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Massive Access in Multi-cell Wireless Networks Using Reed-Muller Codes

Providing connectivity to a massive number of devices is a key challenge in 5G wireless systems. In particular, it is crucial to develop efficient methods for active device identification and message decoding in a multi-cell network with fading and path loss uncertainties. In this paper, we design such a scheme using second-order Reed-Muller (RM) sequences. For given positive integer m, a codebook is generated with up to 2^m(m+3)/2 codewords of length 2^m, where each codeword is a unique RM sequence determined by a matrix-vector pair with binary entries. This allows every device to send m(m+3)/2 bits of information where an arbitrary number of these bits can be used to represent the identity of a node, and the remaining bits represent a message. There can be up to 2^m(m+3)/2 devices in total. Using an iterative algorithm, an access point can estimate the matrix-vector pairs of each nearby device, as long as not too many devices transmit simultaneously. To improve the performance, we also describe an enhanced RM coding scheme with slotting. We show that both the computational complexity and the error performance of the latter algorithm exceed another state-of-the-art algorithm. The device identification and message decoding scheme developed in this work can serve as the basis for grant-free massive access for billions of devices with hundreds of simultaneously active devices in each cell.

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I Introduction

One of the promises of 5G wireless communication systems is to support large scale machine-type communication (MTC), that is, to provide connectivity to a massive number of devices as in the Internet of Things [1, 2, 3]. A key characteristic of MTC is that the user traffic is typically sporadic so that in any given time interval, only a small fraction of devices are active. Also, short packets are the most common form of traffic generated by sensors and devices in MTC. This requires a fundamentally different design than that for supporting sustained high-rate mobile broadband communication.

To extend classical information theory to massive access, an information-theoretic paradigm called many-user access was proposed by Chen, Chen, and Guo in [4, 5], where the number of users grows with the coding blocklength. It was shown to be asymptotically optimal for active users to simultaneously transmit their identifying signatures followed by their message bearing codewords. Meanwhile, massive access signaling techniques have received a lot of attention in the past few years; see examples [10, 12, 13, 6, 11, 7, 20, 24, 21, 22, 23, 16, 14, 17, 18, 19, 8, 9, 15] and references therein.

To discovery and identify the active users, each user must be assigned a unique sequence. Since there are an enormous number of potential users in the system, the signature sequences of most users cannot be orthogonal to each other, so multiaccess interference is unavoidable. The key challenge is to design a large sequence space with a reliable detection method. In [25], the author proposed a random access scheme based on the Zadoff-Chu (ZC) sequence, which allows relatively large number of preambles. However, the collision rate for contention-based random access [26, 27] is still too high in the context of massive access. We note that if the number of users to support is in the millions or more, we must design a highly structured codebook. This is because it is computationally infeasible to visit all user codewords in one frame slot. To address this, Reed-Muller (RM) sequence-based massive access schemes were proposed in [29, 7, 8, 28, 9, 11]. At typical code lengths, the number of RM sequences is several orders of magnitude larger than the number of ZC sequences. In fact the code book size can be so large that every user is assigned a different signature in any practical system. In addition, the RM sequences are well structured to allow very fast detection algorithms. In particular, an asynchronous random access algorithm based on convex optimization was presented in [15], which overcomes the issues of contention-based methods. A structured group sparsity estimation method for device identification and channel estimation was proposed in [16]. Reference [17] proposed an expectation propagation-based joint active user detection and channel estimation technique, and the active device detection for a cloud-radio access network was studied in [18]. To lower the complexity for massive user detection, a dimension reduction-based joint active user detection and channel estimation method was proposed in [19].

Due to the sporadic traffic in MTC, the active user detection problem is usually cast as a compressed sensing problem. Typical compressed sensing algorithms can be applied for detecting active users and channel estimation [10, 12, 13, 6, 14, 30, 31, 32, 35, 34, 33]. With known user channel coefficients, several compressed sensing methods is first applied for active user detection [35, 34, 33]. Then joint active user detection and channel estimation methods for single-cell massive connectivity scenario via approximate message passing (AMP) were proposed in [14, 10, 12, 13]. The authors in [6] further extended the network model to multi-cell massive multiple-input-multiple-output (MIMO) and cooperative MIMO systems. In addition, greedy compressed sensing algorithm was designed for sparse signal recovery based on orthogonal matching pursuit [30, 31]. We note that algorithms using random codes and/or AMP type of decoding do not scale to millions of potential devices.

Different from [4, 5], the author in [36] advocates the unsourced model focusing on how to recover messages from nodes using the same codebook. Identification can still be accomplished by letting all or part of the payload describe the identity. In this case, coded slotted ALOHA schemes were proposed in [37, 38], where each message is sent over multiple slots, and information is passed between slots to recover messages lost due to collision. The authors of [39] proposed the -fold ALOHA scheme such that the decoder can jointly decode all the messages if at most messages are sent in the same slot, whereas nothing is decoded when more than messages are transmitted within the same slot. And the performance of the -fold ALOHA scheme is further improved in [40], and [41] proposed the -ALOHA based random access scheme for handing Rayleigh fading channels. Following the framework in [36], an enhanced chirp reconstruction algorithm based on RM sequences was proposed in [8], assuming identical known channel gain for all users. It is shown in [8] that the worst-case complexity is sub-linear in the number of codewords, which makes it an attractive algorithm for message decoding in MTC.

In this paper, we consider the problem of joint device identification/decoding and channel estimation. We first derive an unambiguous relationship between the RM sequence and its subsequences, where each RM sequence is uniquely determined by a matrix-vector pair. Given this relationship, an iterative RM sequence detection and channel estimation algorithm is proposed. To enhance the performance of the RM detection algorithm, we further divide the codeword into multiple slots and transmit each user’s message in exactly two of the multiple slots [8]

. If a message is successfully decoded in one of the two transmitted slots, the decoded messages are then propagated to the other slots that the messages are transmitted, which can significantly improve the successful decoding probability.

The main differences between our algorithms and the state-of-the-art RM detection algorithms are: 1) The algorithm in [8] works only for the additive white Gaussian noise (AWGN) channel (the channel estimation problem is thus not considered therein), and the authors in [11] considered only the small-scale fading. In contrast, we consider both small-scale fading and large-scale fading in a general network setting; 2) The algorithm in [11] requires the receiver to know the number of active devices in the cell, and its value has a great impact on the decoding result. In addition, the performance of the algorithm in [11] degrades dramatically when the number of active devices is large. In contrast, by adopting slotting and message passing, our algorithm performs gracefully as the number of active devices increases. The main contributions are summarized as follows:

  • We first derive an explicit relationship between an RM sequence and its subsequences. Based on this relationship, we propose an iterative active device detection and channel estimation algorithm, Algorithm 1, for fading channels with path loss, where each device’s matrix-vector pair and the channel coefficient can be estimated successively.

  • Inspired by [8], we further describe an enhanced RM coding scheme with slotting. The corresponding detection algorithm is referred to as Algorithm 2.

  • The computational complexity and performance of Algorithm 1 are comparable to the RM decoding algorithm in [11], while the computational complexity and performance of Algorithm 2 are notably improved, which makes it one important step closer to a practical algorithm.

  • While many papers in the literature study massive access, this work (alone with [11]) is one of the few that can truly accommodate billions of devices and more.

The remainder of this paper is organized as follows. The system model is presented in Section II. Section III outlines the relationship between the RM sequence and its subsequences, which is the basis of the RM decoding algorithm. The device identification and channel estimation algorithm is expressed in Section IV, while the enhanced RM decoding algorithm utilizing slotting and message passing is explained in Section V. Further, the computation complexity analysis is given in Section VI. Section VII presents the numerical results, while Section VIII concludes the paper.

Throughout the paper, boldface uppercase letters stand for matrices while boldface lowercase letters represent column vectors. The superscripts , , and denote the transpose, complex conjugate, and conjugate transpose operator, respectively. The complex number field is denoted by . denotes the 2-norm of a vector and denotes the cardinality of set . denotes an identity matrix. denotes element-wise modulo-2 addition.

Ii System Model

Ii-a Channel Model

We consider a multi-cell network over a large region. User devices are randomly distributed in a similar fashion as was proposed in [42] (for an ad hoc network therein). Let time be slotted. A device may or may not be active in a slot. It is assumed that active devices transmit simultaneously. Each device sends bits, which consists of either an identity or a message or a combination of both. Let denote the set of active devices on the plane, where represents the location of a device. Without loss of generality, we focus on one access point (AP) and assume that the AP is located at the origin of the plane. Let represents the (random) number of active devices on the plane where , and denote the index set of the active devices as without loss of generality. Given this, the AP’s received signal is written as

(1)

where is the codeword corresponding to device with being the codeword length; denotes the complex channel coefficient between device and the AP; denotes the circularly symmetric complex AWGN; denotes the transmit power and we define

as the nominal received signal-to-noise ratio (SNR) for device

.

We further divide the active device set into in-cell device set and out-of-the-cell device set of the AP according to the nominal SNR between the AP and the devices. In other words, if the nominal SNR of a device to the AP is larger than a threshold (to be determined in Section II-B), then this device is considered an in-cell device of the AP. Without loss of generality, denote the index set of the in-cell devices as and denote the index set of the out-of-the-cell devices as . In this case, (1) can be further expressed as

(2)

The purpose of the AP is to identify all in-cell devices and/or decode their messages, where transmissions from out-of-the-cell devices are regarded as interference.

Our approach is to assign each message a length- second-order RM sequence determined by a symmetric binary matrix and a binary vector . Since is determined by bits and is determined by bits, each sequence encodes bits. Given the matrix-vector pair , the -th entry of the RM sequence can be written as [8]

(3)

where , is the -bit binary expression of . Eq. (3) indicates that .

As a toy example, in the special case of , there are altogether 32 RM sequences of length 4. This entire set of sequences is enumerated in Table I. In numerical results in Section VII, we simulate systems with as high as 14, which accommodate up to sequences of length 16,384.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 -1 -1 - - -1 -1 - - 1 1 1 1 -1 -1 - - -1 -1 - -
1 1 1 1 1 1 1 1 -1 -1 -1 -1 - - - - -1 -1 -1 -1 - - - -
1 -1 - - -1 1 -1 1 - - 1 -1 -1 1 - - 1 -1 1 -1 - - -1 1
TABLE I: The entire set of 2nd-order RM sequences in the special case of .

Ii-B Propagation Model and Cell Coverage

Consider a multiaccess channel with devices distributed across the plane according to a homogeneous Poisson point process with intensity . The number of devices in a region with its area equal to

is a Poisson random variable with mean

.

The small-scale fading between device and the AP is modeled by an independent Rayleigh random variable with unit mean. The large-scale fading is modeled by the free-space path loss which attenuates over distance with some path loss exponent .

Let and denotes the distance and the small scale Rayleigh fading gain between device , and the AP, respectively. Then the channel gain is expressed as

(4)

where the phase of

is uniformly distributed on

.

The coverage of the AP can be defined in many different ways. According to [42], device and the AP are neighbors of each other if the channel gain exceeds a certain threshold . Assume device and the AP are neighbors, i.e., , we have . Under the assumption that all devices form a p.p.p., for given , device is uniformly distributed in a disk centered at the AP with radius . According to [42], the distribution of is

(5)

and the average number of neighbors of the AP is

(6)

where is the Gamma function. In addition, the sum power of all out-of-the-cell devices is [42]

(7)

Iii A Property of RM Sequences

In this section, we introduce a fundamental property of RM sequences, which underlies the following detection algorithm.

let be a given positive number. Let be a binary -tuple. For , we have

(8)

Furthermore, let . For , let the binary matrix be defined recursively as

(9)

where is the main diagonal elements of , and is a length column vector.

We have the following new result.

Theorem 1.

Given a length- RM sequence , its order and sub-sequences satisfy

(10)

where

(11)

The vector is a length- Walsh sequence with frequency .

Proof:

Recall is the -bit expression of . For , the vector can be decomposed as

(12)

Consequently

(13)
(14)

Substituting (14) into (3) yields

(15)
(16)

Likewise, the binary vector can be decomposed as

(17)

Then the exponent of is expressed as

(18)
(19)

Substituting (19) into (3) yields

(20)

Iv Device Identification/Decoding and Channel Estimation

In this section, we propose a novel RM detection algorithm for active device detection and channel estimation that leverages Theorem 1.

Specifically, the matrix-vector pair of each message will be estimated recursively. According to (8) and (9), the matrix-vector pair is determined by and . We will show that the algorithm first estimates the triple set , then , and finally the channel coefficient and .

Iv-a Estimation of

From (1) and (10), when , we have

(21)
(22)

and

(23)
(24)

Since , we have . Define , (22) and (24) lead to

(25)
(26)

where

(27)

The first term in the right-hand side of (25) is a linear combination of Walsh functions , with frequency , which can be recovered by applying Walsh-Hadamard Transformation (WHT). The second term, , is a linear combination of chirps, which can be considered to be distributed across all Walsh functions to equal degree, and therefore these cross-terms appear as a uniform noise floor.

Let the Hadamard matrix be and its -th elements are . Denote the WHT transformation as , whose -th entry can be written as

(28)
(29)
(30)
(31)

Equation (31) indicates that, if we have , peaks will appear at frequency , where the maximum value is . On this basis, can be recovered by searching the largest absolute value of . We further have

(32)

Equation (32) indicates that can be estimated by the polarity of the largest value of . For example, if the real part of the maximum value is positive and greater than the absolute value of the imaginary part, then we have .

Further, is recovered through

(33)

Iv-B Estimation of

After recovering , we next estimate in a similar way. Define

(34)

Under the assumption that is correctly estimated, according to (22) and (24), is further expressed as

(35)

where

(36)

i.e., the variance of the channel noise is reduced by half. Since

and are two different Walsh functions, take vales in with equal probability, in other words, . This indicates that the interference from other devices is reduced by a quarter.

For simplicity, let , and we have . Given this, (35) leads to

(37)
(38)

Compared with (1), the device interference from other devices is reduced by a quarter and the variance of the noise is reduced by half.

When , applying Theorem.1 on (38) leads to

(39)
(40)

and

(41)
(42)

Let , we have

(43)

where and

(44)

Similar to (25), applying WHT on (43), can be recovered by searching the maximum value of the result. Comparing (25) and (43), we know that is more likely to be correctly estimated than because the interference terms are reduced by a quarter and the variance of channel noise is reduced by half.

Similar to (32), can be recovered by the polarity of the maximum value.

Iv-C Estimation of Channel Coefficient

We continue these process until all the estimates are obtained. According to (35), the received sequence in the last layer can be written as

(45)

where the term consists of all interferences from other devices which are all second order RM sequences and . Accordingly, we have

(46)
(47)

and

(48)
(49)

Similar to previous processing, define , we have

(50)

where

(51)

According to (50), can be estimated by the polarity of .

Further the channel coefficient can be estimated as

(52)

Note that we can infer from (52) if the AP knows the value of , but we only need to know the product of the two.

Iv-D RM Detection Algorithm

So far, the matrix-vector pair and the corresponding channel coefficient for message has been completely estimated. In addition, can be obtained through (3). Given this, the remaining messages can be estimated iteratively by updating the received signal by removing the interference of device .

The detailed algorithm is summarized in Algorithm 1, where the maximum number of detected devices is limited to . The choice of depends on the expected success rate, miss rate, and false alarm rate. On one hand, a small value of will lead to an unsatisfactory success rate and miss rate. On the other hand, if the value of is too large, it may lead to a higher false alarm rate and computational complexity. Hence, the choice of is determined experimentally by the tradeoff between the success rate, miss rate, and the false alarm rate, as well as the practical complexity constraint.

From the above analysis, on condition that is correctly estimated, , are more likely to be correctly estimated than because both interference and channel noise are reduced. However, if is wrongly estimated, the desired message will also be reduced by half, which will likely cause an error propagation problem that leads to the wrong estimation of .

Algorithm 1: RM detection algorithm.
Input: the received signal , the maximum number of active devices .
.
while and 111Various criteria are possible here. We found works well when there are only in-cell devices transmit, and works well when there are transmission from out-of-the-cell devices. We should emphasize that it’s very much an open question as to how to choose this parameter optimally. Our suggestion would be to try to optimize the value empirically for the regime of interest. do
.
for do
  Split into two partial sequences according to (22) and (24).
  Perform the element-wise conjugate multiplication according to (25).
  Perform WHT and let be the binary index of the largest component.
  Recover according to (32).
  Recover according to (33).
  Calculate according to (34).
end for
 Recover according to (32) and (50).
 Add to the decoded set.
 Calculate the codeword according to and estimate according to (52).
.
end while
Output: , and for .

To avoid error propagation, we propose a list detection algorithm inspired by [11]. Let where , denotes the number of decoding paths in layer . On this basis, when estimating , instead of choosing the largest component, we maintain the largest component as candidates. Thus the total number of decoding paths is . And finally, the path having the minimum residual energy is chosen as the optimal path where the residual energy of path is defined as

(53)

An example of list decoding algorithm is illustrated in Fig.1.

Fig. 1: An example of list decoding with .

V Enhanced RM Decoding Algorithm with Slotting and Message Passing

Inspired by [8], we adopt the idea of slotting and message passing to enhance the performance of Algorithm 1. In general, slotting trades bandwidth for better performance.

V-a Slotting

By slotting, we mean that a codeword of length is divided into slots, and each message of length is transmitted in one or more of the slots. Accordingly, each message is assign a sparse codeword which is zero except for the slots to which the message is sent. Then we estimate the message within each slot, and combine the results.

In practice, we send each message in two out of the slots. We append bits to encode a primary slot location. From the bits used to encode , we use an arbitrary subset of size to encode a translate, which gives the secondary slot location when it is added to the primary slot location. To distinguish the primary and secondary slots, we fix a signal check bit in the matrix to be 0 for the primary slot and 1 for the secondary slot. Thus deducing 1 bit from the total number of bits transmitted. Thus, the number of bits that each message transmitted is

(54)

Given (which determines the codelength), the larger is, the more significant the reduction in the number of information bits, the better the error performance.

V-B Message Passing

By transmitting each message in two slots, the decoded messages in one slot are expected to be propagated to the other slots. The decoding algorithm cycles through all the slots. Whenever a set is found, the algorithm first checks whether the corresponding message has been recorded. If not recorded, the check bit in the matrix reveals whether the current slot is primary or secondary. If the current slot is the primary slot (secondary slot), the location of the secondary slot (primary slot) is determined by the current slot plus the translate that implied in the pair. Then after flipping the check bit in the matrix, the decoded set is propagated to the other slot to which the message was sent.

V-C RM Detection Algorithm

Upon receiving the messages, decoding the slots one by one, i.e., decoding the received signal in slot 1 first, and then slot 2, and so on. The detailed algorithm is summarized as in Algorithm 2.


Algorithm 2: Enhanced RM detection algorithm.
Input: the received signal