Interference Alignment with Diversity for the 2×2 X Network with four antennas
aa r X i v : . [ c s . I T ] A p r Interference Alignment with Diversity for the × X Network with four antennas
Abhinav Ganesan and B. Sundar Rajan
Email: { abhig 88, bsrajan } @ece.iisc.ernet.in Abstract —A transmission scheme based on the Alamouti code,which we call the Li-Jafarkhani-Jafar (LJJ) scheme, was recentlyproposed for the × X Network (i.e., two-transmitter (Tx) two-receiver (Rx) X Network) with two antennas at each node. Thisscheme was claimed to achieve a sum degrees of freedom (DoF) of and also a diversity gain of two when fixed finite constellationsare employed at each Tx. Furthermore, each Tx required theknowledge of only its own channel unlike the Jafar-Shamaischeme which required global CSIT to achieve the maximumpossible sum DoF of . In this paper, we extend the LJJ schemeto the × X Network with four antennas at each node. Theproposed scheme also assumes only local channel knowledgeat each Tx. We prove that the proposed scheme achieves themaximum possible sum DoF of . In addition, we also provethat, using any fixed finite constellation with appropriate rotationat each Tx, the proposed scheme achieves a diversity gain of atleast four.
I. I
NTRODUCTION
The problem of capacity region of Gaussian interferencenetworks has been open for decades except for a few specialcases [1], [2]. In the course of pursuit of capacity region ofgeneral Gaussian interference networks, researchers have beenled into approximating their capacity regions (see for example,[3]) and their sum-capacities. A popular way of approximatingthe sum-capacity of a Gaussian interference network is usingthe concept of degrees of freedom (DoF). The sum DoF ofa Gaussian interference network is said to be d if the sum-capacity can be written as d log SN R + o ( log SN R ) [5]. A K × J MIMO X network is a Gaussian interference networkwhere each of the J receivers (Rx) require one independentmessage from each of the K transmitters (Tx). Henceforth,a K × J MIMO X network with M antennas at each nodeshall be abbreviated as ( K, J, M ) − X Network. The sumDoF of (2 , , M ) − X Network was studied in [4], [5]. In[4], it was shown that a sum DoF of ⌊ M ⌋ is achievable ina (2 , , M ) − X Network while the work in [5] shows thata sum DoF of M is achievable. Furthermore, M was alsoproven to be an outerbound on the sum DoF of (2 , , M ) − X Network [5]. The transmission scheme in [5] that achievedthis sum DoF was based on the idea of interference alignment(IA). We shall henceforth call this scheme as the Jafar-Shamaischeme.The concept of IA for
M > involved linear precodingusing a -symbol extension of the channel in such a waythat the interference subspaces at the receivers overlap whilebeing linearly independent of the desired signal subspace. Thisassumed constant channel matrices and knowledge of all the channel gains at both the transmitters (i.e., global CSIT). Thedesired signals were retrieved by simple zero-forcing.In a recent work by Li et al. [6] an IA scheme for (2 , , − X Network using the Alamouti code and appropriatechannel dependent precoding was proposed. In this scheme,each transmitter needs the knowledge of the channel fromitself to both the receivers (i.e., local CSIT) whereas, in theJafar-Shamai scheme, global CSIT is needed. This scheme,which we call the LJJ scheme, claimed to achieve the sumDoF of (2 , , − X Network which is equal to . However,[6] assumed the channel gains to be independently distributedas circularly symmetric complex Gaussian. Also, the proofof achievability of the sum DoF of (2 , , − X Network isincomplete. We present a complete proof in Section III-B ofthis paper with the assumption that the real and imaginary partsof the channel gains are distributed independently accordingto an arbitrary continuous distribution like in the Jafar-Shamaischeme. Further, the LJJ scheme also achieves a diversitygain of two with node-to-node symbol rate of complexsymbols per channel use (cspcu) where, the complex symbolsare assumed to take values from a fixed finite constellation.In this work, we extend the LJJ scheme to (2 , , − X Network using Srinath-Rajan (S-R) space-time block code(STBC) which was proposed for the asymmetric × singleuser MIMO system [7]. The S-R code possesses a repetitiveAlamouti structure upto scaling by a constant. This makes itconvenient to adapt the LJJ scheme to (2 , , − X Network.We prove that the proposed scheme achieves the sum DoF of (2 , , − X Network which is equal to . This scheme alsorequires only local CSIT like the LJJ scheme. Furthermore,under a more practical scenario of fixed finite constellationinputs, we prove that the proposed scheme achieves a diversitygain of at least four.The contributions of the paper are summarized below. • We provide a complete proof of achievability of sum DoFof by the LJJ scheme (see Theorem 3 in Section III-B). • We extend the LJJ scheme to (2 , , − X Network usingthe S-R STBC. It is proved that this scheme achievesa sum DoF of (see Theorem 5 in Section IV). Theproposed scheme requires only local CSIT while theJafar-Shamai scheme requires global CSIT to achieve thesame sum DoF. • We prove that the proposed scheme also achieves adiversity gain of at least four (see Theorem 4 in SectionIV) when fixed finite constellations are employed at thetransmitters. Simulation results show that the diversityain of the proposed scheme is strictly greater than four.The paper is organized as follows. Section II formallyintroduces the system model. A brief overview of the Jafar-Shamai scheme for (2 , , − X Network and the LJJ schemefor (2 , , − X Network along with a complete proof of thesum DoF achieved by the LJJ scheme is given in Section III.Extension of the LJJ scheme for (2 , , − X Network basedon the S-R STBC is described in Section IV. Simulation resultscomparing the proposed scheme with the Jafar-Shamai schemeand the time division multiple access (TDMA) scheme arepresented in Section V. We conclude the paper with SectionVI.
Notations:
The set of complex number is denoted by C . Thenotation CN (0 , σ ) denotes the circularly symmetric complexGaussian distribution with mean zero and variance σ . For acomplex number x , the notation x denotes the conjugate of x . The real and imaginary parts of a complex number a aredenoted by a R and a I respectively. The trace of a matrix A is denoted by tr ( A ) . For an invertible matrix A , the notation A − H denotes the hermitian of the matrix A − . The i th row, j th column element of a matrix A is denoted by a ij . The i th row and the i th column of a matrix A are denoted by A ( i, :) and A (: , i ) respectively. The Frobenius norm of a matrix A isdenoted by || A || . The identity matrix of size n × n is denotedby I n . The Kronecker product of two matrices A and B isdenoted by A ⊗ B . A diagonal matrix with the diagonal entriesgiven by a , a , · · · , a n is denoted by diag ( a , a , · · · , a n ) .The notation vec ( A ) denotes the vectorized version of thematrix A . II. S YSTEM M ODELFig. 1. System Model.
The (2 , , M ) − X Network is shown in Fig. 1. Eachtransmitter Tx- i has an independent message W ij for eachreceiver Rx- j , where i, j = 1 , . The message generated byTx- i for Rx- j is denoted by W ij . The input symbols and the output symbols over T time slots are related as Y j = r PM X i =1 H ij X i + N j (1)where, Y j ∈ C M × T denotes the output matrix at Rx- j , X i ∈ C M × T denotes the input matrix at Tx- i such that E (cid:2) tr (cid:0) XX H (cid:1)(cid:3) ≤ T M , H ij ∈ C M × M denotes the channelmatrix between Tx- i and Rx- j , N j ∈ C M × T denotes the noisematrix whose entries are i.i.d. distributed as CN (0 , . As in[5], we assume that the entries of all the channel matricesare independent and take values from arbitrary continuousprobability distribution so that they are almost surely fullrank. Specifically, for the diversity gain evaluations, we as-sume that the channel matrix entries are distributed as i.i.d. CN (0 , . The channel gains are assumed to be a constantover the transmitted codeword length. All the channel gainsare assumed to be known to both the receivers (i.e., globalCSIR), and this will not be specifically mentioned henceforth.The average power constraints at both the transmitters areassumed to be equal to P . The achievable rates and sum DoFof (2 , , M ) − X Network are defined in the conventional sense[5].III. B
ACKGROUND - J
AFAR -S HAMAI S CHEME AND
LJJS
CHEME
In the first sub-section we shall briefly review the Jafar-Shamai scheme from [5] and in the second sub-section weshall review the LJJ scheme from [6].
A. Review of Jafar-Shamai Scheme for (2 , , − X Network
The Jafar-Shamai scheme for (2 , , − X Network alignsthe interference symbols by precoding over a -symbol exten-sion of the channel, i.e., T = 3 . Each transmitter transmits complex symbols to each receiver over channel uses so thata sum DoF of is achieved. The input-output relation overa -symbol extension of the channel is given by Y ′ j = r P X i =1 H ′ ij X k =1 V ik tr (cid:0) V ik V Hik (cid:1) X ik ! + N ′ j (2)where, Y ′ j ∈ C × denotes the received symbol vector at Rx- j over channel uses, H ′ ij = H ij H ij
00 0 H ij denotes theeffective channel matrix between Tx- i and Rx- j over channeluses, V ik ∈ C × denotes the precoding matrix, X ik ∈ C × denotes the symbol vector generated by Tx- i meant for Rx- k ,and N ′ j ∈ C × denotes the Gaussian noise vector whoseentries are distributed as i.i.d. CN (0 , . The entries of X ik take values from a set such that E (cid:2) X ik X Hik (cid:3) = I . Theprecoders V ik are chosen as given below. V = E F ′ V F ′ , V = E F ′ V F ′ ,V = H ′− H ′ V , V = H ′− H ′ V We consider a complex random variable to have a continuous probabilitydistribution if its real and imaginary parts are independent and distributedaccording to some continuous distribution. here, E F ′ ∈ C × denotes a matrix whose columns arethe eigen vectors of the matrix F ′ = H ′− H ′ H ′− H ′ , V F ′ = I ⊗ [1 1 0] T , and V F ′ = I ⊗ [1 0 1] T . With the abovechoice of precoders, the interference symbols are aligned and(2) can be re-written as Y ′ = r P H ′ V X + H ′ V X + H V ( X + X )) + N ′ Y ′ = r P H ′ V X + H ′ V X (3) + H V ( X + X )) + N ′ . It is proved in [5] that the above scheme achieves a sumDoF of in the (2 , , − X Network almost surely whenthe channel matrix entries take values from a continuousprobability distribution.
B. Review of LJJ Scheme
In the LJJ transmission scheme for (2 , , − X Network,every transmitter transmits two superposed Alamouti codeswith appropriate precoding in three time slots, i.e., T = 3 .Each Alamouti code corresponds to the symbols meant foreach receiver. The transmitted symbols are given by X = r P V (cid:20) x − x x x (cid:21)| {z } X + V (cid:20) x − x x x (cid:21)| {z } X X = r P V (cid:20) x − x x x (cid:21)| {z } X + V (cid:20) x − x x x (cid:21)| {z } X , where, x kij takes values from a set such that E h(cid:12)(cid:12) x kij (cid:12)(cid:12) i = 1 .The matrices X ij , as defined above, correspond to the symbolsgenerated by Tx- i meant for Rx- j . The matrix entries x kij denote the k th symbol generated by Tx- i for Rx- j . Theprecoders V ij are chosen as V = H − q tr (cid:0) H − H − H (cid:1) , V = H − q tr (cid:0) H − H − H (cid:1) V = H − q tr (cid:0) H − H − H (cid:1) , V = H − q tr (cid:0) H − H − H (cid:1) . (4) The coefficients in the square roots above make sure that thetransmitters meet the average power constraint. Note that allthe channel matrices and the precoders are × matrices. Theabove choice of precoders and the usage of Alamouti codesconcatenated with all zero columns align the interference sym-bols while ensuring that the interference subspace is linearlyindependent of the signal subspace. We briefly describe howthis happens at Rx- . The output symbol matrix at Rx- isnow given by Y = r P H V X + r P H V X + r P (cid:20) ax + bx − ax − bx ax + bx ax + bx (cid:21) + N where, a = q tr ( H − H − H ) and b = q tr ( H − H − H ) . Let the ef-fective channel matrices corresponding to the desired symbolsfrom Tx- and Tx- to Rx- be denoted by ˆ H = H V and ˆ G = H V respectively. Define a × matrix Y ′ whosefirst, second and third columns are given by Y ′ (: ,
1) = Y (: , , Y ′ (: ,
2) = Y (: , , Y ′ (: ,
3) = Y (: , . (5) Similarly, define the matrix N ′ obtained from N . Denotethe i th rows of the × matrices Y ′ and N ′ by Y ′ ( i, :) and N ′ ( i, :) respectively, i = 1 , . The processed output symbolsat Rx- (i.e., Y ′ ) can be written as (cid:20) Y ′ T (1 , :) Y ′ T (2 , :) (cid:21)| {z } Y ′′ = r P ˆ h ˆ h ˆ g ˆ g h − ˆ h ˆ g − ˆ h − h ˆ h ˆ g ˆ g h − ˆ h ˆ g − ˆ h x x x x I I + (cid:20) N ′ T (1 , :) N ′ T (2 , :) (cid:21)| {z } N ′′ (6) where, I = ax + bx and I = ax + bx , and ˆ h ij and ˆ g ij denote the entries of the matrices ˆ H and ˆ G respectively. Notethat, when ˆ h ij and ˆ g ij are non-zero, the interference symbols I and I are aligned in a subspace linearly independent of thesignal subspace. So, pre-multiplying the matrix Y ′′ (definedin (6)) by the zero-forcing matrix given by F = −
10 0 1 0 1 00 0 0 1 0 0 (7)yields
F Y ′′ = r P ˆ h ˆ h ˆ g ˆ g ˆ h − ˆ h ˆ g − ˆ h ˆ h ˆ h ˆ g ˆ g ˆ h − ˆ h ˆ g − ˆ h | {z } R x x x x + F N ′′ . (8) Now, note that decoding the symbols in (8) is similar todecoding symbols in a two user MAC with double antennatransmitters and a double antenna receiver. Hence, [6] makesuse of the interference cancellation procedure for MAC [8]to achieve low complexity symbol-by-symbol decoding. Thisprocedure is described below.Denote the sub-matrices of R , defined in (8), by H = " ˆ h ˆ h ˆ h − ˆ h , ˜ G = (cid:20) ˆ g ˆ g ˆ g − ˆ g (cid:21) (9) ˜ H = " ˆ h ˆ h ˆ h − ˆ h , ˜ G = (cid:20) ˆ g ˆ g ˆ g − ˆ g (cid:21) . (10) Denote the first two entries and the last two entries of the × vector F Y ′′ by ˜ y and ˜ y respectively. Similarly, denote firsttwo entries and the last two entries of the × vector F N ′′ by ˜ n and ˜ n respectively. Let ˜ y = ˜ G H ˜ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ˜ G H ˜ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = r P ˜ G H ˜ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ˜ G H ˜ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | {z } ˜ H (cid:20) x x (cid:21) (11) + ˜ G H ˜ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ˜ G H ˜ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Note that the matrix ˜ H also has an Alamouti structure andhence, x and x are symbol-by-symbol decodable. Simi-larly, x k is decoded at Rx- , and x k and x k are symbol-by-symbol decodable at Rx- , for k = 1 , . The followingtheorem, given as Theorem in [6], states the diversity gainachieved for each symbol. Theorem 1: [6] A diversity gain of is achieved for x kij ,for all i, j, k .A sum DoF of is achieved in the (2 , , − X Networkwith probability one if the effective channel matrix R in (8)and a similar effective channel matrix at Rx- are full rankalmost surely. The following theorem, given as Theorem in[6], claims that matrix R is almost surely full rank. Theorem 2: [6] When the entries of H ij are i.i.d. dis-tributed as CN (0 , , the matrix R defined in (8) is almostsurely full rank.The proof given in [6] for the above theorem goes as follows.“The equivalent channel vectors for x i and x i are orthogo-nal, i.e., the first two columns of R are orthogonal to each otherand so are the last two columns of R . Further, the equivalentchannel vectors of x k (i.e., first two columns of R ) dependon the matrices H and H , while those of x k (i.e., the lasttwo columns of R ) depend on H and H . Almost surely,the equivalent channel vectors of each data stream are linearlyindependent and separable at Rx- (i.e., the matrix R is fullrank almost surely).”Note that the matrix R is full rank iff the subspaces spannedby the first two and the last two columns of R do not intersect.We find that it is not obvious from the facts mentioned in theproof of Theorem 2 in [6] that these subspaces do not intersectalmost surely. This is because the random variables in the firsttwo columns are dependent and so are the random variablesin the last two columns. So, it is not clear what distributionthe determinant of R follows or specifically whether it is continuously distributed or not. Further, note that the Jafar-Shamai scheme assured a sum DoF of when the entries ofthe channel matrices are distributed i.i.d. according to somecontinuous distribution and not necessarily CN (0 , . We nowre-state Theorem 2 and also provide a complete proof. Theorem 3:
When the entries of H ij are distributed i.i.d.according to some continuous distribution, the matrix R de-fined in (8) is almost surely full rank. Proof:
See Appendix A.We propose an extension of the LJJ scheme to (2 , , − X Network in the next section.IV. S-R STBC B
ASED T RANSMISSION S CHEME FOR (2 , , − X N ETWORK
In this section, the LJJ scheme is extended to (2 , , − X Network by exploiting a repetitive Alamouti structure (uptoscaling by a constant) in the S-R STBC. This transmissionscheme is proved to achieve the sum DoF of (2 , , − X Network, and a diversity gain of at least four when fixed finiteconstellations are used at the transmitters. The S-R STBCproposed for × single user MIMO system in [7] is givenby (12) (at the top of the next page) where, s i denotes the i th complex symbol generated by the transmitter, and θ ∈ (0 , π ) .Note that complex symbols are transmitted in channel uses.If complex symbols are transmitted from each transmitterto every receiver in channel uses in the (2 , , − X Network then, a total of complex symbols per channel useis transmitted. This is done using the S-R STBC as follows.The transmitted symbols are given by X = r P V X + V X ) X = r P V X + V X ) where, the matrices X i and X i are given in (13) and (14)respectively, for i = 1 , , and x kij take values from a setsuch that E h(cid:12)(cid:12) x kij (cid:12)(cid:12) i = 1 . The matrices X ij correspond tothe symbols generated by Tx- i meant for Rx- j . The matrixentries x kij denote the k th symbol generated by Tx- i for Rx- j .The choice of precoders V ij is the same as in the LJJ scheme,i.e., given by (4), where the channel matrices H ij are × matrices. The output symbol matrix at Rx- is given by Y = r P H V X + H V X )+ r P q tr (cid:0) H − H − H (cid:1) X + 1 q tr (cid:0) H − H − H (cid:1) X + N where, Y ∈ C × . Note that the third and the sixth columnsof V X + V X are zero. This shall be exploited forinterference cancellation as follows.Define a matrix Y ′ ∈ C × obtained by processing Y asfollows. s R + js I − s R + js I e jθ (cid:0) s R + js I (cid:1) e jθ (cid:0) − s R + js I (cid:1) s R + js I s R − js I e jθ (cid:0) s R + js I (cid:1) e jθ (cid:0) s R − js I (cid:1) e jθ (cid:0) s R + js I (cid:1) e jθ (cid:0) − s R + js I (cid:1) s R + js I − s R + js I e jθ (cid:0) s R + js I (cid:1) e jθ (cid:0) s R − js I (cid:1) s R + js I s R − js I (12) X i = x Ri + jx Ii − x Ri + jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) − x Ri + jx Ii (cid:1) x Ri + jx Ii x Ri − jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri − jx Ii (cid:1) e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) − x Ri + jx Ii (cid:1) x Ri + jx Ii − x Ri + jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri − jx Ii (cid:1) x Ri + jx Ii x Ri − jx Ii (13) X i = x Ri + jx Ii − x Ri + jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) − x Ri + jx Ii (cid:1) x Ri + jx Ii x Ri − jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri − jx Ii (cid:1) e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) − x Ri + jx Ii (cid:1) x Ri + jx Ii − x Ri + jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri − jx Ii (cid:1) x Ri + jx Ii x Ri − jx Ii (14) Y ′ (: ,
1) = Y (: , , (15) Y ′ (: ,
3) = Y (: , , (16) Y ′ (1 ,
2) = Y (1 , − Y (2 , , (17) Y ′ (2 ,
2) = Y (2 ,
2) + Y (1 , , (18) Y ′ (3 ,
2) = Y (3 , − e j θ Y (4 , , (19) Y ′ (4 ,
2) = Y (4 ,
2) + e j θ Y (3 , (20) Y ′ (1 ,
4) = Y (1 , − e j θ Y (2 , , (21) Y ′ (2 ,
4) = Y (2 ,
5) + e j θ Y (1 , , (22) Y ′ (3 ,
4) = Y (3 , − Y (4 , , (23) Y ′ (4 ,
4) = Y (4 ,
5) + Y (3 , . (24) Note that, in (15) and (16), the first and the fourth columnsof Y are retained without further processing because they areinterference free. These are interference free because the firstand fourth columns of X i are zero, for i = 1 , . In (17)-(20),the interference term associated with the second column of Y is canceled using the third column of Y . Similarly, in (21)-(24), the interference term associated with the fifth columnof Y is canceled using the sixth column of Y . Note that theconjugation and scaling of terms in the R.H.S. of (17)-(24) in-volve only the third and sixth columns of Y . This interferencecancellation procedure does not affect the desired symbolsbecause the third and sixth columns of V X + V X arezero. Note that the LJJ scheme for (2 , , − X Network alsoinvolves similar interference cancellation procedure though itwas explained through zero-forcing of aligned interference inSection III-B.Now, the matrix Y ′ can be re-written as Y ′ = H V X ′ + H V X ′ + N ′ (25)where, X ′ i is given by (26) (at the top of the next page), for i = 1 , , and N ′ ∈ C × is a Gaussian noise matrix whosefirst and third column entries are distributed as i.i.d. CN (0 , while the second and fourth column entries are distributed as i.i.d. CN (0 , . The matrices X ′ i is defined in a similar wayas X ′ i , for i = 1 , .We now proceed to evaluate the diversity gain achievedby the above scheme when fixed finite constellation inputsare used at the transmitters. Towards that end, we have thefollowing definition from [10]. Definition 1: [10] The Coordinate Product Distance (CPD)between any two signal points u = u R + ju I and v = v R + jv I ,for u = v , in a finite constellation S is defined as CP D ( u, v ) = (cid:12)(cid:12) u R − v R (cid:12)(cid:12) (cid:12)(cid:12) u I − v I (cid:12)(cid:12) and the minimum of this value among all possible pairs isdefined as the CPD of S .We assume that each symbol x kij takes values from afinite constellation whose CPD is non-zero, for all i, j, k . Asobserved in [10], if a finite constellation has a zero CPD,it can always be rotated appropriately so that the resultingconstellation has a non-zero CPD. Now, define the differencematrix △ X ′ ijk ,k by △ X ′ ijk ,k = X ′ ijk − X ′ ijk where, X ′ ijk and X ′ ij k denote two different realizations (i.e., k = k ) of the matrix X ′ ij .The following lemma shall be useful in establishing thediversity gain of the proposed scheme. Lemma 1:
There exists θ such that the difference matrix △ X ′ ijk ,k is full rank for all k = k and for all i, j . Proof:
See Appendix B.Henceforth, we shall assume that θ is chosen so that thedifference matrix △ X ′ ijk ,k is full rank for all k = k andfor all i, j . We shall assume that ML Decoding of X ′ and X ′ is done from (25) and ML Decoding of X ′ and X ′ is donefrom a similar processed received symbol matrix at Rx- . Thediversity gain of the proposed scheme can be obtained fromthe following theorem. ′ i = x Ri + jx Ii − x Ri + jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) − x Ri + jx Ii (cid:1) x Ri + jx Ii x Ri − jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri − jx Ii (cid:1) e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) − x Ri + jx Ii (cid:1) x Ri + jx Ii − x Ri + jx Ii e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri − jx Ii (cid:1) x Ri + jx Ii x Ri − jx Ii (26) Theorem 4:
The average pair-wise error probability P e forthe pairs of codewords (cid:16) X ′ k , X ′ k (cid:17) and (cid:16) X ′ k ′ , X ′ k ′ (cid:17) isupper bounded as P e (cid:16)(cid:16) X ′ k , X ′ k (cid:17) → (cid:16) X ′ k ′ , X ′ k ′ (cid:17)(cid:17) ≤ cP − . for some constant c > . Proof:
See Appendix C.Hence, using the union bound on the average probability oferror given that a particular symbol is transmitted and usingTheorem 4, we obtain that ML decoding of X ′ and X ′ from(25) gives a diversity gain of four.We shall now evaluate the DoF achievable using the pro-posed scheme. For the DoF evaluation we do not assume anyrestriction on the value of θ . Theorem 5:
The proposed scheme can achieve a node tonode DoF of and hence, a sum DoF of with symbol-by-symbol decoding. Proof:
See Appendix D.Thus, the proposed scheme achieves the sum DoF of (2 , , − X Network using local CSIT while the Jafar-Shamaischeme requires global CSIT.In the following section, we shall present some simulationresults comparing the probability of error performance of theproposed scheme with other schemes using finite constellationinputs. V. S
IMULATION R ESULTS
In this section, we present some simulation results thatinclude comparing the error performance of the proposedscheme for (2 , , − X Network with that of a TDMAscheme, and the Jafar-Shamai scheme. In the TDMA scheme,the channel is used half the time by one transmitter while theother switches off. When Tx- i is switched on, half the time isallocated to transmit to each of the receivers. To ensure a faircomparison, we assume TDMA with CSIT, and the symbolvectors meant to be transmitted are precoded using the fulldiversity precoders proposed in [13] for single user MIMOsystem with square QAM constellation inputs.We shall briefly review the precoding technique proposed in[13] for single user MIMO system. We shall call the precoderas S-R Precoder. Consider a single user MIMO system with M transmit and M receive antennas. Full CSIT and CSIR areassumed. The channel is assumed to be quasi-static and all thechannel gains are distributed as i.i.d. CN (0 , . The channelmodel is given by Y = r SN RM HQX + N (27) where, Y ∈ C M × denotes the output symbol vector, H ∈ C M × M denotes the channel matrix, Q ∈ C M × M denotesthe precoder matrix, X ∈ C M × denotes the transmittedsymbol vector, and N ∈ C M × denotes the Gaussian noisevector with the entries distributed as i.i.d. CN (0 , . Thesignal to noise ratio at each receive antenna is denoted by SN R and E (cid:2) X H X (cid:3) = M . The transmitted symbol vectoris given by X = [ x x · · · x M ] T where the symbols x i take values from a square QAM whose average power istaken to be equal to one, for i = 1 , , · · · , M . Let thesingular value decomposition of H be given by H = U DV H where, U and V are unitary matrices of size M × M , and D = diag ( λ ( H ) , λ ( H ) , · · · , λ M ( H )) with λ ( H ) ≥ λ ( H ) ≥· · · ≥ λ M ( H ) .The precoding matrix Q is given by Q = V P where, P ∈ C M × M . Multiplying the received vector Y by U H we have, Y ′ = U H Y = r SN RM DP X + N ′ where, N ′ = U H N has the same distribution as N . The matrix P for M = 4 is given by P (1 ,
1) 0 0 P (1 , P (1 , P (1 ,
2) 00 P (2 , P (2 ,
2) 0 P (2 ,
1) 0 0 P (2 , where, P i ( j.k ) denotes the j th row, k th column element of thematrix P i given by P i = q τ i (cid:20) cos ψ i cos θ i − cos ψ i sin θ i sin ψ i sin θ i sin ψ i cos θ i (cid:21) , for i = 1 , . The values of τ i , ψ i , and θ i are selected based on the matrix D . The selection of values of these variables is involved andhence, the readers are referred to [13] for details. Similarly,for M = 2 , the matrix P is given by P = q τ (cid:20) cos ψ cos θ − cos ψ sin θ sin ψ sin θ sin ψ cos θ (cid:21) . Among the class of precoders having a real matrix P , theabove choice of P was shown to be approximately optimal inminimizing the ML metric given by min X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y ′ − r SN RM DP X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (28)Further, the precoders were proven to achieve full diversity.We first compare the error probability performance of theLJJ scheme with the TDMA scheme using S-R Precoder inthe (2 , , − X Network. Such a comparison was not donen [6]. The value of
SN R in the S-R precoder is set as P toaccount for time sharing. In the LJJ scheme we perform MLdecoding of the symbols directly from the processed receivesymbol vector F Y ′′ given in (8) rather than symbol-by-symboldecoding as described in Section III-B. The transmitted sym-bols in the LJJ scheme are decoded using the sphere decoder[14]. Since each transmitter achieves a rate of cspcu and cspcu in the LJJ scheme and the TDMA scheme respectively,we use -QAM constellation input for the LJJ scheme and -QAM constellation input for the TDMA scheme usingS-R Precoder so that the spectral efficiency achieved is bits/sec/Hz per transmitter. Fig. 2 compares the Word ErrorProbability (WEP) of the LJJ scheme with -QAM inputwith that of the TDMA scheme using S-R Precoder with -QAM input. The TDMA scheme using S-R Precoder clearlyoutperforms the LJJ scheme inspite of the higher constellationsize because the former has a diversity gain of while thelatter has a diversity gain that is strictly greater than butlesser than . Thus, the sum DoF optimality of the LJJ schemedoes not translate to a better WEP performance compared tothe TDMA scheme with finite constellation inputs, even at lowvalues of P .
16 18 20 22 24 26 28 30 3210 −6 −5 −4 −3 −2 −1 P in dB W EP LJJ Scheme with ML Decoding and 8−QAM InputTDMA usingS−R Precoder with 16−QAM Input c P −2 c P −3 Fig. 2. WEP of LJJ scheme with -QAM input versus WEP of TDMA usingS-R Precoder with -QAM input at a spectral efficiency of bits/sec/Hz pertransmitter. A similar result is observed with the proposed scheme for (2 , , − X Network which we term as the modified S-RSTBC scheme. Here, the TDMA scheme achieves a rate of cspcu per transmitter. Sphere decoder is used to decodethe transmitted symbols from (25) in the modified S-R STBCscheme. We simulate the TDMA scheme using S-R Precoderwith -QAM input and the modified S-R STBC scheme with Here, we take -QAM constellation input to be the Cartesian product of a -PAM constellation that constitutes the real part and a -PAM constellationthat constitutes the imaginary part. -QAM input so that the achieved spectral efficiency is bits/sec/Hz per transmitter. We have set θ = π in the modifiedS-R STBC scheme, and the constellations are rotated by anangle φ = tan − (2)2 to ensure a non-zero CPD [10]. It wasshown in [7] that the difference matrices of the S-R STBCare full rank with θ = π and φ = tan − (2)2 when -QAMinputs are used. Since, the -QAM constellation is a subsetof the -QAM constellation, △ X ′ ijk ,k is full rank for all k , k and for all i, j . Hence, by Theorem 4, a diversity offour is assured for the modified S-R STBC scheme. It canbe observed from Fig. 3 that the TDMA scheme using S-R Precoder with -QAM input outperforms the modified S-R STBC scheme with -QAM input. Hence, like in the LJJscheme, the sum DoF superiority of the modified S-R STBCscheme for (2 , , − X Network over the TDMA schemedoesn’t translate to superiority in terms of WEP when finiteconstellation inputs are used, even at low values of P . Notethat the diversity gain offered by the TDMA scheme using S-R Precoder is whereas the modified S-R STBC scheme hasan assured diversity gain of only . Fig. 3 however shows thatthe diversity gain offered by the modified S-R STBC schemeis strictly greater than .
14 16 18 20 22 24 2610 −5 −4 −3 −2 −1 P in dB W EP Jafar−Shamai Scheme with 8−QAM Input Trivial Alamouti Repetitionwith 8−QAM InputTDMA using S−R Precoder with 16−QAM Input Modified S−R STBC with 8−QAM Input c P −4 Fig. 3. WEP of modified S-R STBC scheme with -QAM input versusWEP of TDMA using S-R Precoder with -QAM at a spectral efficiency of bits/sec/Hz per transmitter. The precoding technique in [13] however applies only tosquare QAM constellations which can be written as a Cartesianproduct of two PAM constellations. Also, optimizing the pre-coder to minimize (28) for a single user MIMO system whileassuring a particular diversity gain for arbitrary constellationsis an open problem. In such a scenario, there is no guaranteethat TDMA with some precoding would surely outperformthe LJJ scheme for (2 , , − X Network or the modified S-RSTBC scheme for (2 , , − X Network at all values of P .Moreover, the TDMA scheme achieves integer rates of cspcund cspcu per transmitter in the (2 , , − X Network andthe (2 , , − X Network respectively whereas the LJJ schemeand the modified S-R STBC scheme achieve fractional rates of cspcu and cspcu per transmitter respectively. So, equatingthe spectral efficiencies for WEP comparison requires the useof higher QAM sizes than what are used in Fig. 2 and Fig. 3.Further, the decoding complexity, even with sphere decoding,is enormous for higher constellation sizes for the LJJ schemeand the modified S-R STBC scheme. Hence, it is not feasibleto compare the WEP performance of the LJJ scheme and themodified S-R STBC scheme with the TDMA scheme usingS-R Precoding with higher QAM sizes.We now compare the WEP performance of the modified S-R STBC scheme with the Jafar-Shamai scheme. We shall alsoobserve the importance of selection of θ so that △ X ′ ijk ,k is full rank for all k , k and for all i, j . Let us call thescheme that uses θ = 0 and φ = tan − (2)2 as the trivialAlamouti repetition scheme. It is easy to observe that, with thesame constellation used for all the symbols and when θ = 0 , △ X ′ ijk ,k is not full rank for some k , k , for all i, j . Thus,Theorem 4 is not applicable for this case. For convenience,the scheme that uses θ = π and φ = tan − (2)2 is termed asthe modified S-R STBC scheme. In the Jafar-Shamai scheme,MAP decoding of the desired symbols from (3) reduces to MLdecoding of all the symbols at high values of P [15], i.e., ( ˆ X , ˆ X ) = arg min X ,X ,X + X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y ′ − r P (cid:0) H ′ V X + H ′ V X (cid:1) + H ′ V ( X + X ) (cid:12)(cid:12)(cid:12)(cid:12) ( ˆ X , ˆ X ) = arg min X ,X ,X + X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y ′ − r P (cid:0) H ′ V X + H ′ V X (cid:1) + H ′ V ( X + X ) (cid:12)(cid:12)(cid:12)(cid:12) . Hence, as noted in [15] sphere decoder can be used whenQAM constellations are employed. Fig. 3 and Fig. 4 comparethe WEP of the modified S-R STBC scheme with that ofthe trivial Alamouti repetition scheme and the Jafar-Shamaischeme, using -QAM inputs and -QAM inputs respectively.It can observed from Fig. 3 and Fig. 4 that the modifiedS-R STBC scheme clearly outperforms the trivial Alamoutirepetition scheme and the Jafar-Shamai scheme.In all the figures, the modified S-R scheme is found to offera diversity gain that is strictly greater than . For additionalclarity, the modified S-R scheme is plotted with BPSK inputsin Fig. 5 which also shows that the diversity gain is strictlygreater than . Intuitively, the modified S-R scheme achievesfull receive diversity while the transmit diversity is affectedbecause of precoding.VI. C ONCLUSION
A new transmission scheme based on the S-R STBC wasproposed for the (2 , , − X Network as an extension ofthe LJJ scheme for the (2 , , − X Network. The proposedtransmission scheme was proven to achieve the sum DoF ofthe (2 , , − X Network which is equal to . In comparison −6 −5 −4 −3 −2 −1 P in dB W EP Trivial Alamouti Repetition Schemewith 4−QAM Input Jafar−Shamai Schemewith 4−QAM InputModified S−R STBC with 4−QAM Input c P −4 Fig. 4. WEP of modified S-R STBC scheme versus Trivial AlamoutiRepetition and Jafar-Shamai scheme with -QAM input at a spectral efficiencyof bits/sec/Hz per transmitter. −5 −4 −3 −2 −1 P in dB W EP Modified S−R Code with 4−QAM Input c P −4 Fig. 5. WEP of modified S-R STBC with BPSK input at a spectral efficiencyof bits/sec/Hz per transmitter. with the Jafar-Shamai scheme, the proposed scheme hasreduced CSIT requirements. Moreover, the proposed schemewas proven to achieve a diversity gain of four when finiteconstellation inputs are used. Simulation results confirmedthat the proposed scheme performs better in terms of errorprobability when compared with the Jafar Shamai scheme.An interesting question that remains to be addressed is- what is the maximum diversity gain achievable at a sumrate of cspcu and cspcu in the (2 , , − X Networkand (2 , , − X Network respectively? Another interestingirection of research is to identify similar schemes for othervalues of M so that the sum DoF of (2 , , M ) − X Networkcan be achieved with lesser CSIT requirement compared tothe Jafar-Shamai scheme along with full receive diversity gainwhen finite constellation inputs are used.A
PPENDIX AP ROOF OF T HEOREM Proof:
We do not attempt a direct proof for showing thatthe matrix R is full rank as the determinant expression iscomplicated. Instead, we shall prove it using some informationtheoretic inequalities and exploit the interference cancellationprocedure given in (11). First, note that the entries of the noisevector F N ′′ in (8) are i.i.d. with the first and last entries beingdistributed as CN (0 , , and the second and third entries beingdistributed as CN (0 , . We now consider a modified systemmodel where, a Gaussian noise vector N ′′′ is added to (8) sothat the entries of the effective noise vector in (8) shall bedistributed as i.i.d. CN (0 , . Henceforth in this proof, (8) isconsidered to be an equation with this extra noise N ′′′ added.The vector ˜ y in (11) is also assumed to be derived from thevector in (8) with the noise N ′′′ added. Define the vector ˜ z ,similar to ˜ y in (11), as ˜ z = ˜ H H ˜ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ H (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ˜ H H ˜ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ H (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = r P ˜ H H ˜ G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ H (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ˜ H H ˜ G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ H (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | {z } ˜ G (cid:20) x x (cid:21) (29) + ˜ H H ˜ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ H (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ˜ H H ˜ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ H (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) We now have the following useful lemmas.
Lemma 2:
The vector norms (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ H (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) arealmost surely non-zero. Proof:
We shall prove the statement only for (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and the proof for (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ H (1 , :) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is similar. To prove this, it issufficient to prove that ˆ g is non-zero almost surely. Notethat ˆ g is given by ˆ g = h v + h v . Conditioned on the random matrix V and the random vari-able h , if v is non-zero then, ˆ g is non-zero almostsurely. This is because the continuously distributed randomvariable h is independent of V and h , and v justscales h while h v shifts the mean. Thus, if v isalmost surely non-zero then, ˆ g is also non-zero almost surely.This is explained as follows. Suppose that v is zero withsome non-zero probability, and consider such events. Since V = H − q tr ( H − H − H ) , we have r tr (cid:16) H − H − H (cid:17) (cid:20) h h h h (cid:21) " h ( − h ( − h ( − = (cid:20) (cid:21) (30) where, h ( − ij denotes the ij th element of H − . Clearly, tr (cid:0) H − H − H (cid:1) is non-zero almost surely because tr (cid:0) H − H − H (cid:1) = 0 would require all the entries of H − to be equal to zero. From (30) we have, h h ( − = 1 q tr (cid:0) H − H − H (cid:1) , and h h ( − = 0 . This necessitates that h = 0 as h = 0 almost surely.However, h = 0 almost surely. Thus, v cannot be equalto zero with non-zero probability. Hence, ˆ g is also non-zeroalmost surely. Lemma 3:
If at least one of the entries in both the matrices ˜ H (defined in (11)) and ˜ G (defined in (29)) are non-zero then,the matrix R is full rank. Proof:
Note that ˜ H and ˜ G are Alamouti matrices. If atleast one of the entries in both these matrices are non-zerothen, both the matrices are full rank. Using chain rule formutual information and data processing inequality, for anyfixed value of channel matrices, we have I (cid:2) x , x , x , x ; F Y ′′ (cid:3) = I (cid:2) x , x ; F Y ′′ (cid:3) + I (cid:2) x , x ; F Y ′′ | x , x (cid:3) ≥ I (cid:2) x , x ; ˜ y (cid:3) + I (cid:2) x , x ; ˜ z | x , x (cid:3) (31) = I (cid:2) x , x ; ˜ y (cid:3) + I (cid:2) x , x ; ˜ z (cid:3) . Assume that the symbols x , x , x , and x are dis-tributed as i.i.d. CN (0 , . Note that the covariance matrixof the noise vectors ˜ G H ˜ n || ˜ G (1 , :) || − ˜ G H ˜ n || ˜ G (1 , :) || and ˜ H H ˜ n || ˜ H (1 , :) || − ˜ H H ˜ n || ˜ H (1 , :) || are given by (cid:18) || ˜ G (1 , :) || + || ˜ G (1 , :) || (cid:19) I and (cid:18) || ˜ H (1 , :) || + || ˜ H (1 , :) || (cid:19) I respectively. From Lemma 2,these covariance matrices are well defined, invertible andhence, can be whitened. Now, if ˜ H and ˜ G are full rank then,following exactly the same steps in Section . of [9] wehave , I (cid:2) x , x ; ˜ y (cid:3) = 2 log ( P ) + o ( log ( P )) , and I (cid:2) x , x ; ˜ z (cid:3) = 2 log ( P ) + o ( log ( P )) . (32)Suppose that the matrix R is not full rank. Then, followingthe same steps in Section . of [9] we have, I (cid:2) x , x , x , x ; F Y ′′ (cid:3) = d log ( P ) + o ( log ( P )) (33)where, d = rank ( R ) is strictly less than . However,from (31) and (32) we have, I (cid:2) x , x , x , x ; F Y ′′ (cid:3) ≥ log ( P ) + o ( log ( P )) . This contradicts (33) which states that I (cid:2) x , x , x , x ; F Y ′′ (cid:3) grows as d log ( P ) , where d < .Hence, the matrix R is full rank.Lemma 3 states that, in order to prove Theorem 3, it issufficient to show that both the matrices ˜ H and ˜ G contain at The effective channel matrices used while following the steps in Section . of [9] should be Σ − ˜ H and Σ − ˜ G , where Σ and Σ are the covariancematrices of the noise vectors associated with ˜ H and ˜ G respectively. east one non-zero entry almost surely. We shall prove thisstatement only for ˜ H and the proof for ˜ G is similar.Since ˜ G is an Alamouti matrix, its columns form a basis forthe two dimensional vector space C over the field of complexnumbers. Hence, the first column of ˜ H can be written as alinear combination of the columns of ˜ G . The entries of thefirst column of ˜ G H ˜ H are equal to the dot product of the twocolumns of ˜ G with the first column of ˜ H . Hence, the firstcolumn of ˜ G H ˜ H is a non-zero vector iff ˜ G and ˜ H are bothnon-zero matrices. From Lemma 2, this is true almost surely.Let ˜ G H ˜ H = (cid:20) a bb − a. (cid:21) where, a = ˆ g ˆ h + ˆ g ˆ h , and b = ˆ g ˆ h − ˆ g ˆ h . Since thefirst column of ˜ G H ˜ H is a non-zero vector almost surely, oneof the following must be true almost surely: (1) a = 0 , b = 0 , (2) a = 0 , b = 0 , or (3) a = 0 , b = 0 . We now considerthe case a = 0 , b = 0 to prove that ˜ H contains at least onenon-zero entry almost surely.Since ˆ H = H V , we have a = ˆ g ( h v + h v ) + ˆ g (cid:0) h v + h v (cid:1) = h R (cid:0) ˆ g v + ˆ g v (cid:1) + jh I (cid:0) ˆ g v − ˆ g v (cid:1) + h R (cid:0) ˆ g v + ˆ g v (cid:1) + jh I (cid:0) ˆ g v − ˆ g v (cid:1) . (34) Clearly, if a = 0 then, at least one among the coefficients of h R , h I , h R , h I in (34) is non-zero. Without loss ofgenerality, consider the coefficient of h R to be non-zero.Now, let ˜ G H ˜ H = (cid:20) c dd − c (cid:21) where, c = ˆ g ˆ h + ˆ g ˆ h , and d = ˆ g ˆ h − ˆ g ˆ h . Substi-tuting for ˆ h and ˆ h , c can be written as c = h R (cid:0) ˆ g v + ˆ g v (cid:1) + jh I (cid:0) ˆ g v − ˆ g v (cid:1) + h R (cid:0) ˆ g v + ˆ g v (cid:1) + jh I (cid:0) ˆ g v − ˆ g v (cid:1) . (35) The first row, first column entry of ˜ H is given by a || ˜ G (1 , :) || − c || ˜ G (1 , :) || . Note that a depends on the random variable h R while c depends on another independent set of random vari-ables h R , h I , h R , and h I . Since h R is continuouslydistributed and independent of other random variables involvedin (34) and (35), a || ˜ G (1 , :) || − c || ˜ G (1 , :) || is non-zero almostsurely. Hence, the first row, first column entry of ˜ H is non-zero almost surely conditioned on the fact that a = 0 . Similarlyit can be proved for the other cases, i.e., a = 0 , b = 0 , and a = 0 , b = 0 , that at least one entry of ˜ H is non-zero almostsurely. The proof that at least one entry of ˜ G is non-zero Note that the set of Alamouti matrices are closed with respect to matrixmultiplication [8]. almost surely is similar to that for ˜ H . Thus, at least one entryof the matrices ˜ H and ˜ G are non-zero almost surely. Hence,from Lemma 3, the matrix R is also full rank.A PPENDIX BP ROOF OF L EMMA Proof:
We shall prove the statement for △ X ′ (i.e., i = j = 1 ) and the proof for other △ X ij are similar. Definethe sub-matrices of X ′ by A = (cid:20) x Ri + jx Ii − x Ri + jx Ii x Ri + jx Ii x Ri − jx Ii (cid:21) B = (cid:20) e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) − x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri − jx Ii (cid:1) (cid:21) C = (cid:20) e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) − x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri + jx Ii (cid:1) e jθ (cid:0) x Ri − jx Ii (cid:1) (cid:21) D = (cid:20) x Ri + jx Ii − x Ri + jx Ii x Ri + jx Ii x Ri − jx Ii (cid:21) so that X ′ = (cid:20) A BC D (cid:21) . Now, consider the difference matrices △ X ′ such that △ A = , △ B = , △ C = , and △ D = .The determinant of △ X ′ can be written as (cid:12)(cid:12) △ X ′ (cid:12)(cid:12) = |△ A | (cid:12)(cid:12) △ D − △ C △ A − △ B (cid:12)(cid:12) (37) Denote the entries of △ A , △ B , △ C , and △ D by △ A = (cid:20) a − a a a (cid:21) , △ B = e jθ (cid:20) a − a a a (cid:21) △ C = e jθ (cid:20) a − a a a (cid:21) , △ D = (cid:20) a − a a a (cid:21) . Now, we have △ A − = | a | + | a | (cid:20) a − a a a (cid:21) , and the productmatrix △ C △ A − △ B is given by (36) (at the top of the nextpage). Note that the product matrix △ C △ A − △ B cannot be azero matrix because each matrix in the product is an Alamoutimatrix.Clearly, |△ A | 6 = 0 . From (37), for |△ X ′ | to be non-zero,there must exist θ such that (cid:12)(cid:12) △ D − △ C △ A − △ B (cid:12)(cid:12) is non-zero.We now prove the existence of such a θ . Denote the elementsof the product matrix △ C △ A − △ B by e j θ (cid:20) a − bb a (cid:21) . We nowhave (cid:12)(cid:12) △ D − △ C △ A − △ B (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) a − e j θ a − a + e j θ ba − e j θ b a − e j θ a (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = | a | + | a | − e j θ (cid:16) aa + aa + ba + ba (cid:17) + e j θ (cid:0) | a | + | b | (cid:1) . The above equation is quadratic in e jθ since △ C △ A − △ B = . Therefore, |△ X ′ | can be equal to zero for at most twodistinct values of e jθ . Since there are infinite possible choicesfor e jθ while there are only a finite number of differencematrices, there always exists θ such that (cid:12)(cid:12)(cid:12) △ X ′ k ,k (cid:12)(cid:12)(cid:12) = 0 , forall k , k .Now, consider the difference matrices △ X ′ k ,k such thatat least one among the difference sub-matrices △ A , △ B , △ C , We have suppressed the superscript k , k for convenience. C △ A − △ B = e j θ | a | + | a | (cid:20) a a a − a a a − a a a − a a a − a a a − a a a − a a a + a a a a a a + a a a + a a a − a a a a a a − a a a − a a a − a a a (cid:21) (36)and △ D is a zero matrix, for k = k . Since we assumedthat each symbol x k takes values from finite constellationswhose CPD is non-zero, △ A = 0 iff △ D = 0 , and △ B = 0 iff △ C = 0 [10]. If △ A = △ D = 0 then, △ X ′ k ,k is fullrank as k = k implies that △ B = 0 , and △ C = 0 . Similarly △ X ′ k ,k is full rank when △ B = △ C = 0 , for k = k .A PPENDIX CP ROOF OF T HEOREM Proof:
Consider a modified system where a Gaussiannoise matrix is added to (25) so that the entries of the effectivenoise matrix in (25) are distributed as i.i.d. CN (0 , . Theaverage pair-wise error probability for this modified system isgiven by P e (cid:16)(cid:16) X ′ k , X ′ k (cid:17) → (cid:16) X ′ k ′ , X ′ k ′ (cid:17)(cid:17) = E (cid:20) Q (cid:18) P ′ q || H V △ X + H V △ X || / (cid:19)(cid:21) (38) where, △ X = X ′ k − X ′ k ′ , △ X = X ′ k − X ′ k ′ , and P ′ = P . Note that either △ X = 0 , △ X = 0 or △ X =0 , △ X = 0 or △ X = 0 , △ X = 0 . We shall prove thestatement of the theorem only for the case △ X = 0 , and theproof for the rest of the cases are similar. The Frobenius normin (38) can be re-written as || H V △ X + H V △ X || = (cid:16) △ X T V T ⊗ I (cid:17) vec ( H ) + (cid:16) △ X T V T ⊗ I (cid:17) vec ( H ) | {z } H ′ H × h(cid:16) △ X T V T ⊗ I (cid:17) vec ( H ) + (cid:16) △ X T V T ⊗ I (cid:17) vec ( H ) i . (39) Note that, conditioned on H and H , the vector H ′ definedin (39) is a Gaussian vector with mean zero and covariancematrix K given by K = (40) (cid:16) △ X T V T (cid:17) (cid:16) △ X T V T (cid:17) H + (cid:16) △ X T V T (cid:17) (cid:16) △ X T V T (cid:17) H | {z } K ′ ⊗ I . In other words, when the successive elements of H ′ aregrouped in blocks of four entries each, the blocks are dis-tributed i.i.d. as Gaussian matrix with zero mean and covari-ance matrix given by K ′ which is defined in the R.H.S of(40). Since K ′ is a positive semi-definite Hermitian matrix, letthe eigen decomposition of the matrix K ′ be given by K ′ = U Λ U H where, U is a × unitary matrix formed by the eigenvectors of K ′ , and Λ = diag ( λ ( K ′ ) , λ ( K ′ ) , λ ( K ′ ) , λ ( K ′ )) denotes the matrix whose diagonal entries are ordered eigenvalues of K ′ with λ ( K ′ ) ≥ λ ( K ′ ) ≥ λ ( K ′ ) ≥ λ ( K ′ ) ≥ .Denote a square-root matrix of K ′ by K ′ , i.e., K ′ = K ′ K ′ H where, K ′ = U Λ . The vector H ′ is now sta-tistically equivalent to the following vector H ′′ = K ′ H K ′ H K ′ H K ′ H where, H i ∈ C × , i = 1 , , , , are Gaussian vectorswhose entries are distributed as i.i.d. CN (0 , . Now, (38)can be successively re-written as in (41)-(47) (given at thetop of the next page) where, (42) follows from the statisticalequivalence between H ′ and H ′′ , (43) follows from the factthat || A || = tr ( A H A ) , and (44) follows from the definitionof K ′ . Now, define K ′ = (cid:0) △ X T V T (cid:1) (cid:0) △ X T V T (cid:1) H and K ′ = (cid:0) △ X T V T (cid:1) (cid:0) △ X T V T (cid:1) H so that K ′ = K ′ + K ′ . Let λ j ( K ′ ) denote the eigen values of K ′ in non-increasing orderfrom j = 1 to j = 4 . Using Weyl’s inequalities (seeSection III.2, pp. of [11]), we have λ j ( K ′ ) ≤ λ j ( K ′ ) , j = 1 , , , . Thus, we have the inequality (46) from (45)where, H i ( j ) denotes the j th entry of the vector H i . Let K ′ = U Λ U H denote the eigen decomposition of K ′ where, Λ = diag ( λ ( K ′ ) , λ ( K ′ ) , λ ( K ′ ) , λ ( K ′ )) , and U is aunitary matrix composed of eigen vectors of K ′ . Equation (47)follows from the fact that the argument inside the Q-functionin (46) is independent of H . Let the singular value decompo-sition of △ X T V T be given by △ X T V T = U Λ V H . Notethat △ X T V T is a square root matrix of K ′ and hence, weshall denote this by K ′ . Now, (48) follows from the fact thatthe distribution of H ′ i is invariant to multiplication by the uni-tary matrix V , and using straight-forward simplifications weobtain (51). Now, let the eigen decomposition of △ X △ X H be given by △ X △ X H = U △ X Λ △ X U H △ X where, Λ △ X denotes the eigen value matrix whose eigen values in non-increasing order are given by λ j ( △ X ) , j = 1 , , , . Notethat λ ( △ X ) > as θ was chosen such that △ X isfull rank. Now, substituting this eigen decomposition in (51)we have (52). The inequality (53) follows from the fact that λ ( △ X ) is the minimum eigen value of △ X , and (54) fol-lows from V being equal to H − q tr ( H − H − H ) and the fact that thedistribution of V is invariant to multiplication by the unitarymatrix U △ X (because H is Gaussian distributed). Usingthe eigen decomposition of (cid:0) V T (cid:1) H V = U V Λ V U V Weyl’s inequalities relate the eigen values of sum two of Hermitianmatrices with the eigen values of the individual matrices. (cid:20) Q (cid:18)q P ′ || H V △ X + H V △ X || / (cid:19)(cid:21) = E H ,H (cid:20) E H ,H | H ,H (cid:20) Q (cid:18)q P ′ || H V △ X + H V △ X || / (cid:19)(cid:21)(cid:21) (41) = E H ,H E H ′′ | H ,H Q s P ′ H ′′ H H ′′ = E H ,H E H ,H ,H ,H | H ,H Q vuut P ′ P i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ′ H i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (42) = E H ,H E H ,H ,H ,H | H ,H Q vuuut P ′ P i =1 tr (cid:18) H Hi K ′ H K ′ H i (cid:19) (43) = E H ,H E H ,H ,H ,H | H ,H Q s P ′ P i =1 tr (cid:0) H Hi Λ H i (cid:1) (44) = E H ,H E H ′ ,H ′ ,H ′ ,H ′ | H ,H Q s P ′ P i =1 P j =1 λ j ( K ′ ) | H i ( j ) | (45) ≤ E H ,H E H ′ ,H ′ ,H ′ ,H ′ | H ,H Q s P ′ P i =1 P j =1 λ j ( K ′ ) | H i ( j ) | (46) = E H E H ,H ,H ,H | H Q s P ′ P i =1 tr (cid:0) H Hi Λ H i (cid:1) (47) = E H E H ,H ,H ,H | H Q vuut P ′ P i =1 tr (cid:16)(cid:0) V H H i (cid:1) H Λ (cid:0) V H H i (cid:1)(cid:17) (48) = E H E H ,H ,H ,H | H Q vuut P ′ P i =1 tr (cid:16) H Hi (cid:16) V Λ U H (cid:17) (cid:16) U Λ V H (cid:17) H i (cid:17) (49) = E H E H ,H ,H ,H | H Q vuut P ′ P i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ′ H i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (50) = E H E H ,H ,H ,H | H Q s P ′ P i =1 H Hi V T H (cid:0) △ X △ X H (cid:1) T V T H i (51) = E H E H ,H ,H ,H | H Q vuut P ′ P i =1 H Hi (cid:16)(cid:0) V U △ X (cid:1) T (cid:17) H Λ △ X (cid:0) V U △ X (cid:1) T H i (52) ≤ E H E H ,H ,H ,H | H Q vuut P ′ λ ( △ X ) P i =1 H Hi (cid:16)(cid:0) V U △ X (cid:1) T (cid:17) H (cid:0) V U △ X (cid:1) T H i (53) = E H E H ,H ,H ,H | H Q s P ′ λ ( △ X ) P i =1 H Hi (cid:0) V T (cid:1) H V T H i (54) = E H E H ,H ,H ,H | H Q s P ′ λ ( △ X ) P i =1 H Hi U V Λ V U HV H i (55) = E H E H ,H ,H ,H | H Q s P ′ λ ( △ X ) P i =1 H Hi U V Λ V U HV H i (56) = E H E H ,H ,H ,H | H Q vuut P ′ λ ( △ X ) P i =1 (cid:16) U HV H i (cid:17) H Λ V U HV H i (57) ( a ) ≤ E H Q j =1 (cid:16) Pλ ( △ X ) λ j ( V )8 (cid:17) ( b ) < (cid:16) Pλ ( △ X )32 (cid:17) c ) ≈ cP − (58) nd some straight-forward techniques involved in evaluatingdiversity as in [12], we obtain (58) ( a ) . Now, note that theeigen values of V are given by λ j ( V ) = λ − j ( H ) P j =1 1 λ j ( H ) where, λ j ( H ) denote the eigen values of H in non-increasing order from j = 1 to j = 4 . Thus, λ j ( V ) canbe lower bounded as λ j ( V ) ≥ λ − j ( H ) P j =1 1 λ ( H ) = λ ( H )4 λ − j ( H ) . For j = 1 , the above lowerbound is equal to , and for j = 2 , , the above lowerbound is in turn trivially lower-bounded by . Hence, we obtain the inequality in (58) ( b ) , andthe approximation in (58) ( c ) holds good at high values of P ,where the constant c = λ ( △ X ) .A PPENDIX DP ROOF OF T HEOREM Proof:
We shall employ an interference cancellation pro-cedure similar to that used in the LJJ scheme in Section III-Bto achieve symbol-by-symbol decoding. The symbols x kij areassumed to be distributed as i.i.d. CN (0 , . We now needto decode X ′ and X ′ from (25) with symbol-by-symboldecoding. We shall decode the first two and the last twocolumns of X ′ i independently.Consider a modified system where a Gaussian noise matrix N ′′ is added to (25) so that the entries of the effective noisematrix in (25) are distributed as i.i.d. CN (0 , . The matrix Y ′ defined in (25) is now taken to be a matrix with the noise N ′′ added. Denote the effective channel matrices from Tx- andTx- to Rx- by ˆ H = H V and ˆ G = H V respectively.Define the matrices ˜ H and ˜ G by ˜ H = q tr (cid:0) H − H − H (cid:1) ˆ H = H H − ˜ G = q tr (cid:0) H − H − H (cid:1) ˆ G = H H − . (59)Define a processed received symbol matrix Y ′′ ∈ C × by Y ′′ (: ,
1) = Y ′ (: , , Y ′′ (: ,
2) = Y ′ (: , Y ′′ (: ,
3) = Y ′ (: , , Y ′′ (: ,
4) = Y ′ (: , . Now, the first two columns of Y ′′ can be re-written as y ′′ y ′′ y ′′ y ′′ y ′′ y ′′ y ′′ y ′′ = H H G G H H G G H H G G H H G G x ′ x ′ x ′ x ′ x ′ x ′ x ′ x ′ + N ′′′ (60) where, H i and G i are defined in (61) (at the top of the nextpage), for i = 1 , , · · · , , and N ′′′ ∈ C × is a Gaussianvector whose entries are distributed as i.i.d. CN (0 , . The symbols x ′ kij are defined in (62). Considering the last twocolumns of Y ′′ , an equation similar to (60) involving thesymbols x ′ ki can be written, for k = 3 , , , and i = 1 , .We however avoid it for the sake of brevity. We now proceedto prove that x ′ i , x ′ i , x ′ i , and x ′ i can be recovered usinginterference cancellation as follows.Let z i = (cid:20) y ′′ i y ′′ i (cid:21) , for i = 1 , , , . The interference cancel-lation is performed in three steps. Step : Define the symbols obtained by eliminating thesymbols x ′ and x ′ from (60) by z ′ = G H z || G (1 , :) || − G H z || G (1 , :) || z ′ = G H z || G (1 , :) || − G H z || G (1 , :) || (63) z ′ = G H z || G (1 , :) || − G H z || G (1 , :) || . The symbols z ′ , z ′ , and z ′ can be written as z ′ z ′ z ′ = H ′ H ′ G ′ H ′ H ′ G ′ H ′ H G ′ x ′ x ′ x ′ x ′ x ′ x ′ + W ′ (64) where, the Alamouti matrices H ′ i ∈ C × , for i = 1 , , · · · , G ′ i ∈ C × , for i = 1 , , , are defined in (65), and W ′ ∈ C × denotes the relevant Gaussian noise matrix. Step : Define the signals obtained by eliminating thesymbols x ′ and x ′ from z ′ i (defined in (63)) by z ′′ = G ′ H z ′ || G ′ (1 , :) || − G ′ H z ′ || G ′ (1 , :) || z ′′ = G ′ H z ′ || G ′ (1 , :) || − G ′ H z ′ || G ′ (1 , :) || . (66) The symbols z ′′ , z ′′ , and z ′′ can be written as (cid:20) z ′′ z ′′ (cid:21) = (cid:20) H ′′ H ′′ H ′′ H ′′ (cid:21) x ′ x ′ x ′ x ′ + W ′′ (67) where, the Alamouti matrices H ′′ i , for i = 1 , , , , are definedin (68), and W ′′ ∈ C × denotes the relevant Gaussian noisematrix. Step : Finally, define the signals obtained by eliminatingthe symbols x ′ and x ′ from z ′′ i (defined in (66)) by z ′′′ = H ′′ H z ′ || H ′′ (1 , :) || − H ′′ H z ′′ || H ′′ (1 , :) || = (cid:20) H ′′ H H ′ || H ′′ (1 , :) || − H ′′ H H ′′ || H ′′ (1 , :) || (cid:21) (cid:20) x ′ x ′ (cid:21) + W ′′′ (69) where, W ′′′ ∈ C × denotes the relevant Gaussian noisematrix. = " ˆ h ˆ h ˆ h − ˆ h , H = " ˆ h ˆ h ˆ h − ˆ h , H = " ˆ h ˆ h ˆ h − ˆ h , H = " ˆ h ˆ h ˆ h − ˆ h ,H = " e jθ ˆ h e jθ ˆ h e − jθ ˆ h − e − jθ ˆ h , H = " e jθ ˆ h e jθ ˆ h e − jθ ˆ h − e − jθ ˆ h , H = " e jθ ˆ h e jθ ˆ h e − jθ ˆ h − e − jθ ˆ h , H = " e jθ ˆ h e jθ ˆ h e − jθ ˆ h − e − jθ ˆ h , (61) G = (cid:20) ˆ g ˆ g ˆ g − ˆ g (cid:21) , G = (cid:20) ˆ g ˆ g ˆ g − ˆ g (cid:21) , G = (cid:20) ˆ g ˆ g ˆ g − ˆ g (cid:21) , G = (cid:20) ˆ g ˆ g ˆ g − ˆ g (cid:21) ,G = (cid:20) e jθ ˆ g e jθ ˆ g e − jθ ˆ g − e − jθ ˆ g (cid:21) , G = (cid:20) e jθ ˆ g e jθ ˆ g e − jθ ˆ g − e − jθ ˆ g (cid:21) , G = (cid:20) e jθ ˆ g e jθ ˆ g e − jθ ˆ g − e − jθ ˆ g (cid:21) , G = (cid:20) e jθ ˆ g e jθ ˆ g e − jθ ˆ g − e − jθ ˆ g (cid:21) .x ′ ij = x Rij + jx Iij , x ′ ij = x Rij + jx Iij , x ′ ij = x Rij + jx Iij , x ′ ij = − x Rij + jx Iij x ′ ij = x Rij + jx Iij , x ′ ij = − x Rij + jx Iij , x ′ ij = x Rij + jx Iij , x ′ ij = − x Rij + jx Iij . (62) H ′ = G H H || G (1 , :) || − G H H || G (1 , :) || , H ′ = G H H || G (1 , :) || − G H H || G (1 , :) || , H ′ = G H H || G (1 , :) || − G H H || G (1 , :) || H ′ = G H H || G (1 , :) || − G H H || G (1 , :) || , H ′ = G H H || G (1 , :) || − G H H || G (1 , :) || , H ′ = G H H || G (1 , :) || − G H H || G (1 , :) || (65) G ′ = G H G || G (1 , :) || − G H G || G (1 , :) || , G ′ = G H G || G (1 , :) || − G H G || G (1 , :) || , G ′ = G H G || G (1 , :) || − G H G || G (1 , :) || . H ′′ = G ′ H H ′ || G ′ (1 , :) || − G ′ H H ′ || G ′ (1 , :) || , H ′′ = G ′ H H ′ || G ′ (1 , :) || − G ′ H H ′ || G ′ (1 , :) || , H ′′ = G ′ H H ′ || G ′ (1 , :) || − G ′ H H ′ || G ′ (1 , :) || , H ′′ = G ′ H H ′ || G ′ (1 , :) || − G ′ H H ′ || G ′ (1 , :) || . (68) A similar interference cancellation algorithm involving thesymbols x k and x k , for k = 3 , , , , can be writtenstarting from the last two columns of Y ′′ . The proof fordecoding these symbols with vanishing probability of error(with respect to the codeword length) is similar to that for x k and x k , for k = 1 , , , , and hence, we avoid thedetails. To prove that the proposed scheme achieves a node-to-node DoF of almost surely, it is sufficient to prove thatat least one of the first column entries of the Alamouti matrix h H ′′ H H ′′ || H ′′ (1 , :) || − H ′′ H H ′′ || H ′′ (1 , :) || i is non-zero almost surely. This isbecause if h H ′′ H H ′′ || H ′′ (1 , :) || − H ′′ H H ′′ || H ′′ (1 , :) || i is a non-zero Alamoutimatrix then, at least one among the matrices H ′′ or H ′′ isa non-zero Alamouti matrix. Hence, if (cid:20) x ′ x ′ (cid:21) can be decodedwith vanishing probability of error then clearly, from (67), (cid:20) x ′ x ′ (cid:21) can also be decoded with vanishing probability of error.We shall now prove that the first row, first column entry of h H ′′ H H ′′ || H ′′ (1 , :) || − H ′′ H H ′′ || H ′′ (1 , :) || i is non-zero almost surely.Substituting for H ′ i in (68), the matrices H ′′ i can be writtenas in (70). Define the matrices E i ∈ C × and F i ∈ C × asin (70). Denote the entries of the matrices E i by E = (cid:20) e e e − e (cid:21) , E = (cid:20) e e e − e (cid:21) E = (cid:20) e e e − e (cid:21) . Similarly, define the entries of the matrices F i , i = 1 , , .Note that the matrices H ′′ and H ′′ depend on ˆ h j through the matrices H and H whereas H ′′ and H ′′ do not dependon ˆ h j , for j = 1 , , , . This crucial observation shall beexploited to show that the first row, first column entry of thematrix h H ′′ H H ′′ || H ′′ (1 , :) || − H ′′ H H ′′ || H ′′ (1 , :) || i is non-zero. The first row, firstcolumn entries of H ′′ H H ′′ and H ′′ H H ′′ are given in (71) and(72) respectively. Since ˆ H = H V , the entries of ˆ H are givenby ˆ h ij = P k =1 h ik v kj , for i, j = 1 , , , . Conditioningon all the random variables except h and substituting for ˆ h ij in (71) we have (73) which is re-written as (74), where c i are functions of the conditioned random variables. Notethat the expression of (cid:2) H ′′ H H ′′ (cid:3) in (72) and || H ′′ (1 , :) || areindependent of h j , for all j . Now, the coefficients of h R and h I in (74) are given by p and − p respectively where, p = e iθ (cid:2)(cid:0) | e | + | e | (cid:1) v v (cid:3) (75) + e − iθ (cid:2)(cid:0) | e | + | e | (cid:1) v v (cid:3) . If p is non-zero then, clearly H ′′ is a non-zero Alamouti matrixand hence, || H ′′ (1 , :) || is also non-zero. We now have thefollowing useful lemmas. Lemma 4:
At least one among e and e (considered nowas random variables) are non-zero almost surely. Proof:
It is easy to prove that G H || G (1 , :) || is anon-zero Alamouti matrix almost surely . Since E = The proof for this is on the same lines as that of Lemma 2 given inAppendix A. ′′ = G ′ H || G ′ (1 , :) || G H || G (1 , :) || | {z } E H − G ′ H || G ′ (1 , :) || G H || G (1 , :) || | {z } E H + (cid:18) G ′ H || G ′ (1 , :) || − G ′ H || G ′ (1 , :) || (cid:19) G H || G (1 , :) || | {z } E H ,H ′′ = G ′ H || G ′ (1 , :) || G H || G (1 , :) || H − G ′ H || G ′ (1 , :) || G H || G (1 , :) || H + (cid:18) G ′ H || G ′ (1 , :) || − G ′ H || G ′ (1 , :) || (cid:19) G H || G (1 , :) || H ,H ′′ = G ′ H || G ′ (1 , :) || G H || G (1 , :) || | {z } F H − G ′ H || G ′ (1 , :) || G H || G (1 , :) || | {z } F H + (cid:18) G ′ H || G ′ (1 , :) || − G ′ H || G ′ (1 , :) || (cid:19) G H || G (1 , :) || | {z } F H , (70) H ′′ = G ′ H || G ′ (1 , :) || G H || G (1 , :) || H − G ′ H || G ′ (1 , :) || G H || G (1 , :) || H + (cid:18) G ′ H || G ′ (1 , :) || − G ′ H || G ′ (1 , :) || (cid:19) G H || G (1 , :) || H . h H ′′ H H ′′ i = (cid:16) e e − jθ ˆ h + e e jθ ˆ h + e e − jθ ˆ h + e e jθ ˆ h + e e − jθ ˆ h + e e jθ ˆ h (cid:17) (cid:16) e ˆ h + e ˆ h + e ˆ h + e ˆ h + e ˆ h + e ˆ h (cid:17) + (cid:16) e e − jθ ˆ h − e e jθ ˆ h + e e − jθ ˆ h − e e jθ ˆ h + e e − jθ ˆ h − e e jθ ˆ h (cid:17) (cid:16) e ˆ h − e ˆ h + e ˆ h − e ˆ h + e ˆ h − e ˆ h (cid:17) (71) h H ′′ H H ′′ i = (cid:16) f e − jθ ˆ h + f e jθ ˆ h + f e − jθ ˆ h + f e jθ ˆ h + f e − jθ ˆ h + f e jθ ˆ h (cid:17) (cid:16) f ˆ h + f ˆ h + f ˆ h + f ˆ h + f ˆ h + f ˆ h (cid:17) + (cid:16) f e − jθ ˆ h − f e jθ ˆ h + f e − jθ ˆ h − f e jθ ˆ h + f e − jθ ˆ h − f e jθ ˆ h (cid:17) (cid:16) f ˆ h − f ˆ h + f ˆ h − f ˆ h + f ˆ h − f ˆ h (cid:17) . (72) h H ′′ H H ′′ i = (cid:16) e e − jθ v h + e e jθ v h + c (cid:17) (cid:16) e v h + e v h + c (cid:17) + (cid:16) e e − jθ v h − e e jθ v h + c (cid:17) (cid:16) e v h − e v h + c (cid:17) (73) = (cid:16) h R h e e − jθ v + e e jθ v i + jh I h − e e − jθ v + e e jθ v i + c (cid:17) (cid:16) h R (cid:2) e v + e v (cid:3) + jh I (cid:2) e v − e v (cid:3) + c (cid:17) + (cid:16) h R h e e − jθ v − e e jθ v i + jh I h − e e − jθ v − e e jθ v i + c (cid:17) (cid:16) h R (cid:2) e v − e v (cid:3) + jh I (cid:2) e v + e v (cid:3) + c (cid:17) (74) G ′ H || G ′ (1 , :) || G H || G (1 , :) || is a product of Alamouti matrices, it isnow sufficient to prove that G ′ is a non-zero matrix almostsurely. Substituting for G , G , G , and G from (61) in thedefinition of G ′ , we have g ′ = 1( | ˆ g | + | ˆ g | ) ( | ˆ g | + | ˆ g | ) × (cid:0)(cid:0) | ˆ g | + | ˆ g | (cid:1) (cid:2) e jθ ˆ g ˆ g + e − jθ ˆ g ˆ g (cid:3) (76) − (cid:0) | ˆ g | + | ˆ g | (cid:1) (cid:2) e jθ ˆ g ˆ g + e − jθ ˆ g ˆ g (cid:3)(cid:1) . Note that the term outside the parenthesis in (76), i.e., ( | ˆ g | + | ˆ g | ) ) ( | ˆ g | + | ˆ g | ) is non-zero almost surely. We shallnow prove that the term inside the parenthesis in (76) isalso non-zero almost surely. Since ˆ G = H V , the entries ˆ g j and ˆ g j are given by ˆ g j = P k =1 h k v kj and ˆ g j = P k =1 h k v kj respectively, for j = 1 , , , . Conditioningon all the random variables except h , we have ˆ g j = h v j + q j where, q j is some function of the conditionedrandom variables. Note that ˆ g j , for all j , are independent of h . Considering the terms inside the parenthesis in (76), thecoefficient of | h | is given by (77) (at the top of the nextpage). If this coefficient is non-zero then, further conditioningon h I , the terms inside the parenthesis in (76) constitutea non-zero polynomial of degree in h R . Since h R iscontinuously distributed, the term inside the parenthesis in (76)is almost surely non-zero.Hence, the proof shall be complete if we prove that the expression in (77) is non-zero almost surely. Substituting for v ij , we have (78) where, h ( − ij denotes the entries of H − .Since ˜ g ij = P k =1 h k h ( − kj , the coefficient of | h | in theterm inside the parenthesis of (78) is given by (cid:18)(cid:12)(cid:12)(cid:12) h ( − (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) h ( − (cid:12)(cid:12)(cid:12) (cid:19) (cid:16) e jθ h ( − h ( − + e − jθ h ( − h ( − (cid:17) (79) − (cid:18)(cid:12)(cid:12)(cid:12) h ( − (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) h ( − (cid:12)(cid:12)(cid:12) (cid:19) (cid:16) e jθ h ( − h ( − + e − jθ h ( − h ( − (cid:17) . Note that the entries of H − are rational polynomial func-tions in the variables h R ij and h I ij , for i, j = 1 , , , . Ifthe expression in (79) is a non-constant rational polynomialfunction in h R ij and h I ij then, clearly (79) is non-zeroalmost surely, for any θ . This is because, under a commondenominator, the numerator of (79) would be a non-constantpolynomial function in h R ij and h I ij which are independentand continuously distributed random variables for all i, j . Toshow that the expression in (79) is a non-constant rationalpolynomial function in h R ij and h I ij for some ( i, j ) and forany θ , it is sufficient to show that (79) evaluates to differentvalues for different choices of H . Choose two values for H to be The coefficient of | h | is equal to zero. So, we consider the coefficientof | h | . | ˆ g | + | ˆ g | (cid:17) e jθ v v + (cid:16) | ˆ g | + | ˆ g | (cid:17) e − jθ v v − (cid:16) | v | + | v | (cid:17) (cid:16) e jθ ˆ g ˆ g + e − jθ ˆ g ˆ g (cid:17) (77) = r tr (cid:16) H − H − H (cid:17)! (cid:20)(cid:16) | ˜ g | + | ˜ g | (cid:17) e jθ h ( − h ( − + (cid:16) | ˜ g | + | ˜ g | (cid:17) e − jθ h ( − h ( − − (cid:16) | h ( − | + | h ( − | (cid:17) (cid:16) e jθ ˜ g ˜ g + e − jθ ˜ g ˜ g (cid:17)(cid:21) (78) H = . − − . − . − , − . − . − . − . . so that for the first matrix, h ( − = h ( − = h ( − = h ( − = 1 ,h ( − = h ( − = h ( − = 2 , h ( − = 3 H − (3 , :) = [1 0 0 0] , H − (4 , :) = [0 1 0 0] . and for the second matrix all the entries of H − are the sameas above except that h ( − = 4 . Thus, for any value of θ , (79)evaluates to − e − jθ and − e − jθ for the two chosen valuesof H . Hence, for any value of θ , the expression in (79) isa non-constant rational polynomial function in the entries of H . Lemma 5:
The random variable p defined in (75) is non-zero almost surely. Proof:
We have p = (cid:0) | e | + | e | (cid:1) (cid:2) e iθ v v + e − iθ v v (cid:3) . From Lemma 4, since e and e are non-zero almost surely, weonly need to need to prove that e iθ v v + e − iθ v v is non-zero almost surely. Since V = H − tr ( H − H − H ) , weonly need to show that e iθ h ( − h ( − + e − iθ h ( − h ( − isnon-zero because tr (cid:0) H − H − H (cid:1) is non-zero almost surely.Using similar arguments as in Lemma 4, it can be shownthat e iθ h ( − h ( − + e − iθ h ( − h ( − is a non-constant rationalpolynomial function in the entries of H , for any θ . Hence, e iθ h ( − h ( − + e − iθ h ( − h ( − is non-zero almost surely.Let us now complete the proof for the statementthat the first row, first column entry of the ma-trix h H ′′ H H ′′ || H ′′ (1 , :) || − H ′′ H H ′′ || H ′′ (1 , :) || i is non-zero almost surely.The coefficients of h R and h I in the expression || H ′′ (1 , :) || h H ′′ H H ′′ − || H ′′ (1 , :) || H ′′ H H ′′ || H ′′ (1 , :) || i can be derivedto be equal to p − (cid:0) | e | + | e | (cid:1) (cid:16) | v | + | v | (cid:17) H ′′ H H ′′ || H ′′ (1 , :) || and − p − (cid:0) | e | + | e | (cid:1) (cid:16) | v | + | v | (cid:17) H ′′ H H ′′ || H ′′ (1 , :) || respectively. Clearly, since p is non-zero almost surely, both ofthe above coefficients cannot be equal to zero simultaneously.Thus, h H ′′ H H ′′ − || H ′′ (1 , :) || H ′′ H H ′′ || H ′′ (1 , :) || i is a quadratic poly-nomial in the continuously distributed random variables h R and h I and hence, non-zero almost surely. R EFERENCES[1] H. Sato, “The Capacity of the Gaussian Interference Under StrongInterference”, IEEE Trans. Info. Theory, Vol. 27, no.6, pp. 786-788,Nov. 1981.[2] R. K. Farsani, “Fundamental Limits of Communications in InterferenceNetworks-Part III: Information Flow in Strong Interference Regime”,Available at: arXiv:1207.3035v2 [cs.IT].[3] R. Etkin, D. Tse, and H. Wang, “Gaussian Interference ChannelCapacity to Within One Bit”, IEEE Trans. Info. Theory, Vol. 54, no.12, pp. 5534-5562, Dec. 2008.[4] M. A. Maddah-Ali, A. S. Motahari, and A. K. Khandani, “Communica-tion Over MIMO X Channels: Interference Alignment, Decomposition,and Performance Analysis”, IEEE Trans. Info. Theory, Vol. 54, no. 8,pp. 3457-3470, Aug. 2008.[5] S. Jafar and S. Shamai, “Degrees of Freedom Region of the MIMO X Channel”, IEEE Trans. Info. Theory, Vol. 54, no. 1, pp. 151-170, Jan.2008.[6] L. Li, H. Jafarkhani, and S. A. Jafar, “When Alamouti codes meetinterference alignment: transmission schemes for two-user X channel”,IEEE ISIT 2011, Jul. 31 - Aug. 5, 2011, pp. 2577-2581.[7] K. Pavan Srinath and B. Sundar Rajan, “Low ML-Decoding Complex-ity, Large Coding Gain, Full-Rate, Full-Diversity STBCs for × and ×2