Wireless Communications and Mobile Computing

Volume 2018, Article ID 3981048, 11 pages

https://doi.org/10.1155/2018/3981048

## Low Complexity Pilot Allocation Scheme for a Large OFDM Block with Null Subcarriers

^{1}Department of Electrical and Electronics Engineering, Konkuk University, Seoul, Republic of Korea^{2}Department of Computer Engineering, Sejong University, Seoul, Republic of Korea^{3}Department of Electronics, Information and Communication Engineering, Sejong University, Seoul, Republic of Korea

Correspondence should be addressed to Eunchul Yoon; rk.ca.kuknok@nooyce

Received 25 December 2017; Revised 25 February 2018; Accepted 4 March 2018; Published 10 April 2018

Academic Editor: Donatella Darsena

Copyright © 2018 Janghyun Kim 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

A low complexity pilot allocation scheme is proposed for a large OFDM block with null subcarriers. The proposed scheme allocates pilots to the edge part of the active subcarrier region according to the 2nd order polynomial and to the middle part of the active subcarrier region according to the 1st order polynomial or the comb-type pilot pattern. To find a parameter of the 2nd order polynomial, the proposed scheme applies exhaustive search of a single parameter by using an integer unit resolution. It is shown by simulation that the proposed scheme is a close-to-optimal pilot allocation scheme yielding better symbol error rate (SER) performance than the traditional 3rd and 5th order polynomial-based schemes although the proposed scheme has lower computation and implementation complexity.

#### 1. Introduction

Orthogonal frequency division multiplexing (OFDM) is an attractive transmission technique for broadband communications. Since OFDM is appropriate for high spectral efficiency and scalable to low complexity receivers, it is well suited for the 5G cellular network [1]. It has been well known that the comb-type pilot pattern with equal pilot spacing leads to minimum mean square error (MSE) of the least-square- (LS-) based channel estimation [2]. The comb-type pilot pattern was applied to OFDM systems with multiple antennas [3]. However, the OFDM block usually includes null subcarriers for the purpose of guard band [4, 5]. Given null subcarriers, the channel estimation with the comb-type pilot pattern incurs severe channel estimation error at the edge part of the active subcarrier region [6, 7]. Recently, the international telecommunications union announced the minimum requirements related to technical performance for IMT-2020 radio interfaces, which indicated that a very large bandwidth up to 1 GHz will be used for future mobile communications [8]. Given a large OFDM block with null subcarriers, allocating pilots to the OFDM block becomes a more crucial issue. The minimum mean square error- (MMSE-) based channel estimation may suppress the channel estimation error at the edge part of the active subcarrier region effectively. However, the static pilot locations cannot be optimized by minimizing the MSE of the MMSE-based channel estimation because the MSE of the MMSE-based channel estimation is dependent on time-varying channel and noise statistics [4, 9]. Therefore, static pilot locations are usually optimized with respect to the LS-based channel estimation. In order to minimize the MSE of the LS-based channel estimation, the 3rd order polynomial-based pilot allocation was suggested in [10], which applied exhaustive search of a single parameter by using a fractional unit resolution. While the 3rd order polynomial-based pilot allocation showed good channel estimation performance for an OFDM block size 512, it incurred considerable performance degradation for an OFDM block size 1024. In [11], the optimal pilot powers were numerically computed by minimizing norm of the channel estimate error, and the optimal pilot locations were iteratively found by symmetrically removing a certain number of insignificant pilot candidates. However, its algorithm was not optimal because it was heuristically chosen and it caused high complexity due to iteration. In [12], the 5th order polynomial-based pilot allocation was suggested. Differently from the 3rd order polynomial-based pilot allocation, the 5th order polynomial-based pilot allocation performed well with both the OFDM block sizes, 512 and 1024. In [13], the 5th order polynomial-based pilot allocation was also applied to MIMO-OFDM systems. However, the 5th order polynomial-based pilot allocation requires more complicated implementation than the 3rd order polynomial-based pilot allocation because the 5th order polynomial-based pilot allocation applies exhaustive search of two parameters by using two fractional unit resolutions. Moreover, according to our simulation results, the performance of the 5th order polynomial-based pilot allocation tends to be degraded as the OFDM block size increases further than 1024. It is because a single polynomial function cannot determine the optimal pilot locations precisely for a large OFDM block.

In this paper, a low complexity pilot allocation scheme is proposed for a large OFDM block with null subcarriers. The proposed scheme allocates pilots to the edge part of the active subcarrier region according to the 2nd order polynomial and to the middle part of the active subcarrier region according to the 1st order polynomial or the comb-type pilot pattern. To find a parameter of the 2nd order polynomial, the proposed scheme applies exhaustive search of a single parameter by using an integer unit resolution. It is shown by simulation that the proposed scheme is a close-to-optimal pilot allocation scheme yielding better symbol error rate (SER) performance than the traditional 3rd and 5th order polynomial-based schemes for all the OFDM block sizes although it has lower computation and implementation complexity.

#### 2. System Model

An OFDM system with an OFDM block size is considered, where the set of the subcarrier indices is given by . Let and define the number of null subcarriers and that of active subcarriers in the OFDM block, respectively. The active subcarriers are divided into pilot subcarriers and data subcarriers. Let and define the index set of pilot subcarriers and that of data subcarriers, respectively. It is assumed that the channel impulse response is composed of multipaths aswhere denotes the transpose operator. The channel frequency response coefficient vectors for the pilot and data subcarriers can be written aswhere and denote two Vandermonde matrices given by for , , and for , , while denotes the element of a matrix in the th row and the th column. To apply the LS-based channel estimation, the number of the pilots should be greater than or equal to the channel length . For the purpose of achieving maximum frequency efficiency, it is assumed that . Let and define the transmitted data symbol vector and the pilot symbol vector, respectively. In addition, let define a diagonal matrix with its diagonal components given by the components of when is a column vector and let define a column vector with its components given by the diagonal components of when is a diagonal matrix. Then, the received signal vector over the pilot subcarriers can be written aswhere denotes a zero mean circularly symmetric complex Gaussian noise vector with component wise variance . Under the assumption that the data and pilot symbols have unitary average power, SNR is defined by . The channel frequency response coefficient vector over the pilot subcarriers is estimated by to giveThe channel frequency response coefficient vector for the data subcarriers is estimated by using the LS method [14] aswhere denotes the transpose complex conjugate operator. Note that as increases, the matrix as shown in (6) becomes ill-conditioned to be a nearly singular matrix. In such a case, the inverse of the matrix can be computed by using the Moore-Penrose inverse [15]. The Moore-Penrose inverse computes a best fit solution to a system of linear equations that lacks a unique solution. A simple way of describing the Moore-Penrose inverse is the singular value decomposition. Let define the singular value decomposition of a square matrix , where and are two unitary matrices and is a diagonal matrix. The Moore-Penrose inverse of is given by , where the diagonal matrix is obtained by changing the diagonal components of smaller than some small tolerance to zeros and taking the reciprocal numbers of the other diagonal components. From (2) and (5), the channel impulse response can be estimated byThen, the channel frequency response coefficient vector for the data subcarriers is estimated byIf the components of have uniform pilot powers, the MSE vector of the LS-based channel estimation for the data subcarriers can be written asThe averaged MSE over all the data subcarriers can be written aswhere denotes the trace operator. Since optimizing the pilot locations through minimization is not affected by the value of , the scaled MSE given bycan be used for finding the optimal pilot locations. The quantities and in (11) are dimensional square matrices, whose diagonal components are given by 1’s. By applying the singular value decomposition, those quantities can be written aswhere and denote unitary matrices and and denote diagonal matrices. By defining with being its th column and for , the scaled MSE reduces towhere is the th diagonal component of . Since is formed by a weighted sum of the eigenvalues of , it is more critical to prevent a very small eigenvalue of than that of in order to reduce the scaled MSE. Due to the existence of the null subcarrier region, the comb-type pilot pattern is prone to induce a small minimum eigenvalue of and incur a large MSE result. Therefore, if the OFDM block includes the null subcarrier region, pilots should be allocated to the active subcarrier region in efforts to maximize the minimum eigenvalue of .

#### 3. Proposed Pilot Allocation Scheme

Figure 1 shows the structure of the OFDM block that consists of the null subcarrier region and the active subcarrier region. The active subcarrier region is divided into a middle part and two edge parts. The edge part represents a boundary part of the active subcarrier region, which incurs comparatively large MSEs when pilots are allocated to the active subcarrier region according to the comb-type pilot pattern. To effectively suppress the MSEs at both the edge and middle parts, the numbers of the pilots allocated to the edge and middle parts should be well balanced. In particular, since the detrimental impact of the null subcarrier region on the pilot-based channel estimation dissipates as the subcarrier distance from the null subcarrier region increases, a gradually changing pilot density should be applied to the edge part and a uniform pilot density should be applied to the middle part for better performance. Therefore, we propose to apply two separate polynomials to the edge and middle parts. Let and define the number of the one-sided edge part subcarriers and that of pilots allocated to the one-sided edge part, respectively. In the proposed scheme, pilots are allocated to the edge part according to the 2nd order polynomial and pilots are allocated to the middle part according to the 1st order polynomial. Therefore, the subcarrier indices of the first pilots are chosen bywhere denotes the pilot index, denotes the subcarrier index of the th pilot, and denotes the floor operator. From Figure 1, it is easy to infer the following five conditions for the 1st and 2nd order polynomials:From (15) and (16), the coefficients of the 1st order polynomial, and , can be found asFrom (17)–(21), the coefficients of the 2nd order polynomial, , , and , can be found asThe subcarrier indices for the remaining pilots with pilot indices, , can be found by using the symmetry of pilot allocation with respect to the center of the OFDM block asTo determine the subcarrier indices of pilots based on (14) and (25), two parameters and , which can effectively suppress the averaged MSE, should be known in advance. In the proposed scheme, is determined by referring to the number of the pilots that would be allocated to the one-sided edge part and the one-sided null subcarrier region if the pilots were uniformly allocated to the entire OFDM block. Intuitively, this idea means that the pilots, which would belong to the null subcarrier region if the comb-type pilot pattern were applied to the entire OFDM block, are added to the edge part. This idea can be formulated asThe left-hand side of (26) means the pilot density when pilots are uniformly allocated to the entire OFDM block, and the right-hand side of (26) means the pilot density when pilots are uniformly allocated to the one-sided edge part and the one-sided null subcarrier region. Given the value of , the value of can be computed based on (26) asTo compute based on (27), should be known in advance. In the proposed scheme, the optimal value of is found by applying exhaustive search through minimization. The set of the candidate values for is chosen by . The 3rd order polynomial-based pilot allocation in [10] applied exhaustive search of a parameter by using a fractional unit resolution, and the 5th order polynomial-based pilot allocation in [12] applied exhaustive search of two parameters by using two fractional unit resolutions. The proposed scheme has lower optimization complexity than both the 3rd and 5th order polynomial-based pilot allocation schemes because the proposed scheme applies exhaustive search of a single parameter (i.e., ) by using an integer unit resolution.