Downlink Performance of Pilot-Reused HetNet with Large-Scale Antenna Arrays
Considering a heterogeneous network (HetNet) where both macro base station (BS) and small cell (SC) nodes are equipped with massive antennas, this paper studies the performance for multiple-input multiple-output (MIMO) downlinks when the macro and small cells share the same spectrum and hence interfere with each other. Suppose that the large-scale antenna arrays at both macro BS and SC nodes employ maximum-ratio transmission (MRT) or zero-forcing transmission (ZFT) precoding, and transmit data streams to the served users simultaneously. A new pilot reuse pattern among SCs is proposed for channel estimation. Taking into account imperfect channel state information (CSI), capacity lower bounds for MRT and ZFT are derived, respectively, in closed-form expressions involving only statistical CSI. Then asymptotic analyses for massive arrays are presented under specific power scaling laws. Subsequently, two user scheduling algorithms, greedy scheduling algorithm and asymptotical scheduling algorithm (ASA), are proposed based on derived capacity lower bounds and asymptotic analyses, respectively. ASA is demonstrated to be a near optimal in the asymptotic regime and has low complexity. Finally, the derived closed-form expressions are verified to be accurate predictors of the system performance by Monte-Carlo simulations. Numerical results demonstrate the effectiveness of asymptotic analysis and proposed user scheduling schemes.
As a viable and cost-effective way to increase network capacity, heterogeneous networks (HetNets) that embed a large number of low-power nodes, called small cells (SCs), into an existing macro network has emerged with the aim to offload traffic from the macro cell (MC) to small cells  in hot spots or to solve coverage holes in MC. Conventionally deploying more macro base stations (BSs) in already dense networks may be prohibitively expensive and result in severe inter-cell interference . However, due to the large number of potentially interfering nodes in the network, mitigating both the inter-cell and intra-cell interference becomes a crucial issue facing HetNet. Interference control has been intensively studied and applied in HetNet , including the coordinated multi-point (CoMP) transmission . Although the CoMP transmission was shown to provide high spectral efficiency  with the backhaul among the coordinated tiers enabling both user data and channel state information (CSI) exchange, the high signaling overhead results in practical implementation limitations.
Recently, multiple-input multiple-output (MIMO) transmission with large-scale antenna arrays at the BS has attracted substantial interest from both academia and industry. Using simple linear processing, such large-scale antenna arrays were proved to be able to substantially reduce the effects of the uncorrelated noise, small-scale fading and intracell interference . Then, the energy and spectral efficiency of very large multiuser MIMO uplink systems were investigated in , which showed that the power radiated by each terminal could be made inversely proportional to either the number of BS antennas or at least its square-root, considering both perfect and imperfect CSI. In , it was stated that the potential benefits have elevated large-scale MIMO to a central position as a promising technology for the next generation of wireless systems. In a HetNet setting,  proposed to use large scale antenna arrays at the BS and limited antennas at the SCs
This paper presents a comprehensive study of a two-tier network with large-scale antenna arrays set at both BS and SCs. In our preliminary literature , maximum-ratio transmission (MRT) precoding was employed based on the estimated channels obtained from the orthogonal training scheme, and downlink capacity lower bounds for a user in the MC and for a user in an SC were derived in closed-form expressions. However, there are still many critical yet unsolved problems. This paper makes the following contributions to address the remaining issues.
It was stated in  that pilot overhead is proportional to the number of user equipment (UE) for the conventional orthogonal training scheme, i.e., the system performance will degrade as the UE number grows due to heavy pilot overhead. In , the pilot reuse (PR) technique is utilized among the macro cells to reduce the pilot overhead, while UEs within a cell use orthogonal pilots.  studies pilot reuse in a dense small cell network. In a two-tier HetNet with multiple small cells, massive antenna arrays and large number of UEs, we propose to apply pilot reuse among the SCs in this paper, i.e., the same set of orthogonal pilots is reused among the small cells in one macro-cell. Thus the number of orthogonal pilots is smaller than the total UE number in the whole network.
We present for the first time the downlink capacity lower bounds of the large-scale HetNet system, where simple linear precoding such as MRT or zero-forcing transmission (ZFT) is employed at each node, followed by detailed asymptotic analysis.
The design of an efficient and practical user scheduler for the large-scale HetNet is an important and challenging problem, because the required CSI exchange becomes prohibitively complicated due to the large-scale antenna arrays and the large number of UEs in the MC and SCs. Based on the obtained capacity bounds and asymptotic analysis, a greedy scheduling algorithm (GSA) and an asymptotic scheduling algorithm (ASA) are proposed, respectively, where GSA requires only statistical CSI (SCSI) shared between BS and SCs, and ASA even removes the need for any CSI exchange among nodes.
The rest of the paper is organized as follows. We briefly describe the system model for HetNet with large-scale antenna arrays in Section 2. In Section 3, lower bounds for the achievable rate are derived with both imperfect CSI based MRT and ZFT, followed by corresponding asymptotic analysis. Then, two user scheduling algorithms are developed in Section 4. Moreover, simulation results under different system configurations are given in Section 5 to demonstrate the effectiveness of both the derived rate expressions and the developed schemes. Finally, conclusions are drawn in Section 6.
For a matrix , we use , , and to denote the transpose, the Hermitian transpose, the conjugate, and the trace, respectively. is an identity matrix and is an zero matrix. Moreover, denotes the expectation operator. The symbol indicates the 2-norm of vector , and denotes a diagonal matrix with being its diagonal entries. is the absolute value of , while represents the number of elements in set . Finally, the notation means almost sure convergence, and represents a circularly symmetric complex Gaussian vector with zero mean and covariance matrix .
Figure 1 shows the considered two-tier network architecture with one cell consisting of one macro BS, which is overlaid with a dense tier of uniformly distributed SCs by sharing the same time-frequency resources. Assume that the BS and SCs are respectively equipped with large-scale arrays of and antennas, where , while each user has only one antenna due to the size or complexity constraint. Notably, uniform user distribution in the cell is focused here. Based on the biased user association , the users served by the macro BS are designated to a macro UE (MUE) set, and those served by each SC are designated to a small cell UE (SUE) set. Furthermore, suppose that the macro BS serves MUEs simultaneously while each SC serves SUEs with and . Denote the MUE and SUE sets as and , respectively, then we have and . The selected subsets of MUE and SUE after user scheduling are denoted by and (), respectively.
For the channel matrices, they account for both small-scale fading and large-scale fading. Here, we assume that all the channels between the users and the nodes follow independent and identically distributed (i.i.d.) Rayleigh fading and time division duplex (TDD) is adopted with channel reciprocity satisfied. Denote the channel matrices from the BS and th () SC to the MUEs as and , respectively, and use and to represent the channel matrices from the BS and the th SC to the SUEs in the th SC, respectively. We have , , and where , the first items , , and include the i.i.d. small-scale fading coefficients, and the second items are the large-scale fading diagonal matrices given by , , , and .
2.1Channel Estimation with Pilot Reuse
Practically, the channel matrix from each node to its corresponding users, i.e., and (), have to be estimated based on the uplink training. At the beginning of each coherence interval , all users simultaneously transmit pilot sequences of length symbols. On account of the slight interferences between low-power SCs which are far away from each other, we present a pilot reuse pattern for small cells in a large-scale HetNet system.
First, we denote the reuse factor as , i.e., all SCs utilize sets of pairwise orthogonal pilot sequences with a total of SCs sharing the same set. This requires to satisfy the orthogonality of the MC and SC pilot sets. Then, we group SCs into sets according to the maximum relative distance criterion and SCs in one set use the same pilot sequences. Since all low-power nodes are modeled as uniformly distributed in a circle with BS at the center as shown in Figure 1, we can denote the th () SC set as . Taking and reuse factor for example, the SC sets are and . The SCs in each set share one pilot set which includes pairwise orthogonal pilot sequences, and there are pilot sets for the total of SCs.
Then the training matrix received at the BS and the th () SC can be written as
respectively, where is the transmit power of each pilot symbol, and are the additive white Gaussian noise (AWGN) matrices with i.i.d. components following , the training vectors transmitted by the th () MUE is denoted by the th row of , satisfying , while the training vector transmitted by the th () SUE of one SC in the th set is represented by the th row of , satisfying . Moreover, since the rows of pilot sequence matrices are pairwise orthogonal, we have and ().
In order to estimate and (), we employ the minimum mean-square-error (MMSE) estimation at each node . The estimated channels are given by
where , , and () are defined. Due to the property of and , and are also composed of i.i.d. elements. Then, we have
where and denote the estimation error matrices which are independent of and from the property of MMSE channel estimation . Hence, we have with , with , with the th column vector denoted by , and with the th column vector denoted by . Here, the estimated large-scale fading factors satisfy , and , where and .
In the downlinks, the received signals at MUEs and SUEs in the th small cell are
respectively, where and represent the linear precoding matrices at the BS and th SC, respectively; and are the complex-valued data symbols from BS to its MUEs and from th SC to its own SUEs, respectively, satisfying and ; and and involves the AWGN of variance .
Aiming to maximize the received signal-to-noise ratio, the MRT technique is utilized at both the BS and SCs to process the transmit signals towards the corresponding users. Given the estimated channel state information, the MRT precoding is expressed as 
where and are normalization constants, chosen to satisfy the transmit power constraints at the BS and SCs, respectively. On the basis of (Equation 6) and , we have
where , and with .
where , , and are normalization constants. Similarly, based on (Equation 8) and , we have
where , and with . The detailed derivation is given in Appendix Section 7.
3Achievable Rate Analysis
The exact rate analysis of the MUE and SUE in the pilot assisted massive MIMO heterogeneous network considered is highly complicated and intractable. In this section, we provide a closed-form capacity lower bound of each user for both MRT and ZFT precoding, respectively. The simple lower bounds can be applied to user scheduling and power allocation optimization as detailed in subsequent sections.
In practice, only imperfect CSI derived from transmitted pilots is available at each node for linear precoding. Utilizing MRT precoding in (Equation 4) and (Equation 5), the received signal of the th () MUE and th () SUE at the th () SC can be rewritten as (Equation 10) and (Equation 11).
Note that both the BS and small cell nodes treat the estimated channels as the true channels , and the first term is the desired signal. The remaining terms are considered as interferences and noise, including estimation error caused interference term. Accordingly, with imperfect CSI, the ergodic achievable rate of MUE () and SUE () in the th () SC are given by
respectively, where , and denote the estimation error induced interference, the intra-MC interference and the cross-tier interference for the th MUE, respectively, given by
and , , and denote the estimation error induced interference, the cross-tier interference, the intra-SC interference and the inter-SC interference for the th SUE in the th SC, respectively, given by
Notably, the inter-SC interference includes the pilot contamination effect caused by pilot reuse. In the above achievable rate expressions, expectations over the estimated instantaneous CSI cannot be further derived into tractable forms. Therefore, we adopt a similar bounding technique of  to obtain closed form rate expressions, the result of which will provide insights on the impact of different system parameters and facilitate further optimizations.
With imperfect CSI based MRT, and , the downlink achievable rate of the th () MUE and th () SUE in the th () SC, for finite and , are lower bounded by
where , , , ,