Pilot design for sparse MIMO-OFDM channel estimation with generalized shift invariance property

Int. J. Electron. Commun. (AEÜ) 96 (2018) 48–57

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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue

Regular paper

Pilot design for sparse MIMO-OFDM channel estimation with generalized shift invariance property Shenyang Xiao, Zhigang Jin, Yishan Su ⇑ School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China

a r t i c l e

i n f o

Article history: Received 16 May 2018 Accepted 23 August 2018 Available online xxxx Keywords: Compressed sensing (CS) Mutual coherence Multiple-input multiple-output (MIMO) Pilot design Channel estimation Generalized shift invariance property

a b s t r a c t Compressed sensing (CS) based channel estimation is greatly bound by the measurement matrix according to CS theory. We design pilot patterns by minimizing the mutual coherence of the measurement matrix with the generalized shift invariance property (GSIP). GSIP and a corollary are firstly proposed. Then two pilot pattern design schemes termed pilot design with GSIP (PDGSIP) and tradeoff pilot design with GSIP (TPDGSIP) are put forward to design orthogonal pilot patterns based on GSIP for a multipleinput multiple-output orthogonal frequency division multiplexing system. In PDGSIP, a collection of pilot patterns are firstly obtained and then pilot patterns having large mutual coherence are replaced with new ones generated with optimal pilot patterns. TPDGSIP directly produces new pilot patterns based on GSIP to fully exploit the pilot distance of the obtained pilot pattern as soon as one pilot pattern is obtained. Simulation results have shown that, the proposed pilot pattern design schemes are able to obtain the best pilot patterns in comparison to existing methods from the perspective of mutual coherence. Channel estimation performance using pilot patterns designed by proposed schemes precedes that using pilot patterns designed by existing schemes in terms of normalized mean square error and bit error rate. Ó 2018 Elsevier GmbH. All rights reserved.

1. Introduction The knowledge of channel state information (CSI), which is obtained via channel estimation techniques, is of vital importance to the precoding matrix design, beamforming, etc. [1,2]. Consequently, the channel estimation is a key technique and an essential part of multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems [3,4]. For pilot-assisted channel estimation adopting frequency domain orthogonal pilot placement in MIMO-OFDM systems, as the number of transmitters increases, the accurate acquisition of CSI of multiple transmitters becomes a challenging task due to a mount of parameters to be estimated with limited available pilot overhead [5]. In broadband wireless communication systems, wireless channels usually exhibit sparsity, where the channel time delays are large yet only a small number of channel fading coefficients are nonzero [6–8]. Conventional channel estimation methods such as least square (LS) and minimal mean square error (MMSE) do not take the inherent sparsity of channels into account and incur bad channel estimation performance [9]. Recently, compressed sensing (CS) which provides an alternative to Nyquist sampling has been demonstrated to be more effi⇑ Corresponding author. E-mail address: [email protected] (Y. Su). https://doi.org/10.1016/j.aeue.2018.08.028 1434-8411/Ó 2018 Elsevier GmbH. All rights reserved.

cient in the application of the sparse recovery of sparse signals [10]. When signals are inherently sparse, CS can sample signals at a rate far less than that required in Nyquist and then enable accurate recovery of sparse signals by means of optimization. Up to date, CS has been applied to a host of scenarios due to the sparstiy in many signal classes of interest [11,12]. The application in CS based channel estimation has been extensively investigated and a great many of recovery algorithms have been applied to saprse channel estimation, e.g., orthogonal matching pursuit (OMP), compressive sampling matching pursuit (CoSaMP), basis pursuit (BP) [6,13–15]. In addition, motivated by the CS algorithms, some efforts are made to incorporate the CS into adaptive filtering methods to enable more accurate or less complex sparse channel estimation [16–19]. By the use of variable-step-size techniques and the parameter adjustment method, an adaptive reweighted zeroattracting sigmoid functioned variable-step-size LMS (ARZASVSS-LMS) algorithm with faster convergence speed and better steady-state performance is proposed for sparse channel estimation in [16]. Ref. [18] proposes a reweighted norm-adaption penalized least mean square/fourth (RNA-LMS/F) algorithm by incorporating a p-norm-like into the cost function of the conventional least mean square/fourth and simulation results verify that RNA-LMS/F is superior to the previously reported sparse-aware LMS/F. By incorporating an l1-norm penalty into the cost function of the conventional sparsity-aware set-membership normalized

S. Xiao et al. / Int. J. Electron. Commun. (AEÜ) 96 (2018) 48–57

least mean square (SM-NLMS) algorithm to exploit the sparsity of the sparse systems, a zero-attracting SM-NLMS (ZASM-NLMS) algorithm is proposed for sparse channel estimation in [19]. Ref. [17] proposes an improved norm-constrained set-membership normalized least mean square algorithm (INCSM-NLMS) for sparse adaptive channel estimation which is implemented by incorporating an lp-norm penalty into the cost function of the traditional setmembership normalized least mean square (SM-NLMS) algorithm, and simulation results show that the convergence speed and the channel estimation steady-state error are superior to existing popular SM-NLMS algorithms. According to CS theory, the successful recovery of sparse signals from measurements with high probability requires that the measurement matrix satisfies restricted isometry property (RIP) [20]. While checking whether a measurement matrix satisfies the RIP is a NP-Complete problem in general, there are no known methods in polynomial time to evaluate whether a given measurement matrix satisfies the RIP [12]. An computation feasible alternative method is to compute the mutual coherence, which is equivalent to RIP and can be connected to RIP by Gershgorin circle theorem [21]. CS theory suggests that the reconstruction accuracy may be improved if the mutual coherence can be decreased [22]. Accordingly, in pilot-assisted sparse channel estimation the equispaced pilot placement which is optimal in conventional channel estimation methods is no longer optimal when CS is used to estimate channels. Besides, in practical communication systems, the deterministic pilot pattern is usually used to reduce the system complexity. Therefore, the deterministic pilot design or pilot pattern design for pilot-assisted channel estimation draws amounts of attentions to improve the channel estimation performance by means of minimizing the mutual coherence of the measurement matrix [23–28]. Pilot design methods can be classified into two categories: the pilot pattern for the frequency domain and the training sequence design for the time domain. Frequency domain pilot design is to select a certain number of subcarrier positions as pilot subcarriers from all available subcarriers and meanwhile assign corresponding pilot symbols so that the mutual coherence of the resulting measurement matrix is as small as possible. As far as frequency domain pilot pattern design, random search procedure [9], estimation of distribution algorithm [26], stochastic search schemes (SSS) [27], cross-entropy optimization [29], modified discrete stochastic approximation [30] and discrete stochastic approximation [31] have been proposed to design the pilot pattern of the single-input singleoutput orthogonal frequency division multiplexing (SISO-OFDM) system whilst the pilot power of pilot symbols is assumed to be equal. While it is common to restrict the design to selecting the pilot position, the joint optimization of the pilot positions and the pilot power is investigated in [28] and [32]. As to communication systems with multiple antennas, an improved shuffled frog leaping algorithm (ISFL) and a genetic algorithm (GA) are proposed to design orthogonal pilot patterns in [33] and [4] for MIMO-OFDM systems respectively. The core idea of these two methods is to find one core pilot pattern whose minimum pilot index distance is set up no smaller than the number of transmitters, and then the remaining pilot patterns are produced with the core pilot pattern plus constants. However, the constraint that the minimum pilot distance is no smaller than the number of transmitters may render the pilot pattern with large mutual coherence and consequently degrade the sparse channel estimation performance. Ref. [27] proposes extension Scheme 1 (ES1) to design pilot patterns therein which designs the pilot pattern sequentially. However, the pilot distance of the obtained optimal pilot pattern is underused which leads to pilot patterns with large mutual coherence. When the received signal at a receiver or a user is formulated as the linear convolution of the time domain training sequence and

49

the channel impulse vector in which the measurement matrix is a Toeplitz matrix formed by the cyclic shift of training sequence, the training sequence need to be designed to reduce the mutual coherence of the measurement matrix for improving the sparse channel estimation performance. A training sequence in the form of inverse discrete Fourier transform with cyclic structure is proposed and then a genetic algorithm is applied to further lower the mutual coherence in [34]. Three criteria to optimize the training sequence and a genetic algorithm to lower the merit factors in three criteria are proposed in [35]. Note that both are the training sequence design methods for the time-domain synchronous OFDM (TDS-OFDM) system. Due to the fact that the antenna elements are usually distributed proximally, the effective channels between antenna elements and a given receiver have similar time delays but the path amplitudes are distinct. These channels characterize the common support, which is referred to as the common sparsity. When all channel impulse responses observed at a receiver are concatenated in a single vector and we rearrange the vector to make nonzero path amplitudes clustered in a block, then the rearranged aggregated channel impulse response exhibits the block sparsity. The block sparsity can be used to improve the spectral efficiency and channel estimation accuracy of the MIMO-OFDM system based on the distributed compressed sensing theory or structured compressed sensing [2,5,21,36]. As to pilot design in the scenario of the MIMO-OFDM system with frequency domain orthogonal pilot placement where channels enjoy the common sparsity, the measurement matrix is designed in perspective of interblock coherence based on the distributed compressed sensing in [25]. By formulating the pilot design task as an optimization problem, the authors apply GA to designing orthogonal pilot patterns. When the system is modeled by time domain linear convolution, the training sequence design is investigated with block coherence and GA is applied to the design of the time domain training sequence for TDS-OFDM systems [37,38]. The pilot overhead can be further reduced by means of sharing pilot subcarriers among all transmit antennas that is the superimposed pilot placement and meanwhile using the block sparsity [23,39–41]. In [39] and [40], the positions of superimposed pilots are uniformly spaced in the frequency domain while the frequency domain pilot symbols of different transmit antennas differ one from another in order to distinguish channels associated with different transmit antennas. In [41], the pilot amplitude is random with fixed phase and GA is applied to pilot pattern design for time- and frequency-domain training OFDM system. In order to avoid employing the spatial or temporal common sparsity of channels to make the design applicable even in the scenario where the common assumption do not hold, a deterministic frequency pilot design method which designs both the pilot subcarrier positions and pilot values jointly is investigated in [23]. The block sparsity based pilot allocation designs pilot patterns based on the interblock coherence theory of compressed sensing. The premise of applying the block sparsity is that channels observed at one receiver enjoy common sparsity. Nonetheless, the common sparsity may disappear when antennas elements are not spaced closely or channels are with highly diffusive multipaths [42,43]. Unlike pilot design based on the block sparsity assumption [25,38,41], we avoid employing the spatial or temporal common sparsity properties of the channel to make our scheme applicable for channels without common support. Therefore, we focus on the scenario of channels without common support to make our design applicable even in cases where such assumptions do not hold. In addition, existing works investigate the time domain training sequence design in the scenario of the TDSOFDM system. To the best knowledge of the authors, in the scenario of MIMO-OFDM systems with orthogonal pilot placement

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and channels without the common sparsity, no other works are found towards orthogonal pilot pattern design for MIMO-OFDM systems except [4,27,33]. In this paper, we consider a MIMOOFDM system with frequency domain orthogonal pilot placement over different transmitters, which means that the pilot index sets of any two distinct transmitters are disjoint. In this paper, we design the orthogonal pilot patterns by making full use of pilot distance of obtained optimal pilot patterns to reduce the sum of mutual coherence of pilot patterns. A theory termed generalized shift invariance property (GSIP) and a corollary termed determined pilot pattern number (DPPN) to determine the maximum number of pilot patterns which have the same value of mutual coherence as a given pilot pattern are firstly presented. Then, two pilot design schemes are proposed to design pilot patterns to make full use of the pilot distance of the obtained optimal pilot patterns based on GSIP. Specifically, in the first scheme, we firstly obtain a collection of pilot patterns. Then the pilot distance of pilot patterns with the smallest mutual coherence are fully used to produce new pilot patterns to substitute the inferior ones. In the second scheme, new pilot patterns are generated immediately based on GSIP as soon as a pilot pattern is obtained. Simulation results show that pilot patterns obtained by the proposed two methods are superior to that designed by other existing methods and the proposed first scheme achieves the best channel estimation performance in terms of normalized mean square error (NMSE) and bit error rate (BER). This paper makes the following specific contributions. First, GSIP is introduced and the strict mathematical proof is also given. GSIP indicates that we can get a set of pilot patterns having the same mutual coherence by means of each entries of the given pilot pattern plus or minus a certain positive integer simultaneously. Next, a corollary named determined pilot pattern number is developed and the mathematical proof is also given. DPPN gives how many pilot patterns which enjoy the same mutual coherence can be determined and how to obtain these pilot patterns. Finally, two pilot design schemes that are PDGSIP and TPDGSIP are proposed to design the pilot pattern of the MIMO-OFDM system. The remainder of this paper is organized as follows. In Section 2, the system model for sparse channel estimation and the pilot optimization problem are formulated. Then, the generalized shift invariance property, a corollary and two pilot design schemes are detailed in Section 3. Numerical simulation results are presented in Sections 4 and 5 concludes this paper. The notations used in this paper are defined as follows. Symbols for matrices (upper case) and vectors (lower case) are in boldface. ðÞH ; diagðÞ; £; cardðÞ and CN denote the matrix conjugate transpose (Hermitian), the diagonal matrix, the null set, the cardinality of a set and the complex Gaussian distribution respectively. Define p þ i, p  i and intersect ðp; XÞ as each entry of the vector p plus i and each entry of p minus i, the set of the same elements between the vector p and the set X respectively.

f

f

t

1 Data symbol

Pilot symbol

f

f

For pilot-assisted channel estimation, we adopt the comb-type pilot placement, where the pilot symbols scatter over timefrequency two-dimension resource units as shown in the left part of Fig. 1. In the left part of Fig. 1, each column of units represents an OFDM symbol over one time slot and each row within one time slot represents one subcarrier in frequency domain. The pilot symbols and the data symbols are carried on units of OFDM symbols to pass frequency selective fading channels. Consider a MIMO-OFDM system with N T transmitters and N R receivers and assume the coherence time of frequency selective fading channels is larger than the MIMO-OFDM symbol period.

Null

Fig. 1. Pilot pattern of the first transmitter (the left portion of the figure) and frequency orthogonal pilot placement for a MIMO-OFDM system (the right portion of the figure).

The frequency selective fading channel between the iT th transmitter and the jR th receiver can be modeled as a finite time delay filter given by

hiT ;jR ðnÞ ¼

XL1 l¼0

aiT ;jR ðlÞdðn  lÞ

ð1Þ

where L and aiT ;jR ðlÞ are the discrete channel length and the complex channel gain of the lth tap respectively. For a K sparse channel, the discrete channel vector can be expressed as T

hiT ;jR ¼ ½hiT ;jR ð0Þ; hiT ;jR ð1Þ; . . . ; hiT ;jR ðL  1Þ and all hiT ;jR ðlÞ’s but K elements are zeros with K  L. Assume the transmitted OFDM symbol consists of N subcarriers, among which N d subcarriers are selected as data subcarriers and the number of pilot subcarriers of each transmitter is N p ¼ N  N d . Denote the pilot index set of the iT th transmitter by KiT ¼ fkiT ;1 ; kiT ;2 ; :::; kiT ;Np g, where kiT ;i satisfies 1 6 kiT ;i 6 N; 1 6 i 6 N p and 1 6 iT 6 N T . Without loss of generality, we assume kiT ;i ’s are sorted in ascending order, i.e., 1 6 kiT ;1 6 ::: 6 kiT ;Np 6 N. According to the current wireless standard [44], the frequency domain orthogonal pilot placement is commonly used at different transmitters as shown in the right portion of Fig. 1, which means the pilot patterns of any two distinct transmitters are disjoint, i.e., KiT \ KjT ¼ £, 1 6 iT –jT 6 N T and the cardinality of fK1 ; K2 ; :::; KNT g is N T N p . Therefore, a MIMOOFDM channel estimate can be decomposed into several SISOOFDM channel estimates [33]. Assume that the guard interval of the OFDM symbol is larger than L and ignore inter-symbol interference as well as intercarrier interference. Based on aforementioned assumptions, the pilot symbol Y iT ;jR received from the iT th transmitter at the jR th receiver can be given by

Y iT ;jR ¼ X iT W iT hiT ;jR þ N iR 2. Problem statement

NT

2

2

1 6 1 1 6 W iT ¼ pffiffiffiffi 6 . N6 4 ..

xkiT ;1 xkiT ;2 .. .

1 xkiT ;Np

ð2Þ 3

xkiT ;1 ðL1Þ 7    xkiT ;2 ðL1Þ 7  ..

.

.. .

7; x ¼ ej2p=N 7 5

ð3Þ

   xkiT ;Np ðL1Þ

where X iT ¼ diag ðX iT ðkiT ;1 Þ; X iT ðkiT ;2 Þ; ::::; X iT ðkiT ;Np ÞÞ is a diagonal matrix whose entries on the main diagonal are the transmitted pilot symbols of the iT th transmitter, W iT is a partial Fourier transform matrix of size N p  L constructed by selecting N p rows indexed by KiT and first L columns from a full N  N discrete Fourier matrix, N iR is a complex additive white Gaussian noise vector with zero

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mean and variance r2iR , i.e., N iR  CNð0; r2iR Þ. In addition, to obtain better estimation performance of every channel alike the reference [33], we assume all other transmitters must remain silent when the iT th antenna transmits signals, i.e., X jT ðnÞ ¼ 0; n 2 X ¼ f1; 2; :::; Ng; 1 6 iT –jT 6 N T . In the case of sparse channels, (2) is in agreement with the compressed sensing model and CS recovery algorithms can be applied to enabling the accurate recovery of hiT ;jR by solving

  minhiT ;jR 1 ; s:t:k AiT hiT ;jR  Y iT ;jR k2 6 e

ð4Þ

where e is error tolerance and AiT ¼ X iT W iT is the measurement matrix in CS theory. According to CS, the recovery accuracy is related to the mutual coherence of the measurement matrix AiT when CS is employed to enable unique identification of a sparse channel from its measurement vector Y iT ;jR [12,45]. The mutual coherence lðAiT Þ is defined as maximum absolute inner product between two distinct columns of AiT given by

jh lðAiT Þ ¼ 16m;n6L max kAi

Ai ðmÞ;Ai ðnÞij

T

T

T

ðmÞk kAi ðnÞk 2

T

2

PNp  ðmnÞ=N  2 j2pk  jX ðk Þj e iT ;l l¼1 iT iT ;l PN p ¼ max ; 2 16m;n6L jX ðk Þj l¼1 iT iT ;l m–n m–n

ð5Þ kiT ;l 2 KiT

where AiT ðmÞ, AiT ðnÞ are the mth and nth column of AiT respectively and h;i denotes inner product operation. (5) indicates that lðAiT Þ depends on both the pilot position KiT and the transmitted pilot symbol vector X iT . For simplicity, we consider pilot powers of pilot symbols at each transmitter are equivalent and assume the pilot power of each pilot symbol is E, i.e.,

      X i ðki ;1 Þ2 ¼ X i ðki ;2 Þ2 ¼    ¼ X i ðki ;N Þ2 ¼ E T T T T T T p

ð6Þ

as small as possible, we use the sum of mutual coherence of N T pilot patterns to evaluate whether N T pilot patterns is good or not. Accordingly, we aim to minimize the sum of mutual coherence of N T pilot patterns namely

min

NT P iT ¼1

!

lðKiT Þ

s:t: ð9Þ

KiT  f1; 2; :::; Ng; KiT ¼ fkiT ;1 ; kiT ;2 ; :::; kiT ;Np g; K1 \ K2 ::: \ KNT ¼ £; card ðK1 [ K2 ::: [ KNT Þ ¼ N T Np : Now, we aim to minimize

PNT

iT ¼1

lðKiT Þ by means of the pilot pat-

tern design to select N p pilot indices for each transmitter. Intuitively, though we can obtain N T orthogonal pilot patterns K1 ; K2 ; . . . ; KNT by exhaustively generating and evaluating all possi  N combinations, it is computationally prohibitive. An ble Np alternative method is to search N T pilot patterns with practical search methods [27,33]. Given a pilot pattern K1 ¼ f k1 ; k2 ; . . . ; kNp g with minðjkiþ1  ki jÞ P N T  1; i ¼ 1; 2; :::; N p  1, the shift mechanism in [33] indicates that K1 and KiT ¼ fk1 þ iT  1; k2 þ iT  1; . . . ; kNp þ iT  1g; iT ¼ 2; 3; :::; N T have the same value of mutual coherence, i.e., lðK1 Þ ¼ lðKiT Þ, due to the periodicity of ej2p=N . The shift mechanism based GA in [33] first finds a core pilot pattern K1 with the imposed constrained condition that minðkiþ1  ki Þ; 1 6 i 6 N p  1, ki 2 K1 called pilot distance is no smaller than N T  1. However, the constrained condition imposed on K1 may render the resultant core pilot pattern with large mutual coherence because there may be pilot patterns which are with smaller mutual coherence but dissatisfy the constrained condition. This motives us to explore new methods to design pilot patterns for a MIMO-OFDM system.

where iT ¼ 1; 2; :::; NT . (5) can be rewritten as

 N p  1 X j2pki ;l ðmnÞ=N  T lðAiT Þ ¼ 16m;n6L max e    N p  l¼1 m–n

3. Pilot optimization

ð7Þ

Obviously, (7) merely depends on KiT and lðAiT Þ can be uniquely determined by KiT . Henceforth, we also use lðKiT Þ to denote lðAiT Þ, i.e., lðKiT Þ ¼ lðAiT Þ. As a matter of fact, lðAiT Þ can be calculated easily because lðAiT Þ also equals the maximum absolute offdiagonal entry of GiT ¼ NNp AHiT AiT and the ðm; nÞth entry of GiT is PNp j2pki ;l ðmnÞ=N T e . Then we have GiT ðm; nÞ ¼ ð1=N p Þ l¼1





lðAiT Þ ¼ maxðGiT ðm; nÞÞ; 1 6 m–n 6 L

ð8Þ

According to CS theory, the smaller the mutual coherence is, the more accurate the reconstructed sparse signals are [27]. Hence, the measurement matrix AiT with small mutual coherence can guarantee more accurate recovery, which means we can carefully select entries of KiT to reduce lðAiT Þ for improving sparse channel estimation performance between the receiver and the iT th transmitter. Aforementioned analysis is based on estimating channels between one receiver and single the iT th transmitter. For a MIMO-OFDM system with N T transmitters, in order to raise average channel estimation performance of MIMO-OFDM systems, we should ensure that every lðAiT Þ; iT ¼ 1; 2; :::; N T is as small as possible to enable accurate sparse channel estimates at each receiver. Moreover, due to the orthogonality of N T pilot patterns, existing pilot design methods for SISO-OFDM systems can not directly applied to MIMO-OFDM pilot design. Because every lðAiT Þ is expected to be

Different from [33], we investigate how to make full use of the pilot distance of optimal pilot patterns obtained in advance without the imposed constrained condition on pilot distance to reduce PNT iT ¼1 lðKiT Þ. For the convenience of elaboration, we first give the proposed generalized shift invariance property and a corollary namely determined pilot pattern number. GSIP and DPPN answer the following questions respectively: (1) Given a pilot pattern K, how to use K to generate new pilot patterns whose the mutual coherence equals lðKÞ. (2) Given a pilot pattern K, what is the maximum number of pilot patterns that can be obtained when we use K to generate pilot patterns to make full use of pilot distance of K. 3.1. Generalized shift invariance property

Theorem 1 (Generalized shift invariance property). Given any a pilot pattern denoted by K ¼ fk1 ; k2 ; :::; kNp g with 1 6 ki 6 N and 1 6 i 6 N p , assume u1 and u2 are positive integers. Then fk1  u1 ; k2  u1 ; :::; kNp  u1 g, fk1 þ u2 ; k2 þ u2 ; :::;kNp þ u2 g and K possess the same value of mutual coherence and u1 and u2 are given as following. (1) Backward shift scheme. If N  kNp and minðkiþ1  ki Þ  1 satisfy N  kNp < minðkiþ1  ki Þ  1 for 1 6 i 6 N p  1, u2 takes Otherwise, if N  kNp and u2 ¼ 1; 2; :::; N  kNp .

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minðkiþ1  ki Þ  1 satisfy N  kNp P minðkiþ1  ki Þ  1 for 1 6 i 6 N p  1, u2 takes u2 ¼ 1; 2; :::; minðkiþ1  ki Þ  1. (2) Forward shift scheme. If k1  1 and minðkiþ1  ki Þ  1 satisfy k1  1 < minðkiþ1  ki Þ  1 for 1 6 i 6 N p  1, then u1 takes u1 ¼ 1; 2; :::; k1  1. Otherwise, if k1  1 and minðkiþ1  ki Þ  1 satisfy k1  1 P minðkiþ1  ki Þ  1 for 1 6 i 6 N p  1, u1 takes u1 ¼ 1; 2; :::; minðkiþ1  ki Þ  1. Note that we can use K to obtain new pilot patterns based on backward shift scheme and forward shift scheme simultaneously as long as the resultant pilot patterns are guaranteed to be disjoint. As a matter of fact, backward shift scheme of Theorem 1 is a strict mathematical description of shift mechanism in [33], we refer readers to [33] for the proof of (1). Now, we give the detailed proof of forward shift scheme as following. Proof:. Assume that the measurement matrix corresponding to the pilot pattern sorted in ascending order K ¼ fk1 ; k2 ; :::; kNp g is A. u1 must be no larger than minðkiþ1  ki Þ  1, otherwise the new generated pilot patterns will overlap with K. In addition, the smallest entry of fk1  u1 ; k2  u1 ; :::; kNp  u1 g must be positive which means u1 must satisfy k1  u1 P 1. Therefore, the value of u1 depends on the relationship of size between k1  1 and minðkiþ1  ki Þ  1. Consequently, if k1  1 and minðkiþ1  ki Þ  1 satisfy k1  1 < minðkiþ1  ki Þ  1 for 1 6 i 6 N p  1, u1 takes u1 ¼ 1; 2; :::; k1  1. If k1  1 and minðkiþ1  ki Þ  1 satisfy k1  1 P minðkiþ1  ki Þ  1, u1 must be less than or equal to minðkiþ1  ki Þ  1 to ensure K and fk1  u1 ; k2  u1 ; :::; kNp  u1 g are disjoint, i.e., u1 ¼ 1; 2; :::; minðkiþ1  ki Þ  1. Next, we will prove fk1  u1 ; k2  u1 ; :::; kNp  u1 g and K possess the same value of mutual coherence. The ðm; nÞth off-diagonal PNp j2pk ðmnÞ=N i entry of G ¼ NNp AH A is Gðm; nÞ ¼ ð1=N p Þ i¼1 e , 1 6 m–n 6 L. Let Au1 express the measurement matrix determined by K  u1 and then the ðm; nÞth entry of Gu1 ¼ NNp AH u1 Au1 can be expressed as

Gu1 ðm; nÞ ¼ ð1=Np Þ

Np P

ej2pðki u1 ÞðmnÞ=N

i¼1

¼ ej2pu1 ðmnÞ=N  ð1=Np Þ

Np P

ð10Þ ej2pki ðmnÞ=N

i¼1

Then we have

    maxðGu1 ðm; nÞÞ ¼ maxðej2pu1 ðmnÞ=N jGðm; nÞjÞ ¼ maxðjGðm; nÞjÞ

ð11Þ

Therefore, lðKÞ ¼ lðfk1  u1 ; k2  u1 ; :::; kNp  u1 gÞ holds, where u1 takes u1 ¼ 1; 2; :::; k1  1 if k1  1 < minðkiþ1  ki Þ  1or u1 ¼ 1; 2; :::; minðkiþ1  ki Þ  1 if k1  1 P minðkiþ1  ki Þ  1. Here, we arrive at forward shift scheme. h When we have obtained N T pilot patterns K1 ; K2 ; :::; KNT , we would like to replace pilot patterns having large mutual coherence with ones generated by pilot patterns having the smallest mutual coherence within obtained N T pilot patterns. In this way the resulPN T tant iT ¼1 lðKiT Þ is expected to be reduced. Theorem 1 just gives

how to use a given pilot pattern K to generate pilot patterns with the same mutual coherence lðKÞ as K. However, the maximum number of pilot patterns that we can obtained using K is not provided in Theorem 1. Besides, though we can calculate corresponding u1 and u2 based on Theorem 1 for a given pilot pattern K, the number of pilot patterns with mutual coherence lðKÞ is usually unequal to u1 þ u2 . The reason is that u1 þ u2 may be larger than the pilot distance minðkiþ1  ki Þ  1 of K leading to nonorthogonal

pilot patterns. In what follows, Corollary 1 gives the maximum number of pilot patterns which has the same value of mutual coherence as K and how to achieve the maximum. According to the relationship of size among k1 ; minðkiþ1  ki Þ  1; N  kNp , there are five cases of the maximum number. Corollary 1 (Determined pilot pattern number). Denote a given pilot pattern in ascending order by K ¼ fk1 ; k2 ; :::; kNp g, assume that the maximum number of orthogonal pilot patterns including K whose the mutual coherence equals lðKÞ is N K . N K and N K pilot patterns are given as following. (1) If k1  1 6 minðkiþ1  ki Þ  1 6 N  kNp , then N K ¼ minðkiþ1 ki Þ and NK  1 new generated pilot patterns are fk1 þ t; k2 þ t; :::; kNp þ tg with t ¼ 1; 2; :::; N K  1. (2) If minðkiþ1  ki Þ  1 6 k1  1 6 N  kNp or minðkiþ1  ki Þ  1 6 N  kNp 6 k1  1 then N K ¼ minðkiþ1 ki Þ and N K  1 new pilot patterns are given by fk1 þ t; k2 þ t; :::; kNp þ tg or fk1  t; k2  t; :::; kNp  tg with t ¼ 1; 2; :::; N K  1. then (3) If N  kNp 6 minðkiþ1  ki Þ  1 6 k1  1, N K ¼ minðkiþ1 ki Þ and N K  1 new pilot patterns are given by fk1  t; k2  t; :::; kNp  tg with t ¼ 1; 2; :::; N K  1. then NK ¼ (4) If k1  1 6 N  kNp 6 minðkiþ1  ki Þ  1, N  kNp þminðminðkiþ1  ki Þ  1  ðN  kNp Þ; k1  1Þ þ 1. new pilot patterns are given by N  kNp fk1 þ t; k2 þ t; :::; kNp þ tg with t ¼ 1; 2; :::; N  kNp and the remaining minðminðkiþ1  ki Þ  1ðN  kNp Þ; k1  1Þ new pilot patterns are given by fk1  s; k2  s; :::; kNp  sg with s ¼ 1; 2; :::; minðminðkiþ1  ki Þ  1ðN  kNp Þ; k1  1Þ. then (5) If N  kNp 6 k1  1 6 minðkiþ1  ki Þ  1, N K ¼ k1  1þminðminðkiþ1  ki Þ  1  ðk1  1Þ; N  kNp Þ þ 1. new pilot patterns are given by k1  1 fk1  t; k2  t; :::; kNp  tg with t ¼ 1; 2; :::; k1  1 and the remaining minðminðkiþ1  ki Þ  1ðk1  1Þ; N  kNp Þ pilot patwith terns are given by fk1 þ s; k2 þ s; :::; kNp þ sg s ¼ 1; 2; :::; minðminðkiþ1  ki Þ  1ðk1  1Þ; N  kNp Þ. Since (2)–(3) and (5) of Corollary 1 can be analogously demonstrated as the proofs of (1) and (4) respectively, we just give proofs of (1) and (4) in detailed. Proof:. Let X ¼ f1; 2;:::;Ng and K ¼ fk1 ; k2 ; :::; kNp g represent the subcarriers index set of an OFDM symbol and a given pilot pattern respectively. The proof of (1) is given as following. If k1  1 6 minðkiþ1  ki Þ  1 6 N  kNp holds, we can produce either k1  1 pilot patterns by fk1  u1 ; k2  u1 ; :::; kNp  u1 g with or N  kNp pilot patterns by u1 ¼ 1; 2; :::; k1  1 fk1 þ u2 ; k2 þ u2 ; :::; kNp þ u2 g with u2 ¼ 1; 2; :::; N  kNp . Since the size of available pilot distance is minðkiþ1  ki Þ  1, the maximum number of pilot patterns possessing the same mutual coherence as K is minðkiþ1  ki Þ  1 except K. Therefore, total N K ¼ minðkiþ1  ki Þ pilot patterns with the mutual coherence lðKÞ can be obtained and the new pilot patterns are fk1 þ t; k2 þ t; :::; kNp þ tg with t ¼ 1; 2; :::; N K  1. The proof of (4) is given as following. If the inequality k1  1 6 N  kNp 6 minðkiþ1  ki Þ  1 is obeyed, we at most yield pilot patterns by fk1 þ t; k2 þ t; :::; kNp þ tg with N  kNp t ¼ 1; 2; :::; N  kNp . Then the size of remaining pilot distance is minðkiþ1  ki Þ  1  ðN  kNp Þ. Meanwhile, we can also continue generating pilot patterns by fk1  s; k2  s; :::; kNp  sg with s ¼ 1; 2; :::; minðminðkiþ1  ki Þ  1ðN  kNp Þ; k1  1Þ according to

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forward shift scheme. In summary, including K we can in total get N K ¼ N  kNp þ minðminðkiþ1  ki Þ  1  ðN  kNp Þ; k1 1Þ þ 1 pilot patterns when k1  1 6 N  kNp 6 minðkiþ1  ki Þ1 holds. h

(1) Initialize X1 ¼ f1; 2; :::; Ng and attain p1 with SSS from X1 .

3.2. Pilot design schemes

pk with SSS within Xk . (3) Output N T pilot patterns, i.e., p1 ; p2 ; :::; pNT .

Theorem 1 and Corollary 1 suggest that, given a pilot pattern K, we can maximum determine N K orthogonal pilot patterns whose the value of mutual coherence equals lðKÞ. Consequently, when we are given N T pilot patterns, we would like to make full use of the pilot distance of the pilot pattern with smallest mutual coherP T ence to generate pilot patterns so as to reduce Ni¼1 lðKi Þ. Based on this idea, we propose the following two pilot pattern design methods namely PDGSIP and TPDGSIP. 3.2.1. Pilot design with GSIP (PDGSIP) Different from [33] which firstly searches a core pilot pattern, 





we first search N T orthogonal pilot patterns K1 ; K2 ; :::; KNT and 





assume lðK1 Þ 6 lðK2 Þ::: 6 lðKNT Þ. Note that there is no the imposed constrained condition that the pilot distance is no smaller 

than N T  1 imposed on Ki ; i ¼ 1; 2; :::; N T so that the mutual coherence of each pilot pattern is as small as possible. Then, we make 

full use of optimal pilot pattern K1 to produce new pilot patterns according to Corollary 1. Assume the maximum number of pilot 

patterns with the same mutual coherence as K1 is d1 and d1 pilot patterns are K1 ; K2 ; :::; Kd1 . Then we successively use 

Ki ; i ¼ 2; :::; NT to generate new pilot patterns until we obtain NT orthogonal pilot patterns. Through the aforementioned procedure the pilot distance of optimal pilot patterns are fully used and the PNT resultant i¼1 lðKi Þ is expected to be smaller than original  PN T l ð K Þ. This is the core idea of the following PDGSIP. i i¼1 According to the previous analysis, PDGSIP needs to obtain N T orthogonal pilot patterns beforehand which can be obtained with any a MIMO-OFDM pilot design method. Without loss of generality, we base PDGSIP on ES1 in [27] for which exhibits the better performance. For the self-containedness of this paper, we give ES1 in detail as following.

Extension Scheme 1: (2) For k ¼ 2; 3; . . . ; N T , update Xk with Xk ¼ Xk1 pk1 and attain

The procedure of PDGSIP is as following. Given the predetermined N T ; L; N p ; N,

we

initialize

P ¼ 0NNp ; X ¼ f1; 2; :::; Ng; r ¼ 0. We first obtain N T pilot patterns p1 ; p2 ; :::; pNT with ES1 and assume lðp1 Þ 6 lðp2 Þ 6 ::: 6 lðpNT Þ. Initialize X0 ( Xp1 ; p2 ; :::;pNT g. For k ¼ 1; 2; :::; N T , we perform the following loops. Update the count variable r as r ( r þ 1 and save pk to the rth row of P namely Pðr; :Þ ( pk . Calculate v 1 ¼ k1  1; v 2 ¼ minðkiþ1  ki Þ  1,v 3 ¼ N  kNp associated with pk according to Corollary 1. Denote q ¼ q1 þ q2 where q1 ; q2 express the number of new pilot patterns generated with pk by means of each entry of pk minus j, j ¼ 0; 1; :::; q1 and each entry of pk plus j, j ¼ 0; 1; :::; q2 respectively. Note that q pilot patterns are orthogonal pilot patterns and q1 as well as q2 are given in Table 1 for (1)–(5) in Corollary 1 respectively. (1) If k ¼ 1 and q P N T  1 are simultaneously satisfied, we straightforward generated N T  1 pilot patterns with p1 plus or minus a constant as shown in line 7 to line 12 in Algorithm 2. We detail this procedure as follows. We firstly obtain q2 new pilot patterns with p1 plus a constant j and then save the new pilot patterns to P, i.e., Pð1 þ j; :Þ ( p1 þ j where j ¼ 0; :::; q2 .Then obtain q1 new pilot patterns with p1 minus a constant j and save the new pilot patterns to P, i.e., Pðq2 þ 1 þ j; :Þ ( p1  j where j ¼ 0; :::; q1 . Then we terminate PDGSIP and output the first N T rows of P as eventual pilot patterns, i.e., Ki ¼ Pði; :Þ; i ¼ 1; :::; N T . (2) Otherwise, we successively continue performing following steps.  We obtain pilot patterns by pk plus a constant. Specifically, for j ¼ 0; 1; :::; q2 , if the cardinality of the intersection of pk þ j and Xr equals N p , which means pk þ j is orthogonal top1 ; p2 ; :::; pNT ; Pð1; :Þ; Pð2; :Þ; :::; Pðr; :Þ; we update r ( r þ 1; Pðr; :Þ ( pk þ j and the available index set Xr ( Xr1 n Pðr; :Þ as shown in line 17.  We obtain pilot patterns by pk minus a constant. Specifically, for j ¼ 0; 1; :::; q1 , if the cardinality of the intersection of pk  j and Xr equals Np , which means pk  j is orthogonal to p1 ; p2 ; :::; pNT ;Pð1; :Þ; Pð2; :Þ; :::; Pðr; :Þ; we update r ( r þ 1; Pðr; :Þ ( pk  j and the available index set Xr (Xr1 n Pðr; :Þ. This procedure is realized in lines 20–24. If r P N T , we terminate loops and output first N T pilot patterns, i.e., Ki ¼ Pði; :Þ; i ¼ 1; :::; N T . Table 1 q1 and q2 for (1)–(5) of Corollary 1. Cases

q1

q2

(1) (2) (3) (4) (5)

0 0

v2 v2

v2

minðv 2  v 3 ; v 1 Þ

v1

0

v3

minðv 2  v 1 ; v 3 Þ

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With the PDGSIP which fully employs the pilot distance, the mutual coherence sum of Ki ¼Pði; :Þ; i ¼ 1; :::; N T are expected to be smaller than that of p1 ; p2 ; :::; pNT . In addition, the reason that N T pilot patterns in line 2 of Algorithm 2 should be attained beforehand is that at least N T pilot patterns are guaranteed to be obtained after performing Algorithm 2. The detailed procedure of PDGSIP is given in Algorithm 2.

3.2.2. Tradeoff pilot design with GSIP (TPDGSIP) Let N, N p , N T express the IFFT size, the pilot number and the transmitter

number

respectively.

Set

P ¼ 0NT Np

and

X ¼ f1; 2; 3; :::; Ng. 0

Denote by the number of obtained pilot patterns r and initialize r ¼ 0. While r < N T , we perform the rth iteration as following. Increase r by one, i.e., r ¼ r þ 1. We first search a pilot pattern p whose entries are selected from the available subcarrier index set

Xr1 by performing SSS, and then save p as Pðr; :Þ ( p . Next, the usable pilot index set for the ðr þ 1Þth transmitter is updated as following.

Xr ¼ Xr1 Pð1; :Þ; Pð2; :Þ; :::; Pðr; :Þg

ð12Þ

After we obtain Pðr; :Þ, we yield new pilot patterns with p by means of increasing or decreasing each entry of p by a constant. Specifically, we first calculate q; q1 ; q2 according to Table 1. Then, the procedure of yielding new pilot patterns are the same as lines 15–24 of Algorithm 2, in which pk is replaced by p . Note that, compared with Algorithm 2, we may yield more pilot patterns with a given pilot pattern because the cardinality of Xr in lines 9–14 of Algorithm 3 is usually larger than that in lines 6 and 21 of Algorithm 2. Therefore, the possibility of pilot pattern intersection is alleviated. We end the iteration until r P N T so as to guarantee obtaining at least N T pilot patterns. In the end, output first N T pilot patterns K1 ; K2 ; . . . ; KNT . The pseudo code of TPDGSIP is given in Algorithm 3. Compared with PDGSIP, the priority of TPDGSIP is that TPDGSIP makes full use of pilot distance of an obtained pilot pattern without first searching a group of N T pilot patterns in advance. Therefore, TPDGSIP alleviates the problem of possible pilot patterns superposition between the acquired pilot patterns and new generated pilot patterns. Moreover, the TPDGSIP is more efficient in comparison to PDGSIP in terms of time cost, which is verified by simulation results in Section 4. Algorithm 3: Tradeoff Pilot Design with GSIP 1: Input:N; N T ; N p ; r ¼ 0; X0 ¼ f1; 2; 3; :::; Ng; P ¼ 0NT Np . 2: while r < N T 3: r ¼ r þ 1. 4: 5:

P T Albeit PDGSIP is efficient in the view of minimizing Ni¼1 lðKi Þ, it is inefficient from the point of time consumption because N T pilot patterns need to be obtained ahead of time. In addition, when we use pk to produce pilot patterns, the derived new pilot pattern pk þ j or pk  j may intersect with the unavailable subcarrier index set X Xr1 ¼ fp1 ; p2 ; :::;pNT ; Pð1; :Þ; :::; Pðr  1; :Þg, namely line 16 and line 21 do not always hold for some j’s, which results in underutilizing the pilot distance of pk . Consequently, we propose the second pilot design scheme namely TPDGSIP, which sequentially searches pilot patterns. Specifically, TPDGSIP will use the 

obtained pilot pattern K1 to generate new pilot patterns 

K1 ; K2 ; :::; Kd1 according to DPPN as soon as K1 is obtained, where d1 is the number of pilot patterns with mutual coherence equaling 

lðK1 Þ. Repeat the aforementioned procedure until NT pilot patterns are obtained. We now elaborate the TPDGSIP in detail as follows.

Obtain p with SSS, where p  Xr1 . Pðr; :Þ ( p .

6: Update Xr ¼ Xr1 Pð1; :Þ; Pð2; :Þ; :::; Pðr; :Þg. 7: Compute q; q1 ; q2 according to Table 1. 8–16: The same as lines 15–24 of Algorithm 2, where the pk is replaced by p . 17: end while 18: Output N T pilot patterns Ki ¼ Pði; :Þ; i ¼ 1; 2; :::; N T .

Remark:. While PDGSIP is based on ES1, PDGSIP is effectively based on SSS because SSS is the main procedure of ES1. Both PDGSIP and TPDGSIP are two universal pilot pattern design frames. In other words, we can acquire new pilot design schemes when SSS in PDGSIP or TPDGSIP is superseded by any other pilot design schemes for SISO-OFDM systems. For example, we can get new pilot pattern design schemes for MIMO-OFDM systems when we replace SSS in PDGSIP or TPDGSIP with cross-entropy optimization in [29], estimation of distribution algorithm in [26] and so on.

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4. Simulation results

10

NMSE ¼ ð1=nr Þ

10

-1

BER

In order to validate our proposed two pilot design schemes, we consider a sparse frequency selective multipath fading channel and the multipath time delay is evenly sampled with equivalent interval. Unless specified, in the following we employ the finite time delay filter model as shown in the equation (1). In the simulation of the following Figs. 2 and 3, we assume that hiT ;jR ; iT ¼ 1; :::; N T ; jR ¼ 1; :::; N R has the equal channel length L ¼ 60, among which six sampled taps are nonzero. We also assume that the amplitudes of nonzero taps are randomly distributed within ½1; L and the tap amplitude hiT ;jR ðlÞ follows Gaussian distribution with zero mean and unit variance, i.e., hiT ;jR ðlÞ  CNð0; 1Þ. Note that there are total N T  N R channel vectors for every MIMO-OFDM transmission and each channel vector is independent identical distribution. We take a MIMO-OFDM system with N T ¼ 4;N R ¼ 2 where the transmitted symbols are modulated with QPSK and the zero-forcing equalizer is used to equalize the received data. The detailed simulation parameters are listed in Table 2, where nr in Table 2 represents the realization times of the MIMO-OFDM transmission. Apart from the mutual coherence, we also evaluate the channel estimation performance of different pilot design methods in terms of NMSE and BER. The NMSE is defined as

0

LS EQ GA ES1

10

T PDGSIP

-2

PDGSIP 0

ð13Þ b i is the estimated channel vector between the iT th transwhere h iT ;jR mitter and jR th receiver at the ith realization. In the simulations for pilot-assisted sparse channel estimation, we compare five pilot patterns and each is designed as following five methods, respectively. (1) GA in [33]. GA aims to first search GA GA KGA 2 ; K3 ; K4 with the shift mechanism. Alike parameters setting for GA in [33], the boundary of pilot distance is within [5,60] in each iteration and we set the population size, chromosome size, generation size, cross rate, mutation rate as 100, 16, 5000, 0.7, 0.6, respectively. In addition, we employ GA in genetic algorithm toolbox released by Andrew Chipperfield of University of Sheffield [46]. Note that GA in this paper also performs the perfection operation described in [33]. (2) ES1 proposed in the reference [27]. ES1

5

NMSE/dB

0

-5

LS EQ

-10

GA -15

ES1 T PDGSIP PDGSIP

-20

0

5

10

20

25

30

Fig. 2. NMSE performance comparisons of different pilot design schemes.

20

25

30

Parameter

Value

N Np L K NT NR nr

512 16 60 6 4 2 1000

sequentially obtains four pilot patterns KES1 i ; i ¼ 1; 2; 3; 4. In ES1, the number of the outer loop and inner loop are set to M 1 ¼ 8 and M2 ¼ 20 respectively. (3) Two proposed pilot design algorithms, i.e., PDGSIP and TPDGSIP. In PDGSIP, we use the output pilot patterns of ES1 in (2) as the input pilot pattern of PDGSIP. (4) Equal-spaced pilot patterns (denoted by EQ). The first position index of four equal-spaced pilot patterns KEQ i ; i ¼ 1; 2; 3; 4 is 1, 2, 3, 4 respectively and the pilot distance of each pilot pattern is 31. In addition, we also simulate the sparse channel estimation using the conventional LS with 32 equal-spaced pilots as comparison. As to the CS reconstruction algorithm, we employ prevalent OMP to enable the recovery of sparse channels. We first need to attain the pilot patterns designed by GA, ES1, PDGSIP and TPDGSIP respectively. The values of mutual coherence of each pilot pattern and the sum of values of mutual coherence of four pilot patterns obtained with aforementioned four pilot design schemes as well as EQ are given in Table 3. The specific pilot patterns designed by five schemes are given in Appendix A. It is shown that the EQ yields the worst pilot patterns with the largest values P of mutual coherence as well as 4i¼1 lðKi Þ, and the values of mutual coherence can be effectively reduced by pilot design schemes. Compared with GA, PDGSIP and TPDGSIP gain an improvement of 0.18 and 0.14 in the mutual coherence sum respectively. ES1 obtains four different pilot patterns and the value of mutual coherence of the best pilot pattern is 0.2861. However, the pilot distance of KES1 with 1

15 SNR/dB

15 SNR/dB

Table 2 Parameters of the MIMO-OFDM system.

T

a core pilot pattern KGA and then obtain the remaining 1

10

Fig. 3. BER performance comparisons of different pilot design schemes.

X NR X NT nr  X 1 b i  hi k2 =k hi k2 kh iT ;jR 2 iT ;jR 2 iT ;jR N N T R i¼1 j ¼1 i ¼1 R

5

lðKES1 1 Þ ¼ 0:2861 is not fully used to generate pilot

patterns with the same value of mutual coherence as KES1 1 . In contrast, PDGSIP gains an improvement of 0.04 over ES1 in the mutual coherence sum because the PDGSIP makes full use of the pilot dis-

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Table 3 Mutual coherence of pilot patterns with different pilot design schemes. Schemes

lðK1 Þ

lðK2 Þ

lðK3 Þ

lðK4 Þ

P4

EQ GA ES1 PDGSIP TPDGSIP

1 0.3305 0.2861 0.2861 0.2956

1 0.3305 0.2961 0.2861 0.2956

1 0.3305 0.3025 0.2861 0.2956

1 0.3305 0.3071 0.2861 0.2956

4 1.3220 1.1918 1.1444 1.1824

tance of optimal pilot pattern namely KES1 to generate the remain1 ing three pilot patterns, which are with the same value of mutual coherence as KES1 1 . Besides, TPDGSIP also gains an improvement of 0.01 in the mutual coherence sum in comparison with ES1. Obviously, the proposed two schemes outperform ES1 because both fully employ the pilot distance as much as possible in designing pilot patterns. Owing to SSS is a part of ES1, both PDGSIP and TPDGSIP in nature are based on SSS. Because of PDGSIP conducting more SSS realization times than TPDGSIP, PDGSIP searches pilot patterns from much more index combinations than TPDGSIP and achieves better pilot patterns. Among five pilot design schemes, the proposed two pilot design schemes achieve the best pilot patterns from the perspective of mutual coherence as well as P4 i¼1 lðKi Þ. The reason is that two proposed pilot design schemes make full use of pilot distance of the pilot pattern. As to computational complexity, we compare the time cost of four pilot pattern design schemes. The running time of four pilot pattern methods are given in Table 4. Note that PDGSIP optimizes the result of ES1 and hence the time cost of PDGSIP involves that of ES1. The time cost of PDGSIP except line 2 in Algorithm 2 is very small and hence we leave out the decimal portion of time cost of PDGSIP. Clearly, the time cost of both ES1 and PDGSIP is equivalent and around five times the time cost of TPDGSIP. The time cost of GA is about two times as long as TPDGSIP. However, TPDGSIP makes a tradeoff between the pilot design performance and time cost. Now, we compare the channel estimation performance using pilot patterns obtained by five pilot design schemes as well as LS channel estimation, where the values of mutual coherence of pilot patterns of each pilot scheme are given in each line of Table 3 respectively. Figs. 2 and 3 show the channel estimation performance using five pilot patterns as well as LS channel estimation in terms of NMSE and BER respectively. Clearly, CS channel estimation performance using evenly distributed pilot patterns is nearly the same as LS channel estimation performance in NMSE and BER due to the measurement matrix determined by EQ failing to provide guarantees of uniqueness with the measurements. Meanwhile, channel estimation using pilot pattern designed by GA gains a marked improvement of around 17 dB in NMSE and more than one order of magnitude in BER over EQ as well as LS in signal to noise ratio at 30 dB. Therefore, CS based channel estimation raises the spectral efficiency compared with LS since the CS with only 16 pilots achieves far better channel estimation performance than LS. It also noticed that the channel estimation using the pilot pattern obtained from PDGSIP outperforms that using pilot patterns designed by TPDGSIP, but both outperform GA and ES1 because pilot patterns designed by both PDGSIP and TPDGSIP enjoy smaller mutual coherence. Specifically, PDGSIP gains around 2.5 dB and 1 dB over GA and ES1 in signal to noise ratio at 30 dB respectively

Table 4 Time cost of different pilot pattern design schemes. Schemes

GA

ES1

PDGSIP

TPDGSIP

Time

1629 s

3713 s

3713 s

703 s

i¼1

lðKi Þ

owing to the pilot patterns designed by PDGSIP with the smallest P4 i¼1 lðKi Þ as shown in Table 3. PDGSIP is slightly superior to TPDGSIP because PDGSIP conducts much more SSS realizations than TPDGSIP. Consequently, PDGSIP guarantees the highest success probability of spare reconstruction and hence achieves the best channel estimation performance among five pilot design schemes in terms of NMSE and BER. 5. Conclusion In this paper, we have investigated the pilot pattern design for MIMO-OFDM systems based on the mutual coherence of compressed sensing. We first derive the GISP to produce pilot patterns having equivalent value of mutual coherence and a corollary which provides the number of pilot patterns with the same value of mutual coherence as a given pilot pattern. Besides, two pilot design schemes called PDGSIP and TPDGSIP have been proposed based on GSIP. We evaluate the performance of pilot pattern design schemes from the perspective of mutual coherence and sparse channel estimation. The simulation results show that both PDGSIP and TPDGSIP are superior to existing pilot schemes and PDGSIP achieves the best pilot patterns but with larger time cost in comparison to TPDGSIP. Simulation results have verified two proposed pilot design schemes outperform ES1 and GA in term of NMSE and BER. Though channel estimation with pilot patterns designed by PDGSIP gains the smallest NMSE and BER, TPDGSIP makes a tradeoff between channel estimation performance and time cost. Since the pilot design is implemented in system design phrase, the computational complexity has no effect on the system performance and hence two proposed methods are suited to pilot pattern design for MIMO-OFDM systems. Acknowledgment This work is supported by the National Natural Science Foundation of China [grant number 61701335, 61571318, 61571323]; Natural Science Foundation of Tianjin [grant number 17JCQNJC01300]; and the Guangxi Science and Technology Project [grant number AC16380094]. Appendix A A.1. Specific pilot patterns corresponding to algorithms in Table 3 The four equal-spaced pilot patterns with l ¼ 1 are K1 ¼ f1, 33, 65, 97, 129, 161, 193, 225, 257, 289, 321, 353, 385, 417, 449, 481}, K2 ¼ K1 þ 1, K3 ¼ K1 þ 2, K4 ¼ K1 þ 3 respectively. The four pilot patterns obtained by GA with l ¼ 0:3305 are K1 ¼ f1, 16, 62, 121, 177, 233, 238, 249, 273, 284, 289, 315, 367, 416, 434, 440}, K2 ¼ K1 þ 1, K3 ¼ K1 þ 2, K4 ¼ K1 þ 3 respectively. The four pilot patterns obtained by ES1 with l1 ¼ 0:2861;l2 ¼ 0:2961; l3 ¼ 0:3025; l4 ¼ 0:3071 are K1 ¼ f81, 97, 105, 146, 181, 201, 249, 311, 387, 438, 444, 469, 476, 484, 500, 511}, K2 ¼ f109, 136, 187, 195, 256, 302, 349, 357, 365, 390, 397, 431, 465, 481, 488, 502}, K3 ¼ f 96, 147, 170, 186, 193, 213, 220, 282, 298, 331, 350,

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364, 408, 451, 506, 512}, K4 ¼ f2, 83, 104, 154, 160, 171, 178, 221, 260, 325, 354, 360, 369, 391, 404, 487} respectively. The four pilot patterns obtained by PDGSIP with l ¼ 0:2861 are K1 ¼ f81, 97, 105, 146, 181, 201, 249, 311, 387, 438, 444, 469, 476, 484, 500, 511}, K2 ¼ K1  1, K3 ¼ K1  2, K4 ¼ K1  3 respectively. The four pilot patterns obtained by TPDGSIP with l ¼ 0:2956 are K1 ¼ f1, 67, 195, 208, 216, 223, 232, 268, 327, 335, 367, 404, 412, 418, 460, 490}, K2 ¼ K1 þ 1, K3 ¼ K1 þ 2, K4 ¼ K1 þ 3 respectively. References [1] Molisch AF, Ratnam VV, Han SQ, Li ZD, Nguyen SLH, Li LS, et al. Hybrid beamforming for massive MIMO: a survey. IEEE Commun Mag 2017;55 (9):134–41. [2] Shen WQ, Dai LL, Shi Y, Gao Z, Wang ZC. Massive MIMO channel estimation based on block iterative support detection. In: IEEE wireless communications and networking conference. IEEE; 2016. p. 1–6. [3] Hari Krishna E, Sivani K, Ashoka Reddy K. On the use of EMD based adaptive filtering for OFDM channel estimation. AEU-Int J Electron Commun 2018;83:492–500. [4] Singh H, Bansal S. Channel estimation with ISFLA based pilot pattern optimization for MIMO-OFDM system. AEU-Int J Electron Commun 2017;81:143–9. [5] Hou W, Lim CW. Structured compressive channel estimation for large-scale MISO-OFDM systems. IEEE Commun Lett 2014;18(5):765–8. [6] Dai LL, Wang JT, Wang ZC, Tsiaflakis P. Spectrum- and energy-efficient OFDM based on simultaneous multi-channel reconstruction. IEEE Trans Commun Signal Process 2013;61(23):6047–59. [7] Choi JW, Shim B, Chang S. Downlink pilot reduction for massive MIMO systems via compressed sensing. IEEE Commun Lett 2015;19(11):1889–92. [8] Gao Z, Dai L, Dai W, Shim B, Wang Z. Structured compressive sensing-based spatio-temporal joint channel estimation for FDD massive MIMO. IEEE Trans Commun 2016;64(2):601–17. [9] He XY, Song RF, Zhu WP. Optimal pilot pattern design for compressed sensingbased sparse channel estimation in OFDM systems. Circuits Syst Signal Process 2012;31(4):1379–95. [10] Qaisar S, Bilal RM, Iqbal W, Naureen M, Lee S. Compressive sensing: from theory to applications, a survey. J Commun Netw 2013;15(5):443–56. [11] Candes EJ, Wakin MB. An introduction to compressive sampling. IEEE Signal Process Mag 2008;25(2):21–30. [12] Baraniuk RG, Cevher V, Duarte MF, Hegde C. Model-based compressive sensing. IEEE Trans Inf Theory 2010;56(4):1982–2001. [13] Bajwa WU, Haupt J, Sayeed AM, Nowak R. Compressed channel sensing: a new approach to estimating sparse multipath channels. Proc IEEE 2010;98 (6):1058–76. [14] Shen W, Dai L, Shi Y, Shim B, Wang Z. Joint channel training and feedback for FDD massive MIMO systems. IEEE Trans Veh Technol 2015;65(10):8762–7. [15] Xie H, Andrieux G, Wang Y, Feng S, Yu Z. A novel threshold based compressed channel sensing in OFDM system. AEU-Int J Electron Commun 2017;77:149–55. [16] Li Y, Hamamura M. Zero-attracting variable-step-size least mean square algorithms for adaptive sparse channel estimation. Int J Adapt Control Signal Process 2015;29(9):1189–206. [17] Li Y, Jin Z, Wang Y. Adaptive channel estimation based on an improved normconstrained set-membership normalized least mean square algorithm. Wirel Commun Mob Comput 2017;2017(1):1–8. [18] Li Y, Wang Y, Jiang T. Norm-adaption penalized least mean square/fourth algorithm for sparse channel estimation. Signal Process 2016;128 (216):243–51. [19] Li Y, Wang Y, Jiang T. Sparse-aware set-membership NLMS algorithms and their application for sparse channel estimation and echo cancelation. AEU-Int J Electron Commun 2016;70(7):895–902. [20] Candes EJ, Tao T. Decoding by linear programming. IEEE Trans Inf Theory 2005;51(12):4203–15.

57

[21] Duarte MF, Eldar YC. Structured compressed sensing: from theory to applications. IEEE Trans Commun Signal Process 2011;59(9):4053–85. [22] Zelnik-Manor L, Rosenblum K, Eldar YC. Sensing matrix optimization for blocksparse decoding. IEEE Trans Commun Signal Process 2011;59(9):4300–12. [23] Mohammadian R, Amini A, Khalaj BH. Deterministic pilot design for sparse channel estimation in MISO/Multi-user OFDM systems. IEEE Trans Wirel Commun 2017;16(1):129–40. [24] Mohammadian R, Amini A, Khalaj BH. Compressive sensing-based pilot design for sparse channel estimation in OFDM systems. IEEE Commun Lett 2017;21 (1):4–7. [25] He XY, Song RF, Zhu WP. Pilot allocation for distributed compressed sensing based sparse channel estimation in MIMO-OFDM systems. IEEE Trans Veh Technol 2016;65(5):2990–3004. [26] Wang H, Guo Q, Zhang GX, Li GX. Pilot pattern optimization for sparse channel estimation in OFDM systems. IEEE Commun Lett 2015;19(7):1233–6. [27] Qi CH, Yue GS, Wu LN, Huang YM. Pilot design schemes for sparse channel estimation in OFDM systems. IEEE Trans Veh Technol 2015;64(4):1493–505. [28] Qi CH, Wu LN, Huang YM, Nallanathan A. Joint design of pilot power and pilot pattern for sparse cognitive radio systems. IEEE Trans Veh Technol 2015;64 (11):5384–90. [29] Qi C, Yue G, Wu L, Nallanathan A. Pilot design for sparse channel estimation in OFDM-based cognitive radio systems. IEEE Trans Veh Technol 2014;63 (2):982–7. [30] Qi C, Wu L. Optimized pilot placement for sparse channel estimation in OFDM systems. IEEE Signal Process Lett 2011;18(12):749–52. [31] Qi CH, Wu LN. A study of deterministic pilot allocation for sparse channel estimation in OFDM systems. IEEE Commun Lett 2012;16(5):742–4. [32] Khosravi M, Mashhadi S. Joint pilot power & pattern design for compressive OFDM channel estimation. IEEE Commun Lett 2015;19(1):50–3. [33] He XY, Song RF, Zhu WP. Pilot allocation for sparse channel estimation in MIMO-OFDM systems. IEEE Trans Circuits Syst II Express Briefs 2013;60 (9):612–6. [34] Ma X, Yang F, Ding W, Song J. A study of training sequence design for channel estimation based on compressive sensing. In: GLOBECOM workshops. IEEE; 2016. p. 1–6. [35] Ma X, Yang F, Ding W, Song J. Novel approach to design time-domain training sequence for accurate sparse channel estimation. IEEE Trans Broadcast 2016;62(3):512–20. [36] Lee D. MIMO-OFDM channel estimation via block stagewise orthogonal matching pursuit. IEEE Commun Lett 2016;20(10):2115–8. [37] Ma X, Yang F, Liu S, Song J, Han Z. Design and optimization on training sequence for mmWave communications: a new approach for sparse channel estimation in massive MIMO. IEEE J Sel Areas Commun 2017;35(7):1486–97. [38] Ma X, Yang F, Liu S, Song J, Han Z. Training sequence design and optimization for structured compressive sensing based channel estimation in massive MIMO systems. In: GLOBECOM workshops. IEEE; 2017. p. 1–6. [39] Gao Z, Dai LL, Wang ZC. Structured compressive sensing based superimposed pilot design in downlink large-scale MIMO systems. Electron Lett 2014;50 (12):896–8. [40] Gao Z, Dai L, Wang Z, Chen S. Spatially common sparsity based adaptive channel estimation and feedback for FDD massive MIMO. IEEE Trans Commun Signal Process 2015;63(23):6169–83. [41] Ma X, Yang F, Liu S, Song J, Han Z. Sparse channel estimation for MIMO-OFDM systems in high-mobility situations. IEEE Trans Veh Technol 2018;67 (7):6113–24. [42] Barbotin Y, Hormati A, Rangan S, Vetterli M. Estimation of sparse MIMO channels with common support. IEEE Trans Commun 2012;60(12):3705–16. [43] Masood M, Afify LH, Al-Naffouri TY. Efficient coordinated recovery of sparse channels in massive MIMO. IEEE Trans Commun Signal Process 2014;63 (1):104–18. [44] Ghosh A, Ratasuk R, Mondal B, Mangalvedhe N, Thomas T. LTE-advanced: nextgeneration wireless broadband technology. IEEE Wirel Commun 2010;17 (3):10–22. [45] Donoho DL. Compressed sensing. IEEE Trans Inf Theory 2006;52(4):1289–306. [46] Chipperfied A, Fleming P, Pohlheim H, FonSeca C. Genetic algorithms toolbox. Version 1.2 [software]; 1994 [cited 2018 May 14]. Available from: http://codem.group.shef.ac.uk/index.php/ga-toolbox.