A Zero-Forcing Linear Precoder for Downlink Multi-user LoS MIMO Channels

Mohammadreza Alimadadi, Abbas Mohammadi, Abdolali Abdipour and Mohammad Dehghani Soltani

Amirkabir University of Technology, Tehran, Iran {alimadadi_m, abm125, abdipour, m_dehghani} @aut.ac.ir

Abstract—In this paper we introduce an algorithm to design a zero-forcing linear precoder for downlink multi-user Line-of-Sight (LoS) MIMO systems. We observe that the existence of LoS component, changes the restrictions of conventional SVD-BD approach on channel dimensions and correlation of users. In addition, we find a new criterion that relates the received signal to noise ratio and optimal form of pure LoS channel matrix. We perceive that this algorithm improves performance nearly 1.5dB in compare to the SVD-BD method for Rayleigh channels in the case of two users.

Index Terms: MIMO systems, linear precoder, block diagonalization, LoS channel, Shannon capacity, uniform linear array.

I. INTRODUCTION Recently multiple-input multiple-output (MIMO)

communications systems have received much attention due to their attractive features that provide increment channel capacity [1]. However, researches done in this area often has been focused on single user case. In this case, a transmitter and receiver equipped with an array of antennas, exchanges data and it is assumed that there is no co-channel user around them. In recent years, another scenario named multi-user MIMO has been proposed in which it is assumed that a central station sends data to a number of co-channel users [2]. Researches in this area have focused more on two optimization problems: one is maximization of the capacity and the other is minimization of transmitted power. In the first issue, it is desirable that the total capacity of all users, under a constraint on the transmitted power, be maximized. In the second problem, we want to minimize the transmitted power under the condition that a minimum level of quality of service for each user to be fulfilled.

One important feature of downlink multi-users MIMO channels is that all transmitted antennas are in a central station and coordination between them is possible, but the receiver antennas is distributed between different users—users that often, there is not the possibility of coordination and cooperation between them [3]. In [4], capacity of downlink multi-user channel in the case that each user is equipped with a receiver antenna has been investigated and in [5] generalized to users with any number of receiver antennas. Common feature of all these studies is utilization of a popular technique called “Dirty Paper Coding” which was introduced by Costa in [6]. The main idea of this technique is that the transmitter can calculate the interference of each user and design a code that compensates this interference by using knowledge of the transmitted data and the channel matrix of each user. So the channel capacity will be thoroughly the same as the capacity of a channel in which there is no inter-user interference. In the multi-user MIMO channel this condition is true and we can compensate interference effects by designing a suitable precoder. The only defect of this approach is the use of

unusual coding techniques that lead to increase complexity of transceivers.

On the other hand, zeroforcing linear precoding method is the easiest way to achieve diversity gain in multi-user channel. In [7], authors have shown that if each user is equipped with just one antenna, as the number of users increases to infinity, using a particular technique called Zero Forcing BeamForming (ZFBF), the capacity will approach to its optimum value (DPC). For a general case where each user has more than one antenna, generalized zero forcing method based on SVD decomposition and block diagonalization (SVD-BD) is used [8, 9]. This method has some restrictions, including that the number of transmitter antennas should be greater than the total number of receiver antennas for different users.

In the all above methods it is assumed that transmitter has complete information on channel matrix. Since, it is difficult to realize such an assumption in practice, new versions of mentioned methods that give more attention to practical constraints, are introduced. For instance, in 3GPP LTE Release 8 one pre-defined precoder codebook is considered and each user transmits only its desired precoding matrix indicator (PMI) with channel quality indicator (CQI) to the central station. Based on this information, scheduling and user classification will be done.

Most of algorithms and methods that were mentioned above are based on modeling MIMO channel to a Rayleigh channel. Although, this model seems to be acceptable for many scenarios, but investigation of the other models can be effective in improving performance of communication systems. On the other hand, one important feature of the new cellular communication generation and network protocols is increasing the portable applications of high-speed data transmission. For example, most of the new laptops and tablets support 3G and 802.11n. This feature necessitates certain limitations and requirements. In this paper, we survey multi-user MIMO channel in case that there is a dominant LoS component between transmitter station and users. We will see that this model under circumstances when user’s location changes slowly, leads to special restriction in the design of zero-forcing based linear precoder. Subsequent sections of this paper are as follows: Section ΙΙ introduces the channel model and desired scenario. In section ΙΙΙ, we study the capacity of single user LoS MIMO system and obtain a new quantitative criterion for connection between received SNR and optimum form of channel matrix. In section ΙV, we focus our attention on the presented scenario and we show that how the physical parameters should be designed to achieve maximum capacity in low SNRs. Section V surveys linear precoders and block diagonalization method. In section VΙ, the proposed algorithm to design zero-forcing precoder for multi-user LoS channels is introduced. In section VΙΙ, the results of simulations performed on the proposed algorithm are presented and finally section VΙΙΙ states the conclusion and summary of our work.

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978-1-4673-2073-3/12/$31.00 ©2012 IEEE

II. CHANNEL MODEL Considered scenario in this paper is as follows: one central

station which is equipped with transmitter antennas sends data to users, each of them has receiver antennas. All antennas are uniform linear array (ULA) and inter-antenna space in transmitter and -th user are and , respectively. Arrays are parallel with each other, located along the axis and their total length are 1 and 1 , respectively. Fig. 1 depicts antennas structure and relative position of transmitter and a typical user.

It is assumed that the LoS component is the dominant component of channel. Nonetheless, as we consider Rayleigh model in section VΙΙ to compare results, it is also necessary to introduce the channel matrix for this model. The R components are modeled as independent symmetric complex Gaussian variables with zero mean and unit variance, i.e. vec R ~ , .

Entries of matrix L S are function of the distance between transmitter and receiver as well as physical parameters and . Transmitted signal from -th transmitter antenna arrives to -th receiver antenna passing the route . Phase shift between transmitter antenna array and -th receiver antenna is presented with the below vector:

, , … , , , 1

where is the wavelength of transmitted signal. According to previous equations, the channel matrix can be shown as follows:

L S H H … H H, 2

where · H is complex transpose and it is obvious that [10] cov vec L S , cov vec R , 1 1, … ,

III. CAPACITY OF SINGLE USER LOS MIMO CHANNEL Let us first consider single user MIMO system. We know

that the capacity of the system can be obtained in general using below equation, ∑ 1 / / , 3

Figure 1. The general structure of the transceiver in the desired scenario for a

LoS channel with dimensions .

where is the average signal to noise ratio and , is -th eigenvalue of matrix W which is defined as below: H ,H , 4

is the normalized transmitted power in the i-th subchannel. Usually there is a constraint on value of such that ∑ 1.

is the number of non-zero eigenvalues of matrix W. For simplicity we define , and , . It is obvious that . On the other hand, we know that the relationship between the eigenvalues of matrix W and the singular values of matrix H is , where is the -th singular value of matrix H. As mentioned in the previous section, entries of matrix H in general are in the form of

and regarding the dependence of and on , we can conclude that ’s and ’s are functions of model parameters. On the other hand, we know that the capacity in (3) can be maximized via utilizing waterfilling to . This maximum value is equal to, ∑ / / , 5

where max 0, and the value of is chosen such that the below equation is established, ∑ 1 , 5

In this section, we want to answer this question that how should be chosen so that the capacity in (5) would be maximized? On the other hand, with regards to 1, we can show that for pure LoS channel, ∑ . . 6

Therefore, the eigenvalue optimization problem can be expressed by the following equation, , , … , max , ,…, ∑ 7

or equivalently, , , … , max , ,…, ∑ 8

and both of them perform under the constraints of (5b) and (6).

In order to solve this problem, we consider two different cases :

a) ∏ 0, means at least one of the singular values of matrix H is zero. Assume that the number of non-zero singular value of this matrix is . It can be easily shown that in this case, capacity in (5) will be maximized when all ’s are equal. As a result, according to (6) it can be deduced, . 1 . 9

By substitution this result in (5b), can be obtained, 10

By substituting this value in (5a), we can express maximum capacity in terms of as below, C log 11

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To calculate a value of that maximizes capacity in (11), we derivative from with respect to and put the result equal to zero. log 0

Where by performing some simple mathematical operations it comes in the below form,

1 12

We define, : 13

so, (12) can be written as below, 1 14

It can be shown by numerical methods that the last equation has only one answer which is almost equal to, 0.255. Now according to (13), we can obtain a value of that applies to (12) as follow, 0.255 15

Since can have just integer values, we must choose the nearest integer to as .

According to above discussion, we can summarize case (a) as follows:

Result 1) In a pure LoS MIMO channel, in the single user mode and assuming that the matrix H has at least one non-zero singular value, then capacity in (5) is maximized when the channel matrix has n non-zero and equal singular values. n is the nearest integer to in the range 1, 1 .

b) ∏ 0 , means none of the singular values of matrix H is equal to zero. We can demonstrate that in this case, maximum capacity is achieved when numbers of singular values of matrix H are equal to in which, 16

and consequently according to (5b), we have,

17

With these two recent equations for we, can obtain in terms of and then by substituting these values in (5a), express capacity in terms of . Calculating the value of for which the capacity would be maximized, is complex and here we ignore it. We only want to show that if value of in (15) becomes less than 0.5 , capacity in the case of (b) for all values of 1 will be less than the capacity of case (a). According to (5b) and (16), it is evident that for number of singular values of matrix H, 0 and for other singular values . Therefore, according to (3) capacity is equal to, ∑ log 1 18

On the other hand, in the case of (a) for we have:

A ∑ log 1 19

and it is obvious that . According to results of (a) and (b), we can conclude the following significant theorem:

Theorem 1) In a pure LoS MIMO channel, capacity in (5) would be maximized if channel matrix has non-zero and equal singular values. is the nearest integer number to in (15). If 0.5 , maximum capacity is obtained as mentioned in section (b).

Theorem (1) forms the basis of our work in this paper. This theorem represents general form of matrix H that is needed to obtain maximum capacity of LoS MIMO channel.

IV. OPTIMUM DESIGN OF LOS CHANNEL IN SINGLE USER MODE

As we saw in the previous sections, entries of channel matrix depend on the model parameters and the distance between transmitter and receiver. On the other hand, theorem (1) expresses the general form of matrix H to maximize capacity. As we saw, in this context the value of in (15) plays an important role. Due to this equation we can say that the value of depends on the average received signal to noise ratio (SNR) and also values of and (or equivalently and ). Here we consider small values of . If is sufficiently small so that 1.5, then according to theorem (1) we have 1, and the maximum capacity of LoS channel would be achieved when matrix H has just one non-zero singular value. In this case, the only non-zero eigenvalue of matrix W is equal to . . This issue is also consistent with the intuitive sight, because in low SNRs, it is better that the total transmitted power focused on one subchannel, in order to maximize the channel capacity. In the following discussion, we assume the received signal to noise ratio is small enough that results to 1.5, in other words, 0.255 1.5 .. 20

According to theorem (1), for maximizing the capacity, matrix W should has just one non-zero eigenvalue equal to . . This case occurs when, . , 21

where is a matrix with entries all equal to one. As it was mentioned in previous sections, entries of matrix H depend on model parameters and the distance between transmitter and receiver. Thus, by choosing these parameters properly, the necessary condition for maximizing capacity which was expressed by (21) can be achieved.

According to content of first section and (1) and (2), it is explicit that the aforementioned condition is equivalent to the fact that inner product of the rows of matrix H should be equal to V, , 0 , 1 22

For this purpose, we must calculate , i.e. the distance between n-th transmitter antenna and m-th receiver antenna, in terms of model parameters and the distance between transmitter and receiver. Fig. 1 shows the desired scenario. Transmitter and receiver use ULA antennas and the inter-antenna space are and , respectively. These antennas are perpendicular to the surface and the radial distance between two arrays is R. By considering cylindrical coordinate system and assuming that the origin is at the place of first transmitter antenna, vectors corresponding to each antenna can be expressed as below, 0,1, … , 1 ,

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0,1, … , 1 , 23

where and represent unique vectors in orientation of r and z axis, respectively. Consequently can be achieved as follows: | | 1 24

By utilizing the approximation √1 1 , that is satisfied for 1, the above equation can be simplified as, 1 25

Above condition is satisfied when , i.e. the distance between transmitter and receiver is much greater than the dimensions of ULA arrays. According to (22), for maximizing capacity we should have, 0 , 1 26

Therefore, according to (2), , ∑ ∑ ∑ 2 0 , 1 27

To establish the above condition, it is sufficient to have, √2 . 28

So according to (3), the channel capacity is, log 1 29

V. LINEAR PRECODER AND BLOCK DIAGONALIZATION ALGORITHM

In [8], block diagonalization algorithm for multi-user MIMO channels has been introduced. Since this algorithm is the basis of our work in subsequent sections, we briefly review it in this section. The relation between input and output of system can be expressed as follows, , 30

where d is the transmitted data vector of length m and M is the precoder matrix of size at the transmitter which covers all the precoding operations. By means of matrix D of size

, received signal vector , becomes an estimation of the transmitted data d.

Suppose a downlink multi-user channel with K users and a transmitter station. Each user has antennas and transmitter station is equipped with antennas. We express such a channel with , , … , . For instance, a 3,3 4 channel has one transmitter station with 4 antennas and two users each equipped with 3 antennas. Received signal at j-th receiver can be expressed as below, ∑ , 31

where and are modulation matrix and transmitter vector of all users except j-th user. … … … … 32

If H is completely known at transmitter, you can maximize capacity by selecting M as right singular vectors of matrix H and utilizing waterfilling technique [3]. To achieve this goal, the multiplication of should be a diagonal matrix. Although this case is not optimum and do not maximize capacity of channel, it is very close to optimum case.

If we define matrices and as follows, … … , 33

When is a block diagonal matrix then the optimal solution obtains under the condition of eliminating inter-user interference. We see that this solution causes some restrictions on the channel dimensions and independence of different matrices and limits the number of user that can be serviced simultaneously. But these constraints are not cumbersome in practice, because MIMO systems usually use the benefit of the SDMA, TDMA and FDMA combination. In practice, users can be divided into groups such that the mentioned restrictions are met in each of them and separate time or frequency intervals are assigned to different groups.

A. Block diagonalization algorithm The purpose of this algorithm, is expression of a specific

procedure to find the optimum matrix S, so that all inter-user interference become zero. Therefore, S S will be a block diagonal matrix. To eliminate all inter-user interference we should have for . By considering the restriction on transmitted power, capacity can be obtained as below, max , ∑ log | | 34

We define matrix as follows, … … 35

Being zero of inter-user interference implies that places in the null space of matrix . This definition is led to a restriction on the channel dimensions. Only if the dimensions of null space of matrix are greater than zero, elimination of interferences is possible. This condition is established when rank . So, for an arbitrary , block diagonalization is possible only when rank , … , rank . We assume that this condition is fulfilled. Define rank 1 and consider SVD decomposition of matrix as below, H , 36

where , contains first right singular vector and possesses the other vectors. Therefore, columns of

form a basis for the null space of matrix and hence any linear combination of its columns can be considered as a candidate for matrix . Now consider the production . We show the rank of this matrix by . Transmission of data under the condition of zero interference is possible only if the condition 1, is satisfied. In other words, we should have

. A sufficient condition for 1 is that at least one row of matrix to be linearly independent of the rows of matrix . To satisfy this condition, we should refuse to put the users that have correlated channel matrix in the same group.

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With the assumption of satisfying of two above conditions, we define matrix as follows,

′ 37

Then, we can write capacity of system under the condition of inter-user interference as below, max log | H H| 38

Hence, aforesaid problem is converted to a single user case and for its solution we should consider as the weighted right singular vectors of .

Instead of computing the SVD decomposition of large matrix , block structure permits us to determine the SVD of each user’s matrix separately and combine them, next. Consider SVD of matrix as below, ∑ H , 39

where ∑ is a matrix of size containing non-zero singular values and includes the first right singular vectors. Product of and utilizing waterfilling method leads to maximize the capacity, under the condition of zero forcing of the inter-user interferences. Therefore, we define

as below, … , 40

where is a diagonal matrix with entries that scales transmitted power by each of columns. Now, we can calculate maximum capacity as below, max log |I ∑ | 41

where:

∑ ∑ ∑ 42

Values of ’s are obtained using waterfilling upon the entries of matrix ∑. Thus, block diagonalization algorithm can be summarized as follows,

1) For each user compute using SVD decomposition of matrix .

2) Calculate 3) After SVD decomposition of , compute non-

zero singular values λ and right singular vectors

corresponding to . 4) Obtain using waterfilling approach upon λ so that ∑ ∑ log 1 is maximized under the condition

of ∑ ∑ 1. 5) Calculate / for each user. 6) Obtain as, … .

VI. LINEAR PRE-CODING ALGORITHM FOR LOS CHANNEL AT LOW SNRS

Now we want to utilize block diagonalization algorithm for LoS channel. For this purpose, two required conditions that

were mentioned in previous section, should be satisfied. These conditions are,

I) rank , … , rank II) At least one row of matrix should be linearly

independent from rows of matrix .

The first condition provides a constraint on the channel dimensions, since rank 1 . Therefore, in general we should have 1 . On the other words, the number of transmitter antennas cannot be smaller than a specified number.

The second condition states that the users whose channel matrices are highly correlated, should not be placed in the same group. In the LoS channel described in the previous sections, it is clear, the users that are in the same distance from central station have identical channel matrix and therefore they should not be multiplexed together. On the other hand, as stated in the previous sections, the values of and should be fulfilled (27). Suppose we want to multiplex the users at the distances of from central station and observe what relation ’s should have with each other. In order to establish (27) for all users, we should have, 2 2 1 , 43

where is an arbitrary integer. So, should be properly chosen so that all the above equations are fulfilled. Putting these equations equal, we obtain, 2 , 44

where is the distance of first user from central station and is considered as a parameter. With this assumption, values of

for 2 are obtained in terms of as follows, 2 45

It is obvious that from each ring with the radius of from the transmitter, just one user can be placed in each group.

According to previous contents, we can express proposed algorithm for multi-user LoS MIMO channel as below:

1) All users continuously monitor the received channel’s SNR and make central station aware of it.

2) Users that their signal to noise ratio satisfy (20), put their inter-antenna space equal to 2 , where is the distance of k-th user from central station.

3) Central station select users that their received SNR is smaller than the required limit. This selection should be done in a way so that constraint I and (45) are fulfilled. Therefore, the inter-antenna space obtains from (44).

4) In Central station by utilizing block diaganalization algorithm, the optimum precoder matrix would be achieved. Consequently, matrices , , … , are obtained and the precoding operations would be done in the transmitter.

VII. SIMULATION RESULTS In this section, according to obtained equations in the

previous sections and performing several simulations, we evaluate the proposed algorithm. Considered system here, is a 3,3 4 pure LoS MIMO channel. This system contains one transmitter station with 4 antennas and two users each equipped with 3 receiver antennas. Obviously, in this case,

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constraint (I) on channel dimensions is satisfied. Here we can hint one important advantage of proposed algorithm in compare to SVD-BD in Rayleigh channels. When 2, with regards to rank 1, constraint (I) comes into n 1. So it is not necessary that the number of transmitter antennas to be more than the total receiver antennas of the two users.

As mentioned in the previous sections, to establish the second condition, two users should not be in the same distance from transmitter. Otherwise, channel matrix will be identical for both users and constraint II will not be established. By utilizing (45), we can obtain relation between and as 4 .

Fig. 2 depicts capacity region for the proposed algorithm in the LoS channel. The average value of received SNR and working frequency are 1 and 30GHz, respectively. For comparison, diagram of capacity for a similar system in the Rayleigh channel has been shown. We should mention that the capacity of Rayleigh channel depends on channel matrix, but here we perform 10000 different Rayleigh channel realizations for averaging on the mutual capacity of two users. The star points shown in the figure, state the sum capacity of channel. It is obvious that the proposed algorithm improves the capacity of LoS channel compared to Rayleigh channel. Indeed, one can show that if signal to noise ratio for Rayleigh channel increases from -1dB to 0.5dB, performance of two systems will be approximately equal. This issue has been depicted in Fig. 3. In this figure sum capacity of the users in two channels, are compared in terms of . As one can see, the proposed algorithm has improved performance of the system approximately 1.5dB with respect to the traditional SVD-BD in the Rayleigh channel.

VIII. CONCLUSION In this paper, we introduced a zero-forcing linear precoder

for multi-user LoS MIMO channels. We showed that in low SNRs, one of the main constraints of SVD-BD approach that exists on the channel dimensions would be resolved. Equation (15) is one of the most significant results of this paper that proposes a quantitative criterion to determine the general structure of optimum channel matrix in LoS condition. Simulation results showed that the algorithm, has a better performance which is approximately 1.5dB in compare to SVD-BD method for Rayleigh channels when two users are considered. Although, we focus our attention just to low SNRs, according to (15) we can obtain similar algorithms for any arbitrary interval of . However, this will lead to greater complexity.

Acknowledgments

This work is supported in part by The Center of Excellence on Radio Communication Systems of Amirkabir University of Technology.

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Figure 2. Capacity region for a , system using proposed algorithm

for a LoS channel and SVD-BD for a Rayleigh channel in .

Figure 3. Sum capacity of all users in terms of for a , system using proposed algorithm for a LoS channel and SVD-BD for a Rayleigh channel.

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