 Research
 Open Access
Eigenvectorbased initial ranging process for OFDMA uplink systems
 Sungeun Lee^{1}Email author and
 Xiaoli Ma^{1}
https://doi.org/10.1186/1687618020151
© Lee and Ma; licensee Springer. 2015
 Received: 15 April 2014
 Accepted: 24 October 2014
 Published: 16 January 2015
Abstract
At the contentionbased synchronization, e.g., initial ranging process, it is crucial to identify multiple users through ranging codes and estimate the corresponding parameters such as timing offset and frequency offset. This paper presents an improved parameter estimation and pairing algorithm for initial ranging process in orthogonal frequencydivision multiple access (OFDMA) uplink systems by exploiting multidimensional harmonic retrieval (HR) technique. Unlike most existing techniques that estimate each parameters independently and associate the ranging codes with the estimated parameters manually, the proposed method estimates multiple user’s ranging code, timing and frequency offsets simultaneously, and pair up all the multiple parameters automatically. The simulation results confirm that the proposed technique not only improves the ranging code detection capability and adjusts the acquisition range of the estimators but also increases the maximum number of resolvable users for given samples.
Keywords
 Initial ranging
 Synchronization
 Contention
 OFDMA
 Eigenvector
 Harmonic retrieval
 Random access
 Uplink
 Identifiability
1 Introduction
In general orthogonal frequencydivision multiple access (OFDMA) uplink systems, to maintain the orthogonality among multiple users, the signals from all active users should arrive at the base station (BS) synchronously [1]. In order to align the signals of multiple users, contentionbased random access procedure, called as initial ranging process, should be performed in the beginning in order to identify multiple users as well as to estimate the corresponding timing and/or frequency offsets for the adjustment/alignment [2, 3]. So, code identification as well as multiuser timing estimation are the main tasks of the BS in contentionbased synchronization processes.
On the contentionbased synchronization process, the same time/frequency resources are shared by multiple users, so the multiple access interference (MAI) limits the performance of ranging detectors. Consequently, it is essential to deal with MAI of ranging subscriber stations (RSSs) to achieve the reliable initial ranging performance.
In the beginning, the existing ranging detection methods simply treat the MAI as noise, which results in the performance degradation as the number of RSSs increases due to the limitation by the amount of MAI [4–6]. To decrease the effect of the accumulated MAI on the ranging process, Ruan et al. [7] proposed a successive multiuser detection (SMUD) algorithm which detects the channel paths of active ranging signals and cancels their interference for further detection. In order to alleviate the high computational complexity of SMUD, the lowcomplexity method is proposed in [8], and in order to increase the resilience to MAI, a generalized likelihood ratio test (GLRT) approach is also exploited to derive a twostage interference cancellation scheme [9]. However, it is shown that still SMUD and interference cancellationbased methods are vulnerable for MAI, and the code detection and parameter estimation performance are severely degraded as the number of active RSSs increases due to accumulated residual MAI [10].
In order to reduce the chance undergoing severe amount of MAI and exploiting the frequency selectivity of the fading channels, subchannelbased frame structure is proposed which allocate a smaller number of subcarriers to each ranging opportunity so that most of the RSSs are expected to transmit on disjoint sets of subcarriers with alleviated interference to each other [11–15].
With the improved parameter estimation capability, subspace decompositionbased approaches are exploited to accommodate the detection of multiple RSSs simultaneously [14, 15]. However, in order to achieve the code detection and parameter estimation at the same time, the allowable parameter estimation range is strictly limited. In order to bind the detected ranging codes with the estimated timing or frequency offsets, the methods either sacrifice the estimation performance by inserting the code sequence in the estimated parameters [14] or perform exhaustive full search by correlating all sets of ranging codes and offsets at the BS [11, 15]. Bacci et al. introduced a gametheoretic approach to derive an energyefficient method for contentionbased synchronization problem [16]. The ranging process is formulated as a noncooperative game for each terminal to determine the transmit power and detection strategy in a selfish way to maximize the detection probability with the minimum energy consumption.
The topic of multidimensional harmonic retrieval (HR) problems are encountered in a variety of signal processing applications including radar, sonar, seismology, communications, MIMO wireless channel, and the different types of schemes are utilized for the application [17–21]. Among these, principalsingularvector utilization for modal analysis (PUMA) is used as the computationally attractive solution in HR [17]; however, it may not work properly when there are identical frequencies in one or more dimensions. On the other hand, the improved multidimensional folding (IMDF) scheme, which uses eigenvector instead of eigenvalue for estimating frequencies, can resolve identical frequency scenarios by introducing the randomness on the sample data [18, 19]. These IMDF algorithms introduced more relaxed identifiability bound, i.e., increased number of resolvable frequencies for the given sample data [19]. In order to improve the performance, the approaches with the use of higherorder singular value decomposition (HOSVD) and/or structured least squares technique are introduced in [20, 21]. Instead of using stacking matrix form, the HOSVD method captured the structure inherent in the received data at the expense of a high computational complexity [20, 21]. It was shown in [20] that the general computational complexity of the tensorbased approach (HOSVD) is higher than that of the matrix based one (IMDF) even with the computationally efficient subspace estimation methods.
In order to improve the ranging process performance as well as to incorporate the superiority of HR technique with reasonable complexity, the preliminary design utilizing IMDF approach was first introduced in [22]. However, since the single snapshot IMDF approach [18] is simply exploited in [22], the automatic pairing property and multiple resource element utilization were not fully exploited in [22], which results in the limited number of resolvable RSSs as well as the limited detection and estimation performance improvement without proper twodimensional code design.
Since the multiple snapshot approach is quite well aligned with the inherent OFDMA subchannel allocation concept, we propose an improved ranging technique using IMDF with finite snapshots [19] for the initial ranging process. It enables for all the detected ranging codes, the estimated timing offsets and frequency offsets of multiple users to be automatically paired, so extra ranging code and offset association process, which usually requires high computational complexity, can be avoided.
Even though our approach is based on the IMDF algorithm, but it is not a trivial application of the IMDF algorithm to the ranging process. On the top of the IMDF methodology, our novel contribution is twofold.
First, we properly formulate initial ranging signals into finite snapshots such that automatic pairing property can be fully exploited during the procedure, which brings the improved performance as well as the reduced pairing complexity. Moreover, the maximum number of resolvable RSSs that the BS can distinguish can be also increased by applying multidimensional folding (MDF) algorithm.
Next, with the virtue of the automatic paring property, twodimensional ranging code is newly proposed. The new code index pair design, ranging code extraction and detection procedure, decision boundary for the circular code index pair, and redefined detection events using the code index pair with multiset theory are introduced. According to the new code index pair concept, the enlarged timing and/or frequency offset acquisition range is also derived.
The rest of this paper is organized as follows: Section 2 explains an OFDMA uplink system model in a contentionbased synchronization (initial ranging) process. Section 3 explains the proposed harmonic retrieval algorithm with multidimensional folding for extracting the code, timing offset, and frequency offset of active RSSs. In addition, the proposed twodimensional code design and code detection/evaluation methodology is described. In Section 4, the statistical characteristics of the proposed algorithm, which utilizes IMDF method, is derived to show the identifiability for the number of resolvable RSSs and to confirm the acquisition range of the proposed algorithms on the timing offset and frequency offset. Section 5 evaluates the performance of the proposed method and investigates the comparison with the existing algorithms. Section 6 presents our conclusion.
Throughout this paper, upper (lower) boldface letters will be used for matrices (column vectors). A^{∗}, A^{ T }, A^{ H }, and A^{ † } denote the conjugate, transpose, Hermitian transpose, and pseudo inverse of A, respectively. We will use ⊗ for the Kronecker product, ⊙ for the KhatriRao (columnwise Kronecker) product [23], I_{ n } for a n×n identify matrix, 0_{m×n} for an m×n zero matrix, [ a]_{ n } for the nth element of a, and [ A]_{m,n} for the (m,n)th element of A, D(a) for a diagonal matrix with a as its diagonal.
2 System model
2.1 Ranging process overview
On the other hand, the absolute frame timing of BS is unknown to each RSS since the propagation delay is unknown to each RSS (refer Figure 1). Therefore, each RSS only estimates its relative downlink frame timing position at first. Next, each RSS transmits its own initial ranging sequence, which is locked on the estimated frame timing, to the BS with the network access request. Once BS receives the sequences from multiple RSSs, the BS should identify multiple RSSs and estimate every RSS’s timing offset and residual frequency offset independently. Finally, from the given estimate, the BS provides the timing and frequency feedback info to the identified RSSs with the network grant message. Consequently, it is crucial on the initial ranging process for BS to identify as many as RSSs and to estimate their timing and frequency offsets as accurate as possible.
2.2 Resource allocation for ranging process
Let us consider an OFDMA uplink system employing N subcarriers. Among the whole subcarriers, virtual subcarriers are placed at both edges of the spectrum to prevent the spectrum aliasing. Except the virtual subcarriers, the useful subcarriers are grouped by multiple subchannels, and these subchannels are assigned to multiple users (subscribers) for transmission. Typically, each subchannel is divided into Q subbands composed of a set of V adjacent subcarriers [14, 15, 24]. When one transmission block consists of M consecutive OFDMA symbols, let us define a bunch of consecutive V subcarriers (one subband) and M OFDMA symbols as a tile.
Now, among multiple subchannels, let us assume that each RSS only uses a set of R subchannels for ranging with M consecutive OFDMA symbols, and then the total number of subcarriers used for ranging is N_{ R }=RQV.
with ∀q∈ [ 1,Q],∀v∈ [ 1,V],∀m∈ [ 1,M]. In (1), K is the number of active RSSs participating the ranging process, and v and m denote the subcarrier and OFDMA symbol index within the tile, respectively. H_{ k }[ x],x∈ [ 0,N1] is the channel frequency response of the kth RSS on the xth subcarrier position.
Since there is no big frequency offset difference between stations after the coarse frequency offset synchronization, the intercarrierinterference (ICI) term caused by the residual frequency offset is ignored for the simplicity in (1) whereas the accumulated phase rotation term caused by frequency offset is still included.
In (1), the length of one OFDMA symbol is N_{ T }=N+N_{ G } with N_{ G } guard interval for ranging symbols, and i_{ q } is the starting subcarrier index for the qth tile, and θ_{ k } and ε_{ k } denote the timing and frequency offsets of the kth RSS, respectively. [W_{ q }]_{v,m} is a complex additive white Gaussian noise (AWGN) with zero mean and variance . C_{ k } is the code sequence matrix for the kth RSS.
Here, let us assume that θ_{ k }∈ [ 0,θ_{max}] and ε_{ k }∈ [ ε_{max},ε_{max}] where θ_{max} and ε_{max} are defined as the maximum allowable timing offset and and absolute frequency offset of the system, respectively.
2.3 Code design
Here, C_{ T } and C_{ F } are defined as the design parameters to determine the twodimensional code index pair grid. Note that C_{ T } and C_{ F } can be properly chosen according to RSS’s timing and frequency distributions. Let us define a_{ k }= [ t_{ k }f_{ k }]^{ T } as a twotuple for the code index pair of the kth user where t_{ k }∈ [ 0,C_{ T }) is the code index at the timing offset domain and f_{ k }∈ [ 0,C_{ F }) is the one at the frequency offset domain, respectively. The tuple a_{ k } is chosen from the whole code index pair set , i.e., .
3 MDF estimation using multiple tiles
In order to exploit automatic pairing property of HR, we formulate the received signal as a twodimensional (2D) mixture form and then apply MDF estimation method [18] for estimating timing and frequency offsets of RSSs as well as detecting RSSs codes simultaneously.
3.1 2D mixture model
where and are the transpose of Vandermonde matrices with the common ratios and for each column to represent the timing and frequency offsets of K RSSs, respectively.
where is the vector form of the noise matrix W_{ q }. Since the received signal is formed as undamped 2D exponentials for each tile, we can apply MDF algorithm [18] for each tile at the initial ranging process.
3.2 MDF algorithm using multiple tiles
3.2.1 Smoothing operation
As shown in (6) and (7), the observed symbols at each tile can be described as the special form of the multiple harmonic combinations. Based on this structure, we even apply smoothing operator in order to fully utilize this special characteristics of the observed symbols and improve the estimation performance.
where V_{1},V_{2} and M_{1},M_{2} are positive integers satisfying V_{1}+V_{2}=V+1 and M_{1}+M_{2}=M+1.
with V_{1}+V_{2}=V+1 and M_{1}+M_{2}=M+1. and , are the transpose of Vandermonde matrices with the common ratios and .
For the noisy case, the perturbation analysis can be applied to derive the theoretical MSEs of one realization of timing and frequency offset estimation [19], Equations (33) to (47). However, since timing offset θ_{ k } and frequency offset ε_{ k } are all unknown random variables in the ranging process, the theoretical performance analysis on the deterministic realization of η_{ k } and ξ_{ k } would be meaningless on the whole. Therefore, for the simplicity, the noiseless case is used on the derivation while the noisy case is revisited on the simulation experiment.
3.2.2 Multiple tile combination
where , . Whereas [22] uses each tile independently for the code detection and timing/frequency offset estimation, now all channel responses and signals from multiple tiles are stacked and utilized together to create just one augmented matrix . Therefore, it would be more robust for the noisy scenario due to multiple tile usage.
3.2.3 Eigenvectorbased estimation
where U_{ s } has K columns that together span the column space of . Since the same space is spanned by the columns of G_{1} from (19), there exists K×K nonsingular matrix L^{1} such that U_{ s }=G_{1}L^{1}.
U_{ s } can be partitioned into two submatrices U_{1} and U_{2} (refer (46) and (47) in Appendix). Then, L can be obtained from the eigenvalue decomposition of up to column scaling and permutation ambiguity, i.e., L_{ sp }=LΛΔ where L_{ sp } is a practically obtained eigenvector including column scaling and permutation ambiguity, Λ is a nonsingular diagonal column scaling matrix, and Δ is a permutation matrix [18, 19]. This part is maybe most computationally complex part of the whole calculation, and the major computational load complexity to acquire L_{ sp } is calculated as [19–21].
where and , i.e., . In P, the phase rotation components caused by timing offset and frequency offset appear in the same column of the P. In other words, for a fixed kth column, and appear simultaneously in the same column of P. Consequently, thanks to this combined structure, we can estimate K RSS’s timing offset and frequency offset at the same time by dividing suitably chosen elements of the P.
The detailed procedure on how to obtain the above P_{ sp } and L_{ sp } from U_{ s } is described in Appendix.
3.3 Effective timing offset and frequency offset estimation
Even though P_{ sp } contains the column scaling ambiguity Λ, it is immaterial for estimating the timing and frequency offset of each user. It is because the timing and frequency offset of each user is obtained by phase differential term. Therefore, the scaling effect on the matrix is diminished. In addition, each timing offset with frequency offset is even automatically paired without heavy computational search for the pairing.
Note that still we have the the permutation ambiguity issue on the estimation, i.e., unknown Δ. Since the user (column) identification along with timing and frequency estimation should be performed accordingly in the ranging process, therefore it is crucial to identify each column correctly and to remove the permutation ambiguity clearly. This can be done by the proposed code extraction and code detection process.
3.4 Code extraction
Recall the effective timing and frequency offsets are composed of the code index as well as actual timing and frequency offsets as shown in (4) and (5). Since and are automatically paired, the paired ranging code, timing offset, and frequency offset of the kth RSS can be detected and estimated simultaneously by using the effective timing and frequency offsets.
where and . Since the timing offset θ_{ k } only has the positive values, by introducing the bias ΔT on the extracted code index pair, the disturbance due to timing offset can be spread out around the original code index point t_{ k }. Consequently, by observing b_{ k }, we can retrieve the code index pair (t_{ k },f_{ k }) accordingly.
3.5 Code detection
Now, the new code detection methodology and the decision boundary/criterion for the circularly repeated code index pair are introduced using multiset theory. In addition, the code detection events are newly redefined for the multidimensional code, e.g., phantom code, collided code, missing code, falsealarmed code, and correctly detected code in this section.
where κ(k) is the detected code index for the kth received tuple (kth extracted column), is the set of all integers, and is the square of the twonorm. Once all of K code index pair is detected, and then, it should be investigated from the detected code index pairs whether or not it contains the phantom code index pair which means the index pair is duplicated more than one time.
where is the nonnegative integer set and μ(x) denotes the number of occurrences of x in the set. Now, let us take a look at different events on the code detection procedure.
3.5.1 Phantom code
with κ(k) defined in (31). Since a_{1} and a_{2} are detected more than once in (34), here, . The phantom code can be detected at the receiver, so this code index can be opted out during the detection procedure.
3.5.2 Collided code
From the example, is defined as . Typically, it is hard to detect the collided code at the receiver side, but with the virtue of automatic pairing property in HR, if each RSS has different timing offsets and/or frequency offsets, it has a chance to remove this collided code as phantom code at the receiver side (like the example). Therefore, HR can avoid the false detection of the collided code effectively.
3.5.3 Missing code
From the example, the missing code is shown as .
3.5.4 Falsealarmed code
Here, .
3.5.5 Correctly detected code
From the example, the correctly detected code is calculated as .
On the whole, the index set of detected codes is described as {3} while the index multiset of misdetected codes including both and is {1,1,2,4,5}. Finally, the index set of falsedetected codes is defined as {6}. Note {1} appears twice on the misdetected multiset since already carries the two collided code.
3.6 Actual timing and frequency estimation
and it would be compared with if it is successfully identified without falsealarm ( ) and the offset is accurate estimated.
and it would be also compared with if it is successfully identified without falsealarm ( ).
4 Statistical characteristics
4.1 Identifiability
Actually, (44) can be easily induced from (19) since the matrix size of G_{1} and would determine the overall rank.
By adjusting the smoothing factors V_{1},V_{2},M_{1}, and M_{2}, the code detection and timing and frequency offset estimation performance can be improved as well as K_{uid} can be increased.
4.2 Acquisition range
Basically, the transmitted code index pair is twisted by the timing and frequency offsets. So, large timing and/or frequency offsets can cause the misdetection and/or false detection. Therefore, for the reliable code detection, it is essential to limit the allowable timing and/or frequency offset ranges.
where θ_{max} and ε_{max} stand for the maximum allowable timing and frequency offsets, respectively. So, from the inequality, it is available to adjust the maximum ranges of timing offset and frequency offset accordingly by enlarging one of the allowable ranges while shrinking another ranges at the same time. Remark that even though the acquisition range is derived as a lower bound by taking only a portion of the decision region in (45), the maximum allowable ranges θ_{max} and ε_{max} are larger than the one introduced in MUSIC [15] and ESPRIT [14]. For example, with the same C_{ T }=C_{ F }=5, the maximum ranges of MUSIC and ESPRIT were defined as and while the ranges of the proposed HR can be picked up as and (whose range is enlarged by more than 10%) when we just treat the importance of both timing offset and frequency offset ranges equivalently with from the Figure 3.
5 Simulation results
The overall system model and simulation parameters are based on the IEEE 802.16m and 3GPP LTE environments in [2, 3]. The total bandwidth is 10 MHz, and N=1024. Let us assume that each ranging subchannel is composed of Q=3 tiles with V=6 consecutive subcarriers, while M=5 OFDMA blocks are presented in ranging subchannels. In order to corporate with wide bandwidth, extended vehicular A (EVA) channel model is applied to evaluate the performance [25].
The timing and frequency offset ranges are designed for θ_{ k }∈ [ 0,114) and ε_{ k }∈[0.02,0.02] which correspond to the system more than 1 km cell radius considering the roundtrip delay, e.g., dense small cell, and 135 km/h Doppler frequency at f_{ c }=2.4 GHz. Note that all the simulation results reflect the ICI caused by residual frequency offset even though it is neglected in (1) for the simplicity.
The code design parameters C_{ T } and C_{ F } are set as C_{ T }=C_{ F }=7 for the proposed HR algorithm.
Parameters  HR, MUSIC, ESPRIT  ZadoffChu sequence 

Sampling frequency  15.36 MHz  15.36 MHz 
Total number of subcarriers  1,024  2,048 
Number of OFDM symbols (M)  5  2 
Each ranging symbol length 


Total ranging symbol length  5.675N_{ T }=425.625 μs  5.5N_{ T }=412.5 μs 
Total number of ranging subchannels (R)  4  1 
Number of tiles/ranging subchannel (Q)  3  1 
Number of subcarriers/tile (V)  6  144 
Total number of ranging subcarriers (RQV)  72  144 
Timing offset range  [0, 114] samples  [0, 114] samples 
Frequency offset range  [ 300, 300] Hz  [ 300, 300] Hz 
Contentionbased initial ranging capability with the number of RSSs
Parameters  MUSIC  ESPRIT  HR  ZadoffChu sequence 

Number of uniquely identifiable RSSs per subchannel, K_{uid}  4  4  16  N/A 
Total number of uniquely identifiable RSSs, R·K_{uid}  16  16  64  64 
Total number of code sequences for RSSs,  16  16  28  28 
Number of active RSSs per subchannel,  3, 6  3, 6  3, 6  N/A 
Total number of active RSSs, R·K  12, 24  12, 24  12, 24  12, 24 
5.1 Smoothing factor decision and identifiability comparison
Let us recall that the smoothing operator enlarges the observed data matrix to the smoothed data matrix by choosing appropriate smoothing factors V_{1},V_{2} and M_{1},M_{2} with the condition V_{1}+V_{2}=V+1 and M_{1}+M_{2}=M+1. Therefore, the size of smoothed data matrix depends on the smoothing factors. In fact, this smoothing factor could affect the code detection and offset estimation performance as well as the maximum number of distinguishable RSSs.
In addition, Figure 5b shows the trend on the maximum number of identifiable RSSs according to the number of subcarriers for fixed number of OFDM symbols M=5. As seen in the figure, the maximum number of RSSs which MUSIC and/or ESPRIT can support is limited by M even though we allocate more subcarriers for the ranging whereas the number of RSSs for HR can keep increasing without limitation. This addresses that the proposed HR technique can be popularly exploited on the scenarios such that high volume of users and/or devices should be separated at the same time at the dense small cell.
5.2 Misdetection probability
Code detection and offset estimation notation
Main algorithm  Code detector (CD)  Timing offset estimator (TE)  Frequency offset estimator (FE) 

ZadoffChu (ZC)  ZCCD  ZCTE  ZCFE 
MUSIC  MCD  AHTE (adhoc TE)  MFE 
ESPRIT  ECD  ETE  EFE 
Harmonic retrieval (HR)  HRCD  HRTE  HRFE 
As seen in the figures, the proposed HRCD shows good P_{miss} performance. It is because automatic pairing property in HR improves the performance of code indices detection. Even though ESPRIT code detector (ECD) also exploits two independent estimates of the code indices to make a decision, the misdetection frequently occurs for ECD due to the discrepant code indices for two independent estimations. In addition, it is shown in Figure 6a,b that ZCCD has very poor performance regardless of SNR. It is because the orthogonality property of ZadoffChu sequence can sustain only with small number of RSSs. Different multipath fading channels as well as asynchronous timing/frequency offsets among multiple RSSs deteriorate the ZCCD detection performance dramatically.
Note that in Figure 6b, MUSIC and ESPRIT misdetection performance cannot be evaluated for R·K=24 since it exceeds the maximum number of RSSs which MUSIC and/or ESPRIT can support, which is the limiting factor of MUSIC and ESPRIT algorithms.
5.3 False detection probability
5.4 Timing and frequency offset estimation accuracy
In Figure 9, it is shown that HRFE outperforms MFE and EFE over the whole range, and HRFE becomes better than ZCFE with SNR increase. Even though HRFE shows worse performance on low SNR ranges compared to ZCFE, but please remark that this estimation error variance is only evaluated for the successfully corrected code sets, i.e., . Therefore, if we consider the poor falsealarm and missing probability performance of ZCCD shown in Figure 7 (more than 10% of falsealarm and missing), the overall ZCFE performance will be severely degraded on the whole.
On the whole, the performance of other algorithms undergo severe performance degradation because wrongly detected code index disturbs timing and frequency offset estimation. However, the proposed HRTE and HRFE algorithm more robust to the wrongly detected code effect.
6 Conclusions
First, the proposed method utilizing IMDF scheme supports the flexible subcarrier allocation usage. Basically, the proposed algorithm can be applied for the subcarrier allocation based on small tile and/or resource block structure whereas ZadoffChu sequence algorithm requires only the subband allocation, which should consist of consecutive subcarriers. Since the proposed algorithm also supports frequency noncontiguous combination of multiple tiles, actually it would even bring the performance improvement on the proposed scheme due to the enriched frequency diversity experience on the sample data. In addition, tile concept is compliant to the stateoftheart wireless cellular systems, e.g., IEEE 802.16m and 3GPP LTEAdvanced [2, 3], so this scheme can be adopted directly to the practical system without any painful remedy on the frame and resource allocation structure.
Next, the proposed method using IMDF scheme fully utilizes the resources to distinguish multiple RSSs as many as possible. In the practical scenario which only a few OFDMA symbols are utilized in the ranging process, the proposed HR algorithm can still increase K_{uid} by adjusting the number of tiles Q and the number of subcarriers V. Note that the number of RSSs in MUSIC and ESPRIT is easily restricted by the limited number of OFDMA symbols. Consequently, it confirms that the HR algorithm can smartly utilize the given data matrix to distinguish many RSSs simultaneously.
From the simulation results, it is shown that the proposed method outperforms the other algorithms in terms of the code detection and offset estimation while supporting more number of RSSs. Especially, the newly designed and introduced twodimensional code index pair with the aid of automatic pairing property enables for the algorithm to increase the minimum Euclidean distance between two code index pairs and improve the acquisition range and detection/estimation performance accordingly. Consequently, this HR method can be proper for the dense network ranging process such as small cell, Machinetype communications, devicetodevice communications.
Appendix
where and .
Declarations
Authors’ Affiliations
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