Study of Adaptive Activity-Aware Iterative Detection Techniques for Massive Machine-Type Communications

TL;DR

This paper proposes adaptive activity-aware iterative detection techniques for massive machine-type communications, improving joint activity and data detection in sporadic uplink scenarios with low pilot symbol requirements.

Contribution

It introduces an adaptive decision feedback detector and an $l_0$-norm regularized recursive least-squares algorithm tailored for mMTC, along with an LDPC-based iterative detection scheme.

Findings

Enhanced detection performance over existing methods

Effective activity detection with minimal pilot symbols

Robustness in sporadic activity scenarios

Abstract

This work studies the uplink of grant-free low data-rate massive machine-to-machine communications (mMTC) where devices are only active sporadically, which requires a joint activity and data detection at the receiver. We develop an adaptive decision feedback detector along with an $l_0$-norm regularized activity-aware recursive least-squares algorithm that only require pilot symbols. An iterative detection and decoding scheme based on low-density parity-check (LDPC) is also devised for signal detection in mMTC. Simulations show the performance of the proposed approaches against existing schemes.

Tables

Algorithm 1 Proposed IDD with AA-RLS-DF algorithm
1.	Initialization: $M$, $N$, $𝝆$, $\beta$, $\gamma$, $\lambda$, ${𝐏}_n=𝝆{𝐈}_{\text{M}}$
2.	Compute the a priori probability with Eqs. (15) and (18);
	% For each metadata sequence ${\widehat{𝐝}}_p\left[i\right]$ and received vector ${𝐲}_{p,n}\left[i\right]$,
3.	Compute the Kalman gain vector
	${𝐤}_n\left[i\right]=\left({𝐏}_n\left[i\right]{𝐲}_{p,n}\left[i\right]\right)/\left(\lambda +{𝐲}_{p,n}^{\text{H}}\left[i\right]{𝐏}_n\left[i\right]{𝐲}_{p,n}\left[i\right]\right)$;
4.	Estimate ${\tilde{d}}_{{\varphi }_n}\left[i\right]={𝐰}_n^{\text{H}}\left[i\right]{𝐲}_{p,n}\left[i\right]$;
5.	Update the error value with ${ϵ}_{{\varphi }_n}\left[i\right]={\hat{d}}_{p,{\varphi }_n}\left[i\right]-{\tilde{d}}_{{\varphi }_n}\left[i\right]$;
6.	Update the filters with Eq. (13);
7.	Update the auxiliary matrix
	${𝐏}_n\left[i\right]={\lambda }^{-1}\left({𝐏}_n\left[i\right]-{𝐤}_n\left[i\right]{𝐲}_{p,n}^{\text{H}}\left[i\right]{𝐏}_n\left[i\right]\right)$;
8.	Update the sequence of detection with Eq.(6);
	% After computing the filter values ${𝐰}_n\left[i\right]$, the algorithm starts the
	direction mode, returning to 3. to soft estimate data and go to 9.
9.	Compute ${\mu }_{{\varphi }_n}\left[i\right]$ and ${\eta }_{{\varphi }_n}^2\left[i\right]$ with Eqs. (19) and (III-C);
10.	Verify the likelihood function $P\left({\hat{d}}_{{\varphi }_n}\left[i\right]|x\right)$ with Eq. (22);
11.	Compute the LLR value according to Eq.(21);

Equations

$$\mathbf{y}\left[i\right]=\mathbf{Hx}\left[i\right]+\mathbf{v}\left[i\right]$$

$$\mathbf{w}_{n}\left[i\right]=\left\{\begin{array}[]{cl}\mathbf{w}^{f}_{n}\left[i\right],&n=1;\\ \left[{\mathbf{w}^{f}_{n}}^{\text{T}}\left[i\right],{\mathbf{w}^{b}_{n}}^{\text{T}}\left[i\right]\right]^{\text{T}},&n=2,\dots,N.\end{array}\right.$$

$$\mathbf{y}_{n}\left[i\right]=\left\{\begin{array}[]{cl}\mathbf{y}\left[i\right],&n=1;\\ \left[{\mathbf{y}}^{\text{T}}\left[i\right],{\hat{\mathbf{d}}^{\text{T}}_{\phi_{n-1}}}\left[i\right]\right]^{\text{T}},&n=2,\dots,N.\end{array}\right.$$

$${\tilde{d}}_{{\varphi }_n}\left[i\right]={\mathbf{w}}_n^{\text{H}}\left[i\right]{\mathbf{y}}_n\left[i\right].$$

$${\mathcal{J}}_{n,j}\left[i\right]$$

$${\varphi }_n=\underset{j\in S_i}{\text{arg min}}{\mathcal{J}}_{n,j}\left[i\right].$$

$${\hat{d}}_{{\varphi }_n}\left[i\right]=\mathcal{Q}\left[{\mathbf{w}}_n^{\text{H}}\left[i\right]{\mathbf{y}}_n\left[i\right]\right].$$

$${\mathcal{J}}_{n,j}\left[i\right]=\sum _{l=0}^i{\lambda }^{i-l}{\left|{\hat{d}}_j\left[l\right]-{\mathbf{w}}_n^{\text{H}}\left[i\right]{\mathbf{y}}_n\left[i\right]\right|}^2+\gamma \Vert {\mathbf{w}}_n\left[i\right]{\Vert }_0$$

$${\mathcal{J}}_{n,j}\left[i\right]$$

$${\mathbf{w}}_n\left[i\right]$$

$$\text{sgn}\left(w_{n,p}\left[i\right]\right)=\left\{\begin{array}[]{rl}w_{n,p}\left[i\right]/|w_{n,p}\left[i\right]|,&w_{n,p}\left[i\right]\neq 0;\\ 0,&\text{otherwise.}\end{array}\right.$$

$$\text{exp}\left(-\beta\,|w_{n,p}\left[i\right]|\right)\approx\left\{\begin{array}[]{rl}1-\beta\,|w_{n,j}\left[i\right]|,&|w_{n,j}\left[i\right]|\leq 1/\beta;\\ 0,&\text{otherwise.}\end{array}\right.$$

$${\mathbf{w}}_n\left[i\right]$$

$$f_{\beta}\left(w_{n,p}\left[i\right]\right)=\left\{\begin{array}[]{rl}\beta^{2}\left(w_{n,p}\left[i\right]\right)+\beta,&-1/\beta\leq w_{n,p}\left[i\right]<0;\\ \beta^{2}\left(w_{n,p}\left[i\right]\right)-\beta,&0<\leq w_{n,p}\left[i\right]\leq 1/\beta;\\ 0,&\text{otherwise.}\end{array}\right.$$

$$\text{Pr}\left(x_n\left[i\right]=x\right)=\sum _{x\in {\mathcal{A}}_0}x\left(\prod _{z=1}^{M_c}{\left[1+\text{exp}\left(-x^z{L}_{e_n}^z\left[i\right]\right)\right]}^{-1}\right),$$

$$\displaystyle\text{Pr}\left(x_{n}{\left[i\right]}=x\right)\left\{\begin{array}[]{rl}\rho_{n}+\left(1-\rho_{n}\right)\text{Pr}\left(x_{n}{\left[i\right]}=x\right),&{\scriptstyle{\text{if}\,\left(x^{1}\text{ and }x^{2}\right)=0,}}\\ \left(1-\rho_{n}\right)\text{Pr}\left(x_{n}{\left[i\right]}=x\right),&\text{{\small{otherwise}}}.\end{array}\right.$$

$${\mu }_n\left[i\right]$$

$${\eta }_n^2\left[i\right]$$

$$L_{c_n}^z\left[i\right]=$$

$$P\left({\hat{d}}_{{\varphi }_n}\left[i\right]|x\right)\approx \frac{1}{\pi {\eta }_n^2\left[i\right]}\text{exp}\left(-\frac{1}{{\eta }_n^2\left[i\right]}|{\hat{d}}_{{\varphi }_n}\left[i\right]-{\mu }_n\left[i\right]x{|}^2\right).$$

Full text

I Introduction

The fifth generation (5G) of mobile radio communication systems should support Internet of Things (IoT), Internet of Everything (IoE), and Industry 4.0, which are referred to as massive machine type communications (mMTC). In contrast to human-type communications (HTC) the traffic of mMTC originates from a large number of devices concentrated in the uplink with small packets of up to a few hundred bits that transmit sporadically [1]. Due to the huge number of mMTC devices it is impractical to pre-allocate resources to them. Therefore, the solution is to employ grant-free random access. As the number of active mMTC devices is random their average value should be considered and the network should maximize the arrival rate that can be supported given the available radio resources. Furthermore, the packet error rate of a unique mMTC transmission is on the order of $10^{-1}$ [2] and each packet is structured in a preamble (metadata) and a payload (data) [3]. In Low-Active Code Division Multiple Access (LA-CDMA) schemes, metadata are used as the ID of each user and, due to the huge number of devices, it is infeasible to provide orthogonal metadata (pilot symbols) for channel estimation and receive processing of active users  [4].

Prior work on channel estimation and detection of devices includes compressed sensing (CS) methods that outperform conventional channel estimation techniques [5]. However, if the device activity detection is accurate then conventional channel estimation techniques may be employed. As device activity detection is an open problem, most of the works to date consider perfect channel state information (CSI) at the BS and investigate different detection approaches [6, 7, 8]. Unlike prior work with successive interference cancellation and list-based detectors [12],[13],[14],[15],[16],[17],[18],[19],[20] detectors for mMTC must exploit the activity of devices.

In this work, we propose a decision feedback detector along with an activity-aware recursive least squares (RLS) algorithm (AA-RLS-DF) based on an $l_{0}$-norm penalty function that does not require explicit channel estimation. The regularization exploits the sparsity in the coefficients update formulation, thus improving the performance of parameter estimation and detection. An iterative detection and decoding (IDD) scheme based on low-density parity-check (LDPC) is also devised for signal detection. Unlike existing approaches, the proposed AA-RLS-DF scheme exploits the activity of devices. Simulations show that the proposed AA-RLS-DF scheme significantly outperforms prior work with a competitive complexity.

The organization of this paper is as follows: Section II briefly describes the LA-CDMA system model and the augmented alphabet specifically designated for mMTC. Section III describes the adaptive implementation, the $l_{0}$ regularization and introduces the iterative detection and decoding, proposed for mMTC. Section IV presents the setup for simulations and the results while Section V draws the conclusions.

Notation: Matrices and vectors are denoted by boldfaced capital letters and lower-case letters, respectively. The space of complex (real) $N$-dimensional vectors is denoted by $\mathbb{C}^{N}\left(\mathbb{R}^{N}\right)$. The $i$-th column of a matrix $\mathbf{A}\in\mathbb{C}^{M\times N}$ is denoted by $\mathbf{a}_{i}\in\mathbb{C}^{M}$. The superscripts $\left(\cdot\right)^{\text{T}}$ and $\left(\cdot\right)^{\text{H}}$ stand for the transpose and conjugate transpose, respectively. For a given vector $\mathbf{x}\in\mathbb{C}^{N},||\mathbf{x}||$ denotes its Euclidean norm. $\mathbb{E\left[\cdot\right]}$ stands for expected value and $\mathbf{I}$ is the identity matrix.

II System model

We assume the uplink of slot-synchronized mMTC devices in which the base station (BS) with multiple antennas receives data from multiple devices with a single antenna. The transmitted data, at each time instant, is composed by symbols from a regular modulation scheme denoted by $\mathcal{A}$, as quadrature phase-shift keying (QPSK) if the device is active, or zero, otherwise. So, each transmitted symbol belongs to the augmented alphabet $\mathcal{A}_{0}$, which consists of a regular modulation scheme and the zero ($\mathcal{A}_{0}=\mathcal{A}\cup\left\{0\right\}$).

In the mMTC system of Fig. 1, the data symbols of the $N$ devices are transmitted over Rayleigh fading channels, grouped into a $M\times N$ channel matrix $\mathbf{H}$ which contains the spreading sequences and channel impulse responses to the BS. The received signal is collected into an $M\times 1$ vector $\mathbf{y}{\left[i\right]}$ as

[TABLE]

where $i$ is the time index, the $M\times 1$ vector $\mathbf{v}{\left[i\right]}$ is a zero mean complex circular symmetric Gaussian noise with covariance matrix $\mathbb{E}\left[\mathbf{v}{\left[i\right]}\mathbf{v}{\left[i\right]}^{\text{H}}\right]=\sigma^{2}_{v}\mathbf{I}$. The $N\times 1$ symbol vector ${\bf x}{\left[i\right]}=[x_{1}{\left[i\right]}\ldots x_{k}{\left[i\right]}\ldots x_{N}{\left[i\right]}]^{\text{T}}$ has zero mean and covariance matrix $\mathbb{E}\left[\mathbf{x}{\left[i\right]}\mathbf{x}{\left[i\right]}^{\text{H}}\right]=\sigma^{2}_{x}\mathbf{I}$, where $\sigma_{x}^{2}$ is the signal power of the active devices.

Since in the mMTC scenario devices have low-activity probability the system can be interpreted as a sparse signal processing system model. In this way, even in an underdetermined system ($N>>M$), it is possible to recover the transmitted vector $\mathbf{x}{\left[i\right]}$. In order to avoid the need to estimate channels, we propose an adaptive scheme which considers the activity of devices, combined with an $l_{0}$-norm penalty function.

III Proposed Iterative Detection and Decoding

The proposed AA-RLS-DF detector is introduced in this section along with the detection ordering algorithm based on the least squares criterion (LSE). We also detail the proposed $l_{0}$-norm regularized RLS parameter estimation algorithm that exploits sparsity to refine the detection and estimation tasks. Lastly, we present an IDD scheme based on the proposed AA-RLS-DF detector and LDPC codes.

III-A Adaptive Detection Design

In order to reduce detection errors, we propose an adaptive AA-RLS-DF detector, which performs parameter estimation using metadata in training mode and then carries out data detection in a decision-directed mode. The receive filters and the ordering are updated at each iteration, as illustrated in Fig. 1. The feedforward and feedback receive filters are written as

[TABLE]

and the augmented received vector,

[TABLE]

where $\mathbf{w}_{n}\left[i\right]=\left[w^{f}_{n,1}\left[i\right],\dots,w^{f}_{n,M}\left[i\right],w^{b}_{n,M+1}\left[i\right],\dots,\right.$

$\left.w^{b}_{n,M+N}\left[i\right]\right]^{\text{T}}$ corresponds to both filters used for the detection of the symbol of the $n$-th device and the $N\times 1$ vector $\hat{\mathbf{d}}_{\phi_{n}}\left[i\right]$ contains the previously detected symbols of the $n$-th device. The soft symbol estimates of the $n$-th device are obtained by

[TABLE]

To determine the detection order $\phi_{n}$, the set of candidate symbols are evaluated. Then, we select the symbols associated with the minimum LSE. The receive filter is computed by the $l_{0}$-norm regularized RLS, whose cost function is given by

[TABLE]

where $0<\lambda\leq 1$ is the forgetting factor which gives exponentially less weight to older error samples. At each symbol detection, the index of the chosen feedback filter is stored in the sequence of detection $\phi_{n}$, represented as

[TABLE]

After this, the filter weights are updated and the result of the decision block, given by

[TABLE]

The received vector is concatenated with the output of (7), as in (3). In the next detection, the new cost functions are computed just for the symbols which have not been detected yet. For this, index $j$ in (5) belongs to the set $S_{i}=\{1,2,\dots,N\}-\{\phi_{1},\phi_{2},\dots,\phi_{n-1}\}$, which contains the index of not yet detected symbols.

After using the metadata to adjust the receive filters, AA-RLS-DF starts the decision-directed mode. The received data $\mathbf{y}{\left[i\right]}$, initially repeats the same sequence of operations in the adaptation mode but, after obtaining all soft estimates $\tilde{\mathbf{d}}{\left[i\right]}$, the vector is reorganized in the original sequence and each soft estimate is converted to a log-likelihood ratio (LLR) ($L_{c_{n}}{\left[i\right]}$). After the SISO decoder block, the extrinsic LLRs ($L_{e_{n}}{\left[i\right]}$) return in the flow to the next IDD iteration, refining the new $L_{c_{n}}{\left[i\right]}$ computation.

III-B $l_{0}$-norm Regularized RLS Algorithm

In order to exploit the sparse activity of devices and compute the parameters of the proposed DF detector without the need for explicit channel estimation, we devise an $l_{0}$-norm regularized RLS algorithm that minimizes the cost function:

[TABLE]

where $\|\cdot\|_{0}$ denotes $l_{0}$-norm that counts the number of zero entries in $\mathbf{w}_{n}$ and $\gamma$ is a non-zero positive constant to balance the regularization and, consequently, the estimation error. We consider an approximation to the regularization term [21],[22],[23],[24].

Approximating the value of the $l_{0}$-norm [26], the cost function in (8) can be rewritten as

[TABLE]

The parameter $\beta$ regulates the range of the attraction to zero on small coefficients of the filter. Thus, taking the partial derivatives for all entries $i$ of the coefficient vector $\mathbf{w}_{n}{\left[i\right]}$ in (9) and setting the results to zero, yields

[TABLE]

where $\mathbf{k}\left[i\right]$ is the gain vector and $\text{sgn}\left(\cdot\right)$ is a component-wise sign function defined as

[TABLE]

In order to reduce computational complexity in (10), the exponential function is approximated by the first order of Taylor series expansion, given by

[TABLE]

As the exponential function is positive, the approximation of (12) is also positive. In this way, (10) becomes

[TABLE]

where the function $f_{\beta}\left(w_{n,p}\left[i\right]\right)$ is given by

[TABLE]

We notice that the function $f_{\beta}\left(w_{n,j}\left[i\right]\right)$ in (13) imposes an attraction to zero in small coefficients. So, if the value of $w_{n,j}\left[i\right]$ is not equal or between to $\left[-1/\beta,1/\beta\right]$, no additional attraction is exerted. Thus, the convergence rate of near-zero coefficients of parameters of devices in mMTC applications that exhibit sparsity will be increased [26]. The computational cost of AA-RLS-DF is $O(N^{2})$, which is comparable with a standard DF detector with an RLS algorithm. Since the other considered algorithms have a computational cost of $O(N^{3})$, AA-RLS-DF outperforms both performance and efficiency. The pseudo-code, which also considers an IDD scheme with AA-RLS-DF, is described in Algorithm 1.

III-C Proposed Soft Information Processing and Decoding

In this section, the structure of the proposed IDD scheme [30] is described. Unlike prior IDD schemes [27], [28], [29] which employ convolutional, Turbo and LDPC [31, 32] codes, our scheme does not require matrix inversions and exploit the activity of devices to refine the iterative processing. We estimate and incorporate the probability of each device being active in the mMTC scenario in the computation of each a priori probability symbol.

Based on $L_{e_{n}}$, provided by the LDPC decoder and assuming the bits are statistically independent of one another [33], the a priori probabilities are calculated as

[TABLE]

where $M_{c}$ represents the total number of bits of symbol $x$, the superscript $z$ indicates the $z$-th bit of symbol of $x$, in $x^{z}$ (whose value is $(+1,-1)$) and the form of extrinsic LLR, $L_{e_{n}}^{z}{\left[i\right]}$. As $\mathcal{A}_{0}$ is the augmented alphabet, the considered symbols are from the modulation scheme chosen and zero. Considering the probability of the $n$-th device being active as $\rho_{n}$, the a priori probabilities can be rewritten as

[TABLE]

Eq.(18) incorporates the a priori probabilities related to the probability that the $n$-th device is active. We consider different $\rho_{n}$ values for each user, drawn uniformly at random in the interval $[0.1,0.3]$. Due to the large number of independent variables considered to compute the filter output, it can be approximated by a complex Gaussian distribution [27]. Hence, we approximate $\hat{d}_{\phi_{n}}{\left[i\right]}$ by the output of an equivalent AWGN channel with $\hat{d}_{\phi_{n}}\left[i\right]=\mu_{n}\left[i\right]x_{n}\left[i\right]+b_{n}\left[i\right]$, where

[TABLE]

Note that $x_{n}$ are the metadata symbols and $b_{n}\left[i\right]$ are zero-mean complex Gaussian variables with variance $\eta^{2}_{n}\left[i\right]$ as

[TABLE]

Then, the extrinsic LLR computed by the AA-RLS-DF detector for the $z$-th bit ($z\in\{1,\dots,M_{c}\}$) of the symbol $x_{n}$ transmitted by the $n$-th device is

[TABLE]

where $\mathcal{A}_{z}^{+1}$ is the set of $2^{M_{c}-1}$ hypotheses of $x$ for which the $z$-th bit is +1 (analogously for $\mathcal{A}_{z}^{-1}$). Therefore, the likelihood function $P\left(\hat{d}_{\phi_{n}}|x\right)$ is approximated by

[TABLE]

IV Simulation Results

We evaluate the simulation results of an under-determinated mMTC system with $N=128$ devices and unit-norm random sequences with length of $M=64$ for spreading. The proposed and existing schemes experience an independent and identically-distributed (i.i.d.) random flat-fading channel model and the values are taken from complex Gaussian distribution of $\mathcal{C}\mathcal{N}\left(0,1\right)$. The active devices radiate QPSK symbols with the same power and the activity probabilities $\left\{\rho_{n}\right\}^{N}_{n=1}$ are drawn uniformly at random in $\left[0.1,0.3\right]$. Each symbol block has 128 symbols, split in to 60 metadata and 68 data. This balance between pilots and data is suggested in [34]. All these assumptions are considered for two scenarios, uncoded and coded systems, in which numerical results are averaged over $10^{5}$ runs. The performance of AA-RLS-DF is compared with other relevant schemes, as the linear mean squared error (LMMSE), unsorted SA-SIC [6], SA-SIC with A-SQRD [7], AA-MF-SIC [8] and a version of AA-RLS-DF without decision feedback, AA-RLS-Linear. As a lower bound, the Oracle LMMSE detector, which has the knowledge of the index of nonzero entries, is considered. For all algorithms that require explicit channel estimation, we considered $\hat{\mathbf{H}}=\mathbf{H}+\mathbf{E}$, where $\mathbf{H}$ represents the channel estimate and $\mathbf{E}$ is a random matrix corresponding to the error for each link. Each coefficient of the error matrix follows a Gaussian distribution, i.e., $\sim\mathcal{C}\mathcal{N}\left(0,{\sigma}^{2}_{e}\right)$, where $\sigma^{2}_{e}=\sigma_{v}^{2}/5$. For uncoded systems, the average SNR is given by $10\log\left(N\,\sigma^{2}_{x}/\sigma^{2}_{v}\right)$, while for coded systems is $10\log\left(NR\,\sigma^{2}_{x}/\sigma^{2}_{v}\right)$.

Fig. 2 shows the net symbol error rate (NSER) which considers only the active devices. LMMSE exhibits poor performance since the system is under-determined. Due to error propagation, the unsorted SA-SIC does not perform well. A-SQRD and AA-MF-SIC are effective since both consider the activity probabilities, but under imperfect CSI conditions, their performance is not so good. In contrast, as AA-RLS-DF does not need a explicit channel estimation, it is more efficient. The decision-feedback scheme provides a NSER gain due to the interference cancellation. For the coded systems with IDD, Fig. 3 shows the BER of the already considered algorithms and scheme of Wang and Poor [27]. The sparsity of mMTC approach degrades the expected efficiency of LMMSE-PIC, obtaining little variation in relation to LMMSE and SA-SIC. The hierarchy of performance of the other considered algorithms is the same as the uncoded case but with better error rate values.

V Conclusion

We have proposed an activity-aware adaptive DF detector, named AA-RLS-DF, for mMTC scenarios, which unlike competing techniques does not need explicit CSI to detect the symbols. Moreover, an IDD scheme equipped with the AA-RLS-DF technique that exploits device activity to refine the iterative processing has been presented. Simulations have shown that AA-RLS-DF significantly surpass existing approaches.