Reconstruction of Finite-Alphabet Block-Sparse Signals From MAP Support Detection

This paper addresses finite-alphabet block-sparse signal recovery by considering support detection and data estimation separately. To this aim, we propose a maximum a posteriori (MAP) support detection criterion that takes into account the finite alphabet of the signal as a constraint. We then incorporate the MAP criterion in a compressed sensing detector based on a greedy algorithm for support estimation. We also propose to consider the finite-alphabet property of the signal in the bound-constrained least-squares optimization algorithm for data estimation. The MAP support detection criterion is investigated in two different contexts: independent linear modulation symbols and dependent binary continuous phase modulation (CPM) symbols. The simulations are carried out in the context of sporadic multiuser communications and show the efficiency of proposed algorithms compared to selected state-of-the-art algorithms both in terms of support detection and data estimation.

SECTION I. Introduction

The number of connected machines will reach 50 billion by 2020 with an ever-increasing demand for wherever and whenever communications [1]. The challenges of future mobile and wireless communication systems are to address these societal needs by improving the energy and the spectral efficiency, while lowering the cost and improving the scalability to deal with a large number of connected devices.

Wireless Machine-to-Machine (M2M) communication is a typical class of wireless communication which is expected to develop tremendously in the future. This is especially related to the emergence of surveillance applications/remote control in various fields such as the environment (to anticipate the volcanic eruptions for example), health (for tracking a patient at home from the surgery) or the industry 4.0 (sensor networks, smart meters). The M2M challenge [2] involves the deployment of a huge number of connected machines which are stand-alone energy devices and with a limited complexity processing capability. Information is transmitted to a concentrator (or fusion center) having a higher processing capability.

M2M communications have low data rates and transceivers are most of the time idle, yielding sporadic traffic with short frames. To reduce the frame overhead, the connected objects directly send their data packets without signaling, in a random access manner. Thus, the receiver has to detect both the activity of the objects and the transmitted data. Due to the low activity rate, the number of simultaneous active objects is rather small compared to the total number of objects. If the number of objects is huge, though, the system of equations to be solved to recover the active users and their data may be underdetermined.

However, assuming that the number of simultaneous active objects is small compared to the total number of objects yields the solution of the system to be sparse. Solving such sparse underdetermined systems can be successfully performed thanks to Compressed Sensing based Multi-User Detection (CS-MUD) [3], where algorithms from sparse signal processing and Compressed Sensing (CS) are utilized. CS has applications in many different research topics such as radar [4], neural network [5], image processing [6], channel estimation [7], etc. CS theory permits the sensing and recovery of sparse signals using a small number of linear measurements, roughly proportional to the number of non-zero values the signals take in the sparse domain [8], [9].

Various CS algorithms apply to solve problems made of an underdetermined set of equations. These algorithms can be classified into two categories. The first one includes all convex optimization algorithms [10]. They involve a sparsity promoting term, using the $\ell _{1}$ -norm for example, to guarantee the convergence and to find the sparsest solution to the underdetermined system [11]. The second category corresponds to iterative greedy algorithms, which are commonly used owing to their low complexity and simple geometrical interpretation. The most prominent one is the Orthogonal Matching Pursuit (OMP) [12], [13].

In addition to sparsity and depending on the application, other constraints can be considered in the recovery problem formulation. For example in multi-band signals [14], face recognition [15], spectrum sensing, equalization of sparse communication channels, the non-zero values are not only few, but they are also clustered in blocks [16]–​[18]. Such a signal is said to be block-sparse. Non-zero values usually take values in a specific set that can be either bounded to some continuous convex region or discrete (alphabet). Both features (structure and set) can define additional constraints. However a discrete set (non-zero values) means a non-convex solution space, which complicates its consideration in the problem formulation. This is the reason why most published CS algorithms, which are dedicated to finite-alphabet recovery, propose to relax the discrete set constraint to convex constraints [18]–​[21].

In this paper, we consider the detection of block-sparse signals whose non-zero values belong to a finite alphabet. We apply a two-step signal recovery, carrying out the support detection first (localization of non-zero blocks), followed by the data estimation (evaluation of non-zero values). The support corresponds to the non-zero positions in the signal vector. The separation between support detection and data estimation has already been considered in [3] and will be referred to as ED-LS. It proposes an energy detection (ED) algorithm to identify the support and then, a Least Squares (LS) estimation to recover the data using the previously estimated support. However, the ED-LS performance exhibits an error floor when the signal-to-noise ratio increases. Maximum A Posteriori (MAP) support detection alone was proposed in [22] but without considering the assumptions of ${\boldsymbol (i)}$ block-sparsity and $\boldsymbol (ii)$ finite alphabet. Support detection of block-sparse signals is considered in [23], [24]. However, no a priori on the signal distribution is taken into account in [23], whereas a continuous a priori signal distribution is used in [24] to develop a MAP support detection criterion.

A. Contributions

In this paper, ${\boldsymbol (i)}$ we compute a maximum a posteriori (MAP) support detection criterion that takes into account the finite alphabet constraint and we approach it from a continuous mixture of Gaussian distributions. $\boldsymbol (ii)$ We incorporate the MAP criterion in a new CS detector based on a greedy algorithm for support estimation. $\boldsymbol (iii)$ We develop a MAP-based support detection in the context of CS-MUD for M2M sporadic communications that use Continous Phase Modulations (CPM). Indeed, due to their high energy efficiency, CPM are very attractive for M2M communication schemes. But their non-linearity can hardly be considered in CS-MUD. We thus propose to use a linear decomposition of the CPM and to apply the support detection as the first CS-MUD step.

B. Paper Organization and Notations

The paper is organized as follows: In Section II, we first present a state-of-the-art of greedy block-sparse CS algorithms where the support detection and the data estimation are jointly performed. Since the alphabet constraint is hardly considered in the literature, we then modify the best CS algorithms so as to incorporate the finite-alphabet constraint. Their performance will be used as a baseline of our proposed algorithms. The detection problem in sporadic multi-user communications is investigated in Section III. The two-step approach (MAP-based support detection followed by data estimation) is detailed, first for linear modulations and then for CPM. The simulation results are given in Section IV. Finally a conclusion and some perspectives are given in Section V.

Throughout this paper, lower-case bold letters ${\boldsymbol x}$ , upper-case bold letters ${\boldsymbol X}$ and regular letters $x$ denote a column vector, a matrix and a scalar value respectively. ${\boldsymbol X}^{T}$ is the transpose of matrix ${\boldsymbol X}$ , ${\boldsymbol X}^{H}$ is its Hermitian matrix and ${\boldsymbol X}^{\dagger }$ its Moore-Penrose pseudo-inverse. ${\boldsymbol X}_{\mathcal {F}}$ is a submatrix of $\boldsymbol X$ that contains columns indexed by the set $\mathcal {F}$ whose cardinal is denoted by ${\mathcal {F}}$ . ${\boldsymbol 0}_{N}$ is the null vector of length $N$ , ${\boldsymbol 1}_{N}$ is the all-one vector and $\otimes$ denotes the Kronecker product. $\hat {\boldsymbol x}$ stands for the estimated value of ${\boldsymbol x}$ in a continuous domain included in $\mathbb {R}$ or $\mathbb {C}$ . $Q_{\Xi }({\boldsymbol x})$ denotes the quantizer operator which maps ${\boldsymbol x}$ to its closest values in a discrete alphabet denoted by $\Xi$ . $\tilde {\boldsymbol x}$ stands for the estimated values of ${\boldsymbol x}$ in $\Xi$ .

SECTION II. Block-Sparse Compressed-Sensing With Alphabet Constraints for Joint Detection and Estimation

In most state-of-the-art CS algorithms that are applied to recover block-sparse signals whose non-zero values belong to a discrete set, the support detection and the data estimation are jointly performed, and the alphabet constraint is hardly taken into account. In this section, we propose to modify some greedy CS algorithms in order to incorporate the alphabet constraint. Their performance will serve as a benchmark for the algorithms developed in Section III.

A. Notations

We consider the observation of a block-sparse signal in a noisy environment from $n$ measures. We denote by $\boldsymbol {y} \in \mathbb {C}^{n}$ the length-$n$ observation vector which can be defined as $$\begin{equation*} {\boldsymbol y} = {\boldsymbol A} {\boldsymbol x} + {\boldsymbol \eta },\tag{1}\end{equation*}$$ with $\boldsymbol A$ the measurement matrix, ${\boldsymbol x}$ the block-sparse signal vector and ${\boldsymbol \eta }$ the noise vector.

${\boldsymbol \eta }$ is assumed to be a complex Gaussian-distributed vector of length $n$ with zero mean and covariance matrix $\sigma _\eta ^{2} {\boldsymbol I_{n}}$ , i.e., ${\boldsymbol \eta } \sim \mathcal {CN}(0, \sigma _\eta ^{2} {\boldsymbol I_{n}})$ .

${\boldsymbol x}$ is a concatenation of $K$ length-$N$ blocks (yielding a total length of $K N$ ). The $i$ -th block is denoted by ${\boldsymbol x}_{i}$ so that $\boldsymbol x$ can be expressed as: $$\begin{equation*} {\boldsymbol x} = [\underbrace {x_{1},\ldots,x_{N}}_{\boldsymbol x^{T}_{1}}, \ldots, \underbrace {x_{ (K - 1)N +1},\ldots, x_{KN}}_{\boldsymbol x^{T}_{K}}]^{T}.\tag{2}\end{equation*}$$ A vector ${\boldsymbol x}$ is called $K_{a}$ block-sparse if it has $K_{a}$ blocks different from the all-zero vector ${\boldsymbol 0}_{N}$ . The support of ${\boldsymbol x}$ , denoted by ${\boldsymbol s}$ , is defined as the vector of binary components such that $s_{i}=1$ if the $i$ -th block of ${\boldsymbol x}$ is different from ${\boldsymbol 0}_{N}$ and $s_{i}=0$ otherwise. Let ${\boldsymbol x}_{\boldsymbol s}$ stand for the vector gathering all the non-zero blocks. $\boldsymbol {x}_{s}$ is a length-$K_{a}N$ vector. The components of non-zero blocks take values in the alphabet $\Xi = \{\xi _{1},\xi _{2}, {\dots },\xi _{M}\}$ .

Similarly to (2), we can represent the measurement matrix ${\boldsymbol A}$ as a concatenation of sub-matrices ${\boldsymbol A}_{\Gamma (i)}$ of size $n \times N$ : $$\begin{equation*} ~{\boldsymbol A} \!=\! [\underbrace {\boldsymbol a_{1},\ldots,{\boldsymbol a}_{N}}_{\boldsymbol A_{\Gamma (1) }},\underbrace {\boldsymbol a_{N+1},\ldots,{\boldsymbol a}_{2N}}_{\boldsymbol A_{\Gamma (2) }}, \ldots, \underbrace {\boldsymbol a_{ (K \!-\!\, 1)N \!+\!\,1},\ldots, {\boldsymbol a}_{KN}}_{\boldsymbol A_{\Gamma (K)}}],\end{equation*}$$ where $\Gamma (i)$ denotes the set of indices $\{(i-1)N+1,\ldots,iN\}$ for $i\in \mathcal {K}$ and $\mathcal {K}=\{1,\ldots,K\}$ . Let ${\boldsymbol A}_{\boldsymbol s}$ denote the concatenation of the sub-matrices ${\boldsymbol A}_{\Gamma (i)}$ having the index $i$ of the non-zero blocks (i.e. $s_{i}=1$ ). Then (1) can be rewritten as: $$\begin{equation*} {\boldsymbol y} = {\boldsymbol A}_{\boldsymbol s} {\boldsymbol x}_{\boldsymbol s} + {\boldsymbol \eta }.\tag{3}\end{equation*}$$

B. Block-Sparse CS Algorithms Without Alphabet Constraints

1) GOMP

One of the most popular CS algorithms of the literature is the Orthogonal Matching Pursuit (OMP) which iterativeley determines the sparse vector. The extension of this algorithm to the block-sparse case is the Group OMP (GOMP) [16] reminded in Algorithm 1, where no assumption is made regarding the alphabet of the components of $\boldsymbol x$ . The GOMP algorithm combines support detection (step 3) and data estimation (step 5). In the context of CS-MUD, the GOMP algorithm has been exploited in e.g. [3].

Algorithm 1 Group Orthogonal Matching Pursuit (GOMP)

1:

Initialize:

$\mathcal {G}^{(0)} = \emptyset$ , ${\boldsymbol r}^{(0)}={\boldsymbol y}$ , $\hat {\boldsymbol x}={\boldsymbol 0}_{KN}$

2:

for $i =1$ to $K_{a}$ do

3:

$\displaystyle k^{*} = \mathop{\mathrm {arg\,max}}\limits_{k \in {\mathcal {K}\setminus \mathcal {G}}^{(i-1)}} \sum _{j \in \Gamma (k)} \left |{{\boldsymbol a}_{j}^{H} {\boldsymbol r}^{(i-1)}}\right |$

4:

$\mathcal {G}^{(i)} = \mathcal {G}^{(i-1)} \cup \{k^{*} \}$

5:

$\hat {\boldsymbol x}_{\Gamma (\mathcal {G}^{(i)})} = ({\boldsymbol A}_{\Gamma (\mathcal {G}^{(i)})})^{\dagger } {\boldsymbol y}$ (LS estimation)

6:

${\boldsymbol r}^{(i)} = {\boldsymbol y} - {\boldsymbol A} \hat {\boldsymbol x}$

7:

end for

8:

return $\hat {\boldsymbol x}$

2) GMMP

One drawback of GOMP is the selection of only one non-zero block position at each iteration. To reduce the risk caused by a wrong decision on one support position, the authors of [25] proposed an extended version of OMP algorithm called Multipath Matching Pursuit (MMP) which relies on an iterative list construction. Given an iteration, from each temporary list element, $m$ candidates are generated and the new intermediate list is built by collecting all candidates (duplications are eliminated). The support decision is taken from the final list whose maximum size is $m^{2}$ , by selecting the candidate associated to the minimal residual error. The MMP algorithm of [25] is described in Algorithm 2 for a block-sparse signal and named accordingly Group MMP (GMMP).

Algorithm 2 Group Multipath Matching Pursuit (GMMP)

1:

Initialize:

$\boldsymbol {\xi }^{\ast }=\mathop{\mathrm {arg} \max }\limits_{\boldsymbol \xi \subset \mathcal {K}}\sum _{k\in \boldsymbol \xi }\sum _{t\in \Gamma (k)}\left |{\boldsymbol {a}_{t}^{H}\boldsymbol {y}}\right |\quad$ subject to $\quad \mathrm {card}{(\boldsymbol \xi)}=m$ ,$\mathcal {S}^{(1)}=\{s^{(1) }_{1}, {\dots },s^{(1) }_{m}\}$ with $s_{u}^{(1) }=\{\xi _{u}^{\ast }\}$ ,$\hat {\boldsymbol x}_{u}^{(1) } = \left ({{\boldsymbol A}_{\Gamma \left ({s_{u}^{(1) }}\right)}}\right)^{\dagger } {\boldsymbol y}$ and $\boldsymbol r_{u}^{(1) } = {\boldsymbol y} - {\boldsymbol A}_{\Gamma \left ({s_{u}^{(1) }}\right)} \hat {\boldsymbol x}_{u}^{(1) }$ , $\forall u\in \{1, {\dots },m\}$

2:

for $i = 2$ to$K_{a}$ do

3:

$u=0$ , $\mathcal {S}^{(i)} = \emptyset$

4:

for $j = 1$ to $\mathrm {card}(\mathcal {S}^{(i-1)})$ do

5:

$\begin{array}{rcl} \boldsymbol {\xi }^{\ast } & = & \mathop{\mathrm {arg} \max }\limits_{\boldsymbol \xi \subset \mathcal {K}}\sum _{k\in \boldsymbol \xi }\sum _{t\in \Gamma (k)}\left |{\boldsymbol {a}_{t}^{H}\boldsymbol {r}_{j}^{(i-1)}}\right | \\ & & \text {subject to } \quad \mathrm {card}{(\boldsymbol \xi)}=m \end{array}$

6:

for $w = 1$ to $m$ do

7:

$s_{temp} = s^{(i-1)}_{j} \cup \{ \xi ^{\ast }_{w} \}$

8:

if $s_{temp} \notin \mathcal {S}^{(i)}$ then

9:

$u=u+1$

10:

$s_{u}^{(i)} =s_{temp}$ and $\mathcal {S}^{(i)} = \mathcal {S}^{(i)} \cup \{s^{(i)}_{u}\}$

11:

$\hat {\boldsymbol x}_{u}^{(i)} = \left ({{\boldsymbol A}_{\Gamma \left ({s_{u}^{(i)}}\right)}}\right)^{\dagger } {\boldsymbol y}$ (LS estimation)

12:

$\boldsymbol r_{u}^{(i)} = {\boldsymbol y} - {\boldsymbol A}_{\Gamma \left ({s_{u}^{(i)}}\right)} \hat {\boldsymbol x}_{u}^{(i)}$

13:

end if

14:

end for

15:

end for

16:

end for

17:

$u^{\ast } = \mathop{\mathrm {arg\,min}}\limits_{u} ||{\boldsymbol r}_{u}^{(K_{a})}||_{2}^{2}$

18:

$\hat {\boldsymbol x}=\boldsymbol {0}_{KN}$ and $\hat {\boldsymbol {x}}_{\Gamma \left ({s_{u^{\ast }}^{(i)}}\right)}=\hat {\boldsymbol x}_{u^{\ast }}^{(K_{a})}$

19:

return $\hat {\boldsymbol x}$

C. Block-Sparse CS Algorithms With Alphabet Constraint Using BLS

We propose hereinafter simple extensions of the two previous popular block-sparse CS algorithms to take into account the alphabet constraints. The aim is to come up with a reasonable benchmark for the performance comparison in Section IV.

We will introduce the alphabet knowledge by the substitution of the LS estimation step of Algorithms 1 and 2 by another estimation step that takes into account the finite data set. In order to reduce the error propagation and to take into account the modulation alphabet, we propose to bound the estimation using fixed bounds according to the modulation alphabet. Since the introduction of a discrete set constraint on non-zero data values is not feasible, we relax it to a convex constraint. More precisely, we restrict the solution space to a convex region that includes the discrete alphabet.

The LS estimation step will be replaced by the Bound-constrained LS estimation first proposed in [26], mathematically detailed in [27, Chapter 5] and [28] and described in Algorithm 3.

Algorithm 3 Bound-Constrained Least-Squares: BLS$(\boldsymbol y, {\boldsymbol H},\Xi)$

1:

$\begin{cases} \hat {\boldsymbol \chi } = \mathop{\mathrm {arg\,\min } }\limits_{\boldsymbol {\chi } }||{\boldsymbol y}-{\boldsymbol H}{\boldsymbol \chi }||_{2} \\ \text {subject to} \\ \min \Re \{\Xi \} \leq \Re \{\boldsymbol \chi \} \leq \max \Re \{\Xi \} \\ \min \Im \{\Xi \} \leq \Im \{\boldsymbol \chi \} \leq \max \Im \{\Xi \} \end{cases}$

2:

return $\hat {\boldsymbol \chi }$

Replacing Line 5 in Algorithm 1 by $\hat {\boldsymbol x}_{\Gamma (\mathcal {G}^{(i)})} = \text {BLS}(\boldsymbol y,{\boldsymbol A}_{\Gamma (\mathcal {G}^{(i)})},\Xi)$ defines the algorithm referred to as BLS-GOMP. Likewise, replacing in Lines 1 and 11 of Algorithm 2 the calculation of $\hat {\boldsymbol x}_{u}^{(i)}$ by $\hat {\boldsymbol x}_{u}^{(i)} = \text {BLS} \Big (\boldsymbol y, {\boldsymbol A}_{\Gamma \left ({s_{u}^{(i)}}\right)},\Xi \Big)$ defines the algorithm referred to as BLS-GMMP.

SECTION III. MAP Support-Detection and Application to Compressed-Sensing Multiuser Detection

In Section II, we introduced the finite-alphabet constraint in CS algorithms that jointly detect and estimate block-sparse signals. In this section, we will focus on CS algorithm where the support detection is separated from the symbol estimation and we propose a MAP criterion for support detection exploiting the finite alphabet property of the signal.

We illustrate it in the context of sporadic communications for which we consider two modulation schemes: first a linear modulation with independent symbols and then a non-linear modulation with dependent symbols. For each of them, we proceed into two steps: we start by providing an explicit expression of the measurement matrix $\boldsymbol A$ in Equation (1), since it also depends on the considered transmission scheme; then we investigate a MAP criterion to compute the support estimate $\hat {\boldsymbol {s}}$ , which will be used to estimate the transmitted vector $\boldsymbol x$ . Finally, data estimation is achieved thanks to a BLS algorithm.

A. Sporadic Communications Model

M2M communications or Machine-Type-Communications (MTC) can often be represented by a sporadic communication model for which compressed sensing-based multi-user detection (CS-MUD) is well suited [3], [21]. In this setup, a large number $K$ of users (sensors for example) may randomly transmit a data frame to a single aggregation node (concentrator). We also assume that users are synchronized as illustrated in Figure 1 and that frames are independent (uncoded-slotted ALOHA). The probability of a user transmitting a frame is called the activation probability and is denoted by $p_{a}$ . Let us denote by $K_{a}$ the number of users that simultaneously transmit (active users). Then $K_{a}=K p_{a}$ . As $p_{a}$ is usually quite small, the number of active users $K_{a}$ is low compared to the total number of users $K$ . The aggregation point is supposed to ignore whether the $k-$ th user is active or not during a given slot. Therefore, it has to detect both the node activity and the transmitted data.

FIGURE 1.

M2M sporadic random access communication scenario involving $K$ synchronous users. On this illustration, User 1 sends a frame on time-slot 3, user 2 on time-slot 1 and 4, etc.

Sparse signal detection with high successful recovery probability requires a minimum number of observations [29]. This number can be increased with the help of either spreading sequences as proposed in [30] or multiple receive antennas. In this paper, we choose the first solution. Pseudo-random sequences of length $N_{s}$ are used to spread each symbol period and thus increase the number of observations per transmitted symbol. The value of $N_{s}$ satisfies the following assumptions: it is large enough to ensure a satisfactory successful recovery probability and smaller than the total number of users, which yields an underdetermined multiuser detection problem.

B. Linear Modulation of Independent Symbols

As a first example, we assume that independent symbols from a linear modulation are sent by the users over a Rayleigh fading channel. This assumption isn’t strictly exact since an error correction code is usually implemented yielding dependent symbols. However, its accuracy depends on the code length and the use of an interleaver.

1) Measurement Matrix

The $i$ -th symbol of the $k$ -th user, denoted by $x_{k,i}$ , is spread into chips by a real-valued pseudo-random sequence ${\boldsymbol c}_{k} = [c_{k,0},c_{k,1},\ldots,c_{k,N_{s}-1}]^{T}\in \{\pm 1\}^{N_{s}}$ . The chips of the $k$ -th user are then transmitted over a time-invariant frequency-selective Rayleigh fading channel whose length-$L_{g}$ channel tap vector is ${\boldsymbol g}_{k}$ . Thus the aggregation node receives the synchronous superposition of modulated frames corrupted by additive noise. The observation vector reads $$\begin{equation*} {\boldsymbol y} = \sum _{k=1}^{K} {\boldsymbol G}_{k} {\boldsymbol C}_{k} {\boldsymbol x}_{k} + {\boldsymbol \eta },\tag{4}\end{equation*}$$ where ${\boldsymbol y} \in \mathbb {C}^{n}$ , $n=N N_{s}$ , ${\boldsymbol C}_{k}$ is the $(n \times N)$ medium access matrix defined from the spreading sequence of the $k-$ th user ${\boldsymbol c}_{k}$ : $$\begin{equation*} {\boldsymbol C}_{k} = {\boldsymbol I}_{N} \otimes {\boldsymbol c}_{k}.\tag{5}\end{equation*}$$ The noise vector $\boldsymbol \eta$ is independent of the signal vector. It is a complex circularly symmetric Gaussian-distributed vector with zero mean and covariance matrix $\sigma ^{2}_{\eta } {\boldsymbol I}_{n}$ : ${\boldsymbol \eta } \sim \mathcal {CN}(0,\sigma ^{2}_{\eta } {\boldsymbol I}_{n})$ . Given the $k$ -th user, ${\boldsymbol G}_{k}$ is the $(n \times n)$ convolution matrix that corresponds to the channel tap vector ${\boldsymbol g}_{k}$ :

Based on [31], (4) can be written as: $$\begin{equation*} {\boldsymbol y} = {\boldsymbol A} {\boldsymbol x} + {\boldsymbol \eta },\tag{7}\end{equation*}$$ where matrix ${\boldsymbol A} = [{\boldsymbol G}_{1} {\boldsymbol C}_{1}, {\boldsymbol G}_{2} {\boldsymbol C}_{2}, {\dots }, {\boldsymbol G}_{K} {\boldsymbol C}_{K}] \in \mathbb {C}^{n \times K N}$ combines the spreading and the channel coefficients. ${\boldsymbol x}$ is a block-sparse vector as defined in (2).

2) MAP Criterion for Support Detection (Independent Symbols)

The key idea of our approach relies on the approximation of the discrete distribution of ${\boldsymbol x}$ by a continuous complex Gaussian Mixture Model [32], i.e., the discrete uniform distribution of non-zero components $x_{i}$ : $$\begin{equation*} \Pr \big (x_{i}=\xi \big) = \begin{cases} 1/M & \text {if} \quad \xi \in \Xi \\ 0 & \text {otherwise} \end{cases}\tag{8}\end{equation*}$$ is approximated by the following complex continuous Gaussian Mixture distribution with means $\xi _{j}$ with $j= \{1,2,\ldots, M\}$ and variance $\sigma ^{2}_{\epsilon }$ : $$\begin{equation*} \Pr \big (x_{i}=\xi \big) \approx \frac {1}{ M \pi \sigma _{\epsilon }^{2}}\sum _{j=1}^{M} e^{- \frac {|\xi -\xi _{j}| ^{2} }{ \sigma _{\epsilon }^{2}}}\tag{9}\end{equation*}$$ The value of $\sigma _{\epsilon }$ has to be low in order to closely approach the exact discrete distribution by the mixture of complex Gaussian distributions. The elements of ${\boldsymbol x}_{s}$ , (which are the non-zero components of ${\boldsymbol x}$ ) are assumed to be independent. Thus for a $K_{a}$ -block sparse vector ${\boldsymbol x}$ , the approximation of $\Pr ({\boldsymbol x}_{s}|{\boldsymbol s})$ by a complex circularly symmetric Gaussian distribution yields $$\begin{equation*} \Pr \big ({\boldsymbol x}_{s}| {\boldsymbol s} \big) \approx \frac {1}{(M\pi \sigma _{\epsilon }^{2})^{K_{a}N}} \sum _{\boldsymbol \zeta \in \Xi ^{K_{a}N} } e^{-\frac {||{\boldsymbol x}_{s} - {\boldsymbol \zeta } ||_{2}^{2}}{\sigma _{\epsilon }^{2}}}.\tag{10}\end{equation*}$$ The noise is assumed zero mean and complex circularly symmetric Gaussian distributed. Then, conditionally to ${\boldsymbol s}$ and ${\boldsymbol x}_{s}$ , ${\boldsymbol y}$ is a length-$n$ complex circularly symmetric Gaussian vector with mean ${\boldsymbol A}_{s} {\boldsymbol x}_{s}$ and covariance matrix $\sigma _{\eta }^{2} {\boldsymbol I}_{n}$ . Thus, the conditional distribution for the measurement vector ${\boldsymbol y}$ given its sparse representation is: $$\begin{equation*} p({\boldsymbol y}|{\boldsymbol x}_{s},{\boldsymbol s}) = \frac {1}{(\pi \sigma _{\eta }^{2})^{n}} \exp \left ({- \frac {1}{\sigma _{\eta }^{2}} ||{\boldsymbol y} - {\boldsymbol A}_{s} {\boldsymbol x}_{s} ||_{2}^{2} }\right)\tag{11}\end{equation*}$$

To compute the MAP estimation of ${\boldsymbol x}$ given ${\boldsymbol y}$ , we suggest to first perform the MAP estimation of ${\boldsymbol s}$ given ${\boldsymbol y}$ , and then to proceed with the MAP estimation of ${\boldsymbol x}$ given ${\boldsymbol y}$ and the estimated support $\hat {\boldsymbol s}$ .

Given the observation vector $\boldsymbol {y}$ , the MAP estimation $\hat {\boldsymbol s}$ of the support ${\boldsymbol s}$ is defined by $$\begin{align*} \hat {\boldsymbol s}=&\underset {\boldsymbol u}{\arg \,\max } \Pr (\boldsymbol {s}={\boldsymbol u}|{\boldsymbol y})\\=&\underset {\boldsymbol u}{\arg \,\max } \; p({\boldsymbol y}|\boldsymbol {s}={\boldsymbol u}) \Pr (\boldsymbol {s}={\boldsymbol u}).\end{align*}$$ Assuming a uniform distribution of the support vector, we obtain the MAP criterion (see Appendix V): $$\begin{align*}&\hspace {-1.6pc}\hat {\boldsymbol s} = \underset {\boldsymbol u}{\arg \,\max } \frac {1}{|\det ({\boldsymbol Q}_{u})|} \sum _{\boldsymbol \zeta \in {\Xi ^{K_{a} N}}} \exp \Bigg (\frac {1}{\sigma _\eta ^{2}} \bigg [\bigg ({\boldsymbol y}^{H} {\boldsymbol A}_{u} \\&\quad +\, \left.{\left.{ \frac {\sigma ^{2}_\eta }{\sigma ^{2}_{\epsilon }} {\boldsymbol \zeta }^{H}}\right)\! {\boldsymbol Q}_{u}^{-1} \!\left ({{\boldsymbol A}_{u}^{H} {\boldsymbol y} + \frac {\sigma ^{2}_\eta }{\sigma ^{2}_{\epsilon }} {\boldsymbol \zeta } }\right) \!- \!\frac {\sigma ^{2}_\eta }{\sigma ^{2}_{\epsilon }} ||{\boldsymbol \zeta }||_{2}^{2} }\right] \Bigg) \tag{12}\end{align*}$$ where ${\boldsymbol Q}_{u} = {\boldsymbol A}_{u}^{H} {\boldsymbol A}_{u} + \frac {\sigma _\eta ^{2}}{\sigma _\epsilon ^{2}} {\boldsymbol I}_{K_{a}N}$ and $\boldsymbol {u}$ belongs to the set of supports of length-$K$ vectors with exactly $K_{a}$ non-zero components (equal to one).

In practice, the cardinality of $\Xi ^{K_{a} N}$ is so high that the calculation of the summation in (12) is infeasible. In order to reduce the computational complexity, a tight approximation based on a simplified max-log-MAP algorithm [33] reads $$\begin{equation*} \hat {\boldsymbol s} = \underset {\boldsymbol u}{\arg \,\max } \quad f({\boldsymbol u})\tag{13}\end{equation*}$$ where $$\begin{align*}&\hspace {-1.9pc}f({\boldsymbol u}) = \underset {\boldsymbol \zeta \in {\Xi ^{K_{a} N}}}{\max } \Biggl ({\frac {1}{\sigma _\eta ^{2}}\left ({{\boldsymbol y}^{H} {\boldsymbol A}_{u} + \frac {\sigma ^{2}_\eta }{\sigma ^{2}_{\epsilon }} {\boldsymbol \zeta }^{H}}\right) {\boldsymbol Q}_{u}^{-1} \bigg ({\boldsymbol A}_{u}^{H} {\boldsymbol y} } \\&\qquad +\, {{ \frac {\sigma ^{2}_\eta }{\sigma ^{2}_{\epsilon }} {\boldsymbol \zeta } }\bigg) \!-\! \frac {\sigma ^{2}_\eta }{\sigma ^{2}_{\epsilon }} ||{\boldsymbol \zeta }||_{2}^{2} }\Biggr) \!- \!\log \Big (\left |{\det \left ({{\boldsymbol Q}_{u}}\right)}\right | \Big) \tag{14}\end{align*}$$ Let us mention that depending on the alphabet, the cost function can be simplified which contributes to reduce the computational cost. For example, in the case of BPSK or QPSK modulations, $||{\boldsymbol \zeta }||_{2}^{2}$ can be removed from the maximization.

3) Sub-Optimal Implementation

The max-log-MAP criterion developed in (13) requires an exhaustive search over all of the $\binom {K}{K_{a}}$ possible supports and over all of the possible $\zeta$ . In most cases, the resulting complexity is so high that the optimum solution of (13) is unfeasible. To circumvent this difficulty, we propose to apply a suboptimum iterative greedy pursuit algorithm that exploits the sparsity of ${\boldsymbol s}$ and whose cost function is given in (14). Several algorithms can be used such as the Iterative Hard Thresholding algorithm [34], Model-CoSaMP [35], the Multipath Matching Pursuit (MMP) [25], the Orthogonal Matched Pursuit (OMP) [13] etc. Without loss of generality (since the extension to the other aforementioned greedy algorithms is straightforward), we consider hereinafter an OMP algorithm.

The iterative suboptimum solution of (13) is described in Algorithm 4. We start with an empty support $\hat {\boldsymbol s} = {\boldsymbol 0}_{K}$ (all positions have an inactive default status). At the first iteration, we check each of the $K$ possible positions that can be added to the empty support and evaluate the cost function in (13). Then, we select the entry $i^{\star }$ that gives the maximum value and set $s_{i^{\star }}$ to be 1. Next, at each iteration and given the updated support, we test all remaining positions (with inactive default status) and we choose the one that leads to the maximal value in (13). We proceed exactly in the same manner until the support estimate has exactly $K_{a}$ components equal to 1 that is to say, the algorithm stops after $K_{a}$ iterations. When $K_{a}$ is unknown, a stopping criterion can be applied. For example, the iterative process can go on as long as the cost function (14) decreases.

Algorithm 4 MAP-Based OMP Support Detection

1:

Initialize:

$S^{(0)} = \emptyset$ , $\hat {\boldsymbol {s}} = {\boldsymbol 0}_{K}$

2:

for $l=1$ to $K_{a}$ do

3:

for $i \in \mathcal {K}/ S^{(l-1)}$ do

4:

$\boldsymbol {u}=\hat {\boldsymbol {s}}$ , $u_{i}=1$

5:

Compute $f_{i}=f(\boldsymbol {u})$

6:

end for

7:

$i^{\ast } = \mathop{\mathrm {arg\,max}}\limits_{i\in \mathcal {K}/ S^{(l-1)}}\,\,\,\,f_{i}$

8:

$\hat {s}_{i^{\ast }}=1$ , $S^{(l)}=S^{(l-1)}\cup \{i^{\ast }\}$

9:

end for

10:

return $\hat {\boldsymbol {s}}$

It is important to mention the main difference between the classic OMP algorithm and the proposed version. In fact, the classic OMP also allows to detect the support, but based on a previous data estimation. The $l$ -th OMP iteration consists of a non-zero position detection followed by an LS estimation of the data, which is used to update the residue. It means that the next active element in the support depends on the estimated data of the current iteration. Traditionally, sparse recovery pursues the objective of reconstructing an information source progressively and jointly with the support. The two major drawbacks of classic OMP are: $\boldsymbol (i)$ it estimates data without taking into account their belonging to a finite alphabet and, $\boldsymbol (ii)$ data estimation errors at one iteration will propagate to the next support estimation, which will necessarily affect decision of the active elements of the support. On the contrary, our proposed algorithm focuses entirely on the MAP support detection. Reliable recovery of the support of the sparse signal is critical because missing any point in the support dramatically penalizes the quality of the reconstructed signal. So, our first objective with Algorithm 4 is not to reconstruct an information source but rather to recover the sparse support only. Here there is no need to have a data estimation to detect the active element of the support since the data are introduced through their statistical model. One may wonder about the usefulness of approximating the discrete alphabet by a continuous mixture of Gaussian distributions centered on the alphabet elements and having a small variance: an intuitive understanding is that a continuous model limits with high probability the variation of $x$ around its possible alphabet values, whereas a convex constraint on the data only bounds the belonging region and tolerates its variation inside, without restricting the data around their possible values. Let us consider the alphabet $\mathcal {A} = \{-1,+1,-j,+j\}$ as an example to illustrate this comment. A convex constraint such as $|x| \leq 1$ relaxes the feasible set to the unit circle. The best convex constraint is $|\Re (x)|+|\Im (x)| \leq 1$ which relaxes the feasible set to a diamond whose vertices are the elements of the alphabet. But both relaxations are too loose. Hence the importance of our continuous mixture Gaussian model for discrete alphabets, which, though probabilistic, offers a more accurate constraint on the alphabet.

Once the support has been estimated, a BLS optimization (Algorithm 3) including the alphabet constraints can be applied to estimate the transmitted symbols. Denoting the support estimate by $\hat {\boldsymbol s}$ , the BLS algorithm is applied to estimate $\hat {\boldsymbol x} = \text {BLS}({\boldsymbol y},{\boldsymbol A}_{\hat {\boldsymbol s}},\Xi)$ .

C. Continuous-Phase Modulation

Some M2M applications involving e.g. sensing require battery powered transceivers without the possibility to change nor charge the battery. The energy efficiency is then one of the key performance indicator for such systems. When a massive number of such transceivers is to be deployed, low-cost is another stringent specification. Constant envelope modulations enable both low-cost and energy-efficient transceivers: the signal is not distorted by highly non-linear amplifiers which are cheaper and more energy-efficient.

Continuous phase modulations (CPM) [36] are a particular case of constant envelope modulations which exhibit a more confined power spectrum density compared to their non-continuous phase counterpart.

Several applications show the potential of CPM including early wireless standards (GSM), satellite communications [37], deep-space communications or low-rate long-distance microwave radio links for cellular backhauling [38], Bluetooth, telemetry [39] etc.

The symbols sent through a CPM are dependent, unlike in Section III-B. We propose to take into account this dependence to compute the MAP criteria of support detection in a frequency-selective fading channel. As for linear modulations, there exist binary or non-binary CPM formats. In the remainder of this paper, we will focus on binary CPMs for which the information symbols belong to a binary alphabet, as described in the following subsection.

1) CPM Reminder

The complex envelope of a binary CPM signal can be expressed as [36] $$\begin{equation*} \Psi (t,{\boldsymbol \alpha },h) = \sqrt {\frac {2E_{s}}{T}} \exp \left ({j 2 \pi h \sum _{i=0}^{J-1} \alpha _{i+1} w(t-iT) }\right)\!,\tag{15}\end{equation*}$$ where $E_{s}$ is the energy per symbol assumed without loss of generality identical for every user, $T$ the symbol duration and $h$ the modulation index. The function $w(t)$ is the phase-smoothing response and its derivative is the frequency pulse, assumed to be of duration $LT$ . The information symbols ${\boldsymbol \alpha }=[\alpha _{1},\ldots,\alpha _{J}]$ are assumed independent and belong to the alphabet {±1} for active users and are equal to 0 for inactive users. The phase-smoothing response is a continuous function satisfying the following property: $$\begin{equation*} w(t) = \begin{cases} 0 & \text {when} ~t \leq 0 \\ \dfrac {1}{2} & \text {when} ~t \geq LT \end{cases}\tag{16}\end{equation*}$$

Based on Laurent representation [40] which provides a decomposition of the complex envelope of the CPM over a set of pulses $\varrho _{\ell }(t)$ , $0\leq \ell \leq 2^{L}-1$ , and considering that most of the signal power is concentrated in the first pulse $\varrho _{0}(t)$ , which is called principal component [40], a tight approximation [41], [42] of (15) is given by: $$\begin{equation*} \Psi (t,{\boldsymbol \alpha },h) \approx \sum _{i=0}^{J-1}\beta _{i+1} \varrho _{0}(t-iT),\tag{17}\end{equation*}$$ where $\beta _{i}$ are the pseudo-symbols associated to the principal component: $$\begin{equation*} \beta _{i} = \beta _{i-1} e^{j\pi h \alpha _{i}}.\tag{18}\end{equation*}$$ When the modulation index is rational, that is to say when it can be expressed as $h=m/q$ , where $m$ and $q$ are relatively prime integers, the symbols $\beta _{i}$ can take $q$ values [40].

2) System Model Under Spread Binary CPM Assumption

In the sporadic communication model, the sequence of symbols of the $k-$ th user, denoted by ${\boldsymbol \alpha }_{k}=[\alpha _{k,1},\ldots,\alpha _{k,J}]$ , is spread symbol-wise by a binary pseudo-random spreading sequence ${\boldsymbol c}_{k}$ of length $N_{s}$ . Defining the chip duration $T_{c}=T/N_{s}$ and the spread symbol $\tilde {\alpha }_{k,(i-1)N_{s}+l} = \alpha _{k,i} c_{k,l-1}$ , the spread binary CPM signal is then expressed as: $$\begin{align*}&\hspace {-2.1pc}{\Psi }(t,\tilde {\boldsymbol \alpha }_{k},h) = \sqrt {\frac {2E_{s}}{T}} \exp \Big (j 2 \pi h \\&\qquad \,\times \sum _{i=0}^{J-1} \sum _{l=0}^{N_{s}-1} \tilde {\alpha }_{k,iN_{s}+l+1} w(t-(i N_{s} + l) T_{c}) \Big),\end{align*}$$

The approximation of Laurent decomposition of ${\Psi }(t,\tilde {\boldsymbol \alpha }_{k},h)$ is given by: $$\begin{equation*} {\Psi }(t,\tilde {\boldsymbol \alpha }_{k},h) \approx \sum _{i=0}^{J-1}\sum _{l=0}^{N_{s}-1}\tilde {\beta }_{k,iN_{s}+l+1} \tilde {\varrho }_{0}\big (t-(iN_{s}+l)T_{c}\big),\tag{19}\end{equation*}$$ where $\tilde {\beta }_{k,iN_{s}+l}$ are the pseudo-symbols associated to the principal component $\tilde {\varrho }_{0}(t)$ : $$\begin{equation*} \tilde {\beta }_{k,iN_{s}+l} = \tilde {\beta }_{k,iN_{s}+l-1} e^{j\pi h \tilde {\alpha }_{k,iN_{s}+l}}.\tag{20}\end{equation*}$$

In order to obtain a vectorial formulation of the MUD problem, we first need to overcome the non-linearity induced by the pseudo-symbols $\tilde {\beta }_{k,iN_{s}+l}$ . When the $k$ -th user is active (otherwise $\tilde {\beta }_{k,iN_{s}+l}=0$ in case of $k$ -th user inactivity), by using the definition of $\tilde {\boldsymbol {\alpha }}_{k}$ as well as (20), we can express the pseudo-symbols associated to an active user as: $$\begin{equation*} \tilde {\beta }_{k,(i-1)N_{s}+l} = \delta _{k,i} e^{ j \pi h \kappa _{k,l} \alpha _{k,i}}\tag{21}\end{equation*}$$ where $\kappa _{k,l} = \sum _{\ell =0}^{l-1} c_{k,\ell }$ and $\delta _{k,i} = e^{j \pi h \kappa _{k,N_{s}} \sum _{m=1}^{i-1} \alpha _{k,m} }$ . $\delta _{k,i}$ conveys the memory effect of the CPM signal. As $\kappa _{k,l}$ is an integer and $\alpha _{k,i} \in \{\pm 1\}$ when the user is active, we can further develop (21) as $$\begin{align*} \tilde {\beta }_{k,(i-1)N_{s}+l}=&\frac {\delta _{k,i}}{2} \left ({(1 \!+\! \alpha _{k,i})e^{j \pi h \kappa _{k,l}} }\!+\!\, { (1 \!-\! \alpha _{k,i})e^{-j \pi h\kappa _{k,l}} }\right) \\=&\delta _{k,i} \left ({\cos (\pi h \kappa _{k,l}) + j \alpha _{k,i}\sin (\pi h \kappa _{k,l})}\right).\tag{22}\end{align*}$$ We can thus deduce an expression of $\tilde {\beta }_{k,(i-1)N_{s}+l}$ that takes into account the activity ($\alpha _{k,i} \in \{\pm 1\}$ ) or inactivity ($\alpha _{k,i} =0$ ) of the user: $$\begin{align*}&\hspace {-2.6pc}\tilde {\beta }_{k,(i-1)N_{s}+l} =\delta _{k,i} \Biggl ({\alpha _{k,i}^{2} \cos (\pi h \kappa _{k,l}) } \\&\qquad \qquad \qquad \qquad \qquad { +\, j \alpha _{k,i}\sin (\pi h \kappa _{k,l})}\Biggr) \tag{23}\end{align*}$$ Let us introduce $\theta _{k,i}=\delta _{k,i}\alpha _{k,i}$ , $\bar {\theta }_{k,i}=\delta _{k,i}\left ({\alpha _{k,i}}\right)^{2}$ , $d_{k,l}=j \sin (\pi h \kappa _{k,l})$ and $v_{k,l}= \cos (\pi h \kappa _{k,l})$ . Then (23) becomes $$\begin{equation*} \tilde {\beta }_{k,(i-1)N_{s}+l} = \theta _{k,i} d_{k,l} + \bar {\theta }_{k,i} v_{k,l}\tag{24}\end{equation*}$$ We can go further in the model by first defining ${\boldsymbol \theta }_{k} = \left [{\theta _{k,1},\cdots,\theta _{k,J} }\right]^{T}$ , $\bar {\boldsymbol \theta }_{k} = \left [{\bar {\theta }_{k,1},\cdots,\bar {\theta }_{k,J}}\right]^{T}$ , ${\boldsymbol d}_{k} = \left [{d_{k,1}, {\dots }, d_{k,N_{s}} }\right]^{T}$ and ${\boldsymbol v}_{k} = \left [{v_{k,1}, {\dots }, v_{k,N_{s}} }\right]^{T}$ . Then we introduce ${\boldsymbol D}_{k}={\boldsymbol I}_{J} \otimes {\boldsymbol d}_{k}$ , ${\boldsymbol V}_{k}={\boldsymbol I}_{J} \otimes {\boldsymbol v}_{k}$ and $\tilde {\boldsymbol \beta }_{k}=\left [{\tilde {\beta }_{k,1},\cdots,\tilde {\beta }_{k,JN_{s}}}\right]^{T}$ . By using (24), we obtain an expression of $\tilde {\boldsymbol \beta }_{k}$ which reads $$\begin{equation*} \tilde {\boldsymbol \beta }_{k} = {\boldsymbol D}_{k}{\boldsymbol \theta }_{k} + {\boldsymbol V}_{k} \bar {\boldsymbol \theta }_{k}\tag{25}\end{equation*}$$

Let us now derive the matrix formulation of the problem. We assume a frequency-selective fading channel whose discrete impulse response is sampled at chip period $T_{c}$ and is defined by the channel tap vector ${\boldsymbol g}_{k} = [g_{k,0}, g_{k,1}, \ldots, g_{k,L_{g}-1}]^{T}$ . We assume that ${\boldsymbol g}_{k}$ is constant over the whole frame. The aggregation node receives the synchronous superposition of the modulated frames corrupted by additive noise. At time instant $t$ , the received signal reads $$\begin{equation*} y(t) = \sum _{k=1}^{K}\sum _{j=0}^{L_{g}-1} g_{k,j} {\Psi }(t-jT_{c},\tilde {\boldsymbol \alpha }_{k},h) + b(t),\tag{26}\end{equation*}$$ where $b(t)$ is a white Gaussian noise process with variance $\sigma _{b}^{2}$ . Given the Laurent decomposition approximation (19), the matched-filter computes the correlation using the adaptive filter $\check {\varrho }_{0}(t) \triangleq \tilde {\varrho }_{0}(-t)$ over each chip duration. For the $(i-1)N_{s}+l$ -th chip interval, we obtain the following sample: $$\begin{align*} y_{i,l}=&\int y(t) \tilde {\varrho }_{0}\left ({t-\left ({(i-1)N_{s}+l-1}\right)T_{c}}\right) dt \\=&z_{i,l} + \eta _{i,l}. \tag{27}\end{align*}$$ Collecting all $JN_{s}$ correlation samples, we define ${\boldsymbol y}_{i}=[y_{i,1},\cdots,y_{i,N_{s}}]^{T}$ , ${\boldsymbol z}_{i}=[z_{i,1},\cdots,z_{i,N_{s}}]^{T}$ , ${\boldsymbol \eta }_{i}=[\eta _{i,1},\cdots, \eta _{i,N_{s}}]^{T}$ and then ${\boldsymbol y} = [{\boldsymbol y_{1}}^{T},\cdots,{\boldsymbol y}_{J}^{T}]^{T}$ , ${\boldsymbol z} = [{\boldsymbol z_{1}}^{T},\cdots,{\boldsymbol z}_{J}^{T}]^{T}$ , ${\boldsymbol \eta } = [{\boldsymbol \eta _{1}}^{T},\cdots,{\boldsymbol \eta }_{J}^{T}]^{T}$ . We also fix $n=JN_{s}$ .

The noise coefficient $\eta _{i,l}$ is given by: $$\begin{equation*} \eta _{i,l} = \int b(t) \tilde {\varrho }_{0}\left ({t-\left ({(i-1)N_{s}+l-1}\right) T_{c}}\right) dt.\tag{28}\end{equation*}$$

The sample $z_{i,l}$ introduced in (27) results from the contribution of the $K$ user signals only. Using the approximation (19), it can be modeled as: $$\begin{equation*} z_{i,l} \approx \sum _{k=1}^{K} \sum _{j=0}^{L_{g}-1} g_{k,j} \sum _{u=-LN_{s}+1}^{LN_{s} - 1} \lambda _{u} \tilde {\beta }_{k, (i-1) N_{s} + l + j - u }\tag{29}\end{equation*}$$ with $$\begin{equation*} \lambda _{u} \triangleq \int \tilde {\varrho }_{0}(t) \tilde {\varrho }_{0}(t-u T_{c}) dt = \lambda _{-u}.\tag{30}\end{equation*}$$ Using both (25) and (29), the vector ${\boldsymbol z}$ can be written as follows: $$\begin{align*} {\boldsymbol z}=&\sum _{k=1}^{K} {\boldsymbol G}_{k} {\boldsymbol \Lambda } {\boldsymbol \beta }_{k} \\=&\sum _{k=1}^{K} {\boldsymbol G}_{k} {\boldsymbol \Lambda } {\boldsymbol D}_{k} {\boldsymbol \theta }_{k} + {\boldsymbol G}_{k} {\boldsymbol \Lambda } {\boldsymbol V}_{k} \bar {\boldsymbol \theta }_{k} \tag{31}\end{align*}$$ where

and

both of size $JN_{s} \times JN_{s}$ . Let us now introduce ${\boldsymbol x} = \left [{ {\boldsymbol x}_{1}^{T}, {\boldsymbol x}_{2}^{T}, {\dots }, {\boldsymbol x}_{K}^{T} }\right]^{T}$ with ${\boldsymbol x}_{k} = \left [{ \left ({{\boldsymbol \theta }_{k}}\right)^{T}, \left ({\bar {\boldsymbol \theta }_{k}}\right)^{T}}\right]^{T}$ and ${\boldsymbol A}\,\,= \left [{{\boldsymbol G}_{1} {\boldsymbol \Lambda } {\boldsymbol D}_{1},{\boldsymbol G}_{1} {\boldsymbol \Lambda } {\boldsymbol V}_{1}, {\dots }, {\boldsymbol G}_{K} {\boldsymbol \Lambda } {\boldsymbol D}_{K},{\boldsymbol G}_{K} {\boldsymbol \Lambda } {\boldsymbol V}_{K} }\right]$ . The final mathematical model is thus given by $$\begin{equation*} {\boldsymbol y} = {\boldsymbol A} {\boldsymbol x} + {\boldsymbol \eta }.\tag{34}\end{equation*}$$ The vector $\boldsymbol {x}$ is block-sparse. The length of ${\boldsymbol x}_{k}$ equals $N=2J$ as it gathers both $\theta _{k,i}=\delta _{k,i} \alpha _{k,i}$ and $\bar {\theta }_{k,i}=\delta _{k,i} \left ({\alpha _{k,i}}\right)^{2}$ . The linearization of the CPM provides a linear formulation of the observation, but the system is still not linear as far as the original transmitted symbols are concerned. However, the cardinality of the support is unchanged: this means that the number of non-zero blocks is the same while the number of unknown symbols is doubled in the case of the proposed CPM linearization. This justifies our approach and the importance of first estimating the support.

3) MAP Criterion for Support Detection (Dependent Symbols)

In the model described above, the parameter $\kappa _{k,l} = \sum _{\ell =0}^{l-1} c_{k,\ell }$ depends on the spreading sequences used to get enough observations to recover the signal. Let us denote by $\Upsilon _{k}$ (respectively $\bar {\Upsilon }_{k}$ ) the set of values of $\theta _{k,i}$ (respectively $\bar {\theta }_{k,i}$ ) when the $k$ -th user is active. Then we denote by $\Xi _{s}$ the set of all realizations of $\boldsymbol {x}_{s}$ . If $\mathcal {S}$ stands for the active user index set, $\Xi _{s}$ is defined as the Cartesian product of the sets $\Upsilon _{k}\cup \bar {\Upsilon }_{k}$ , $k \in \mathcal {S}$ .

Let ${\boldsymbol \Omega }$ stand for the $JN_{s} \times JN_{s}$ covariance matrix of the filtered noise $\boldsymbol {\eta }$ . Given $\ell =(i-1)N_{s}+l$ and $j=(i'-1)N_{s}+l'$ , the entry of ${\boldsymbol \Omega }$ at the $\ell$ -th row and $j$ -th column is denoted by $\omega _{\ell,j}$ and is computed as follows: $$\begin{align*} {\omega }_{\ell,j}=&\mathbb {E}(\eta _{i,l} \eta _{i',l'}^{*}) \tag{35}\\=&\iint _{\mathbb {R}^{2}} \mathbb {E}(b(t) b(t')^{*}) \tilde {\varrho }_{0}(t-\ell T_{c}) \tilde {\varrho }_{0}(t'-jT_{c}) dt dt' \\=&\sigma _{b}^{2}\int _{\mathbb {R}} \tilde {\varrho }_{0}(t-\ell T_{c}) \tilde {\varrho }_{0}(t-jT_{c}) dt \\=&\begin{cases} \sigma _{b}^{2} \lambda _{\ell -j} & \text {if }~|\ell -j| \leq LN_{s}-1\\ 0 & \text {otherwise.} \end{cases}\tag{36}\end{align*}$$ where $\mathbb {E}(\cdot)$ denotes the mathematical expectation. Thus ${\boldsymbol \Omega } = \sigma _{b}^{2} {\boldsymbol \Lambda }$ . As the filtered noise is a complex circularly symmetric Gaussian vector with zero mean and covariance matrix ${\boldsymbol \Omega }$ , the conditional distribution for the filtered measures vector ${\boldsymbol y}$ given its sparse representation becomes: $$\begin{equation*} p({\boldsymbol y}|{\boldsymbol x}_{s},{\boldsymbol s}) = \frac {1}{\pi ^{n} |{\boldsymbol \Omega }| } \exp \Big (-({\boldsymbol y} - {\boldsymbol A}_{s} {\boldsymbol x}_{s})^{H} {\boldsymbol \Omega }^{-1} ({\boldsymbol y} - {\boldsymbol A}_{s} {\boldsymbol x}_{s}) \Big)\end{equation*}$$

Following the same approach as in Section III-B.2, we obtain the max-log-MAP-based approximation of the support given by Equation (51), as shown at the bottom of page 12,

in Appendix B.

It is possible to simplify the cost function in (51). First, CPM signals have a constant modulus, and so $||{\boldsymbol \zeta }||_{2}$ is a constant whatever ${\boldsymbol \zeta }$ . Second, given $\boldsymbol {u}$ , the form of the random matrix ${\boldsymbol A}_{u}$ allows to have a product ${\boldsymbol A}_{u}^{H}\boldsymbol {\Omega }^{-1} {\boldsymbol A}_{u}$ with dominant values on the diagonal. Thus for two different realizations of ${\boldsymbol \zeta }$ , we have a slight variation of the term ${\boldsymbol \zeta } \left ({{\boldsymbol Q}_{u}'}\right)^{-1} {\boldsymbol \zeta }$ compared to the variation undergone by the term $\Re \{ {\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{u} {\boldsymbol Q_{u}'}^{-1} {\boldsymbol \zeta } \}$ . As a consequence, we propose the following simplified support estimation criterion: $$\begin{align*}&\hspace {-0.2pc}\hat {\boldsymbol s} = \underset {\boldsymbol u}{\arg \,\max } \Biggl \{{ {\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{u} {\boldsymbol Q_{u}'}^{-1} {\boldsymbol A}_{u}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol y} } \\&\qquad { + \frac {2}{\sigma _\epsilon ^{2}} \underset {\boldsymbol \zeta }{\max } ~\Re \{ {\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{u} {\boldsymbol Q_{u}'}^{-1} {\boldsymbol \zeta } \} - \log \left ({\left |{\det \left ({{\boldsymbol Q}_{u}'}\right)}\right | }\right) }\Biggr \}. \\ {}\tag{37}\end{align*}$$ Depending on the CPM, the alphabet $\Xi$ (see Section IV-C for an example) may have some properties that allow the simplification of (37), which makes the computation cost far less expensive.

4) Data Estimation

Once the support has been estimated, the data estimation is carried out in two steps. The linear decomposition of the CPM signal involves the vectors $\boldsymbol {x}_{k}$ whose component vectors $\boldsymbol {\theta }_{k}$ and $\bar {\boldsymbol {\theta }_{k}}$ are mutually dependent. The first step aims to estimate the elements of vector ${\boldsymbol { x}_{\hat {\boldsymbol {s}}}}$ without considering their dependence. Therefore using the estimated support $\hat {\boldsymbol s}$ , we perform the data detection taking into account the finite alphabet constraints as $\hat {\boldsymbol x} = \text {BLS}({\boldsymbol y},{\boldsymbol A}_{\hat {\boldsymbol s}},\Xi)$ .

The solution of this optimization problem gives the estimated values of $\theta _{k,i}$ and $\bar {\theta }_{k,i}$ which will be denoted respectively by $\widehat {\theta _{k,i}}$ and $\widehat {\bar {\theta }_{k,i}}$ .

Then the second step aims to determine the transmitted data symbols $\alpha _{k,i}$ from the estimated coefficients $\widehat {\theta _{k,i}}$ and $\widehat {\bar {\theta }_{k,i}}$ . We can exploit the following relation $$\begin{equation*} \delta _{k,i+1}=\delta _{k,i}e^{j\pi h \kappa _{k}\alpha _{k,i}}\tag{38}\end{equation*}$$ and apply the Viterbi algorithm on a trellis whose states belong to the set of values of $\delta _{k,i}$ . We denote and define this set of states by $\Delta _{k}$ . The transition from one state to another is ruled by (38). At time instant $i$ , the transition from the state $\upsilon _{i} \in \Delta _{k}$ to the state $\upsilon _{i+1}\in \Delta _{k}$ is valid if there exists $\alpha$ such that $\upsilon _{i+1}=\upsilon _{i}e^{j\pi h \kappa _{k}\alpha }$ . The associated branch metric is computed as $$\begin{equation*} \gamma (\upsilon _{i},\upsilon _{i+1})=\frac {1}{2} \left ({{\left ({\upsilon _{i}\alpha }\right)}^{\ast } \widehat {\theta _{k,i}} + {\left ({\upsilon _{i}\alpha ^{2}}\right)}^{\ast } \widehat {\bar {\theta }_{k,i}} }\right).\tag{39}\end{equation*}$$ Once the Viterbi algorithm is over, given the survivor path $\{\widehat {\upsilon _{0}}, \widehat {\upsilon _{1}}, {\dots },\widehat {\upsilon _{N}}\}$ , we decide $\widehat {\alpha _{k,i}}=\alpha (\widehat {\upsilon }_{i},\widehat {\upsilon }_{i+1})$ .

SECTION IV. Performance Evaluation

In this section, the algorithms proposed in Section III are compared with some state-of-the-art algorithms. First, the complexity of the considered algorithms is estimated. Then, the case of independent symbols is studied through linear BPSK modulation and the case of dependent symbols is studied through MSK modulation, that is a binary CPM modulation with $h=1/2$ and a linear phase smoothing response.

A. Complexity Analysis

To give a rough estimation of the complexity, we only kept the most complex computational operations and we neglected the others.

For the GOMP algorithms, the overall complexity is dominated by the LS or BLS estimation step, maximum at last iteration and given by $\mathcal {O}((K_{a}N)^{3})$ . Hence, the complexity for the $K_{a}$ iterations is upper-bounded by $\mathcal {O}(K_{a}^{4}N^{3})$ .

For the GMMP algorithms, the overall complexity is also dominated by the LS or BLS estimation step which has to be performed at most $m^{2}$ times at each iteration, where $m$ is the number of generated candidates. Therefore, considering a constant list size $m$ , the overall complexity is upper-bounded by $\mathcal {O}(K_{a}^{4}N^{3})$ .

The complexity of the proposed algorithm is dominated by the complexity of the support detection (Algorithm 4) and particularly Step 5 dominated by $\mathcal {O}((K_{a}N)^{3})$ which has to be performed $K K_{a}$ times. Considering a constant activation probability $p_{a}$ the overall complexity of Algorithm 4 is given by $\mathcal {O}(K_{a}^{5}~N^{3})$ .

B. Linear Modulation (Independent Symbols)

We chose the Binary Phase Shift Keying (BPSK) modulation. Activity-detection error rate for BPSK modulation with $K=80$ users, $N_{s}=25$ and $p_{a}=10$ % In Figure 2, the ratio of erroneous activity detection is illustrated as a function of $E_{s}/N_{0}$ where $E_{s}$ is the average symbol energy and $N_{0}$ is the one-sided white noise spectral density. The ED-LS algorithm from [3] performs poorly compared to the greedy GOMP and GMMP ($m=3$ ) algorithms, and taking into account the alphabet constraint by replacing the LS estimation step by the a BLS estimation improves consequently the performance of the greedy algorithm (BLS-GOMP and BLS-GMMP). Compared to BLS-GOMP and BLS-GMMP, the proposed algorithm clearly outperforms them even at low signal-to-noise ratios (SNR), thus illustrating the benefit of both constraining the alphabet and also performing activity detection before data estimation.

FIGURE 2.

Activity-detection error rate for BPSK modulation with frames of $N=50$ symbols for $K=80$ users, $N_{s}=25$ and $p_{a}=5$ %.

The symbol error rate performance for the same algorithms is illustrated in Figure 3. It is also compared to oracle bounds obtained by assuming the perfect knowledge of the activity of each user. Thus only the data estimation has to be performed either using an LS algorithm (Oracle-LS) or a BLS algorithm (Oracle-BLS). We can observe that the proposed algorithm performs very close to the lower bound (Oracle BLS) whereas the other ones exhibit an error floor in the high SNR regime.

FIGURE 3.

Symbol error rate for BPSK modulation with frames of $N=50$ symbols for $K=80$ users, $N_{s}=25$ and $p_{a}=5$ %.

C. CPM (Dependent Symbols)

For the simulations of CPM, we chose the MSK (Minimum Shift Keying) modulation which is a popular CPM modulation defined by a rectangular frequency pulse of length $L=1$ and by a modulation index equal to $h=1/2$ . In this case the elements of the vector ${\boldsymbol x}$ belong to the alphabet $\Xi =\{\pm 1, \pm j\}$ .

The alphabet $\Xi$ has some properties that allow the simplification of (37) and thus offers a reduction of the computation cost. Defining ${\boldsymbol v} = {\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{u} {\boldsymbol Q_{u}'}^{-1}$ , the max-log-MAP estimation of the support can be approximated by: $$\begin{align*}&\hspace {-0.4pc}\hat {\boldsymbol s} \approx \underset {\boldsymbol u}{\arg \,\max } \Bigg \{ {\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{u} {\boldsymbol Q_{u}'}^{-1} {\boldsymbol A}_{u}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol y} \\&\quad +\, \left.{\frac {2}{\sigma _\epsilon ^{2}} \sum _{m=1}^{K_{a} N} \max (|\Re \{ v_{m} \}|,|\Im \{v_{m} \}|) - \log \left ({\left |{\det \left ({{\boldsymbol Q}_{u}'}\right)}\right | }\right) }\right \}.\end{align*}$$

The comparison of the activity detection error rate is illustrated in Figure 4 using the same algorithms as for the linear modulation case. We can clearly see that the algorithms which are based on joint data and support detection fail to recover the signal. The BLS greedy algorithms, despite incorporating the finite alphabet constraint, exhibit a noticeable error floor after 20dB. On the contrary, our proposed algorithm allows a significant gain of nearly 6 dB over BLS-GMMP ($m=3$ ) without exhibiting any observable error floor. This allows us to deduce that first the consideration of convex constraints on the alphabets only slightly improves the performance, since convex constraints do not necessarily give a fine description of the data but only bound them. Moreover, estimating the support thanks to estimated data is prompt to error propagation.

FIGURE 4.

Activity-detection error rate for MSK modulation with frames of $N=50$ symbols for $K=80$ users, $N_{s}=25$ and $p_{a}=5$ %.

The symbol error rate (SER) performance is illustrated in Figure 5 where a comparison is also performed with oracle bounds as for the linear modulation. The conclusions drawn for the activity detection error rate simulations are still observable in the symbol error rate simulations. Our proposed algorithm converges to the lower Oracle-BLS bound.

FIGURE 5.

Symbol error rate for MSK modulation with frames of $N=50$ symbols for $K=80$ users, $N_{s}=25$ and $p_{a}=5$ %.

Scaling the detection to a higher number of users does not degrade the performance, on the contrary: the activity detection error rate as a function of the number of users $K$ is depicted in Figure 6 for 2 signal-to-noise ratios. Note that to perform this comparison, the ratio $K/N_{s}$ is kept constant, and the frame are made of $N=30$ symbols.

FIGURE 6.

Activity-detection error rate as a function of the number of user $K$ for $N=30$ and $p_{a}=5$ %.

SECTION V. Conclusions

This paper addressed the problem of block-sparse finite-alphabet signal recovery. Usual greedy-based approaches jointly detect the signal support and data, and do not consider the finite-alphabet in the optimization constraints. In this paper, we rather focus on algorithms where support detection and data estimation are performed successively. We propose to take into account the finite alphabet both in the cost function for support detection and in the bound-constrained least-squares optimization algorithm for data estimation. The proposed support detection greedy algorithms rely on a maximum-a-posteriori-like cost function based on a Gaussian mixture distribution approximation model for the discrete data symbol distribution. We considered two cases: linear modulation with independent symbols and binary continuous phase modulation (CPM) where the symbols are dependent. Simulations were carried out in the context of sporadic multiuser communications for which CPM is a good candidate combined with energy-efficient power amplifiers. First, the proposed algorithms significantly enhance the user activity detection probability compared to usual greedy algorithms. Besides, scaling up the number of users doesn’t degrade the performance. Secondly, the proposed algorithms strongly improve the data estimation probability with convergence towards the performance of their corresponding oracle. Future work will consider the extension to block-sparse $M$ -ary CPM signals.

Appendix A MAP Estimation of the Support (Linear Modulation)

All the integrals and all the sums in this Appendix will be evaluated for ${\boldsymbol x}_{s} \in \mathbb {C}^{K_{a}N}$ and over the set ${\boldsymbol \zeta } \in \Xi ^{K_{a}N}$ respectively. Hence for compactness purpose, we will omit to specify the integration space and the summation set in the upcoming equations.

We begin by developing an expression for $\Pr ({\boldsymbol y}|{\boldsymbol s})$ by integrating over all possible values of ${\boldsymbol x}_{s} \in \mathbb {C}^{K_{a}N}$ using the expression of equations (10) and (11): $$\begin{align*} p({\boldsymbol y}|{\boldsymbol s})=&\int \Pr ({\boldsymbol y}|{\boldsymbol x}_{s},{\boldsymbol s}) \Pr ({\boldsymbol x}_{s}|{\boldsymbol s}) d {\boldsymbol x}_{s} \\=&\frac {1}{ (M \pi \sigma _{\epsilon }^{2})^{ K_{a}N}} \frac {1}{(\pi \sigma _{\eta }^{2})^{m}} \\&\times \sum \int \exp \left ({- \frac {||{\boldsymbol y}-{\boldsymbol A}_{s} {\boldsymbol x}_{s}||_{2}^{2}}{\sigma _{\eta }^{2}} - \frac {||{\boldsymbol x}_{s}-{\boldsymbol \zeta }||_{2}^{2}}{\sigma _{\epsilon }^{2}} }\right) d {\boldsymbol x}_{s} \\ {}\tag{40}\end{align*}$$

The exponential term can be developed by considering its variable $$\begin{align*} G({\boldsymbol x}_{s})=&\frac {||{\boldsymbol y} - {\boldsymbol A}_{s} {\boldsymbol x}_{s}||_{2}^{2}}{\sigma _{\eta }^{2}} + \frac {||{\boldsymbol x}_{s}-{\boldsymbol \zeta }||_{2}^{2}}{\sigma _{\epsilon }^{2}} \\=&\frac {||{\boldsymbol y}||_{2}^{2}}{\sigma _{\eta }^{2}} + \frac {1}{\sigma _{\eta }^{2}} {\boldsymbol x}_{s}^{H} {\boldsymbol A}_{s}^{H} {\boldsymbol A}_{s} {\boldsymbol x}_{s} - \frac {2}{\sigma _{\eta }^{2}} \Re \left ({{\boldsymbol y}^{H} {\boldsymbol A}_{s} {\boldsymbol x}_{s}}\right) \\&+ \frac {1}{\sigma _{\epsilon }^{2}} \left ({{\boldsymbol x}_{s}^{H} {\boldsymbol x}_{s} + {\boldsymbol \zeta }^{H} {\boldsymbol \zeta } - 2 \Re ({\boldsymbol \zeta }^{H} {\boldsymbol x}_{s}) }\right)\tag{41}\end{align*}$$ Let us introduce ${\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s} = \frac {1}{\sigma _\eta ^{2}} {\boldsymbol A}_{s}^{H} {\boldsymbol A}_{s} + \frac {1}{\sigma _\epsilon ^{2}} {\boldsymbol I}_{K_{a}N}$ . ${\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s}$ is positive-definite matrix and can thus be inverted. $$\begin{align*} G({\boldsymbol x}_{s})=&{\boldsymbol x}_{s}^{H} {\boldsymbol R}_{s}^{H}{\boldsymbol R}_{s} {\boldsymbol x}_{s} + \frac {1}{\sigma _\eta ^{2}} ||{\boldsymbol y}||_{2}^{2} \\&-\, \frac {2}{\sigma _\eta ^{2}} \Re \left ({{\boldsymbol y}^{H} {\boldsymbol A}_{s} ({\boldsymbol R}_{s}^{H}{\boldsymbol R}_{s}) ^{-1} {\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s} {\boldsymbol x}_{s} }\right) \\&+ \frac {1}{\sigma _\epsilon ^{2}} \left ({{\boldsymbol \zeta }^{H} {\boldsymbol \zeta } - 2 \Re ({\boldsymbol \zeta }^{H} ({\boldsymbol R}_{s}^{H}{\boldsymbol R}_{s})^{-1} {\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s} {\boldsymbol x}_{s}) }\right) \\=&||{\boldsymbol R}_{s} {\boldsymbol x}_{s} - {\boldsymbol R}_{s} {\boldsymbol \nu }_{\boldsymbol \zeta }||_{2}^{2} + \gamma _{\boldsymbol \zeta }\tag{42}\end{align*}$$ with $$\begin{align*} {\boldsymbol \nu }_{\boldsymbol \zeta }=&({\boldsymbol R}_{s}^{H}{\boldsymbol R}_{s}) ^{-1} \left ({\frac {1}{\sigma _\eta ^{2}}{\boldsymbol A}_{s}^{H} {\boldsymbol y} + \frac {1}{\sigma _\epsilon ^{2}} {\boldsymbol \zeta } }\right) \tag{43}\\ \gamma _{\boldsymbol \zeta }=&\frac {1}{\sigma _\eta ^{2}} ||{\boldsymbol y}||_{2}^{2} + \frac {1}{\sigma _\epsilon ^{2}} ||{\boldsymbol \zeta }||_{2}^{2} - {\boldsymbol \nu }_{\boldsymbol \zeta }^{H} {\boldsymbol R}_{s}^{H}{\boldsymbol R}_{s} {\boldsymbol \nu }_{\boldsymbol \zeta }\tag{44}\end{align*}$$ The probability density function of ${\boldsymbol y}$ given ${\boldsymbol s}$ can thus be written as $$\begin{align*}&\hspace {-2.4pc}p({\boldsymbol y}|{\boldsymbol s}) = \frac {1}{ (M \pi \sigma _{\epsilon }^{2})^{K_{a}N}} \frac {1}{(\pi \sigma _{\eta }^{2})^{m}}\sum \exp (-\gamma _{\boldsymbol \zeta }) \\[3pt]&\quad \times \int \exp \left ({- ({\boldsymbol x}_{s} - {\boldsymbol \nu }_{\boldsymbol \zeta })^{H}{\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s} ({\boldsymbol x}_{s} - {\boldsymbol \nu }_{\boldsymbol \zeta }) }\right) d {\boldsymbol x}_{s}\tag{45}\end{align*}$$ Using the predefined multidimensional Gaussian form $$\begin{align*}&\hspace {-4.8pc}\int \exp \left ({- ({\boldsymbol x}_{s} - {\boldsymbol \nu }_{\boldsymbol \zeta })^{H}{\boldsymbol R}_{s}^{H}{\boldsymbol R}_{s}({\boldsymbol x}_{s} - {\boldsymbol \nu }_{\boldsymbol \zeta })) }\right) d {\boldsymbol x}_{s} \\[3pt]=&\int \exp \left ({- ||{\boldsymbol R}_{s} ({\boldsymbol x}_{s} - {\boldsymbol \nu }_{\boldsymbol \zeta })||_{2}^{2} }\right) d {\boldsymbol x}_{s} \\[3pt]=&\left |{\mathrm {det}\left ({({\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s})^{-1}}\right)}\right | \pi ^{K_{a}N} \\[3pt]&\times \, \int \frac { \exp \left ({-|| {\boldsymbol R}_{s} ({\boldsymbol x}_{s} - {\boldsymbol \nu }_{\boldsymbol \zeta })||_{2}^{2} }\right)}{\left |{\mathrm {det}(({\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s})^{-1})}\right | \pi ^{K_{a}N}}d {\boldsymbol x}_{s} \\=&\frac {\pi ^{K_{a}N}}{|\mathrm {det}({\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s})|}\tag{46}\end{align*}$$ We thus obtain $$\begin{align*}&\hspace {-2.2pc}p({\boldsymbol y}|{\boldsymbol s}) = \frac {C}{|\mathrm {det}({\boldsymbol Q}_{s})|} \sum \exp \Biggl ({\frac {1}{\sigma _\eta ^{2}} \Biggl [{ \left ({{\boldsymbol y}^{H} {\boldsymbol A}_{s} + \frac {\sigma _\eta ^{2}}{\sigma _\epsilon ^{2}} \,{\boldsymbol \zeta }^{H}}\right)}} \\&\qquad \times \, {{ {\boldsymbol Q}_{s}^{-1} \left ({{\boldsymbol A}_{s}^{H} {\boldsymbol y} + \frac {\sigma _\eta ^{2}}{\sigma _\epsilon ^{2}} \,{\boldsymbol \zeta } }\right) - \frac {\sigma _\eta ^{2}}{\sigma _\epsilon ^{2}} ||{\boldsymbol \zeta }||_{2}^{2} }\Biggr] }\Biggr)\tag{47}\end{align*}$$ with $C = \frac {\exp (-||{\boldsymbol y}||_{2}^{2} / \sigma _\eta ^{2})}{(M \sigma _{\epsilon }^{2})^{K_{a}N} (\pi \sigma _{\eta }^{2})^{m} }$ is a constant independent of the sparsity support and ${\boldsymbol Q}_{s} = \sigma _\eta ^{2} {\boldsymbol R}_{s}^{H} {\boldsymbol R}_{s} = { \boldsymbol A}_{s}^{H} {\boldsymbol A}_{s} + \frac {\sigma _\eta ^{2}}{\sigma _\epsilon ^{2}} {\boldsymbol I}_{K_{a}N}$ . When considering an uncoded slotted ALOHA scheme, $\Pr ({\boldsymbol s})$ is constant so the estimator for the support is given by: $$\begin{align*} \hat {\boldsymbol s}=&\underset {\boldsymbol u}{\arg \,\max } \,\Pr ({\boldsymbol s}={\boldsymbol u}|{\boldsymbol y}) \\=&\underset {\boldsymbol u}{\arg \,\max }\,\, p({\boldsymbol y|{\boldsymbol s}={\boldsymbol u}})\, \Pr ({\boldsymbol s}={\boldsymbol u})\end{align*}$$ Hence: $$\begin{align*} \hat {\boldsymbol s}=&\underset {\boldsymbol u}{\arg \,\max }\, \frac {1}{|\mathrm {det}({\boldsymbol Q}_{u})|} \sum \exp \Biggl ({\frac {1}{\sigma _\eta ^{2}} } \\&\times \, {\!\! \left [{\! \left ({{\boldsymbol y}^{H} {\boldsymbol A}_{u} \!+\!\, \!\!\frac {\sigma _\eta ^{2}}{\sigma _\epsilon ^{2}} {\boldsymbol \zeta }^{H}}\right )\! {\boldsymbol Q}_{u}^{-1} \!\left ({{\boldsymbol A}_{u}^{H} {\boldsymbol y} \!+\!\, \frac {\sigma _\eta ^{2}}{\sigma _\epsilon ^{2}} {\boldsymbol \zeta } }\right )\! }{ - \frac {\sigma _\eta ^{2}}{\sigma _\epsilon ^{2}} ||{\boldsymbol \zeta }||_{2}^{2} }\right ] }\Biggr )\end{align*}$$

Appendix B MAP Estimation of the Support (CPM)

All the integrals and all the sums in this Appendix will be evaluated for ${\boldsymbol x}_{s} \in \mathbb {C}^{K_{a}N}$ and over the set ${\boldsymbol \zeta } \in \Xi ^{K_{a}N}$ respectively. Hence for compactness purpose, we will omit to specify the integration space and the summation set in the upcoming equations. $$\begin{align*} p({\boldsymbol y}|{\boldsymbol s})=&\int \Pr ({\boldsymbol y}|{\boldsymbol x}_{s},{\boldsymbol s}) \Pr ({\boldsymbol x}_{s}|{\boldsymbol s}) \,d {\boldsymbol x}_{s} \\=&\frac {\sum \int \exp \left ({- ({\boldsymbol y}-{\boldsymbol A}_{s} {\boldsymbol x}_{s})^{H} {\boldsymbol \Omega }^{-1} ({\boldsymbol y}-{\boldsymbol A}_{s} {\boldsymbol x}_{s}) }\right) }{ (M \pi \sigma _{\epsilon }^{2})^{K_{a}N} \pi ^{N_{s} N}\det \left ({{\boldsymbol \Omega }}\right) } \\&\times \, \exp \left ({- \frac {||{\boldsymbol x}_{s}-{\boldsymbol \zeta }||_{2}^{2}}{\sigma _{\epsilon }^{2}} }\right) d {\boldsymbol x}_{s}\tag{48}\end{align*}$$

The exponential term can be developed by considering its variable $$\begin{align*} G({\boldsymbol x}_{s})=&({\boldsymbol y}-{\boldsymbol A}_{s} {\boldsymbol x}_{s})^{H} \,{\boldsymbol \Omega }^{-1} \,({\boldsymbol y}-{\boldsymbol A}_{s} {\boldsymbol x}_{s}) +\, \frac {||{\boldsymbol x}_{s}-{\boldsymbol \zeta }||_{2}^{2}}{\sigma _{\epsilon }^{2}} \\=&{\boldsymbol x}_{s}^{H} \left ({{\boldsymbol A}_{s}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{s} + \frac {1}{\sigma _\epsilon ^{2}} {\boldsymbol I}_{K_{a}N} }\right) {\boldsymbol x}_{s} \\&-\, 2 \Re \left [{ {\boldsymbol x}_{s}^{H} ({\boldsymbol A}_{s}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol y} + {\boldsymbol \zeta }) }\right] + {\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol y} + \frac {1}{\sigma _\epsilon ^{2}} ||{\boldsymbol \zeta }||_{2}^{2} \\=&{\boldsymbol x}_{s}^{H} \left ({{\boldsymbol A}_{s}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{s} + \frac {1}{\sigma _\epsilon ^{2}} {\boldsymbol I}_{K_{a}N} }\right) {\boldsymbol x}_{s} \\&-\, 2 {\boldsymbol x}_{s}^{H} ({\boldsymbol A}_{s}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol y} + {\boldsymbol \zeta }) - 2 ({\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{s} + {\boldsymbol \zeta }^{H}) {\boldsymbol x}_{s} \\&+ {\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol y} + \frac {1}{\sigma _\epsilon ^{2}} ||{\boldsymbol \zeta }||_{2}^{2} \\=&\left ({{\boldsymbol x}_{s} - {\boldsymbol \nu _\zeta }' }\right)^{H} {\boldsymbol Q}_{s}' \left ({{\boldsymbol x}_{s} - {\boldsymbol \nu _\zeta }' }\right) + \gamma _\zeta '\tag{49}\end{align*}$$ with: $$\begin{align*} {\boldsymbol Q}_{s}'=&{\boldsymbol A}_{s}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol A}_{s} +\, \frac {1}{\sigma _\epsilon ^{2}} {\boldsymbol I}_{K_{a}N} \\ {\boldsymbol \nu }_\zeta '=&{\boldsymbol Q_{s}'}^{-1} ({\boldsymbol A}_{s}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol y} +\, {\boldsymbol \zeta }) \\ \gamma _\zeta '=&{\boldsymbol y}^{H} {\boldsymbol \Omega }^{-1} {\boldsymbol y} +\, \frac {1}{\sigma _\epsilon ^{2}} ||{\boldsymbol \zeta }||_{2}^{2} - {\boldsymbol \nu _\zeta '}^{H} {\boldsymbol Q}_{s}' {\boldsymbol \nu }_\zeta '\end{align*}$$ Therefore, $$\begin{equation*} p({\boldsymbol y}|{\boldsymbol s}) = \frac {C^{st}}{ \det \left ({{\boldsymbol Q}_{s}'}\right)} \sum \exp {(- \gamma _\zeta ') }\tag{50}\end{equation*}$$ Thus, the estimator for the support given by: $$\begin{align*} \hat {\boldsymbol s}=&\underset {\boldsymbol u}{\arg \,\max } \,\Pr ({\boldsymbol s}={\boldsymbol u}|{\boldsymbol y}) \\=&\underset {\boldsymbol u}{\arg \,\max }\,\, p({\boldsymbol y|{\boldsymbol s}={\boldsymbol u}})\, \Pr ({\boldsymbol s}={\boldsymbol u})\end{align*}$$ leads to Equation (51).