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Feedback-aided complexity reductions in ML and Lattice
decoding
Arun Kumar Singh, Petros Elia
To cite this version:
Arun Kumar Singh, Petros Elia. Feedback-aided complexity reductions in ML and Lattice decoding.
IEEE International Symposium on Information Theory (ISIT’12), Jul 2012, United States. pp.5.
�hal-00707823�
Feedback-aided complexity reductions in ML and Lattice decoding
Arun Singh and Petros Elia
Mobile Communications Department
EURECOM, Sophia Antipolis, France
Email: {singhak, elia}@eurecom.fr
Abstract—The work analyzes the computational-complexity
savings that a single bit of feedback can provide in the compu-
tationally intense setting of non-ergodic MIMO communications.
Specifically we derive upper bounds on the feedback-aided com-
plexity exponent required for the broad families of ML-based
and lattice based decoders to achieve the optimal diversity-
multiplexing behavior. The bounds reveal a complexity that is
reduced from being exponential in the number of codeword bits,
to being at most exponential in the rate. Finally the derived
savings are met by practically constructed ARQ schemes, as
well as simple lattice designs, decoders, and computation-halting
policies.
I. INTRODUCTION
The current work is a continuation of studies on the rate-
reliability-complexity limits of non-ergodic MIMO commu-
nications
1
. In this setting, computational complexity and rate-
reliability performance are highly intertwined, in the sense that
limitations to computational resources (commonly measured
by floating point operations - flops), bring about substantial
degradation in the system performance. In the high rate setting
of interest, the lion’s share of computational costs is due to
decoding algorithms, on which we here focus, specifically
considering the broad family of ML-based and regularized
(MMSE-preprocessed) lattice decoding algorithms.
a) Error and complexity exponents: In terms of relia-
bility, the diversity multiplexing tradeoff (DMT, cf. [1]) has
been extensively used to quantify the relationship between the
rate, denoted as R, and the probability of error P
err
. In the
high SNR regime (SNR will be henceforth denoted as ρ),
this relationship was described in [1] using the high SNR
measures of multiplexing gain r := R/ log ρ and diversity
gain d(r) := − lim
ρ→∞
log P
err
/ log ρ. As a result the same
work revealed, for the case of no feedback, the optimal DMT
in the form of the maximum possible diversity gain d
∗
(r) for
a given r.
The work in [2]–[4] provided a similar treatment for
complexity. Specifically for N
max
denoting the amount of
computational reserves, in flops per duration of one codeword,
that the transceiver is endowed with
2
, the work in [2], [3]
introduced the complexity exponent to take the form
c(r) := lim
ρ→∞
log N
max
log ρ
, (1)
where the value of the above exponent was derived as a
function of the desired r and d(r). In the specific setting of
1
By non-ergodic MIMO we refer to the setting where there is considerable
channel state information at the receiver (CSIR), and very little if any channel
state information at the transmitter (little or no CSIT).
2
In the sense that after N
max
flops the transceiver must simply terminate
potentially prematurely and before completion of its task.
quasi-static MIMO, without feedback and in the absence of
lattice reduction techniques
3
, the above work revealed that to
achieve the optimal DMT d
∗
(r), the complexity exponent is
upper bounded by a piecewise linear function which, at integer
r takes the form
c(r) = r(n
T
− r). (2)
This bound was shown to be tight for a broad range of
practical settings, and it also revealed a complexity that scales
exponentially with the number of codeword bits.
This very considerable complexity brought to the fore the
need for methods that manage to achieve the same near-
optimal performance, but do so with much reduced compu-
tational resources.
b) Feedback gains: In terms of feedback, the work in
[5] utilized the DMT machinery to analyze the reliability
gains of feedback, and to specifically show that an L-round
ARQ scheme can provide for a much increased feedback-aided
DMT which
4
was shown to take the form d
∗
(r/L).
Motivated by the considerable magnitude of the complexity
exponent in (2), we here seek to understand the role of
feedback in reducing complexity, rather than in improving
reliability. For this we seek to quantify the feedback-aided
complexity exponent required to achieve the original d
∗
(r) in
the presence of a modified version of the above mentioned L-
round MIMO ARQ. Specifically we will derive upper bounds
on the minimum
5
complexity exponent required by ML and
regularized (MMSE-preprocessed) lattice based sphere de-
coders (SD) to achieve the optimal DMT d
∗
(r) . We will
focus on the family of minimum delay ARQ schemes (to
be described later on). The derivations focus on ML-based
decoding, but given the equivalence of ML and regularized
lattice based decoding shown in [3], these same results extend
automatically to the regularized lattice decoding case. We
note that the validity of the presented bounds depends on the
existence of actual schemes that meet them. These schemes
will be here provided, together with the associated lattice
designs, decoders, as well as halting and ordering polices.
3
We here note that while lattice reduction (LR) indeed allows here for near-
optimal behavior at very manageable complexity, it is the case that there exist
scenarios for which these same LR methods cannot be readily applied. Such
problematic cases include the ubiquitous scenario where outer binary codes
are employed and decoded using soft information. It is for this exact reason
that we focus on the complexity analysis of non LR-aided schemes which
remains of strong interest for many pertinent communication scenarios.
4
This held for the setting of quasi-static fading and no power adaptation -
which is the setting of interest here.
5
By minimum we refer to a minimization over all lattice code designs
(which must vary accordingly depending on the setting), all policies of
computational halting, and all policies on decoding ordering. A decoding
ordering policy describes the order in which the transmitted information
symbols are decoded by sphere decoding algorithm.
The analysis and the constructed feedback schemes tell us
how to properly utilize a single bit of feedback to alleviate the
adverse effects of computational constraints, as those seen in
the derived rate-reliability-complexity tradeoffs of [4].
Before proceeding to a brief description of the MIMO ARQ
signaling, we quickly note that we here employ an ARQ
variant which reduces the L-round scheme to a two-round
scheme with uneven but fixed durations, and we do so by
disregarding all but the first and last rounds. Such a scheme
requires just one bit of feedback. We will however, for clarity
of exposition, maintain use of the notation of the better known
L-round scheme.
A. MIMO-ARQ signaling
We here present the general n
T
×n
R
MIMO-ARQ signaling
setting, and focus on the details which are necessary for
our exposition. For further understanding of the MIMO-ARQ
channel, the reader is referred to [5] as well as [6].
Under ARQ signaling, each message is associated to a
unique block [X
1
C
X
2
C
· · · X
L
C
] of signaling matrices, where
each X
i
C
∈ C
n
T
×T
, i = 1, · · · , L, corresponds to the n
T
× T
matrix of signals sent during the ith round. The accumulated
code matrix at the end of round `, ` = 1, · · · , L, takes the form
X
ARQ,`
C
= [X
1
C
X
2
C
· · · X
`
C
] ∈ C
n
T
×`T
. We note that the
signals X
ARQ,L
C
are drawn from a lattice design that ensures
unique decodability at every round
6
.
In the quasi-static case of interest, the received signal
accumulated at the end of the `-th round takes the form
Y
`
C
= θH
C
X
ARQ,`
C
+ W
`
C
, ` = 1, · · · , L, (3)
where H
C
∈ C
n
R
×n
T
, where the scaling factor θ is chosen
such that E(kθX
i
C
k
2
) ≤ ρT, 1 ≤ i ≤ `.
We proceed with quantifying the complexity reductions due
to ARQ feedback.
II. COMPLEXITY REDUCTION USING ARQ FEEDBACK
We here seek to analyze the complexity reductions due to
MIMO ARQ feedback. Specifically for d
∗
(r) denoting the
optimal DMT of the n
T
× n
R
MIMO channel in the absence
of feedback, we here seek to describe the feedback-aided
complexity exponent required to meet the same d
∗
(r) with
the assistance now of an L-round ARQ scheme. As stated
before, our analysis focuses on the setting of L ≤ n
T
and of
minimum-delay ARQ schemes, corresponding to T = 1. The
derived exponent is to be compared with the exponent in (2)
(cf. [2, Theorem 6] and [3, Corollary 1b]) corresponding to no
feedback. The following holds for the n
T
× n
R
(n
R
≥ n
T
),
i.i.d. regular fading
7
MIMO channel.
Theorem 1: Let c(r) be the minimum complexity exponent
required to achieve d
∗
(r), minimized over all lattice designs,
all ARQ schemes with L ≤ n
T
rounds of ARQ, all halting
6
Loosely speaking, unique decodability means that, for any ` = 1, · · · , L,
the corresponding X
ARQ,`
C
carries all bits of information.
7
The i.i.d. regular fading statistics satisfy the general set of conditions as
described in [7], where a) the near-zero behavior of the fading coefficients h
is bounded in probability as c
1
|h|
t
≤ p(h) ≤ c
2
|h|
t
for some positive and
finite c
1
, c
2
and t, where b) the tail behavior of h is bounded in probability
as p(h) ≤ c
2
e
−b|h|
β
for some positive and finite c
2
, b and β, and where c)
p(h) is upper bounded by a constant K.
policies and all decoding order policies. Then c(r) ≤ c
red
(r)
where
c
red
(r) ,
1
n
T
r(n
T
− brc − 1) + (n
T
brc − r(n
T
− 1))
+
,
which is a piecewise linear function that, for integer r, takes
the form
c
red
(r) =
1
n
T
r(n
T
− r), for r = 0, 1, · · · , n
T
.
The proof of the above theorem will be presented in
Appendix A, together with the proofs for the upcoming
Propositions 1 and 2, and it will include the derivation of
the upper bound, and the constructive achievement of this
bound which is presented in Propositions 1, 2. The constructive
part of the proof is based on designing ARQ schemes and
implementations (lattice designs and halting policies) that meet
the bound. We proceed with these propositions where we
identify cases for which the above complexity bound suffices
to achieve d
∗
(r) with the help of feedback.
An important aspect in ARQ schemes is knowing when to
decode and when not to decode across the different rounds.
Towards this we have the following definition.
Definition 1 (Aggressive intermediate halting policies):
We define aggressive intermediate halting policies to be
the family of policies that halt decoding in the first round
whenever the minimum singular value of the channel scales
as ρ
−
for some > 0, which do not decode in the second to
the L-1 round, and which decode at the last round iff a) they
have not decoded in the first round and b) the channel is not
in outage with respect to the effective rate of ARQ scheme.
Given such aggressive halting policies, the L round scheme
reduces to a two round scheme where the second round
comprises of (L − 1)T channel uses. As noted before, for
notational uniformity with earlier works in [5], [6], we will
continue to use the notation of the L-round schemes but again
clarify that only one bit of ARQ feedback is needed.
Furthermore we will henceforth use the term ARQ-
compatible, minimum delay, NVD, rate-1 lattice designs to
refer to the family of n
T
× n
T
lattice designs X
ARQ,L
C
with
total number of transmitted integers κ = 2n
T
, with non-
vanishing determinant (NVD)
8
for r ≤ 1, and with all the
information appearing in all rounds.
Proposition 1: A minimum delay ARQ scheme with L =
n
T
rounds achieves d
∗
(r) with c(r) ≤ c
red
(r), irrespective
of the ARQ-compatible, minimum delay, NVD, rate-1 lattice
design, for any aggressive intermediate halting policy, and any
sphere decoding order policy.
The following describes a very simple MIMO ARQ coding
implementation that achieves d
∗
(r) with c(r) ≤ c
red
(r). The
proof of this proposition will appear later on, and is crucial in
the achievability part of the proof of Theorem 1.
Proposition 2: The minimum delay ARQ scheme with L =
n
T
rounds, implemented with any aggressive intermediate
halting policy, any sphere decoding order policy, and a rate-1
8
A code has a non-vanishing determinant if, without power normalization,
there is a lower bound on the minimum determinant that does not depend
on the constellation size. The determinant of any non-normalized difference
matrix is lower bounded by a constant independent of ρ (see [8]).
lattice design X
ARQ,L
C
drawn from the center of perfect codes
(cf. [8], [9])
9
, achieves d
∗
(r) with c(r) ≤ c
red
(r).
Theorem 1 has quantified the computational reserves that
are sufficient to achieve DMT optimality. These computational
reserves can be seen to be smaller than those required to
achieve the same optimal DMT d
∗
(r) without feedback. For
example, given any known minimum-delay DMT optimal
design which remains fixed for all r, in the absence of
feedback, the exponent needed to achieve d
∗
(r) is that in (2)
(cf. [2, Theorem 6] and [3, Corollary 1b]) and takes the form
c(r) = r(n
T
− r), (5)
(for integer r = 0, 1, · · · , n
T
), whereas as we have just seen,
for L = n
T
, T = 1 this exponent reduces to a much smaller
c(r) ≤
1
n
T
r(n
T
− r).
We proceed with a few examples.
Example 1 (Corresponding to Theorem 1 and Proposition 2):
For the general n
T
× n
R
setting with n
R
≥ n
T
, and for
r = n
T
/2, the computational resources required to achieve
the optimal d
∗
(r) with existing DMT optimal (minimum
delay) non-feedback schemes (cf. [2, Theorem 6]), scales as
10
N
max
.
= ρ
n
2
T
/4
.
= 2
Rn
T
/2
,
whereas the feedback aided complexity required by the feed-
back scheme in Proposition 2 scales as
N
max
.
= ρ
n
T
/4
.
= 2
R/2
.
Generally, given a rate that scales linearly with n
T
, in the
absence of feedback the complexity exponent of achieving
d
∗
(r) scales with n
2
T
, whereas the feedback aided complexity
exponent scales with n
T
.
Example 2: Figure 1 considers the case of n
T
= 4 ≤ n
R
and Rayleigh fading, and compares the above complexity
upper bound in the presence of feedback (L = 4, T = 1),
to the equivalent complexity exponent in (5) of achieving the
same optimal DMT d
∗
(r) without ARQ feedback (Perfect
codes and natural, fixed decoding ordering (cf. [2])). The
feedback-aided complexity exponent reveals an exponential
reduction by a factor of n
T
= 4.
9
For general lattice designs derived from cyclic division algebra (CDA) (cf.
[8], [9]), F and L are number fields, with L a finite, cyclic Galois extension
of F of degree n. Let σ denote a generator of the Galois group Gal(L/F).
Let z be an indeterminate satisfying lz = zσ(l), ∀ l ∈ L and z
n
= γ for
some non-norm element γ ∈ F
∗
. Then the set of all elements of the form
P
n−1
i=0
z
i
l
i
forms a CDA D(L/F, σ, γ) with center F and maximal subfield
L. The mentioned codes are limited in the center of the division algebra and
take the simple form
X
ARQ,L
C
=
f
0
γf
n
T
−1
· · · γf
1
f
1
f
0
· · · γf
2
.
.
.
.
.
.
.
.
.
.
.
.
f
n
T
−1
f
n
T
−2
· · · f
0
∈ C
n
T
×n
T
, (4)
where f
i
belong to the QAM constellation.
10
We use
.
= to denote the exponential equality, i.e., we write f (ρ)
.
= ρ
B
to denote lim
ρ→∞
log f (ρ)
log ρ
= B, and
.
≤,
˙
<, and
.
≥,
˙
> are defined similarly.
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
3.5
4
Multiplexing Gain (r)
Complexity exponent c(r)
Complexity reduction via feeback in 4x4 MIMO
minimum delay ARQ scheme
Perfect code
Fig. 1. Complexity reduction with minimum delay ARQ schemes.
A. Feedback reduction for asymmetric channels: n
R
≤ n
T
We now consider the case of n
R
≤ n
T
, and specifically
the case where n
R
|n
T
(i.e., n
T
is an integer multiple of n
R
),
to observe again how simple implementations offer substantial
reductions in complexity. In terms of statistics, the results hold
for any i.i.d. regular fading distribution.
Theorem 2: In the MIMO ARQ channel with n
R
|n
T
, the
minimum complexity exponent c(r) required to achieve d
∗
(r),
minimized over all lattice designs, all halting policies, and all
minimum delay ARQ schemes with L ≤ n
T
rounds of ARQ,
is bounded as c(r) ≤ c
red
(r) where
c
red
(r) ,
1
n
R
r(n
R
− brc − 1) + (n
R
brc − r(n
R
− 1))
+
,
which is a piecewise linear function that, for integer r, takes
the form
c
red
(r) =
1
n
R
r(n
R
− r), for r = 0, 1, · · · , n
R
.
Applying as the constructive part of the proof of the above
theorem, the following describes a very simple MIMO ARQ
block-diagonal repetition coding implementation that achieves
d
∗
(r) with a much reduced c(r) ≤ c
red
(r).
Proposition 3: A minimum delay ARQ scheme with L =
n
T
, T = 1, implemented with any aggressive intermediate
halting policy, any sphere decoding order policy, and a rate-
n
R
n
T
block-diagonal repetition lattice design X
ARQ,L
C
where the
(rate-1) block component code is drawn from the center of
n
R
× n
R
perfect codes, achieves d
∗
(r) with c(r) ≤ c
red
(r)
from Theorem 2.
The proof of Theorem 2 and Proposition 3 will be presented
in Appendix B. Of interest is the special MISO-ARQ case of
n
R
= 1, where the above described scheme will allow for a
zero complexity exponent, and for a complexity that scales as a
subpolynomial function of ρ and as a subexponential function
of the number of codeword bits and of the rate.
Corollary 2a: Over the n
T
×1 MISO channel, the minimum
delay ARQ scheme with L = n
T
rounds, implemented with a
rate-
1
n
T
repetition QAM design X
ARQ,L
C
, achieves d
∗
(r) with
c(r) = 0.
This corollary follows directly from Theorem 2.
We proceed with a few examples.
Example 3 (Corresponding to Theorem 2 and Proposition 3):
For the 4 × 2 MIMO channel with L = 4, T = 1, applying a
lattice design of the form
X
ARQ,L
C
=
f
0
γf
1
0 0
f
1
f
0
0 0
0 0 f
0
γf
1
0 0 f
1
f
0
∈ C
4×4
,
where f
0
, f
1
∼ QAM, together with an aggressive inter-
mediate halting policy for the first round decoder, and with
any sphere decoding ordering policy, can achieve the optimal
d
∗
(r) of the 4 × 2 channel, and can do so with computational
resources of N
max
.
= ρ
c
red
(r)
flops, which for integer r
translates to N
max
.
= ρ
1
n
R
r(n
R
−r)
= ρ
r
2
(2−r)
.
Example 4 (Corresponding to Theorem 2 and Proposition 3):
Figure 2 compares two schemes: the 2 × 2 MIMO channel
(minimum delay, DMT optimal lattice design), and the 4 × 2
minimum delay MIMO-ARQ channel with L = n
T
= 4, 1
bit of feedback, and the implementation of Proposition 3.
We see a considerably reduced complexity of the feedback
aided scheme (Fig. 2(a), lower line) which, at the same time,
achieves a much higher DMT performance (Fig. 2(b), upper
line) than its non-feedback counterpart.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Multiplexing Gain (r)
Complexity exponent c(r)
Complexity reduction via ARQ feeback
minimum delay ARQ scheme
2 × 2 Perfect code
(a) Complexity exponent
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
1
2
3
4
5
6
7
8
Multiplexing Gain (r)
Diversity Gain d(r)
Diversity Performance
minimum delay ARQ scheme
2 × 2 Perfect code
(b) DMT
Fig. 2. Complexity reduction n
R
|n
T
i.i.d. Rayleigh channel with ARQ
feedback.
APPENDIX A
PROOF OF THEOREM 1
The proof follows from the footsteps of the [2, proof of
Theorem 2]. Due to space limitations we restrict this expo-
sition to the proof steps that are necessary to understand the
complexity exponent for the novel ARQ schemes discussed in
this paper. For further understanding of encoders and decoders
considered here, the reader is referred to [2], [4]. We begin by
establishing necessary conditions for L-round ARQ scheme to
achieve d
∗
(r) over an n
T
× n
R
(n
R
≥ n
T
) MIMO.
Condition 1: To achieve maximum diversity gain of n
R
.n
T
the total number of channel uses LT ≥ n
T
. For minimum
delay (T = 1) L-round ARQ schemes with L ≤ n
T
, it then
follows that L = n
T
.
Condition 2: To achieve maximum multiplexing gain of n
T
the total number of integers transmitted κ ≥ 2n
T
T = 2n
T
.
It can be seen from [2] that the complexity of sphere decoder
increases with κ. Thus, for minimum delay L-round ARQ
schemes with L ≤ n
T
, the tightest upper bound on the
complexity exponent is established by considering T = 1,
L = n
T
and κ = 2n
T
. Setting T = 1 and L = n
T
implies
use of at least rate-1 lattice designs for L-round ARQ scheme
for achieving DMT performance of d
∗
(r) for 0 ≤ r ≤ n
T
.
We will later show that encoding-decoding policy described
in Proposition 1 and Proposition 2 achieve DMT performance
of d
∗
(r) for T = 1, L = n
T
and κ = 2n
T
, which in turn
implies that lim
ρ→∞
r = r
1
, where r
1
is the multiplexing
gain for the first round of ARQ.
Having established the necessary parameters we proceed to
prove the claim of Theorem 1. Following the footsteps of the
[2, proof of Theorem 2] we can show that in the presence
of aggressive halting policy and SD with search radius ξ >
p
d
∗
(
r
L
) log ρ, an upper bound on the complexity exponent
for first round decoder can be obtained as the solution to a
constrained maximization problem according to
c
1
(r) , max
{µ
1
<,
µ
1
≥···≥µ
n
T
≥0}
n
T
X
j=1
min
r
n
T
− (1 − µ
j
),
r
n
T
+
,
where µ
j
, −
log σ
j
(H
H
C
H
C
)
log ρ
, j = 1, · · · , n
T
with µ
1
≥ · · · ≥
µ
n
T
, where σ
j
denotes j-th singular value of H
H
C
H
C
and
where we have made use of the fact that lim
ρ→∞
r = r
1
. In
the limit → 0 this upper bound simplifies to
c
1
(r) = 0.
To establish the L-th round complexity exponent, we proceed
with the L-th round system model given by
Y
L
C
= θH
C
X
ARQ,L
C
+ W
L
C
,
where for rate-1 lattice designs we have θ
2
= ρ
1−r
L
, where
r
L
=
r
L
denotes multiplexing gain for L-th round of ARQ.
The vectorized real valued representation of L-th round system
model takes the form
y
L
= θH
L
x
L
+ w
L
, where (6)
H
L
= I
L
⊗ H
R
, with H
R
=
Re{H
C
} −Im{H
C
}
Im{H
C
} Re{H
C
}
,
x
L
= (x
T
1
, · · · , x
T
L
)
T
∈ R
2n
T
L
with x
t
=
[Re{X
ARQ,L
t,C
}
T
, Im{X
ARQ,L
t,C
}
T
]
T
for t = 1, · · · , L,
where X
ARQ,L
t,C
is t-th column of X
ARQ,L
C
, w
L
and y
L
can
be defined similarly. The vectorized codeword x
L
takes the
form (cf. [3])
x
L
= Gs, s ∈ S
κ
r
, Z
κ
∩ ρ
r
L
2
R, (7)
where G ∈ R
2Ln
T
×κ
is the lattice generator matrix, where
κ = 2n
T
and where R ⊂ R
κ
is a natural bijection of the code
shaping region that preserves the code, and contains the all
zero vector 0. For simplicity we consider R ,[−1, 1]
κ
to be
a hypercube in R
κ
, although this could be relaxed. Combining
(6) and (7) yields the equivalent system model
y
L
=M
L
s + w
L
, (8a)
where M
L
,ρ
1
2
−
r
L
2
H
L
G ∈ R
2n
R
L×κ
. (8b)
Let G = [Γ
T
1
Γ
T
2
· · · Γ
T
L
]
T
, where Γ
i
∈ C
2n
T
×2n
T
, for i =
1, · · · , L.
Then the equivalent code-channel matrix (M
L
) takes the
form
M
L
=ρ
1
2
−
r
L
2
H
R
· · · 0
.
.
.
.
.
.
.
.
.
0 · · · H
R
Γ
1
.
.
.
Γ
L
, (9)
=ρ
1
2
−
r
L
2
[Γ
T
1
H
T
R
· · · Γ
T
L
H
T
R
]
T
. (10)
