STBC Identification for Multi-User Uplink SC-FDMA Asynchronous Transmissions Exploiting Iterative Soft Information Feedback of Error Correcting Codes

With the advancement and widespread implementation of multiple-input multiple-output (MIMO) wireless communication systems over the last decade, space-time block coding (STBC) identification has become a critical task for intelligent radios. Previous examinations of STBC identification were focused on single-user transmissions over single-carrier and multi-carrier systems in combination with uncoded broadcasts. Practical systems, on the other hand, contain many users and employ error-correcting codes. For the first time in literature, this work explores the problem of STBC identification for multi-user uplink transmissions in single-carrier frequency division multiple access (SC-FDMA) systems. We take another step closer to real systems by addressing asynchronous transmissions and by conducting multi-user channel estimation. We also exploit the outputs of the channel decoder, which is usually used in many practical systems, to improve the identification and estimation processes. The mathematical analysis demonstrates that the maximum-likelihood (ML) solution of STBC identification, channel estimation, and synchronization can be executed by an iterative approach. The space-alternating generalized expectation-maximization (SAGE) algorithm is used to separate the overlaid signals arriving at the base-station (BS). The parameters under consideration for each user are then updated using an expectation-maximization (EM) processor. Simulation results show that the proposed architecture outperforms other identification methods published in the literature while maintaining a reasonable level of computational complexity.


I. INTRODUCTION
A NALYSIS of wireless signals, aimed at determining the specific transmission parameters of the transmitter used, has been an prominent research area for decades. This analysis is generally referred to as signal identification with military and civilian implications. This has long been used in military applications such as signal interception, radio surveillance, interference detection and mitigation, jamming detection, and electronic warfare [1], [2]. The advent of intelligent radios, reconfigurable transceivers having the ability to alter their transmission settings such as modulation format [3]- [6] and channel coding rate [7]- [11], has heightened interest in signal identification systems in the context of recent civilian applications such as cellular mobile systems and WiFi networks [12], [13].
Signal identification for multiple-input multiple output (MIMO) systems poses unique technical issues that must be taken into account during identifier development. Not only is it difficult to estimate the number of transmit antennas and the type of transmit-side antenna arrangement in such systems, but there is also the issue of estimation and/or tracking of the signal characteristics and the wireless channel. Space-time block coding (STBC) is a MIMO approach in which many copies of a data stream are broadcasted in different time slots via multiple transmit antennas, achieving diversity with a simple receiver structure.
In single-carrier transmissions contexts, a group of methods relying on the fourth-order moment is suggested to recognize between two STBC signals, Alamouti (AL) and spatial multiplexing (SM) over Nakagami frequency-flat channels in [14]. The authors of [15] investigate the usage of second-order cyclostationarity of two different received signals to classify among several STBC signals. The dispersive characteristics of multipath fading channels is utilized in [16] to discriminate between AL and SM STBC signals. A maximumlikelihood (ML) technique [17] and a Frobenius norm [18] are designed to distinguish between STBC signals. Recently, a convolutional neural network is used to create an STBC classification method [19]. The most significant drawback of [19] is the requirement for a large amount of data in order to accomplish training. In fact, it is not always possible to obtain training data from a source. For example, the categorization of military signals is an excellent illustration of this. Additionally, identifiers are typically implemented on small portable devices with less processing capability. Therefore, if and when it becomes required, retraining will be extremely difficult. This demonstrates the urgent necessity for proposing non-machine learningbased identification methods.
The existing standards' high data rate requirements demand transmissions over wide-band frequencyselective channels. The combination of MIMO systems and multi-carrier (MC) transmissions offers a fascinating solution to the problem of inter-symbol interference, which is a key concern in these conditions. Several wireless communication standards, including WiMAX, LTE, 5G cellular communications, IEEE 802.11n, IEEE 802.11ac, and IEEE 802.11ax, have used MC-MIMO transmissions [20]- [22]. In the framework of MC transmissions, the authors of [23]- [26] use the time-domain correlation functions of two different received signals to classify between AL and SM signals.
Although the earlier studies focused on single-user transmissions, in most real communications systems, the STBC identification process should be carried out in the presence of many users' signals. The main challenge in these multi-user scenarios is that different signals experience different unknown STBC signals, propagation delays, and channel coefficients. Also, the STBC identification process is hampered by multiple access interference (MAI). To the best of the authors' knowledge, this work is the first of its kind to investigate the problem of STBC classification in an uplink multi-user scenario of single-carrier frequency division multiple access (SC-FDMA) systems. We provide a novel strategy in which the proposed identifier benefits from the soft information outputs of channel decoders, which are used in a variety of real SC-FDMA systems. The proposed identification algorithm operates also in uplink orthogonal frequency division multiple access (OFDMA) scenarios. Simply one disconnects the FFT and IFFT units from the transmitter and receiving sides, respectively.
The mathematical study in this work reveals that the actual ML solution to STBC identification of multi-user SC-FDMA uplink scenarios is too sophisticated for real applications. Therefore, we resort to a new technique that acts iteratively. The overlaid signals arriving at the base-station (BS) are detached using the space-alternating generalized expectationmaximization (SAGE) algorithm at each iteration [27], [28]. The SAGE algorithm's expectation step uses the soft information of the channel decoders to reduce MAI created by other asynchronous users. This replaces the sophisticated multi-dimensional search with a series of one-dimensional searches. The resulting design is evocative of parallel STBC identification for multiple users, in which MAI is re-constructed and eliminated from the received signal to optimize each user's identification process. Channel estimation and timing synchronization algorithms are also designed to complement the proposed identification technique. Notably, the feedback provided by an uncoded data detector has been used to solve several difficulties that have arisen in multi-carrier systems, such as equalization [29], [30] and frequency synchronization [27], [28]. However, this is the first time it has been used in conjunction with coded transmissions in STBC classification for such systems.
The remainder of the work is broken down into the following sections. The problem formulation and system model are discussed in Section II. The proposed identification algorithm is described in Section III. Practical considerations and interpretations are reported in Section IV. Simulation results are discussed in Section V. Concluding remarks are presented in Section VI.

II. SIGNAL MODEL AND PROBLEM FORMULATION
We consider a wireless uplink multi-user SC-FDMA system with K active users as shown in figure 1. The total number of subcarriers M is splitted into U ≥ K subgroups. Each subgroup has M s = M/U subcarriers which are uniquely allocated to an active user, not to other users in the same time slot. Mathematically speaking, the set of subcarriers reserved to user k, S (k) , satisfies K−1 k=0 S (k) = {0, 1, · · · , M − 1} and S (k) S (k ′ ) = ϕ for k ̸ = k ′ . Here ϕ refers to the null set. Each user k has P (k) transmit antenna elements.

STBC Identification
Data detection Synchronization Channel estimation A sequence of binary information digits of user k passes through a channel encoder of rate c (k) which adds redundancy bits to correct errors produced in the transmission. The coded bits are interleaved and then mapped to complex data symbols which are selected from a given signal constellation Φ (k) of unit energy. Here, we do not impose any constraints on modulation, coding, and interleaving parameters of each user. A few pilots are encapsulated into data symbols to initialize the identification process as shown later on. The resulting sequence is split into Z (k) vectors, each has M s symbols. Let a where d ing to interleaved mapping defined as and localized mapping characterized as into an SC-FDMA symbol is performed by using an M−point inverse FFT (IFFT) with including a cyclic prefix of length ν. We write the m 1 th sample of zth SC-FDMA symbol x Each user k selects a STBC scheme, denoted as ϖ (k) , from a pool of candidates. The transmitted signal from antenna p is created by concatenating all time-domain vectors broadcasted in different time slots,c (k,p) as a subscript to emphasize that the structure of vectorc (k,p) ϖ (k) depends on STBC ϖ (k) . Finally, each transmit antenna of user k communicates with the BS through unknown L-path wireless channel, h (k,p) = h (k,p) (0), · · · , h (k,p) (L − 1) .

B. Receiver
Because users are placed at different positions from the BS, their received signals are subjected to distinct propagation delays. The propagation delay of each user is divided into an integer part and a fractional part with respect to the sampling interval. The fractional part can be incooperated into the channel impulse response (CIR) of each user as reported in [28], therefore, it does not be included in the following analysis. Denoting µ (k) as the integer part of the propagation delay of user k, the received signal at the BS is expressed as where c (k,p) being all zero sequence of length µ (k) , ⊙ refers to the convolution operation, and w ′ is the corresponding additive white Gaussian noise (AWGN). The aim is to utilize the received signal r to identify the type of STBC ϖ (k) under unavailability of h (k,p) and propagation delay µ (k) for k = 0, · · · , K − 1, and p = 0, · · · , P (k) − 1. This is a prerequisite for performing multi-user data detection.

III. PROPOSED ALGORITHM
For the sake of mathematical convenience, the expression of (5) is written in a matrix form as where r = r ′T and w = w ′T . Here (·) T refers to vector transpose operator andC (k,p) µ (k) is given as where µ max is the maximum possible integer propagation delay 1 Bearing in mind (9), one writes the ML estimates of the unknown parameters as shown in (6), (7), and (8).
Here⋄ is the trial value of variable ⋄ and Pr (• |⋄ ) is the probability density function of • given ⋄. A closer look at (7) reveals that the ML algorithm performs 1 In practice, µ max is expressed as a function of the cell radius as µ max ≈ cell radius speed of light . Since µ max does not depends on users locations, user superscript k is dropped from µ max without the loss of generality. π (0) , · · · ,π (K−1) ,μ (0) , · · · ,μ (K−1) ,ĥ (0,0) , · · · ,ĥ (K−1,P (K−1) ) = arg max where, and Pr r C (0,0) ____________________________________________________________________________________________________ averaging over the transmission matrices of all users. This is because the original data symbols are unknown at the BS. However, the real implementation of (7) is not possible because it demands huge computations, which are highly undesirable in practical systems.
The expectation-maximization (EM) procedure is useful in this context as it provides a iterative technique to estimate the ML solution in the presence of nuisance parameters. The procedure updates all unknown variables simultaneously, resulting in a time-consuming and complicated search procedure due to the large number of dimensions involved. In contrast, the SAGE methodology separates the unknown variables into numerous non-overlapping groups and then utilizes the EM algorithm to modify each group one at a time. Thus, the SAGE technique can be thought of as an upgraded version of the EM algorithm, which improves the convergence rate significantly with preserving the advantages of numerical simplicity and stability. The SAGE approach has been widely utilized to tackle parameter estimate problems in multicarrier systems, such as synchronization and channel estimation [27], [28]. It is the first time that the SAGE algorithm has been utilized to identify STBC for uplink SC-FDMA systems using channel coding outputs, which is a departure from its traditional context of parameter estimation with uncoded transmissions.
Each iteration of SAGE comprises of cycles rather than estimating all parameters at once. By maximizing the conditional expectation of the log-likelihood of the augmented data corresponding to a cycle, the parameter subset associated with this cycle is updated. As a result, the complex multidimensional search problem that is required to maximize the likelihood function is reduced to several one-dimensional effortless search problems.
The mathematical details of the proposed SAGE procedure for computing the parameters under consideration are provided as follows. We divide the unknown parameters into K non-overlapping subgroups A single user's parameters are updated at a time. This means that an iteration is made up of K cycles, each of which updates the user's settings. Given initial estimates, the (ι + 1)th iteration consists of the following steps.
• During the k ′ th cycle, we update the parameters of user k ′ while the other users' parameters remain unaltered. • Subtracting all other users' MAI from the total received signal produces where y k ′ is the received signal component of user k ′ , Ω (k,p) π (k) (ι) ,μ (k) (ι) is the a posteriori expectation of the transmission matrixC (k,p) given in (10), andP (k) (ι),π (k) (ι),μ (k) (ι) and h (k,p) (ι) are the estimates of P (k) , ϖ (k) , µ (k) , and h (k,p) , respectively, at iteration ι. Note that the transmission matrix ofC (k,p) ϖ (k) (ι) μ (k) (ι) is inaccessible since the information symbols are unknown at the BS. As a result, Ω (k,p) π (k) (ι) ,μ (k) (ι) is used in (14) instead ofC This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
________________________________________________________________________________________________ cally, Ω (k,p) π (k) (ι) ,μ (k) (ι) is expressed as where E [·] is the statistical expectation over the transmitted data symbols of user k. One rewrites (15) as Bearing in mind that then, after eliminating the useless elements, we rewrite (17) as ) • In the EM processor's expectation step, given the existing estimates, the expected value of L with regard to transmitted data symbols is calculated as shown in (12), where ℜ (·) denotes the real value of a complex argument. • In the EM processor's maximization step, the parameters of user k ′ is updated as indicated in (13).
To simplify the multidimensional optimization problem shown in (13), we decompose the joint problem into simple dimensional search problems as follows. For each discrete pair of µ (k ′ ) , ϖ (k ′ ) , the updated value of h (k ′ ,p) is computed by setting the derivative of the objective function in (12) to zero aŝ Using (20) into (12), the updated values of µ (k ′ ) and ϖ (k ′ ) are computed as

IV. PRACTICAL CONSIDERATIONS AND INTERPRETATIONS
The following practical concerns with the suggested iterative identification, estimation, and data detection structure are worth mentioning.

A. EXPECTATION OF TRANSMISSION MATRICES
As observed from (15), (20), and (21), the proposed design relies on determining the expectation of each user's transmission matrix Ω π (k) (ι) ,μ (k) (ι) . How to compute this matrix in practice is a question that emerges. According to (15), Ω (k,p) π (k) (ι) ,μ (k) (ι) can be calculated by substituting each matrix element with its a posteriori expectation. Bearing (4) in mind, (23) As noted from (23), the key issue is to compute the a posteriori probability of Pr d (k) z (m 2 ) y k ,π (k) (ι),μ (k) (ι) . Fortunately, the decoders of modern error-correcting codes include convolutional, turbo, and low-density parity check codes computes this probability during their iterative nature [32], [33]. As a result, we exploit this probability to support the proposed identification and estimation algorithm without causing extra overhead on the decoding process. The conceptual block diagram of the proposed design is shown in figure 2.

B. CHANNEL DECODER UPDATE
We must re-compute the a posteriori probability Pr d (k) z (m 2 ) y k ,π (k) (ι) ,μ (k) (ι) every time we update ϖ (k) (ι) , µ (k) (ι), and h (k,p) for every user k. This necessitates the resetting of the channel decoder, which results in numerous iterations. To reduce this overhead, we employ the embedded estimation technique [34], in which the channel decoder is not reset when the parameters ϖ (k) (ι) , µ (k) (ι), and h (k,p) are updated, but the extrinsic and a priori probabilities from the previous iteration of the channel decoder are kept unchanged. The overhead associated with the suggested iterative VOLUME 4, 2016 procedure becomes tolerable in this instance.

C. COMPUTATIONAL COMPLEXITY
We analyze the computational complexity of the proposed SAGE-based identification algorithm in terms of the number of floating operations (flops), with 6 and 2 flops required for multiplication and addition of two complex numbers, respectively. We also assume that the M-point FFT algorithm requires 5M log 2 (M) flops, that multiplication of two complex matrices with dimensions of ϱ 1 × ϱ 2 and ϱ 2 × ϱ 3 requires 8ϱ 1 ϱ 2 ϱ 3 flops, that addition/subtraction of two complex matrices with dimensions of ϱ 1 × ϱ 2 each requires 2ϱ 1 ϱ 2 flops, and that the inverse of a complex matrix with dimension of ϱ 1 × ϱ 1 requires ϱ 3 1 flops [35], [36]. As shown in Table 1, the precise computations of the required computational complexity ψ per iteration per user is where λ = (M + ν) g + L − 1 with g being the maximum number of input STBC blocks and Z is the number of STBC candidates. As a numerical example, we consider system specifications of M = 1024, ν = 7, g = 10, L = 6, µ max = 30, Z = 5. This provides 8.3(10) 6 flops, which yield a run-time of 66.4 µsec with a central processing unit of 1 Teraflops per second [3]. This runtime is clearly appropriate in terms of actual execution.

D. INITIAL ESTIMATES
Users send a few pilot symbols to the BS in order to initialized the proposed SAGE-based algorithm. As the number of pilot symbols grows, the first estimateŝ ϖ (k) (0) ,μ (k) (0), andĥ (k,p) (0) improve. Increasing the number of pilot symbols, on the other hand, reduces the amount of energy available for data symbols and increases the needed bandwidth. As a result, the number of pilot symbols to data symbols must be remain as low as feasible. In the sense that it produces good identification performance with minimal throughput loss, the suggested algorithm takes advantage of the data symbols' soft information supplied by the channel decoders. Without the use of supplemental pilot symbols, this iteratively improves the first estimates. The starting values ofπ (k) (0) ,μ (k) (0), andĥ (k,p) (0) are extracted from (15) by setting the entries in Ω µ (k) , ϖ (k) to simply the contribution of pilot symbols.

V. SIMULATION RESULTS
The proposed STBC identification technique was investigated using Monte Carlo simulations. If not stated differently, we considered a SC-FDMA system with the following parameters.
• The number of active users was K = 8.
• The number of total subcarriers was M = 1024.
• The number of allocated subcarriers per user was M s = 128. • The number of cyclic prefix samples was ν = 16. • The interleaved sub-carrier assignment was used. • The allocated signal constellation Φ (k) for each user was randomly selected from a pool of eight higher order QAM constellations, 4-QAM, 8-QAM, 16-QAM,..., 512-QAM. Similar results can be accomplished with ease for PSK signals. • A convolutional code of rate 1/2, constraint length 5, and generator polynomials (23) 8 and (35) 8 was employed for each user. • Pilots symbols of length P s = 40 were inserted to initialized the identification process. • Each wireless channel, h (k,p) between antenna p of user k and the BS was generated using 15 paths where each one has an exponential power delay profile as [24], [37]: where Ξ ch was selected in such a way that the average energy was equal to one. • The maximum propagation delay normalized to the sampling duration was µ max = 50, and each user's propagation delay was chosen at random. • Each user was assigned an STBC at random from the list of {ST1, ST2, ST3, ST4, ST5} where those candidates' transmission matrices are shown in (26a-26e) [15], [22]. It is worth mentioning that the proposed identifier can be employed with any number of STBCs. Those five codes are offered solely for the purpose of simulating various scenarios.

Equation Required flops
Channel update (20) 16λ (L − 1) 2 + 8 (L − 1) 3 + 8λ (L − 1) Code and synchronization update (21) 8µ max Z (L − 1)(M + 2) (26e) • The probability of incorrect identification P f was utilized as a figure of merit for the suggested identifier, probability mass function P m was employed to evaluate the proposed synchronizer, and the mean square estimation error (MSE) was used to assess channel estimation performance. Figure 3 compares the STBC identification performance of the proposed algorithm to that of [38] for a singleuser transmission over a wide range of signal-to-noise ratios (SNR). As far as the authors' knowledge, the mentioned reference is the sole study in the literature dedicated to STBC identification for SC-FDMA systems, and it is restricted to the classification of AL and SM STBC signals over a single-user transmission. To be fair, the proposed algorithm's identification performance is also limited to these signals. As can be observed, the proposed algorithm's identification improves with iterations. This is in line with the theo-  retical analysis presented in Section III. Furthermore, the suggested approach significantly outperforms [38]. This is due to the fact that we employ the channel decoder outputs to refine the identification process, whereas [38] does not. Figure 4 describes the proposed algorithm's STBC identification performance as a function of the number of users at different values of SNR, with the number of iterations being seven. Hereafter, each user selects a STBC signal among the five candidates shown in (26a-26e). It is worth noting that the values of P f at K = 1 of this figure are slightly greater than that in the preceding one. This is due to the fact that we are classifying among five STBC signals in this figure. However, in the prior one, we were limited to only two STBC signals. It has been observed from figure 4 that there is a performance loss in P f when compared to a single-user transmission. This is the result of MAI associated with the circumstance of multiple users. Despite the fact that we developed a method to eliminate this interference as indicated in (14). This deterioration, which is caused by residual interference, has a slight detrimental influence at high values of SNR and a large number of users. However, it almost vanishes otherwise. This is due to the fact that in the former case, the residual interference dominates with a lesser influence of AWGN.
In order to access iterative algorithm's convergence rate, figure 5 evaluates the identification performance  as a function of the number of iterations for eight users at different values of SNR. In general, the number of iterations required to reach convergence depends on many factors such as the operating SNR, number of active users, number of pilots, and the type of the channel coding used. More iterations are needed when the initial values are far away from the actual values. Figure 5 concludes that the proposed iterative structure is essentially converged at 7 iterations at most. Figure 6 depicts the identification performance of the proposed design in four scenarios at iteration 7. The first scenario includes STBC identification as well as propagation delays and channel impulse responses estimation. The second one involves STBC identification and channel estimation with perfect estimation of propagation delays. Third scenario performs STBC identification and propagation delays with perfect channel estimation. The last one has STBC identification with perfect estimation of propagation delays and channel impulse responses. The results show that there are no significant variations in the identification performance of the four scenarios. This validates the proposed STBC identification algorithm with channel estimates for asynchronous uplink SC-FDMA transmissions.
The mean square error of the proposed channel estimator is shown in Figure 7 as a function of SNR and the number of iterations. The MSE performance is low in the initialization step. However, by utilizing soft information of channel decodes as mentioned in (20), the suggested estimator's performance improves until it converges at the seventh iteration. Figure 8 illustrates the probability mass function of the propagation delay error, τ ∆ = 1 of the proposed synchronizer at SNR = 14 dB. As previously explained, the performance improves with iterations. This is consistent with the theoretical conclusions in Section III.

VI. CONCLUSIONS
This work investigated the problem of space-time block coding (STBC) identification for multi-user uplink asynchronous transmissions in single-carrier frequency division multiple access (SC-FDMA) systems. The mathematical analysis revealed that a space-alternating expectation-maximization (SAGE) approach can be used to implement the maximum-likelihood (ML) solution of STBC identification, channel estimation, and synchronization. The channel decoder's a posteriori This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  probabilities were exploited to improve the quality of identification and estimation processes in an iterative manner. Simulation results indicated that the proposed design outperforms the existing identification algorithms reported in the literature, with a reasonable processing time. Despite the fact that the presented strategy has recognized to be an effective technique for STBC identification, it is constrained by the requirements of the modulation type and error correcting codes being used. The process of jointly identifying all of these factors will be undertaken in the future.