Interference Aligned Space-Time Transmission with Diversity for the 2\times 2 X-Network

# Interference Aligned Space-Time Transmission with Diversity for the 2×2 X-Network

\authorblockNAbhinav Ganesan and K. Pavan Srinath Abhinav Ganesan is with the Institute of Network Coding, The Chinese University of Hong Kong (e-mail: abhinav@inc.cuhk.edu.hk). K. Pavan Srinath is with the Department of Engineering, University of Cambridge, UK (e-mail: pk423@cam.ac.uk). A preliminary version of a part of this paper was presented at the 2014 IEEE International Symposium on Information Theory (ISIT). Another part of a content of this paper has been submitted for possible publication to the 2015 IEEE ISIT.
###### Abstract

The sum degrees of freedom (DoF) of the two-transmitter, two-receiver multiple-input multiple-output (MIMO) X-Network ( MIMO X-Network) with antennas at each node is known to be . Transmission schemes which couple local channel-state-information-at-the-transmitter (CSIT) based precoding with space-time block coding to achieve the sum-DoF of this network are known specifically for . These schemes have been proven to guarantee a diversity gain of when a finite-sized input constellation is employed. In this paper, an explicit transmission scheme that achieves the sum-DoF of the X-Network for arbitrary is presented. The proposed scheme needs only local CSIT unlike the Jafar-Shamai scheme which requires the availability of global CSIT in order to achieve the sum-DoF. Further, it is shown analytically that the proposed scheme guarantees a diversity gain of when finite-sized input constellations are employed.

{keywords}

Interference Alignment, X-Channels, X-Networks, Diversity, Space-time Block Codes, Degrees of Freedom.

## I Introduction

The advent of smart phones has led to an explosion in mobile data demand. But a limited spectrum calls for a better spectrum management that incorporates techniques beyond conventional approaches like orthogonalization of spectrum. A further increase in the number of mobile users and data demand means that cell edge users are susceptible to interference from the neighbouring base stations and vice-versa. These issues have instigated research on better transmission techniques in interference networks, with information-theoretic rate tuples often used as the metric for designing better schemes. Since the capacity of interference networks is unknown in general, degrees of freedom (DoF) [1] is the widely targeted metric due to its relative ease of characterization. The sum-DoF of a Gaussian network is said to be if its sum-capacity (in bits per channel use) can be approximated as .

Availability of channel-state-information at the transmitters (CSIT) is an important assumption in the characterization of the approximate capacity of Gaussian interference networks. Availability of perfect global CSIT111Global CSIT means that all the channel gains of the Gaussian network are available a priori at all the transmitters. often enables one to design precoders that cast interference onto subspaces independent of the desired signal space at the receivers. This technique, termed interference alignment (IA), was first used implicitly in [2, 3], and explicitly appeared in [4, 5] in the context of multiple-input multiple-output (MIMO) X-Networks. A X-Network is a Gaussian interference network with transmitters and receivers and a total of independent messages meant to be sent over the network, one from every transmitter to every receiver. A X-Network with antennas at each node is referred to as the X-Network. A lower bound on the sum-DoF was shown to be for such a network in [3], and it was proven in [5] that the sum-DoF equals , achieved using an IA scheme. All the aforementioned works assume the availability of perfect global CSIT.

The concept of DoF assumes the use of a codebook with unconstrained alphabet size as well as unlimited peak power, but with an average power constraint. The channel is assumed to be static during the transmission of an entire codeword. Further, information-theoretic rate definitions also assume the usage of unlimited coding length. Clearly, all these assumptions are infeasible in practice. In practical communication, the coding length and the codebook size are constrained by factors such as delay requirement and computational complexity. Moreover, the practically used input constellations like QAM and PSK have limited peak power. So, these issues222In the context of multiuser communication, these issues have motivated the study of the effects of constellation constraints on information-theoretically achievable rates in the two-user multiple access channel [6] and the Gaussian Interference Channel [7, 8]. However, these works do not take into account limited coding length. have motivated the research on high reliability communication in MIMO systems under practical constraints like limited coding length, constrained alphabet size, and limited peak power, thus leading to the development of space-time block codes (STBCs) for the single user MIMO systems [9]. The theory of STBCs makes the assumption that the channel is constant during the transmission of an entire codeword block but changes independently after every codeword transmission, i.e., the channel is a block fading one. A metric of significant interest in the design of STBCs is the diversity gain which indicates the nature of the fall in error probability with SNR. Most of the literature on STBCs is on linear STBCs [10] (see Definition 1 and Definition 2 in Section II-A for a formal definition of “STBC” and “linear STBC”, respectively) primarily due to the ease of symbol encoding and, to an extent, decoding (using the sphere decoder [11]). Associated with such linear STBCs is the notion of symbol rate which is the number of linearly and statistically independent complex symbols transmitted per channel use (see Definition 3 in Section II-A for a formal definition of “STBC rate”). It is known that for a single user MIMO system with transmit antennas and receive antennas, the maximum possible STBC rate (in complex symbols per channel use) is , which equals the DoF333For a general MIMO system, i.e., a MIMO system with transmit antennas and receive antennas, the DoF is . For the case where , it is currently not known if the best STBC with a rate of complex symbols per channel use (cspcu) offers any advantage over the best STBC in the comparable class with a rate of cspcu. (DoF is the maximum achievable multiplexing gain [1]) of the single user MIMO system.

The above notion of rate (henceforth in this paper, “rate” refers to the rate of the STBC unless otherwise mentioned) can be extended to the multiuser setting as follows. Analogous to rate (in a single user MIMO system using STBCs) is the “sum-rate” of a linear transmission scheme444A linear transmission scheme is one where the vectorized version of the symbols received across all the antennas and time instants spanning the codeword length can be expressed as a linear combination of the statistically independent input symbols. In a single user MIMO system, a linear transmission scheme is equivalent to a linear STBC. in a Gaussian interference network. This sum-rate is a measure of the total number of linearly and statistically independent complex symbols transmitted per channel use (see Definition 9 in Section II-A for a formal definition of the sum-rate) and is related to the number of independent complex symbols that can be recovered at the receiver by simple zero-forcing. Note that the definition of sum-DoF applies to non-linear transmission schemes while the sum-rate applies strictly to linear schemes with limited coding length and with finite input constellation. However, it is trivially true that the sum-rate cannot exceed the sum-DoF. Therefore, for the X-Network, the maximum sum-rate is cspcu, achieved by an IA scheme that is linear [5]. The primary goal of this paper is to look for linear transmission schemes for the X-Network that achieve the maximum sum-rate along with a non-trivial guaranteed diversity gain when finite and fixed input constellations are employed.

### I-a Prior Works on Diversity Gain in Interference Networks

A linear transmission scheme (Definition 8, Section II-A) based on the quasi-orthogonal STBC [12] was proposed for the X-Network for different configurations of the number of transmit and receive antennas in [13]. There are several drawbacks with this transmission scheme, though full transmit and receive diversity gains are guaranteed. The transmission scheme requires at least six transmit antennas, and has a sum-rate of 4 cspcu, which does not scale with the number of transmit and receive antennas. Further, the work aims for orthogonality of the desired signals from the two transmitters to a single receiver as well as orthogonality between the desired signal sub-space and the interference sub-space, with the assumption of global CSIT. However, such an orthogonality can easily be achieved without global CSIT using the time division multiple access scheme (TDMA). Another linear transmission scheme achieving an (asymptotic) sum-rate of four cspcu was proposed in [14] for the X-Network equipped with transmit and receive antennas, without the assumption of channel-state-information at any of the transmitters. Clearly, the sum-rate does not scale with the number of transmit or receive antennas, though full transmit and receive diversity gains are guaranteed. Moreover, better sum-rate can be achieved with TDMA along with full transmit and receive diversity gains. Nevertheless, TDMA cannot achieve the maximum sum-rate of cspcu for the X-Network.

Linear transmission schemes for the X-Network and the X-Network were proposed in [15] and [16]. The first linear transmission scheme with a guaranteed diversity gain of 2 with fixed finite input constellations for the X-Network that achieves the maximum sum-rate of cspcu was proposed in [17, 15]. This transmission scheme couples the Alamouti STBC [18] with channel-dependent precoding and achieves IA. The same (structure-wise) IA precoding matrices were coupled with the Srinath-Rajan STBC [19] to guarantee a diversity gain of 4 with fixed finite input constellations at the maximum sum-rate of cspcu for the X-Network [16]. In general, STBC designs for single user MIMO systems assume only the availability of perfect channel-state-information at the receivers (CSIR) but not CSIT. However, since the channel matrices are random, CSIT in the X-Network is inevitable in order to achieve IA, and hence the maximum sum-rate transmission. Moreover, the assumption of CSIT is not an impractical one, since a few state-of-the-art wireless systems support CSIT (for example, the Wi-Fi 802.11ac standard [20]). The precoders of [15], which we call the LiJ precoders, assume the availability of local CSIT, i.e., each transmitter is aware of only its own channel matrices to both the receivers, and global CSIR, i.e., all the channel matrices are known to all the receivers. This is in contrast to the assumption of global CSIT (i.e., all the channel matrices are known to all the transmitters) in [5] to achieve IA.

Furthermore, the transmission schemes in [15, 16] also achieve the sum-DoF of the X-Network, for , when the input constellation is Gaussian distributed. In this work, we generalize the above schemes for arbitrary values of . We identify a class of STBCs which when coupled with LiJ precoders achieve the maximum sum-rate (and hence, the sum-DoF555Throughout the paper, the term “sum-rate” pertains to the case where finite input constellations are employed while achievability of “sum-DoF” holds relevance when the input constellations are Gaussian distributed. when utilizing Gaussian distributed input constellations) of cspcu for the X-Network666This absence of reduction in the DoF upon the introduction of an STBC is analogous to information-losslessness due to certain STBCs in single-user MIMO systems [10, 21].. The Alamouti STBC and the Srinath-Rajan STBC used in [15, 16] are special cases of the class we propose in this paper. Moreover, with fixed finite input constellations, a diversity gain of is proven to be guaranteed, and this also establishes that the linear transmission schemes of [15, 16] achieve a diversity gain of and respectively for the X-Network and the X-Network . It must be noted that a straightforward generalization of the proof of diversity gain given in [16] to the transmission scheme proposed in this paper can guarantee a diversity gain of only . So, the result in this paper on the diversity gain is an improvement over existing ones in the literature.

The contributions of this paper may be summarized as follows.

• A class of STBCs, namely STBCs with the column-cancellation property (see Definition 7 in Section II-A), when coupled with the LiJ precoders is shown to achieve the sum-DoF of the X-Network. These STBCs are based on STBCs obtained from cyclic division algebras (CDA) [22], the explicit construction of which is available in the literature for arbitrary . Since LiJ precoders are used in this work, the sum-DoF is achieved using local CSIT whereas the Jafar-Shamai scheme [5] assumes global CSIT.

• We prove that when fixed finite input constellations are employed, a diversity gain of is guaranteed with the proposed transmission scheme.

• For , we propose a new STBC with the column-cancellation property and having the minimum possible delay. We show that upon using this STBC in the X-Network, the maximum sum-rate of cspcu and a diversity gain of with fixed finite input constellations is achieved.

The rest of the paper is organized as follows. Section II provides the signal model and relevant definitions. In Section III, the proposed linear transmission scheme for the X-Network is presented and it is shown to achieve the maximum sum-rate of cspcu and also a guaranteed diversity gain of (with fixed finite input constellations) for arbitrary values of . Section IV provides a novel low-delay linear transmission scheme for the X-Network which achieves the maximum sum-rate of cspcu and a guaranteed diversity gain of 4. The simulation results are presented in Sub-section IV-A and the concluding remarks constitute Section V.

Notation: Throughout the paper, the following notation is employed.

• Bold, lowercase letters denote vectors, and bold, uppercase letters denote matrices.

• , , , , and denote the conjugate transpose, the transpose, the determinant, the trace, the rank, and the Frobenius norm of , respectively. Further, denotes the entry-wise conjugation of the elements of , i.e., .

• denotes a block diagonal matrix with matrices , , , on its main diagonal blocks.

• The real and the imaginary parts of a complex-valued vector are denoted by and , respectively.

• For a set , denotes its cardinality while for a complex number , denotes its absolute value.

• denotes the identity matrix of size , and denotes the null matrix whose dimensions, unless specified in the subscript, are understood from context.

• For a complex random matrix , denotes the expectation of a real-valued function over the distribution of .

• and denote the field of real and complex numbers, respectively.

• Unless used as an index, a subscript or a superscript, denotes .

• Unless otherwise specified, for a matrix , denotes the column of , , and for a set , denotes the matrix whose columns are the columns of indexed by the elements of . Further, denotes the submatrix of consisting of the elements of from Row to Row , Column to Column , with , .

• For a complex variable , is defined as

 ˇx:=[xI−xQxQxI],

and for any matrix , the matrix belonging to is obtained by replacing each entry with , .

• The operator acting on a complex vector is defined as follows. For , .

• denotes the vector obtained by stacking the columns of the matrix one below the other so that . It follows that, .

• The Q-function of is denoted by and given as

 Q(x)=∫∞x1√2πe−t22dt.
• Throughout the paper, denotes the logarithm of to base 2.

• The notation denotes that has the standard complex normal distribution.

• denotes that , and is similarly defined.

• denotes that .

• .

• For a real number , denotes the smallest integer not lower than while denotes the largest integer not greater than .

## Ii Signal Model and Definitions

The X-Network is depicted in Fig. 1. Two transmitters and two receivers seek to communicate with each other in the presence of additive white Gaussian noise at the receivers. Transmitter (Tx-) generates an independent message intended for Receiver (Rx-), . The messages are mapped to a signal matrix , . Denoting the output signal matrix at Rx- by , and the channel matrix from Tx- to Rx- by , the input-output relation over time slots is given by

 Yj=√ρ2∑i=1HijXi+Nj,

where denotes the noise matrix whose entries are independent and identically distributed (i.i.d.) standard complex normal random variables. The average power constraint at each of the transmitters is , and hence , . The channel gains are assumed to be constant during the transmission of an entire signal matrix. For the sum-DoF evaluation, the real and imaginary parts of the channel gains are assumed to be distributed independently according to some arbitrary continuous distribution. For the diversity gain evaluation, the channel gains are assumed to be i.i.d. standard complex normal random variables, and experience block-fading. Local CSIT and global CSIR is assumed throughout the paper.

### Ii-a Definitions

A few of the definitions presented below are already available in the literature, while a few other terms are introduced in this paper.

###### Definition 1 (Space-Time Block Code [22])

For an transmit antenna MIMO system, an space-time block code (STBC) is a finite set of complex matrices of size . The block length of the STBC is channel uses.

###### Definition 2 (Linear STBC [10])

An STBC is called a linear STBC if it can be expressed as

 X={X=k∑i=1AiIxiI+AiQxiQ ∣∣ AiI,AiQ∈CM×T,xi:=xiI+ixiQ∈Qi},

where the matrices are called weight matrices [23], and , , are complex constellations with finite cardinality.

In the literature, it is generally assumed that where is either a QAM or a PSK constellation. Linear STBCs are particularly of interest because of the ease of encoding and to an extent, decoding (using the sphere decoder [11]).

###### Definition 3 (Rate of a linear STBC)

The rate of an linear STBC given by

 X={X=k∑i=1AiIxiI+AiQxiQ ∣∣ AiI,AiQ∈CM×T,xi=xiI+ixiQ∈Q}

is said to be complex symbols per channel use (cspcu) if the weight matrices are linearly independent over .

Note that rate is not defined to be the number of statistically independent symbols encoded per channel use because an arbitrary number of statistically independent symbols could be packed even in a single dimension. Definition 3 is inspired by the general design principle that it is more energy-efficient to pack a given number of constellation points in a higher dimensional space than in a lower dimensional space [24, Chapter ]. An implication of Definition 3 is that is a linearly independent set over . Associated with every linear STBC is its generator matrix which is defined as follows.

###### Definition 4 (Generator matrix of a linear STBC [19])

For an linear STBC given by

 X={X=k∑i=1AiIxiI+AiQxiQ ∣∣ AiI,AiQ∈CM×T,xi=xiI+ixiQ∈Q},

its generator matrix is given by

 G=[˜vec(A1I) ˜vec(A1Q) ˜vec(A2I) ˜vec(A2Q) ⋯  ˜vec(AkI) ˜vec(AkQ)]

so that where . For those linear STBCs of the form

 X={X=k∑i=1Aixi ∣∣ Ai∈CM×T,xi∈Q},

we prefer to use the complex version of the generator matrix , which is defined as

 GC=[vec(A1) vec(A2) ⋯ vec(Ak)] (1)

so that .

###### Definition 5 (Full-rank STBC [9])

An STBC is said to be full-ranked if

 Rank(X1−X2)

In other words, full-rankness of an STBC means that the difference matrix of any two distinct codewords of the STBC must be full-ranked.

###### Definition 6 (Gaussian-stabilizer function)

A function is said to be a Gaussian-stabilizer (GS) function if for .

Examples of GS-functions are for any unitary matrix , and . Also, if and are two GS-functions, then so is , where .

###### Definition 7 (Column-Cancellation (CC) Property of an STBC)

Consider an STBC . Let . Then, is said to possess the column-cancellation property if there exist a permutation and GS-functions , , such that for every ,

 X(π(i))+fi(X(π(i+T)))=gi(X(π(i)))+X(π(i+T))=0,     ∀i=1,⋯,T.

In other words, the CC-property ensures that upon permuting the columns of the codewords of the STBC, the first columns can be respectively canceled using the last columns and vice-versa using GS-functions.

###### Example 1

The Alamouti STBC whose codeword matrix is of the form

 X=[x1−x∗2x2x∗1]

has the CC-property with . On choosing , , where

 P1=[0−110],   P2=[01−10],

it is clear that the first column of the STBC can be canceled using the second and vice-versa, i.e.,

 X(1)+f1(X(2))=g1(X(1))+X(2)=0.

Note that both and are GS-functions.

###### Example 2

The Srinath-Rajan STBC whose codeword matrix is of the form

 X=⎡⎢ ⎢ ⎢ ⎢ ⎢⎣x1I+ix3Q−x2I+ix4Qeiθ(x5I+ix7Q)eiθ(−x6I+ix8Q)x2I+ix4Qx1I−ix3Qeiθ(x6I+ix8Q)eiθ(x5I−ix7Q)eiθ(x7I+ix5Q)eiθ(−x8I+ix6Q)x3I+ix1Q−x4I+ix2Qeiθ(x8I+ix6Q)eiθ(x7I−ix5Q)x4I+ix2Qx3I−ix1Q⎤⎥ ⎥ ⎥ ⎥ ⎥⎦

for some , also possesses the CC-property with . Choosing , , , and GS-functions , , , , where

 P1=⎡⎢ ⎢ ⎢ ⎢⎣0−1001000000−e2iθ00e2iθ0⎤⎥ ⎥ ⎥ ⎥⎦,   P2=⎡⎢ ⎢ ⎢ ⎢⎣0−e2iθ00e2iθ000000−10010⎤⎥ ⎥ ⎥ ⎥⎦,
 P3=⎡⎢ ⎢ ⎢ ⎢⎣0100−1000000e2iθ00−e2iθ0⎤⎥ ⎥ ⎥ ⎥⎦,   P4=⎡⎢ ⎢ ⎢ ⎢⎣0e2iθ00−e2iθ000000100−10⎤⎥ ⎥ ⎥ ⎥⎦,

it is clear that the conditions necessary for the CC-property to hold are satisfied.

###### Definition 8 (Linear Transmission Scheme)

Consider a Gaussian interference network777It must be noted that the terminology “Gaussian network”, by default, refers to linear channels. A Gaussian interference network has a linear channel with arbitrary (fixed) number of transmitters and an arbitrary (fixed) number of receivers with arbitrary (fixed) message demands. with transmitters each having antennas. Let be the signal matrix that is transmitted over uses of the channel by Tx-, , with , where and . Here, represents the information bearing symbol vector that Tx- intends to transmit over the channel and is its encoding function. This transmission scheme is said to be linear if for every , , , for some complex constants and .

Note that in practice, the symbol vectors with having finite cardinality. So, it might well be that for , , but this has no bearing on Definition 8.

###### Definition 9 (Sum-rate of a linear transmission scheme)

Consider a Gaussian interference network with transmitters and receivers, each having antennas. For a linear transmission scheme , the received signal matrix at Rx-, , is

 Yj=√ρK∑i=1Hijfi(xi)+Nj,

where denotes the noise matrix with its entries being i.i.d. standard complex normal random variables, and the channel matrix from Tx- to Rx- (constant during the transmission of an entire signal matrix). Let be the desired symbol vector at Rx-. Then, the sum-rate of is said to be complex symbols per channel use if there exist functions and positive integers , , which satisfy

 gj:CM×T ⟶ Cpj×1 Yj ⟼ Ajx′j+n

where which is dependent on has rank almost surely, and .

###### Remark 1

It is easy to see that the maximum sum-rate (in cspcu) that a linear transmission scheme can achieve equals the sum-DoF of the network. Using standard information-theoretic arguments, it follows that a maximum-sum-rate achieving linear transmission scheme achieves the sum-DoF of the network when the input constellations are Gaussian distributed and the coding length is unlimited.

## Iii Linear Transmission Scheme for the (2×2,M) X-Network

We now describe the linear transmission scheme for the general X-Network that achieves the sum-rate of cspcu. We make use of STBCs from cyclic division algebras (CDA) [22]. It is well known that STBCs from CDA exist for any number of transmit antennas [25]. For a detailed understanding of STBCs from CDA, one can refer to [25], [26], and references therein. Two key properties of STBCs from CDA that we need in this paper are as follows. Let be an STBC from CDA for transmit antennas.

1. For any , if and only if . In other words, is a full-rank STBC (Definition 5, Section II-A).

2. is an linear STBC that encodes linearly and statistically independent complex symbols in channel uses. Therefore, is a rate- STBC of block length . The complex generator matrix of , as defined in (1), is of size [25].

Now, for reasons that are made clear in Theorem 1 and Theorem 2 that are stated in the following part of this section, we seek full-rank STBCs that have a rate of cspcu and are further equipped with the CC-property. In view of this, we make use of the following lemma.

###### Lemma 1

For every , there exist full-rank, rate- STBCs of block length for some that have the CC-property.

{proof}

Let be an STBC from CDA. Then, the STBC given by

 ¯X:={[X  PX] | X∈X},

where is any unitary matrix, has a rate of cspcu and is of block length . It is easy to check that has the CC-property. Since is full-ranked, so is .

###### Remark 2

It is not necessary that the block length of a full-rank, rate- STBC with the CC-property be at least . For , we have already shown that the Alamouti STBC and the Srinath-Rajan STBC, which are both full-rank STBCs, have the CC-property and both of them have a rate of cspcu. It turns out that is the lower bound on the block length of full-rank, rate- STBCs with the CC-property. A general method to construct such minimum-block length STBCs is an open problem.

Let denote an STBC equipped with the CC-property888Henceforth in this paper, it is assumed without loss of generality that the first columns of the STBC with the CC-property can be canceled using the last columns and vice-versa. If not, the columns of the STBC can always be permuted to achieve the same.. The messages , with reference to the signal model in Section II, are mapped to the signal matrices as follows. Each is mapped to as

 W11↦X11=[¯X11  0M×T],  W21↦X21=[¯X21  0M×T], W12↦X12=[0M×T  ¯X12],  W22↦X22=[0M×T  ¯X22], (2)

where . We assume that with the codewords being uniformly drawn from . We observe that there is a “non-zero overlap” from column to between the matrices and , as also indicated by the hatched regions at the transmitters in Fig. 2.

The transmitted symbols from Tx- and Tx- are now (with the average transmit power at each transmitter being limited by ) given by

where , , are the LiJ precoders [15] given by

 V11:=H−112∥H−112∥, V21:=H−122∥H−122∥, V12:=H−111∥H−111∥, V22:=H−121∥H−121∥.

The LiJ precoders ensure that the effective channel matrices faced by the interference symbols are scaled identity matrices, and hence are aligned in the same subspace at the unintended receivers. The normalizing factors999Note that if existed and equalled (for some positive real number ) for a random matrix whose entries are i.i.d. standard complex normal random variables, we could have simply used as the normalizing factor for each . But this unfortunately is not the case [27]. for are chosen to satisfy the power constraint which is , for .

The received symbol matrix at Rx- is given by

 Y1=√3ρ4 (H11V11X11+H21V21X21+X12∥H−111∥+X22∥H−121∥)+N1. (3)

It can be observed from the structure of the zero and non-zero columns of defined in (2) that only the received symbols from time instants to face interference, as also indicated by the hatched regions at the receivers in Fig. 2. These interfering symbols can be canceled on account of the CC-property of the STBC used. Define the processed received symbol matrix, obtained after interference cancellation, by

 Y′1(t):=Y(t), for t=1,⋯,T, Y′1(t):=Y(t)+ft−T(Y(t+T)), for t=T+1,⋯,2T,

where are GS-functions (Definition 6, Section II-A). Note that the received symbols from time instants to are interference-free because of the presence of zero columns in . We thus have an interference-free processed received symbol matrix given by

 Y′1=√3ρ4(H11V11¯X11+H21V21¯X21)+N′1, (4)

where is a noise matrix whose entries are independent but not identically distributed. We have , , and , . Since increasing the noise variance affects neither the achieved DoF nor the diversity gain, we assume that , .

Similarly, exploiting the CC-property of (where we make use of the GS-functions , ), the interference-free processed received symbols at Rx- is given by

 Y′2=√3ρ4(H12V12¯X12+H22V22¯X22)+N′2,

where has the same distribution as . Hereafter, we shall focus only on the symbol matrix at Rx- and any claims about decoding the desired symbols hold good at Rx- also. Let denote the probability of error in decoding at Rx-. The diversity gain is given by [9]

 dg=−limρ→∞logPelogρ.

We now show that a diversity gain of is achievable if the following maximum-likelihood (ML) decoding rule is used.

 (^X11,^X21)=argmin(¯X11,¯X21∈¯X)∥∥ ∥∥Y′1−√3ρ8(H11V11¯X11+H21V21¯X21)∥∥ ∥∥. (5)

It is well-known that a diversity gain of is achieved in a single user MIMO system with Gaussian distributed channel coefficients when a full-rank STBC is employed [9]. Here, we show that when the underlying STBC is full-ranked, a diversity gain of is guaranteed (it goes without saying that the input constellation is of fixed finite cardinality). The loss in the diversity gain relative to the single user MIMO setting is due to the fact that the effective channels seen by the STBCs are not Gaussian distributed due to channel-dependent precoding at the transmitters. Full receive diversity gain is obtained whereas the transmit diversity gain is affected by precoding.

###### Theorem 1

If the STBC is full-ranked, then the diversity gain obtained in the X-Network by ML decoding of using (5) is at least .

{proof}

The pair-wise codeword error probability that the transmitted codeword pair is erroneously decoded to the codeword pair , denoted by , is given by

 Pe(△¯X)≤EHij,i,j=1,2⎡⎣Q⎛⎝√3ρ8∥∥H11V11△¯X11+H21V21△¯X21∥∥⎞⎠⎤⎦,

where and . Note that we can have the following three possibilities; 1) and , 2) and , 3) and . We shall prove the statement of the theorem only for the case , and the proofs for the rest of the cases follow similarly. Let

 h′ = vec(H11V11△¯X11+H21V21△¯X21) (6) = (△¯XT11VT11⊗IM)vec(H11)+(△¯XT21VT21⊗IM)vec(H21).

Note that (6) is due to the simple observation that for , it follows that . So, we now have

 ∥H11V11△¯X11+H21V21△¯X21∥=∥h′∥.

Conditioned on the random matrices and which the precoders and respectively depend on, has the same distribution as where

with (since is non-negative definite) and .

Using eigen-decomposition101010Any eigen-decomposition that appears in this proof assumes that the eigenvalues are arranged in non-ascending order along the main diagonal of the diagonal eigenvalue matrix. of to obtain with , we have . We now have

 Pe(△¯X) ≤ = EH12,H22⎡⎣Eh|H12,H22⎡⎣Q⎛⎝√3ρ8∥∥∥(K12⊗IM)h∥∥∥⎞⎠⎤⎦⎤⎦.

Now, denoting the entries of by , , let be such that . Then,

 ∥∥∥(K12⊗IM)h∥∥∥2 = ∥∥ ∥∥vec(H(K12)T)∥∥ ∥∥2 =

where , , are the columns of . Therefore,

 Pe(△¯X) = EH12,H22⎡⎢⎣Ehi,i=1,⋯,M|H12,H22⎡⎢⎣Q⎛⎜⎝ ⎷3ρ8M∑i=1∥∥∥K12hi∥∥∥2⎞⎟⎠⎤⎥⎦⎤⎥⎦ ≤ EH12,H22⎡⎢⎣Ehi,i=1,⋯,M|H12,H22⎡⎢⎣Q⎛⎜⎝ ⎷3ρ8M∑i=1hHiK12HK12hi⎞⎟⎠⎤⎥⎦⎤⎥⎦ = EH12,H22⎡⎢⎣Ehi,i=1,⋯,M|