# Multi-User Detection for Sporadic IDMA Transmission Based on Compressed Sensing

^{*}

## Abstract

**:**

## 1. Introduction

**d**denote vectors, whereas upper case bold characters

**A**denote matrices. Moreover, scalar values are given by regular letters, such as m; ${x}_{k}(f)$ denotes the f-th chip of user k; d

_{k}(i) denotes the i-th symbol of user k.

## 2. Sporadic IDMA Transmission System Model and Chip-by-Chip Multi-User Detection

#### 2.1. Sporadic IDMA Transmission System Model

_{a}in a frame time, i.e., a simple Bernoulli traffic model. In a practical IoT, the average the number of active users K

_{act}is small due to a large number of sensors connecting with the aggregation point.

_{s}-length spreading sequence, where F is the length of the chip vector and $F=M{N}_{s}$. Finally, a chip interleaver ${\phi}_{k}$ specific to each user is applied and the chip vector is permutated to an interleaved chip vector ${\mathbf{x}}_{k}={[{x}_{k}(1),\mathrm{\dots},{x}_{k}(f),\mathrm{\dots},{x}_{k}(F)]}^{T}$. To detect both node activity and data, the IDMA system model contains all possible frames from K sensors. Further, we model that the transmitted symbols for active nodes are taken from the discrete finite modulation alphabet $\mathcal{A}$, while all zeros symbols for inactive users. Thus, the signal which be detected is from the augmented modulation alphabet ${\mathcal{A}}_{0}=\{\mathcal{A}\cup 0\}$.

**y**is modeled as:

_{k}. The total influence of transmission can be represented as one matrix $\mathbf{A}\in {\mathbb{R}}^{F\prime \times M}$, which is the concatenation of ${\mathbf{H}}_{k}{\mathbf{\Pi}}_{k}\mathbf{S}$. Vector $\mathbf{d}\in {\mathcal{A}}_{0}^{L}$ is the stacked vector of all

**d**

_{k}, where $L=KM$. Further, the noise vector $\mathbf{n}\in {\mathbb{R}}^{F\prime}$ is i.i.d. zero-mean Gaussian distributed, i.e., $\mathbf{n}~\mathcal{N}(0,{\sigma}_{\mathrm{n}}^{2}\mathrm{I})$. Herein, the symbols of

**d**are taken from the discrete augmented alphabet ${\mathcal{A}}_{0}$.

#### 2.2. Chip-by-Chip Iterative Multi-Users Detection

#### 2.2.1. The Basic ESE Function

_{k}is the a priori channel coefficient at the receiver side, and n

_{f}is the sample of additive white Gaussian noise (AWGN) with variance ${\sigma}^{2}={N}_{0}/2$.

#### 2.2.2. The DEC Function

_{APP}(d

_{k}(i)) of each sensor after several iterations. The DEC calculates L

_{APP}(d

_{k}(i)) as following:

## 3. Activity and Data Detection Based on Compressed Sensing

#### 3.1. CS Detection

#### 3.1.1. Compressed Sensing

**A**is considered to be a fat matrix, that is, M is smaller than N.

**y**, CS mainly focuses on the recovery of the sparse signal x, instead of z. For a noise model setting, we conclude the model of measurement vector

**y**from Equation (9):

#### 3.1.2. Application to Multi-User Detection

**d**appear in groups or blocks forming a fixed length. This feature is also known as block sparsity or group sparsity [20]. For the sake of uniform expression in this paper, we choose block sparsity in the following. The MUD problem can be treated as a block-sparse signal recovery, inherently, which naturally incorporates the powerful tool CS into the joint user activity and data detection problem.

**A**is that CDMA blocks the non-zero elements together, whereas IDMA distributes them randomly. That is to say, in IDMA, the individual symbols are interconnected with many more symbols than in CDMA. This randomness feature in IDMA conforms to the matrix selection of CS, which results in better performance of multi-user signal detection [21], which will be proved by simulations in Section 4.

**d**. From the perspective of bandwidth constrained wireless communication, this leads to a high resource efficiency, since the number of measurements of

**y**is strikingly smaller than the number of transmitted symbols. On the other hand, in order to recover the sparse signal, CS provides various detectors in the case of different structures of the sparse signal

**d**.

#### 3.1.3. GOMP

**d**by simply selecting the indices of those columns of matrix

**A**. This method has the maximum usefulness, but neglects the additional information that the user activity symbols of one user are the same. In order to utilize the additional information, during each iteration, the GOMP algorithm, builds support by adding indices of groups to a group index set G, instead of a single index in OMP.

Algorithm 1: Group orthogonal matching pursuit. |

Input: y, A, ${K}_{act}$Output: $\tilde{\mathbf{d}}$1: initialize ${G}^{0}=\varnothing ,l=0,{\mathrm{r}}^{0}=\mathrm{y}$ |

2: repeat |

3: $l=l+1$ |

4: ${k}_{\mathrm{max}}=\underset{k}{\mathrm{arg}\mathrm{max}}{\displaystyle \sum _{j\in \gamma (k)}\left|{\mathrm{A}}_{j}^{H}{\mathrm{r}}^{l-1}\right|}\mathrm{with}k\in {\overline{G}}^{l-1}$ |

5: ${G}^{l}={G}^{l-1}\cup {k}_{\mathrm{max}}$ |

6: ${\tilde{d}}_{\{\gamma ({G}^{l})\}}={A}_{\{\gamma ({G}^{l})\}}^{\u2020}y\mathbf{a}\mathbf{n}\mathbf{d}{\tilde{d}}_{\{\gamma ({\overline{G}}^{l})\}}=0$ |

7: ${r}^{l}=y-A{\tilde{d}}^{l}$ 8: until 9: $l={K}_{act}.$ |

**d**. Additionally, the superscript t always denotes the t-th iteration, e.g.,

**d**

^{t},

**A**

^{t}, and G

^{t}. Further, ${\mathbf{A}}^{\u2020}$ is the Moore-Penrose pseudo-inverse of matrix

**A**, and

**A**

^{H}denotes the Hermitian matrix of

**A**.

- (1)
- (Step 4 in Algorithm 1) Group support selection: To be specific, the GOMP chooses the group ${k}_{\mathrm{max}}$, which has the highest correlation to the previous residual ${\mathbf{r}}^{l-1}=\mathbf{y}-\mathbf{A}{\tilde{\mathbf{x}}}^{l-1}$ as the active support.
- (2)
- (Step 5 in Algorithm 1) Update the group-indices set G: The set of the newly-selected group index obtained in the previous step is added to the set ${G}^{l-1}.$
- (3)
- (Steps 6 and 7 in Algorithm 1) LS estimation is executed by using the sub-matrix ${\mathbf{A}}_{\{\psi ({G}^{l})\}}$ of all columns in $\psi ({G}^{l})$ to determine the values of nonzero elements in $\tilde{\mathbf{d}}$ and renew the current residual.

**r**

^{l}

^{− 1}. An ideal stopping criterion is adopted to terminating the algorithm after a number of iterations, e.g., l = Kact.

#### 3.2. CBC-AD

_{APP}(b

_{k}(i)) is only treated as the input information of the hard decision, whereas the L

_{APP}(b

_{k}(i)) contains the activity information, but not been utilized [24]. In order to exploit the activity information, we adopt the activity detection strategy on the basis of CBC that summing up a frame of the absolute value of a posteriori LLRs produced in DEC for each user to determine active users. User-k activity information can be computed by:

_{APP}(b

_{k}(i)) denotes the a posteriori LLRs of the i-th symbol in the k-th user. Principally, L

_{APP}(b

_{k}(i)) contains activity information of user-k. Therefore, active users are identified on the basis of a threshold given by the assumed noise model. However, to provide a fair comparison with CS detection and avoid threshold dependencies, we assume that the activity detector has perfect knowledge of the instantaneous number of active users K

_{act}. Thus, the nodes with the highest K

_{act}values are collected in the set of active nodes G. Subsequently, hard decisions on active set G can be applied to estimate the transmitted symbols of active users.

#### 3.3. CS-CBC

**d**, rather than the values of non-zero elements.

**y**, which composes of the symbols transmitted by active sensors and modulated by interleaved spreading sequences, can be expressed as:

**d**

_{active}contains the transmitted symbols of active sensors,

**A**

_{2}has the same form as

**A**in Equation (1) except that it only includes interleaved spreading sequences and channel influence for active sensors. Then, CBC algorithm can be implemented to realize active data detection.

**d**has the structure of block sparse, we take GOMP algorithm as the choice of CS detection part. The GOMP algorithm can provide good performance in detecting the support of sparse signal even in a high-noise environment. Compared to CS detection, CS-CBC only increases the complexity of MUD based on CBC, which is upper bounded by O(K

_{act}).

## 4. Simulation Results and Discussions

_{s}is 64. Therefore, the overloading factor is 156%. Binary phase shift keying (BPSK) signaling is always considered. In many sporadic communication scenarios, such as machine-to-machine communication and sensor networks, only low data rates are required and, thus, small bandwidth is used. For a small requirement of bandwidth, a non-frequency selective channel is more practical most of the time, which means multi-path propagation has little influence on the frame transmission. Thus, we consider a single-tap channel, in which the real part coefficients are generally correspondent to Gaussian distribution. The estimation of received power and the phase are assumed to be well prepared for each node before detection. Therefore, we adopt the AWGN channel as a simplified channel model, such that the channel matrix

**H**

_{k}for each user is an identity matrix.

#### 4.1. The SER Performance Comparison between IDMA and CDMA System

**A**will vary for different multiple access technologies, and further distinguish the symbol error rate (SER) performance of multi-user detection based on CS. For fair comparison, we adopt the GOMP algorithm to implement MUD for both multiple access system. Figure 5 shows the SER performance comparison between IDMA and CDMA system, with two active probabilities setting, i.e., p

_{a}= 0.1 and p

_{a}= 0.3.

#### 4.2. SER Performance with Different E_{b}/N_{0}

_{a}. Active probability p

_{a}reflects the frequency of sensors’ activity in the whole wireless sensor networks. The larger the active probability p

_{a}is, the more data sensors will be transmitted. Assuming the active probabilities are p

_{a}= 0.1, p

_{a}= 0.2, p

_{a}= 0.3 and p

_{a}= 0.4, respectively, the simulation results are illustrated in Figure 6.

_{b}/N

_{0}and the SER performance gap is more pronounced with increasing active probability. The CBC-AD takes the advantage of full frame activity knowledge, which results in a good performance in a high noise levels due the averaging. Two obvious characteristics should be noted when E

_{b}/N

_{0}increases: on one hand, due to the high correlation of the interleaves in overloaded system, the performance of the CBC-AD quickly hits an error floor; on the other hand, CS-CBC is superior to CS detection and CBC-AD and their gaps become larger with higher E

_{b}/N

_{0}. Furthermore, it is close to the performance of Genie-knowledge CBC under high E

_{b}/N

_{0}. Thanks to the reliable performance of CS detection, CS-CBC has perfect knowledge of active sensors to position non-zero items, and makes data recovery better than CS detection.

#### 4.3. SER Performance with Different Active Probability p_{a}

_{a}and a given ${E}_{b}/{N}_{0}=9\hspace{0.17em}\mathrm{dB}$. Simulation results are illustrated in Figure 7.

_{a}, where the E

_{b}/N

_{0}is 9 dB. As can be seen, CBC-AD and CS detection deteriorate with increasing p

_{a}and the difference between them becomes smaller. This is because the number of active sensors becomes larger, and the number of errors caused by highly correlated sensors become larger. The CS-CBC is superior than CS detection and CBC-AD and the difference between them becomes larger with increasing p

_{a}. It should be noted that CS-CBC can maintain the same SER performance with Genie-knowledge CBC before p

_{a}reaches 0.4. When p

_{a}is greater than 0.4, CS-CBC deteriorates more quickly than Genie-knowledge CBC. This means that there exists incorrect detections in active sensors when p

_{a}is too high, which results in the decay of CS-CBC.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**IoT uplink scenario with K sensors sporadically transmitting frames to an aggregation node.

**Figure 2.**Transmitter structure of an IDMA scheme with K sensors, including a multiple access channel.

**Figure 6.**SER performance with different E

_{b}/N

_{0}. (

**a**) p

_{a}= 0.1; (

**b**) p

_{a}= 0.2; (

**c**) p

_{a}= 0.3; and (

**d**) p

_{a}= 0.4.

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Li, B.; Du, R.; Kang, W.; Liu, G.
Multi-User Detection for Sporadic IDMA Transmission Based on Compressed Sensing. *Entropy* **2017**, *19*, 334.
https://doi.org/10.3390/e19070334

**AMA Style**

Li B, Du R, Kang W, Liu G.
Multi-User Detection for Sporadic IDMA Transmission Based on Compressed Sensing. *Entropy*. 2017; 19(7):334.
https://doi.org/10.3390/e19070334

**Chicago/Turabian Style**

Li, Bo, Rui Du, Wenjing Kang, and Gongliang Liu.
2017. "Multi-User Detection for Sporadic IDMA Transmission Based on Compressed Sensing" *Entropy* 19, no. 7: 334.
https://doi.org/10.3390/e19070334