Multiple-Antenna Interference Channel with Receive Antenna Joint Processing and Real Interference Alignment

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by  Mahdi Zamanighomi

Multiple-Antenna Interference Channel with
Receive Antenna Joint Processing and Real
Interference Alignment
Mahdi Zamanighomi and Zhengdao Wang
Dept. of Electrical & Computer Eng.
Iowa State University
Ames, Iowa, USA
Email: {mzamani,zhengdao}@iastate.edu
3
1
0
2 Abstract—We consider a constant K-user Gaussian interfer- In this paper, we consider a constant K-user Gaussian

n ence channel with M antennas at each transmitter and N interferencechannel with M antennasat each transmitter and
a antennas at each receiver, denoted as a (K,M,N) channel. N antennasateachreceiver,denotedasa(K,M,N)channel.
J RelyingonaresultonsimultaneousDiophantineapproximation,
We develop a real alignment scheme for MIMO interference
a real interference alignment scheme with joint receive antenna
7
processingisdeveloped.Theschemeisusedtoprovidenewproofs channelthatemploysjointreceiveantennaprocessing.Relying
2
for two previously known results, namely 1) the total degrees of on the recent results on simultaneous Diophantine approxi-

] freedom(DoF)ofa(K,N,N)channelisNK/2;and2)thetotal mation, we are able to obtain new proofs of two previously
T DoFofa(K,M,N)channelisatleastKMN/(M+N).Wealso knownresults, and derive two new results on the DoF region;
I derive the DoF region of the (K,N,N) channel, and an inner see Sec. III.
. bound on the DoF region of the (K,M,N) channel.
s
c I. INTRODUCTION II. SYSTEM MODEL
[
Interference channel is an important model for multi-user Notation: K, D, D′, and N are integers and K =
1 communicationsystems.InaK-userinterferencechannel,the {1,...,K}, N ={1,...,N}. We use k and kˆ as transmitter
v
5 k-th transmitter has a message intended for the k-th receiver. indices,andj asreceiverindices.Superscriptstandrareused
1 At receiver k, the messages from transmitters other than the fortransmitterandreceiverantennaindices.Thesetofintegers
3 k-th are interference. Characterizing the capacity region of andrealnumbersaredenotedasZandR,respectively.Theset
6 a general interference channel is an open problem, although of non-negativerealnumbersis denotedas R+. Letter i and l
1. results for some specific cases are known. are usedas the indicesof directionsandstreams, respectively.
0 To quantifythe shape of the capacity region at high signal- Vectors and matrices are indicated by bold symbols. We use
3 to-noiseratio(SNR),theconceptofdegreesoffreedom(DoF) kxk to denote infinity norm of x, (·)∗ matrix transpose, and
1 has been introduced [1], [2]. The DoF of a message is its ⊗ the Kronecker product of two matrices.
:
v rate normalized by the capacity of single-user additive white Consider a multiple-antenna K-user real Gaussian interfer-
Xi Gaussian noise channel, as the SNR tends to infinity. ence channel with M antennas at each transmitter and N
To achieve the optimal DoF, the concept of interference antennas at each receiver. At each time, each transmitter, say
ar alignment turns out to be important [3]. At a receiver, the transmitter k, sends a vector signal xk ∈ RM intended for
interference signals from multiple transmitters are aligned in receiver k. The channel from transmitter k to receiver j is
the signalspace, so thatthe dimensionalityofthe interference represented as a matrix
sinpathceesiisgninaltesrpfaecreenccaenfbreeemainnidmiczaend.bTeheurseefdoref,otrhethreemdaeisniriendg Hj,k ..=[hj,k,r,t]Nr=,M1,t=1 (1)
signals. Two commonly used alignment schemes are vector where k ∈ K, j ∈ K, and H ∈ RN×M. It is assumed
j,k
alignment and real alignment. In real alignment, the concept that the channel is constant during all transmissions. Each
of linear independenceover the rational numbersreplaces the transmitteris subjected to a powerconstraint P. The received
morefamiliarvectorlinearindependence.AndaGroshevtype signal at receiver j can be expressed as
of theoremis usuallyused to guaranteethe requireddecoding
y = H x +ν , ∀j ∈K (2)
performance [4]. j j,k k j
So far the real alignment schemes have been mainly de- kX∈K
veloped only for scalar interference channels. For multiple- where{ν |j ∈K}isthe setofindependentGaussianadditive
j
input multiple output (MIMO) interference channels, antenna noises with real, zero mean, independent, and unit variance
splitting argument has been used in [4] and [5] to derive the entries. Let H denote the (KN) × (KM) block matrix,
totalDoF.Insuchantennasplittingarguments,nocooperation whose (j,k)th block of size N × M is the matrix H .
j,k
isemployedeitheratthetransmittersideoratthereceiverside. The matrix H includes all the channel coefficients. For the
interference channel H, the capacity region C(P,K,H) is x11 y11
defined in the usual sense: It contains rate tuples R (P) =
K
[R1(P),R2(P),...,RK(P)] such that reliable transmission x2 y2
from transmitter k to receiver k is possible at rate R for all 1 1
k
k ∈ K simultaneously, under the given power constraint P.
| {z } |{z}
Reliable transmissions mean that the probability of error can usefulsignals interference
be made arbitrarily small by increasing the encoding block
length while keeping the rates and power fixed. x12 y21
ADoFvectord=(d ,d ,...,d )issaidtobeachievable
1 2 K
if for any large enough P, the rates R = 0.5log(P)d ,
i i x2 y2
i=1,2,...,K,aresimultaneouslyachievablebyallK users, 2 2
namely 0.5log(P) · d ∈ C(P,K,H), for P large enough.
| {z } |{z}
The DoF region for a given channel H, D(K,H), is the usefulsignalsinterference
closure of the set of all achievable DoF vectors. The DoF
Fig. 1. 2-user Gaussian interference channel with 2 antennas at each
region D(K,M,N) is the largest possible region such that transmitter receiver
D(K,M,N) ⊂ D(K,H) for almost all H in the Lebesgue
sense. The totalDoF ofthe K-userinterferencechannelH is
defined as Remark 4: Theorem 1 follows from Theorem 2 by setting
K M =N and the outer bound for K-user interference channel
d(K,H)= max d .
k that has been obtained before in [2].
d∈D(K,H)kX=1 From the above remarks, it is only necessary to prove
The total DoF d(K,M,N) is defined as the largest possible Theorem 4. However, we will first prove the achievability of
realnumberµsuchthatforalmostall(intheLebesguesense) Theorem 1 in the next section, which serves to introduce the
realchannelmatricesHofsize(KN)×(KM),d(K,H)≥µ. joint antenna processing at the receivers, and the application
of the result in simultaneous Diophantine approximation on
III. MAINRESULTS
manifolds. Theorem 4 will be proved in Sec. V.
The following two theorems have been proved before, in
[4] and [5], respectively: IV. ACHIEVABILITYPROOF FORTHEOREM1
Theorem 1: d(K,N,N)= NK. One important technique for proving achievability result is
2
Theorem 2: d(K,M,N)≥ MN K. the real interference alignment [4] which seeks to align the
M+N
The main contributions of the paper are 1) providing al- dimensions of interferences so that more free dimensions can
ternative proofs of the above two theorems, and 2) prove the beavailableforintendedsignals.Thedimensions(alsonamed
following theorems. directions) are represented as real numbers that are rationally
Theorem 3: The DoF region of a (K,N,N) interference independent (cf. Appendix A).
channel is the following We will denote set of directions, a specific direction, and
vector of directions using T, T, and T respectively.
D(K,N,N)={d∈RK×1 d +maxd ≤N,∀k ∈K}.
+ k kˆ
(cid:12) kˆ6=k ENCODING: Transmitter k sends a vector message xk =
Theorem 4: The DoF regio(cid:12)n of a (K,M,N) interference (x1,...,xN)∗ where xt, ∀t ∈ N is the signal emitted by
k k k
channel satisfies D(K,N,N)⊃D(in) where antenna t at transmitter k. The signal xt is generated using
k
transmit directions in a set T = {T ∈ R|1 ≤ i ≤ D} as
i
D(in) ..={d∈RK+×1 Mdk+Nmaxdkˆ ≤MN, follows
(cid:12) kˆ6=k
(cid:12)
∀k ∈K}. xt =Tst (3)
k k
Remark 1: The DoF region of K-user time-varying in- where
terference channel with N antennas at each node has been
obtainedbeforein[6].ThefacttheDoFregionofa(K,N,N) T..=(T1,...,TD), stk ..=(stk1,...,stkD)∗ (4)
channel in Theorem 3 is the same as that of a time-varying
and ∀1≤i≤D,
MIMOinterferencechannelwiththesamenumberofantennas
indicates that the DoF region for this channel is an inherent st ∈{λq q ∈Z,−Q≤q ≤Q}. (5)
ki
spatial property of the channel that is separate from the time (cid:12)
The parameters Q and λ(cid:12) will be designed to satisfy the rate
or frequency diversity.
and power constraints.
Remark 2: Theorem 3 follows from Theorem 4 by setting
M =N,andtheMultipleAccessChannel(MAC)outerbound ALIGNMENTDESIGN:Wedesigntransmitdirectionsinsucha
[7, Sec. 14.3]. waythatatanyreceiverantenna,eachusefulsignaloccupiesa
Remark 3: Theorem 2 follows from Theorem 4 by setting setofdirectionsthatarerationallyindependentofinterference
d =MN/(M +N), ∀k ∈K. directions (cf. Appendix A).
k
Tomakeitmoreclear,considerFig.1.x1 andx2 areshown and
1 1
by white triangle and square. In a similar fashion, x1 and x2 s1
2 2 k
ainresiuncdhictartaendsmwiitthdbirleacctkiotnrisanthgaletaantdeasqchuarreec.eWiveeraraenitnetnenraestthede sk ..=s...2k, uk = sλk, (14)
interferences,forinstanceblacktriangleandsquareatreceiver  
sN
1, are aligned while the useful messages, white triangle and  k 
square, occupy different set of directions.
suchthatBisaN×ND matrixwith (N−1)D zerosateach
TRANSMIT DIRECTIONS: Our scheme requires all transmitter row. Using above definitions, yj can be rewritten as
antennas to only contain directions of the following form
T = (hj,k,r,t)αj,k,r,t (6) yj =λHj,jBuj + Hj,kBuk+νj. (15)
jY∈Kk∈KY,k6=jrY∈NtY∈N k∈XK,k6=j
 
where
The elements of u are integers between −Q and Q, cf. (5).
k
0≤α ≤n−1, (7)
j,k,r,t We rewrite
∀j ∈ K, k ∈ K, k 6= j, r ∈ N, t ∈ N. It is easy to see that
the total number directions is H Bu =(H ⊗T)u =
j,j j j,j j
D =nK(K−1)N2. (8) hj,j,1,1T hj,j,1,2T ... hj,j,1,NT T1j
hj,j,2,1T hj,j,2,2T ... hj,j,2,NT T2j
We also assumethatdirectionsin T areindexedfrom 1 toD.  ... ... ... ... uj ..= ... uj
The exact indexing order is not important here.    
hj,j,N,1T hj,j,N,2T ... hj,j,N,NT TNj 
ALIGNMENT ANALYSIS: Our design proposes that at each (16)
antenna of receiver j, j ∈ K, the set of messages {xt|k ∈
k
K,k 6=j,t∈N}arealigned.Toverify,considerallxtk,k 6=j where ∀r ∈N, Trj is the rth row of Hj,jB. Also,
that are generated in directions of set T. These symbols are
interpreted as the interferences for receiver j. Let
H Bu = (H ⊗T)u =
j,k k j,k k
D′ =(n+1)K(K−1)N2. (9) k∈XK,k6=j k∈XK,k6=j
(h Tut) T′u′1
k∈K,k6=j t∈N j,k,1,t k j
aarnedidneffirnoemaosfeTt Tas′ =in{(6T)i′b∈utRw|1ith≤ais≤maDll′c}hsauncghetahsatfoallllowTis′ PPk∈K,k6=jPPt∈..N (hj,k,2,tTutk)(a=)T′..u′j2 (17)
 .   . 
   
0≤αj,k,r,t ≤n. (10)  k∈K,k6=j t∈N (hj,k,N,tTutk) T′u′jN
P P
Clearly, all xtk, k 6=j arrive at antenna r of receiver j in the where ∀r ∈ N, u′r is a column vector with D′ integer ele-
j
directionsof{(hj,k,r,t)T|k∈K,k 6=j,t∈N,T ∈T}which ments, and(a) followssince the set T′ containsalldirections
is a subset of T′. of the form (h )T where k6=j; cf. the definition of T′.
j,k,r,t
This confirms that at each antenna of any receiver, all
Considering (16) and (17), we are able to equivalently
the interferences only contain the directions from T′. These
denote y as
j
interference directions can be described by a vector
DECODINGSCHEME:TI′n..=thi(sTp1′a,r.t.,.w,TeD′fi′r)s.trewritetherecei(v1e1d) yj =λTT...1j2j T0...′ T0...′ ......... 00... uu...′jj1+νj. (18)
  
signals. Then, we prove the achievability part of Theorem 1 TN 0 0 ... T′u′N
 j  j 
using joint antenna processing.
The received signal at receiver j is represented by We finally left multiply yj by an N ×N weighting matrix
y =H x + H x +ν . (12) 1 γ ... γ
j j,j j j,k k j 12 1N
k∈XK,k6=j γ21 1 ... γ2N
Let us define W= ... ... ... ...  (19)
 
T 0 ... 0 γN1 γN2 ... 1 
 
0 T ... 0
B..=... ... ... ... (13) usuncifhortmhalyt cahllosienndefrxoemd iγntearrvealr[a1n,d1o]m. Tlyh,isinpdroepceesnsdecnautlsye,satnhde
  2
0 0 ... T zeros in (18) to be filled by non-zero directions.
 
After multiplying W, the noiseless received constellation then the power constraint is satisfied (ζ is a constant here).
belongs to a lattice generated by the N ×N(D+D′) matrix Moreover, similar to [4], we choose
T1 T′ 0 ... 0 1−ǫ
j Q=P2(D+D′+1+ǫ) for ǫ∈(0,1) (25)
T2j 0 T′ ... 0
A=W ... ... ... ... ... . (20) assuring that the DoF per direction is D+D1−′+ǫ1+ǫ. Since, we
  are allowed to arbitrarily choose ǫ within (0,1), 1 is
TN 0 0 ... T′ D+D′+1
 j  also achievable.
The above matrix has a significant property that allows us Using (22)–(25) and the performanceanalysis describedby
to use Theorem 5 (cf. AppendixC). Theorem 5 requireseach [4], the hard decoding error probability of received constella-
rowofAtobeanondegeneratemapfromasubsetofchannel tion goesto zero as P →∞ and the total achievable DoF for
coefficients to RN(D+D′). The nondegeneracy is established almost all channel coefficients in the Lebesgue sense is
because (cf. Appendix B):
1) allelementsofT′andTtj,∀t∈N areanalyticfunctions NKD = NKnK(K−1)N2 (26)
of the channel coefficients; D+D′+1 nK(K−1)N2 +(n+1)K(K−1)N2 +1
2) all the directionsin T′ and Tt, ∀t∈N togetherwith 1
are linearly independent overjR ; and as n increases, the total DoF goes to N2K which meets
the outer bound [2].
3) all indexed γ in W have been chosen randomly and
independently.
V. PROOF OFTHEOREM4
Hence, using Theorem 5, the set of H such that there exist
Notation: Unless otherwise stated, all the assumptions and
infinitely many
definitions are still the same.
uj Consider the case where the number of transmitter and
q=u.′j1∈ZN(D+D′) receiver antennas are not equal. This can be termed the K-
. user MIMO interference channel with M antennas at each
 . 
  transmitter and N antennas at each receiver. Hence, for all
u′N
 j  j ∈ K and K ∈ K, H is a N ×M matrix. We prove that
j,k
with for any d∈D(in), d is achievable.
kAqk<kqk−(D+D′)−ǫ for ǫ>0 (21) Underthe rationalassumption,itis possibletofindan inte-
has zero Lebesgue measure. In other words, for almost all gerρ suchthat∀k ∈K, d¯k =ρdMk where d¯k isa non-negative
H, kAqk > kqk−(D+D′)−ǫ holds for all q ∈ ZN(D+D′) integer. The signal xtk is divided into d¯k streams. For stream
except for finite number of them. By the construction of A, l, l ∈ {1,...,maxd¯k}, we use directions {Tl1,...,TlD} of
k∈K
all elements in each row of A are rationally independent the following form
with probability one, which means that Aq 6= 0 unless
T = (h δ )αj,k,r,t (27)
q = 0. Therefore, almost surely for any fixed channel l j,k,r,t l
(hence fixed A), there is a positive constant β such that jY∈Kk∈KY,k6=jrY∈NtY∈N
kAqk>βkqk−(D+D′)−ǫ holds for all integer q6=0. Since where 0 ≤ α ≤ n − 1 and δ is a design parameter
j,k,r,t l
that is chosen randomly, independently, and uniformly from
kqk≤(K−1)NQ,
the interval [12,1]. Let Tl ..= (Tl1,...,TlD). Note that, at
the distance between any two points of received constellation any antenna of transmitter k, the constants {δ } cause the
l
(without considering noise) is lower bounded by streams to be placed in d¯ different sets of directions. The
k
−(D+D′)−ǫ alignment scheme is the same as before, considering the fact
βλ (K−1)NQ . (22)
that at each antenna of receiver j, the useful streams occupy
Remark 5: The(cid:0)noiseless rec(cid:1)eived signal belongs to a Md¯ separate sets of directions. The interferences are also
j
constellation of the form aligned at most in max d¯ sets of directions independent
k
k∈K,k6=j
y=λAq¯ (23) from useful directions.
By design, xt is emitted in the following form
where q¯ is an integer vector. Then, the hard decision maps k
the received signal to the nearest point in the constellation. d¯k D
NotethattheharddecoderemploysallN antennasofreceiver xtk = δl Tlistkli =Tkstk (28)
j to detect signals emitted by intended transmitter. In other Xl=1 Xi=1
words, our decoding scheme is based on multi-antenna joint where
proWceessnionwg.design the parameters λ and Q. With reference to Tk ..=(δ1T1,...,δd¯kTδd¯k), (29)
[4], if we choose P12 stk ..=(stk11,...,stkd¯kD)∗, (30)
λ=ζ , (24)
Q and all stkli belong to the set defined in (5).
Pursuingthe samestepsof theprevioussectionforreceiver B. Nondegenerate Manifolds [9]
j, B becomes a M × MDd¯j matrix and A will have N Consider a d-dimensional sub-manifold M = {f(x)|x ∈
rowsandMDd¯j+ND′k∈mKa,kx6=jd¯k columns.Thetotalnumber U}ofRn,whereU ⊂Rd isanopensetandf =(f1,...,fn)
directions G of both useful signals and the interferences at is a Ck embedding of U to Rn. For l ≤ k, f(x) is an l-
j
receiver j satisfies nondegenerate point of M if partial derivatives of f at x
of order up to l span the space Rn. The function f at x is
Gj ≤MDd¯j +ND′ max d¯k. (31) nondegenerate when it is l-nondegenerateat x for some l.
k∈K,k6=j
If the functions f ,...,f are analytic, and 1,f ,...,f
1 n 1 n
For any DoF points in D(in) satisfying Theorem 4, we have are linearly independent over R in a domain U, all points of
M=f(U) are nondegenerate.
ρ
Gj ≤ Md¯j +N max d¯k D′ ≤ NMD′ =ρND′ C. Diophantineapproximationforsystemsoflinearforms[8]
(cid:18) k∈K,k6=j (cid:19) M
(32) Consider a m×n real matrix A and q ∈ Zn. The theory
and as n increases, the DoF of the signal x intended for of simultaneousDiophantineapproximationtries to figure out
j
receiver j, ∀j ∈K can be arbitrarily close to how small the distance from Aq to Zm could be. This can
be viewed as a generalization of estimating real numbers by
lim MDd¯ N = rationals [8].
n→∞ jρND′ Theorem 5: [8] : Let fi, i=1,...,m be a nondegenerate
M d¯nK(K−1)N2 M map from an open set Ui ⊂Rdi to Rn and
lim j = d¯ =d (33)
n→∞ ρ (n+1)K(K−1)N2 ρ j j f1(x1)
.
where N istheDoFperdirectionforlargeD′.Thisproves F :U1×...×Um →Mm,n, (x1,...,xm)7−→ .. 
TheoreρmND4.′  fm(xm) 
As a special case, it is easy to see when all dk are equal, where Mm,n denotes the space of m×n real matrices.
thetotalachievableDoFis MN K.Moreover,whenM =N, Then, for ǫ>0, the set of (x1,...,xm) such that for
M+N
the achievable DoF region meets the outer bound [2]. f (x )
1 1
VI. CONCLUSIONS ANDFUTUREWORKS A= ...  (34)
 fm(xm) 
We developed a new real interference alignment scheme  
there exist infinitely many q∈Zn with
for multiple-antenna interference channel that employs joint
receiver antenna processing. The scheme utilized a result kAq−pk<kqk−mn−ǫ for some p∈Zm (35)
on simultaneous Diophantine approximation and aligned all
has zero Lebesgue measure on U ×...×U .
interferencesateachreceiveantenna.Wewereabletoprovide 1 m
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