\section{Introduction}
Any overdense region in the Universe is driven by the competition between its
self-gravity and the cosmic expansion, and therefore can be characterized by an
idealized zero-velocity surface that separates these zones. De Vaucouleurs
(1958, 1964, 1972) presupposed systematic deviations from linearity in the
velocity-distance relation and interpreted these deviations as a local
phenomenon caused by the Virgo complex. The expected effect has only
subsequently been supported by observations. Peebles (1976) found the
virgocentric infall signal using the field galaxy data available at that time
(Sandage \& Tammann 1975).

Lynden-Bell (1981) and Sandage (1986) focussed on the Local Group of galaxies.
They showed that, in the simplest case of the spherically symmetric system in
the empty Universe with $\Lambda=0$, the radius of the zero-velocity surface
$R_0$ and the total mass of the group $M^0_T$ are related as

\begin{equation}
M^0_T= (\pi^2/8G)\times R^3_0\times T^{-2}_0,
\end{equation}
where $G$ is the gravitational constant and $T_0$ is the age of the Universe
(Lynden-Bell 1981, Sandage 1986). In the standard cosmological $\Lambda$CDM
model, where $\Omega_m$ is the mean cosmic density of matter and $H_0$ the
Hubble parameter, the relation between $R_0$ and $M_T$ becomes

\begin{equation
M_T=(\pi^2/8G)\times R^3_0\times H^2_0/f^2(\Omega_m),
\end{equation}
where the dimensionless parameter

\begin{equation
f(\Omega_m)=(1-\Omega_m)^{-1}-\frac{\Omega_m}{2} (1-\Omega_m)^{-\frac{3}{2}} \cosh^{-1}(\frac{2}{\Omega_m}-1)
\end{equation}
changes in the range from 1 to 2/3 while varying $\Omega_m$ from 0 to 1. Taking
the Planck model parameters $\Omega_m =0.315, \Omega_{\lambda} = 0.685$ and
$H_0$ = 67.3 km\,s$^{-1}$\,Mpc$^{-1}$ (Planck Collaboration 2014), we obtain the
relation

\begin{equation
(M_T/M_{\odot})_{0.315} = 1.95\times 10^{12} (R_0/Mpc)^3,
\end{equation}
which is by 1.50 times more than the classical estimate from equation (1).

This method was sucessfully applied to determine masses of the Local Group
(Ekholm et al. 2001, Karachentsev et al. 2002, Teerikorpi et al. 2005,
Karachentsev et al. 2009), M81 group (Karachentsev \& Kashibadze 2006), CenA
group (Karachentsev et al. 2006), as well as the Virgo cluster (Tully \& Shaya
1984, Karachentsev \& Nasonova 2010, Karachentsev et al. 2014) and the Fornax
cluster (Nasonova et al. 2011).

It is important to stress that the $R_0$ method estimates the total mass of a
group independently of mass estimates based on virial motions. Notably, the
corresponding total mass $M_T$ is confined on the linear scale of $R_0$, which
is three to four times as large as the virial radius of a group or cluster,
$R_{vir}$.

\begin{figure*}
\centering
\includegraphics[width=17cm]{synthetic_mw.eps}
\caption{Distribution of 45 Milky Way satellites by their spatial distances and
radial velocities relative to the Milky Way. Dashed lines correspond to the
parabolic velocity for a point mass of $1\times10^{12}M_{\odot}$.}
\label{fig1}
\end{figure*}

The implementation of the $R_0$ method became possible with wholesale
measurements of distances to nearby galaxies from luminosities of the red giant
branch stars (TRGB) with accuracy of $\sim5$\% attainable by the Hubble Space
Telescope. In the Local Volume, limited to 11 Mpc, there are about a thousand
known galaxies; most of these galaxies have measured radial velocities with a
typical accuracy less than 5 km\,s$^{-1}$. About one-third of the Local Volume
population already has accurate TRGB distance estimates. The compilation of
observational data on these objects is presented in the Updated Nearby Galaxy
Catalog (Karachentsev et al. 2013) and its latest electronic version:
http://sao.ru/lv/lvgdb/ (Kaisina et al. 2012). For a typical galaxy of the Local
Volume with a distance of $\sim6$ Mpc, the TRGB distance error of $\sim300$ kpc
is comparable with a virial radius of the group, thus its location can be
confidently fixed relative to the group centroid and zero velocity surface.
Other methods of secondary importance are the Tully \& Fisher (1977) relation
distances or the brightest stars distances with an accuracy of $\sim(20-30)$\%
do not provide an opportunity to determine $R_0$ value even for the nearest
groups.

Below we use the most complete data on distances and radial velocities of the
Local Volume galaxies to estimate the zero-velocity radius around the local
massive galaxies.

\section{Galaxy motions around the Milky Way and M31}

\begin{figure*}
\centering
\includegraphics[width=17cm]{synthetic_m31_projected.eps}
\caption{Distribution of 52 test particles by their differential radial
velocities and projected distances from M31. Dashed lines correspond to the
parabolic velocity for a point mass of $1\times10^{12}M_{\odot}$.}
\label{fig2}
\end{figure*}

The recent surveys of large sky areas (Abazajian et al. 2009, Tonry et al. 2012,
Koposov et al. 2015) led to the discovery of new Milky Way (MW) dwarf
satellites with low luminosities and extremely low surface brightnesses. The
recent overview by McConnachie (2012) reports 29 MW satellites with measured
radial velocities and accurate distances. In recent years, this list has
been expanded up to 45 objects. The corresponding data are presented in Table~1.
The Table columns contain (1) galaxy name; (2) equatorial coordinates
J2000.0; and (3) tidal index,

\begin{equation}
TI = \max[\log(M^*_n/D^3_n)]-10.96, \,\,\,\, n=1, 2, \ldots N,
\end{equation}
distinguishing the most significant galaxy (main disturber = MD) among $N$
neighbouring galaxies, whose tidal force dominates the remaining galaxies with
masses $M^*_n$ and spatial separations $D_n$. The constant, --10.96, is chosen
in such a way that $TI=0$ corresponds to a significant neighbour located on the
zero velocity surface with $TI<0$ galaxies ranked as isolated. Finally, Col. (4)
lists the main disturber name, Col. (5) distance to a galaxy in Mpc, and Cols.
(6, 7) radial velocity of a galaxy (in km\,s$^{-1}$) relative to the Sun and
relative to the Milky Way centre with apex parameters adopted in NASA
Extragalactic Database (NED). References to the used values of distances and
velocities of galaxies are presented in the Local Volume Galaxies Database
(http://sao.ru/lv/lvgdb/).

The distribution of 45 satellites of the MW by their Galactocentric distances
and radial velocities is shown in Figure~1. The dashed lines correspond to the
parabolic velocity for a point mass of $1\times10^{12}M_{\odot}$. The velocity
distribution of satellites looks symmetrical relative to the MW centre, although
two satellites with near-parabolic velocities~--- Tucana and LeoI~--- are close
to the upper escape limit. Three MW satellites, Sag~dIr, DDO\,210, and Tucana
with distances $D\sim 1$~Mpc and negative $\Theta_1$, belong to field galaxies.
However, the MW is dynamically the most significant neighbour for each of these.

Specialized searches for faint satellites in the outskirts of the spiral galaxy
M31 in the Andromeda constellation (Ibata et al. 2007, Martin et al. 2009, Ibata
et al. 2014) has proved to be notably productive. While the sample by
McConnachie (2012) included 23 satellites, now their number is roughly doubled
amounting up to 44. The data on these satellites are presented in Table~2, where
the first six columns have the same meaning as in Table~1. The seventh column of
Table~2 contains spatial distances of satellites (in Mpc) relative to M31, while
eighth and ninth list the projected separation of satellites in the sky (in Mpc)
and their differential radial velocities relative to M31 (in km\,s$^{-1}$).
Aside from dwarf galaxies, we tabulate also the data on eight distant globular
clusters from PAndAS survey (Huxor et al. 2014) with measured radial velocities.
Their spatial distances still remain unknown, and we set them equal to 0.78 Mpc.

The distribution of $44+8$ test particles by their differential radial
velocities and projected separations relative to M31 is presented in Figure~2.
The dashed lines also mean the parabolic velocity for a point mass of
$1\times10^{12}M_{\odot}$. Similar to the MW case, the distribution of M31
satellites by their relative velocities seems to be very symmetrical; two
satellites~--- And~XIV and And~XII~--- have near-parabolic velocities that are
close to the lower escape limit.

\section{Orbital masses of the Milky Way and M~31}

For a massive galaxy surrounded by small satellites, the orbital mass estimate
is expressed as

\begin{equation
M_{orb}=(32/3\pi)\times(1-2e^{2}/3)^{-1}\times G^{-1}\times\langle\Delta V^2\times R_p\rangle,
\end{equation}
where $\langle\Delta V^2 \times R_p\rangle$ is the mean product of squared
radial velocity difference of a satellite with its projected distance from the
main galaxy and $e$ is the orbit eccentricity (Karachentsev \& Kudrya 2014).
This relation is obtained under the assumption of uniformly random orientation
of satellite orbits relative to the line of sight. With the typical eccentricity
value of $\langle e^2\rangle = 1/2$ (Barber et al. 2014) the relation (6)
becomes

\begin{equation
M_{orb}=(16/\pi)\times G^{-1}\times \langle\Delta V^2 \times R_p\rangle.
\end{equation}

Applying eq. (7) to the assembly of the MW and M31 satellites, we get values for
orbital masses $M_{orb}$(MW)$ = 1.51\times10^{12}M_{\odot}$ and $M_{orb}$
(M31)$ = 1.69\times10^{12}M_{\odot}$. Since in the case of Milky Way satellites
we observe their 3D distances, then project distances, the orbital mass
estimation should be reduced by a factor of $(\pi/4)$ yielding $M_{orb}$(MW)$ =
1.18\times10^{12}M_{\odot}$. Hence, the ratio of mass estimates for these two
galaxies reaches

\begin{equation
M_{orb}(MW)/M_{orb}(M31)\simeq0.70.
\end{equation}

This value is quite close to the ratio $M_{orb}(MW)/M_{orb}$(M31)=0.82 obtained
by Karachentsev \& Kashibadze (2006) from a minimum value for scatter of
peculiar velocities with respect to the Hubble regression line, while varying
the centroid position between the MW and M31.

A comparison of the derived total masses of the MW and M31, their combined mass,
and the mass ratio with other mass estimates in the recent literature is
presented in Table 3. These estimates were based on kinematics of satellites and
globular clusters assuming that the MW and M31 haloes follow the standard NFW
profile or fit the kinematics of high-velocity stars and blue horizontal branch
stars.  Our present measurements are in good agreement with the median values
given in the last line of Table 3. An essential part of the mismatch between the
different estimates in Table 3 may arise from the observed orbital anisotropy of
the MW and M31 satellites (Ibata et al. 2013, Pawlowski et al. 2014) and from
the uncertain dynamical status of two Milky Way satellites, Leo~I and Tucana,
and  the two M31 satellites, And~XIV and And~XII. Excluding these objects
reduces the mass estimates by 14--15\% in both cases.

\section{Hubble flow around the Local Group}

The proximate velocity field around the Local Group was considered in most
detail by Karachentsev \& Kashibadze (2006) and Karachentsev et al. (2009). For
a sample of 30 galaxies with TRGB distances from 0.7 to 3.0 Mpc with respect to
the Local Group centre, it was shown that the Hubble flow is characterized by
the local Hubble parameter $H_{loc}=(78\pm2)$ km\,s$^{-1}$ Mpc$^{-1}$, the
radial velocity dispersion $\sigma_v\simeq25$ km\,s$^{-1}$, and the radius of
zero-velocity surface $R_0=(0.96\pm0.03)$ Mpc. The minimal value of $\sigma_v$
corresponded to the barycentre position of $D_c=(0.55\pm0.05) D_{M31}$ = 0.43
Mpc, determining the mass ratio of $M_{MW}/M_{M31}\simeq0.8$ stated above.

\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{geometry.eps}}
\caption{Models of major and minor attractor. O represents an observer, C
represents the centre of a galaxy group, and G represents a test particle (a
galaxy).}
\label{fig3}
\end{figure}

\begin{figure*
\vspace{0.2cm
\includegraphics[width=12cm]{synthetic_regression_lg_binary_minor.eps}\\%
\vspace{0.2cm}\\%
\sidecaptio
\includegraphics[width=12cm]{LGcentroid_lg.eps
\caption{Distribution of isolated galaxies by distances and velocities relative
to the Local group barycentre assuming the minor attractor model.
{\itshape{}Upper panel:} The barycentre position locates at $x = D_c/D_{M31} =
0.55$ towards the M 31. The solid circles indicate galaxies that have the MW or
M31 as the main disturber. {\itshape{}Lower panel:} The Hubble flow around the
LG at the barycentre position of $x = D_c/D_{M31} = 0.43$. The grey wedges trace
the companion positions under different $x$, where their thin end  corresponds
to the barycentre coinciding with M~31.Solid and dotted lines in the insert
indicate the velocity dispersion as a function of $x$ with and without the
Leo~A, respectively.\vspace{0.8cm}}
\label{fig4}
\end{figure*}

In that approach, we considered the so-called minor attractor model, illustrated
by the upper panel of Figure~3. Here, a galaxy group with centre, $C,$ is
separated by a distance, $D_c$, from the observer, $O$, and moves away along the
line of site with the velocity, $V_c$. In the outskirts of the group there is a
galaxy, $G$, with distance, $D_g$, and radial velocity, $V_g$. If the angle
between $C$ and $G$ is $\theta$, then their mutual separation is expressed as

\begin{equation
R^2=D_g^2+D_c^2 -2D_g\times D_c\times\cos \theta,
\end{equation}
and the projected differential velocity is given by

\begin{equation
V_{gc}=V_g\times\cos\lambda-V_c\times \cos\mu,
\end{equation}
where $\mu=\lambda+\theta$, and

\begin{equation
\tan\lambda = D_c \times \sin\theta/(D_g-D_c \times \cos\theta).
\end{equation}
In this scheme we assumed peculiar velocities of galaxies in the vicinity
of a group to be small compared with velocities of the regular Hubble flow.

Yet, there is another possibility, which is the major attractor case (see lower
panel of Figure~3); this case is characterized by predominating infall towards
the centre of a group or a cluster. If $V_i$ is the infall velocity than

\begin{equation
V_g=V_c\times\cos\theta-V_i\times\cos\lambda,
\end{equation}
and the velocity of a galaxy relative to the group centre is expressed as

\begin{equation
V_i=[V_c\times\cos\theta-V_g]/\cos\lambda.
\end{equation}

Evidently the difference between these two models would be insignificant if the
galaxy lays almost strictly behind ($\lambda\simeq0$) or in front
($\lambda\simeq180^{\circ}$) of the group centre.

The last few years astronomers have detected some new dwarf galaxies in the
vicinity of the Local Group (KKs3, LeoP, and KK258) and measured their accurate
TRGB distances and radial velocities. For some galaxies (KKR25 and Tucana), old
inexact values of radial velocities were corrected and distances were refined.
This circumstance has motivated us to redefine parameters of the local Hubble
flow.

To reduce the role of virial motions, we excluded galaxies with $TI>0$ from
consideration; thus, the MW and M31 satellites with distances $D_{MW} <0.8$ Mpc
were consequently excluded. The data on the rest field galaxies with $D_{MW}
<3.5$ Mpc are presented in Table~4. The columns of Table 4 contain (1) galaxy
name; (2) distance (in Mpc) from the Milky Way; (3) heliocentric radial
velocity( in km\,s$^{-1}$); (4) distance from the Local Group barycentre located
at $D_c= 0.43$ Mpc; (5, 6) velocity (in km\,s$^{-1}$) relative to the barycentre
in the case of minor and major attractor, respectively; (7) tidal index; (8) the
main disturber name; and (9) $\lambda$ in degrees (see Figure~3).

The distribution of 35 isolated galaxies by distances and velocities relative to
the Local Group barycentre for the case of minor attractor is presented in the
upper panel of Figure~4. As shown in the Table~4 data, only 14 galaxies of 35
have the MW or M31 as the main disturber; they are denoted by solid circles.
With reference to these objects, the zone affected gravitationally by the Local
Group reaches $R_c\simeq2.5$ Mpc, while more distant field galaxies are
influenced by other massive neighbours of the Local Group, such as M81,
NGC\,253, and NGC\,5128.

According to Peirani \& de Freitas Pacheco (2008), Falco et al. (2014), and
Penarrubia \& Fattahi (2017), the radial velocity profile around the spherically
symmetrical group or cluster can be expressed as

\begin{equation
V(R)=H_0\times R-H_0\times R_0\times (R_0/R)^{1/2},
\end{equation}
where $R_0$ is the radius of the zero-velocity surface to be found. The solid
line in Figure~4 corresponds to equation (14) with parameters defined from the
least squares method, $R_0=0.95\pm0.03$ Mpc, $H_0=90\pm2$ km\,s$^{-1}$
Mpc$^{-1}$, and $\sigma_v=42$ km\,s$^{-1}$. The errors of $R_0$ and $H_0$
parameters were estimated using the Monte Carlo method, assuming that
distance errors for galaxies are distributed normally with a typical value of
$\sim5$\%. The peculiar velocity dispersion in the upper panel of Figure~4 is
contributed mostly by distant galaxies, which are disturbed by the neighbouring
groups. Considering the only 14 galaxies in the zone affected gravitationally by
the Local Group, we obtain the following parameters for the surrounding Hubble
flow: $R_0=0.85\pm0.03$ Mpc, $H_0=79\pm3$ km\,s$^{-1}$ Mpc$^{-1}$, and
$\sigma_v=23$ km\,s$^{-1}$. In the major attractor model,
these parameters vary slightly, since $\lambda$ values for these 14 galaxies are
small (see the last column in Table~4).

Three parameters, i.e. $R_0$, $H_0$ and $\sigma_v$, characterizing the local
cosmic expansion, moderately depend on the position of the LG barycentre. Above,
we used the barycentre location at the distance of $D_c= 0.55 D_{M31}$ = 0.43
Mpc, corresponding to the mass ratio of $M_{M31}/M_{MW}$= 1.2. This ratio
matches well with the medians in Table~3. However, Penarrubia et al.(2014) found
that the minimal scatter of nearby galaxies within 3 Mpc around the LG is
achieved with $M_{M31}/M_{MW}$ = 0.75. The authors have concluded that their
analysis rules out models in which M31 is more massive than our Galaxy with
about 95\% confidence. To check this statement, we calculated $\sigma_v$ for 14
nearest isolated galaxies as a function of the position of the LG barycentre $ x
= D_c/D_{M31}$ on the line connecting the MW with M31. The data on $\sigma_v$
and $R_0$ are presented in Table~5. The lower panel of Figure~4 shows the local
Hubble diagram for 14 galaxies at various M31-to-MW mass ratios. Each galaxy is
drawn by grey wedge with caliber inversely related to the dispersion $\sigma_v$
at given barycentre position; thus its thinner end indicates the barycentre
position at M31. The insert in the figure shows the velocity scatter of galaxies
respect to the best-fitting regression line. The solid and dotted lines in the
insert represent the behaviour of $\sigma_v$ for a case of included or excluded
Leo~A, respectively. This dwarf galaxy is a marginally isolated object with the
tidal index TI= +0.03. The derived minimums of these two lines fix the M31-to-MW
mass ratio near 0.7 and 1.0, respectively, not allowing a firm assessment of
which galaxy mass is dominated. Over the range of $M_{M31}/M_{MW}= [1/3 - 3]$
the value of the zero velocity radius is changing within $R_0 =0.86 - 0.96$ Mpc.
Thus, the observed coldness of the local Hubble flow leads us to measure the
radius of the sphere separating the Local Group from the global cosmic expansion
with $\sim5\%$ error. According to (4), the radius $R_0=0.91\pm0.05$ Mpc yields
the total mass estimate  for the Local Group $M_T=(1.5\pm0.2)\times
10^{12}M_{\odot}$ with an unprecedented accuracy, although this quantity lies
below all values of M(MW+M31) in Table 3. The mismatch becomes slightly less
dramatic when the Planck model parameters in (4) replace the WMAP parameters as
follows:\ $\Omega_m =0.24, \Omega_{\lambda} = 0.76$ and $H_0$ = 73
km\,s$^{-1}$\,Mpc$^{-1}$ (Spergel et al. 2007)); this increases the coefficient
in (4) from 1.95 to 2.12.

As noted by Chernin et al. (2004), the actual deviation of the binary
shape of the Local Group from the spherical symmetry produces a minor bias in
the $R_0$ and mass estimate. According to N-body simulations by Penarrubia
et al. (2014), neglecting the quadrupole potential overestimates the Local Group
mass up to $\sim30\%$.

\section{Other massive galaxies in the Local volume}

Considering the Hubble flow around other giant galaxies of the Local Volume, we
selected 15 galaxies with stellar masses $M^*>3\times 10^{10}M_{\odot}$ and
accurate distances. Their overview is presented in Table~6 with objects ranging
by their distances from the observer. For each of these 15 galaxies, surrounded
by a suite of satellites, the second most massive member of its group is also
indicated. In some cases, i.e. M31 and the Milky Way, NGC\,5128 (CenA) and
NGC\,5236, Maffei~2, and IC\,342, the second galaxy is comparable in mass with
the first galaxy and acts itself as the centre of a dynamically separated
subgroup.

The columns of Table~6 contains (1) galaxy name; (2, 3) its supergalactic
coordinates; (4) the galaxy distance from the MW; (5) its radial velocity
relative to the Local Group centroid; (6) logarithmic stellar mass; (7)
logarithmic orbital mass according to Karachentsev \& Kudrya (2014); (8) number
of satellites of the main galaxy with measured radial velocities and accurate
distances.

Aside from the galaxies presented in Table~6, the Local Volume contains another
two massive galaxies~--- NGC\,2903 ($\log M^*=10.82)$ and NGC~6946 ($\log
M^*=10.76)$. But their distances measured from the luminosity of brightest stars
are not yet sufficiently  accurate. In total, the 15 giant galaxies have about
500 satellites in their suites, but, as shown in the last column of Table~6,
only 102 satellites outside the Local Group have accurate estimates of distances
and velocities. Among the second most massive members of 15 groups, three
galaxies~--- NGC\,4242, NGC\,4597 and NGC\,6684~--- have Tully-Fisher distances
with accuracy of $\sim20$\% (denoted with column signs). In 11 of 15 groups, the
main galaxy exceeds its satellites twice or more in mass, allowing us to
estimate its halo mass from the orbital motions. This approach is not worthwhile
in the case of the rich group Leo~I, where NGC\,3379, NGC\,3368, and several
other bright members have compatible luminosities.

Despite the great efforts to measure highly accurate TRGB distances of nearby
galaxies from Hubble Space Telescope data, many neighbouring groups stay still
poorly explored. For example, in the outskirts of giant galaxies NGC\,4594
(Sombrero), NGC\,5055, and  NGC\,3115, no satellites have reliable
distance estimates.

\section{Cosmic flow around the synthetic (stacked) nearby group}

\begin{figure*}
\vspace{0.2cm
\includegraphics[width=12cm]{synthetic_regression_lv_main_minor.eps}\\%
\vspace{0.2cm}\\%
\sidecaptio
\includegraphics[width=12cm]{synthetic_regression_lv_binary_minor.eps}
\caption{Hubble diagram for the synthetic group of the Local Volume, assuming
the minor attractor model. {\itshape{}Upper panel:} The distances and velocities
of satellites are calculated relative to the main galaxy in a group.
{\itshape{}Lower panel:} The distances and velocities of satellites are
calculated relative to the barycentre of a pair of the most massive galaxies in
each group.\vspace{0.8cm}}
\label{fig5}
\end{figure*}

Seeking to use as much information as possible about companion motions around
the nearby massive galaxies outside their virial zones, we combined the data on
companions of various galaxies into the single synthetic group. To be included
into the consolidated group, a galaxy should satisfy the following four
conditions: (1) a companion has accurate estimates of distance and radial
velocities; (2) the companion distance from the main galaxy, $R_{MG}$, is less
than 3.5 Mpc; 3) the companion belongs to field galaxies, having $TI<0$; and 4)
the companion has a proper aspect, when its position angle $\lambda$ between the
vector of companion radial velocity and the line joining it with the main galaxy
(see Figure~3) lays within $\lambda<45^{\circ}$ or $\lambda>135^{\circ}$.

These conditions are satisfied for 66 galaxies of the Local Volume; the
corresponding data are presented in Table~7. Its columns contain: (1) name of
the main galaxy acting as the centre of its suite; (2) name of a companion
galaxy; (3) companion galaxy distance from the group barycentre; (4, 5)
companion galaxy velocity relative to the group barycentre in the case of minor
or major attractor; (6, 7) tidal index of the galaxy and the name of its main
disturber; and (8) position angle of the companion as indicated in Figure~3.

The Hubble diagram for the synthetic group of the Local Volume for the minor
attractor model with distances and velocities calculated relative to the main
galaxy is shown in the upper panel of Figure~5. The cosmic flow around the
synthetic group is characterized by the Hubble parameter $H_0=(76\pm2)$
km\,s$^{-1}$/Mpc, velocity dispersion $\sigma_v=62$ km\,s$^{-1}$, and radius of
the zero velocity surface $R_0=0.83\pm0.03$ Mpc. As one can see, the radius
$R_0$ turned out to be quite small, corresponding to the effective mass of the
synthetic group of $\sim1.1\times10^{12}M_{\odot}$. To estimate how various
factors influence $R_0$, we constructed another series of Hubble diagrams.
An alternative Hubble diagram with distances and velocities calculated relative
to the group barycentre rather than from the main galaxy itself is presented in
the lower panel of Figure~5. The barycentre is supposed to lie between the two
most massive galaxies of each group given in Table~6. In this case the local
Hubble parameter is $H_0=(85\pm2)$ km\,s$^{-1}$ Mpc, peculiar velocity
dispersion  $\sigma_v=57$ km\,s$^{-1}$, and radius $R_0$ reaches
$R_0=0.93\pm0.02$ Mpc.

As follows from the data on Table~6, the nearby galaxy groups differ
substantially in their stellar and virial masses, $M^*$ and $M_{orb}$, which can
lead to a systematic bias in the averaged $R_0$ estimate. To verify this effect,
we normalized distances of companions around each group to its individual
radius $R_0$, assuming $R_0\propto M_*^{1/3}$ or $R_0\propto M_{orb}^{1/3}$.
After that we did not find any decrease in peculiar velocity dispersion in the
synthetic Hubble diagram.

The resulting values of $H_0, \sigma_v$ and $R_0$ parameters for all discussed
cases are presented in Table~8; i.e. distances and velocities calculated
relative to the  main galaxy or group barycentre and the minor or major
attractor model. These data allow us to conclude, first, that changing a model
from a minor attractor to major attractor increases the $R_0$ estimate and
causes a significant increment in  dispersion, and, second, that accounting for
the second most massive galaxy in a group leads to a notable growth of the $R_0$
estimate.

\section{Discussion}

\begin{figure*}
\vspace{0.2cm}\sidecaptio
\includegraphics[width=12cm]{synthetic_fig6.eps}
\caption{Distribution of barycentres of 15 nearby groups by their radial
velocities and distances from the Local Group centre. The solid line corresponds
to the regular Hubble flow with Hubble parameter $H_0
=73$~km\,s$^{-1}$Mpc$^{-1}$.\vspace{0.8cm}}
\label{fig6}
\end{figure*}

As our estimates suggest, galaxies in the infall zone between the virial radius
and the $R_0$ are relatively small in number, $\sim15$\%. This circumstance,
inherent for the Local Group and for other nearby groups, opts for
estimating $R_0$ value within minor attractor model. The low value of peculiar
velocity dispersion resulting in this case is also an oblique argument for such
a choice.

The second most massive galaxy plays an essential role in the kinematics of
several nearby groups, often forming a dynamical subsystem. So, deciding on a
barycentre of the two most bright galaxies as the reference point for distances
and velocities of companions seems to be more preferable than the main galaxy
itself. Hence, we adopt the value of $0.93\pm0.02$ Mpc as the optimal estimate
for $R_0$ radius of the cumulative group (see the lower panel of Figure~5). The
corresponding mass is $\log(M_T/M_{\odot})=12.20$ with a formal error of
$\sim0.04\,$dex. Averaging orbital mass estimates from Table~6 and considering
the representation of each group in the Hubble diagram, we obtain the mean
logarithmic mass $\log(M_{orb}/M_{\odot}) =12.42\pm0.07$. So, the mass of the
synthetic group derived from outer motions of surrounding galaxies turned out to
be $\sim$60\% of the expected mass from inner orbital motions of satellites. A
probable source of this discrepancy was discussed by Chernin et al. (2013) and
Karachentsev \& Kudrya (2014).

As noted by Chernin et al. (2013), the estimate of the total
mass of a group includes two components,
$M_T = M_m + M_{DE}$, where $M_m$ is the mass of dark and baryonic matter
and $M_{DE}$ is the mass, negative in magnitude,  which is determined by the dark
energy with the density of  $\rho_{DE}$,

\begin{equation
M_{DE}= (8\pi/3)\times\rho_{DE}\times R^3.
\end{equation}
On the scale of virial radius, the contribution of this component in the group
mass does not exceed 1\%, but in the sphere of $R_0$  radius, the role of this
kind of a mass defect becomes significant. In the standard $\Lambda$CDM  model
with $\Omega_m =0.24$ and $H_0 =73$ km\,s$^{-1}$ Mpc$^{-1}$ the contribution of
dark energy is

\begin{equation
(M_{DE}/M_{\odot})= -0.85\times 10^{12} \times (R_0/ {\rm Mpc})^3,
\end{equation}
i.e. about 30\% of the Local Group mass determined by orbital motions. This
correction essentially reduces the observed discrepancy between the mass
estimates for the Local Group, as well as for other nearby groups, derived via
internal (virial) and external galaxy motions.

Another possible explanation might be caused by the existence of unrelaxed
(tidal) thin planar structures of satellites seen around the Milky Way and M~31
(Kroupa 2014), which are at variance with the assumption of spherical symmetry
case.

The peculiar velocity dispersion in the vicinity of the synthetic group, 57
km\,s$^{-1}$, is twice as large as in the outskirts of the Local Group. This
difference might originate from bulk motions of galaxies, which become
perceptible on the scale of $\sim5-10$ Mpc. A giant galaxy is not necessarily
the main disturber for neighbouring field galaxies. Indeed, this is the case for
only a portion of companion objects presented in Table~7. Another portion, which
are comprised of mostly distant field galaxies (shown by open circles), are
gravitationally influenced by a massive galaxy from another neighbouring group.

Figure~6 reproduces the distribution of barycentres of 15 nearby groups listed
in Table~5 by distances and radial velocities relative to the Local Group
centre. The straight line corresponds to the regular Hubble flow with $H_0 =73$
km\,s$^{-1}$ Mpc$^{-1}$. Barycentres of the groups with $D_{LG}<6$ Mpc situated
in the supergalactic plane (i.e. in the Local Sheet; Tully et al. 2016)
demonstrate a small scatter of radial velocities. More distant groups, around
M~101, NGC~5055, NGC~2683, NGC~3115, and NGC~3379 at supergalactic latitudes
$|SGB|>10^{\circ}$ (denoted with asterisks), exhibit negative peculiar
velocities about $-200$~km\,s$^{-1}$. These velocities are caused by the
observed expansion of the Local Void with an amplitude of $\sim
260$~km\,s$^{-1}$(Tully et al. 2016). Also, the group around Sombrero (NGC~4594)
is located just near the zero-velocity surface of the Virgo cluster. Its
positive peculiar velocity reflects the group fall towards the cluster.
Apparently, some portion of these bulk motions manifest themselves as extra
peculiar velocities of the Local Volume galaxies in the panels of Figure~5.
Ignoring these non-virial coherent motions may lead to the overestimation of
galaxy masses based on the Numerical Action Method (Peebles, 2017).

Our conclusion that the peripheral regions of the Local Group and other
neighbouring groups do not contain a large amount of dark matter seems to be the
most important result of this work. The bulk of mass is concentrated within the
virial radius of these groups. The same inference was made for the nearest Virgo
cluster (Karachentsev et al. 2014) from the observed infall of galaxies towards
the cluster centre. Yet further evidence is provided by Kourkchi \& Tully
(2017), who have considered infall zones and collapsed cores of halos in the
Local Universe.

A review of available observational data on distances and radial velocities of
the Local Volume galaxies shows that the population of outskirts of the nearby
groups has not yet been covered with highly accurate distance measurements.
There are groups, for example around the giant Sombrero galaxy, totally lack
reliable distance estimates, even for close probable satellites. The
systematical measurements of TRGB distances with the Hubble Space Telescope
within the Local Volume have the potential to provide meaningful data on the
distribution of the dark matter on the scales of $\sim1$ Mpc.

\begin{acknowledgements}
The authors are grateful to Elena Kaisina for updating the Local Volume Galaxies
Database (https://www.sao.ru/lv/lvgdb/), Dmitry Makarov for the renewed data on
galaxy tidal indices, and Brent Tully for reviewing the manuscript and for his
helpful comments. This work is supported by the Russian Science Foundation grant
No. 14--12--00965.
\end{acknowledgements}


\section{Introduction} \label{introduction}

The computational complexity associated with maximum-likelihood (ML)
detection (MLD) in multiple-input multiple-output (MIMO) communication
systems poses significant challenges for corresponding VLSI
implementations~\cite{burg05}. Promising approaches for complexity
reduction make explicit use of the structure in the problem by
transforming it into an equivalent lattice decoding\footnote{In the
  context of this paper it would be more appropriate to use the term
  ``lattice detection''.}  problem, where the underlying lattice is
finite. Some of the most prominent techniques for efficient lattice
decoding were, however, developed for infinite lattices, e.g., sphere
decoding~(SD)~\cite{FP1985} or lattice-reduction (LR)-aided detection,
e.g.,~\cite{YW02,WF03}.

The goal of this paper is to investigate the fundamental differences,
in terms of complexity and performance, between MIMO detection on
finite and on infinite lattices. We show that relaxation of the MIMO
detection problem to infinite lattices followed by LR can reduce the
complexity of SD at the cost of a significant performance loss in
terms of error rate.  This performance loss is caused by symbol
estimates that do not belong to the finite lattice and hence require
remapping (onto the finite lattice). We demonstrate that (near-)ML
achieving remapping strategies entail a complexity which is comparable
to that of SD operating directly on the finite lattice, thus rendering
LR in conjunction with SD unattractive for practical applications.

\subsubsection*{Notation}

Matrices are set in boldface capital letters, vectors in boldface
lowercase letters. The superscripts~$^T$ and~$^H$ stand for transpose
and conjugate transpose, respectively. We write~$A_{i,j}$ for the
entry in the~$i$th row and~$j$th column of the matrix~$\ensuremath{\mathbf{A}}$,~$b_i$ for
the~$i$th entry of the
vector~\mbox{$\ensuremath{\mathbf{b}}=[\,b_1\,\,b_2\,\,\cdots\,\,b_N\,]^T$}, and we
denote the~\mbox{$\ell^2$-norm} of \mbox{$\ensuremath{\mathbf{b}}\in\ensuremath{\mathbb{C}}^N$} as~$\|\ensuremath{\mathbf{b}}\|$.
$\ensuremath{\mathbf{I}}_N$ stands for the $N\times N$ identity matrix and $\ensuremath{\mathbf{1}}_{N}$
denotes the $N$-dimensional all-ones vector. $|\ensuremath{\mathcal{O}}|$ designates the
cardinality of the set~$\ensuremath{\mathcal{O}}$ and~$\ensuremath{\mathbb{CZ}}$ stands for the set of
Gaussian integers, i.e.,~$\ensuremath{\mathbb{CZ}} = \ensuremath{\mathbb{Z}} +\sqrt{-1}\,\ensuremath{\mathbb{Z}}$.  The real and
imaginary part of~$x\in\ensuremath{\mathbb{C}}$ is denoted by $\Re\{x\}$ and $\Im\{x\}$,
respectively.

\section{MIMO Detection as Lattice Decoding}
\label{latticedetection}

Consider a coherent MIMO system, i.e., the receiver has perfect
channel state information (CSI) while the transmitter does not have
CSI, with~${\ensuremath{M}}$ transmit and~${\ensuremath{M}}$ receive antennas\footnote{The
  restriction to the number of transmit antennas being equal to the
  number of receive antennas is made for simplicity of exposition. The
  results are also valid for the number of receive antennas being
  larger than the number of transmit antennas.}. The bit-stream to be
transmitted is mapped to ${\ensuremath{M}}$-dimensional transmit symbol
vectors~$\ensuremath{\mathbf{s}}\in\ensuremath{\mathcal{O}}^{\ensuremath{M}}$, where~$\ensuremath{\mathcal{O}}$ corresponds to the
underlying scalar QAM constellation. The associated complex baseband
input-output relation is given by
\begin{align} \label{eq:MIMOchannelmodel}
\ensuremath{\mathbf{y}} = \ensuremath{\mathbf{H}}\ensuremath{\mathbf{s}} + \ensuremath{\mathbf{n}}
\end{align}
where~$\ensuremath{\mathbf{H}}$ stands for the ${\ensuremath{M}}\times{\ensuremath{M}}$ channel matrix,~$\ensuremath{\mathbf{y}}$ is
the~${\ensuremath{M}}$-dimensional received signal vector, and~$\ensuremath{\mathbf{n}}$ is an i.i.d.\
circularly symmetric complex Gaussian distributed noise vector of
dimension~${\ensuremath{M}}$ with variance $N_o$ per complex entry. The
signal-to-noise ratio is defined as~$\ensuremath{\mathrm{SNR}}=E_s/N_o$, where~$E_s$
denotes the average signal power per receive antenna.

\subsection{Transformation to Lattices}

In order to see that the MIMO detection problem is equivalent to a
lattice decoding problem, we start by mapping the elements~$s\in\ensuremath{\mathcal{O}}$
to elements~$x\in\ensuremath{\mathbb{CZ}}$ using the transformation~$x = as+c$.  The
constants~$a,c\in\ensuremath{\mathbb{R}}$ with~$a>0$ and~$c>0$ are independent of~$s$ and
are chosen such that~$x\in\ensuremath{\mathcal{X}}\subset\ensuremath{\mathbb{CZ}}$
with~$\abs{\ensuremath{\mathcal{X}}}=\abs{\ensuremath{\mathcal{O}}}$ and
\begin{align} \label{eq:formalsetdefinition} \ensuremath{\mathcal{X}} = \big\{ x\in\ensuremath{\mathbb{CZ}}
  \mid & (k_\mathrm{min} \leq \Re\{x\} \leq k_\mathrm{max}) \nonumber
  \\ & \land
  (k_\mathrm{min} \leq \Im\{x\} \leq k_\mathrm{max}) \big\}
\end{align}
where~$k_\mathrm{min},k_\mathrm{max}\in \ensuremath{\mathbb{Z}}$. Note
that~\eqref{eq:formalsetdefinition} can be used for square QAM
constellations to check whether~$x'\in\ensuremath{\mathbb{CZ}}$ is in~$\ensuremath{\mathcal{X}}$ by performing
separate boundary checks for the real and imaginary part of~$x'$. In
the case of non-square QAM constellations the boundary checks take a
slightly more complicated form. The transmit
vectors~$\ensuremath{\mathbf{s}}\in\ensuremath{\mathcal{O}}^{\ensuremath{M}}$ can be mapped to
vectors~$\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}\subset\CZMT$ according to
\begin{align} \label{eq:constellationtrafo}
  \ensuremath{\mathbf{x}}=a\ensuremath{\mathbf{s}}+\ensuremath{\mathbf{c}}
\end{align}
where~$\ensuremath{\mathbf{c}}=c\ensuremath{\mathbf{1}}_{{\ensuremath{M}}}$. The inverse transformation associated
with~\eqref{eq:constellationtrafo} is given by~$\ensuremath{\mathbf{s}} =
a^{-1}(\ensuremath{\mathbf{x}}-\ensuremath{\mathbf{c}})$. The input-output
relation~\eqref{eq:MIMOchannelmodel} can now be transformed into
\begin{align} \label{eq:MIMOlatticechannelmodel}
\ensuremath{\mathbf{r}} = \ensuremath{\mathbf{G}}\ensuremath{\mathbf{x}} + \ensuremath{\mathbf{n}}
\end{align}
where~$\ensuremath{\mathbf{G}}=a^{-1}\ensuremath{\mathbf{H}}$ and~$\ensuremath{\mathbf{r}}=\ensuremath{\mathbf{y}}+\ensuremath{\mathbf{G}}\ensuremath{\mathbf{c}}$ is a translated version
of the received vector~$\ensuremath{\mathbf{y}}$.  The essence of the transformation
of~\eqref{eq:MIMOchannelmodel} into~\eqref{eq:MIMOlatticechannelmodel}
is that now the received vector~$\ensuremath{\mathbf{r}}$ can be interpreted as a lattice
point~$\ensuremath{\mathbf{u}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ that has been translated by the additive
Gaussian noise vector~$\ensuremath{\mathbf{n}}$. Here,
\begin{align} \label{eq:finitelattice}
  {\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}}) \define \Big\{ \ensuremath{\mathbf{G}}\ensuremath{\mathbf{x}} \mid \ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}} \Big\}
\end{align}
denotes the finite lattice generated by~$\ensuremath{\mathbf{G}}$. In the remainder of the
paper, we shall work with the input-output
relation~\eqref{eq:MIMOlatticechannelmodel} exclusively.

\subsection{ML Detection}

MLD in MIMO systems computes the estimate
\begin{align} \label{eq:constrainedMLrule}
  \ensuremath{\hat{\bmu}}^\mathrm{ML} = D^{\ensuremath{\mathrm{ML}}}(\ensuremath{\mathbf{r}}) = \argmin_{\ensuremath{\mathbf{u}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})}\|\ensuremath{\mathbf{r}}-\ensuremath{\mathbf{u}}\|
\end{align}
which amounts to solving a closest-vector problem (CVP) in the finite
lattice~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$. Since each lattice point in~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$
is associated with a transmit vector in~$\ensuremath{\mathcal{X}}^{\ensuremath{M}}$ according to the
relation~\mbox{$\ensuremath{\mathbf{u}}=\ensuremath{\mathbf{G}}\ensuremath{\mathbf{x}}$}, the ML-estimate~$\ensuremath{\hat{\bmu}}^{\ensuremath{\mathrm{ML}}}$ obtained
by solving~\eqref{eq:constrainedMLrule} can be transformed
into\footnote{Note that we implicitly assume that~$\ensuremath{\mathbf{G}}$ has full
  rank; this is satisfied, for example, with probability 1, if the
  entries of~$\ensuremath{\mathbf{G}}$ are i.i.d.\ circularly symmetric complex Gaussian
  distributed.  Furthermore, in practice~$\ensuremath{\mathbf{G}}^{-1}$ does not have
  to be computed explicitly as~$\ensuremath{\mathbf{u}}$ in~\eqref{eq:constrainedMLrule}
  can be replaced by~$\ensuremath{\mathbf{G}}\ensuremath{\mathbf{x}}$ and the minimization can be performed
  over~$\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}$.} \mbox{$\ensuremath{\hat{\bmx}}^{\ensuremath{\mathrm{ML}}}=\ensuremath{\mathbf{G}}^{-1}\ensuremath{\hat{\bmu}}^{\ensuremath{\mathrm{ML}}}$}, which upon
inversion of~\eqref{eq:constellationtrafo} yields
the ML-estimate~$\ensuremath{\hat{\bms}}^{\ensuremath{\mathrm{ML}}}$.  Solving~\eqref{eq:constrainedMLrule}
through an exhaustive search over all lattice points
in~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ typically results in high computational complexity
as $\abs{{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})}=\abs{\ensuremath{\mathcal{X}}}^{\ensuremath{M}}$ may be large.

\subsection{Relaxation and Lattice Reduction}
\label{relaxationandLR}

A promising approach to reducing the computational complexity
associated with solving~\eqref{eq:constrainedMLrule} is to relax the
finite lattice in~\eqref{eq:finitelattice} to the infinite
lattice\footnote{In the remainder of the paper, underlined quantities
  always refer to the infinite-lattice case.}
\begin{align} \label{eq:infinitelattice}
  \LatticeREL(\ensuremath{\mathbf{G}}) \define \Big\{ \ensuremath{\mathbf{G}}\bmxREL \mid
  \bmxREL\in\CZMT \Big\}
\end{align}
and to solve a relaxed version of~\eqref{eq:constrainedMLrule} by
computing
 \begin{align} \label{eq:unconstrainedMLrule} \bmuhatREL^{\ensuremath{\mathrm{ML}}} =
    {\ensuremath{\underline{D}}}^{\ensuremath{\mathrm{ML}}}(\ensuremath{\mathbf{r}}) =
    \argmin_{{\ensuremath{\underline{\mathbf{u}}}}\in\LatticeREL(\ensuremath{\mathbf{G}})}\|\ensuremath{\mathbf{r}}-{\ensuremath{\underline{\mathbf{u}}}}\|
\end{align}
i.e., by searching over the infinite lattice generated by~$\ensuremath{\mathbf{G}}$. This
search can be carried out more efficiently by using algorithms from
lattice theory, e.g., SD~\cite{FP1985} possibly in conjunction with LR
through the LLL algorithm~\cite{LLL82}. As the resulting
estimate~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}$ is not necessarily in~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$,
remapping of~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}$ onto the finite lattice~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$
is required whenever \mbox{$\bmuhatREL^{\ensuremath{\mathrm{ML}}}\notin{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$}.  In
the remainder of the paper, the term ``MLD'' always refers to the
finite-lattice MLD problem~\eqref{eq:constrainedMLrule}, whereas
``relaxed MLD'' is used to refer to the infinite-lattice MLD
problem~\eqref{eq:unconstrainedMLrule}.

The main motivation for relaxation resides in the fact that LR can be
applied only to infinite lattices. LR computes an equivalent and
``more orthogonal'' basis (for the lattice~$\LatticeREL(\ensuremath{\mathbf{G}})$) with
the generator matrix \mbox{$\ensuremath{\mathbf{B}}=\ensuremath{\mathbf{G}}\ensuremath{\mathbf{T}}$}, where~$\ensuremath{\mathbf{T}}$ is an
${\ensuremath{M}}\times{\ensuremath{M}}$ unimodular matrix, i.e., $\abs{\det(\ensuremath{\mathbf{T}})}=1$
with~$T_{i,j}\in\ensuremath{\mathbb{CZ}}$ ($\forall i,j$). Thanks to the unimodularity
of~$\ensuremath{\mathbf{T}}$, we have \mbox{$\LatticeREL(\ensuremath{\mathbf{B}})=\LatticeREL(\ensuremath{\mathbf{G}})$}.  It is
important to note that this equivalence holds only for infinite
lattices, in contrast to finite lattices where \mbox{${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{B}})
  \neq {\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$}, in general.  We therefore conclude that
relaxation to an infinite lattice is essential for the application of
LR to MIMO detection.

We emphasize that LR, in general, reduces the complexity of a
subsequent SD step and results in (often significant) performance
improvements when followed by sub-optimal detectors. In particular,
in~\cite{TMK07} it is shown that LR followed by linear detection
achieves full diversity.

\subsection{Schnorr-Euchner Sphere Decoding}
\label{spheredecoder}

SD was initially designed for the efficient solution of CVPs on
infinite lattices~\cite{FP1985}, but has also proven very efficient
for MLD~\cite{Mow91,VB93}. In the following, we briefly review the
main ingredients of Schnorr-Euchner SD (SESD) with radius reduction
for the finite-lattice case~\cite{AEVZ02}.

The algorithm starts by performing a QR decomposition (QRD) of~$\ensuremath{\mathbf{G}}$
according to~$\ensuremath{\mathbf{G}}=\ensuremath{\mathbf{Q}}\ensuremath{\mathbf{R}}$, where the ${\ensuremath{M}}\times{\ensuremath{M}}$ matrix~$\ensuremath{\mathbf{Q}}$ is
unitary and the upper-triangular~${\ensuremath{M}}\times{\ensuremath{M}}$ matrix~$\ensuremath{\mathbf{R}}$ has
real-valued non-negative entries on its main diagonal.
Left-multiplying~\eqref{eq:MIMOlatticechannelmodel} by~$\ensuremath{\mathbf{Q}}^H$ leads
to the modified input-output relation $\ensuremath{\tilde{\bmr}} = \ensuremath{\mathbf{R}}\ensuremath{\mathbf{x}} +
\ensuremath{\tilde{\bmn}}$, where $\ensuremath{\tilde{\bmr}}=\ensuremath{\mathbf{Q}}^H\ensuremath{\mathbf{r}}$ and $\ensuremath{\tilde{\bmn}}=\ensuremath{\mathbf{Q}}^H\ensuremath{\mathbf{n}}$ is
again (thanks to~$\ensuremath{\mathbf{Q}}$ being unitary) i.i.d.\ circularly symmetric
complex Gaussian distributed with variance~$N_o$ per complex entry.
We arrange the partial symbol vectors (PSVs)
\mbox{$\ensuremath{\mathbf{x}}^{(i)}=[\,x_i\,\,x_{i+1}\,\,\cdots\,\,x_{\ensuremath{M}}\,]^T$} in a
tree that has its root just above level~$i={\ensuremath{M}}$ and leaves on
level~$i=1$; the leaves correspond to vectors~$\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}$.
Each PSV is associated with a partial distance~(PD) according to
\begin{align} \label{eq:partialdistance}
  d\big(\ensuremath{\mathbf{x}}^{(i)}\big) = \big\|\ensuremath{\tilde{\bmr}}^{(i)}-\ensuremath{\mathbf{R}}_i\ensuremath{\mathbf{x}}^{(i)}\big\|,
  \quad i=1,2,\ldots,{\ensuremath{M}}
\end{align}
where~$\ensuremath{\tilde{\bmr}}^{(i)}=[\,\tilde{r}_i\,\,\tilde{r}_{i+1}\,\,\cdots\,\,\tilde{r}_{\ensuremath{M}}\,]^T$
and~$\ensuremath{\mathbf{R}}_i$ contains the lower-right $({\ensuremath{M}}-i+1)\times
({\ensuremath{M}}-i+1)$-dimensional sub-block of~$\ensuremath{\mathbf{R}}$. The MLD problem has
therefore been transformed into a weighted tree-search problem, where
PSVs and PDs are associated with nodes in the tree. For brevity, we
shall often say ``the node~$\ensuremath{\mathbf{x}}^{(i)}$'' meaning the node
corresponding to the PSV~$\ensuremath{\mathbf{x}}^{(i)}$. Each path through the tree
(from the root node down to a leaf node) corresponds to a symbol
vector~\mbox{$\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{{\ensuremath{M}}}$}.  Note that in the infinite-lattice
case, the tree-search problem needs to be relaxed to
vectors~$\bmxREL\in(\ensuremath{\mathbb{CZ}})^{\ensuremath{M}}$.  The solution
of~\eqref{eq:constrainedMLrule} is given by the leaf associated with
the smallest metric.

The essence of SD is to constrain the search in the tree through a
sphere constraint (SC), realized by searching only those nodes that
satisfy~$d\big(\ensuremath{\mathbf{x}}^{(i)}\big)\leq r$, where~$r$ is the search radius
associated with a hypersphere centered in the received point.  SESD
refers to conducting the tree search in a depth-first manner and
visiting the children of a given node in ascending order of their
PDs~\cite{AEVZ02}. Since the PDs~$d\big(\ensuremath{\mathbf{x}}^{(i)}\big)$ are
non-decreasing as a function of~$i={\ensuremath{M}},{\ensuremath{M}}-1,\ldots,1$, a
node~$\ensuremath{\mathbf{x}}^{(i)}$ violating the SC can be pruned along with the entire
subtree emanating from that node. The algorithm initializes~$r=\infty$
and performs the update~$r\gets d(\ensuremath{\mathbf{x}})$ whenever a valid leaf
node~$\ensuremath{\mathbf{x}}$ has been found. This approach is known in the literature
as radius reduction~\cite{AEVZ02,burg05}.

The complexity measure employed in this paper is given by the number
of nodes visited by SESD in the tree search, including the leaf nodes,
but excluding the root node. This complexity measure is directly
related to the VLSI implementation complexity for the finite-lattice
case\footnote{We expect this complexity measure to be equally relevant
  for the infinite-lattice case.}~\cite{burg05}.  The complexity
required for preprocessing (i.e., QRD or LR) will be ignored
throughout the paper. We note, however, that in latency-critical
applications, the preprocessing complexity can be critical.

In the remainder of the paper, ``SESD'', ``relaxed SESD'', and
``LR-aided SESD'' refer to SESD on finite lattices, on infinite
lattices, and to LR followed by SESD on infinite lattices,
respectively.

\section{Complexity of SESD and of Relaxed SESD}
\label{detectioncomplexity}

We next analyze the differences in the complexity behavior of SESD and
of relaxed SESD.

\subsection{Low-SNR Regime}
\label{lowsnrcomplexity}

\subsubsection*{Finite-lattice case}

SESD visits a node~$\ensuremath{\mathbf{x}}^{(i)}$ if the SC $d\big(\ensuremath{\mathbf{x}}^{(i)}\big) \leq
r$ is satisfied. Since the minimum search radius of SESD (guaranteeing
that the corresponding hypersphere contains at least one lattice
point) is given by the Euclidean distance between the received vector
and the ML solution, we can conclude that SESD visits at least all
nodes~$\ensuremath{\mathbf{x}}^{(i)}$ satisfying
\begin{align} \label{eq:sufficientconditionfiniteSESD}
  d\big(\ensuremath{\mathbf{x}}^{(i)}\big) \leq
  \min_{\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}}\|\ensuremath{\tilde{\bmr}}-\ensuremath{\mathbf{R}}\ensuremath{\mathbf{x}}\| = r_\mathrm{min}.
\end{align}
Thanks to radius reduction, the final (with respect to the repeated
tree-traversals as described above) search radius of SESD corresponds
to~$r_\mathrm{min}$. We can therefore conclude that the genie-aided
choice \mbox{$r=r_\mathrm{min}$} for the search radius initialization,
indeed, yields a lower bound on the number of nodes visited by SESD.
Denoting the transmitted data vector as~$\ensuremath{\mathbf{x}}'\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}$, we obtain
a lower bound on~$r_\mathrm{min}$ as follows:
\begin{align}
  r_\mathrm{min} & \stackrel{a)}{=} \min_{\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}}
  \big\|\ensuremath{\tilde{\bmn}}+\ensuremath{\mathbf{R}}(\ensuremath{\mathbf{x}}'-\ensuremath{\mathbf{x}})\big\| \geq \nonumber \\
  & \stackrel{\mathrm{b)}}{\geq} \min_{\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}} \Big(
  \|\ensuremath{\tilde{\bmn}}\|-\big\|\ensuremath{\mathbf{R}}(\ensuremath{\mathbf{x}}-\ensuremath{\mathbf{x}}')\big\| \Big)= \nonumber \\
  & = \|\ensuremath{\tilde{\bmn}}\|-
  \max_{\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}}\big\|\ensuremath{\mathbf{R}}(\ensuremath{\mathbf{x}}-\ensuremath{\mathbf{x}}')\big\| \geq \nonumber \\
  & \stackrel{\mathrm{c)}}{\geq}
  \|\ensuremath{\tilde{\bmn}}\|-\sigma_\mathrm{max}(\ensuremath{\mathbf{R}})K
  \label{eq:SESDcomplxlowerbound}
\end{align}
where a) results from~$\ensuremath{\tilde{\bmr}} = \ensuremath{\mathbf{R}}\ensuremath{\mathbf{x}}' + \ensuremath{\tilde{\bmn}}$, b) follows
from the inverse triangle inequality, and c) is a consequence of the
Rayleigh-Ritz theorem~\cite{HJ85} ($\sigma_\mathrm{max}(\ensuremath{\mathbf{R}})$ denotes
the largest singular value of $\ensuremath{\mathbf{R}}$) and the fact that the
constant~\mbox{$K=\max_{\ensuremath{\mathbf{x}},\ensuremath{\mathbf{x}}'\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}}\|\ensuremath{\mathbf{x}}-\ensuremath{\mathbf{x}}'\|$} is
finite in the finite-lattice case.
From~\eqref{eq:SESDcomplxlowerbound} we can infer that SESD visits at
least all nodes~$\ensuremath{\mathbf{x}}^{(i)}$ satisfying
\begin{align} \label{eq:sufficientconditionfiniteSESDLB}
  d\big(\ensuremath{\mathbf{x}}^{(i)}\big) \leq \|\ensuremath{\tilde{\bmn}}\|-\sigma_\mathrm{max}(\ensuremath{\mathbf{R}})K.
\end{align}
Note that setting the search radius equal to the right-hand side (RHS)
of~\eqref{eq:sufficientconditionfiniteSESDLB} will no longer guarantee
that the corresponding hypersphere contains at least one lattice
point.  This is, however, irrelevant for the point we want to make
next.

Defining~\mbox{$\ensuremath{\mathbf{v}}=\ensuremath{\tilde{\bmn}}/\sqrt{N_o}$}, we can
rewrite~\eqref{eq:sufficientconditionfiniteSESDLB} as
\begin{align}  \label{eq:sufficientconditionfiniteSESDLBnormalized}
  \Bigg\|\ensuremath{\mathbf{v}}^{(i)} +
  \frac{\ensuremath{\mathbf{R}}_i\big(\ensuremath{\mathbf{x}}'^{(i)}-\ensuremath{\mathbf{x}}^{(i)}\big)}{\sqrt{N_o}}\Bigg\| \leq
  \|\ensuremath{\mathbf{v}}\|-\frac{\sigma_\mathrm{max}(\ensuremath{\mathbf{R}})K}{\sqrt{N_o}}.
\end{align}
We can now see that for~$N_o\to\infty$ and hence for~$\ensuremath{\mathrm{SNR}}\to0$, the
condition in~\eqref{eq:sufficientconditionfiniteSESDLBnormalized}
reduces to~$\big\|\ensuremath{\mathbf{v}}^{(i)}\big\|\leq\|\ensuremath{\mathbf{v}}\|$, which is trivially
satisfied for~$i>1$ for every realization of~$\ensuremath{\tilde{\bmn}}$ and for
given~$\ensuremath{\mathbf{R}}$. In this case, SESD visits at least all nodes down to and
including the level just above the leaf level.  An intuitive
explanation for this effect is that the noise can shift the transmit
vector arbitrarily far away from the finite lattice and hence, the RHS
of~\eqref{eq:SESDcomplxlowerbound} and consequently~$r_\mathrm{min}$
can become arbitrarily large, which inhibits efficient tree pruning.

\subsubsection*{Infinite-lattice case}

The fundamental difference between the complexity of SESD and of relaxed
SESD is due to the fact that in the infinite-lattice case, the maximum
distance between the received vector~$\ensuremath{\mathbf{r}}$ and the nearest lattice
point is bounded for a given (full rank)~$\ensuremath{\mathbf{R}}$. The complexity of
relaxed SESD can therefore be lower than that of SESD.  The downside
is that relaxed SESD is not guaranteed to find the ML solution, which
necessarily has to be a point in the finite lattice~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$.
Specifically, if relaxed SESD delivers a lattice point
outside~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$, this point needs to be remapped onto the
finite lattice.  Corresponding remapping approaches are described
in~\secref{relaxedMLDperformance}.  In the following, we derive an
analytic upper bound on the complexity of relaxed SESD.

The number of nodes~$\bmxREL^{(i)}$ on tree level~$i$ that satisfy the
SC with radius~$r$ is denoted by~${\ensuremath{\underline{C}}}_{i}(r)$ and corresponds to the
number of lattice points in~$\LatticeREL(\ensuremath{\mathbf{R}}_i)$ that are within a
$2({\ensuremath{M}}-i+1)$-dimensional (real-valued) hypersphere of radius~$r$
centered at~$\ensuremath{\mathbf{r}}^{(i)}$. This number can be upper-bounded by dividing
the volume of a $2({\ensuremath{M}}-i+1)$-dimensional hypersphere of
radius~$r+\mu_i$ by the volume of a Voronoi cell
in~$\LatticeREL(\ensuremath{\mathbf{R}}_i)$~\cite{BK98}, where~$\mu_i$ corresponds to the
covering radius of the lattice~$\LatticeREL(\ensuremath{\mathbf{R}}_i)$~\cite{CS99}.  More
specifically, we have~\cite{Wills91}
\begin{align} \label{eq:complexityupperboundunbestimmt}
  {\ensuremath{\underline{C}}}_{i}(r) \leq \frac{V_{({\ensuremath{M}}-i+1)}(r+\mu_i)}{\mathrm{Vol}\big(\LatticeREL(\ensuremath{\mathbf{R}}_i)\big)}
\end{align}
where~$V_k(r) = \frac{\pi^k}{k!}r^{2k}$ denotes the volume of
a~$2k$-dimensional hypersphere with radius~$r$ and the volume of a
Voronoi cell of~$\LatticeREL(\ensuremath{\mathbf{R}}_i)$ is
\mbox{$\mathrm{Vol}\big(\LatticeREL(\ensuremath{\mathbf{R}}_i)\big)=\det(\ensuremath{\mathbf{R}}_i^H\ensuremath{\mathbf{R}}_i)$}.

In order to derive an upper bound on the total complexity
\mbox{${\ensuremath{\underline{C}}}=\sum_{i=1}^{{\ensuremath{M}}}{\ensuremath{\underline{C}}}_i(r)$} of relaxed SESD, we consider
the case where only the first radius update is performed. The first
leaf node found by relaxed SESD corresponds to the Babai
point~$\bmxhatREL^{\ensuremath{\mathrm{B}}}$~\cite{BABAI86} and hence the radius after
the first update from the initial value~$r=\infty$ is given
by~$d\big(\bmxhatREL^{\ensuremath{\mathrm{B}}}\big)$, which can be upper-bounded
as~\cite{BABAI86,AEVZ02}
\begin{align} \label{eq:babaibound}
  d\big(\bmxhatREL^{\ensuremath{\mathrm{B}}}\big) = \big\|\ensuremath{\tilde{\bmr}}-\ensuremath{\mathbf{R}}\bmxhatREL^{\ensuremath{\mathrm{B}}}\big\| \leq
  \sqrt{\frac{1}{2}\sum_{j=1}^{{\ensuremath{M}}}R^2_{j,j}} = \beta.
\end{align}
Since the covering radius~$\mu_1$ is also upper-bounded by~$\beta$
\cite{Kannan87,BK98} and, more generally, $\mu_i$ can be upper-bounded
as~$\mu_i\leq\sqrt{\frac{1}{2}\sum_{j=i}^{{\ensuremath{M}}}R^2_{j,j}}=\gamma_i$, we
have $r+\mu_i\leq\beta+\gamma_i$, which yields an upper bound on the
total complexity of relaxed SESD according to
\begin{align} \label{eq:totalcomplexityupperbound}
  {\ensuremath{\underline{C}}} \le
  \sum_{i=1}^{{\ensuremath{M}}}
  \frac{V_{({\ensuremath{M}}-i+1)}(\beta+\gamma_i)}{\det(\ensuremath{\mathbf{R}}_i^H\ensuremath{\mathbf{R}}_i)}.
\end{align}

\subsection{High-SNR Regime}
\label{highSNRcomplexity}

In the high-SNR regime, relaxed SESD is likely to return the ML
solution.  Moreover, if~$\|\ensuremath{\tilde{\bmn}}\|$ is sufficiently small, relaxed
SESD and SESD find the same leaf nodes, which will belong to the
finite lattice~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$. Consequently, relaxed SESD and SESD
will operate with the same search radii and hence the same SCs.
However, the two detectors will, in general, not exhibit the same
complexity, as the SESD additionally (to the SC) takes into account
the finite alphabet nature of~$\ensuremath{\mathcal{X}}\subset\ensuremath{\mathbb{CZ}}$, whereas relaxed SESD
visits \emph{all} points in~$\ensuremath{\mathbb{CZ}}$ satisfying the SC.  Therefore, in
the high-SNR regime, SESD tends to result in smaller complexity than
relaxed SESD.

\subsection{Numerical Complexity Assessment}
\label{numericalcomplexityassessment}

A simple numerical result serves to demonstrate the potential
complexity savings of relaxed SESD over SESD in the low-SNR regime. We assume a MIMO channel
with~$\ensuremath{\mathbf{R}}=\ensuremath{\mathbf{I}}_{\ensuremath{M}}$ and ${\ensuremath{M}}=4$. Counting all the nodes down to
and including the level just above the leaf level and noting that only
one leaf node is visited by SESD if~$\ensuremath{\mathbf{R}}$ is a diagonal matrix, leads
to the corresponding complexity
\begin{align*}
  C = \frac{\abs{\ensuremath{\mathcal{X}}}^{\ensuremath{M}}-1}{\abs{\ensuremath{\mathcal{X}}}-1}
\end{align*}
of SESD for~$\ensuremath{\mathrm{SNR}}\to0$ (see the discussion in the last paragraph of
\secref{lowsnrcomplexity}). For a 16-QAM alphabet, this leads to
\mbox{$C=4369$} nodes. The total complexity of relaxed SESD is
upper-bounded by (16), which can be refined by noting that relaxed SESD
visits only one leaf node in the case of~\mbox{$\ensuremath{\mathbf{R}}=\ensuremath{\mathbf{I}}_{\ensuremath{M}}$}. Consequently, we have
\begin{align*}
  {\ensuremath{\underline{C}}} \leq 1+ \sum_{i=2}^{{\ensuremath{M}}}
  V_{({\ensuremath{M}}-i+1)}\Bigg(\!\sqrt{\frac{{\ensuremath{M}}}{2}}+\sqrt{\frac{{\ensuremath{M}}-i+1}{2}}\Bigg)
\end{align*}
which shows that at most~1928 nodes will be visited, irrespectively of
the SNR. We can therefore conclude that relaxation can result in a
significant complexity reduction at low SNR values.

\figref{fig:CPXremapping} shows the complexity\footnote{All simulation
  results are for an uncoded \mbox{${\ensuremath{M}}=4$} MIMO system with 16-QAM
  symbol constellation using Gray labeling. The entries of~$\ensuremath{\mathbf{H}}$ are
  i.i.d.\ circularly symmetric complex Gaussian distributed with unit
  variance.  For (relaxed) SESD, we use sorted QR-decomposition (SQRD)
  according to~\cite{wuebben01}, for LR, a complex-valued variant of
  the LLL algorithm with~$\delta=3/4$ and SQRD preprocessing as
  described in~\cite{GW05,WBKK04}. All complexity and performance
  simulations are averaged over 640,000 channel realizations and for
  each channel realization a single noise realization has been
  generated.}  corresponding to SESD (which yields ML performance) and
to relaxed SESD (denoted by R-SESD). Consistent with the observations
made in Sections~\ref{lowsnrcomplexity} and~\ref{highSNRcomplexity},
we see that relaxation leads to complexity savings in the low-SNR
regime, but results in higher complexity in the high-SNR regime. We
can furthermore see that relaxation followed by LR (denoted by
LR-SESD) leads to lower complexity than SESD for all SNR values.

The complexity reductions of R-SESD and LR-SESD over SESD come,
however, at a significant performance loss in terms of error-rate and
in the case of LR-SESD also at increased computational complexity in
the preprocessing stage caused by the need for LR (realized through
the LLL algorithm~\cite{LLL82}, for example). For naive lattice
decoding, i.e., estimates~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}\notin {\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ are
simply discarded and an error is declared,~\figref{fig:BERremapping}
shows a 3\,dB~SNR loss of LR-SESD compared to SESD (MLD). We emphasize
that the performance of R-SESD, not shown
in~\figref{fig:BERremapping}, is identical to that of LR-SESD; the
associated complexities are, however, different
(cf.~\figref{fig:CPXremapping}). Note that naive lattice decoding was
shown to achieve full diversity, while leading to an unbounded SNR gap
(growing logarithmically in SNR) for MIMO systems with an equal number
of transmit and receive antennas~\cite{TK07}. In the next section, we
discuss several approaches to the mitigation of this performance loss,
i.e., we consider algorithms for remapping
estimates~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}\notin {\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ onto~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$.

\section{Remapping Methods}
\label{relaxedMLDperformance}

If the estimate delivered by relaxed SESD satisfies
\mbox{$\bmuhatREL^{\ensuremath{\mathrm{ML}}} \in {\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$}, the lattice
point~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}$ corresponds to the ML
solution~\eqref{eq:constrainedMLrule},
i.e.,~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}=\ensuremath{\hat{\bmu}}^{\ensuremath{\mathrm{ML}}}$. If~$\bmuhatREL^{\ensuremath{\mathrm{ML}}} \notin
{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ and the performance of naive lattice decoding is not
sufficient, the solution delivered by relaxed SESD needs to be
remapped onto the finite lattice~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$. Since all
constellation points~$\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}$ (corresponding to
elements~$\ensuremath{\mathbf{u}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ through the relation\footnote{In the
  remainder of the paper, we treat vectors~$\bmxREL\in(\ensuremath{\mathbb{CZ}})^{\ensuremath{M}}$ and
  \mbox{${\ensuremath{\underline{\mathbf{u}}}}\in\LatticeREL(\ensuremath{\mathbf{G}})$} as well as~$\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}$
  and~$\ensuremath{\mathbf{u}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ as interchangeable as they are related
  through the one-to-one mappings ${\ensuremath{\underline{\mathbf{u}}}}=\ensuremath{\mathbf{G}}\bmxREL$ and
  $\ensuremath{\mathbf{u}}=\ensuremath{\mathbf{G}}\ensuremath{\mathbf{x}}$, respectively. Of course, this assumes that~$\ensuremath{\mathbf{G}}$ has
  full rank.} $\ensuremath{\mathbf{u}}=\ensuremath{\mathbf{G}}\ensuremath{\mathbf{x}}$) satisfy~\eqref{eq:formalsetdefinition},
determining whether~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}$ is in the finite
lattice~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ can be performed efficiently through
element-wise boundary checks of the real and imaginary parts
of~$\bmxhatREL^{\ensuremath{\mathrm{ML}}}$.  In the following, we briefly review an existing
remapping method known as quantization and describe novel remapping
approaches that realize different performance/complexity trade-offs.

\subsection{Quantization}

The standard remapping approach known in the literature is
quantization and amounts to remapping
\mbox{$\bmxhatREL^{\ensuremath{\mathrm{ML}}}\notin\ensuremath{\mathcal{X}}^{\ensuremath{M}}$} to the closest (in Euclidean
distance) symbol vector~$\ensuremath{\hat{\bmx}}_\mathrm{q} \in \ensuremath{\mathcal{X}}^{\ensuremath{M}}$ according
to~\cite{WF03,WBKK04}
\begin{align} \label{eq:quantization}
  \ensuremath{\hat{\bmx}}_\mathrm{q} =
  \argmin_{\ensuremath{\mathbf{x}}\in\ensuremath{\mathcal{X}}^{\ensuremath{M}}} \big\|\bmxhatREL^{\ensuremath{\mathrm{ML}}}-\ensuremath{\mathbf{x}}\big\|.
\end{align}
As quantization corresponds to element-wise slicing of the entries
of~$\bmxhatREL^{\ensuremath{\mathrm{ML}}}$ to the nearest candidates in~$\ensuremath{\mathcal{X}}$, it requires
low computational complexity. The resulting
estimate~$\ensuremath{\hat{\bmx}}_\mathrm{q}$ does, however, not necessarily
correspond to the ML solution~\eqref{eq:constrainedMLrule} and hence
leads to a (potentially significant) performance loss.

\figref{fig:BERremapping} shows the performance obtained by LR-aided
SESD followed by remapping through quantization (denoted by LR-SESD,
quant.).  The numerical results indicate a slight improvement over
naive lattice decoding, but still show a 2.5\,dB SNR loss compared to
ML performance. Finally, \figref{fig:BERremapping} shows the
performance obtained by LR-aided successive interference cancellation
(SIC) on the infinite lattice with remapping through naive lattice
decoding (denoted by LR-SIC, naive). The resulting performance loss,
compared to relaxed SESD with remapping through naive lattice
decoding, is low (approximately 0.25\,dB SNR), whereas the complexity
of LR-SIC is significantly lower than that of LR-SESD.

\subsection{Optimal Lattice Remapping} \label{optimumlatticeremapping}

The discussion above brings out the importance of remapping.  It is
therefore natural to ask for the optimum lattice remapping strategy in
the sense of maximizing the probability of the result of remapping to
correspond to the transmitted symbol vector given the result of the
relaxed detector~$\bmuhatREL$. Let~$\ensuremath{\mathbf{u}}'\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ be the
transmitted vector and assume that the relaxed detector yields
\begin{align*}
  {\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})={\ensuremath{\underline{D}}}(\ensuremath{\mathbf{u}}'+\ensuremath{\mathbf{n}})=\bmuhatREL, \quad
  \bmuhatREL\in\LatticeREL(\ensuremath{\mathbf{G}})\!\setminus\! {\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}}).
\end{align*}
Optimal lattice remapping according to the criterion described above
now amounts to identifying the lattice point $\ensuremath{\mathbf{u}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$
which maximizes $\Pr{\ensuremath{\mathbf{u}}'= \ensuremath{\mathbf{u}}|{\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})=\bmuhatREL}$, i.e.,
\begin{align*}
  \ensuremath{\hat{\bmu}}_{\mathrm{opt}} = \argmax_{\ensuremath{\mathbf{u}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})}
  \Pr{\ensuremath{\mathbf{u}}'=\ensuremath{\mathbf{u}}|{\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})=\bmuhatREL}\!.
\end{align*}
We emphasize that the optimum remapping rule depends on~${\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})$
and hence, on the relaxed MIMO detector employed. Using Bayes's
theorem and assuming that all transmit vectors are equally likely, we
get
\begin{align} \label{eq:remapperoptimum} \ensuremath{\hat{\bmu}}_{\mathrm{opt}} =
  \argmax_{\ensuremath{\mathbf{u}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})} \Pr{{\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})=\bmuhatREL|\ensuremath{\mathbf{u}}'=\ensuremath{\mathbf{u}}}
\end{align}
which amounts to solving a finite-lattice MLD
problem~\eqref{eq:constrainedMLrule} based on the relaxed
estimate~$\bmuhatREL$ instead of the received vector~$\ensuremath{\mathbf{r}}$. We stress
that relaxed MLD followed by optimum lattice remapping according
to~\eqref{eq:remapperoptimum} will, in general, not achieve ML
performance as remapping is conducted based on~$\bmuhatREL$ rather
than on~$\ensuremath{\mathbf{r}}$. For the sake of simplicity of exposition, in the
following, we write $\Pr{{\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})=\bmuhatREL|\ensuremath{\mathbf{u}}}$ instead of
$\Pr{{\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})=\bmuhatREL|\ensuremath{\mathbf{u}}'=\ensuremath{\mathbf{u}}}$.

\subsubsection*{Closest-vector remapping}

The optimal remapping rule in~\eqref{eq:remapperoptimum} can, in
general, not be expressed in closed form, as it depends on the
decision region of the relaxed MIMO detector. We can, however, obtain
a simple remapping strategy that achieves near-optimal performance by
computing an approximation of~\eqref{eq:remapperoptimum} as follows.
We start by noting that
\begin{align} \label{eq:latticeprobability}
  \Pr{{\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})=\bmuhatREL|\ensuremath{\mathbf{u}}} =
  \int_{\ensuremath{\mathcal{D}}(\bmuhatREL)} \!
  f_G( \ensuremath{\mathbf{w}} - \ensuremath{\mathbf{u}} ) \,\mathrm{d}\ensuremath{\mathbf{w}}
\end{align}
where
\begin{align*}
f_G(\ensuremath{\mathbf{w}}) = (\pi N_o)^{-{\ensuremath{M}}}\exp\big(\!-N_o^{-1}\|\ensuremath{\mathbf{w}}\|^2\big)
\end{align*}
is the joint probability density function of a multi-variate complex
Gaussian with i.i.d.\ circularly symmetric components each of which has
variance~$N_o$ and
\begin{align*}
\ensuremath{\mathcal{D}}(\bmuhatREL)
  = \Big\{ \ensuremath{\mathbf{w}} \in \ensuremath{\mathbb{C}}^{\ensuremath{M}} \mid {\ensuremath{\underline{D}}}(\ensuremath{\mathbf{w}}) = \bmuhatREL\Big\}
\end{align*}
denotes the decision region of the relaxed MIMO detector employed.
Next, we consider optimal remapping for relaxed MLD, i.e.,
${\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})={\ensuremath{\underline{D}}}^{\ensuremath{\mathrm{ML}}}(\ensuremath{\mathbf{r}})$ as defined
in~\eqref{eq:unconstrainedMLrule} and $\bmuhatREL=\bmuhatREL^{\ensuremath{\mathrm{ML}}}$, and
note that in this case~\eqref{eq:latticeprobability} formally
corresponds to the probability of mistakenly decoding the transmitted
point~$\ensuremath{\mathbf{u}}$ for~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}$.  This probability can be
upper-bounded as
\begin{align} \label{eq:Prupperbound}
  \Pr{{\ensuremath{\underline{D}}}^{\ensuremath{\mathrm{ML}}}(\ensuremath{\mathbf{r}})=\bmuhatREL^{\ensuremath{\mathrm{ML}}}\big|\ensuremath{\mathbf{u}}}  \leq
  Q\Bigg(\frac{\big\| \bmuhatREL^{\ensuremath{\mathrm{ML}}} -
    \ensuremath{\mathbf{u}}\big\|}{\sqrt{2N_o}}\Bigg)
\end{align}
where~$Q(a)=(2\pi)^{-\frac{1}{2}}\!\int_{a}^\infty\exp(-\frac{x^2}{2})\,\mathrm{d}x$.
Replacing the function to be maximized over
in~\eqref{eq:remapperoptimum} by the upper bound
in~\eqref{eq:Prupperbound} leads to a novel remapping rule, which we
refer to as closest-vector remapping (CVR) and which is given by
\begin{align} \label{eq:cvremapper} \ensuremath{\hat{\bmu}}_{\mathrm{cv}} =
  \argmin_{\ensuremath{\mathbf{u}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})}\big\|\bmuhatREL^{\ensuremath{\mathrm{ML}}}-\ensuremath{\mathbf{u}}\big\|.
\end{align}
The solution of~\eqref{eq:cvremapper} corresponds to the
point~$\ensuremath{\hat{\bmu}}_{\mathrm{cv}}\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ that is closest (in
Euclidean distance) to the lattice
point~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}\in\LatticeREL(\ensuremath{\mathbf{G}})\!\setminus\! {\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$.
We emphasize that CVR amounts to solving a finite-lattice CVP and can,
hence, be carried out by (non-relaxed) SESD with the relaxed
ML-estimate~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}$ taking the role of the received point.
Note that CVR according to~\eqref{eq:cvremapper} exhibits structural
similarities to quantization according to~\eqref{eq:quantization}.
Formally, quantization can be seen as CVR with~$\ensuremath{\mathbf{G}}=\ensuremath{\mathbf{I}}_{\ensuremath{M}}$, which
amounts to ignoring the structure of the lattice generated by~$\ensuremath{\mathbf{G}}$.

\subsubsection*{Numerical performance and complexity results}

\begin{figure}[tb]
  \centering
  \includegraphics[width=\figscale\columnwidth]{CPXremapping.pdf}
  \figspacepre
  \caption{Complexity of SESD (MLD), relaxed SESD (R-SESD), and
    LR-aided SESD (LR-SESD).}
  \figspacepost
  \label{fig:CPXremapping}
\end{figure}

\figref{fig:BERremapping} compares the performance of relaxed SESD
followed by CVR to that obtained by relaxed SESD followed by
quantization. We observe that CVR significantly outperforms
quantization (and naive lattice decoding) and yields close-to-ML
performance.  This improvement comes, however, at the cost of having
to solve a finite-lattice CVP, namely~\eqref{eq:cvremapper}, in
contrast to simple element-wise slicing in the case of quantization
and to simply discarding estimates that do not belong
to~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ in the case of naive lattice decoding, both of
which exhibit significantly smaller complexity.

\begin{figure}[tb]
  \centering
  \includegraphics[width=\figscale\columnwidth]{BERremapping.pdf}
  \figspacepre
  \caption{Bit error rate (BER) performance comparison of SESD (MLD), LR-SESD, and LR-SIC.}
  \figspacepost
  \label{fig:BERremapping}
\end{figure}

We next compare the overall complexity (i.e., the total complexity of
both SESD runs) of LR-aided SESD followed by CVR (denoted by LR-SESD,
CVR) to that of SESD operating directly on the finite lattice, i.e.,
SESD (MLD).  \figref{fig:CPXremapping} shows that LR-SESD with CVR
outperforms SESD (MLD) \emph{only} in the very-high-SNR regime with
the corresponding complexity savings being minor. In addition, this
comparison favors \mbox{LR-SESD} with CVR as the additional complexity associated
with LR has been neglected.

\subsection{Two-Stage Detection}

The discussion above leads us to the question of whether
using~${\ensuremath{\underline{D}}}(\ensuremath{\mathbf{r}})=\bmuhatREL$ for remapping instead of~$\ensuremath{\mathbf{r}}$ is
sensible. After all, by mapping~$\ensuremath{\mathbf{r}}$ onto~$\bmuhatREL$ we are
potentially discarding useful information. In the following, we show
that, indeed, performing remapping based on~$\ensuremath{\mathbf{r}}$ rather than
on~$\bmuhatREL$ leads to a performance improvement (compared to, e.g.,
CVR) without requiring additional complexity.

To this end, we propose a new detector structure, which consists of
two stages. The first stage corresponds to relaxation
from~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ to~$\LatticeREL(\ensuremath{\mathbf{G}})$ (possibly followed by LR)
and using a relaxed MIMO detector. The second stage is invoked
\emph{only} if the first stage yields a result that does \emph{not}
belong to the finite lattice~${\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ and corresponds to a MIMO
detector operating on the finite lattice.  This second-stage detector
\emph{uses the received vector}~$\ensuremath{\mathbf{r}}$ instead of~$\bmuhatREL$. We
emphasize that if both stages employ MLD, the two-stage detector
achieves ML performance, since in the case of
\mbox{$\bmuhatREL^{\ensuremath{\mathrm{ML}}}\notin{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$}, the second stage performs
non-relaxed MLD on the basis of~$\ensuremath{\mathbf{r}}$.

\subsubsection*{Numerical performance and complexity results}

\figref{fig:CPXremapping} shows the overall complexity of two-stage
detection (i.e., the total complexity across the two stages) using
LR-aided SESD in the first stage and SESD in the second stage (denoted
by LR-SESD, MLD). Again, the complexity associated with LR is ignored.
Consulting the corresponding performance results
in~\figref{fig:BERremapping}, we can conclude that two-stage detection
employing SESD in both stages and LR in the first stage outperforms
\mbox{LR-SESD} with CVR both in terms of complexity \emph{and}
performance and, hence, the performance loss associated with
\mbox{LR-SESD} followed by CVR (compared to ML performance) must
result from the fact that~$\bmuhatREL^{\ensuremath{\mathrm{ML}}}$ rather than~$\ensuremath{\mathbf{r}}$ is used
for remapping. We can furthermore conclude that in the high-SNR regime
the two-stage approach exhibits slightly lower complexity than SESD,
while realizing the same (i.e., ML) performance.  We finally note that
the computational complexity of LR and the complexity of checking
whether~$\bmuhatREL\in{\ensuremath{\ensuremath{\mathcal{L}}}}(\ensuremath{\mathbf{G}})$ has been neglected in this
comparison.

\vspace{-0.3cm}

\section{Conclusions}

We demonstrated that relaxation from finite to infinite lattices
combined with lattice reduction (LR) can reduce the (tree-search)
complexity of Schnorr-Euchner sphere decoding (SESD), but generally
leads to a significant performance loss. This performance loss is
caused by the fact that the relaxed detector does not necessarily
deliver estimates that belong to the finite lattice and hence does not
realize ML performance. Remapping methods that yield (close-to) ML
performance necessitate solving finite-lattice closest vector problems
(CVPs).  Hence, if ML performance is required, using SESD directly on
the finite lattice seems to require smaller complexity, as remapping
is avoided and only one (finite-lattice) CVP needs to be solved. We
proposed an ML-optimal two-stage detector which was shown to require
lower complexity than SESD on the finite lattice provided i) the SNR
is high enough and ii) the complexity corresponding to LR and to
checking whether estimates correspond to the finite lattice is
neglected.  Generally speaking, we conclude that relaxation and LR are
better suited for use in conjunction with low-complexity detectors
(such as linear detection or SIC) with overall sub-optimal
performance.

\bibliographystyle{IEEEtran}