Degrees of Freedom Rate Region of the K-user Interference Channel with Blind CSIT Using Staggered Antenna Switching

Degrees of Freedom Rate Region of the -user Interference Channel with Blind CSIT Using Staggered Antenna Switching

Milad Johnny and Mohammad Reza Aref
Information System and Security Lab (ISSL),
Sharif Universiy of Technology, Tehran, Iran
E-mail: Johnny@ee.sharif.edu, Aref@sharif.edu
Abstract

In this paper, we consider the problem of the interference alignment for the -user SISO interference channel with blind channel state information at transmitters (CSIT). Our achievement in contrast to popular user interference alignment (IA) scheme has more practical notions. In this case every receiver is equipped with one reconfigurable antenna which tries to place its desired signal in a subspace which is linearly independent from interference signals. We show that if the channel values are known to the receivers only, the sum degrees-of-freedom (DOF) rate region of the linear BIA with staggered antenna switching is , where . The result indicates that the optimum DoF rate region of the user interference channel is to achieve the DoF of for an asymptotically large network. Thus, the DoF of the -user interference channel using staggered antenna switching grows sub-linearly with the number of the users, whereas it grows linearly in the case where transmitters access the CSI. In addition we propose both achievability and converse proof so as to show that this is the DoF rate region of blind interference alignment (BIA) with staggered antenna switching.

Blind CSIT, degrees-of-freedom (DoF), blind interference alignment (BIA), staggered antenna switching, multi-mode switching antenna.

I Introduction

The new increasing demand for higher data rate communication motivates researchers to introduce new tools to reduce channel constrains such as interference in the transmission medium. In the network area, due to high speed of progressing, the opportunities for innovation and creativity increases. Interference channel due to its important role in today’s communication systems has been the focus of attention in today’s wireless networks. The importance of the problem of finding the capacity of interference channel is so essential that after point-to-point communication scenario it is the second problem which was introduced by Shannon [1] and it has many applications in today’s communication networks. Unfortunately finding the exact capacity of the interference channel is so hard that it is still open for near half of the century. While finding the exact capacity of many networks is still open, DoF can analyze capacity characteristics of such networks at high regions. Recently [2], by the basic idea of IA with some constraints shows that one can achieve DoF for the fast fade interference channel. This method for practical cases where transmitters do not have access to channel values fail to get any achievement. The CSI was not the only barrier for implementation of such a method, the long precoder size at transmitters and the high speed channel changing pattern show further impractical aspects of such a method. Such assumption is hard to materialize under any practical channel feedback scheme. To combat the CSIT problem, there are two different strategies which are related to blind CSIT and outdated CSIT (delay CSIT). Moreover, as a first step to study the impact of the lack of channel knowledge, [3] shows that with some conditions on the direct and interference channels, one can perfectly or imperfectly align interference; if half of the interference channel values are not available at both the transmitters and receivers, one can achieve the DoF of . In [4], the authors show that artificially manipulating the channel itself to create the opportunities, one can facilitate BIA. They equip each user with simple staggered antenna which can switch between multi-mode reception paths. In this work by the use of staggered antenna switching one can achieve DoF for the well-known MISO broadcast channel where each receiver is equipped with multi-mode antenna. After finding DoF rate region of MISO broadcast channel in the case of delay CSIT [5] there are several works characterizing the DoF of the interference channel with the delayed CSIT. In [6], with the assumption of delay CSIT it is shown that the DoF of the -user interference channel can achieve the value of as . The BIA scheme with staggered antenna switching only requires multi-mode antenna switching at the receivers, which does not need any significant hardware complexity [7]. In [8], for the 3-user interference channel Wang shows that using staggered antenna switching one can achieve the sum DoF of in the case of blind CSIT. Alaa and Ismail in [9], trie to generalize the DoF rate region of 3-user interference channel with staggered antenna switching to the user interference channel but to some extend it has contradiction with our work for the and evident case of where DoF is one. In this paper, we generalize Wang problem for the case of user interference channel and we show that with the aid of multi-mode antenna switching at receivers, the sum DoF is . This result indicates that when the number of the users limits to infinity, BIA can achieve DoF which is in contradiction with the DoF upper-bound of , thus the sum DoF does not scale linearly with as in the case when CSIT is available, but rather scales sub-linearly with the number of users.

I-a Organization

This paper is organized as follows. The next section describes the system model. In Section III, we explore overviews of the main result. In section IV, by providing both achievability and converse proofs we show that is the sum DoF rate region of the user interference channel with staggered antenna switching. Finally, we draw our conclusions in Section V.

Ii System Model

Fig. 1: Structure of the two-mode staggered antenna switching. In this case every receiver equipped with two antennas and a switch which can select between two different modes.

Consider the user interference channel, where each receiver has more than one receiving antenna. In this case at each time snapshot all the receivers can switch to one of the receiving antennas to receive its desired signal from corresponding transmitter and all other transmitters as interferences (see Figure 1). This channel consists of transmitters and receivers . Let a discrete interference channel be tuple , where and are finite input and output of the channel respectively; in the interference channel, the input of transmitter is represented by . Similarly the output of the channel can be represented by column matrix of . For a specific case where the thermal noise power is zero, is a collection of such a diagonal matrices which maps to received signal at receiver and represents channel model. Therefore, the received signal at the receiver consisting of time snapshot channel uses can be represented as follows:

(1)

where represents the received signal over channel uses (time or frequency slots), is the transmitted signal vector by the transmitter subject to average power constraint of , is an additive white Gaussian noise in which , and is a diagonal matrix representing the channel model between the and . The channel matrix can be written as:

(2)

where depending on the number of antenna modes . In other words the diagonal matrix can be represented as follows:

(3)

where , shows the switching pattern matrix at . This switching pattern for all channels which end in the same destination e.g. have the same effect. We assume all the channel links between different transceivers are constant for channel uses. Also, is a column matrix with the size of and can be represented as follows:

(4)

where is the number of symbols transmitted by the user over channel uses, is the transmitted symbol and is an transmit beamforming vector for the symbol. The equation of (4) can be defined and simplified as follows:

(5)

where and . Also, is one of the basic vectors of designed precoder at .

Ii-a Degrees of Freedom for the user Interference Channel

In the -user BIA interference channel using staggered antenna switching, we define the degrees of freedom region as follows[11]:

(6)

Iii Overview of the Main Result

In this paper we explore interference alignment for the user interference channel with blind CSIT. We provide both achievability and the DoF upper bound by the linear interference alignment. The summary of the results can be expressed by the following theorem.

Theorem 1

The number of DoFs for the -user SISO interference channel with BIA using staggered antenna switching is .

The result indicates that when the number of users limits to infinity and there is not any information at transmitters about CSI, the number of DoFs goes to .

Iv Outer Bound on the Degrees of Freedom for the BIA user Interference Channel Using staggered Antenna Switching

In this section, we derive an upper bound on the sum DoF of the interference channel with BIA using staggered antenna switching at the receivers. In the next theorem, we assume no CSIT, each receiver is equipped with a reconfigurable antenna with an arbitrary number of antenna modes, and each transmitter has a conventional antenna.

RX set

TX set

TX

RX

Fig. 2: In this figure we show transceivers number of the set with the closed circular shape. The complimentary transceivers out of this circular shape can be modeled by the set . Also there is a connection between all transmitters and receivers but to avoid being so crowded we show a few of them.

Now consider the set where and . We assume every basic vector from each transmitter aligns with interference generated from transmitters at receivers. In other words, if is one of the basic vectors of transmitter, we have:

(7)

Where, and .
Remark: if not, the desired signal space is polluted by interference.
Lemma 1: If is aligned with interference of transmitters (in the set of ) at the members of the set receivers, it can not be aligned with the interference generated from the transmitters set of at receivers of the set .
Proof: Suppose that and are transmitters in the set of . Also, is a receiver in the set of and is a receiver in the complimentary set of (in the set of ). From the assumption of this lemma we can assume:

(8)

From Lemma 2 of [8], since and are diagonal and have the same changing pattern, .
Suppose not. We take the negation of the given statement and suppose it to be true. Assume, to the contrary, that:

(9)

From this assumption we have:

(10)

Since and have similar changing pattern, we get:

(11)

Therefore, since , we have:

(12)

and finally we get:

(13)

The above relation shows that the desired signal at receiver has been polluted by the interference of transmitter. Hence by the assumption of we have a contradiction. This contradiction shows that the given assumption is false and the statement of the lemma is true. So, this completes the proof.

  • missingi_2…i_r

, i_1 ≠i_2 ≠…≠i_ri_1i_2i_rj^thj∉{i_1,i_2,…,i_r}i’_1,…,i’_r ∈{i_1,i_2,…,i_r}

Iv-a Converse Proof:

The converse proof follows from the following upper bound on the DoF of the user interference channel with BIA. At receiver the interference signal from transmitters ,,… and , where occupy dimensions. In other words, every shared vectors between different users (,,… and users) occupy just only one dimension at receiver. On the other hand the total number of dimensions is . Therefore, at receiver we have:

(15)

where, the coefficient comes from this fact that just only occupy one dimension at receiver while it counts times when we calculate . Similarly at all the receivers we have:

(16)

Adding all the above relations we conclude that:

(17)

in addition, it is clear that:

(18)

Since for the value of we have:

(19)

which shows that:

(20)

Therefore from (17) we have:

(21)

After simplifying (21) we get:

(22)

thus, we complete the converse proof.
In order to find the maximum value of we analyze the continuous function of . The first derivation of this function has just one positive root of which shows that it has just only one extremum point. Also it can easily be shown that for the function is greater than or equal to zero. Since and the function for is something like Figure 2.

Fig. 3: The function versus continuous variable of for .

Therefore, the maximum value of the can be achieved by finding out the minimum value of such that:

(23)

In order to find to satisfy condition we have:

(24)
(25)
(26)
(27)

Therefore, the minimum value of which satisfies above equation is . The exact value of has been shown for different values of in Figure 2. Thus, for a large number of users, the sum DoF of BIA in the -user interference channel approaches . In the following section, we propose an algorithm to systematically generate the antenna switching patterns and the beamforming vectors such that the sum DoF is achieved.

V Achievable DoF Using Staggered Antenna Switching

V-a Beamforming vectors generation

To design beamforming vectors, we assume all the elements of the beamforming vectors are binary, thus . Let’s design the precoder matrices and switching pattern from the basic matrix of . The basic matrix can be expressed as follows:

(28)

where, , and is a matrix with distinct rows and each row containing exactly ones. Also is an all-ones square matrix and is an identity matrix. For instance, in the case of and , the matrix can be represented as follows (take note is not the optimum value for the ):

(29)

The matrix consists of columns where column of this matrix is expressed by . In this case all the basic column vectors of the precoder matrix at is chosen from the following set:

(30)

It means that all the precoder matrices have the size of or equivalently have the size of . Thus every different transmitter like and have exactly one shared basic vector which can be represented as follows:

(31)

in other words:

(32)

V-B Antenna Switching Pattern at the Receivers

As it was declared in section II, each receiver equipped with a multi-mode antenna can select among different receiving paths. Therefore, for the switching pattern where we should find proper among different switching patterns to satisfy the following conditions:

  • The shared basic vector which is used commonly at after being multiplied by should be aligned at their complimentary receivers .

  • The shared basic vector which is used commonly at after being multiplied by channel matrices should be linearly independent of each other at their corresponding receivers .

Assume that the matrix is an matrix which is defined as follows:

(33)

Now, let the be the antenna switching pattern at which is represented by column of the matrix as follows:

(34)

where shows row and column of the matrix . Therefore, in our designed switching pattern each receiver has been equipped with single antenna by different receiving mode. Every basic vector like can be equivalently expressed by sub-matrices as follows:

(35)

where and are two and column matrices, respectively. Now we must show that all the basic vectors generated at the specific transmitter like are linearly independent. The following lemma shows that every generated basic vector from (31) are linearly independent.

Lemma 1

For all the values of the number of the users and , every generated basic vector at a specific transmitter e.g. from Hadamard product of all the combination of column vectors of the are linearly independent.

Proof:

Consider , all the basic vectors of this transmitter chosen from the following set:

(36)

We must show that at where , all the sub-matrices are linearly independent. Since the matrix generates the elements of these vectors we have to analyze it. Each row of the matrix contains exactly ones. If , it means that for generating nonzero element e.g. in the especial position of the sub-matrix which is shared among transmitters we must have:

(37)

In appendix I we show that for every value of and the value of . For the case of except one of which is all zero matrix, referring to equation (37), all the sub-matrices have a nonzero element in the unique position. For the case of , it is clear from (37) that all the sub-matrices have at least one nonzero element in the unique position. Therefore, for all values of and all the generated are linearly independent. Since are the sub-matrices of the basic vectors of are linearly independent too. Therefore, the proof was completed.       

Lemma 2

Using at for every basic vector , the received vectors are aligned with each other.

Proof:

The proof was provided by analyzing both nonzero elements of the basic vector and the diagonal matrix of . Similar to (35), the basic vector of can be represented by the sub-matrices as follows:

(38)

From (32), for the matrix we have:

(39)

It means that the only nonzero elements of is its elements. Similarly for the switching pattern at receiver e.g. we have:

(40)

similarly for nonzero elements e.g. the value of is equal to 1. Therefore, at