Rethinking Grant-Free Protocol in mMTC
Minhao Zhu Yifei Sun Lizhao You Zhaorui Wang Ya-Feng Liu Shuguang Cui

Fast photon-mediated entanglement of continuously-cooled trapped ions 
 for quantum networking
    Christopher Monroe
    April 29, 2024
==============================================================================================






  

This paper revisits the identity detection problem under the current grant-free protocol in massive machine-type communications (mMTC)  by asking the following question: for stable identity detection performance, is it enough to permit active devices to transmit preambles without any handshaking with the base station (BS)? Specifically, in the current grant-free protocol, the BS blindly allocates a fixed length of preamble to devices for identity detection as it lacks the prior information on the number of active devices K. However, in practice,  K varies dynamically over time, resulting in degraded identity detection performance especially when K is large. Consequently, the current grant-free protocol fails to ensure stable identity detection performance. To address this issue, we propose a two-stage communication protocol which consists of estimation of K in Phase I and detection of identities of active devices in Phase II. The preamble length for identity detection in Phase II is dynamically allocated based on the estimated K in Phase I through a table lookup manner such that the identity detection performance could always be  better than a predefined threshold. In addition, we design an algorithm for estimating K in Phase I, and  exploit the estimated K to reduce the computational complexity of the identity detector in Phase II. Numerical results demonstrate the effectiveness of the proposed two-stage communication protocol and algorithms. 








§ INTRODUCTION

Massive machine-type communications (mMTC) is envisioned as one of the candidates in 6G. mMTC provides efficient random access communications for a large number of devices, out of which only a small number of them are active. To reduce communication latency, a grant-free communication protocol is adopted where devices can transmit signals without permission from the base station (BS)<cit.>. Each device under the grant-free protocol is pre-assigned with a unique and nonorthogonal preamble that is served as identity of the device. A significant challenge posed by the grant-free protocol is to detect the identities of active devices through the transmission of  preambles<cit.>. The identities of active devices are useful in subsequent stages, e.g., resource allocation and channel estimation<cit.>. 

This paper revisits the identity detection problem under the current grant-free protocol in mMTC by asking the following question: for stable identity detection performance, is it enough to permit the active devices to transmit preambles without any handshaking with the BS? Specifically, in the current grant-free protocol, since the BS has no prior information on the number of active devices K, it just blindly allocates a fixed length of preamble L_ II for identity detection, no matter how large the actual value K is. However, in practice, K varies dynamically over time.
Theoretical findings from <cit.> suggest that maintaining a fixed L_ II at the BS significantly compromises identity detection performance when K is large. This scenario could lead to identity detection performance falling below a predefined threshold, which is undesirable in practical communication systems.

We take Fig. <ref> as an example for further illustration, where  a covariance-based approach in <cit.> is applied to detect the identities of active devices. Fig. <ref> shows that for fixed L_ II=100, the identity detection performance decreases significantly when the number of active devices K increases from 100 to 200. If we improve the preamble length to L_ II=180, the identity detection performance when K=200 approaches to that when K=100. This phenomenon underscores the necessity of dynamically allocating the preamble length L_ II according to the dynamic value of K, rather than blindly assigning a fixed length.

 

Building upon this observation, we argue that the current grant-free protocol is inadequate for ensuring stable identity detection performance. In particular,  the number of active devices K is required for the allocation of preamble length L_ II. To solve this problem, we first propose a two-stage communication protocol, which consists of estimation of K in Phase I and detection of identities of active devices in Phase II. The preamble length L_ II for identity detection in Phase II is dynamically allocated based on the estimated K in Phase I through a table lookup manner, such that the identity detection performance could  always be  better than a predefined threshold. Second, in Phase I of the protocol, we propose an efficient algorithm to estimate K at a negligible cost of preamble symbols. Third, in addition to the allocation of preamble length L_ II, we found that the estimated K can also be exploited to reduce the computational complexity of the identity detector in Phase II. We propose an efficient algorithm to achieve this goal. Numerical results demonstrate the effectiveness of the proposed two-stage communication protocol and algorithms. 

Previous works, including <cit.>, have primarily focused on the design of identity detector under the grant-free protocol but have overlooked the importance of ensuring stable identity detection performance, which is crucial in practical systems.  In contrast to these works, we study a two-stage communication protocol that estimates K before the identity detector in order to achieve stable identity detection performance. In addition,  <cit.> has proposed an algorithm to reduce the computational complexity of the identity detector. Compared with <cit.>, our proposed algorithm is much more efficient to reduce the computational complexity by leveraging the knowledge of the estimated K.



§ TWO-STAGE COMMUNICATION PROTOCOL

We consider a narrowband uplink system which consists of a BS equipped with M antennas and N single-antenna devices. At each time slot, only K ≪ N devices are active. Denote γ_n an activity indicator of device n, i.e., 

    γ_n={[ 1,  if device  n  is active,;               0,  otherwise, ].
    	   n=1,2, … ,N.

In the traditional grant-free protocol, each device is pre-assigned with a unique and nonorthogonal preamble with a fixed length of L_ II  to detect the identities of active devices through the estimation of γ_n, n=1, 2, …, N. 



 §.§ Problem of Existing Grant-Free Protocol

In this paper, we take the covariance-based approach as an example to show the problem of identity detection under the current grant-free protocol. Specifically, <cit.> studied under what conditions the identities of all K devices can be correctly detected by the BS. It has been proved in <cit.> that when M is sufficient large,  the maximum number of active devices that can be correctly detected scales as

    K∼𝒪(L_ II^2),

where 𝒪 denotes the order of approximation. Eq. (<ref>) clearly shows that the maximum number of active devices K that can be correctly detected highly depends on the preamble length L_ II, although the exact expression of  K with respect to L_ II is not clear far from now. On the one hand, under the current grant-free protocol, since there is no handshaking between the devices and the BS before the signal transmission, the BS has no prior information of K at each time slot. In this case, the BS can only blindly allocate a fixed length of preamble L_ II for identity detection. On the other hand, the number of active devices K varies at different time slots. In the heavy-load case where K is far larger than L_ II^2, the identity detection performance would be much worse. 

In practical scenarios, to ensure system-level performance, a practical system typically defines a threshold that delineates the minimum acceptable performance for identity detection across the entire system. Through the aforementioned analysis, it becomes evident that the current grant-free protocol may yield identity detection performance inferior to the predefined threshold, thereby degrading the overall system performance.



 §.§ Proposed New Communication Protocol

To solve the above problem, the BS should have the prior information on the number of active devices K. In this case, the BS can allocate the preamble length L_ II for identity detection according to the estimated K. However, the existing literature has not established an explicit relationship between K and L_ II for a specific identity detection error rate when M is finite.

To solve the problem of allocating L_ II according to the estimated K, we propose to create a table that specifies this relationship through simulations. Once we have the knowledge of estimated K, we can determine the preamble length L_ II through a table lookup manner. The table lookup approach has been widely applied in wireless systems when the explicit relationship among the parameters is hard to characterize, e.g., parameter selection of  channel coding and modulation in 5G<cit.>.  

The overall proposed two-stage communication protocol is summarized as follow: 

	
  * Phase I: Estimation of the number of active devices K. Device n is allocated L_ I preamble symbols to estimate the number of active devices K at the BS, n=1,2,…, N. Then, based on the estimated K, the BS allocates L_ II preamble symbols through a table lookup manner, in order to detect the identities of active devices.
	
  * Phase II: Detection of the identities of active devices. The BS detects the identities of active devices based on the received L_ II preamble signals from all active devices.

The overall preamble length is

    L=L_ I+L_ II.


There are several technical problems to be addressed in the proposed communication protocol. First, how to efficiently estimate K? That is, if we spend too many preamble symbols on the estimation of K in Phase I, the time for identity detection in Phase II would be quite limited. Second, the proposed estimation method for K in Phase I should not be sensitive to K itself; otherwise, we should also allocate L_ I dynamically according to the dynamic value of K, which further complicates the design of the communication protocol. Third, although the primary goal of estimating K is to determine the length of preamble L_ II for identity detection, it's important to explore how to further exploit the estimated K  to improve the performance of the identity detector in Phase II? We will address the above problems in the rest of this paper.



§ ESTIMATION OF NUMBER OF ACTIVE DEVICES
 
In this section, we show how to estimate the number of active devices K. In addition, through numerical results, we show that the parameter K can be estimated through only a few preamble symbols, and the proposed estimator is not sensitive with K, i.e., the estimator keeps roughly the same performance under different K.



 §.§ Problem Formulation and Estimator Design

In Phase I, in order to estimate K, all devices apply a same preamble s∈ℂ^L_ I×1. In addition,  we assume block-fading channels, i.e., the channels among the devices and BS remain roughly constant within each coherence block, but may vary among different coherence blocks. The channel from device n to BS is denoted by √(β_n)h_n∈ℂ^M×1, where β_n denotes the large-scale fading component, and h_n∼𝒞𝒩(0,I) denotes the independent and identically distributed (i.i.d) Rayleigh fading coefficient. Assume active devices transmit their preambles at a predefined time slot such that the received signals are perfectly synchronized at the BS. In this case, the received signal at Phase I becomes 

    Y̅=∑_n=1^Nγ_n√(p_n)s√(β_n)h^T_n+Z̅(a)=√(β)s∑_n=1^Nγ_nh^T_n+Z̅,

where p_n is the transmit power of device n, and Z̅ is the additive white Gaussian noise,  (a) is because we do power-control  on each device, i.e., 

    p_nβ_n=β, n=1, 2,…, N.

The received signal Y̅ is then normalized by √(β), and is expressed as

    Y=Y̅/√(β)=[y_1, y_2, …,y_M]=s_∑_n=1^Nγ_nh^T_n+Z,

where y_m is the received signal at the m-th antenna,  Z=Z̅/√(β)=[z_1, z_2, …, z_M]∈ℂ^L_ I×M is the normalized additive white Gaussian noise with z_m∼𝒞𝒩(0, σ^2 I), m=1, 2, …, M, and σ^2 is the normalized noise power. In (<ref>), since the channels follow the i.i.d. Rayleigh distribution, we have

    ∑_n=1^Nγ_nh^T_nd=√(K)h^T_1∼𝒞𝒩(0,KI),

where d= denotes equal in  distribution. In this case,  we have 

    Yd=√(K)sh^T_1+Z.

That is, the received signal Y is a function of K in terms of its distribution. Denote p(Y| K,s) the probability density function of Y given K and s, and it is expressed as

    p(Y|K,s)=∏_m=1^M p(y_m| K,s) =1/|Σπ|^M∏_m=1^M exp(-y_m^H Σ^-1y_m),

where 

    Σ=𝔼(y_my_m^H)  
    	=Kss^H+σ^2 I.

Note that maximizing p(Y| K,s) is equivalent to minimizing -log(p(Y| K,s)). In this case, the estimation problem is thus formulated as

	   min_K         log|Σ|+Tr(Σ^-1Σ̂) 

	     s.t.                                       N≥K≥0,
where 

    Σ̂=1/M∑_m=1^My_my_m^H.

By taking derivative on the objective function in (<ref>) with respect to K, we can get the estimated K as

    K̂=s^H Σ̂s/s_2^4-σ^2/s_2^2.





 §.§ Performance of Designed Estimator

We present the performance of the proposed estimator when N=1000. Denote E_K as the normalized estimation error, i.e.,

    E_K=𝔼(|K-K̂/K|),

where 𝔼(x) denotes the expectation of x. The noise power spectrum density is -169 dBm/Hz, the bandwidth is 10 MHz, the transmit power of each device is set to be 23 dBm, and  the large-scale fading component is modeled as  128.1+37.6log_10(d), where d is the distance between the BS and devices in km<cit.>.  In the simulations, we set d=1 km. 

The numerical results are shown in Fig. <ref>.  First, Fig. <ref> shows that the estimation error decreases slowly with the preamble length L_ I. In this case, we can set a small value of L_ I for the estimation of K, e.g., L_ I=4. Second, Fig. <ref> shows that the estimation error keeps roughly the constant under different K, which suggests that our estimator is robust to K. In this case, we can set a same L_ I to estimate K, regardless the value of K. Third, increasing the number of antennas M at BS can reduce the estimation error significantly. 

We will have a further evaluation of the estimation error of K on the two-stage protocol and the identity detector in Section <ref>. 



§ DETECTION OF IDENTITIES OF ACTIVE DEVICES


In this section, we show how to reduce the computational complexity of the identity detector in Phase II by exploiting the information K̂ obtained in Phase I.



 §.§ Problem Formulation

In Phase II,  device n is allocated a preamble s_n∈ℂ^L_ II×1 to report its activity to the BS, n=1,2,…, N. The channel  and noise models are the same as those in (<ref>). The received signal at the BS is expressed as

    Y=[y_1, y_2, …,y_M]=∑_n=1^Nγ_ns_nh^T_n+Z(b)=Sγ^1/2H+Z,

where y_m is the received signal at the m-th antenna. In (<ref>), we reuse the notations Y, y_m, and Z in this subsection in order to ease the exposition; (b) in (<ref>) follows as we denote S=[s_1, s_2, …, s_N] ∈ℂ^L_ II× N  the preamble sequence matrix; γ=diag(γ_1, γ_2, …,γ_N) ∈ℝ^N × N; and H=[h_1, h_2, …, h_N]^T ∈ℂ^N × M. 
 
Denote p(Y|γ, S) the PDF of Y given γ and S. Then, we get

    p(Y|γ, S)
    	    = 1/|Σπ|^M∏_m=1^M exp(-y_m^H Σ^-1y_m) 
        = 1/|Σπ|^Mexp(-Tr(Σ^-1YY^H)),

where 

    Σ=SγS^H+σ^2 I=∑_n=1^N γ_n s_ns_n^H+σ^2 I.

Note that maximizing p(Y|γ, S) is equivalent to minimizing -log(p(Y|γ, S)). In this case, the detection problem is thus formulated as

	   min_γ         log|Σ|+Tr(Σ^-1 Σ̂)  

	     s.t.                                       γ_n∈[0,1],  n=1, 2, …, N,

where Σ̂=1/MYY^H is the sample covariance matrix of the received signal. In the above, we reuse the notations Σ and Σ̂ in this subsection in order to ease the exposition. 

Our goal is to find a feasible point that satisfies
the first-order optimality condition of problem in (<ref>). To this end, denote the objective function in (<ref>) by

    f(γ)=log |Σ|+Tr(Σ^-1Σ̂).

Then, the gradient of f(γ) with respect to γ_n is given by

    [∇ f(γ)]_n=s_n^H Σ^-1s_n-s_n^H Σ^-1Σ̂Σ^-1s_n.

The first-order optimality condition of problem (<ref>) is 

    [∇ f(γ)]_n{≥ 0,       if γ_n=0, 
    ≤ 0,       if γ_n=1, 
    
    		=0,       if 0<γ_n<1. 
    	.

To quantify the degree to which the coordinate γ_n violates the optimality condition in (<ref>), in this paper, we define a nonnegative metric as follows:

    V(γ_n)={   |P_0^+∞(-[∇ f(γ)]_n)|,         if γ_n=0, 
       |P_-∞^1(1-[∇ f(γ)]_n)-1|,  if γ_n=1, 
       |[∇ f(γ)]_n|,                      if  0<γ_n<1, 
    	.

where 

    P_a^b(x)={   a,    if x<a, 
       b,   if x>b, 
       x,   if  a≤ x≤ b. 
    	.

Note that the first-order optimality is equivalent to

    V(γ_n)=0,   n=1,2,…, N,

where a larger V(γ_n) indicates a larger violation of the first-order optimality.
 




 §.§ Review of Coordinate Descent (CD) Algorithm in <cit.>
 

The activity detection problem in (<ref>) has been popularly solved by the CD algorithm. In this subsection, we first review an existing CD algorithm in <cit.> since our proposed algorithm by exploiting K̂ is building on that CD algorithm.

In the CD algorithm, at the i-th iteration, the CD algorithm updates each coordinate as follows:

    γ_n←γ_n+η,   n=1, 2,…, N,

where 

    η=min{max{s^H_nΣ^-1Σ̂Σ^-1s_n-s^H_nΣ^-1s_n/(s^H_nΣ^-1s_n)^2,-γ_n},1-γ_n}.

The algorithm is terminated when 

    V(γ)_∞≤ε,

where V(γ)=[V(γ_1),V(γ_2),…,V(γ_N)]^T, V(γ)_∞ denotes the infinity norm of V(γ), and ε denotes the error tolerance.  
The CD algorithm is summarized in Algorithm <ref>.



 §.§ Proposed K-Coordinate Descent Algorithm
  
A potential problem in the CD algorithm is its high complexity of updating all coordinates at each iteration (i.e.,  Lines <ref>–<ref> in Algorithm <ref>). Specifically, the complexity of updating one coordinate is in the order of 𝒪(L_ II^2). Therefore, the total per-iteration complexity of updating N coordinates is in the order of 𝒪(NL_ II^2). In particular, when N is large, the CD algorithm will suffer from a huge complexity.

In the following, we exploit the information K̂ obtained in Phase I to reduce the per-iteration computational complexity in CD. Specifically, through the estimation of K in Phase I, we have high confidence in believing that there are K̂ active devices, and updating the coordinates in γ corresponding to these K̂ active devices can mostly reduce the objective function value of problem  (<ref>). In this case, we only need to update at most K̂ coordinates in the CD algorithm at each iteration. As a result, the per-iteration complexity of our proposed CD algorithm  is in the order of 𝒪(K̂L_ II^2), which is significantly smaller than that in <cit.> when K̂ is far smaller than N (a very common scenario in mMTC). Below we show the details of our proposed CD algorithm by judiciously exploiting the information K̂ obtained in Phase I.

Selecting only K̂ coordinates to update: We propose a criterion based on the first-order optimality metric in (<ref>) to update at most K̂ coordinates among the total N coordinates at each iteration.  Specifically, as introduced before, a larger V(γ_n) in (<ref>) indicates a larger violation of the first-order optimality condition, n=1, 2, …, N. In this case, at the i-th iteration of Algorithm <ref>, we first sort the components in V(γ) in the descending order as follows:

    V(γ_i_1)≥ V(γ_i_2)≥…≥ V(γ_i_N);

Then, we define an importance index set 𝒜^(i) that contains the indices of the K̂ largest V(γ_n) as follows:

    𝒜^(i)={i_1,i_2,…,i_K̂};

Finally, we only update the coordinates with n∈𝒜^(i). Thus, the per-iteration complexity of Algorithm <ref> can be reduced significantly. 

Reducing the complexity in computing V(γ): In order to get the importance set 𝒜^(i) in (<ref>), we need to compute V(γ_n) in (<ref>) and [∇ f(γ)]_n in (<ref>), n=1,2,…, N. The computation of [∇ f(γ)]_n will cause additional complexity,  which is actually in the same order of the complexity when updating a coordinate in CD.  The motivation here  is to spend less computational cost while achieving the coordinate selection as in  𝒜^(i). In this way, we can further reduce the per-iteration computational complexity of the CD  algorithm.


Observe that, if V(γ_n) is below a threshold for several iterations, the coordinate γ_n has likely converged. In this case, there is no need to update γ_n in  subsequent iterations. To leverage the above observation, we define a quantity b_n for coordinate γ_n, which indicates how many times the violation V(γ_n^(i)) of the coordinate is below a predefined threshold α, i.e., 

    b_n={   b_n+1,   if V(γ_n^(i))<α, 
       b_n,        otherwise, 
    		. n=1,2,…, N,

where γ_n^(i) denotes the value of  γ_n at the i-th iteration. The initial value of b_n is set to be zero. Then, the candidate coordinate set to be updated at the i-th iteration is defined as

    𝒞^(i)={n | b_n≤ D},

where D is an integer that controls how many times that V(γ_n) falls below the threshold α is allowed.  If n∉𝒞^(i), in the subsequent iterations, we no longer compute the gradient [∇ f(γ)]_n and 	V(γ_n).

Refining set 𝒜^(i) in (<ref>): Through the design of coordinate set 𝒞^(i) in (<ref>),  the coordinates n∉𝒞^(i) will not be updated anymore in the rest of the iterations. Accordingly, we only need to compute and sort V(γ_n) for n∈𝒞^(i), i.e.,

    V(γ_i_1)≥ V(γ_i_2)≥…≥ V(γ_i_E),

where 

    E=min{|𝒞^(i)|,K̂},

and |𝒞^(i)| denote the cardinality of 𝒞^(i). Then,  𝒜^(i) in (<ref>) is refined as follows:

    𝒜^(i)={i_1,i_2,…,i_E}.


Based on the above discussion, we can see that only at most K̂ coordinates are updated at each iteration,  we name this algorithm as K-coordinate descent (K-CD). The K-CD algorithm is summarized in Algorithm <ref>.

 



 §.§ A Related Work in <cit.>

Prior to our work, the idea of reducing the per-iteration computational complexity by updating a selected subset of coordinates at each iteration of CD was developed in <cit.>, where the algorithm is referred to as Active Set CD. In Active Set CD, the importance index set is 

    𝒜^(i)={n  |  V(γ_n^(i))≥ω},
 
where ω is a pre-selected constant, and the coordinate set is  𝒞^(i)={1, 2,…, N}.

There are two key differences between our proposed K-CD algorithm and the Active Set CD algorithm in <cit.>. First, the criterion for selecting which coordinate to update in Active Set CD is through the preset parameter ω. In sharp contrast, our K-CD algorithm selects coordinates based on an estimation of K. Numerical results in Section <ref> will demonstrate that K-CD significantly reduces the number of coordinate updates compared with Active Set CD. Second, unlike Active Set CD which needs to compute the full gradient at each iteration, K-CD only needs to compute a partial gradient at each iteration, whose cardinality is generally significantly less than the problem's dimension N. This reduction plays a significant role in reducing the computational complexity, as the computation of the gradient itself is a major source of complexity.



 §.§ Complexity Analysis
 
We compare the per-iteration complexity of the proposed K-CD algorithm with that of  CD and Active Set CD algorithms in terms of floating point operations (FLOPs). The per-iteration complexity of the proposed K-CD algorithm is

    5L_ II^2E+4L_ II^2|𝒞^(i)|,

where  5L_ II^2 is the complexity of one coordinate update, and 4L_ II^2 is the complexity  of one gradient component computation (i.e., the complexity of computing [∇ f(γ)]_n in (<ref>)), E is given  in (<ref>) and 𝒞^(i) is given in (<ref>);  the per-iteration complexity of the
CD algorithm is 5L_ II^2N; the per-iteration complexity of the Active Set CD algorithm is 5L_ II^2|𝒜^(i)|+4L_ II^2N, where 𝒜^(i) is given in (<ref>).



 






§ NUMERICAL RESULTS

In this section, we show the performance of the identity detector for active devices, and the overall performance of the proposed two-stage communication protocol in terms of the identity detection error rate, which is the error rate when the missed detection probability (MDP) equals the false alarm probability (FAP). The MDP is the probability that an active device is detected to be inactive; and the FAP is the probability that an inactive device is detected to be active. In the simulations, we set N=1000 and ε=10^-3. The channel setup is the same as that in Section <ref>. 



 §.§ Performance of K-CD Algorithm

We have two benchmarks to evaluate the identity detection performance of the proposed K-CD algorithm, which are CD algorithm<cit.> and Active Set CD algorithm<cit.>.


In Figs. <ref> and <ref>, we show the performance of the proposed K-CD identity detector designed in Phase II when K=100 and M=15. In the simulations, we first allocate L_ I=4 symbols to estimate K by applying the designed estimator in Phase I. The output K̂ is then fed to the detector in Phase II, where we allocate L_ II=100 symbols to detect the identities of active devices. In addition, to have a fair comparison, we do not count the CPU runtime when computing V(γ) in the CD algorithm while we do count the CPU runtime when computing V(γ) in the K-CD and Active Set CD algorithms. Fig. <ref> shows the average total number of coordinate updates and gradient component computation when the algorithms meet the termination condition in (<ref>). 

First, as shown in Fig. <ref>, with a sufficient runtime, K-CD with D=2 and α=0.01, alongside CD and Active Set CD all achieve the same detection performance. This shows that although K-CD reduces the computational complexity, K-CD does not compromise the detection performance. Second, compared with the CD algorithm, our K-CD algorithm notably reduces the convergence time by a factor of three. The reason behind is shown in Fig. <ref>. Specifically, K-CD reduces the total number of updated coordinates significantly compared with CD. Although  K-CD needs to compute some partial gradients additionally, the overall complexity is reduced in the end. Third, the computational complexity of the K-CD algorithm is significantly less than that of Active Set CD. This is due to the fact that, as shown in Fig. <ref>, K-CD can reduce the number of coordinate updates and gradient component computation significantly at the same time, while Active Set CD only reduces the number of coordinate updates. Last, if we fix the runtime to be 0.02s, the detection performance of K-CD outperforms CD and Active Set CD by nearly two orders of magnitude.  









 §.§ Performance of Two-Stage Communication Protocol

We compare the proposed two-stage communication protocol with the existing grant-free protocol, which applies the CD algorithm (Algorithm <ref>) for activity detection.

In the two-stage protocol, we apply Table <ref> to determine L_ II based on K̂. The table is constructed according to a  predefined performance threshold, i.e., the maximum identity detection error rate of the two-stage protocol, which is set as 10^-2. Even though K̂ is only an estimate value of K, we proceed as if it were accurate when referring Table <ref>. The table covers a range of K from 10 to 300 with increments of 10. If K̂ does not exactly match a table boundary but falls within an interval, L_ II is assigned based on the higher boundary of that interval. For example, when  K̂=10, L_ II=15, and when 10<K̂<20, L_ II=25. In addition, since the two-stage protocol allocates  L_ II dynamically, in Fig. <ref>, we still use L_ II to denote the average allocated preamble length in Phase II.

First, Fig. <ref> shows that with a sufficient runtime the proposed two-stage protocol achieves a stable identity detection performance as K increases from 100 to 200, keeping the detection error rate below the predefined threshold 10^-2. This is in sharp contrast to the grant-free protocol, where increasing K from 100 to 200 significantly degrades the detection performance. Second, for  K=200, when the grant-free protocol is allocated L = L_ II = 177 preamble symbols which is equal to the total average preamble symbols L=177 in the two-stage protocol, Fig. <ref>  shows that the performance of the proposed two-stage protocol is much better than that of the grant-free protocol when the runtime is smaller than 0.15s. The performance gain comes from the exploitation of K̂ in Phase II to reduce  complexity.

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