includes the data at all the tones,
$\boldsymbol{X}[1],\cdots,\boldsymbol{X}[4096]$, where we have re-introduced
the omitted tone index of $\boldsymbol{X}$. We assume that the impulse noise
$\boldsymbol{U}$ obeys the complex Gaussian distribution associated with a
zero-mean vector and the covariance matrix $\sigma^{2}_{u}\boldsymbol{I}_{L}$.
Typically, the impulse noise is 20 dB stronger than the additive Gaussian
noise $\boldsymbol{W}$ [55, 56], that is,
$10\log_{10}\left(\sigma_{u}^{2}/\sigma_{w}^{2}\right)=20$ dB. In our
simulations, we consider both $\kappa=0.01$ and $\kappa=0.1$, which can be
viewed as the lower bound and the upper bound of the probability that an OFDM
symbol is infested by impulse noise.

It can be seen from Fig. 12 that the impulse noise degrades the achievable
detection performance, particularly for high $\kappa$, where the case of
$\kappa=0$ corresponds to no impulse noise. It can also be seen that at
$\kappa=0.01$, both the ML-MUD and the DEA-MUD exhibit almost an identical
performance, while at $\kappa=0.1$, the DEA-MUD is slightly inferior to the
ML-MUD. Not surprisingly, the DEA-MUD considerably outperforms the ZF-MUD.

Figure 12: Impact of impulse noise on the achievable detection performance of
four idealized MUDs associated with perfect CSI. Figure 13: Comparison of the
achievable detection performance of four MUDs based on LS-CE acquired by
training as well as based on perfect CSI.

###  V-G Impact of channel estimation error

In Fig. 13, we compare the BER performance of the four MUDs based on the LS-CE
acquired by training to those associated with perfect CSI. Clearly, the
channel estimation error has a significant impact on the achievable detection
performance of an MUD. Explicitly, for both the DEA-MUD and the ML-MUD, there
exists an SNR gap of around 10 dB between the idealized performance associated
with the perfect CSI and the performance associated with the estimated CSI,
which can also be seen from Fig. 10 (b). It can also be seen from Fig. 13,
that the SNR gap is approximately 9 dB between the idealized ZF-MUD associated
with perfect CSI and the ZF-MUD based estimated CSI.

###  V-H Computational complexity comparison

As demonstrated by the aforementioned results, given the same CSI, the DEA-MUD
is capable of attaining the optimal detection performance of the ML-MUD. We
now compare the computational complexity of the DEA-MUD to that of the ML-MUD.
Again, the ML-MUD finds the optimal solution by evaluating the CF values of
all the $M^{L}=16^{4}=65536$ potential candidates on each single tone. By
contrast, the discrete DEA evolves a population of ‘candidates’, as detailed
in Section III-C, based on the CF values of the population. Let $N_{\rm ML}$
be the total number of CF evaluations imposed by the ML-MUD and let $N_{\rm
DEA}$ be the total number of CF evaluations required by the DEA to converge to
the optimal ML solution. We can compare the computational complexities of both
the DEA-MUD and of the ML-MUD by calculating the ratio

$\displaystyle\text{Complexity of DEA-MUD}=$ $\displaystyle\frac{N_{\rm
DEA}}{N_{\rm ML}}~{}[\%],$ (43)

which we use to quantify the computational complexity of the DEA-MUD, in
comparison to the ML-MUD.

Figure 14: Computational complexity of the DEA-MUD expressed as the ratio of
its required CF evaluations over the total CF evaluations imposed by the ML-
MUD. The perfect CSI is assumed.

Fig. 14 compares the complexities of the DEA-MUD to those of the ML-MUD for
the same four systems as specified in Section LABEL:S4.3 at both
$E_{b}/N_{0}=20$ dB and 30 dB. It can be seen from Fig. 14 that the DEA-MUD
only requires 5% of the computational complexity imposed by the ML-MUD, while
still attaining the same optimal solution as the ML-MUD.

##  VI Conclusions

We have proposed a DEA aided turbo CE and MUD for mitigating the adverse
effects of FEXT encountered by the G.Fast systems caused by the utilization of
high frequencies up to 212 MHz. The proposed DEA aided turbo CE and MUD is
constituted by a continuous DEA aided CE and a discrete DEA aided MUD,
exchanging extrinsic information between them. We have demonstrated that our
DEA-MUD significantly outperforms the widely adopted low-complexity ZF-MUD,
also known as the ZF-FEXT canceller. More remarkably, we have shown that given
the same CSI, our discrete DEA aided MUD is capable of attaining the optimal
performance of the ML-MUD, while only imposing 5% of the computational
complexity associated with the ML-MUD. In our simulation study, we have also
investigated the impact of CE error, of the impulse noise and of the loop
length. Most importantly, in this paper, we have demonstrated that by
iteratively exchanging information between the continuous DEA aided CE and the
discrete DEA aided MUD, the DEA-CE is capable of approaching the optimal CRLB
of the channel estimate, while the DEA-MUD based on the estimated CSI is
capable of attaining the optimal detection performance of the idealized ML-MUD
associated with perfect CSI. Specifically, we have shown that 18 dB of the
NMSE gain is attained by the channel estimator and 10 dB of the SNR gain is
gleaned by the MUD by exploiting iteration gains. This study therefore has
demonstrated that the proposed DEA aided turbo CE and MUD is capable of
offering near-capacity performance at an affordable complexity for the
emerging G.fast systems.

## Acknowledgements

The authors would like to thank Tong Bai from Next Generation Wireless group
for his discussions and contributions about impulse noise.

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