International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 6142574, 8 pages

http://dx.doi.org/10.1155/2016/6142574

## Pilot Design for Sparse Channel Estimation in Large-Scale MIMO-OFDM System

Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 8 January 2016; Revised 24 March 2016; Accepted 20 April 2016

Academic Editor: Larbi Talbi

Copyright © 2016 Chao Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The pilot design problem in large-scale multi-input-multioutput orthogonal frequency division multiplexing (MIMO-OFDM) system is investigated from the perspective of compressed sensing (CS). According to the CS theory, the success probability of estimation is dependent on the mutual coherence of the reconstruction matrix. Specifically, the smaller the mutual coherence is, the higher the success probability is. Based on this conclusion, this paper proposes a pilot design algorithm based on alternating projection and obtains a nonorthogonal pilot pattern. Simulation results show that applying the proposed pattern gives the better performance compared to applying conventional orthogonal one in terms of normalized mean square error (NMSE) of the channel estimate. Moreover, the bit error rate (BER) performance of the large-scale MIMO-OFDM system is improved.

#### 1. Introduction

Over the past decade, multiple input multiple output (MIMO) wireless communication has gained great popularity due to its substantial capabilities of improving the transmission rate and reliability [1]. With the MIMO transmission scheme being incorporated into the long-term evolution advanced (LTE-A) standard, recent research efforts have been devoted to MIMO systems using more antennas which further improve transmission performance. Inspired by this, large-scale MIMO systems equipped with many antenna elements are of great research interest and considered one of the key enabling technologies for future broadband wireless communications [2].

In MIMO system, channel estimation is of crucial importance to the performance of coherent demodulation. It also helps to obtain channel state information (CSI) to support precoding and resource allocation [3]. However, when the number of antennas becomes large-scale, the conventional channel estimation methods such as the least squares (LS) and the linear minimal mean square error (LMMSE) need a mass of pilots, which seriously waste the bandwidth resources. By taking the inherent sparsity of the wireless channel into account, the compressed sensing (CS) channel estimation methods [4, 5] depend on much less pilots than conventional estimation methods. So, in large-scale MIMO system, this kind of channel estimation obtains more attention [6, 7]. Now, the key challenge of using CS in channel estimation of large-scale MIMO systems lies in the following aspects: seeking the sparsest representation of CSI and designing the optimal pilot pattern to ensure estimation success. Our work concentrates on the latter one and tries to seek an optimal pilot pattern.

As early as the year of 2011, papers [8–10] studied the pilot design problem in CS channel estimation of single input single output (SISO) system and made an explicit research direction, that is, minimizing the mutual coherence of the reconstruction matrix. For the MIMO system, paper [11] proposed a random generation method and paper [12] puts forward two outperforming methods: the first is to minimize the largest element in the mutual coherence set whose elements are the values of mutual coherence corresponding to the pilot patterns for all multiple antennas; the second obtained a pattern in which the pilot of each antenna is orthogonal with others based on genetic algorithm (GA) and shifting mechanism. However, all the three methods above are essentially about the pilot location optimization. And the pilot value optimization is not considered although it is intuitively reasonable that the better pilot pattern can be obtained through joint optimization of pilot location and value. In this paper, inspired by the papers [13–15] which all study how to optimize the measurement matrix of signal reconstruction based on CS, a pilot value optimization method is proposed. Specifically, the main contributions of this paper are listed as follows:(1)Studying an alternating projection method denoted in [13] and solving the proprietary matrix projection problem in MIMO-OFDM system;(2)Using the alternating projection method to obtain a nonorthogonal pilot pattern in MIMO-OFDM system;(3)Proposing a pilot design algorithm based on alternating projection and grouping shifting mechanism for CS channel estimation in large-scale MIMO-OFDM system.

The reminder of this paper is organized as follows. Section 2 gives the channel estimation model and the optimization target of the pilot design. The proposed pilot design algorithm is presented in Section 3. Simulation results are provided in Section 4. Finally, conclusions are drawn in Section 5.

For the notations used in the paper, boldface uppercase and boldface lowercase letters separately represent matrices and column vectors, while , , , , and denote the transpose, the conjugate, the Hermitian transpose, the diagonal matrices, and the trace of the matrix, respectively. Finally, and represent the -order norm and the Frobenius norm of “”, respectively.

#### 2. Problem Statement

##### 2.1. Channel Estimation Model

Consider an OFDM downlink transmission where the base station and the terminal are equipped with transmitting antennas and one single receiving antenna, respectively. The wireless channel between the th transmitter and the receiver is frequency-selective and has the coherence time larger than the OFDM symbol duration. And it can be modeled as length finite impulse response (FIR) filter: where is the complex weight of the th tap and the channel impulse response (CIR) vector has only nonzero elements and is -sparse.

Assume that the MIMO-OFDM system has subcarriers, among which subcarriers are reserved for pilots, the cyclic prefix has the length, , not less than , the parallel stream of th transmitter, , , is modulated with Inverse Fast Fourier Transform (IFFT), and, within the stream, the pilot location index set is .

Let ; then the received signal samples in one OFDM symbol after FFT, , can be represented as follows: where is the additive white Gaussian noise (AWGN) at the receiver distributed with , denotes the partial FFT matrix composed of only the first columns in the standard -order FFT matrix, and all the CIR vectors stack to an aggregate vector .

Let , and the rows in form the . Consequently, the received pilots can be expressed as follows:where denotes the reconstruction matrix and .

Notably, for received pilots model (3), when the number of antennas is large, the number of pilots may be smaller than , the length of the aggregate channel vector, which leads the reconstruction matrix to be undetermined. As a result, it is infeasible to obtain a unique solution of by the conventional channel estimation methods. However, since is -sparse, estimating in (3) can be seen as a typical sparse signal reconstruction problem which can be solved in the framework of the CS theory. And, according to this theory, can be recovered from and the deliberately designing reconstruction matrix by solving the -minimization problem [9]: where denotes the error tolerance of reconstruction. At present, many approaches, that is, orthogonal matching pursuit (OMP) and basis pursuit (BP), have been proposed to solve this problem.

##### 2.2. Mutual Coherence

Now, for the mutual coherence of reconstruction matrix , there are two types of definition: one is the maximum absolute value of normalized inner products between every two columns in [4] as follows: where represents the th column of , and the other named as “-average coherence” is put forward by Elad in paper [13], where the coherence greater than is averaged as follows: where denotes the th row and th column element in the Gram matrix and is the column-normalized version of .

Notably, in order to make these two types of definition have a unified form, in this paper, an alternating way to describe the first definition is shown as follows:

According to the CS theory, the success probability of estimation is highly dependent on the mutual coherence of the reconstruction matrix. Specifically, suppose that is a necessarily sparsest reconstructed signal whose sparsity satisfies the following condition: Then, when and are known, both OMP and BP are guaranteed to succeed in solving problem (4) [16] and the deviation of from can be bounded by where is the variance of noise in (3) and is not necessarily equal to . The aforementioned discussion indicates that, for a -sparse vector , the smaller the is, the better the approximation of can be obtained. And because is determined by the pilots value and the pilots location , the aim of pilot designing is minimizing to improve the performance of CS channel estimation, which can be described mathematically as follows:

Notably, in this paper, we address this problem by optimizing and fixing . The joint optimal solution can be then obtained by traversing all of the pilot location cases.

#### 3. Proposed Pilot Design Scheme

According to the definition of , optimization problem (10) can be transformed into the question of how to construct the Gram matrix with the following properties:(i) has or small enough;(ii) has a special structure that , where is the column-normalized version of .

For solving this matrix construction problem, in this paper, an algorithm named alternating projection is investigated. In the following, this algorithm is introduced; then, based on this algorithm, a pilot design scheme is proposed.

##### 3.1. Statement of Algorithm

The alternating projection algorithm which attempts to construct a matrix that satisfies the properties (i) and (ii) simultaneously can be described in Algorithm 1.