Given the architecture of radio stripe, where APs are connected sequentially and CPU being connected to last AP (daisy chain architecture) the sequential processing becomes obvious choice. Therefore, authors [13] propose sequential uplink processing for cell-free massive MIMO based on radio stripe. Here, each AP computes the local channel estimate in the uplink, make soft estimate of the transmitted signal and thereby forwards local CSI estimate, soft estimate and error statistic to next AP. Upon receiving the information from the previous AP, the AP improves its own soft estimate. This sequential processing goes on till the last AP, the final estimate is forwarded to CPU for final decoding. Authors [13] claim the proposed processing mechanism achieves comparable performance to ‘Level 4’, while using 90% less signaling in the front-haul. In [8], authors propose Optimal Sequential Linear Processing (OSLP), an uplink sequential processing algorithm which they prove optimal both in the maximum spectral efficiency sense and minimum Mean Square Error (MSE) sense.

However, sequential processing leads to higher processing latency [14]. Therefore, authors in [14] propose two parallel processing schemes in the Radio Stripe network, namely interference-aware MR processing and distributed regularized zero-forcing (D-RZF). Authors [15], propose Q-LMMSE, which implements MRC processing leading to low front-haul loading, simple AP hardware and fast parallel computations.

All works discussed so far in cell-free massive MIMO based on radio stripe [8,13–15] has huge computational needs (for example channel estimation), since next generation network will involve large numbers of APs and UEs. These high computational costs can only be reduced by the application of Deep Learning (DL). DL replaces the complex computations with the trained model, which reduces the complexity considerably.

Authors in [16] propose DeepSIC, which is a data-driven technique for detecting symbols in the Multi-User MIMO (MU-MIMO) scenario. It is based on an iterative SIC algorithm, a MIMO-based detection scheme. Results show it outperforms Iterative SIC in the presence of CSI uncertainty. Since it is an iterative approach, detection is carried out in iteration in a single base station (BS). Our other works on interference cancellation are for Software Defined Radio (SDR) platform [17]. Motivated by the above [16], in our previous work [18], we propose DNN-based distributed sequential uplink processing (DBDSUP) for detecting symbols in the uplink of Cell-Free massive MIMO System Based on Radio Stripe. Here, each AP consists of K (numbers of UEs) parallel Soft Estimate Network (SEN), which computes the conditional distribution of each user at each AP (each AP carries out a single iteration). The conditional distribution of each user consists of the estimation of each symbol in the constellation. The Symbol Error Rate (SER) of the proposed mechanism ‘DBDSUP’ is evaluated. However, the processing here is sequential.

In order to take advantage of DL (to reduce computational complexity) and parallel processing (to reduce latency) there is need to develop a mechanism which is based on DL and does parallel processing.

Moreover, next-generation radio stripe network would involve large numbers of APs (typically in the range of 1000 s in particular give area like train, the bus as shown in Fig. 2), if all APs sequentially decodes transmitted symbol of given user, it would impose huge complexity on the system. Therefore, it becomes extremely important to develop a mechanism for users to select APs in Cell-free massive MIMO based on radio stripe.

Authors in [19] propose an effective channel gain-based algorithm in massive cell-free MIMO to assign users to each AP sequentially. Here author considers two metrics, first measure the channel quality of each user to all APs, and another one is to calculate the effective channel gain between every user and AP in the system, which reduces interferences and leads to a higher sum rate. Other works on AP selection focus on Machine learning (ML) [20].

This paper aims to propose a deep learning-based parallel decoding mechanism and AP selection algorithm for cell-free massive MIMO based on radio stripe.

The contribution is stated in the points below.

The rest of the paper is organized as follows. In Section 2, we discuss the network model of radio stripe . Section 3, elaborate about different aspect of proposed DNN based Parallel Decoding in Radio Stripe (DNNBPDRS). In Section 4, a detailed description of the proposed AP selection algorithm, Channel link-based AP selection (CLBAPS), is detailed. Simulation setup, training of neural network and numerical results are discussed in Section 5. Finally, the article is concluded in Section 6.

In this subsection, we briefly discuss about iterative Soft Iterative Cancellation (SIC) [22]. Iterative Soft Interference Cancellation (SIC) is a multi-user detection technique that combines parallel (multi-stage) interference cancellations with soft decoding. In a general sense, iterative SIC is an iteration-based detection scheme wherein each iteration, two operations are carried out in parallel for each user, namely Interference cancellation and soft decoding. Considering K^th user and c^th iteration, during the interference cancellation stage, the expected values and variances of {s_u}_{u ≠ k} (interfering symbols) are computed based on { P^u(c−1) }_{u ≠ k}. P_k^(c) is the estimated conditional distribution of k^th user symbol s_k. Entries of P_k^(c) are the estimate of the distribution of s_k for each possible symbol in the constellation. { P^u(c−1) }_{u ≠ k} is the estimated conditional distribution of interfering symbols obtained in the previous iteration. The expected value of interfering user in cth iteration is computed as

(5) eu(c−1)=∑rw∈S⁡rw(P^u(c−1))w

where {r_w}^W_{w = 1} are the indexed elements of the constellation set CS. The variance of interfering symbols is given by,

(6) Vu(c−1)=∑rw∈CS⁡(rw−eu(c−1))2 (P^u(c−1))w

The contribution of the interfering symbols from the received signal y is canceled by replacing it with mean {e_u^(c−1)} and subtracting the resultant term. Considering h_u the u^th column of H the elements of H with respect to user u. where H is channel matrix of all K user where each column represents one user channel vector. Interference canceled output of the channel is given by,

(7) Gk(c)=y−∑u≠k⁡hueu(c−1)

(8) =hksk+∑u≠k⁡hu(su−eu(c−1))+N

The second step carried out in parallel for each user is soft decoding. The bijective transformation of y is Gk(c) , such that each possible value of y there is one unique value in Gk(c). Therefore, Psk|y(rw|y)=Psk|Gk(c)(rw|Gk(c)) for each r_w ∈ CS. Therefore, the conditional distribution of s_k given y is approximated from the conditional distribution of Gk(c) given s_k using Bayes theorem. Each symbol is equiprobable. Conditional distribution is computed as,

(9) (P^k(c))w=PGk(c)|sk(Gk(c)|rw)∑r′w∈S⁡PGk(c)|sk(Gk(c)|r′w)

Symbols are decoded after the final iteration such that symbol which maximizes estimated conditional distribution is decoded as the transmitted symbol for each user.

(10) s^k=arg⁡max(P^k(C))w

where w ∈ {1, 2, …….., W}

The algorithm in steps is given in our previous works [18].

Iterative SIC is leveraged to perform data-driven detection “DeepSIC” in Multi-User MIMO (MU-MIMO) [16], motivated by this work, in our last work, we proposed DNN-based distributed sequential uplink processing (DBDSUP) in cell-Free massive MIMO based on radio stripe by exploiting data-driven iterative SIC. However, it considers sequential processing leading to higher latency, which can be reduced given the processing is carried out in a parallel fashion. Thus, we propose DNN based Parallel Decoding in Radio Stripe (DNNBPDRS), which carries out parallel processing in radio stripe networks.

Fig. 3 shows a simplified proposed parallel processing network model.

As shown in Fig. 3, each AP consists of K-parallel Soft Estimate Network (SEN) (which is discussed in detail in Section 3.4). As opposed to sequential processing, here each AP process the received signal simultaneously. The signal received at AP₁, AP₂, ……….., AP_L is given by y₁, y₂, …….., y_L.

As shown in Fig. 4, Soft Estimate Network (SEN) is a classification-based Neural Network. As explained above, iterative SIC carries out two operations in parallel for each user, namely, interference cancellation and soft decoding. In particular, for c^th iteration and k^th user soft decoding estimate P^K(c) is a classification task since entries of P^K(c) is an estimation of the distribution of k^th user symbols for each symbol in the constellation with the dimension of P^K(c) being R^B.

SEN replaces these complex computations for each iteration. Moreover, interference cancellation and soft decoding are dependent on the underlying channel model. However, SEN does not require any knowledge of the channel model since it is based on dedicated neural network which is trained for different channel scenarios.

Moreover, the inputs to SEN are y_l and Ik(c−1) . Here y_l and Ik(c−1) are the received signal at the l^th AP and interference coefficient of the k^th user during c^th iteration, respectively.

The last layer of SEN is the Softmax layer; it consists of neurons equal to the numbers of symbols in the constellation. For example, if Binary Phase shift Keying (BPSK) is employed, then the Softmax layer consists of 2 (B) neurons.

In this subsection, we discuss about the proposed parallel decoding scheme, DNN based Parallel Decoding in Radio Stripe (DNNBPDRS). Fig. 5 shows the K parallel SENs in each AP. As shown in Fig. 5, AP₁ receives y₁, which forms one of two inputs to SEN_1,1, SEN_2,1, ……….., SEN_K,1 (first subscript denotes the k^th user and the second subscript denotes the l^th AP) while the second input to SENs are interference coefficients I1(0) , I2(0) , …………, IK(0) and takes the value of 1B [16], which are given by

(11) I1(0)=[P^2(0),P^3(0),.........,P^K(0)]

(12) I2(0)=[P^1(0),P^3(0),.........,P^K(0)]

Considering SEN_1,1, the SEN belonging to UE 1 at AP₁, it receives y₁ and I1(0) to produce P^1(1) , the conditional distribution of UE 1. Similarly, SEN_2,1 and SEN_3,1, produces P^2(0)andP^3(0) respectively as shown in Fig. 5. Likewise, SEN_1,2 in AP₂ accepts inputs y₂ and I1(1) to produce P^1(2) , where I1(1) is given by

(13) I1(1)=[P^2(1),P^3(1),..........P^K(1)]

In a similar way, all SENs in all L APs compute respective conditional distribution. To sum the conditional distribution of L APs, L−1 numbers of User-Wise Conditional Distribution Combiner (UWCDC) is required.

Finally, the UWCDC performs the following operations,

(14) S^1t=P^1(1)+P^1(2)+……….+P^1(L)

(15) S^2t=P^2(1)+P^2(2)+……….+P^2(L)

and,

(16) S^kt=P^k(1)+P^k(2)+……….+P^k(L)

where, S^1t , S^2t and S^kt are the total added conditional distribution of 1^st, 2^nd and k^th user and has the dimension of 1 × B (B is the size of constellation). For instance, if BPSK is considered then B = 2.

For each user the CPU performs the operation given below

(17) S^1=maxcs∈{+1,−1}⁡(S^1t)

(18) S^2=maxcs∈{+1,−1}⁡(S^2t)

and,

(19) S^k=maxcs∈{+1,−1}⁡(S^kt)

Above max operations, choose the symbol from the constellation which has the maximum value. For example, if BPSK constellation is considered and S^1t=[0.15,0.85] , then S^1=0.85 and the decoded symbol is +1.

The total numbers of SENs present in radio stripe is given by N_SEN.

(20) NSEN=KL

The proposed DNN based Parallel Decoding in Radio Stripe (DNNBPDRS) is given in steps below.

This subsection discusses and compares the complexity of proposed DNNBPDRS, DNNBDSD [18] and iterative SIC.

For large-scale fading, we assume that all users are randomly distributed over a stadium of circular shape with a radius 10 m which corresponds to the total coverage area of 314 m². Radio stripe is installed across the circumference of the stadium at a height of 5 m from the ground.

The symbols are randomly derived from BPSK Constellation Set (CS) ∈ {0, 1}, which undergoes operation as given by Eq. (2). For each particular value of y_l at all the APs, SENs across all APs are trained in a parallel manner. The SENs across all APs are trained on 20 K samples. At l^th AP, the received signal is y₁, based on the initial estimate {P^k(0)}k=1K all K SENs in each AP are trained in a parallel fashion for all 20 K samples.

For a given transmitted symbol vector S, all SENs across all APs are trained in a parallel fashion. Since the proposed DNNBPDRS is trained via supervised learning, the label (actual conditional distribution) corresponding to the transmitted symbol vector is available to compute loss.

Since, SEN_1,1 has a softmax layer as the output layer, the output at each neuron of the softmax layer is given as

(48) exp(xi)∑j=1B⁡(xj)

where (xi) and (xj) are elements of P^1(1) .

Let ∅ represents weight and biases of SEN_1,1. Since, P1(1) consists of B elements, P1(1)=[O1,O2,……….,OB] . The loss function at the output of SEN_1,1 is defined as

(49) L(∅)=−∑i=1B⁡(P1(1))ilog⁡(P^1(1))i

where P1(1) and P^1(1) are the actual conditional distribution and estimated conditional distribution respectively. The parameter ∅ are updated using Adam optimizer and is defined as

(50) ∅f+1=∅f−ηmfvf+∈

η is Learning rate. ∈ is a smoothing term which prevents divisions by 0. mf and vf are the estimate of the mean (first moment) and uncentered variance (second moment) of the gradients and is given as,

(51) mf=β1mf−1+(1−β1)∇L(∅f)

(52) v=β2vf−1+(1−β2)∇[L(∅f)]2

β1 and β2 are the decay rates of the moving average. Similarly, all SENs across all APs are trained in a sequential manner.

In this subsection, we analyze the performance of the proposed DNNBPDRS algorithm and compare its performance with Cell-Free massive MIMO ‘Level 4’ implementing iterative SIC [11] under CSI uncertainty and our proposed sequential processing in radio stripe [18]. It is quite evident from Fig. 6, the best performance is achieved by the proposed DNNBPDRS algorithm.

Considering the numerical difference in Symbol Error Rate (SER) performance of the three frameworks are compared, the percentage difference of DNNBPDRS-iterative SIC and DNNBPDRS-DBDSUP are 99.05% and 32.72%, respectively for 2dB Signal-to-Noise Ratio (SNR).

For 8dB SNR, the percentage difference of DNNBPDRS-iterative SIC and DNNBPDRS-DBDSUP are 191.35% and 132.53%, respectively. For all SNRs the difference between DNNBPDRS-iterative SIC is much higher than DNNBPDRS-DBDSUP.

The better performance of DNNBPDRS then DBDSUP is attributed to the fact that in DNNBPDRS, each AP is trained for five iterations, while in DBDSUP each AP is trained for a single iteration. Moreover, in DBDSUP there are chances that errors in conditional distribution estimation may propagate from one AP to next AP, which is eliminated completely in DNNBPDRS.

Since, we consider radio stripe to be deployed across the walls of room/stadium as indicated in Fig. 2, it becomes important to determine the decoding performance of the proposed DNNBPDRS framework for different numbers of AP L, given the numbers of UEs present in the radio stripe network. Therefore, we analyze the performance of DNNBPDRS with different numbers of APs L for a given number of UEs K in the network.

Simulation is carried out for three different numbers of APs L using the simulation parameters given in Tab. 1. In particular, simulation is carried out for L = 5, L =10 and L = 20. Tab. 2 shows the percentage of difference in obtained SER.

As the number of AP L increases, the performance of DNNBPDRS improves. Tab. 2 shows the percentage difference in SER achieved for different combinations. It can be clearly noted for the second combination L = 5, L = 20 (the difference being 4 times), the percentage difference is highest.

Moreover, it can be clearly observed from Fig. 7, that the performance of the proposed DNNBPDRS improves as L > K inequality increases.

The above four figures (Figs. 8–11) compare the CDF of proposed Channel-link based AP selection (CLBAPS) with CDF of Random AP selection (RAPS) (here AP are selected randomly). Analysis is carried out for different numbers of AP L in the radio stripe network, keeping La constant. It can be clearly observed the gap between the proposed scheme and random selection increases as the numbers of AP in the network increases. In particular, from Fig. 8, the proposed CLBAPS algorithm gains 0.960 bit/s/Hz as compared to the RAPS with 50% probability. While with the proposed algorithm the SE of the UEs with 50% probability have 5.261 bit/s/Hz gain over random selection, when L = 200. The reason for these results is quite obvious, as the numbers of APs in the network increases there are more chances of UE being close to AP with excellent channel link. However, there is insignificant impact of increase in numbers of AP L for random AP selection as it gains 0.0383 bit/sec/Hz when increasing the APs in the network from 10 to 200.