JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

133

Low-Complexity Integer-Forcing Methods for Block
Fading MIMO Multiple-Access Channels

arXiv:1711.07977v1 [] 21 Nov 2017

Ricardo Bohaczuk Venturelli and Danilo Silva

Abstract—Integer forcing is an alternative approach to conventional linear receivers for multiple-antenna systems. In an
integer-forcing receiver, integer linear combinations of messages
are extracted from the received matrix before each individual
message is recovered. Recently, the integer-forcing approach
was generalized to a block fading scenario. Among the existing
variations of the scheme, the ones with the highest achievable
rates have the drawback that no efficient algorithm is known
to find the best choice of integer linear combination coefficients.
In this paper, we propose several sub-optimal methods to find
these coefficients with low complexity, covering both parallel
and successive interference cancellation versions of the receiver.
Simulation results show that the proposed methods attain a performance close to optimal in terms of achievable rates for a given
outage probability. Moreover, a low-complexity implementation
using root LDPC codes is developed, showing that the benefits
of the proposed methods also carry on to practice.
Index Terms—Block fading, integer-forcing linear receivers,
lattices, root LDPC codes, successive interference cancellation.

I. I NTRODUCTION
Integer-forcing (IF) receivers are an alternative to conventional methods of equalization, such as zero-forcing (ZF) and
minimum-mean-squared-error (MMSE) equalization [2] for
multiple-input and multiple-output (MIMO) channels. The IF
approach follows from the compute-and-forward framework
[3], [4] for relay networks, where the receivers attempt to
extract integer linear combinations of the transmitted messages
from the received signals, before recovering the messages
themselves.
Although joint maximum likelihood (ML) receivers achieve
the best performance among all methods, by searching over
all possible transmitted codewords [5], their complexity is
prohibitively high, increasing exponentially with the number of
users. In contrast, IF receivers have a much lower complexity
and can approach the ML performance in many situations [2].
Moreover, the performance of an IF receiver can be further
improved in some situations by successive computation, leading to the so-called successive IF (SIF) receiver, analogously
to the successive interference cancellation (SIC) technique for
conventional linear receivers.
The Ad Hoc Associate Editor coordinating the review of this manuscript
and approving it for publication was Prof. Cristiano Magalhães Panazio.
Ricardo Bohaczuk Venturelli and Danilo Silva are with the Department of Electrical and Eletronic Engineering, Federal University of Santa
Catarina, Florianopolis-SC, Brazil (e-mails: ricardo.bventurelli@gmail.com,
danilo.silva@ufsc.br).
This work was partially supported by CAPES under Grant 1345815 and by
CNPq under Grants 153535/2016-4, 429097/2016-6 and 310343/2016-0.
A preliminary version of this paper was presented at the XXXIV Simpósio
Brasileiro de Telecomunicações e Processamento de Sinais (SBrT’16), Santarém, PA, Brazil, August 30-September 4, 2016 [1].
Digital Object Identifier: 10.14209/jcis.2017.14

Recent works on IF include the design of practical channel
codes compatible with IF [6], [7], efficient methods to select
the coefficients of the integer linear combinations [8], [9], as
well as its application to relay networks [10]. Moreover, the
IF principle has been used not only as receive method in the
MIMO uplink scenario, but also for precoding in the MIMO
downlink scenario [11], [12] and even for source coding [13].
The main results about IF receivers consider static fading,
where all symbols of a codeword are subject to the same
channel fading. However, in a practical situation where a
powerful code with large blocklength is used, it may not be
realistic to assume that all symbols of the codeword are subject
to the same channel fading. Therefore, channels that allow
block fading [14], where the channel fading can vary during
the transmission of a codeword, seem to be a more realistic
model.
In a recent work, El Bakoury and Nazer [15] generalize
the IF approach to a block fading scenario. They described
two decoding methods for block fading, which are called AM
(arithmetic mean) and GM (geometric mean) decoding. The
AM decoding method approximates the effective noise seen
on all blocks (after equalization) as having the same variance,
as in the case of static fading. On the other hand, the GM
decoding method optimally exploits the diversity inherent in
the channel variation, allowing to achieve higher rates than
with AM decoding. Both decoding methods are applicable to
both (non-successive) IF and SIF receivers.
The rates achievable by all these receivers depend on the
choice of an integer matrix A specifying the coefficients of
the linear combinations that should be decoded. However,
finding the optimal choice of A for GM-IF, AM-SIF and
GM-SIF appears to be a hard problem for which no efficient
approximation algorithm is known, making these receivers
currently infeasible to implement in practice. Nevertheless,
results obtained in [15] by exhaustive search demonstrate that
optimal GM-IF and AM-SIF significantly outperform AM-IF
in terms of achievable rates, while GM-SIF outperforms all of
them.
Our main contribution in this paper is to develop lowcomplexity optimization methods for these receivers (GMIF, AM-SIF and GM-SIF) which can closely approach the
performance obtained by an optimal search. Each proposed
method is based on optimizing a more tractable lower bound
on the achievable rate, which can be performed very efficiently.
Simulation results show that the proposed method for GM-IF
has a small gap compared to optimal performance, while the
remaining two methods, at least for the scenarios tested, have
performance almost indistinguishable from the optimal one.
Another contribution of this paper is to develop a low-

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

complexity implementation of these receivers using finitelength codes over a low-order constellation, in order to validate
the information-theoretic results under practical constraints.
Attaining the GM performance requires the use of fulldiversity codes, from which we have adopted root low-density
parity-check (LDPC) codes. Simulation results are shown to
be consistent with the theoretical ones, with an expected gap
due to the finite codeword length.
The remainder of the paper is organized as follows. The
system model is described in Section II. Section III reviews
integer forcing for static fading as well as block fading, while
Section IV reviews successive integer forcing, again for both
fading types. In Section V, we present our proposed methods
for selecting A. Section VI describes our low-complexity
implementation under practical constraints. Simulation results
are shown in Section VII, covering the information-theoretic
performance as well as the performance with practical codes.
Lastly, our conclusions are presented in Section VIII.
A. Notation
For any x > 0, define log+ (x) , max(log(x), 0). We
denote row vectors as lowercase bold letters (e.g., x) and
matrices as uppercase bold letters (e.g., X). The `2 -norm
of a vector x is denoted by kxk. The matrix XT denotes
the transpose of X. We use I and 0, respectively, to denote
an identity and an all-zero matrix of appropriate size, which
should always be clear from the context. The set of all m × n
matrices with entries from the set A is denoted Am×n .
II. S YSTEM M ODEL
Consider a discrete-time, real Gaussian MIMO multipleaccess channel (MAC) with NT single-antenna transmitters
and one NR -antenna receiver, subject to block fading with F
independent fading realizations per codeword.
Specifically, let n be the codeword length and assume, for
simplicity, that F divides n. For ` = 1, . . . , NT , let x` ∈ Rn
denote the vector transmitted by the `th transmitter, which
can be represented by the `th row of a matrix X ∈ RNT ×n .
Similarly, let Y ∈ RNR ×n be a matrix whose jth row
represents the vector received by the jth receive antenna.
Assuming fading realizations of equal length, the received
matrix can be expressed as


Y = Y(1) · · · Y(F )
(1)
where, for i = 1, . . . , F ,
Y(i) = H(i) X(i) + Z(i)

134

We assume that the receiver has perfect knowledge of the
channel realization H(1:F ) , while the transmitters do not have
this knowledge and are only aware of the channel statistics.
Each transmitted vector x` is assumed to be the encoding
of a message w` ∈ W produced by the `th transmitter, where
W is the message space. The encoder rate, which is the same
for all transmitters, is defined as
1
(5)
R = log2 |W|.
n
Moreover, the transmitted vectors must satisfy a (symmetric)
power constraint
1
kx` k2 ≤ P.
(6)
n
These assumptions on equal power and equal rate are reasonable since the transmitters do not have knowledge of the
channel matrix. For convenience, we denote SNR = P/σ 2 .
At the receiver, the decoder attempts to recover all messages, producing estimates ŵ1 , . . . , ŵNT . An error is said to
occur if ŵ` 6= w` for any `. The error probability of the
scheme (encoder/decoder pair) is Pe = E[Pe (H(1:F ) )], where
Pe (H(1:F ) ) denotes the error probability for a fixed channel
realization.
For any fixed H(1:F ) , a rate R is said to be achievable if,
for any , δ > 0 and sufficiently large n, there exists a scheme
of rate at least R − δ such that Pe (H(1:F ) ) ≤ .
For a given family of schemes (indexed by n), let
Rscheme (H(1:F ) ) denote its maximum achievable rate under a
fixed channel realization H(1:F ) . For a target rate R, the outage
probability is defined as pout (R) , P[Rscheme (H(1:F ) ) < R],
and for a fixed probability ρ ∈ (0, 1], the outage rate is defined
as Rout (ρ) , sup{R : pout (R) ≤ ρ}.
Remark: We have adopted a real-valued channel in order
to facilitate comparison with optimal IF methods that require
exhaustive search—whose complexity becomes prohibitively
large over a complex-valued channel even for low dimension—
as well as with the existing literature, which mostly considers
real-valued channels. There is no loss of generality, since it is
always possible to express a complex-valued channel Yc =
Hc Xc + Zc as a real-valued channel

 

 

Re(Yc )
Re(Hc ) −Im(Hc ) Re(Xc )
Re(Zc )
=
+
.
Im(Yc )
Im(Hc ) Re(Hc )
Im(Xc )
Im(Zc )
Alternatively, all expressions presented here can be straightforwardly generalized to the complex case by replacing transpose
with conjugate transpose.

(2)

H(i) ∈ RNR ×NT is the matrix of channel fading coefficients
for the ith block, X(i) ∈ RNT ×(n/F ) is such that


X = X(1) · · · X(F )
(3)
and Z(i) ∈ RNR ×(n/F ) is a Gaussian noise matrix with i.i.d.
entries of zero mean and variance σ 2 . Note that H(i) remains
constant throughout the transmission of a block of n/F
symbols, but can vary independently between realizations.
For convenience, let

H(1:F ) = H(1) , . . . , H(F ) .
(4)

III. I NTEGER F ORCING
To aid the understanding, we first review integer forcing for
static fading (F = 1), before describing its extension to block
fading (F ≥ 2). For simplicity, the superscript indicating the
block index is omitted when F = 1.
A. Static Fading
An integer-forcing receiver [2] shares the same basic structure of a conventional linear receiver, as illustrated in Fig. 1:
first, the received matrix is linearly transformed by an equal-

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

y1
y2
..
.
yNR

ŵ1

SISO decoder
B
(over R)

−1

SISO decoder

..
.

SISO decoder

A
(over Z)

where
2
σeff,m
= bm H − am

ŵ2
ŵNT

conventional linear receiver

135

2

P + bm

2

σ2

is the per-component variance of the vector zeff,m , and am ,
bm and zeff,m are the mth row of A, B and Zeff , respectively.
The optimal equalization matrix B, for a given integer
matrix A, can be found using MMSE estimation [2] as
B = AHT (SNR−1 I + HHT )−1

Fig. 1. Integer-forcing receiver.

ization matrix; then, each resulting stream is individually
processed by a channel decoder. The difference, in the case
of the integer-forcing receiver, is that the equalizer attempts
to estimate not the transmitted signals directly but rather an
integer linear transformation of the transmitted signals, which
is then inverted after noise is removed.
Crucial to this noise removal step at the channel decoders
is the use of a lattice code common to all transmitters [2],
[3]. A lattice Λ ∈ Rn is a discrete subgroup of Rn , i.e., it is
closed under integer linear combinations [16]. In particular, a
lattice can be expressed as Λ = {x ∈ Rn : x = uG, u ∈ Zn }
where G ∈ Rn×n is the generator matrix of Λ. It follows that
if every x` ∈ Λ, then aX ∈ Λ, for all a ∈ Z1×NT [16]. In
other words, since aX is also a codeword from the same code
Λ, then a decoder for Λ is able to decode it (i.e., “denoise”
it), regardless of the value chosen for a [3].
In more detail, consider the received matrix in (2),
Y = HX + Z.

(7)

NT ×NT

be a full-rank integer matrix. The receiver
Let A ∈ Z
applies the equalization matrix B ∈ RNT ×NR to create an
effective channel output
Yeff = BY
= BHX + BZ

(8)
(9)

= AX + Zeff

(10)

= V + Zeff

(11)

Zeff = (BH − A)X + BZ

(12)

where
is the so-called effective noise [2] and
V = AX

(13)

is an integer linear transformation of X.
Assuming that each x` is chosen from the same lattice Λ ⊆
Rn , we have that each row of V is also a lattice point from Λ
[3]. Thus, if decoding is successful and V is recovered, then
X can also be recovered, provided A is full-rank.1
As a consequence, it is shown in [2], [3] that the following
rate is achievable
!
1
P
+
(14)
RIF (H, A, B) , min log
2
m 2
σeff,m
1 One might think that a stricter condition is required, namely, that A
is invertible. This would be true if any element of Λ could be chosen for
transmission without a power constraint. However, when nested lattice shaping
is used to satisfy the power constraint, it is possible to show that requiring
rank A = NT is enough. See [2], [17] for details.

(15)

(16)

In this case, we have [4]
2
σeff,m
= σ 2 am MaT
m

(17)

M = (SNR−1 I + HT H)−1

(18)

where

and the achievable rate becomes
1
RIF (H, A) , min log+
m 2



SNR
am MaT
m


.

(19)

Optimizing the choice of A, we have the achievable rate
RIF (H) ,

max

A:rank A=NT

RIF (H, A).

(20)

As can be seen, the optimal matrix A solving (20) consists
of a set of linearly independent integer vectors minimizing
(17). Since M is symmetric and positive definite [4], it admits
a Cholesky decomposition M = GGT , where G ∈ RNT ×NT ,
leading to
2
σeff,m
= σ 2 kam Gk2 .

(21)

Thus, the optimal solution can be described as the integer
coefficients (under the basis G) of a set of NT shortest
linearly independent vectors in the lattice generated by G.
This is known as the Shortest Independent Vector Problem
(SIVP) [18], which is believed to be NP-Hard [19]. However,
suboptimal algorithms for basis reduction2 exist that can find
an approximation in polynomial time, such as the LenstraLenstra-Lovasz (LLL) algorithm [21].
Note that, if A = I is chosen, then the scheme reduces
to MMSE equalization [2]. Thus, integer forcing generalizes
linear equalization, providing potentially higher achievable
rates.

B. Block Fading
In the case of block fading, a complication arises, since now
each block of the transmitted matrix experiences a different
channel realization [14]. While it is possible to independently
equalize each block of the received matrix, the resulting
integer linear transformation A must be the same for all
blocks, in order for the rows V to remain lattice codewords
[15].
2 For the special case of N
T = 2, an optimal basis can be found very
efficiently using Lagrange’s algorithm [20].

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

More precisely, for each ith block, let A(i) ∈ ZNT ×NT and
B(i) ∈ RNT ×NR be its corresponding integer and equalization matrices, respectively, and compute the effective channel
output for the block as
Yeff,(i) = B(i) Y(i)

(22)

= B(i) H(i) X(i) + B(i) Z(i)

(23)

= A(i) X(i) + Zeff,(i)

(24)

= V(i) + Zeff,(i)

(25)

where V(i) = A(i) X(i) and
Zeff,(i) = (B(i) H(i) − A(i) )X(i) + B(i) Z(i) .

(26)

It follows that
Yeff = V + Zeff

(27)

where Yeff , V and Zeff denote the horizontal concatenation of
Yeff,(i) , V(i) and Zeff,(i) , respectively, for i = 1, . . . , F .
However, we can see that, in general, the rows of


V = A(1) X(1) · · · A(F ) X(F )
(28)
are not guaranteed to be lattice points, since we cannot
generally express V as an integer linear transformation of X.
For instance, the first half of a codeword concatenated to the
second half of another codeword is not guaranteed to form a
codeword in a general code.3
Thus, for integer forcing to work over a block fading
channel, we should require all A(i) to be equal [15],
A(i) = A,

i = 1, . . . , F

(29)

so that
V = AX.

(30)

For a given A, the optimal equalization matrix for each ith
block can be obtained as

−1
−1
T
B(i) = AHT
SNR
I
+
H
H
(31)
(i)
(i)
(i)
and the per-component variance of the vector zeff,m,(i) , the
m-th row of the Zeff,(i) , is given as

where

2
σeff,m,(i)
= σ 2 am M(i) aT
m

(32)


−1
M(i) = SNR−1 I + HT
(i) H(i)

(33)

and am is the mth
 row of A.

Let zeff,m = zeff,m,(1) · · · zeff,m,(F ) denote the mth
row of Zeff . Since the noise variance (32) may now be different
for each block, an achievable rate expression is not immediate
to obtain and may, in fact, depend on the type of decoder used
[15]. Specifically, it depends on whether the decoder properly
exploits the diversity inherent in the block fading channel.
Two decoding methods are proposed and analyzed in [15]:
the Arithmetic Mean (AM) and the Geometric Mean (GM)
decoders discussed below.

136

1) AM decoder: This decoder does not attempt to exploit
the channel variation and instead treats each component of the
effective noise vector zeff,m as having the same variance [15],
2
denoted by σAM,m
. This variance is given by the arithmetic
mean (hence the decoder name) of the variance of the effective
noise on each block,
2
σAM,m
=

F
1 X 2
.
σ
F i=1 eff,m,(i)

(34)

It is shown in [15] that the AM-IF receiver achieves the
following rate
!
1
P
+
RAM-IF (H(1:F ) , A) = min log
2
m 2
σAM,m


1
SNR
+
= min log
(35)
m 2
am MAM aT
m
where
MAM =

1 X
M(i) .
F i

(36)

Thus, the maximum rate achievable by this method is
RAM-IF (H) ,

max

A:rank A=NT

RAM-IF (H(1:F ) , A).

(37)

Note that (35) is identical to (19) with M replaced by MAM .
Thus, (37) can be solved in the same way as in the case of
static fading [15]. In particular, the LLL algorithm (or similar)
may be used to find an approximately optimal solution in
polynomial time.
A special case of AM-IF consists of choosing A = I, which
corresponds to conventional MMSE equalization followed by
an AM decoder [15]. This scheme is referred to as AM-MMSE
and its achievable rate denoted as
RAM-MMSE (H(1:F ) ) = RAM-IF (H(1:F ) , I).

(38)

2) GM decoder: This decoder optimally exploits the fact
that the effective noise variance is not constant across the
blocks [15]. The rate achievable by this method can be understood as the average achievable rate among all the individual
blocks, as if they could be treated as parallel channels (which
is clearly an upper bound). More precisely, the rate achievable
by GM-IF is proven in [15] to be
!
P
1 X1
+
log
(39)
RGM-IF (H(1:F ) , A) = min
2
m F
2
σeff,m,(i)
i
!
1
P
+
= min log
(40)
2
m 2
σGM,m
where
! F1
2
σGM,m

=

Y

2
σeff,m,(i)

(41)

i

! F1
2

=σ ·

Y

am M(i) aT
m

.

(42)

i
3 Unless the code can be expressed as the Cartesian product of shorterlength codes. But this effectively violates the assumption of block fading,
corresponding to static fading with a shorter codeword length.

Note that (41) is the geometric mean (hence the decoder
name) of the variance of the effective noise in each block.

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

Therefore, the maximum achievable rate with GM-IF is
RGM-IF (H) ,

max

A:rank A=NT

RGM-IF (H(1:F ) , A).

(43)

It is useful pointing out that, for the same matrix A, the GM
decoder always achieves a rate at least as high as that of the
2
2
AM decoder, since σGM
≤ σAM
due to AM-GM inequality
[22]. However, there is currently no known efficient method
to find an optimal (or approximately optimal) solution for A
in (43) [15], making optimal GM-IF currently infeasible to
implement in practice, especially as NT grows.
As before, a special case of GM-IF consists of choosing
A = I, which corresponds to conventional MMSE equalization followed by a GM decoder [15]. This scheme is referred
to as GM-MMSE and its achievable rate denoted as
RGM-MMSE (H(1:F ) ) = RGM-IF (H(1:F ) , I).

(44)

137

Note that the generalized covariance matrix of N is the identity
matrix [24].
The effective channel output (11) can be rewritten as
Yeff = V + LN

(48)

where V = AX. Since L is a lower triangular matrix, we
have that
m
X
yeff,m = vm +
`m,j nj
(49)
j=1

where `m,j denotes the (m, j) entry of L. Note that yeff,m is
not affected by nm0 for any m0 > m.
The decoder acts in each row of Yeff in a successive
way. Suppose that v1 is successfully recovered. Then we can
compute
1
(yeff,1 − v1 )
(50)
n1 =
`1,1
and remove its influence on the second row of Yeff ,

IV. S UCCESSIVE I NTEGER -F ORCING
One way to improve the performance of a conventional
linear receiver is to apply successive interference cancellation
(SIC) [23]: after a codeword is successfully decoded, the
receiver can use it as side information in order to cancel
part of the interference, reducing the variance of the effective
noise. This principle can be applied to integer-forcing as
well [24]. Specifically, in a successive integer-forcing (SIF)
receiver, each integer linear combination of codewords that is
successfully decoded is used to cancel its contribution to the
effective noise affecting the remaining linear combinations,
potentially enabling a higher achievable rate.
Note that SIF decoding must be done sequentially, in
contrast to conventional IF, which may be done in parallel.
Thus, the decoding order is relevant. We assume that decoding
follows the index of am , m = 1, . . . , NT . Thus, in contrast to
conventional IF, the ordering of the rows of A may have an
impact on the achievable rates for SIF.
We start by reviewing SIF for static fading, followed by its
extension to block fading.

y20 , yeff,2 − `2,1 n1 = v2 + `2,2 n2

(51)

so that v2 can be decoded in the presence of less noise.
Generalizing, suppose that v1 , . . . , vm−1 have been successfully recovered, providing the estimates n1 , . . . , nm−1 .
We can remove the influence of this noise to obtain
0
ym
, yeff,m −

m−1
X

`m,j nj = vm + `m,m nm

(52)

j=1

so that vm can be decoded under noise of variance `2m,m .
Then, the corresponding noise vector can be estimated as
nm =

1
(y0 − vm ).
`m,m m

(53)

Proceeding this way, it can be shown that the following rate
is achievable [24]


P
1
+
RSIF (H, A) = min log
.
(54)
m 2
`2m,m
By choosing the optimal A, the maximum achievable rate is

A. Static Fading

RSIF (H) =

Recall the effective channel (11) described in Section III.
The SIF receiver exploits the fact that the rows Zeff are
correlated in general and starts by performing a whitening
transformation.
Consider the generalized covariance matrix of Zeff , defined
as
1
KZeff , E[Zeff ZT
(45)
eff ].
n
Assuming that the optimal equalization matrix B is used, it is
possible to show that [2], [24]
KZeff = σ 2 AMAT

(46)

where M is defined in (18).
Since KZeff is a symmetric positive definite matrix [24], it
admits a Cholesky decomposition KZeff = LLT , where L is a
lower triangular matrix with strictly positive diagonal entries.
Define
N , L−1 Zeff .
(47)

max

A:rank A=NT

RSIF (H, A).

(55)

However, finding the optimal matrix for the SIF decoder is
a different (and harder) problem than that for the IF decoder.
In particular, each row permutation of A may give a different
achievable rate. As shown in [24], it is possible to restrict
the choice of A to the class of unimodular matrices and the
optimal solution is obtained by finding a Korkin-Zolotarev
(KZ) basis for the lattice generated by G ∈ RNT ×NT , obtained
from the Cholesky decomposition of M = GGT .
Finding a KZ basis for a lattice involves finding a shortest
lattice vector and is therefore an NP-hard problem [20].
Suboptimal algorithms can be used, for example, applying the
LLL algorithm NT successive times, where in each iteration
the dimension of the underlying lattice decreases [20].
Note that it is possible to choose A = π(I), where π(I)
denotes a row permutation of the identity matrix. In this case,
the method reduces to conventional SIC decoding, which is referred to as MMSE-SIC. In principle, all possible permutations

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

could be tested, however this quickly becomes unattractive as
the number of users increases. Heuristic methods [23] can be
applied to find a good decoding order, for instance, decoding
2
first the user with the highest SNR (i.e., lowest σeff,m
), at the
expense of some performance degradation.
B. Block Fading
As in the case of parallel IF, for successive IF in the block
fading scenario it is required that A remain the same for all
blocks so that the rows of V in (48) are still lattice points and
lattice decoding and subsequent inversion is possible. All the
other steps are the same as for static fading applied separately
to each ith block, namely: equalization by B(i) given in (31),
Cholesky decomposition of
1
KZeff,(i) =
E[Zeff,(i) ZT
eff,(i) ]
n/F
= σ 2 AM(i) AT
= L(i) LT
(i)

(56)

where M(i) is defined in (33), and successive noise cancellation and estimation from
m−1
X
0
ym,(i)
, yeff,m,(i) −
`m,j nj,(i)
(57)
j=1

= vm,(i) + `m,m,(i) nm,(i) .

(58)

Note that the effective channel after cancellation can be
expressed more simply as
0
ym
= vm + z0m

(59)

0
0
and z0m are the horizontal concatenation of ym,(i)
where ym
0
and zm,(i) = `m,m,(i) nm,(i) , respectively, for i = 1, . . . , F .
In particular, each ith block of the reduced effective noise z0m
has a possibly different variance `2m,m,(i) .
For the lattice decoding step, either of the two decoding
methods discussed before, namely AM and GM decoding, may
be used [15]. Their generalization to the case of block fading
is straightforward.
1) AM decoder: This decoder treats z0m as white noise of
variance
F
1 X 2
2
σAM-SIF,m =
`
.
(60)
F i=1 m,m,(i)

It follows that the achievable rate of AM-SIF is given by
!
1
P
+
RAM-SIF (H(1:F ) , A) , min log
(61)
2
m 2
σAM-SIF,m
max

A:rank A=NT

RAM-SIF (H(1:F ) , A).

π

where

! F1
2
σGM-SIF,m

=

Y

`2m,m,(i)

.

(66)

i

Note that, due the AM-GM inequality, for the same matrix A,
the GM decoder achieves a rate at least as high as that of the
AM decoder.
The maximum achievable rate for GM-SIF is then given by
RGM-SIF (H(1:F ) ) =

max

A:rank A=NT

RGM-SIF (H(1:F ) , A).

(67)

However, there is currently no known efficient method to find
an optimal or approximately optimal solution for A, making
optimal GM-SIF infeasible to implement in practice. Even an
exhaustive search is more costly for GM-SIF than for GMIF, since now all row permutations of the same A must be
considered.
As a special case, it is always possible to choose A = π(I),
which corresponds to conventional SIC with a GM decoder.
This method is referred to as GM-SIC and its achievable rate
is given by
RGM-SIC (H(1:F ) ) = max RGM-SIF (H(1:F ) , π(I)).
π

(68)

V. P ROPOSED M ETHODS
Although AM-SIF, GM-IF and GM-SIF all have higher
achievable rates than AM-IF, the fact that no efficient algorithm is known to find even an approximately optimal A can
undermine the practical applicability of these methods.
In this section, we propose four suboptimal, low-complexity
methods for choosing the integer matrix A. The first two are
applicable to GM-IF, the third is applicable to AM-SIF, while
the fourth applies to GM-SIF.
A. Proposed Method 1 (GM-IF)
Let
(69)

A:rank A=NT

(62)

However, there is currently no known efficient method to
find an optimal or approximately optimal choice of A, which
should be chosen to minimize (60).
Note that, if we choose A = π(I), then the method reduces
to conventional SIC with an AM decoder. This method is
referred to as AM-SIC and its achievable rate is given by
RAM-SIC (H(1:F ) ) , max RAM-SIF (H(1:F ) , π(I)).

2) GM decoder: As before, this decoder attempts to optimally exploit the variation of the noise statistics across blocks.
The achievable rate for GM-SIF can be computed as the
average achievable rate all blocks, given by [15]
!
SNR
1 X1
+
log
(64)
RGM-SIF (H(1:F ) , A) = min
m F
2
`2m,m,(i)
i
!
1
SNR
+
= min log
(65)
2
m 2
σGM-SIF,m

AAM-IF , arg max RAM-IF (H(1:F ) , A)

which, after optimizing over A, becomes
RAM-SIF (H(1:F ) ) ,

138

(63)

be the optimal matrix A for AM-IF. We propose to use either
this matrix or the identity matrix, depending on which one
gives the highest rate under GM-IF decoding.
Let A1 = {I, AAM-IF }. The rate achievable by this method
is given by
Rprop1 (H(1:F ) ) , max RGM-IF (H(1:F ) , A).
A∈A1

(70)

Note that, if matrix AAM is chosen, then a rate at least
as high as that of AM-IF is achieved, since for the same

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

choice of A, the GM decoder always outperforms the AM
decoder. On the other hand, if the identity matrix is chosen,
then the proposed method becomes the same as GM-MMSE
and, therefore, achieves the same rate. Thus, the proposed
method achieves rates as high as both GM-MMSE and AM-IF.
The complexity of this method is dominated by that of finding an optimal matrix for AM-IF, which can be approximated
in polynomial time with the LLL algorithm. Therefore, the
complexity is the same as that of AM-IF.
B. Proposed Method 2 (GM-IF)
In addition to the choices discussed above, we propose to
test also the optimal matrix A(i) for each ith block that would
be obtained with IF under static fading, namely
A(i) , arg max RIF (H(i) , A)

(71)

A:rank A=NT

for i = 1, . . . , F .
Let A2 = {I, AAM-IF , A(1) , . . . , A(F ) }. The rate achievable
by this method is given by
Rprop2 (H(1:F ) ) , max RGM-IF (H(1:F ) , A).
A∈A2

(72)

Note that Proposed Method 1 chooses a matrix which may
be “reasonably good” for all blocks simultaneously but which
is not necessarily optimal for any block. Proposed Method 2
expands this choice by including matrices which are optimal
for at least one block, even if it they are worse for the others
blocks. A reasoning behind this approach is that, in contrast
to the AM decoder, the GM decoder is not limited by the
performance of the worst block, since individual rates are
added in (39).
The complexity of Proposed Method 2 is higher than that
of Proposed Method 1 since it is necessary run the LLL
algorithm F +1 times. Since the LLL algorithm can be done in
polynomial time, this proposed method still viable in practice,
especially for small F .
C. Proposed Method 3 (AM-SIF)
Recall that the AM-SIF receiver correctly takes into account the fact that the effective noise matrix has a different
generalized covariance matrix KZeff,(i) for each block, using
each corresponding L(i) for noise cancellation; only the lattice
decoding step treats the reduced effective noise z0m as having
2
equal variance σAM-SIF,m
across blocks.
An upper bound on this variance can be obtained by treating
each block of the effective noise matrix Zeff as having the same
generalized covariance matrix, given by

1 
KZeff = E Zeff ZT
eff
n
i
1X h
=
E Zeff,(i) ZT
eff,(i)
n i
1 X
=
KZeff,(i)
(73)
F i
1 X 2
=
σ AM(i) AT
(74)
F i
= σ 2 AMAT

(75)

139

P
where M = F1 i M(i) . From this point on, we can proceed
similarly to the case of static fading (Section IV-A), obtaining
a reduced effective noise z0m with variance `2m,m , where L
is a lower triangular matrix with positive diagonal entries
given by the Cholesky decomposition KZeff = LLT . It follows
2
that σAM-SIF,m
≤ `2m,m , since a suboptimal noise cancellation
scheme is used.4
To be clear, the scheme described above uses block IF
equalization, followed by static noise cancellation (SNC) and
AM decoding, in contrast to AM-SIF, which uses block noise
cancellation. To distinguish it from AM-SIF, we refer to this
scheme as AM-SIF-SNC. Its achievable rate is then given by


P
1
(76)
RAM-SIF-SNC (H(1:F ) , A) = min log+ 2
m 2
`m,m
which is a lower bound on RAM-SIF (H(1:F ) , A).
Let
AAM-SIF-SNC , arg max RAM-SIF-SNC (H(1:F ) , A)

(77)

A:rank A=NT

be the optimal matrix A for AM-SIF-SNC. We propose to use
this matrix for AM-SIF. The rate achievable by this method is
given by
Rprop3 (H(1:F ) ) , RAM-SIF (H(1:F ) , AAM-SIF-SNC ).

(78)

Note that optimizing (76) is exactly the same problem as optimizing (54). Thus, as discussed in Section IV-A, AAM-SIF-SNC
can be computed by KZ reduction of the lattice generated by
G ∈ RNT ×NT , obtained from the Cholesky decomposition
of M = GGT , and this procedure can be approximated by
applying the LLL algorithm NT times.
D. Proposed Method 4 (GM-SIF)
We can extended the same ideas of Proposed Method 2 to
the case of successive decoding, simply by redefining (71)
and (72) with their SIF counterparts, while using AAM-SIF-SNC
instead of AAM-SIF , as in Proposed Method 3. However, since
the decoding order is important, in principle we would have to
test all permutations of the identity matrix, but that number of
permutations grows exponentially with the number of users. To
avoid this complexity, we simply exclude the identity matrix
(and all its permutations) from the set of possible choices
for A.
Let A4 = {AAM-SIF-SNC , ASIF,(1) , . . . , ASIF,(F ) }, where, for
i = 1, . . . , F ,
ASIF,(i) , arg max RSIF (H(i) , A)

(79)

A:rank A=NT

is the optimal matrix for the ith block that would be obtained
with SIF under static fading.
The rate achievable by this method is given by
Rprop4 (H(1:F ) ) = max RGM-SIF (H(1:F ) , A).
A∈A4

(80)

Similarly to Proposed Method 2, the complexity of this
method grows linearly with the number of blocks. However,
finding AAM-SIF-SNC as well as each ASIF,(i) requires running
the LLL algorithm NT times, for a total of NT (F + 1) runs.
4 This upper bound can also be derived directly by diagonalizing the positive
P
T
definite matrix LLT = F1
i L(i) L(i) .

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

TABLE I
C OMPLEXITY OF I NTEGER -F ORCING M ETHODS
Method

Complexity

AM/GM-MMSE
AM-IF
Proposed Method 1
Proposed Method 2
GM/AM-SIC
Proposed Method 3
Proposed Method 4
Optimal GM/AM-SIF

O(NT4 log(NT ))
O(NT4 log(NT ))
O((F + 1)NT4 log(NT ))
O(NT !)
O(NT4 log(NT ))
O((F + 1)NT4 log(NT ))
√
O(( 1 + SNR)NT )

E. Summary of Complexity
Table I
shows the complexity of finding A for each method discussed. For AM/GM-MMSE methods, since A = I is known
a priori, the complexity is negligible. The AM-IF and Proposed
Method 1 have the same complexity, corresponding to a single
run of the LLL algorithm [9]. Note that, for fixed F , Proposed
Method 2 approaches the complexity of AM-IF and Proposed
Method 1. For the successive scenario, the optimal choice
of A in AM/GM-SIC is found by testing all permutations
of the identity matrix. Proposed Method 3 and Proposed
Method 4 have similar complexity as Proposed Method 1 and
2, respectively. Finally, the complexity of optimal GM/AMSIF by exhaustive search follows from the bound in [2] (see
also [2], [9]).
Note that, besides finding A, the receiver operation includes
also the tasks of equalization and channel decoding, whose
complexity scales with the blocklength n and is identical for
each method in the same category (parallel or successive).
Thus, the task of finding A tends to take a smaller fraction of
the overall decoding time as n grows.
VI. I MPLEMENTATION WITH P RACTICAL C ODES
All the achievable rate results discussed above assume the
use of lattice codes of asymptotically high dimension over
an asymptotically large constellation. However, in practice, a
finite-length code and a finite-order modulation must be used.
In this section, we discuss how practical lattice encoders and
decoders for an IF receiver may be implemented with low
complexity. We focus on the use of binary LDPC codes with
2-PAM modulation.
A. Encoding and Decoding
We start with conventional IF; the extension to successive IF
is straightforward. To simplify the description, with a slight
abuse of notation, we consider the finite field of size 2, denoted
Z2 , as a subset of the integers, Z2 = {0, 1} ⊆ Z. Suppose the
`th user encodes its message as a codeword c` ∈ C from
a linear (n, k) block code C over Z2 . The codeword c` is
then modulated into a vector x` ∈ {− 21 , 12 }n from a 2-PAM
constellation, computed as
x` = c` + d` mod 2

(81)

140

where d` ∈ {− 12 , 12 }n is a (discrete) dither vector independent
from x` and known to the receiver, and the x mod 2 operation
is applied element-wise and assumed to return a real-valued
number in the interval (−1, 1].
Note that the dither vector must be used in order to
reduce the transmit power from the {0, 1} constellation to
the {− 21 , 12 } constellation, and it could be interpreted more
simply as a modulation map. In this case, a simple choice
would be d` = (− 21 , . . . , − 12 ). However, using a random
dither uniformly distributed over {− 12 , 12 }n is more convenient
for our purposes since, as we shall see, it makes the error
probability independent from the transmitted codeword. As a
consequence of the use of dithers, the transmitted vector x`
is not a lattice point anymore, in contrast to the description is
sections III and IV. However, dithers can be easily removed
at the receiver with a simple modification, re-enabling the
results of the those sections. More precisely, let C and D
be matrices whose `th row is c` and d` , respectively. The
receiver computes the effective channel output as


Yeff = B(1) Y(1) · · · B(F ) Y(F ) − AD mod 2 (82)
= AC + Zeff mod 2

(83)

= V + Zeff mod 2

(84)

where Zeff is defined as in Section III and V = AC mod 2.
Note that each row vm of V is a codeword from C. Thus, we
recover the same effective channel (11), except for the mod-2
operation.5
For j = 1, . . . , n, let yeff,m [j], vm [j], zeff,m [j] denote the
jth component of yeff,m , vm and zeff,m , respectively. In order
to use a belief propagation decoder, it is necessary to compute
the log-likelihood ratio (LLR) at the channel output, defined
as


P[yeff,m [j] | vm [j] = 0]
(85)
LLR[j] = log
P[yeff,m [j] | vm [j] = 1]
for j = 1, . . . , n, where P(·|·) denotes conditional probability.
To simplify the decoder, the (non-Gaussian) effective noise
component zeff,m [j] is treated as a Gaussian random variable
2
with zero mean and variance σeff,m
[j], where
(
2
σAM,m
,
for AM decoding
2
σeff,m [j] =
(86)
2
σeff,m,(I[j]) , for GM decoding
and I[j] denotes the index of the block to which the jth
component belongs. However, since (84) is a mod-2 channel,
there is an infinite number of noise realizations that lead to
any given channel output, resulting in the expression
!
X
(yeff,m [j] − vm [j] − k)2
P[yeff,m [j]|vm [j]] =
exp −
.
2
2σeff,m
[j]
k∈2Z
(87)
In order to further simplify the LLR computation, we can
approximate the above expression by keeping only the largest
5 Actually, a modulo-lattice channel is required for the results in [2], which
is omitted in the IF overview given here and in [2]. Thus, some modulo
operation must be used at the receiver if the transmitted vectors satisfy a
power constraint.

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

8

It is well-known that an important parameter characterizing
the performance of a code for a fading channel is its diversity
order, defined as [25]

Real LLR
Approximated LLR

6

141

4

d,−

LLR[j]

(91)

where Pe is the error probability of the decoder. For a block
Rayleigh fading channel, the diversity order of a q-ary code
is known to satisfy a Singleton-like bound [25]

 
R
(92)
d≤1+ F 1−
log2 q

2
0
-2
-4
-6
-8
-1

log Pe
SNR→∞ log SNR
lim

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

yeff,m [j]
Fig. 2. LLR in the mod-2 channel for SNR = 5 dB.

term, corresponding to the element vm [j] + 2Z nearest to
yeff,m [j] in Euclidean distance. It follows that

1 + 2yeff,m [j]


, −1 < yeff,m [j] ≤ 0

2
2σeff,m
[j]
LLR[j] ≈ 1 − 2y
(88)

eff,m [j]

,
0
≤
y
[j]
≤
1

eff,m
2
2σeff,m
[j]

1

where b·c is the floor function and R is the code rate in bits per
channel use. Thus, codes that achieve full diversity (d = F )
are limited by R ≤ (log2 q)/F , or R ≤ 1/F for binary codes.
A family of rate-1/F binary LDPC codes that achieve
full diversity under belief propagation and have performance
close to theoretical limits are the so-called root LDPC codes
[26]. These codes are systematic, with the information bits
corresponding to the first n/F 2 positions of each block, and
(F −1)n/F ×n
have a parity-check matrix H ∈ Z2
satisfying the
following structure




H11 · · · H1F
Hij1


..  , H = 
..
..
H =  ...

 (93)
ij
.
. 
.
HF 1

···

Hijk




0,
h
=
I


0,

1 − 2|yeff,m [j]|
2σeff,m [j]2

(89)

since yeff,m [j] ∈ (−1, 1].
Fig. 2 shows the exact LLR in comparison with its approximation for a channel with SNR = 5 dB. As we can see, for
most of the input range the approximation is indistinguishable
from the exact value. The approximation is slightly less
accurate when the input is close to an integer, but since
these correspond to the peak LLR values, i.e., when there
is a high degree of certainty about the value of vm [j], this
loss of accuracy should not degrade the performance of belief
propagation.
The above decoding procedure can be easily extended to
0
successive IF by replacing yeff,m with ym
, zeff,m with z0m ,
2
and σeff,m [j] with
(
σ2
, for AM decoding
2
σSIF,m [j] = 2AM-SIF,m
(90)
`m,m,(I[j]) , for GM decoding.
B. Code Construction
In practice, approaching the GM performance (be it for
MMSE, SIC, IF or SIF reception) requires not only a suitable
decoder but also well-designed codes that allow the decoder to
exploit diversity. This issue is not apparent in [15] since their
achievable rate results (even for AM) are based on asymptotically good lattices that are already optimal for exploiting
diversity. Designing such codes under finite-length and lowcomplexity constraints, however, is far from trivial.

Hi,j,F −1

where

or more simply
LLR[j] ≈

HF,F

j < i and k 6= j
i

0 ,

j=i

(94)

j > i and k 6= j − 1

for all 1 ≤ i, j ≤ F and all 1 ≤ k ≤ F − 1. This structure
implies that, for each information bit from each ith block,
there is one parity-check equation relating it to the bits from
the jth block (and no other blocks), for all j 6= i.
It is worth mentioning that root LDPC codes under belief
propagation guarantee full diversity only over the information
bits. Thus, if one is interested in recovering the entire codeword, it must be regenerated by re-encoding the information
bits after decoding.
VII. S IMULATION R ESULTS
In this section we present simulation results comparing
the outage rate performance of our proposed methods with
the optimal performance obtained by exhaustive search. For
comparison, we include the performance of AM-IF (previously
the only low-complexity IF receiver), as well as that of ML and
conventional linear receivers. In our simulations, we specify
an outage probability ρ = 0.01, estimated by 104 channel realizations. In each realization, the channel fading coefficients are
drawn independently from a real-valued Gaussian distribution
with zero mean and unit variance.
For the optimal GM-IF, AM-SIF and GM-SIF, the matrix A
was obtained by an exhaustive search over all matrices vectors
whose `1 -norm of each row does not exceed 15. Fig. 3 shows
the outage rate for all these receivers on a 2 × 2 channel
with F = 2 blocks. As can be seen, Proposed Method 1 and

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

3

3
ML
Optimal GM-IF
Prop. 2 (GM-IF)
Prop. 1 (GM-IF)
GM-MMSE
AM-IF
AM-MMSE

2.5

ML
Optimal GM-SIF
Prop. 4 (GM-SIF)
GM-SIC
Optimal AM-SIF
Prop. 3 (AM-SIF)
AM-SIC

2.5

2

Rate (bits/dim)

2

Rate (bits/dim)

142

1.5

1.5

1

1

0.5

0.5

0

0
0

5

10

15

20

25

30

SNR (dB)

(a)

0

5

10

15

20

25

30

SNR (dB)

(b)

Fig. 3. Outage rate on a 2 × 2 channel with F = 2 blocks. (a) Parallel decoding methods. (b) Successive decoding methods.

2 achieve performance close to optimal GM-IF and strictly
higher than the maximum between AM-IF and GM-MMSE. In
particular, for an outage rate of 1.5 bits/dim, the performance
of Proposed Method 2 is within 1.8 dB of optimal GM-IF
and outperforms Proposed Method 1 by 2.4 dB. On the other
hand, Proposed Method 3 and 4 appear to have performance
indistinguishable from optimal AM-SIF and optimal GM-SIF,
respectively. Note that Proposed Method 4 outperforms GMSIC by approximately 3.2 dB for an outage rate of 2 bits/dim,
while AM-SIF has a much lower performance in this case.
Fig. 4 shows the outage rate for a scenario with F =
2 blocks and SNR = 25 dB, varying the number of users
NT , while assuming the same number of receive antennas,
NR = NT . Due to the complexity of exhaustive search, which
grows exponentially with NT , the performance of optimal
GM-IF is shown only for NT = 2 and NT = 3, while
that of optimal AM-SIF and GM-SIF is shown only for
NT = 2. As can be seen, as NT increases, the performance
of both Proposed Methods 1 and 2 appears to converge and is
approached by that of AM-IF. Similarly, the performance of
Proposed Method 3 significantly improves as NT increases,
outperforming GM-SIC for NT ≥ 4 and approaching that
of Proposed Method 4. Nevertheless, Proposed Method 4 still
outperforms all other methods by a visible margin.
Fig. 5 considers the same scenario as Fig. 4, but with
F = 4 blocks. Similar observations can be made, except that,
comparatively to Proposed Methods 1 and 2, the performance
of AM-IF has worsened and that of GM-MMSE has improved,
while still being significantly outperformed by the proposed
methods. For the successive methods, a behavior similar to the
F = 2 case is observed, except that now GM-SIC outperforms
Proposed Method 3 for all NT ≤ 6 and achieves a smaller
gap to Proposed Method 4. While one might expect this gap
to vanish for large F , it should be noted that, due to rate
limitations, constructions of full-diversity codes are typically
restricted to the small F case.

Lastly, Figs. 6 and 7 show the frame-error rate (FER) on
a 2 × 2 channel with F = 2 and F = 4, respectively,
using 2-PAM modulation. A regular, rate-1/F root-LDPC
code [26] of length n = 208, constructed using a PEG-based
technique [27], is used in each simulation. As can be seen, the
simulations are consistent with the theoretical results, with a
performance gap due to the small constellation size and the
non-optimality of the channel code. In particular, for a FER
of 1%, both proposed methods are within 3.7 dB of their
theoretical FER for F = 2 and within 3.4 dB for F = 4.
The FER for successive decoding is not shown, since for
R ≤ 1/2 all methods have performance similar to their non-IF
counterpart. For the benefits of SIF to become salient, lattice
codes with higher spectral efficiency are needed. The design
of such codes, however, is outside the scope of this paper.
VIII. C ONCLUSIONS
In this paper, we propose four suboptimal methods for
selecting an integer matrix A for IF reception in a block fading
scenario, two of them applicable to GM-IF and the other two
applicable to AM-SIF and GM-SIF, respectively. The main
idea behind these methods is to use a matrix A optimized for
a lower-performance scheme with a simpler objective function
for which an approximately optimal solution can be found
in polynomial time. For AM-SIF, the corresponding simpler
scheme is the proposed AM-SIF-SNC scheme, while, for GMIF and GM-SIF, it is the best choice among their respective
AM counterpart and certain static fading solutions, all of which
can be found very efficiently.
As shown by simulations, the proposed methods for GMIF achieve outage rates strictly higher than both GM-MMSE
and AM-IF (until now the best low-complexity methods),
regardless of the number of blocks and users, while being
only slightly more complex than AM-IF. Exactly the same

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

4.5

4.5
ML
Optimal GM-IF
Prop. 2 (GM-IF)
Prop. 1 (GM-IF)
AM-IF
GM-MMSE
AM-MMSE

4
3.5

4
3.5
3

Rate (bits/dim)

3

Rate (bits/dim)

143

2.5
2

2.5
2

1.5

1.5

1

1

0.5

0.5

0

ML
Optimal GM-SIF
Prop. 4 (GM-SIF)
Prop. 3 (AM-SIF)
GM-SIC
Optimal AM-SIF
AM-SIC

0
2

3

4

5

6

2

3

Nt (and Nr)

4

5

6

Nt (and Nr)

(a)

(b)

Fig. 4. Outage rate for a channel with F = 2 blocks and SNR = 25 dB. (a) Parallel decoding methods. (b) Successive decoding methods.

4.5

4.5
ML
Optimal GM-IF
Prop. 2 (GM-IF)
Prop. 1 (GM-IF)
AM-IF
GM-MMSE
AM-MMSE

4
3.5

3.5
3

Rate (bits/dim)

Rate (bits/dim)

3

4

2.5
2

2.5
2

1.5

1.5

1

1

0.5

0.5

0

ML
Optimal GM-SIF
Prop. 4 (GM-SIF)
GM-SIC
Prop. 3 (AM-SIF)
Optimal AM-SIF
AM-SIC

0
2

3

4

5

6

Nt (and Nr)

(a)

2

3

4

5

6

Nt (and Nr)

(b)

Fig. 5. Outage rate for channel with F = 4 blocks and SNR = 25 dB. (a) Parallel decoding methods. (b) Successive decoding methods.

observations hold for GM-SIF in comparison with GM-SIC
and AM-SIF-SNC.

particular at outage rates for which these schemes are much
superior to their AM or non-IF counterparts.

We also show that AM-(S)IF and GM-(S)IF schemes can be
realized in practice with low complexity, under finite codeword
length and constellation constraints. Simulation results using
full-diversity root LDPC codes are found to agree with theoretical ones, confirming the superiority of GM-IF in comparison
with GM-MMSE and AM-IF.

The authors would like to thank Islam El Bakoury and
Bobak Nazer for valuable discussions.

An interesting avenue for future work is the development of
low-complexity, full-diversity lattice codes with higher spectral efficiency (for instance, full-diversity q-ary linear codes
with q > 2 for use in Construction A [2]). Such codes would
be directly applicable to the GM-IF and GM-SIF schemes,
allowing a wider and more interesting operating range, in

[1] R. Venturelli and D. Silva, “Low-complexity integer forcing for block
fading MIMO multiple-access channels,” in XXXIV Simpósio Brasileiro
de Telecomunicações e Processamento de Sinais (SBrT’16), Aug 30-Sep
4 2016, pp. 110–114.
[2] J. Zhan, B. Nazer, U. Erez, and M. Gastpar, “Integer-Forcing Linear
Receivers,” IEEE Trans. Inf. Theory, vol. 60, no. 12, pp. 7661–7685,
Dec. 2014. doi: 10.1109/TIT.2014.2361782

ACKNOWLEDGMENTS

R EFERENCES

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

10 0

144

10.1109/TWC.2015.2462844
[9] W. Liu and C. Ling, “Efficient Integer Coefficient Search for Computeand-Forward,” IEEE Trans. Wirel. Commun., vol. 15, no. 12, pp. 8039–
8050, Dec. 2016. doi: 10.1109/TWC.2016.2611580
[10] S. M. Azimi-Abarghouyi, M. Nasiri-Kenari, B. Maham, and M. Hejazi,
“Integer Forcing-and-Forward Transceiver Design for MIMO Multipair
Two-Way Relaying,” IEEE Trans. Veh. Technol., vol. 65, no. 11, pp.
8865–8877, Nov. 2016. doi: 10.1109/TVT.2016.2518667
[11] W. He, B. Nazer, and S. Shamai, “Uplink-downlink duality for integerforcing,” in 2014 IEEE International Symposium on Information Theory,
10 -1
Jun. 2014. doi: 10.1109/ISIT.2014.6875293 pp. 2544–2548.
[12] D. Silva, G. Pivaro, G. Fraidenraich, and B. Aazhang, “On IntegerForcing Precoding for the Gaussian MIMO Broadcast Channel,” IEEE
Trans. Wirel. Commun., vol. 16, no. 7, pp. 4476–4488, Jul. 2017. doi:
10.1109/TWC.2017.2699178
[13] O. Ordentlich and U. Erez, “Integer-Forcing Source Coding,” IEEE
Trans. Inf. Theory, vol. 63, no. 2, pp. 1253–1269, Feb. 2017. doi:
10.1109/TIT.2016.2629491
[14] R. Knopp and P. A. Humblet, “On coding for block fading channels,”
IEEE Trans. Inf. Theory, vol. 46, no. 1, pp. 189–205, Jan. 2000. doi:
10 -2
10.1109/18.817517
0
5
10
15
20
25
30
[15] I. E. Bakoury and B. Nazer, “The impact of channel variation on
SNR (dB)
integer-forcing receivers,” in 2015 IEEE International Symposium on
Information Theory (ISIT), Jun. 2015. doi: 10.1109/ISIT.2015.7282520
Fig. 6. FER on a 2 × 2 channel using parallel decoding methods. R =
pp. 576–580.
1/2 bits/dim and F = 2 blocks.
[16] R. Zamir, Lattice Coding for Signals and Networks.
Cambridge:
Cambridge University Press, 2014. ISBN 978-0-521-76698-2
[17] B. Nazer, V. R. Cadambe, V. Ntranos, and G. Caire, “Expanding the
10 0
Compute-and-Forward Framework: Unequal Powers, Signal Levels, and
AM-MMSE
Multiple Linear Combinations,” IEEE Trans. Inf. Theory, vol. 62, no. 9,
AM-IF
pp. 4879–4909, Sep. 2016. doi: 10.1109/TIT.2016.2593633
GM-MMSE
[18] J. Blömer, “Closest Vectors, Successive Minima, and Dual HKZ-Bases
Prop. 1 (GM-IF)
Prop. 2 (GM-IF)
of Lattices,” in Automata, Languages and Programming. Springer,
Optimal GM-IF
Berlin, Heidelberg, Jul. 2000. doi: 10.1007/3-540-45022-X 22 pp. 248–
Prop. 1 (GM-IF) (theo.)
259.
Prop. 2 (GM-IF) (theo.)
[19] O. Regev, “On Lattices, Learning with Errors, Random Linear Codes,
Optimal GM-IF (theo.)
and Cryptography,” in Proceedings of the Thirty-Seventh Annual ACM
Symposium on Theory of Computing, ser. STOC ’05. New York, NY,
-1
10
USA: ACM, 2005. doi: 10.1145/1060590.1060603. ISBN 978-1-58113960-0 pp. 84–93.
[20] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, “Closest point search in
lattices,” IEEE Trans. Inf. Theory, vol. 48, no. 8, pp. 2201–2214, Aug.
2002. doi: 10.1109/TIT.2002.800499
[21] A. K. Lenstra, H. W. Lenstra, and L. Lovasz, “Factoring polynomials
with rational coefficients,” Math. Ann., vol. 261, pp. 515–534, 1982. doi:
10.1007/BF01457454
[22] J. M. Steele, The Cauchy-Schwarz Master Class: An Introduction to
the Art of Mathematical Inequalities.
Cambridge; New York: The
10 -2
Mathematical Association of America, Apr. 2004. ISBN 978-0-521-6
-4
-2
0
2
4
6
8
10
12
14
54677-5
SNR (dB)
[23] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela,
“V-BLAST: An architecture for realizing very high data rates over the
Fig. 7. FER on a 2 × 2 channel using parallel decoding methods. R =
rich-scattering wireless channel,” in 1998 URSI International Symposium
1/4 bits/dim and F = 4 blocks.
on Signals, Systems, and Electronics. Conference Proceedings (Cat.
No.98EX167), Sep. 1998. doi: 10.1109/ISSSE.1998.738086 pp. 295–
300.
[24] O. Ordentlich, U. Erez, and B. Nazer, “Successive integer-forcing and
[3] B. Nazer and M. Gastpar, “Compute-and-Forward: Harnessing Interits sum-rate optimality,” in 2013 51st Annual Allerton Conference on
ference Through Structured Codes,” IEEE Trans. Inf. Theory, vol. 57,
Communication, Control, and Computing (Allerton), Oct. 2013. doi:
no. 10, pp. 6463–6486, Oct. 2011. doi: 10.1109/TIT.2011.2165816
10.1109/Allerton.2013.6736536 pp. 282–292.
[4] C. Feng, D. Silva, and F. R. Kschischang, “An Algebraic Approach
[25] E. Biglieri, Coding for Wireless Channels, ser. Information technology–
to Physical-Layer Network Coding,” IEEE Trans. Inf. Theory, vol. 59,
transmission, processing, and storage. New York: Springer, 2005. ISBN
no. 11, pp. 7576–7596, Nov. 2013. doi: 10.1109/TIT.2013.2274264
978-1-4020-8083-8 978-1-4020-8084-5
[5] D. Tse and P. Viswanath, Fundamentals of Wireless Communication,
[26] J. J. Boutros, A. Guillen i Fabregas, E. Biglieri, and G. Zemor, “Low1st ed. Cambridge, UK ; New York: Cambridge University Press, Jul.
Density Parity-Check Codes for Nonergodic Block-Fading Channels,”
2005. ISBN 978-0-521-84527-4
IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4286–4300, Sep. 2010. doi:
[6] O. Ordentlich and U. Erez, “Achieving the gains promised by Integer10.1109/TIT.2010.2053890
Forcing equalization with binary codes,” in 2010 IEEE 26-Th Convention [27] A. G. D. Uchoa, C. T. Healy, and R. C. de Lamare, “Structured rootof Electrical and Electronics Engineers in Israel, Nov. 2010. doi:
LDPC codes and PEG-based techniques for block-fading channels,”
10.1109/EEEI.2010.5662120 pp. 000 703–000 707.
J Wireless Com Network, vol. 2015, no. 1, p. 213, Dec. 2015. doi:
[7] S. H. Chae, M. Jang, S. K. Ahn, J. Park, and C. Jeong, “Multilevel
10.1186/s13638-015-0439-6
Coding Scheme for Integer-Forcing MIMO Receivers With Binary
Codes,” IEEE Trans. Wirel. Commun., vol. 16, no. 8, pp. 5428–5441,
Aug. 2017. doi: 10.1109/TWC.2017.2711002
[8] L. Ding, K. Kansanen, Y. Wang, and J. Zhang, “Exact SMP Algorithms for Integer-Forcing Linear MIMO Receivers,” IEEE Trans.
Wirel. Commun., vol. 14, no. 12, pp. 6955–6966, Dec. 2015. doi:
FER

FER

AM-MMSE
AM-IF
GM-MMSE
Prop. 1 (GM-IF)
Prop. 2 (GM-IF)
Optimal GM-IF
Prop. 1 (GM-IF) (theo.)
Prop. 2 (GM-IF) (theo.)
Optimal GM-IF (theo.)

JOURNAL OF COMMUNICATION AND INFORMATION SYSTEMS, VOL. 32, NO.1, 2017.

Ricardo Bohaczuk Venturelli received the B.Sc.
degree and M.Sc. degree from the Federal University
of Santa Catarina (UFSC), Florianópolis, Brazil, in
2014 and 2016, respectively, all in electrical engineering. Currently, he is Ph.D. student in electrical
engineering at the Federal University of Santa Catarina. His research interests include channel coding,
network coding, information theory and wireless
communications. He is a member of the Brazilian
Telecommunications Society (SBrT) since 2014.

145

Danilo Silva (S’06–M’09) received the B.Sc. degree
from the Federal University of Pernambuco (UFPE),
Recife, Brazil, in 2002, the M.Sc. degree from the
Pontifical Catholic University of Rio de Janeiro
(PUC-Rio), Rio de Janeiro, Brazil, in 2005, and
the Ph.D. degree from the University of Toronto,
Toronto, Canada, in 2009, all in electrical engineering. From 2009 to 2010, he was a Postdoctoral
Fellow at the University of Toronto, at the École
Polytechnique Fédérale de Lausanne (EPFL), and at
the State University of Campinas (UNICAMP). In
2010, he joined the Department of Electrical Engineering, Federal University
of Santa Catarina (UFSC), Brazil, where he is currently an Assistant Professor.
His research interests include wireless communications, channel coding,
information theory, and machine learning.
Dr. Silva is a member of the Brazilian Telecommunications Society (SBrT).
He was a recipient of a CAPES Ph.D. Scholarship in 2005, the Shahid
U. H. Qureshi Memorial Scholarship in 2009, and a FAPESP Postdoctoral
Scholarship in 2010.

