An improved reconstruction algorithm based on multi-user detection for uplink grant-free NOMA

For the traditional orthogonal matching pursuit(OMP) algorithm in multi-user detection(MUD) for uplink grant-free NOMA, here is a poor BER performance, so in this paper we propose an temporal-correlation orthogonal matching pursuit algorithm(TOMP) to realize muli-user detection. The core idea of the TOMP is to use the time correlation of the active user sets to achieve user activity and data detection in a number of continuous time slots. We use the estimated active user set in the current time slot as a priori information to estimate the active user sets for the next slot. By maintaining the active user set l T̂ of size K(K is the number of users),but modified in each iteration. Specifically, active user set is believed to be reliable in one iteration but shown error in another iteration, can be added to the set path delay l T̂ or removed from it. Theoretical analysis of the improved algorithm provide a guarantee that the multi-user can be successfully detected with a high probability. The simulation results show that the proposed scheme can achieve better bit error rate (BER) performance in the uplink grant-free NOMA system.


Introduction
With the rapid development of mobile communication, the spectrum resources become increasingly scarce, in order to meet the needs of future 5G in spectrum efficiency and the number of connections; Non-orthogonal multiple access (NOMA) as a candidate for 5G technology has been widely concerned [1].The future 5G wireless communication network will meet the wireless networking needs of a variety for network equipment, especially for the scene of 5G Massive Machine Type Communication(MMTC), the sporadic overload systems are some small transmission packets, continue along by orthogonal multiple access (OMA) will be bound to cause the waste of resources and excessive scheduling overhead, therefore, in order to avoid the waste of resources, we need to study the new non-orthogonal multiple access(NOMA) technology to improve the efficiency of small packet transmission.The existing multi-user detection of NOMA system is based on the premise that all users are active, however, not all users are active at the same time in the system, a large number of statistical results show that even in the busy time, active users generally do not exceed 10% of the total users, which means that although there are massive user connections in 5G, access to the system is only a small part of the user at the same time, so active users are sparse [2].

Compressive sensing
The premise of compressive sensing (CS) [3] is that the signal is sparse, but, most of the natural signal N R f  is not sparse in the time domain, but f can be sparse by a sparse basis Ψ transform, where ) ,..., , ( so the signal f can be expressed as: The sparse vector x is k-order sparse, so indicates that the number of nonzero values are equal to .If we apply a corresponding linear transformation to the coefficient x with measuring matrix The vector x is k-order sparse, so the initial signal can be accurately recovered from the linear measurement set using the optimal method, the observation matrix Φ and the sparse base Ψ follow the constraint isometric (RIP) condition, so reconstruct the signal x , i.e., to find the minimum 0  norm [4]: ,for the general A and y, there are nondeterministic polynomials for the equivalent constraint problem of 0  , so the following method is proposed: The optimal sparse solution requires a very sparse premise, and the perceived matrix A must satisfies the incoherent condition,however, in practice, these conditions cannot be satisfied, 1  cannot find the optimal solution, we must turn to other nonconvex problems, which needs weaker conditions to guarantee the accuracy of the successful recovery [5]: ，it can be seen from the above formula that when p is close to zero.The solution of p  will be closer to the solution of 0  .

System model
This paper considers a typical uplink NOMA system with one base station(BS) and K users,here the single antenna scheme is considered without loss of generality.The transmitted symbol k x for user k is modulated onto a spreading sequence k s of length N, in particular,we consider the case of K N  ,that is, the overloaded system, where the number of users can be larger than the length of spreading sequences; After that, signals from all active users are superimposed,and then are transmitted over N orthogonal OFDM subcarriers [6].The received signal on subcarrier n at the BS is denoted as:

TOMP algorithm
The core idea of the TOMP is to use the time correlation of the active user sets to achieve user activity and data detection in a number of continuous time slots.This is the biggest difference with OMP.The main steps of the algorithm are as follows: Maximum sparsity level: K . Initialization: 1) The initial support is set as 3) The estimated signal: 4) User estimated support: 5) Residual: The number of elements in the estimated support do not exceed the sparsity level K: 6) The estimate of the support set at the end of the iteration by 0 T ,which also serves as the initial estimate support for the following operation: ,searching for the most relevant atoms and indexs to join set 1 ˆ l T ,in each iteration , the user that correlates best with the residual signal will be included in the active user set at first.So, the active user set can be calculated by: is the residual signal.
8) Update: The sparse transmitted signal vector in this lth time slot can be updated as follows: 9) Modify the estimated support: Estimated support set is constantly being modified in each iteration.
For example, in an iteration, the active user set is reliable, but the active user set is not accurate at the next iteration,and then it should be removed from the l T ˆ.
The estimated support is revised by: 10) Update the residual signal: 11) When the energy of the residual error signal does not decrease,the iteration will be terminated with the following condition: calculating the active user set in each time slot, the sparse transmission signal vector can be restored in the tth time.The simulation platform is the MATLAB R2014a version.Let's assume that the number of active users are overloading factor is 200%.The simulation results are shown in the following Fig.1 and Table.

1 :Fig. 1 . 6 Conclusions
Fig.1.Comparison of performance between BER and SNR in uplink grant-free NOMA system