Performance analysis of low-complexity dual-cell random beamforming transmission with user scheduling
© Zhu and Yang; licensee Springer. 2011
Received: 26 January 2011
Accepted: 1 December 2011
Published: 1 December 2011
In this paper, we study three low-complexity random beamforming transmission schemes for dual-cell multiuser multi-input-single-output systems. Among them, selfish random beamforming and interference-aware random beamforming need no information exchange between cells, while random beamforming with limited coordination (LC-RB) selects the beamforming vector and users with the help of a small amount of overhead signaling. We develop the exact analytical expressions of the ergodic sum-rate for the resulting systems, based on which, we compare the performance of the proposed schemes in dual-cell environment. We show through selected numerical examples that LC-RB achieves tremendous performance gain over the other schemes, especially when users are populated along the cell boundary, while only requiring beam index sharing between base stations. Furthermore, we propose an adaptive implementation strategy for the more general scenario, where users are arbitrarily distributed within the cell.
In order to meet the increasing demand for high data rate multimedia wireless services, future wireless systems are evolving toward universal frequency reuse, where neighboring cells may utilize the same radio spectrum. Such scenario also applies to the emerging femtocell systems. As such, the performance of future wireless systems will be mainly limited by intercell interference . In parallel, multiple antenna techniques (MIMO) can improve the spectral efficiency of wireless communication systems and provide significant throughput gains. In addition, multiple antennas can also be exploited to suppress intercell interference through coordination among multiple base stations (BSs) . Therefore, the resulting coordinated multicell transmission, also known as network-MIMO, has drawn significant research attention recently.
With conventional network-MIMO approach, multiple coordinated BSs effectively constitute a 'super-BS', which transforms several interfering channels into a MIMO broadcast channel [3–5]. The optimal dirty paper coding (DPC)  and sub-optimal linear precoders have been studied for network-MIMO scenario [7–12]. With some simplified network models, analytical results have appeared in [13–15]. These coordination strategies require, however, the complete channel state information and sometimes, even the user data to be shared among coordinating BSs, which introduce huge load of overhead signaling . Note that, although BSs are usually connected with wired connections with each other through the switching center, these connections are already fully loaded with the increasing amount of multimedia data traffics. Recently, an adaptive strategy was proposed which cancels intercell interference between scheduled user using joint beamforming only when the interference was significant . But user selection was not considered there.
Unlike previous works in the literature, we focus on more practical coordinated beamforming transmission schemes for dual-cell MIMO systems based on random beamforming in this paper. For MIMO systems with random beamforming, in order to achieve good performance, proper user selection is essential. That also applies to coordinated beamforming transmission. To limit the amount of overhead signal between BSs and minimize the additional burden to the back haul connections, we consider the user selection schemes that exchange no or limited amount of control information to achieve coordinated beamforming. Specifically, we present and study selfish random beamforming (SRB), interference-aware random beamforming (IA-RB), both of which require no information exchange between cells, and random beamforming with limited coordination (LC-RB), where only the selected beam index is shared among BSs. We would like to point out that some of these schemes have already been discussed in certain standard activities, such as 3GPP framework . Instead of asymptotic analysis according to most of the literature, which assumes that the number of users is very large, our contribution is to accurately quantify their performance through statistical analysis. We firstly derive the exact analytical expression for the sum-rate of the resulting systems assuming that all the users are located along the cell boundary and average intercell interference power at mobile users can be considered approximately identical. Selected numerical examples show that LC-RB can offer significant sum-rate capacity gain with low system complexity. During the sum-rate performance analysis, we develop the exact statistics of users' SINRs based on some new statistic results of projection norm squares, which can be broadly applied into the performance analysis of other related systems.
We then extend the study to the more practical scenario, where the users are randomly distributed within the whole cell, and average intercell interference power can no longer be regarded as identical, due to the different distances from the neighboring BS to the users. In this case, we propose an adaptive coordinated beamforming scheme and evaluate its performance and complexity. Specifically, the BS can decide whether to perform LC-RB to mitigate the intercell interference, or just to perform SRB, based on the distance information gathered from the mobile users. Note that our scheme differs from the adaptive scheme in  in that we consider user selection in each cell. Selected numerical examples show that LC-RB can offer significant sum-rate capacity gain with low system complexity.
The rest of the paper is organized as follows. In the next section, the system and channel models are introduced. Section 3 presents the proposed transmission strategies. The sum-rate performance analysis of the proposed systems is given in Section 4 (for identical interference power case) and in Section 5 (for non-identical interference power case). In Section 6, we investigate the adaptive implementation strategy for the general case. The paper concludes in Section 7. This paper generalizes the conference version in  by considering the analysis of all three schemes and extending the design to non-identical interference power case.
2. System and channel models
h1iis the N × 1 channel vector from the base station 1 to the i th user in cell 1, i.e..
h2iis the N × 1 channel vector from the base station 2 to the i th user in cell 1, i.e. .
h1jis the N × 1 channel vector from the base station 1 to the j th user in cell 2, i.e..
h2jis the N × 1 channel vector from the base station 2 to the j th user in cell 2, i.e. .
We assume that, with proper power control mechanism, the users experience homogeneous Rayleigh fading with respect to their target BS. Thus, each component of h1iand h2jis modeled as independent and identically distributed (i.i.d) complex Gaussian random variables with zero mean and unit variance. When mobile users are randomly populated in their specific cell coverage area, the average received interference power is dynamic, due to the various distances from the neighboring BS to the users. Each component of the interference channel vector h1jand h2iis modeled as independent complex Gaussian random variables with zero mean and variance δ j (resp. δ i ) with respect to user j (resp. i). As will be seen in later section, we will focus mostly on the interference channel from BS2 to the selected user in cell 1, denoted by h2i*. We assume that each component of h2i*is modeled as i.i.d. complex Gaussian random variables with zero mean and a common variance δi*. For the special case that the mobile users are distributed along the cell boundary, and thus all the users have approximately the same distance with the neighboring BS, we can assume each component of h1jand h2iis modeled as i.i.d. complex Gaussian random variable with zero mean and variance δ, i.e. δ i = δ j = δ for all i and j.
respectively, where s i (i = 1, 2) are data symbols to selected users and w i (i = 1, 2) are the corresponding beamforming vectors. We generally have ||w i ||2 = 1, i = 1, 2. P1 and P2 are the corresponding transmit powers for cell 1 and 2, n i and n j are the additive Gaussian noise.
3. Transmission strategies
In this section, we present the fundamental principles and the mode of operations of several reduced-complexity dual-cell beamforming transmission strategies. For analytical tractability, we focus on dual-cell scenario.
3.1. Selfish random beamforming (SRB)
where N0 denotes the Gaussian noise power.
3.2. Interference-aware random beamforming (IA-RB)
3.3. Random beamforming with limited coordination (LC-RB)
The scheme differs from SRB and IA-RB as it achieves coordinated beamforming transmission with limited overhead signaling. Without loss of generality, we assume that BS1 starts its user selection for beamforming transmission first. In particular, BS1 performs exactly the same as SRB to complete the beam and user selection for first cell.
The selected user by BS1, referred as user i*, will estimate its MISO channel from the interfering base station BS2, denoted by h2i*. With this channel state information, user i* will determine the beamforming vector that leads to the smallest amount of interference to itself and should be used by BS2, and feed its index back. Mathematically speaking, the beamforming vector w2 should satisfy .
BS1 will inform BS2 the desired beamforming vector to use through the wired backhaul connection. BS2 will broadcast training symbol using the selected beamforming vector for its own user selection. Every user in the coverage area of BS2 will estimate and feedback its received SINR. BS2 will select the user that achieves the maximum SINR among all users to serve, i.e. user j* where j* = arg max j γ2,j.
According to the transmission schemes described above, it has been observed that all the strategies can be smoothly extended into the general multi-cell cases. For instance, SRB and IA-RB can be directly applied in multi-cells, due to its non-coordination between BSs. And for LC-RB, all the participating cells can work sequentially in order to complete the scheduling. Indeed, a 3-cell or 7-cell setup will be more practical, however, despite the number of interfering sources and geometric distributions, the strategies keep similar from the interference control perspective.
It is also worth noting that the similar design have been considered in the standard activities for LTE Advanced and IEEE 802.16 m . In this work, we complement those simulation studies of such designs with the exact sum-rate capacity analysis.
4. Sum-rate analysis for identical average interference power case
This section provides the sum-rate analysis assuming that the average interference power is identically distributed, i.e. δ i = δ i = δ. Essentially, we consider the scenario that mobile users are distributed along the cell boundary. Meanwhile, the transmitted signal energies for both cells have been defined as Es 1= P1T, Es 2= P2T, in which T denotes the transmit time duration. Notice that in our later work, without loss of generality, we all assume P1 = P2 = 1. That is why all the power term has been dismissed in the later derivations.
4.1. Common analysis
We first present some statistical results on the ordered projection norm squares, which will be broadly applied in the later analysis. Noting that each component of vectors discussed in this section is modeled as i.i.d. complex Gaussian random variable with zero mean and unit variance.
where A = (N - 1)(B - 1 - i) + N - 2.
4.2. Sum-rate analysis
where and are the PDF of received SINR of the selected users in cell 1 and 2, respectively. We now derive the exact statistics of the selected users' SINRs.
and q j term are i.i.d. with distribution over , but with variance δ2, whose PDF is the same as (15).
respectively. Note that the element of vector has variance δ here.
The statistics of the received SINR at the selected user by BS2 is exactly the same as that of IA-RB scheme presented previously, with PDF given in (22).
4.3. Numerical examples
In this section, we present and discuss selected numerical examples to illustrate the mathematical formalism on the sum-rate analysis of the proposed coordinated beamforming schemes. Noting that all the analytical results in this paper have been verified through Monte-Carlo simulation.
For comparison purpose, we also provide the simulation results of one of the popular conventional coordinated beamforming techniques with user selection, with CSI exchange between cells, which is called coordinated zero-forcing beamforming (CZF). Specifically, the CZF option relies on a simple multiuser scheduling method, i.e. to select the user with the largest channel vector norm square. After the full CSI sharing between two cells, the new 'super-BS' uses zero-forcing method to transform the interference channel into a MIMO broadcast channel [3–5]. Suppose that h1i*and h2j*are the two selected user's channel vectors respectively in cell 1 and 2. Then, the beamforming vector w1 needs to satisfy the orthogonality condition to cancel its interference for cell 2.
5. Extension to non-identical average interference power case
As stated before, the identical average intercell interference power assumption only applies to the case that users are distributed along the cell boundary. In this section, we extend to the more general scenario, where users are randomly distributed in the cell coverage.
5.1. SINR analysis
where I(·; ·) was defined in (12). Applying (24) and (32) into (26), we can obtain the PDF of first selected user's SINR. Give the expression of after the substitution. That would make it easier to follow. Similar to identical interference case, the SINR PDF of the second selected user shares exactly the same expression as that of IA-RB scheme as given in (31).
5.2. Numerical examples
6. Adaptive implementations
As stated above, there is a tradeoff between sum-rate performance versus coordination overhead between LC-RB and SRB scheme, especially when the interference is severe, i.e. the selected user is close to the neighboring BS. Specifically, if the neighboring BS is far away from the selected user, the BS may decide only to perform SRB without coordination. Later simulation results will address that through adaptive implementation, we have managed to further reduce the coordination load with only little rate performance loss for compensation. Noting that the decision-making process only depends on the distance information from the BS to the selected mobile user, the adaptive scheme is easy to implement. Note that the distance information is assumed to be shared between BSs during scheduling.
6.1. Mode of operations
With adaptive implementation, the selected user of cell 1 will firstly estimate its distance to the neighboring BS based on the average interference power. If the distance is larger than a threshold, denoted by dTH, and as such, the interference can be viewed as negligible, the user will suggest BS1 to perform SRB. Otherwise, BS1 will perform LC-RB so as to control the intercell interference. Note that only in the later case, the selected user of BS1 needs to estimate the channel from the neighboring BS. Also, with the adaptive implementation, the coordination overhead is reduced and only used if necessary.
6.2. Coordination overload
We now quantify the average signaling overhead for coordinated beamforming with adaptive implementation. For the adaptive implementation, there are two styles of signaling message between the two BSs, depending coordination is needed or not. Specifically, if d > dTH, BS1 sends one bit of information to BS2 to indicate no coordination is needed, else if d < dTH, BS1 sends the index of w2 in the codebook, plus one bit of coordination indicator, which leads to 1 + log2 B bits of overhead signaling.
6.3. Numerical examples
In this paper, we studied the ergodic capacity of dual-cell MISO broadcast channels with low-complexity random beamforming. In particular, we derived the exact analytical expressions of the ergodic sum-rate for three schemes with the help of some new statistical results and compared their performance in dual-cell environment. We showed through selected numerical examples that the LC-RB scheme achieves tremendous performance over SRB and IA-RB for any volume of active users, with only a beam index sharing between cells. Moreover, we have extended the scenario to the more practical case, where users are arbitrarily distributed within the overall cell coverage and proposed an adaptive coordination scheme. The generalization of the proposed schemes for multiuser parallel transmission is under investigation.
1Other codebook such as Grassmannian codebook may lead to better performance. But for analytical tractability, we limit ourself to random codebook.
This work was supported in part by startup funds from the University of Victoria and in part by a Discovery Grant from NSERC. Part of the work in this paper has been presented in IEEE Global Communications Conference 2010 (Globecom'10).
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