# On Non-Integer Linear Degrees of Freedom of Constant Two-Cell MIMO Cellular Networks

###### Abstract

The study of degrees of freedom (DoF) of multiuser channels has led to the development of important interference managing schemes, such as interference alignment (IA) and interference neutralization. However, while the integer DoF have been widely studied in literatures, non-integer DoF are much less addressed, especially for channels with less variety. In this paper, we study the non-integer DoF of the time-invariant multiple-input multiple-output (MIMO) interfering multiple access channel (IMAC) in the simple setting of two cells, users per cell, and antennas at all nodes. We provide the exact characterization of the maximum achievable sum DoF under the constraint of using linear interference alignment (IA) scheme with symbol extension. Our results indicate that the integer sum DoF characterization achieved by the Suh-Ho-Tse scheme can be extended to the non-integer case only when for the circularly-symmetric-signaling systems and for the asymmetric-complex-signaling systems. These results are further extended to the time-invariant parallel MIMO IMAC with independent subchannels.

Interfering multiple access channel, degrees of freedom, linear interference alignment, symbol extension, multiple-input multiple-output (MIMO).

## I Introduction

Multiple-input multiple-output (MIMO) systems are capable of providing remarkably higher capacity compared to traditional single-input single-out (SISO) systems. The multiple antennas provide the extra dimensions to multiplex signals in space or cancel the interference from multiple unintended transmitters. The number of degrees of freedom (DoF), also known as multiplexing gain or capacity prelog, is a high signal-to-noise ratio (SNR) capacity approximation and characterizes the resolvable signal dimensions of the system. Interference alignment (IA) is an interference managing scheme developed from the study of the degrees of freedom of the time-invariant two-transmitter MIMO channel [1, 2]. The IA scheme is shown to be DoF-optimal for the channel, and the concept of IA has been later applied to many fundamental channels, including the time-varying -user SISO interference channel (IC) that provides sum DoF [3], establishing the unbounded multiuser DoF gain of the channel. Also in [3], Cadambe and Jafar develop a closed-form IA scheme to achieve sum DoF for the time-invariant -user MIMO IC with antennas at each node, and their DoF-optimal IA scheme can be implemented simply by linear precoder and combiner.

Interfering multiple access channel (IMAC) consists of several traditional multiple access channels (MAC) that interfere with each other, and IMAC is of practical importance because it models the environment of the uplink communications of several adjacent cells. Suh and Tse [4] develop a linear IA scheme for the time-invariant two-cell MIMO IMAC, where there are users in each cell and all nodes are equipped with antennas, to achieve sum DoF, under the requirement that The promising result indicates that the same DoF of the two isolated MACs, i.e., , can be realized when approaches infinity, demonstrating the multiuser DoF gain of the channel.

The application of the IA schemes to the two-cell IMAC has been extended in many directions, including the dual interfering broadcast channel (IBC) in [5, 6, 7, 8, 9, 10], the more general antenna settings in [6, 7, 8, 9, 10, 11, 12, 13, 14], and the more general -cell settings in [7, 8, 9, 10, 11]. Suh, Ho, and Tse study the dual time-invariant two-cell MIMO IBC in [5], and show that the same sum DoF can be achieved by linear IA scheme with less exchange of channel state information compared to that of the two-cell IMAC [4]. The time-invariant IMAC with users per cell, antennas at each transmitter, and antennas at each receiver, which will be referred to as the IMAC later in this paper, is studied by Kim et al in [11], and they provide an upper bound for the sum DoF of the channel. Liu and Yang study the time-invariant IBC in [7] and [8], where [7] derives the feasibility condition of the linear IA scheme and [8] obtains the characterization of the sum DoF. However, while significant progress has been made, the issue of the non-integer DoF is addressed neither in [7] due to the assumption of no symbol extension to provide the generic channel matrices required by the algebraic structure nor in [8] due to idea of the spatial extension [15] that avoids the DoF rounding. Although a standard method to provide non-integer DoF is through symbol extension, the limitation of using symbol extension is in general still unknown and plays a key role in the study of non-integer DoF.

The non-integer DoF of time-invariant channels achieved by linear IA schemes with symbol extensions is considered in [15, 16, 17], and their results show that the block-diagonal structure of the symbol-extended channel matrices, where all blocks are the same, provides extra constraints on linear precoding and combining. More specifically, Li, Jafarkhani, and Jafar show that the number of independent variables in the symbol-extended channel matrix, which is termed channel diversity in [18], limits the resolvability of the desired signal subspace and the interference subspace [16]. Under the assumptions of linear processing, each transmitter sending the same number of data streams, and antennas at each node, DoF upper bounds for both the channel and the -user IC are obtained in [16] based on the channel diversity. However, achievability of their DoF upper bounds is not addressed and therefore still an open problem. The non-integer DoF of the time-invariant IMAC is also still an open problem, except for the the special case of studied in [17].

As a stepping stone to explore the non-integer DoF of the time-invariant IMAC, we study the model in the simple setting of two cells, users per cell, and antennas at all nodes. Since is fixed, no spatial extension [15] is allowed. Moreover, all nodes are constrained to use linear pre- and post- processing due to the implementation issue, and symbol extension with arbitrary numbers of time slots is allowed. We apply the idea of channel diversity to this setting, and by exploring the block-diagonal structure of the symbol-extended channel matrices, we propose a novel upper bound and a modified lower bound for the sum DoF of the channel. In particular, the converse is developed by deriving a rank ratio inequality, which is originally proposed for the time-varying channel [19], for the the time-invariant, symbol extended IMAC. And the achievability is obtained by utilizing the generic structure imposed by the scheme design. The tightness of the upper bounds is shown, and we obtain the exact characterization of the maximum linearly achievable sum DoF. Our results indicate that the integer sum DoF characterization achieved by the linear IA scheme [4, 5] can be extended to the non-integer case only when for the traditional circularly-symmetric-signaling (CSS) systems and for the less-traditional asymmetric-complex-signaling (ACS) [20] systems due to the channel diversity constraint. These results are further extended to the time-invariant parallel IMAC with independent subchannels.

The rest of the paper is organized as follows. Section II describes the models. Section III summarizes our main results. In Sections IV and V, we prove the theorems for the CSS and ACS systems, respectively. Section VI extends the results to the parallel channels and Section VII concludes the paper.

Regarding notation usage, we use , , , and to respectively denote the zero matrix, the identity matrix, the zero vector, and the elementary column vector whose elements are all zero except that the element is 1. , , , and denote the inverse, the transpose, the conjugate transpose, and the vectorization operation of a matrix , respectively. We use to denote the block-diagonal matrix with blocks , and to denote the element of a vector .

## Ii System Model

Consider the time-invariant two-cell MIMO interfering multiple access channel with users in each cell and antennas at each node. The channel is described by the input-output equation given as

(1) |

where at the channel use, , are the vectors representing the channel output and additive white Gaussian noise at receiver , is the channel matrix from transmitter in cell to receiver , and is the channel input from transmitter in cell , for and . The elements of are assumed to be outcomes of independent and identically distributed (i.i.d.) continuous random variables and do not change with . The elements of , , are i.i.d. (both across space and time) circularly symmetric complex Gaussian random variables with zero mean and unit variance. Following the existing works in literature, we assume that all channel matrices are known by all nodes in the channel. Note that the value of is part of the model description, and therefore the idea of spatial extension [15] by adding more antennas as an achievable scheme of the channel is not allowed. The transmit power constraint is expressed as

(2) |

There are independent messages , and , associated with rates to be communicated from transmitter in cell to receiver . The capacity region is the set of all rate tuples for which the probability of error can be driven arbitrarily close to zero by using suitably long codewords. The DoF region is defined as

where and . The sum DoF is defined as

(3) |

We include the indices , to denote the for different and .

In this paper, we study the sum DoF achieved by IA scheme in signal space with symbol extension, where an arbitrary number of time slots is allowed. We consider both the traditional CSS system and the ACS system described respectively in the following two subsections.

### Ii-a Circularly-Symmetric-Signaling System

For a CSS system with -symbol extension, the input-output relationship of the extended channel is given as

(4) |

where the matrix is given as

(5) |

and

(6) |

Similar notation applies to and . Linear precoding and combining in extended signal space are described as

(7) |

where is the precoding matrix and is the combining matrix for user in cell that sends data streams described by the source vector . For simplicity, we assume that the data streams are independent by requiring . To introduce the flexibility of a transmitter not to send any information, which is allowed in both the practical operation and the theoretic analysis of the capacity and DoF regions, to the feasibility analysis, we let when . The feasibility condition of the linear IA scheme [7] in the -symbol extended signal space is

(8) | ||||

(9) |

for all and . We further define the feasible sum DoF that represents the largest number of sum degrees of freedom achieved by linear IA scheme in signal space with finite symbol extension for the CSS system as

(10) |

where is the set of all satisfying the IA condition (8), (9) with -symbol extension. We include the indices , to denote the for different and . Note that these definitions imply that .

### Ii-B Asymmetric-Complex-Signaling System

The main idea of ACS is to separate the real and imaginary parts of the transmit and receive signals [15]. The input-output relationship of the extended channel for an ACS system with -symbol extension is given as

(11) |

where is given as

(12) |

where

(13) |

where is the element of , and is given as

(14) |

where

(15) |

Similar notation applies to and . To introduce the notation, linear precoding and combining for ACS systems, which are similar to those for CSS systems, are given as follows.

(16) |

where , , and , whose detail descriptions, along with their feasibility condition of IA scheme, are omitted for brevity.

The feasible sum DoF for the ACS system is defined as

(17) |

where is the set of all satisfying the IA condition with -symbol extension, and where we use coefficient , instead of in (10), because of the fact that one real data stream only provides DoF. Note that these definitions imply that .

## Iii Main Results

We present our main results in this section. For the ease of comparison, we first summarize the important result in the literature as follows. The sum DoF and the feasible sum DoF of the time-invariant two-cell -user IMAC with antennas at each node, as defined in Section II, satisfy

(18) |

which is obtained by combining the lower bound from [4, 5], where CSS systems are considered, and the upper bound from [11]. This result can be easily extended to ACS systems by combining the ACS scheme with the IA scheme in [4, 5], and the extended result is

(19) |

Note that in (18) and (19), when and are integers, the lower bounds meet the upper bound. Otherwise, the upper bound is not tight due to the floor operations.

Our main results are the exact characterizations of and provided in the following theorems, whose proofs are deferred in Sections IV and V, respectively.

###### Theorem 1

(20) |

###### Theorem 2

(21) |

###### Remark 1

###### Remark 2

There are two different regimes of for . When , increases as increases. However, when , is not feasible and the multiuser DoF gain disappears when using CSS linear IA scheme. Similar observation can be made for . These observations are summarized as

(22) | ||||

(23) |

where and represent the degrading factors of the two-cell interfering MAC from the two isolated MACs for CSS systems and ACS systems, respectively.

###### Remark 3

With a slight notation abuse, we can combine the expressions for and as follows. Let be the feasible sum DoF that includes and , understood by context. Then we can combine (20) and (21) as

(24) |

where , which as explained later in Sections IV and V represents the number of active users in each cell, and is the channel diversity, which is for CSS systems and for ACS systems. Now we can clearly see how translates into , which in turn translates into .

###### Remark 4

Comparing , , and , we can divide parameters into three different regimes as follows. For the first regime, where , both CSS and ACS linear IA schemes with finite symbol extension achieve the DoF upper bound of the channel, i.e.,

(25) |

For the second regime, where , only ACS linear IA scheme with finite symbol extension achieves the DoF upper bound, i.e.,

(26) |

For the last regime, where , the characterization of and whether or not ACS linear IA scheme with finite symbol extension achieves the information-theoretic DoF are both still open problems for the considered time-invariant two-cell MIMO IMAC. However, we would like to mention that, on the contrary, for the time-varying setting, where the channel diversity constraint does not hold, the characterization of the sum DoF for all can be shown to be achieved by the CSS scheme given in [4, 5] with symbol extension.

## Iv Proof for CSS Systems

In this section, we prove Theorem 1, whose achievability and converse are stated separately in the following two theorems.

###### Theorem 3

(Achievability:) If , then

(27) |

Proof: The proof is given in Section IV-A.

###### Theorem 4

(Converse:) The feasible sum DoF satisfies

(28) |

Proof: The proof is deferred in Section IV-B.

Theorem 1 is then proved by combining Theorem 3, Theorem 4, and the upper bound in (18), and by communicating only to users in each cell when by letting some be zero matrices. Now we proceed to prove the achievability and converse.

### Iv-a Proof of Achievability

We provide a constructive proof that focuses on the unsolved case, where is not an integer. The scheme is described as follows. The first step is to choose , resulting in an extended signal space of . The second step is to choose the reference vectors by letting the elements of the vectors be outcomes of independent continuous random variables. The third step is to choose the precoding vectors. Let the precoding vector of user in cell be

(29) |

for and . Note that the construction ensures that all , align on in the extended receive signal space at receiver 2. The fourth step is to choose the combining vectors. Let the combining vector of user in cell be

where and for . The last step is to construct the precoding matrix and combining matrix as

(30) | ||||

(31) |

Now we proceed to show the achievable sum DoF of the scheme is . Since alignment of inter-cell interference is ensured by (29), the main task of the proof is to show that all signals are distinguishable at the intended receiver, despite the block-diagonal structure of the channel matrix given in (5). The following lemma establishes the linear independence required by the proposed scheme.

###### Lemma 5

Let

(32) | ||||

(33) |

Then the matrix is full rank with probability one.

Proof: We first show that the column vectors of are linearly independent with probability one. Consider the vector equation

(34) |

where are scalars. We aim to show that all scalars in (34) are zero. Using the property of block-diagonal matrices, we can write as

(35) |

where is the segment of at time as the similar notation given in (6) for . Substituting (35) into (34), we obtain the equivalent condition of (34) for each time slot as

(36) |

for . Note that the normalization terms in (29) and (35) are ignored in (36) for simple exploration without effecting the result. Defining as

(37) |

and rearranging (36) in matrix form, we have

(38) |

where is given as

(39) |

Let’s first consider the cases of and . Since is a generic matrix by design, it is invertible and right invertible with probability one for and , respectively. Multiplying both sides of (39) from the right by the inverse or the right inverse of gives us

(40) |

which implies

(41) |

for . By the fact that the set of all matrices can be considered as an -dimensional vector space, and by the assumption that all channel matrices are generic, the matrices , where , are linearly independent with probability one. Thus, (41) implies that . Thus, all column vectors of are linearly independent with probability one.

Consider the remaining case that , where is not invertible. Let be a basis of the orthogonal space of the column space of , implying Then (38) implies that there exist such that

(42) |

where

(43) |

and satisfies . Applying the operation to both sides of (42) and with some manipulations, we have (IV-A), which is given at the bottom of this page.

(44) |

By the fact that , and are outcomes of (statistically) independent random matrices, the fact that are linearly independent, and the fact that there are total vectors in the -dimensional vector space, we obtain that . Thus, the rank of is with probability one. Since the column vectors of are generic vectors in , along with the fact that and are outcomes of two (statistically) independent random matrices, we have .

The following remark discusses the key ideas in the proof.

###### Remark 5

The relation between the size of the channel matrix and the maximum number of the active users can be observed in (41). Note that here the idea of channel diversity is related directly to the channel matrix and is utilized to provide the achievability proof, while the extended channel matrix and converse proof are considered in [16].

### Iv-B Proof of Converse

The following lemma applies to all possible precoding matrices operated in the extended signal space with an arbitrary number of time slots .

###### Lemma 6

For all precoding matrices of arbitrary sizes , , in cell 1, if

(45) |

then with probability one

(46) |

Proof: Equation (45) implies that there exist and such that the precoding vector of user in cell one, i.e., the column of , satisfies

(47) |

Applying (47) to all column vectors of , we could write as

(48) |

Since is invertible with probability one, we can use the property of block-diagonal matrices to obtain

(49) |

where is given in (37) and is the segment of at time as the similar notation given in (6) for . Using (49), we can write

(50) |

where is the matrix given at the bottom of this page.

(51) |

(52) |

Let denote the element of . Then can be expressed as