Irregular Repetition Slotted ALOHA Over Rayleigh Block Fading Channels: Bounds and Threshold Saturation via Spatial Coupling

In irregular repetition slotted ALOHA (IR-SA) systems, a population of devices transmits their packets to an access point (AP) within a frame of slots. The AP decodes these packets by iterative processing between intra- and inter-slot successive interference cancellations. The average normalized offered traffic, as a performance metric, represents the number of packets transmitted per slot when the packet loss rate approaches zero. Such asymptotic types of traffic as the belief propagation (BP) threshold, the maximum a posteriori (MAP) threshold, and the converse bound of IR-SA systems have been analyzed over various channel models. However, over fading channels, the MAP threshold and the converse bound have not yet been investigated. This paper derives an MAP threshold and a converse bound of the systems over Rayleigh block fading channels. The derivations are based on two extrinsic information transfer (EXIT) curves, which are associated with two iterative density evolution equations to analyze the BP threshold of the IR-SA systems. First, since an open decoding tunnel exists in an EXIT chart, the sum of the two areas below two EXIT curves is smaller than the area of the entire domain. This provides the traffic’s converse bound, which is tight. Second, a coincidence of the BP EXIT and MAP EXIT curves makes it possible to derive the traffic’s MAP threshold. Third, a density evolution for a spatially-coupled scheme is formulated and gives a BP decoding threshold of the traffic. Numerical results show that the spatially-coupled scheme achieves a threshold saturation effect where the BP threshold approaches the MAP threshold.


I. INTRODUCTION
Massive Machine Type Communications (mMTC), which is a potential application of the sixth generation (6G) of wireless technology, is anticipated to provide wireless connectivity to a large number of devices [1].Random access schemes [2], which evolved from slotted ALOHAs (SAs) [3], are regarded as a class of promising solutions for mMTC since such devices transmit their data without pre-establishing connections and pre-requesting channel resources.Among these The associate editor coordinating the review of this manuscript and approving it for publication was Wei Wei .random access schemes, irregular repetition slotted ALOHA (IR-SA) [4] achieves a theoretical bound of the slotted ALOHA-type schemes over the collision channel model [5].In this paper, we further investigate IR-SAs and derive their theoretical bounds over Rayleigh block fading channels.The bounds are approached by a practical, iterative decoding process.

A. PROBLEM DESCRIPTION AND OUR OBJECTIVE
In IR-SA systems [4], [6], [7], [8], [9], [10], [11], N T devices transmit their packets to an access point (AP) within a frame of M time slots.Each device generates a single packet with activation probability π ≪ 1 and transmits its replica a variable number of times (irregular repetition) to several slots.Repetition times l of an active device is drawn from probability distribution { l }.
In the receiver, the AP recovers the packets obtained in the frame.Within a slot, the AP may successfully decode some replicas, for example, with a fading channel model, if their signal-to-interference-plus-noise ratio (SINR) exceeds a certain threshold, and the decoded replicas are removed by intra-slot successive interference cancellation (SIC).When a replica has been decoded successfully, all other replicas from the same device are removed from their accessed slots by inter-slot SICs.AP repeats these intra-and inter-slot SICs until no packets can be decoded.
IR-SA systems have a special case of d = 1, referred as to d-regular repetition slotted ALOHA (d-RR-SA) systems.Moreover, IR-SA systems, where each device repeatedly transmits its replicas, are the repetition code case of a generic linear block code in coded slotted ALOHA (C-SA) systems [12], [13], [14], [15], [16], [17], [18].In this paper, we focus on IR-RA systems and evaluate their asymptotic performance.
The average normalized offered traffic, denoted by G = πN T /M , is one performance metric commonly adopted in IR-SA systems.Traffic G represents the average number of packets transmitted per slot when the packet loss rate (PLR) of the systems approaches 0 or lower than a target value for fading channels.Offered traffic G is equivalent to a throughout.Let N T /M be constant.We are interested in the asymptotic (M → ∞) decoding performance of G, which depends on probability distribution { l }, which must be optimized.
Threshold value G th exits such that when G < G th , all the transmitted packets are successfully decoded.Conversely, if G > G th , then a fraction of the devices' packets will undoubtedly fail to be successfully delivered [12].
Resembling the decoding analysis of low-density paritycheck (LDPC) code over a binary erasure channel (BEC) [19], for IR-SA systems, such analysis of the following is interesting: asymptotic decoding thresholds G th as a belief propagation (BP) threshold, a maximum a posteriori (MAP) threshold, and a converse bound.Density evolution [4] describes the iterative procedure between the erasure probabilities (or beliefs) of devices and slots during iterative intra-and inter-slot decoding.The threshold, obtained by the density evolution, is referred to as BP threshold G BP [4] that can be achieved in practice.Packetwise (bit-wise in an LDPC code context) MAP decoding maximizes the posterior probability of each packet and gives MAP threshold G MAP [5].The MAP threshold, which is the highest performance when not addressing the complexity of decoding, is the upper limit of practical decoding, such as the BP threshold.Converse bound G C [5], [13], [17] expresses an impossibility threshold to be approached and is tight if the MAP or BP thresholds are close to the bound.Although it is comparatively simple to get a converse bound, calculating a MAP threshold is usually difficult.If the converse bound is tight, it can be used as an upper bound for the BP threshold, especially when the MAP threshold is unknown.
Over Rayleigh fading channels, the BP threshold of IR-SA systems was investigated [6].In this paper, our objective is to derive the MAP threshold and the converse bound of the systems over Rayleigh fading channels.We show that the BP threshold of a spatially-coupled scheme approaches the MAP threshold.

B. RELATED WORKS
The extensively studied traditional SA [3] has been implemented in various commercial communication systems and standards [2].Nevertheless, its maximum offered traffic is only 1/e.The offered traffic was improved by the diversity of transmission, which transmits a packet multiple times [20].Noteworthy advances have been achieved by combining transmission diversity and SIC at the AP [4], [12], [21].Such a combination enables us to constructively exploit collisions by canceling interference due to correctly received packets to allow the possible recovery of other initially collided packets.
The collision channel model without erasures [4], [5], [7], [13], [21] is a simple and popular channel model that mainly analyzes the asymptotic decoding performance of IR-SA systems.The collision model assumes that a) noise and fading can be neglected, such that a transmission can be decoded from a singleton slot by default, and b) no transmission can be decoded from a collision slot.With the collision model, Casini, Gaudenzi, and Herrero proposed a 2-RR-SA system called the contention diversity slotted ALOHA (CRDSA) scheme [21], where an active device transmits twin replicas of each packet within a frame, and the AP adopts an inter-slot SIC for resolving collisions.It promotes maximum offered traffic G to about 0.55.Liva identified the key analogies of inter-slot SIC with iterative erasure recovery in channel coding theory [19] and extended d-RR-SA to an irregular approach [4].The inter-slot SIC process can be well modeled by a bipartite graph [4] and resembles the density evolution of LDPC codes over BEC [19].By exploiting design techniques for the LDPC code context, device node degree distribution { l } is optimized by differential evolution [22], and the asymptotic BP decoding threshold is given by the density evolution.When the order of the distribution polynomial is asymptotic, the optimal distribution is the well-known soliton distribution and resulting traffic G can be arbitrarily close to 1 [7].
Over the collision channel model, the converse bound and the MAP decoding threshold of IR-SA systems were given [5], [13].The former was established by observing that the sum of the two areas below the two extrinsic information transfer (EXIT) curves is less 1 [13].The converse bound is tight in the sense that the BP thresholds are close to it.The MAP threshold [5] was derived by the area theorem in the context of the LDPC code [23].However, we observe that the value of the MAP threshold is approximate when normalized population size α = N T /M is large enough.A spatially-coupled scheme was proposed, and the corresponding schemes are referred to as spatially-coupled slotted ALOHA (SC-SA) systems.The BP threshold under iterative inter-slot SIC processing with the spatially-coupled scheme saturates towards the MAP threshold [5].
The collision channel model considers only collisions for transmitted packets or replicas.A generalization of the collision channel model is represented by multi packet reception (MPR) channels [9], [15].In a basic K -MPR channel model, an AP is assumed to be able to successfully decode all the packets in the slots where no more K colliding replicas are present and to extract no information from slots where over K packets are interfering with each other.Note that the K -MPR channel model with K = 1 is reduced to the collision channel model.Over the K -MPR channel model, a BP threshold analysis of the IR-SA systems was conducted [9].Subsequently in the C-SA systems, the converse bound was established, and the numerical results in K = 1, 2, 3 show that the BP threshold of the spatially-coupled scheme saturates towards the bound [15].
Coding for the noiseless binary adder channel model [8], [11] gives a practical implementation of the intra-slot SIC for IR-SA systems over the K -MPR channels.For the binary adder channel, the channel's output is the sum (being over the reals) of the multiple binary inputs [24], [25], [26].In such systems, a codebook is common and shared with all the devices.Each message is encoded, and the corresponding codeword of the codebook is transmitted as a packet or its replicas.Within a slot, the AP can resolve the collisions of up to K packet replicas.In a regime of practical interests, a codebook is constructed as a parity-check matrix of BCH code [8], [24].Currently, a random codebook [11] enhances the MPR capability by resolving more colliding packets.
The basic K -MPR channel model assumes that an intra-SIC can always be applied perfectly.The probability that a packet cannot be correctly subtracted in a slot due to an imperfect intra-SIC was taken into account in the IR-SA [27].The optimal device node distribution that maximizes a BP decoding threshold was derived by density evolution with an imperfect intra-SIC.Moreover, IR-SA's random nature results in a large dynamic range of received power that cannot be recovered under a practical quantizer.An SIC limit exists, i.e., a limit on the maximum number of packets that can be recovered in each slot, and was resolved in IR-SA analysis [28].
Erasure channel models, including both packet and slot erasure channels, were considered in the design of C-SA systems [16].When a transmitted packet or replica experiences deep fading, in the packet erasure channel model, the packet is assumed to be erased and the corresponding erasure probability is ϵ PE = Pr(|h| < η 0 ), where h is the fading coefficient and η 0 is a given threshold.The slot erasure channel model assumes that all the packets transmitted are erased with slot erasure probability ϵ SE in the slot within which a stronger external interference may overwhelm all the transmitted packets.Over the two erasure channels, the device node degree distribution is optimized by linear programming to maximize their corresponding BP thresholds.Moreover, converse bounds for the erasure channels were derived [29].In the erasure channel models, however, erasure probabilities ϵ PE and ϵ SE are constant and no practical fading channels are addressed.
The packet erasure channel is also called an on-off fading channel [29], [30].A K -MPR channel model with on-off fading [17] was introduced to perform contention within a slot in C-SA systems.Each transmitted replica is erased, i.e., faded by the channel at a certain probability.If the number of non-erased replicas is smaller than or equal to K in a slot, then all are perfectly decoded; otherwise, no replicas can be recovered from it.The BP threshold and the converse bound are derived, and the numerical results show that the former with a spatially-coupled scheme is close to the bound [17].
Although the analysis on collision channels [4], [5], [13] and MPR channels [9], [15] is helpful for basic insights into IR-SA and C-SA systems, unfortunately, it neglects the impact of fading and noise in wireless transmission.Although erasure channels [16], [17] consider fading to a certain extent, they fail to describe adequately its impact in wireless scenarios.In addition, a power-domain non-orthogonal multiple access (NOMA) was recently incorporated into both IR-SA [10], [31] and C-SA [18] systems.By setting the target level of the received power, the power loss due to channel fading is compensated.In these systems, however, channel state information (CSI) is assumed to be perfectly known by the devices.The path loss (correlated to distance) channel model that neglects the fading and shadowing effect was considered for IR-SA systems [32], where devices are uniformly distributed on a disk of radius 1. Error probability εm that decodes the packet replica from the m-collision signal is calculated by Monte-Carlo estimations and combined with density evolution to evaluate system performance.
In wireless scenarios, the impact of fading must be addressed.In the context of slotted ALOHA, the MPR capability is enhanced by incorporating intra-SIC processing over fading channels [33], since fading causes power variations among signals observed in collision slots, and sufficiently strong signals may be decoded.
The Rayleigh block fading channel is one standardly used model for evaluating the performance of wireless systems [34].Over such a channel, decoding is considered where m packets or replicas are received within a slot.A reference packet with SINR η is successfully decoded if η > η 0 , where η 0 is a decoding threshold according to Shannon's theorem.The intra-SIC further enhances the decoding capacity within the slot.Clazzer et al. [6] derived an exact expression of average error probability with threshold decoding and intra-SIC.The average error probability, denoted by εm , for decoding a randomly chosen packet replica under m − 1 interferences is expressed as a function of decoding threshold η 0 and the average received SNR of the signal from each device (Appendix A).With average error probability εm , the density evolution for IR-SA systems gives a BP threshold with an optimized device node degree distribution { l } [6].Moreover, an IR-SA system was analyzed that uses multiple antennas at the AP over Rayleigh block fading channels [35].When a perfect CSI is available at the AP, average error probability εm is approximately expressed in a closed form.The BP threshold obtained by density evolution with εm provides an upper bound on the thresholds with an estimated CSI.
For IR-SA systems, this paper focuses on the maximization of the offered traffic.The minimization of the age of information over collision channels can be found [2], [36], [37], and the maximization of device energy efficiency over 1-and 2-MPR channels has been studied [38], [39].Interested readers are referred to these works and references cited therein for more detailed description.

C. OUR CONTRIBUTIONS
In this paper, we perform an asymptotic analysis of IR-SA systems with the Rayleigh fading channel model and make the following contributions: 1) We establish a converse bound of average normalized offered traffic G C .An open decoding tunnel exists in the EXIT chart, and the sum of the two areas below the two EXIT curves is smaller than the entire domain's area.This situation helps us derive the converse bound.The bound is tight in the sense that both the MAP decoding and the explicit spatially-coupled schemes approach closer to the bound.2) We derive a MAP decoding threshold of the average normalized offered traffic.The coincidence of the BP and MAP EXIT curves makes it possible to derive a MAP threshold, which is exact for any value of normalized population size α and serves as a guideline for practical access schemes.3) We present a density evolution algorithm for an explicit spatially-coupled scheme and derive its BP decoding threshold.We observe from the numerical results that the spatially-coupled scheme achieves a threshold saturation effect through which the BP decoding threshold approaches the MAP threshold.
The rest of our paper is organized as follows.Section II reviews IR-SA systems, including a preliminary graph-based analysis and a density evolution.We derive the converse bound over the Rayleigh block fading channels in Section III-A.Next, following a definition of a BP EXIT function in parametric form, we formulate an analysis of the MAP threshold in Section III-B.A spatially-coupled scheme over fading channels is presented and its BP decoding threshold is given in Section IV.Numerical and simulation results are shown in Section V, and finally Section VI concludes the paper.

II. IRREGULAR REPETITION SLOTTED ALOHA: PRELIMINARIES
This section reviews IR-SA systems and their preliminaries.The density evolution, which plays a key role in deriving the converse bound and the MAP threshold in Sections III-A and III-B, will be extended to a spatially-coupled scheme in Section IV.

A. TRANSMISSION AND GRAPH REPRESENTATION [4]
We consider graph-based IR-SA systems [4], [5] where a total of N T devices want to transmit their packets to an AP.A medium access control frame consisting of M time slots is assumed to be a contention duration, where the length of each slot corresponds to a packet's duration.At the beginning of a frame, each device generates with activation probability π ≪ 1 a message, which is encoded into a codeword, called a packet, with common channel code C ch at rate R ch .The l replicas of the packet with probability l , selected from probability distribution { l } l max 1 , are transmitted to l slots within the frame, where each replica embeds pointers to slots containing the other replicas.The average number of replicas per device is d = l max l=1 l l , and its inverse ζ = 1/ d is a measure of the energy efficiency of IR-SA systems [8], [17].Note that in d-RR-SA systems, d = 1.
Devices attempting transmission within a frame are described as active, and those that are idle within a frame are called inactive.Let N a be a random variable representing the number of active devices whose expectation is E(N a ) = πN T .Let α = N T /M be the normalized population size, which is the ratio of the total number of devices to the frame size.The average normalized offered traffic is defined as G = πN T /M = π α, which represents the average number of packet transmissions per slot.
The above transmission frame status is also described as bipartite graph G = (D, S, E), where D is a set of N T device nodes (DNs), S is a set of M slot nodes (SNs), and E is a set of edges.An edge connects DN d i ∈ D to SN s j ∈ S if and only if the j-th slot was selected by the i-th device at the beginning of the frame.In addition, a residual graph (denoted by G a ) is one by removing the DNs associated with inactive devices and their adjacent edges from G [5], [17].In this paper, we use the graph representation of G by treating activation probability π as the erasure probability of BEC in the LDPC context.

B. DEGREE DISTRIBUTIONS 1) DEGREE DISTRIBUTIONS: BASIC [40]
We briefly review the node-perspective degree distributions of graph G and represent them by the following polynomials [40]: where l ( l ) is the probability that a DN (SN) has l edges connected to SNs (DNs).A node (DN or SN) is degree-l if l edges are connected to it.The following are the average numbers of edges per DN and per SN: where ′ (x) ( ′ (x)) is the first-order derivative of function (x) ( (x)).Moreover, the number of edges connected to DNs is identical to that of SNs: Degree distributions are also defined from an edge perspective.The polynomials are [40] where λ l (ρ l ) is the fraction of edges that are connected to a degree-l DN (SN).By definition, edge-perspective degree distributions can be represented by node-perspective distributions [40]: 2) DEGREE DISTRIBUTIONS: GRAPH-BASED IR-SAS [4] In graph-based IR-SA systems, DN degree distribution (x) can be optimized.This section describes the derivation of SN degree distribution (x) for a given (x).Since ′ (1) is the average number of packet collisions per slot, the probability that a device transmits a replica of its packet within a given slot is N T .Thus, the probability that an SN possesses l edges is [4] The corresponding pronominal is When N T → ∞, the number of edges l follows a Poisson distribution.With e −x = lim n→∞ (1 where d ≜ ′ (1) and α = N T /M .From ( 6), it follows that with Poisson distribution C. FADING CHANNELS AND DECODING PROCESSING 1) DECODING WITHIN A SLOT: INTRA-SLOT SICS Consider a generic slot in a frame, e.g., the j-th slot, at some point during decoding.For compact notation, we ignore index j in this section.Assuming m packet collisions, the superimposed signal in the slot is Here x i ∈ C ch is a codeword (or packet) of channel code C ch with rate R ch and length n, transmitted by device i, assuming it accessed the slot.Set R is the collection of devices with packets or replicas that access the slot at this point and |R| = m.Vector z ∼ CN (0, σ 2 I ) is additive, circularly symmetric complex Gaussian (CSCG) noise with mean 0 and diagonal covariance matrix σ 2 I with noise power σ 2 and identity matrix I .Coefficient h i ∼ CN (0, σ 2 h = 1) is channel fading for device i to the AP at the slot, and the channel coefficient remains constant for a slot (i.e., a block of n consecutive coded symbols) and changes to another slot.All devices are subject to the same power constraint ∥x i ∥ 2 /n ≤ P, and the received SNR of the signal from device i For the Rayleigh distribution of |h i |, the SNR of i is exponentially distributed [34]: where h is the average received SNR of the signal from each device.
We next consider the decoding in the slot of a reference codeword, e.g., x i 0 .The SINR is Shannon's theorem [41] shows that x i 0 is successfully decoded if the SINR exceeds a certain SNR threshold η 0 .
Error probability ϵ is where the SNR threshold satisfies R ch = log 2 (1+η 0 ) for code C ch of rate R ch [41].Note that in our decoding, we assume that the AP has perfect channel state information and has a number of packet collisions.When codeword x i 0 is successfully decoded, h i 0 x i 0 is removed from y.This processing, referred to as intra-slot SIC, 106532 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
is repeated until no codewords exist whose SINRs exceed threshold η 0 .For intra-slot SICs, the average error probability of decoding a randomly chosen packet replica from the mcollision signal of ( 12) is exactly expressed in a closed-form (see Appendix A) [6].

2) DECODING BETWEEN SLOTS: INTER-SLOT SICS
When some packets (codewords) are successfully decoded within a slot, their replicas are removed from those slots indicated by those packets.This processing is referred to as an inter-slot SIC.Note that we assume that the AP possesses perfect channel state information, which is required to remove these replicas.
The intra-and inter-slot SIC processings can be described as a successive removal of edges in bipartite graphs G and G a .When a packet from a device (or a DN) is successfully decoded within a slot (or an SN), the corresponding edge connected to the SN is removed due to the intra-slot SIC, and all the l edges connected to the same DN are removed due to the inter-slot SIC.

D. ASYMPTOTIC ANALYSIS: DENSITY EVOLUTION
We now review the density evolution algorithm [6] with constant α = N T /M to evaluate the asymptotic (M → ∞) performance of graph-based IR-SA systems over Rayleigh block fading channels.
As N T and M approach infinity, bipartite graph G becomes a protograph in the form of probability [4], [40].Now we review the density evolution algorithm [6] from the protograph.
We refer to the DNs (SNs) as the DN (SN) group.For the ℓ-th iteration, let ℓ p be the probability that an edge incident on the DN group carries an erasure message towards SNs.Similarly, let ℓ q be the probability that an edge incident on the SN group carries an erasure message towards the DN groups.
First, consider the DN group, which has degree distribution λ(x).For an DN with degree-l, the probability that an edge emanating from it carries an erasure message towards the SN group is π • (ℓ−1) q l−1 , where (ℓ−1) q l−1 is the probability that the other l − 1 incoming messages to the DN are erased, and π is the activation probability of the devices.Averaging over all the edges in the protograph gives the average probability that a DN-to-SN message is erased [4], [5]: Second, consider the SN group, which has degree distribution ρ(x).The probability that an edge carries an erasure message from the SN group to the BN group in the ℓ-th iteration is [4], [6] Here ρ k is the fraction of edges connected to an SN of degreek (11), and ℓ q (k) is the probability that an edge carries an erasure message, given that it is connected to the SN of degree-k.The erasure probability of the degree-k SN is [6] ℓ q where the summation is over all possible values 1 ≤ m ≤ k.
Next we explain the terms in the summation: 1) Term k−1 m−1 (1 − ℓ p) k−m ( ℓ p) m−1 corresponds to the probability that the degree of the SN is reduced to m.In other words, among k − 1 edges, m − 1 edges are unknown, and the remaining k − m edges have been revealed due to inter-slot SICs.
2) (1 − εm ) is the probability that the SN corresponding to the outgoing edge is decoded successfully when the reduced degree of the SN is m.In other words, within a slot of k packets, after removing k − m packets with the help of inter-slot SICs, we decode a randomly chosen packet replica among the remaining m packets.The error probability that decodes the packet replica from the m-collision signal is (Appendix A) [6]: Inserting ( 11) into (17), we have where we used in (b).We rewrite the second term of the summation: p , where we used (11) in (c) and the Maclaurin series of exponential function e x = ∞ i=0 x i i! in (d).Therefore, we have [6] From (10), we observe that in (20), term α dx = − ln ρ(1−x), which means that ℓ q can be rewritten as a function of (1− ℓ p): where Finally, substituting for ( 16) we have [6] ℓ q = 1− e −G dλ( (ℓ−1) q) ∞ m=0 where G = π α is the average normalized offered traffic of the systems.Let (∞) q(G, d, ¯ , η 0 ) = lim ℓ (ℓ) q be the convergence value of the iteration.The PLR of the systems is [4], [6] The asymptotic iterative decoding threshold, also called the BP decoding threshold, is defined as the supremum offered traffic value [4], [6]: Remark 1: Our derivation of density evolution here is slightly different from a previous work [6], where all the N devices were active at the beginning of a frame, i.e., π = 1, and the frame status was described as residual graph G a .In our work, we consider activation probability π and use graph G.The activation probability is analogous to the erasure probability of the BEC in LDPC decoding, like another previous work [5].If the expectation of the number of devices, E(N a ) = πN T , is seen as N , the iteration equation of (23), where G = N /M , is identical as in the previous work [6].
2 Remark 2 (κ-MaxDecoding): In ( 18), the intra-slot SICs are carried out for every possible number of collisions, i.e., 1 ≤ m ≤ k, where the number of occurrences k in the Poisson distribution of (11) tends to infinity.For a practical reason, we considered a κ-MaxDecoding where the intra-slot SICs are carried out only when the number of collisions is equal to or less than given constant κ (1 ≤ κ ≤ k).Under κ-MaxDecoding, the range of the summation in ( 22) Consider the special case of κ = 1, where no intra-slot SIC is performed in the AP.Probability ℓ q j is reduced to For the additive white Gaussian noise (AWGN) channel model, εAWGN For the collision channel model [4], it follows that εColi 1 = 0. 2

III. CONVERSE BOUND AND MAP THRESHOLD
We describe a converse bound and a MAP threshold of the IR-SA systems in Sections III-A and III-B.

A. CONVERSE BOUND OVER RAYLEIGH FADING
We are now ready to describe an asymptotic threshold upper bound on average offered traffic G C of graph-based IR-SA systems.The upper bound is referred to as a converse bound in information theory expressing an impossibility value to be approached.
By omitting the iterative index for simplification, we rewrite the iteration equations ( 16) and ( 21) of the density evolution: Note that the initial value of p is activation probability π.
An example of an EXIT chart is shown in Fig. 1.Function f d (q) (f s (p)) is also called the average EXIT function of DNs (SNs) [19], [40].Let the areas below the DN (SN) EXIT functions over interval [0, 1] ([0, π]) be A necessary condition for successful decoding is an open tunnel between the two curves in the EXIT chart [19] [23], which displays f d (q) and f −1 s (q).Such area matching means that (Fig. 1) The area below the DN EXIT function is On the other hand, the area below the SN EXIT function is 106534 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where is expressed in a closed form.In (f) we used Gaussian integral [42, Eq. (2.321.1)]: x n e ax dx = e ax n k=0 Combining ( 29) and ( 30), we have With G = π α, the solution of equation πα = α dτ (π), i.e., gives the converse bound of asymptotic threshold G C,blk , which is a function of the average number of transmissions d and is independent on a concrete DN distribution.Remark 3: We rewrite (32) as When we use κ-MaxDecoding and set ε1 = • • • = εκ = 0, the solution of (33) gives the converse bound of κ-MPR [15, Eq. ( 17)]. 2

B. MAP THRESHOLD OVER RAYLEIGH FADING
We derive threshold G MAP,blk under MAP decoding for graph-based IR-SA systems over Rayleigh block fading channels.MAP decoding resembles that of LDPC code over BEC where activation probability π is analogous to the channel erasure probability in the LDPC context [5].
Recall the equations of density evolution (26).Erasure probability p emitted by the DNs tends to a limit value, called x, which is a fixed point of equations Solving this fixed point equation for π ∈ (0, 1], we get where π(x) = 1 and 0 < x < 1.In the LDPC context [19,Theorem 3.59], π(x = 1) = 1, since every check node always has two more edges.Thus the constant term in the polynomial of the node-perspective distribution of the check nodes is ρ l=1 = 0.It follows that ρ(0) = 0.In IR-SA systems, however, ρ l=1 ̸ = 0 (11), and thus ρ(0) ̸ = 0, and χ(0) ̸ = 0 (22).At the fixed point, if the erasure probability emitted from the DN is x, then the PLR of the systems equals (1 − χ(1 − x)).Similar to MAP decoding in the LDPC context [19,Lemma 3.116], we define BP EXIT function h BP (π ) in parametric form by the following curve (see an example in Fig. 2): The BP EXIT curve is a trace of this parameter equation for x starting at x = x until x = x BP .Indeed, function π(x) has a unique minimum, which determines threshold π BP = π(x BP ) [19,Lemma 3.116] [43].
The integral under the curve (π(x), (1 − χ(1 − x))) is called a trial entropy associated with (λ, χ) [19, Definition 3.119]: where we used ( 6), (35), and τ (x) in a closed form (31).Note that MAP threshold π MAP is the solution of P(x) = 0 and defines π MAP = π(x MAP ), and the BP and MAP EXIT curves coincide above π MAP TABLE 1. MAP thresholds of 3-RR-SA with collision channel model and approximate values [5].[19,Theorem 3.120].Therefore, given λ(x) and χ(x), we find the MAP threshold by computing (Fig. 2) As a result, the MAP threshold of graph-based IR-SA systems can be upper bounded: Especially for graph-based d-RR-SA systems where λ(x) = x d−1 , the trail entropy becomes: Using P reg (x = x) above, we obtain the MAP threshold of graph-based d-RR-SA systems, G MAP,blk reg .

Remark 4:
With the collision channel model, the value of the MAP threshold of the d-RR-SA was obtained from [5]: We claim that the value of threshold GMAP,blk reg = α πMAP is unfortunately approximate for a large α.
In an LDPC context, the MAP threshold over BEC is computed with (40), where 1 − 1 α is the nominal rate of the LDPC code [19, Theorem 3.120] [5].Note that this is true under the assumption that ρ(x = 0) = 0.

IV. SPATIALLY-COUPLED SLOTTED ALOHA SYSTEMS
In this section, we investigate a spatially-coupled scheme using a convolutional-oriented super-frame [5], [15], [17].We refer to slotted ALOHA systems with such a scheme as SC-SA systems and formulate a density evolution to give a BP threshold of them over Rayleigh block fading channels.

A. TRANSMISSION PROTOCOL
We consider a spatially-coupled scheme and summarize it as follows [5] (Fig. 3).In the systems, N T devices with activation probability π ≪ 1 want to transmit their messages to a single AP.A super-frame is divided in to M f = L + d − 1 frames, each of which consists of M slots.At the beginning of the i-th frame (i = 0, 1, . . ., L − 1), among N T devices, N i devices become active with probability π, and each active device transmits a single packet in a slot picked uniformly at random within the i-th frame.Furthermore, d − 1 replicas of the packet are sent in the following (i + 1)-to (i + d − 1)-th frames, and each replica selects a slot uniformly at random within the corresponding frame.
In the SC-SA, using the same number of repetitions for each device transmission prevents uneven energy consumption among sensor devices, whereas in the IR-SA [6], sensor devices with a higher number of repetitions may experience faster battery depletion.
The expectation of the active devices in each frame is E(N i ) = πN T .The normalized average offered traffic of a super-frame becomes where G = πN T M = π α.

B. GRAPH REPRESENTATION
The transmission process outlined in Section IV-A can also be described as a bipartite graph, an example of which is shown in Fig. 3.We referred to the DNs in the i-th frame as the i-th DN group and the SNs in the j-th frame as the j-th SN group.Since each DN transmits its single packet and d − 1 replicas, the probability that a DN group has d edges connected to BN groups is 1, i.e., d = 1 for each DN group.Moreover, all the edges connected to the j-th SN group are from δ j (≤ d) DN groups, where Similar to (9), when N T → ∞, the number of edges l connected to the j-th SN group follows a Poisson distribution.Therefore, the node-perspective distribution polynomials are [5] ( Similarly, the edge-perspective distribution polynomials are [5] with Poisson distribution (see (11)):   In SC-SA systems, the channels are Rayleigh distributed and the intra-and inter-slot SICs are employed as described in Section II-C.

C. DENSITY EVOLUTION
We next propose a density evolution algorithm to evaluate the asymptotic (M → ∞ with constant α = N T /M ) performance of the SC-SA systems.
As N T and M approach infinity, the bipartite graph (Fig. 3) becomes a protograph in the form of probability (Fig. 4).
DNs are neighbors of an SN when they are connected to one, and vice versa.The set of DN groups neighboring the j-th SN group is N j = {j − δ j + 1, j − δ j + 2, . . ., j}.Analogously, the set of SN groups neighboring the i-th DN group is M i = {i, i + 1, . . ., i + d − 1}.The cardinalities of the sets are We describe the density evolution algorithm as follows.For the ℓ-th iteration, let ℓ p i be the probability that an edge incident on the i-th DN group carries an erasure message towards its neighboring SN groups.Similarly, let ℓ q j be the probability that an incident edge on the j-th SN group carries an erasure message towards its neighboring DN groups.
First, consider the i-th DN group, which has degree d for all i (Fig. 5).Let ℓ p i j be the probability that an edge emanating from the i-th DN group carries an erasure message towards the j-th SN group with j ∈ M i .
Based on message-passing decoding over BEC [40], an edge is revealed whenever at least one of the other d − 1 edges has been uncovered: Second, consider the j-th SN group (Fig. 5).The probability that an edge carries an erasure message from the j-th SN group to the BN groups in the ℓ-th iteration is Here ρ k is the fraction of the edges connected to an SN of degree-k in the j-th SN group (46), and ℓ q (k) j is the probability that an edge carries an erasure message, given that it is connected to the degree-k SN in the SN group.Similar to (18), the erasure probability of the degree-k SN is where ℓ p j is the average erasure probability towards the j-th SN group from its neighbors: end for 8: return { ℓ q j } 9: end procedure Inserting (46) into (48), similar to (20) we have In (51), from ( 47) and (50), term δ j α ℓ p j can be rewritten: where G † is the normalized average offered traffic of the super-frame (42).
From initial value (ℓ=0) q j = 1 for each SN group, by the iteration shown in Algorithm 1, we obtain convergence values which is the PLR of the j-th SN group.
Similar to (47), the following is the probability that a packet is lost for the active devices in the i-th DN group: The average packet loss probability of all the DN groups is The asymptotic iterative (BP) decoding threshold of the SC-SA systems, denoted by G † BP,Conv , is defined as the supremum offered traffic value:

V. NUMERICAL RESULTS AND SIMULATIONS
In this section, we provide the numerical results of the converse bounds and the MAP and BP thresholds proposed in Sections III-A, III-B, and IV-C.We also provide the simulation results to verify the effectiveness of these thresholds.Although PLR approaches 0 in our definition of BP decoding thresholds in (25) and (54), in our numerical analysis and simulations, the target PLR is set to 10 −2 due to channel fading.As mentioned in Remark 2, we conduct κ-MaxDecoding where the summation range was truncated to 0 ≤ m ≤ κ − 1 in ( 22) and (51).Throughout this section, without specific declarations, the specifications in our numerical analysis and simulations are shown in Table 2.In our simulations, we set the number of fading samples to H = 10000, where one fading sample denotes the set of fading coefficients with which all the active devices transmit their packets or replicas within one superframe.
In Fig. 6, we provide the numerical results of converse bounds G C,blk obtained from (32) at various values of parameter κ and MAP thresholds G MAP,blk reg of the d-RR-SA systems obtained from (37) at d = 2, 3, . . ., 6.The regular MAP thresholds are close to their corresponding converse bounds, a result that can also be confirmed from the numerical values at both κ = 15 and κ = 21 (Table 3).Indeed, our converse bound is tight.This result can be attributed to the   fact that the two EXIT functions (Fig. 1) match well and the open tunnel between two curves is very narrow.3).This means that κ = 15 adequately gives a convergent result of these thresholds.In the remaining part of this section, we set κ = 15 without specific declarations.
In Fig. 7, we give the average PLR of the SC-SA obtained from (53) at d = 3 using density evolution.The average normalized offered traffic G † at PLR = 10 −2 , i.e., BP threshold G † BP,Conv = 3.2906, approaches MAP thresholds G MAP,blk reg = 3.2909 and G MAP,blk = 3.2932 and is near converse bound G C,blk = 3.2934 (Table 3).For comparison, Fig. 7 also shows the average PLRs of the 3-RR-SA and the optimal IR-SA obtained from (24) [6].The optimized distributions of IR-SAs are listed in Table 4, which we optimize by differential evolution [22], similar to  a previous work [6].At PLR = 10 −2 there is a large gap between the BP thresholds of the SC-SA and the optimal IR-SA, perhaps due to the restriction of the maximum order of polynomial (x) to 10 during the optimization of the distribution (Table 4).Compared with the optimal IR-SA [6], the SC-SA has higher BP threshold and uniform energy consumption at a cost of multiple frames.This makes it attractive in practical applications.called the threshold saturation effect in the LDPC code context [44].In other word, practical iterative decoding provides an optimal decoding performance for the SC-SA in the sense that the BP threshold approaches the theoretical upper bound.
In Fig. 9, at κ = 15 we show various thresholds for comparison, where the device distributions of the IR-SAs are given in Table 4.The numerical values of these thresholds are also illustrated in Table 3, supplied with κ = 21.We conclude that over the Rayleigh block fading channels, the BP threshold of the spatially-coupled scheme achieves a regular MAP threshold and is close to the tight converse bound.b) (K = 2)-MPR (i.e., ε1 = ε2 = 0)) [15] and (K = 1)-MPR (ε 1 = 0, i.e., the collision channel) [5], c) 2-MPR with on-off fading [17], and 1-MPR with on-off fading (i.e., packet erasure channel) [29].The converse bounds with the collision model, i.e., 1-MPR, were given [5] and extended to MPR [15] and MPR with on-off fading [17], [29].In Fig. 10, we present our converse bounds in comparison to prior results.The bound with K -MPR is larger than those of the other converse bounds and can be seen as an upper bound, since the K -MPR channel model ignores impact of channel fading and noise.On the other hand, there is a cross between the two bounds: K -MPR with on-off fading and our κ-MaxDecoding.The constant packet erasure probability ϵ PE in K -MPR with on-off fading seems an oversimplification of the effect of channel fading.Our work on fading is more practical and includes K -MPR as a special case of εm = 0 for m = 1, 2, . . ., K .
Figure 11 gives the MAP thresholds of the IR-SAs, the BP thresholds of the SC-SAs, and the converse bounds of IR-SAs with various channel models.In the past, the MAP threshold was derived only for the collision channel (i.e., 1-MPR) and was approached by its BP threshold [5].In our Rayleigh fading, this threshold saturation effect was also confirmed by observing that the BP threshold approaches the MAP threshold with 2-MaxDecoding.Furthermore, this effect was also observed in 2-MPR.Note that the MAP threshold in (37) can be determined for 2-MPR by setting ε1 and ε2 to zero.
Finally, we provide the simulation results of the SC-SA systems to verify the effectiveness of the BP thresholds.Figures 12 and 13 show the PLRs and the normalized throughputs of SC-SA with α = 100 and M = 200, 500, 100, and 10000 at κ = 2 and κ = 3.As expected, a rise in the number of slots per frame M yields an increase of the traffic for which the target PLR, e.g., 10 −2 , is achieved and is close to the asymptotic BP threshold.Our analysis, which uses an asymptotic approach, is in agreement with the simulations that were done on a finite number of slots as the number of slots increases.

VI. CONCLUSION
We asymptotically analyzed the decoding performance of IR-SA systems over Rayleigh block fading channels.We derived a converse bound based on the fact that the sum of the two areas below two EXIT curves is smaller than the entire domain's area.The converse bound is tight.Moreover, we derived an MAP threshold, following the coincidence of the BP and MAP EXIT curves.We also produced a BP decoding threshold for a spatially-coupled scheme.Numerical results show that it achieves the threshold saturation effect.
Although we limited this paper to IR-SA systems, the converse bound can be straightforwardly extended to C-SA systems.A future challenge will give a close-formed expression of trail entropy for C-SA systems, which play a key role in computing the MAP threshold.

APPENDIX A ERROR PROBABILITY OF DECODING m-COLLISION WITHIN A SLOT [6]
We now briefly review the decoding probability [6] of an m-collided signal of (12).
Recall that in the intra-slot SIC within a slot, decoding and removing interference is performed step-by-step in descending order from the largest SNR to the lowest.There are at most m decoding steps and m − 1 SIC steps to decode a randomly chosen packet replica.Let D(m, t), (1 ≤ t ≤ m), be the probability that among the m-collided packet replicas, the t − 1 packet replicas at the first t − 1 steps as well as the chosen packet replica at the t-th step are successfully decoded.Thus the error probability of decoding the chosen packet replica, under m − 1 interferences within the slot, is εm = 1 − m t=1 D(m, t).We now give probability D(m, t).The fact that the chosen replica was successfully decoded at the t-th step means that there are t − 1 replicas whose SNRs exceed that of the chosen replica.The number of arrangements for these t −1 replicas is (m−1) P (t−1) = (m−1)!(m−t)! .From these arrangements, the first t − 1 largest SNRs are labeled by replica indices 1 to t − 1 in descending order, i.e., 1 ≥ 2 ≥ . . .t−1 .For this arrangement, the following is the probability that at least t packets were successfully decoded [6]: (1 + η 0 ) t(m−(t+1)/2) .

FIGURE 5 .
FIGURE 5. Message passing at j -th coupled position.

FIGURE 8 .
FIGURE 8. Threshold saturation: BP thresholds of SC-SA approach MAP thresholds of d -RR-SA.

Fig. 6
Fig. 6 also shows that the converse bounds (or the MAP thresholds) at κ = 15 are almost consistent with those at κ = 21, which is due to 1 ≥ εm ≥ 0.9999 for m ≥ 15.Both numerical values are almost identical (Table3).This means that κ = 15 adequately gives a convergent result of these thresholds.In the remaining part of this section, we set κ = 15 without specific declarations.In Fig.7, we give the average PLR of the SC-SA obtained from (53) at d = 3 using density evolution.The average normalized offered traffic G † at PLR = 10 −2 , i.e., BP threshold G † BP,Conv = 3.2906, approaches MAP thresholds GMAP,blk

Figure 8
compares the SC-SA's BP thresholds and the d-RR-SA's MAP thresholds at κ = 2, 3, 4, and 15.The actual values of the BP thresholds achieve MAP thresholds,

TABLE 2 .
Specifications in analysis and simulations.

TABLE 4 .
[6]imized device node degree distribution and its corresponding BP threshold (PLR=10 −2 and κ = 15)[6].Thresholds versus inverse of average number of transmissions d : Thresholds are converse bound, MAPs of d -RR-SA and IR-SA, and BP of SC-SA.Irregular and regular BP thresholds[6]are also given for comparison.Device distributions of IR-SAs are given in Table4and κ = 15.