Table Of ContentLattice coding for Rician fading channels from
Hadamard rotations
Alex Karrila, Niko R. Va¨isa¨nen, David Karpuk, Member, IEEE, and Camilla Hollanti, Member, IEEE
Department of Mathematics and Systems Analysis
Aalto University School of Science, Finland
Email: firstname.(initial.)lastname@aalto.fi
Abstract—In this paper, we study lattice coding for Rician what kind of modulation and encoding of the data will
fading wireless channels. This is motivated in particular by be most useful for energy efficiency and signal robustness.
preliminary studies suggesting the Rician fading model for
Nevertheless, tentative studies [6] show that the scale channel
millimeter-wavelength wireless communications. We restrict to
lattice codes arising from rotations of Zn, and to a single-input coefficients are Rician distributed.
7 single-output (SISO) channel. We observe that several lattice In this paper, we show that several alternative design
1 design criteria suggest the optimality of Hadamard rotations. approaches suggest the optimality of Hadamard rotations in
0 For instance, we prove that Hadamard rotations maximize RicianSISOchannels.WeshowexperimentallythatHadamard
2 the diamond-packing density among all rotated Zn lattices. rotations outperform the algebraic rotations of [1] and the
Finally,weprovidesimulationstoshowthatHadamardrotations
n cross-packing lattices of [7] over the Rician channel. The
outperformoptimalalgebraicrotationsandcross-packinglattices
a in the Rician channel. Rician channelisindexedbya parameterK >0,with K =0
J
Index Terms—algebraic rotations, diamond packings, beingtheRayleighchannelandK theGaussianchannel.
6 Hadamard rotations, lattice code design, orthogonal lattices, Whilethealgebraiclatticesofferbe→tte∞rperformanceatK =0,
2 reliability, Rician fading, single-input single-output (SISO)
they are outperformed by the Hadamard rotations already at
] channels small K. We present results only for K =20 for the sake of
T compactness, but similar results were obtained for several K.
I I. INTRODUCTION
Lattices from Hadamard rotations has previously been pro-
.
s Reliability is a key issue in designing wireless communi- posedforcertainfadingchannelsin[8],andasanalternativeto
c
[ cations, since the channels are vulnerableto distortions. Reli- OFDM for optical channels in [9], to give a few applications.
ability is typically improved by simultaneous error-correction Nevertheless,itseemsthattheirsurprisinglygoodperformance
1
codingandphysical-layerdesign,withthetradeoffofdecoding in the Rician case has not been noticed before.
v
5 complexity and information rate. Organization: In Section II, we provide the necessary
0 Orthogonallatticecodesareahighlyconventionalphysical- background on lattices, Hadamard rotations, and the Rician
6 layer design in all types of wireless channels. Such codes channel. The design approaches based on error probability
7 provide for fast vector decoding based on solving a closest- estimates and sphere packings are presented in Sections III–
0
vectorproblem.Inaddition,theGraymappingfrombitvectors IV, and approachesbased on diamond packings and diversity
.
1 to lattice vectors guarantees a beneficial conversion of vector estimates in Section V. Simulation results are provided in
0
decoding errors to bit decoding errors. Thus, the fundamental Section VI.
7
question in physical-layer reliability is to find the orthogonal
1
: lattice, or equivalently, the rotation, that provides a low rate II. PRELIMINARIES
v
of vector decoding errors. Algebraic rotations are known to
i A. Lattices
X provide a solution in Rayleigh fading single-input single-
r output (SISO), see [1], [2] for a good overview, as well as A lattice is a discrete additive subgroupof Rn. We assume
a in multiple-input multiple-output (MIMO) channels [3]. familiaritywiththebasicconceptsrelatedtolatticesandlattice
The case is neverthelessnot closed:research on future fifth codes, and refer the unaccustomed reader to [1] or [10]. We
generation (5G) communication networks calls for studying point out that we consistently work in the column vector
optimal rotations for the Rician channel. Namely, instead of convention. We are going to be interested in the following
the usual 700MHz–2.6GHz,an extensionto millimeter waves class of lattices.
(e.g. 28GHz) is anticipated [4], [5]. Using millimeter waves
Definition II.1. A full-rank lattice Λ Rn is well-rounded
provides both advantages and challenges. Using traditional (WR) if its minimal vectors span Rn. ⊂
spectrum allows for the transmission of data over a longer
distancebutatlowercapacity,whereasmillimeterwaveoffers TheminimalvectorsofaWRlatticeΛarenotguaranteedto
greater bandwidth, but over shorter distances. However, the generateΛ[11,Ch.2].WRlatticesareofinterestheremainly
channel models for such new frequency spectra are not yet due to their relation to the sphere-packing problem. Namely,
fully understood. For this reason, it is unclear at the moment all local maxima of the sphere packing density and hence
in particular the sphere-packing optimal lattices are well- III. LATTICEDESIGN CRITERIA IN RICIAN CHANNELS
rounded: in a non-WR lattice, one can shrink the orthogonal
In this section, we provide some motivating computations
complement of the minimal vectors to obtain a lattice with
for the optimality of Hadamard rotations in physical-layer
same minimal norm but smaller volume. The following is a
designs. The design criteria rely on the h having small
i
partial converse to this statement:
variance,andthusbeingconcentratedaroundtheirmean.This
Lemma II.2. Let Λ Rn be a WR full lattice, scaled to unit isincontrastwiththeRayleighfadingchannel,inwhichdeep
volume. Then, the sph⊂ere packing density of Λ is minimized if fades (some hi 0) are a major cause of decoding errors.
≈
and only if Λ is a rotation of Zn. The two most conventional ways to design physical-layer
reliability in fading channels are minimizing the PEP bound
Proof. Any n linearly independent minimal vectors of Λ
(4) or,in the AWGN channel,maximizingthe sphere-packing
generate a WR sublattice Λ′ whose sphere-packing density
density of the lattice. We study analogous design approaches
is smaller or equal to that of Λ. Thus, it suffices to prove
inchannelmodelswheredeepfadesarenottheprimarycause
the claim for the WR lattice Λ′ generated by its minimal of decoding errors, such as the Rician channel1, by studying
vectors. But the claim is then immediate from Hadamard’s
the pairwise error probability (PEP) estimate and the sphere-
inequality.
packing at a near-average fade. We find an agreement of the
B. Hadamard matrices two approaches, both suggesting the optimality of Hadamard
Definition II.3. A (real) Hadamardmatrix is a square matrix rotations.
whose all entries are 1 and whose column vectors are
± A. Pairwise error probability
orthogonal.
The standard PEP bound states that the probability P of a
The orthogonality condition can be equivalently cast as
vector decoding error is bounded by
WTW = nI. Thus, W/√n is an orthogonal matrix, called
1 diag(h)t 2
a Hadamard rotation. Hadamard matrices are conjectured to P E exp k k , (4)
exist in all dimensions divisible by four, and known to exist ≤ 2 (cid:26) (cid:18)− 8σ2 (cid:19)(cid:27)
t∈XΛ\{0}
in all such dimensions relevant for lattice coding purposes.
The Kronecker product of two Hadamard matrices yields a where Λ is the code lattice, σ2 the noise variance, and the
third one. Based on this fact, Sylvester’s construction is the expectation is over h. In the deep-fade dominated Rayleigh
simplest way to obtain Hadamard matricesin dimensionsthat fading channel, this expectation is analyzed by fixing the
are powers of two, defined inductively by distribution of the fading hi and expanding around σ2 = 0.
To study a noise dominated channel, let us fix the noise σ2
1 1
W2 =1 −1, W2k+1 =W2⊗W2k. (1) aEndd“ieaxgp(ahn)dta2ro)u/n8dσV2a=r(ǫh,2it)h=e e0x”p:odneennotitainlgin(k(4d)iabge(cho)mtke2s−
{k k }
We denoteHadamardmatricesby W andHadamardrotations
diag(h)t 2 E diag(h)t 2
byU,oftenworkingwithW toavoidnormalizationconstants. exp k k =exp {k k } e−ǫ
(cid:18)− 8σ2 (cid:19) (cid:18)− 8σ2 (cid:19)
C. Fading SISO channels and the Rice distribution
E diag(h)t 2
We consider a single-input-single-output (SISO) wireless =exp {k k } (1 ǫ+ǫ2/2 ...) (5)
(cid:18)− 8σ2 (cid:19) − −
channel. We assume perfect channel state information (CSI)
at the receiver and no CSI at the transmitter. Such a channel Neglecting the higher-order terms represented by the ellipses
is modeled by the real channel equation and substituting E diag(h)t 2 = E h2 t 2, we approxi-
{k k } { }k k
mate the PEP bound as
y=diag(h)x+v, (2)
1 diag(h)t 2
where x Rn and y Rn are the transmitted and received E exp k k (6)
vectors, r∈espectively, a∈nd h Rn and v Rn are mutually 2t∈XΛ\{0} (cid:26) (cid:18)− 8σ2 (cid:19)(cid:27)
∈ + ∈
isnpdeecptievnedlye.nWt reanadsosummeveGctaoursssimanodneoliinseg,fvading a(n0d,σn2oIis)e,,arned- ≈ 21 e−E{h82σ}2ktk2 (cid:20)1+ V2(a8rσ(h2)22)ktk44(cid:21), (7)
an interleaved channel where the h are i.i.∼d.N t∈XΛ\{0}
i
Our primaryinterest lies in Rician distributed hi, modeling where p denotesthe usualLp vectornorm.Since Var(h2)
afadingwithalineofsightandscatteringroutes.Thestrength was asksu·mked small, the minimal vectors of Λ dominate the
of the line of sight is captured by the Rician factor K 0 series above,and their L2 norm shouldbe maximizedand L4
≥
indexingthedifferentRicedistributionsdefinedbythedensity normminimizedin orderto minimizethe error probability.In
ρ(h)=2h(1+K)e−K−h2(1+K)I (2h K2+K), (3) other words, we should first maximize the packing density of
0
p Λ and then rotate it so that the minimal vectors are parallel
where I0 is the zeroth-orderBessel functionof first kind.The to [ 1,..., 1]T. If Λ is a rotation of Zn this condition is
caseK =0isthewell-studiedRayleighfadingchannel,while satis±fied if a±nd only if it is a Hadamard rotation.
inthelimitK ,h becomesdeterministicallyoneandwe
i
→∞
obtain the additive white Gaussian noise (AWGN) channel. 1Thisassumptionisvalidated fortheRician channel inSectionIV.
B. Sphere packings 0
Regarding the fading channelas an instantaneousGaussian
channel, we should maximize the packing density of the
-0.5
randomlyfaded lattice diag(h)Λ. First, the averagenorm of a
given lattice vector t Λ is after fading E diag(h)t 2 =
E h2 t 2. This tells∈us to maximize the p{akcking radkiu}s of C}) -1
{ }k k P{
Λ, but does not differentiate between rotations of Λ. Next, 1-
sthheourladndboemstanboirlmizesdkdariaogu(nhd)tthke2i,reesxppeeccitaaltliyonthEe[shdoiartge(sht)otne2s]., og(10-1.5
l
k k
Hence, we should minimize the variance
Var(h2) t 4 -2
Var(kdiag(h)tk2/E[kdiag(h)tk2])= E[h2]i2 ktk44. (8) n = 2
i k k2 n = 4
Theconclusionsareidenticaltothosereachedfromthecriteria -2.5
0 5 10 15 20
derived from the PEP; specifically, if Λ is a rotation of Zn Rician factor K
and t a minimal vector of Λ, then the above quantity will be
minimized exactly when it is a Hadamard rotation. Fig. 1. The probability (1−P{C}) of Corollary IV.2, when hi are taken
RicianwithparameterK,fortheHadamardmatricesinn=2,4dimensions
Remark III.1. More generally, we could expand the class of fromSylvester’s construction.
latticesweareinterestedintoincludenon-orthogonallattices.
In particular, there exist many well-rounded lattices (which
where the last step used the fact that W/√n is a rotation
necessarily have good sphere packings) all of whose minimal
matrix.Now diag(h)Wω mustsatisfy ω 2 <n to be shorter
vectorsare parallelto [ 1,..., 1]T. Forexample, thebody- k k
centeredcubiclatticein±R3,gene±ratedbythevectors[1,1,1]T, than diag(h)Wwj.
[1, 1, 1]T, and[ 1, 1,1]T,orits tensorproductwith any CorollaryIV.2. LetusdenotebyC theeventthatthe natural
− − − −
Hadamard lattice. generators diag(h)We of a faded Hadamard lattices Λ =
i h
IV. SPHERE-PACKING DENSITY OF HADAMARD LATTICES diag(h)WZn are minimal vectors. Then, P C is given by
{ }
IN RICIAN FADING CHANNELS integrating the joint density of h21,...,h2n over a cone.
In this section, we provide a probabilistic estimate for the Proof. For fixeddimension,the previouslemma givesfinitely
sphere-packingdensityofaHadamardrotatedunitlatticeafter many lattice coordinates ω that can yield minimal vectors.
fading. We work with the unnormalized Hadamard matrices Then,diag(h)We areminimalvectorsif andonlyif allsuch
j
W Rn×n, and denote the lattices of interest by Λ= WZn ω satisfy
and∈Λ =diag(h)WZn.
h n
Notice that the natural generators of the faded lattice Λh diag(h)Wω 2 h2, (13)
are of the form [±h1...,±hn]T. If they are minimal vectors k k ≥ Xk=1 k
of Λ , then Λ is well-rounded and, by Lemma II.2, Λ
h h h a linear inequality in h2,...,h2.
has a good sphere packing. In this section, we compute the 1 n
probability of this event in low dimensions. This corollary allows for numerical computations, as illus-
trated in Figure 1. In particular, when h are taken Rician
Lemma IV.1. For any realization of h, there is a minimal i
with parameter K, the probability (1 P C ) of Λ not
vector diag(h)Wω of the faded lattice Λh, where the integer − { } h
lattice coordinates ω Zn either satisfy ω 2 <n, or ω is a being WR decays exponentially in K. In four dimensions,
∈ k k the exponential decay is faster than in two. This leads to two
row of the matrix W.
conclusions. First, especially in larger dimensions, one can
Proof. Let wj be the jth row of W, and ω Zn some arbi- expectHadamardrotationstoperformwellinRicianchannels
∈
trarylattice coordinates.We comparethelengthsofthelattice whereK issufficientlyawayfromzero.Second,thedecoding
Λh vectorsdiag(h)Wwj anddiag(h)Wω. First, Wwj =nej, errors in such a setup mainly occur due to large noise, rather
so for some j, we have than deep fades. This supports the standing assumption of
diag(h)Ww 2 =n2 min h2. (9) “noise-dominated errors” in the design approach suggesting
k jk 1≤i≤m i Hadamard rotations in Section III.
Next, denote Wω =z and note that
V. CONNECTIONS TO OTHERDESIGN CRITERIA
n
diag(h)Wω 2 = h2z2 (10) Aside from the above criteria derived from the PEP and
k k k k
Xk=1 sphere packing density, there have recently been more subtle
z 2 min h2 (11) design criteria introduced for lattice codes over fading chan-
≥ k k 1≤k≤n k nels, wherein deep fades are not the primary cause of errors.
= nkωk21≤mki≤nnh2k, (12) Theseincludethecross-packingdensity[7]andlocaldiversity
[12]. In this section, we present results which study how well PropositionV.2. LetU Rn×n bea Hadamardrotationand
Hadamard rotations of Zn satisfy these design criteria. let t=Uω be of diversi∈ty k >0. Then,
A. Diamond packings k t 2 n. (18)
k k ≥
Recently, [7] designed lattices for Rician channelsby max-
imizing their cross-packing density (equivalently, maximizing Proof. Since t has diversity k, we have t 2 t 1/√k.
k k ≥ k k
cross-packing radius), where the crosses consist of axis- Substituting the minimal L1 norm t 1 √n, we obtain
k k ≥
aligned line segments. This was motivated by approximating ktk22 ≥n/k as desired.
the shape of a contour surfaces of the terms in the Rician
PEP estimate (4) by a cross polyomino, i.e. an appropriately VI. SIMULATIONS
thickened cross. Analogously,approximatingthe shape of the
We now present simulation results to confirm our previous
contour surfaces by the convex hulls of the n-dimensional
findings. The key parameters for our simulations are the
crosses, also known as L1 norm balls or diamonds, one ends
dimension n of the signal constellation and its order q per
up designing lattices based on their diamond packing density.
dimension (so that the number of constellation points is qn),
We find that the Hadamard rotations maximize the diamond
packing density of the rotated Zn lattices. the parameter K of the Rician distribution, and the volume-
to-noise ratio (VNR) which defines the variance of the noise
Proposition V.1. Let R Rn be a rotation matrix. Then, the σ2 by the formula VNR = Vol(Λ)2/n/(8σ2). In simulations
minimal L1 norm of the∈rotated Zn lattice RZn satisfies fonr n=2 we used q =8, and for n=4 wenused q =4.
min t √n, (14)
t∈RZnk k1 ≤ A. Setup details
t6=0
Let M denote the generator matrix of the simulated code
with equality if and only if R is a Hadamard rotation.
latticeΛandS thesignalconstellation.Toconstructthesignal
Proof. To prove the inequality, let e be any elementary basis constellation,we start froma finite regionof Zn describedby
vector. Take t = Re a rotated basis vector. Recall now the S′′ = (x ,...,x ) x Z, 0 x < q i . Then, we
1 n i i
relation of L2 and L1 norms on Rn: center s{uch a region b|y set∈ting S′ =≤S′′ (q ∀1})/2. Finally,
− −
oursignalconstellationS isaimageofS′ underthegenerator
t √n t , (15)
k k1 ≤ k k2 matrix, S =MS′.
withequalityifandonlyiftisparallelto[ 1,..., 1]T.This The channel simulations were based on first generating a
implies that ± ± uniform random constellation vector x, then generating the
min t √n, (16) Rician fading and Gaussian noise vectors of the channel
t∈RZnk k1 ≤ equation(2)toobtainthereceivedvectory,andfinallysolving
t6=0
the closest vector problem xˆ =argmin y diag(h)t 2.
t∈S
and the equality is possibly reached only if R is Hadamard. The decoding is correct if xˆ =x. || − ||
ToprovethattheinequalityissharpforHadamardrotations,
notice that t a if and only if t s a for some
1 B. Results
k k ≥ · ≥
sign vector s = [ 1,..., 1]T. Now, let t be any nonzero
vectorofaHadam±ardrotate±dlatticeUZn,andletubeabasis Rotations: WefirstcomparedthreedifferentrotationsofZn
vectorofthelattice UZn (i.e.,a columnofU)witha nonzero in dimension n=4. These rotations are the identity rotation,
(integer) coefficient in t UZn. Choosing s = √nu, the the best known algebraic rotation [2], and the Hadamard
orthogonality of the Hada∈mard basis implies ± rotation from Sylvester’s construction. We simulated the per-
formanceof these rotationsin the Rician channelovera large
t s √n, (17) range of K values.
· ≥
so indeed t √n for all nonzero t UZn. In Fig. 2 we plot error rate as a function of VNR, for the
1
k k ≥ ∈ Rician parameter K = 20. The Hadamard rotations perform
B. Local diversity slightlybetterthanthealgebraicrotationsoverthewholeVNR
range, which supports the results of Section III. Similar sim-
In [12] the design of reliable lattices in low signal-to-
ulations for other values of K produced comparable results.
ratio(SNR)Rayleighfadingchannelswasconsidered,andthe
To see how the performance of the different lattices varies
authorsdeducedthatHadamardrotationsare optimalwithin a
with the Rician parameter K, we plot in Figure 3 the error
certain one-parameter Lie group of rotations. They explained
rates as a function of K for fixed VNR. We see that in
the appearance Hadamard rotations, which contrary to con-
dimensionn=4thereisavalueforK,namelyK 4.4,after
ventional algebraic rotations are not fully diverse, by local
≈
which the Hadamard rotation performs better than the other
diversityofHadamardlattices,i.e.,atradeoffbetweendiversity
twosimulatedrotations.Ananalogoussimulationindimension
andlengthofthelatticevectors.Weprovethefollowingsharp
n = 2 produced similar results, with the critical value of K
local diversity estimate for Hadamard rotations.
being K 7.2.
≈
10-1 Cross-packing lattices: Recently, [7] designed a family
of cross-packing lattices for Rician channels, indexed by a
parameter t, of which t = 1 and t = 2 performed best.
Let us compare the Hadamard rotations against these lattices
10-2 (normalized to unit volume). Note that the cross-packing
lattices are not orthogonal, so comparing vector error rates
or in the codes, as plotted, is not equivalent to comparing bit
err error rates.
P
10-3 In Fig. 4 we present simulation results which compare the
performance of the Hadamard rotation to the cross-packing
lattices in dimension n = 2, which show that the Hadamard
Identity rotation rotation offers a modest improvement over the cross-packing
10-4 Algebraic rotation lattices.
Hadamard rotation
5 6 7 8 9 10 VII. CONCLUSIONS
VNR (dB)
Motivated by applicationsto 5G and millimeter wave com-
munications,westudiedlatticecodebookdesignfortheRician
Fig.2. ErrorratesintheRician fadingchannelwithparameter K=20for
latticecodesfromtheidentityrotation,thebest-knownalgebraicrotation,and fading channel. It was found that Hadamard rotations of Zn
theHadamardrotationinn=4dimensions. satisfythedesigncriteriaderivedfromthecorrespondingPEP
and the sphere-packing density. Two particularly attractive
100 features of Hadamard lattices are that they often retain good
sphere-packing properties after experiencing fading, and that
they are maximizers of the diamond packing density among
rotationsofZn.Simulationswereprovidedwhichdemonstrate
10-1 that Hadamard lattices outperform other lattice constructions
in the Rician channel, such as algebraic rotations and cross-
or packing lattices.
err
P
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