A novel random access scheme for M2M communication in crowded asynchronous massive MIMO systems

A new random access scheme is proposed to solve the intra-cell pilot collision for M2M communication in crowded asynchronous massive multiple-input multiple-output (MIMO) systems. The proposed scheme utilizes the proposed estimation of signal parameters via rotational invariance technique enhanced (ESPRIT-E) method to estimate the effective timing offsets, and then active UEs obtain their timing errors from the effective timing offsets for uplink message transmission. We analyze the mean squared error of the estimated effective timing offsets of UEs, and the uplink throughput. Simulation results show that, compared to the exiting random access scheme for the crowded asynchronous massive MIMO systems, the proposed scheme can improve the uplink throughput and estimate the effective timing offsets accurately at the same time.

the long term evolution (LTE) network may induce excessive signaling overhead and cannot support massive access [7].
Researchers are exploring new random access schemes for M2M communication in massive MIMO systems, and the grant-based random access schemes have been proposed in recent years. E. Björnson et al. proposed a strongest-user collision resolution (SUCRe) scheme, which allocates the pilot to the UE with largest channel gain among the contenders [8]. However, the number of successful accessing UEs decreases with the increase of the number of contenders [9]. To improve the pilot resource utilization of the SUCRe scheme, SUCR combined idle pilots access (SUCR-IPA) scheme was proposed in [10], where the weaker UEs randomly select idle pilots to increase the number of successful accessing UEs. A user identity-aided pilot access scheme was proposed for massive MIMO with interleave-division multiple-access systems, where the interleaver of each UE is available at the BS according to the one-to-one correspondence between UE's identity number (ID) and its interleaver [11]. However, since the grant-based random access schemes require two handshake processes between the base station (BS) and UEs, considering the small packet transmission in M2M communication, such kind of random access scheme will introduce heavy signaling overhead and low data transmission efficiency. To address this problem, the grant-free random access schemes have attracted much attention in recent years, which allow active UEs to transmit their pilots and uplink messages to the BS directly and performs activity detection, channel state information (CSI) estimation, and uplink message decoding in one shot. J. Ahn et al. proposed a Bayesian based random access scheme to detect the UE's activity and estimate the CSI jointly by utilizing the expectation propagation algorithm, considering the BS with one antenna [12]. L. Liu et al. proposed a approximate message passing (AMP) based grant-free scheme to achieve the joint activity detection and CSI estimation for massive MIMO systems [13]. However, this AMP-based grant-free random access scheme requires long pilot sequence to achieve better performance, resulting in heavy pilot over-head. These grant-free random access schemes considers the single pilot structure, and Jiang et al. proposed to concatenate several multiple orthogonal sub-pilots into one pilot sequence, where different UEs are allocated different pilot sequences and the pilot sequence is utilized for activity detection and CSI estimation [14]. The performance comparison between these two kinds of pilot structures are made in [15].
The above-mentioned two kinds of random access schemes are based on the assumption that the BS has performed accurate time-frequency synchronization. However, in practice, there are frequency errors caused by the Doppler shifts and/or frequency estimation errors during the initial downlink synchronization, and timing errors caused by the locations of UEs in the cell, which impair the pilot orthogonality and further degrade the access performance [16]- [18]. Considering the time-frequency asynchronous massive MIMO systems, L. Sanguinetti et al. proposed a random access scheme based on orthogonal frequency division multiplexing (OFDM) to solve the pilot collision by exploiting timing offsets and the large number of antennas [19]. However, the number of successful detected UEs is less than or equal to the number of subcarriers of the pilot, which cannot meet the massive access requirements of the M2M communication.
To further resolve the pilot collision for the M2M communication in crowded asynchronous massive MIMO systems, we propose a novel random access scheme, where the BS employs a proposed estimation of signal parameters via rotational invariance technique enhanced (ESPRIT-E) method to estimate the effective timing offsets of UEs, and UEs judge whether it is detected in a distributed manner. Then, the detected UE obtains its timing error from the effective estimated timing error, and further compensates the timing error for uplink message transmission. Furthermore, we analyze the mean squared error (MSE) of the estimated effective timing offsets of UEs and the uplink throughput. Numerical results show that, the proposed random access scheme significantly improves the uplink throughput, and provide accurate value of the effective timing offset.
The remainder of this paper is organized as follows. System model is given in Section II. Section III describes the proposed random access process. We present the performance analysis in Section IV. Simulation results and the conclusion are given in Section V and VI, respectively.
Notation: In this paper, we use the superscript 'T', '*', and 'H' to denote the transpose, complex conjugate, and conjugate transpose of a vector or a matrix, respectively. We use CN (a, b) to denote a circularly-symmetric complex Gaussian distribution with mean a and variance b. Let || · || indicate the Euclidean norm. We use [x] n and X(i) to denote the n th element of vector x and the i th column of matrix X. Let 'round' denote the rounding operation. We utilize arg(d) to denote the phase of the complex d.

II. SYSTEM MODEL
We consider the time-division duplexing (TDD) massive MIMO communication system based on OFDM. There are a BS with M antennas located at the center of the cell and K single-antenna UEs uniformly distributed in the cell. We assume that each UE becomes active with probability p a . The number of UEs residing in the cell is K, and the number of active UEs is N a .
The pilot with symbol length τ consists of Q consecutive OFDM symbols in the time domain and N adjacent subcarriers in the frequency domain (i.e., τ = QN ). We use to represent the frequency domain code set and time domain code set, respectively. The time domain code set C Q can be any orthogonal sequence set, and the frequency domain code f i is the Fourier basis, which is given by [19] [ The received pilot signal of UE k at the BS will introduce frequency error w k and timing error θ k . Since the value of w k is very small in general and its impact can reasonably be neglected if the pilot contains only a few consecutive OFDM symbols [19], [20], we only consider the timing error is the sampling period where N FFT is the number of subcarriers with frequency spacing ∆f . Note that, timing error θ k appears as phase shifts at the output of the receive discrete Fourier transform (DFT) unit [19].   1 shows the four steps of the proposed random access scheme, and the details are described as follows.

A. Step 1: Pilot transmission
Each UE randomly selects a frequency domain code from set C N and a time domain code from set C Q . We use l k ∈ (0, . . . , N − 1) and i k ∈ (0, . . . , Q − 1) to represent the indexes of the frequency domain code and time domain code selected by UE k, respectively. Thus, the pilot transmitted by UE k is f l k t T i k . Then, the received pilot signal at the m th antenna over subcarrier n is given by is the additive noise and any future instances of the matrix or vector W with different sub-or superscripts will take the same distribution, and h m k denotes the channel response between UE k and the m th antenna of the BS which is given by where g m k ∼ CN (0, 1) is the small-scale fading coefficient between UE k and the m th antenna of the BS, β k is the channel gain of UE k which is known to UE k. The channel between UE k and the BS is denoted by h k = (h 1 k , . . . , h M k ) T . Then, the received pilot signal at the m th antenna of the BS can be written as where c( k ) = [1, · · · , e j2π(N −1) k ] stands for the effective frequency domain code of UE k, and j is the unit imaginary number. Furthermore, throughout this paper, we consider that ρ k β k = 1, which can be achieved by the power control mechanism [21]. This ensures that the received signals from UEs have the same power, and thus obtains a fair estimation performance.

B. Step 2: Precoded random access broadcasting
Based on the received pilot signal at the m th antenna of the BS Y m , the BS sends the precoded random access response to active UEs. The procedure is described as follows.

1) The number of active UEs estimation
where A i is the set of UEs selecting t i , and W m = Then, by correlating the first column in Z i with the matrix Z i , we have Eq. (6) is obtained based on the propagation of the massive MIMO channel [22], i.e., Based on Eq. (6), we observe that, when M goes into infinity, the n th element in z i c is indeed the sum of the effective frequency domain code over the n th subcarrier of active UEs selecting the time domain code t i . Actually, the first element in z i c is the number of active UEs selecting the time domain code t i . Therefore, we can utilize the first element in z i c (i.e., [z i c ] 1 ) to estimate the number of UEs selecting the time domain code t i , denoted by N i a .
2) The effective timing offsets estimation By utilizing N i a , we utilize the proposed ESPRIT-E method to estimate the effective timing offsets of UEs, which is described as follows. The sample covariance matrix R i z associated to z i m is computed by By utilizing the eigenvectors of R i z associated to where I s i is the index of UE among the N a active UEs, {ψ i 1 , ψ i 2 , · · · , ψ d i 2 } are eigenvalues of matrix The received signal R i,j k at UE k is written as UE k first correlates the received signal R i,s k with its selected time domain code to obtain (13) Then, UE k uses the following rule to judge whether it is detected in a distributed manner, which is given by If UE k is a detected UE, it uses the effective timing error I j i (i.e., I s i = k) corresponding to h I s i that makes round(r i,s k ) = 1 to obtain its timing error as follows Then, UE k employs θ k to compensate its timing error for uplink message transmission. Otherwise, UE k randomly reselects a frequency domain code from set C N and a time domain code from set C Q to obtain its pilot, and send it to the BS, as we described in step 1.

D. Step 4: Precoded random access response broadcasting again
Similar to step 2, based on the received pilot signal, the BS generates and broadcasts the precoded random access responses again, and each UE reselecting its pilot during step 3 employs the rule in Eq. (14) to judge whether it is detected. If UE p is a detected UE, it compensates its timing error for uplink message transmission. In the following, all detected UEs send their uplink messages to the BS, and thus the BS can utilize any blind detection method to obtain their uplink messages, such as the proposed EICA method proposed in [24], which is not the focus of this paper. round(r i,s k ) = 1 indicates that there are no similar effective timing offsets with UE k, and thus we can obtain its timing error based on Eq. (15).

IV. PERFORMANCE ANALYSIS
In this section, we analyze the performance of the proposed random access scheme, including the MSE of the estimated effective timing offset, and the uplink throughput.

A. MSE of the estimated effective timing offset
In the proposed random access scheme, we utilize the ESPRIT-E method to estimate the effective timing offsets of UEs. Specifically, for the active UEs selecting the same time domain code, we first utilize the ESPRIT method to estimate the effective timing offsets of N − 1 active UEs. Then, after subtracting the N − 1 estimated effective timing offsets from the sum of effective timing offsets of all active UEs, we obtain the effective timing offsets of the remaining active UEs by solving the polynomial equations in Eqs. (10). Obviously, the procedure of Eqs. (10) will introduce noise. Furthermore, based on Eq. (6), the sum of effective timing offsets of all active UEs selecting the same time domain code is accurate when M goes to infinity. Hence, the MSE of the estimated effective timing offset of the proposed ESPRIT-E method is greater than or equal to that of the estimated effective timing offset when M goes to infinity, which is given by where N s is the number of UEs being detected. We utilize the Monte Carlo simulation method to obtain the value of MSE M →∞ , which is the lower bound of the MSE of the estimated effective timing offset of the ESPRIT-E method.

B. Uplink throughput analysis
We define the uplink throughput as the number of the successful detected active UEs. When given the number of active UEs selecting the same time domain code, based on Eqs. (10), the larger the value of M , the more accurate the estimated effective timing offsets. This leads to the increase of the number of the detected active UEs. Hence, we can obtain the upper bound of the uplink throughput of the proposed random access protocol by setting the value of M to infinity. Since it is hard to derive the analysis results of the uplink throughput for any values of M , we utilize the Monte Carlo simulation method to obtain the upper bound, denoted by T u .

V. SIMULATION RESULTS
In this section, we compare the performance of the proposed random access scheme with the random access scheme in [19], including the MSE of the estimated effective timing offset and the uplink throughput. In addition, terms "the proposed random access scheme" and "the random access scheme in [19]" are abbreviated as "The proposed RA" and "RA in [19]" in the result figures, respectively. In the simulation, we consider a cellular network operating over a bandwidth B=20 MHz and the radius of the cell is 250 meters and all UEs locate uniformly at the place farther than 25 meters from the BS. Table I shows the simulation parameters setting.   The reason is that, the increase of the number of subcarriers N , means the increase of the number of pilots, resulting in the decrease of the pilot collision probability and the interference between UEs. We can also note that, the MSE of the estimated effective timing offset is close to the lower bound. Fig.3 shows how the uplink throughput changes with the number of antennas at the BS under Q = 2, N = 8, and N a = 14. We can observe from the simulation results that the uplink throughputs of the proposed random access scheme and the random access scheme in [19] increase dramatically from M = 20 to M = 50, and increases at a slower pace when M ≥ 50. We also note that, the uplink throughput of the proposed random access is significantly higher than that of the random access scheme in [19], and much close to the upper bound T u . The reason is that, the random access scheme in [19] utilizes the ESPRIT method to estimate the effective timing offsets, and thus the number of detected effective timing offsets is limited by the number of subcarriers N . However, to address this problem, our proposed random access scheme proposed an ESPRIT-enhanced (i.e., ESPRIT-E) method to estimated the effective timing offsets.  We can observe from the simulation results that the uplink throughput of the proposed random access scheme is significantly higher than the random access scheme in [19] with the increase of the number of active UEs, and close to the upper bound T u . We also see that, with the increase of the number of active UEs, the uplink throughput of the proposed random access scheme increases almost linearly, whereas that of the random access scheme in [19] increase almost linearly from N a = 6 to N a = 8, and increases at a slower pace when N a ≥ 8. The reason is the same as we described for Fig.3, i.e., the number of detected UEs is limited by the number of subcarriers N .

VI. CONCLUSION
In this paper, we proposed a new random access scheme for M2M communication in crowded asynchronous massive MIMO systems to resolve the intra-cell pilot collision. The proposed random access scheme estimates the effective timing offsets by utilizing the proposed ESPRIT-E method, and then the UE can obtain its timing errors for uplink message transmission. We also analyze the performance of the proposed random access scheme, including the MSE of the estimated effective timing offset and the uplink throughput. Simulation results show that, compared to the exiting random access scheme for the crowded asynchronous massive MIMO systems, the proposed random access scheme can improve the uplink throughput and provide accurate effective timing offsets at the same time.