I Introduction
Nonorthogonal multiple access (NOMA) is envisaged as a promising technique for future radio access [2]. Traditional orthogonal multiple access (OMA) techniques allocate orthogonal resources to different users, e.g., orthogonal time resources in the time division multiple access (TDMA) scheme. Differently, the NOMA provides the multiuser access by allocating nonorthogonal resources to users [3]. For example, in the powerdomain NOMA scheme, the signals for multiple users are superposed at different power levels using superposition coding [4], and the multiuser detection method, such as successive interference cancellation (SIC) [5], is employed at the receiver. The advantages of the NOMA over the OMA have been extensively studied in [2] and the references therein, e.g., providing higher system throughput compared to OMA and supporting massive connectivity.
Another line of research is to study the effects of asynchronous transmission on the performance of the wireless communication systems. Asynchronous transmission refers to the case where the symbol epochs of the signals transmitted by the users are not aligned at the receiver
[6]. In particular, [6] first pointed out the potential advantages of symbolasynchronous communications in terms of increasing the capacity of a multipleaccess channel. The work in [7]applied the symbolasynchronous channel estimation method to tackle the pilot contamination problem in massive multipleinput multipleoutput (MIMO) systems. Asynchronous transmission was studied in
[8, 9] as a tool to mitigate or cancel the interuser interference. In addition, the nonzero symbol offset was used to reduce the interantenna interference in MIMO systems in [10]. Moreover, an asynchronous analog network coding scheme for multiuser cooperative communications was proposed in [11] to provide a greater diversity order compared with that of synchronous analog network coding. Also, adding intentional timing mismatch was proposed in [12] to improve the performance of a relay network. The authors of [13, 14] further proposed several differential decoding schemes for asynchronous multiuser MIMO systems based on orthogonal spacetime block codes (OSTBCs) and for differential distributed spacetime coding systems with imperfect synchronization.Applying the symbolasynchronous transmission to NOMA, a scheme named asynchronous NOMA (ANOMA) was studied in [15]. In fact, an idea similar to the ANOMA in [15], i.e., applying asynchronous transmission for multiple access, has also been proposed and investigated in, e.g., [12, 8, 9]. Specially, a timing mismatch between signals for different users is intentionally added as an additional resource to address the problem of interuser interference. It has been shown using the numerical simulation in [15] that the ANOMA outperforms the conventional (synchronized) NOMA by achieving a larger throughput.
However, the work in [15]
has several limitations. While addressing those limitations is important to understand the ANOMA systems, to the best of our knowledge, no existing paper has tackled the following issues. First, there is no analytical result on the comparison between the performance of the ANOMA and that of the NOMA in terms of the throughput, although numerically it is shown that ANOMA outperforms NOMA in certain scenarios. This is probably because the existing expression for the throughput of the ANOMA system is given as a function of the channel matrix but not the signaltonoise ratio (SNR). The lack of such an expression in terms of SNR makes the analytical comparison between NOMA and ANOMA almost intractable. Second, the optimal design of ANOMA has not been investigated. Despite the fact that the performance of ANOMA is directly affected by important design parameters such as the transmit power and the timing mismatch, existing papers mainly focused on the performance demonstration only. Third, the impact of timing error on the ANOMA systems has not been studied. Existing studies ideally assumed that the timing information was perfectly known. However, the timing information in practice cannot always be perfectly obtained, and the timing error is often inevitable
[16]. The timing information plays a vital role in ANOMA systems, since oversampling is designed using the timing information [7, 8].In this paper, we comprehensively investigate the ANOMA in a twouser uplink system. The primary contributions of the paper are summarized as follows:

For the first time, we analytically prove that the ANOMA with a sufficiently large frame length can always outperform the NOMA in terms of the system sum throughput. To this end, we derive the expression for the sum throughput of the ANOMA system as a function of SNR, frame length, and normalized timing mismatch. A simplified throughput expression is further obtained for the asymptotic case of infinite frame length.

We investigate the optimal design of the twouser uplink ANOMA system aiming at maximizing the sum throughput. We find that each user should transmit at full power despite the negative effect of interuser interference. In addition, we prove that the optimal timing mismatch converges to one half of a symbol time as the frame length goes to infinity.

We analyze the impact of timing error on the performance of the uplink ANOMA system. Two types of timing error are taken into consideration, i.e., the synchronization timing error and the coordination timing error, which account for the timing error caused in signal synchronization and the coordination of the timing mismatch between asynchronous signals, respectively. We derive the expressions for the throughput loss of the ANOMA system with respect to both types of timing error, and analyze how the synchronization timing error and the coordination timing error individually and jointly affect the system performance.
The remainder of the paper is organized as follows. The twouser uplink system model is presented in Section II. The performance of the ANOMA system is analyzed in Section III. We discuss the optimal design of the ANOMA system in Section IV. We analyze the outputs of ANOMA matched filters with timing error and the throughput loss incurred by timing error in Section V. Numerical results are presented in Section VI. Finally, we draw the conclusions in Section VII.
Notations: denotes the Hermitian transpose, denotes the transpose, denotes the trace operation, denotes the inverse operation, denotes the absolute value of , denotes the expectation operation,
denotes the complex normal distribution with zero mean and unit variance, and
denotes the unit step function whose value is zero for negative arguments and one for positive arguments.Ii System Model
In this paper, we consider an uplink system which consists of a single base station (BS) and two users, as shown in Fig. 1. The two users share the same frequencytime resource to transmit signals to the BS. We assume that perfect channel state information (CSI) is known at the BS via the uplink channel training.
Iia ANOMA System
For the ANOMA, a timing mismatch is intentionally introduced between the symbols from two users. As shown in Fig. 2, the intended timing mismatch between the symbols of Users 1 and 2 is denoted by , where is the duration of each symbol and , , is the normalized timing mismatch. Note that the ANOMA system becomes a synchronous NOMA system when . In this section, we assume that is perfectly known at BS via timing offset estimation and uplink timing control techniques, such as the timing advance [17]. We will study the ANOMA system with timing error in Section V.
Let and , where the subscripts and denote the parameters for Users 1 and 2, respectively, () denotes the th normalized transmitted symbol, denotes the channel coefficient in the block of transmission, and denotes the transmit power. The BS’s received signal at time is then given by
(1) 
where denotes the number of symbols in a frame, i.e., the frame length, denotes the time duration of one symbol, denotes the pulseshaping filter with nonzero duration of , and denotes the normalized additive white Gaussian noise (AWGN). Without loss of generality, the rectangular pulse shape is adopted, i.e., when and when . With the block fading model, we assume that the channels remain static during the transmission of consecutive symbols.
The BS uses the oversampling technique to take advantage of sampling diversity in the asynchronous systems [7, 1]. As shown in Fig. 2
, the BS obtains two sample vectors, denoted by
and . Specifically, the th element in the first sample vector is given by(2) 
where denotes the additive noise in the first sampled vector. The th element in the second sample vector is given by
(3) 
where denotes the additive noise in the second sampled vector.
We can write the outputs at the BS in a matrix form as
(4) 
where
(5)  
(6)  
(7)  
(8) 
and
(9) 
We assume that the transmitted symbols are independent, such that . Note that the noise terms in (4) are colored due to the oversampling, and we have
(10) 
Thus, the covariance matrix of is given by
(11) 
IiB Benchmark System – NOMA
Iii Performance Analysis of ANOMA Systems
In this section, we analyze the throughput performance of the ANOMA system. From (4), the sum throughput of the twouser uplink ANOMA system can be written as
(13) 
We note that some existing papers define the throughput of ANOMA as
(14) 
which is slightly different from (13). Although (13) and (14) converge to the same expression as , we highlight that our adopted expression in (13) is more accurate than (14) for evaluating the throughput of ANOMA with finite frame length , since the system actually spends instead of symbol times to transmit symbols for each user.
It is worth mentioning that the practical transmission scheme may even simply allocate instead of symbol times for the transmission, and the throughput becomes . Our analysis is still applicable to that case, since one can simply revise (most) results according to .
In the following theorem, we derive the sum throughput of the twouser uplink ANOMA system in terms of the transmit SNRs, and , the normalized timing mismatch, , and the frame length, .
Theorem 1
The sum throughput of the twouser uplink ANOMA system is derived as
(15) 
where
(16)  
(17)  
(18)  
(19) 
Proof:
See Appendix A.
Based on Theorem 1, we present the throughput of the twouser uplink ANOMA system for the asymptotic case of in the following corollary, which characterizes the limiting performance of the system when the frame length is large.
Corollary 1
The throughput of the twouser uplink ANOMA system in the asymptotic case of is given by
(20) 
Proof:
See Appendix B.
Iiia Comparison with NOMA
According to (12), with perfect SIC at BS, the sum throughput of the two users in the uplink NOMA system can be written as [18]
(21) 
which can also be obtained from (15) by setting .
Due to the complicated expression for the throughput of ANOMA in (15), it is difficult to analytically compare the NOMA with the ANOMA for a general value of . Instead, we provide numerical results in Section VI and consider the asymptotic case of for an analytical comparison in the following theorem.
Theorem 2
The twouser uplink ANOMA system as achieves an equal or higher throughput compared with the NOMA system, i.e.,
(22) 
where if and only if the normalized timing mismatch .
Proof:
See Appendix C.
We further conclude the following corollary.
Corollary 2
With a sufficiently large frame length, the ANOMA outperforms the NOMA for the twouser uplink system in terms of the sum throughput.
Iv Design of ANOMA Systems
From the analysis in Section III, we note that the throughput performance of the uplink ANOMA system is directly determined by the transmit powers and the normalized timing mismatch, i.e., , and . In this section, we investigate the optimal , and that maximize the throughput of the system.
The design problem is formulated as follows:
(23) 
where and are the maximum available powers at which Users 1 and 2 can transmit, respectively. Note that the transmit powers are coupled together in a complicated way in the expression for the throughput of the ANOMA system in (15), which is different from the case of NOMA in (21). Thus, the optimal transmit powers for the ANOMA system are not easy to determine, while it is easy to find that we shall use the maximum available transmit powers at users for the NOMA system.
It is worth mentioning that the performance of the uplink ANOMA system is also affected by the frame length, . However, the frame length is constrained by the channel condition, i.e., the length of each block of the block fading channel, and the acceptable transceiver complexity. Hence, we do not investigate the design of in this work and assume that it is fixed based on the channel conditions and the overall system design.
Iva Optimal Transmit Power
We first obtain the optimal transmit power scheme. We summarize the optimal transmit powers for the ANOMA system as follows.
Theorem 3
For the twouser uplink ANOMA system with any frame length, , and the normalized timing mismatch, , the optimal transmit powers at Users 1 and 2, and , are equal to the maximum available powers at which Users 1 and 2 can transmit, and , i.e., and
Proof:
See Appendix D.
From Theorem 3, we find that the optimal design of transmit powers for the twouser uplink ANOMA system is the same as that for the NOMA system.
IvB Optimal Normalized Timing Mismatch
We now study the optimal normalized timing mismatch, . The optimal normalized timing mismatch, , is analytically intractable for a general finite frame length , while we can numerically obtain for a given finite by simply searching in the range of . In addition, we present in the asymptotic case of in the following theorem.
Theorem 4
For the twouser uplink ANOMA system with the frame length , the optimal normalized timing mismatch to maximize the sum throughput is given by .
Proof:
See Appendix E.
V Impact of Timing Error on ANOMA systems
The analysis in the previous sections is based on the assumption that the BS perfectly knows the timing information. However, the timing information cannot always be perfectly obtained in practice, and the timing error is often inevitable. In this section, we study the impact of timing error on the ANOMA system.
Va Timing Error
We consider two types of timing error for the ANOMA system, i.e., the synchronization timing error and the coordination timing error.
VA1 Synchronization Timing Error
To synchronize the signals, we need a reference signal. Without loss of generality, we use the signal from User 1 as the timing reference (the timing offset is 0). This requires a symbollevel timing synchronization with User 1 at the BS, as it is also done in NOMA. The normalized synchronization timing error, denoted by in Fig. 3, is due to the imperfect timing synchronization. Without loss of generality, we assume that . With the synchronization timing error, is taken from the time to and is taken from the time to , although the BS intends to take from the time to and from the time to . We will study the effect of this timing error later.
VA2 Coordination Timing Error
In order to achieve the desired timing mismatch between the two signals, the BS coordinates the uplink transmission timing of the two users to add the intended timing offsets at each transmitter. For example, the timing advance is the technique employed in long term evolution (LTE) systems to estimate and adjust the timing offsets among uplink signals at BS [19, 17]. The normalized coordination timing error, denoted by in Fig. 3, results from the imperfect coordination between the users. With the coordination timing error, the actual timing mismatch becomes , while the intended timing mismatch is . In addition to the synchronization timing error , the sample is taken from to , although the BS intends to take from to . Without loss of generality, we assume that .
Fig. 3 illustrates the sampling for an ANOMA system with timing error. It is worth mentioning that the sign of the timing error stands for the direction in which the function of the matched filter is shifted. For example, as shown in Fig. 3, the matched filter is shifted to the right by if compared to the matched filter designed with no timing error in Fig. 2. Fig. 3 only presents the case when and , while our analysis works for any values of and .
VB Outputs of ANOMA Matched Filters with Timing Error
We now derive the outputs of the matched filters at the BS with both the synchronization timing error and the coordination timing error.
In the presence of timing error, the th element of the first sample vector is given by
(24) 
and the th element of the second sample vector is given by
(25) 
where and .
We note from (VB) and (VB) that the first sample vector is affected by the normalized synchronization timing error only, while the second sample vector is affected by the sum of the normalized synchronization timing error and the normalized coordination timing error .
With (VB) and (VB), we obtain the outputs of the two matched filters at the BS subject to the timing error in the matrix form as
(26) 
where , , and is given by
(27) 
We note from (VB) that the expression for is related to the signs of and . For the sake of brevity, we present the analytical results for the case of and in the rest of the paper, while our analytical method and findings are applicable to all cases. In addition, we will present the numerical results in Section VI for all possible cases of and . With and , the expression for is rewritten as
(28) 
The covariance matrix of is given by
(29) 
where can be rewritten as
(30) 
We note from (30) that the covariance matrix of the noise terms is affected by the normalized coordination timing error , while it is not related to the normalized synchronization timing error .
VC Impact of Timing Error on Throughput Performance
According to (26), the throughput of the ANOMA system with timing error is given by
(31) 
When there is no timing error, i.e., , we have . Hence, substituting into (VC), we obtain the throughput of the ANOMA system without timing error, which is the same as (13).
In what follows, we separately analyze the throughput loss incurred by the synchronization timing error and the coordination timing error with the practical consideration that these two types of timing error both are relatively small. For each case, we show that the throughput loss is approximately linear to the timing error by omitting the highorder terms of the timing error.
VC1 Impact of Synchronization Timing Error
We first investigate the impact of synchronization timing error on the throughput loss and consider the practical scenario where the error is relatively small, such that and .
In this case, by omitting highorder terms of , we obtain the throughput loss incurred by the synchronization timing error from (VC) as
(33) 
where , , is approximated by using as , is derived using the special case of Jacobi’s formula [20], i.e., , and is derived by omitting the highorder terms of and applying the approximation when . From (VC1), we note that the throughput loss is approximately linear to when and .
VC2 Impact of Coordination Timing Error
We now investigate the impact of the coordination timing error on the throughput loss and still consider the practical scenario where the error is relatively small, such that and .
Vi Numerical Results
In this section, we present numerical results to compare the throughput performances of NOMA and ANOMA systems and illustrate the impact of timing error on the performance of the ANOMA system. Figs. 5, 6, and 7 show the ANOMA system without timing error while the other figures are for the impact of timing error. In our simulations, we set the symbol length and the AWGN with unit power. If not specified, the normalized timing mismatch between the two signals is set to 0.5.
At first, we present the throughput performances of NOMA and ANOMA systems under different channel conditions in Fig. 4. In Fig. 4, the curves of “ANOMA in (13)” are derived directly from the definition in (13), and the curves of “ANOMA in (15)” are obtained from our result in Theorem 1. It is shown that the throughput computed by Theorem 1 completely aligns with that calculated by (13) for different combinations of channel conditions, which confirms the correctness of Theorem 1. Besides, Fig. 4 demonstrates that the throughputs of ANOMA and NOMA systems increase with the channel gains and . It is also shown that the throughput of the ANOMA system is better than that of the NOMA system when the frame length () under different channel conditions.
Then, we compare the throughput performances of NOMA and ANOMA systems. Fig. 5 shows the throughput as a function of the frame length . In Fig. 5, it is demonstrated that as increases, the throughput of ANOMA systems converges to the result in Corollary 1, which provides the throughput in the asymptotic case of . Furthermore, we note from the figure that the throughputs of ANOMA systems for different s as are greater than that of the NOMA system, which is consistent with our analytical results in Theorem 2 and Corollary 2.
In addition, we illustrate the optimal parameter design of the ANOMA system in Figs. 6 and 7. Fig. 6 demonstrates the sum throughput of two users as a function of their transmit powers. It is shown that the maximal sum throughput is reached when the transmit powers are equal to the maximum available powers, which aligns with Theorem 3. We present the optimal normalized timing mismatch found by exhaustive search to maximize the sum throughput of two users in Fig. 7. As shown in Fig. 7, starts with 0, and then increases with , finally converges to 0.5 as grows, which verifies the correctness of Theorem 4.
In what follows, we evaluate the impact of timing error on the throughput of ANOMA systems. In the following figures, the throughput loss ratio is defined as the ratio of the throughput loss in (VC) and the throughput of the ANOMA system without timing error in (13), i.e.,
(35) 
In Fig. 8, we present the throughput loss ratio as a function of and ranging from 0.1 to 0.1. As shown in Fig. 8, the throughput loss ratio increases with both the synchronization timing error and the coordination timing error. We also find that the throughput loss ratio is a continuous function with respect to and but nondifferentiable when or . This is because there are nonlinear step functions in the expression for in (VB).
We also study the individual effects of the timing synchronization error and the coordination timing error on the throughput of ANOMA systems. In Fig. 9, we show the throughput loss ratio as a function of when and when . Note that the curves of “impact of ” and “impact of ” are the slices of Fig. 8 when and , respectively. The approximated results are calculated by and using (VC1) and (VC2). It is demonstrated that the expressions in (VC1) and (VC2) are good approximations of (VC) when and , respectively. Besides, causes almost twice throughput loss compared to for the same value of error. This phenomenon reveals that the performance of ANOMA is more sensitive to the synchronization timing error than the coordination timing error.
Finally, we compare the performances of OMA, NOMA, ANOMA without and with timing error in Fig. 10. In our simulation, the conventional TDMA is adopted as the OMA scheme. As shown in the figure, the throughput curve of OMA is a single point because it is not a function of timing error. The throughput of the NOMA system is calculated under the assumption that perfect SIC is realized at BS. It is demonstrated that the rate performance for ANOMA without timing error is better than that of NOMA with SIC which is further greater than that of OMA. Also, ANOMA always outperforms a perfectly synchronized OMA. We note that for small values of timing error, ANOMA outperforms even a perfectly synchronized NOMA. For the same timing error, the performance of ANOMA is better than that of NOMA. Besides, as shown in Fig. 10, the throughput decreases with the absolute value of
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