Eigenvector-based initial ranging process for OFDMA uplink systems

2.1 Ranging process overview

Figure 1 shows the brief signal flow on the initial ranging process. First of all, each RSS catches the downlink synchronization signal to detect its relative frame timing position and to estimate its absolute carrier frequency offset. Since every RSS can compare and compensate its frequency to the reference one from the BS, the coarse frequency synchronization can be completed before the initial ranging trial is started. Therefore, the residual frequency offset error due to the imperfect estimation is only remained for each RSS as shown in Figure 1.

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.

The whole transceiver block diagram of the ranging process is shown in Figure 2, and the detailed procedure on how to construct the ranging sequence with the code and to perform the code detection and timing/frequency estimation is described in the following.

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.

After the fast Fourier transform (FFT) at the receiver, the received data at the qth tile of the ranging subchannel is written as:

(1)

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,N-1] 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 inter-carrier-interference (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

Let us design the code sequence for the kth RSS user as:

(2)

Here, C _T and C _F are defined as the design parameters to determine the two-dimensional 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 two-tuple 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., .

Since the subband size in one tile is designed to be smaller than the channel coherence bandwidth [14, 15], H _k [ x]≈H _k [ x+1] within the tile. Therefore, by averaging V subcarriers, the received output can be expressed as an average channel response as follows:

(3)

where [h _q ] _k is the average of the phase-rotated channel response for the kth user at the qth tile written as:

In (3), the code sequence of the kth user, [ C _k ]_v,m, is absorbed into the effective timing and frequency offsets as η _k and ξ _k , and this effective timing and frequency offsets of the kth user are given by:

(4)

(5)

3.1 2-D mixture model

Let us recall the received symbols at the qth tile is . Given (3), X _q can be written as 2-D mixture matrix form as:

(6)

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.

For simplicity, let us define the (v,m)th element of the matrix X _q as x_q,v,m, i.e., x_q,v,m= [ X _q ]_v,m. Then, from the given Eq. 6, we can define the sample vector form , which is:

(7)

where is the vector form of the noise matrix W _q . Since the received signal is formed as undamped 2-D 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.

Let us define a set of two-dimensional selection matrices given by:

(8)

where V₁,V₂ and M₁,M₂ are positive integers satisfying V₁+V₂=V+1 and M₁+M₂=M+1.

Using (7) and (8), let us define smoothing operator and the smoothed matrix for the sample vector as:

(9)

where . Here, we show an explicit expression of using x_q,v,m in order to figure out the smoothing operator:

with :

In the absence of noise, it is already verified in [18], Lemma 2 that (9) can be decomposed into harmonic matrices as follows:

(10)

where:

(11)

(12)

with V₁+V₂=V+1 and M₁+M₂=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.

In (10), in order to further explore the data structure, we operate the backward smoothing as well as forward smoothing on the sample vector x _q in (7). Let us define a backward sample vector where Π _n is an n×n backward permutation matrix. By using the property of harmonics, it is easily verified that:

(13)

where and with β _k =(V-1)η _k +(M-1)ξ _k [19]. Since (13) has the same harmonic matrix form as (7), we can use both forward and backward sample vectors to estimate the phase. In the similar way like (9), can be also obtained by backward smoothing operation:

(14)

3.2.2 Multiple tile combination

Let us define the forward and backward channel matrices as:

(15)

(16)

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.

After calculating the smoothed matrices and for each tile, we collect all the smoothed matrices in order to perform MDF algorithm [19]:

(17)

(18)

(19)

where and as follows:

3.2.3 Eigenvector-based estimation

is of rank K almost surely. Therefore, the singular value decomposition (SVD) of yields:

(20)

where U _s has K columns that together span the column space of . Since the same space is spanned by the columns of G₁ from (19), there exists K×K nonsingular matrix L^-1 such that U _s =G₁L^-1.

U _s can be partitioned into two submatrices U₁ and U₂ (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].

Once we obtain L _sp , we can retrieve up to column scaling and permutation ambiguity as:

(21)

where P _sp is a practically calculated matrix which consists of Khatri-Rao product of two Vandermonde matrices with are K columns up to column scaling and permutation ambiguity. Here, the original stacking matrix P is defined as:

(22)

(23)

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.

The effective timing offset on the kth column can be estimated as:

(24)

and the effective timing offset from the kth column can be obtained by:

(25)

In the similar way, the effective frequency offset on the kth column can be estimated as:

(26)

and the effective frequency offset from the kth column can be obtained by:

(27)

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.

Followed by (4) and (5), let us define the extracted code index pair b _k as:

(28)

where . By plugging (4) and (5) in (28), b _k is derived as follows:

(29)

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, false-alarmed code, and correctly detected code in this section.

Figure 3 illustrates the example of code index pair set , decision boundary, and decision region for C _T =C _F =5. As discussed before, from the automatic pairing property of HR, it is possible to build up the two-dimensional code index pair.

From b _k , the code index pair can be extracted as follows:

(30)

where is the decision region for the code index pair . Since the code index is circularly repeated per design parameter C _T and C _F , the modified maximum likelihood approach which reflects the repeated lattice structure can be applied for extracting the code index pair as follows:

(31)

(32)

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 two-norm. 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.

In order to redefine the code detection events using code tuple, let us assume and as the transmitted and the detected code index pair multisets (or bag), respectively, i.e., and . In contrast to set concept, the members of the multiset are allowed to appear more than once. Here is an example to figure out the different events on the detection procedure:

(33)

(34)

This illustrates the case that six active RSSs are participated on ranging process, and two of the RSSs pick up the same code index pair a₁ and transmit ( ), and the BS detects the six code index pairs from the received tuple ( ). The multiset indicator function of a subset of a multiset is:

(35)

defined by:

(36)

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

Let us define phantom code that one code index pair is observed more than once at . The index set of the phantom code is defined as:

(37)

with κ(k) defined in (31). Since a₁ and a₂ 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

Collided code is defined such that one code index pair is utilized more than once at , which is the case when more than two RSSs choose the same code for the ranging. The index set of the collided code is defined as:

(38)

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

Let us redefine missing code such that the transmitted code index pair is not detected at the receiver side except collided code, which is denoted as:

(39)

From the example, the missing code is shown as .

3.5.4 False-alarmed code

False-alarmed code means that the wrong code index pair, which is never transmitted, is detected at the receiver side. There is no way to classify the false-alarmed code at the receiver, so it usually degrades the detection and estimation performance of the system. The false-alarmed code can be calculated as:

(40)

Here, .

3.5.5 Correctly detected code

After excluding the phantom code, all the remained codes are treated as detected codes including the false-alarmed code. However, in order for the accurate evaluation of the code detection performance, let us newly define the correctly detected code which only counts the code truly transmitted and accurately received as:

(41)

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 false-detected 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

Among K extracted code index pairs, the phantom code is firstly excluded, and only remaining code index pairs, , participate the actual timing and frequency estimation. So, the timing and frequency offsets are calculated only for the κ(k^∗)th RSSs . Based on (4) and (25), the estimated actual timing offset for the κ(k^∗)th RSS from the k^∗th received tuple is given by:

(42)

and it would be compared with if it is successfully identified without false-alarm ( ) and the offset is accurate estimated.

In the same way, the estimated actual frequency offset for the kth RSS is written by:

(43)

and it would be also compared with if it is successfully identified without false-alarm ( ).

4.1 Identifiability

When the estimator uses the timing offset dimension as a primary dimension, the maximum number of RSSs, which is almost surely uniquely identifiable, is derived as [19], Equation (31):

(44)

Actually, (44) can be easily induced from (19) since the matrix size of G₁ and would determine the overall rank.

By adjusting the smoothing factors V₁,V₂,M₁, and M₂, 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.

In principle, once resides on the correct decision region, i.e., , the timing and frequency offsets of the κ(k^∗)th RSS can be successfully recovered from the k^∗th received tuple, (refer Figure 3), and depending on the individual , and , the maximum allowable timing and frequency offset ranges, θ_max and ε_max, can be changed. Meanwhile, from (4) and (5), the guaranteed acquisition range not to make ambiguity at the code detection without any noise term can be derived as:

(45)

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.

5.1 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 V₁,V₂ and M₁,M₂ with the condition V₁+V₂=V+1 and M₁+M₂=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.

As shown in (44), K_uid is determined by (V₁,V₂) and (M₁,M₂) pair values, and Figure 5a shows the variation of this K_uid according to V₁ and M₁ for given Q=3, V=6, and M=5. First, we investigated most popularly obtained number in K_uid distribution, which is K_uid=12 as shown in Figure 5a. Then, from the given four cases with K_uid=12, i.e., {(V₁,M₁)|(4,4),(5,3),(6,4),(5,5)}, the smoothing factor combination which shows better performance on the whole is picked up, which is (V₁,M₁)=(4,4).

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

Figure 6 describes the misdetection probability for four code detection algorithms with respect to signal-to-noise ratio (SNR) with the number of active RSSs R·K=12 and 24. The notation for each code detector and offset estimator is expressed in the Table 3.

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 Zadoff-Chu 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

Figure 7 depicts the false detection probability of different algorithms with RSSs R·K=12 and 24. The performance trend is almost the same as the misdetection probability. Since the false detection probability of MCD and ZCCD is very poor, we expect that the overall estimation performance can be also degraded by the false detection accordingly.

5.4 Timing and frequency offset estimation accuracy

Figures 8 and 9 show the error variance performances of the timing and frequency offset estimation given the correctly detected codes, respectively, i.e., and . The performance is evaluated with different numbers of active RSS users in the system. Note that the performance evaluation on the timing and frequency estimation is only performed for the corrected detected code index pairs, i.e., . Unlike other methods, the proposed algorithm with HR can link the k^∗th received tuple to the code index κ(k^∗) directly without any brute-force code index search effort due to automatic pairing property.As seen in the Figure 8, HRTE outperforms other timing estimators over the whole SNR ranges regardless of the number of active users. Since it is crucial to obtain the timing misalignment information of each RSS on the initial ranging process, these curve confirm that the proposed HRTE can sustain the great performance even with high contention volume.

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 false-alarm and missing probability performance of ZCCD shown in Figure 7 (more than 10% of false-alarm 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.