compressive sensing based multi-user detection for machine ... · proposed activity detection on...

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEur. Trans. Telecomms. 2013; 00:1–12

DOI: 10.1002/ett

RESEARCH ARTICLE

Compressive Sensing based Multi-User Detection forMachine-to-Machine CommunicationC. Bockelmann∗, H. F. Schepker, A. Dekorsy

Department of Communications Engineering, University of Bremen, Germany

ABSTRACT

With the expected growth of Machine-to-Machine (M2M) communication, new requirements for future communicationsystems have to be considered. More specifically, the sporadic nature of M2M communication, low data rates, smallpackets and a large number of nodes necessitate low overhead communication schemes that do not require extendedcontrol signaling for resource allocation and management. Assuming a star-topology with a central aggregation node thatprocesses all sensor information one possibility to reduce control signaling is the estimation of sensor node activity.In this paper, we discuss the application of greedy algorithms from the field of Compressive Sensing in a channel codedCode Division Multiple Access context to facilitate a joint detection of sensor node activity and transmitted data. To thisend a short introduction to Compressive Sensing theory and algorithms will be given. The main focus, however, willbe on implications of this new approach. Especially, we consider the activity detection, which strongly determines theperformance of the overall system. We show that the performance on a system level is dominated by the missed detectionrate in comparison to the false alarm rate. Furthermore, we will discuss the incorporation of activity-aware channel codinginto this setup to extend the physical layer detection capabilities to code-aided joint detection of data and activity. Copyrightc© 2013 John Wiley & Sons, Ltd.

∗Correspondence

C. Bockelmann, University of Bremen, Department of Communications Engineering, Otto-Hahn-Allee 1, D-28359 Bremen, Germany.

Email: [email protected]

1. INTRODUCTION

1.1. M2M communication

Machine-to-Machine (M2M) communication is expectedto grow tremendously in the next years [1], thereby posingnew challenges for both existing communication systems,designed with human communication in mind, and existingM2M solutions alike. On the one hand, standards likeLTE have to be extended to cope with the requirementsof M2M communication, which often differ stronglyfrom requirements for speech communication and humandata access. For example, management overhead, such asmobility management or sophisticated resource allocation,is not well suited for low-rate communication of sensornodes that are only transmitting sporadically. In the currentframe structure of LTE-Advanced, at most a few tensof M2M devices can be supported with control channelelements (CCEs) allocated within a sub-frame [2]. This isfar below the target number of M2M devices envisionedfor the future. On the other hand, a growing number ofnodes will be challenging for existing M2M standardslike the IEEE standards 802.15.4 and 802.15.1. Both are

based on Carrier Sense Multiple Access with CollisionAvoidance (CSMA/CA) and spread spectrum techniques.A thorough overview of current activities in differentstandardization bodies like ETSI and 3GPP can be foundin [1]. Furthermore, detailed requirements on future M2Msystems and an assessment of current technologies canbe found in the deliverable D2.1 of the European projectEXALTED [3].

In order to efficiently cope with a high number of sensornodes, which are expected to only sporadically transmitsmall packets with low data rates, new approaches on thephysical layer are required. Low control signal overheadand increased power efficiency are a major concern, forlong battery life of low-cost sensors. Especially the latteris of great importance for future sensor networks, and isresearched in several areas from energy efficient circuitsto routing protocols [4]. Here, we focus on the physicallayer aspects. One approach for decreasing the overhead isto avoid control signaling regarding the activity of sensornodes before transmission, e.g., mobility tracking and dutycycle controlling. Another approach is to facilitate anautomatic detection of active nodes. Accordingly, a reliablejoint data and node activity detection has to be achieved,

Copyright c© 2013 John Wiley & Sons, Ltd. 1Prepared using ettauth.cls [Version: 2010/06/21 v2.00]

CS-based Multi-User Detection for M2M C. Bockelmann, H. F. Schepker, A. Dekorsy

which substitutes functionality usually implemented onhigher layers. A fundamentally new approach to thisproblem statement is Compressive Sensing Multi-UserDetection (CS-MUD).

Considering a star topology, where sensors commu-nicate with a central aggregation node, high processingpower allows for Multi-User detection. Due to the as-sumption of sporadic transmission, only a small numberof sensors are active at any given time. Therefore, theMulti-User signal composed of signals from all sensors inthe network is sparse. Compressive Sensing (CS) theorydescribes the estimation of sparse signals, or equivalentlythe solution of linear equation systems [5, 6] for sparsesignals. With respect to communications, two propertiesof CS are especially interesting in M2M scenarios: bothdata and activity can be reconstructed at the same time andreconstruction is possible for under-determined equationsystems. The first point is very interesting to implement theproposed activity detection on the physical layer, whereasthe importance of the second point becomes clear in thecontext of the Code Division Multiple Access (CDMA)system that will be assumed as an example for M2M com-munication in this paper. Any CDMA Multi-User chip-rate model with sporadic activity can be approached bymeans of CS algorithms [7]. Regarding the second point,however, overloaded CDMA systems, i.e., the number ofnodes in the network is larger than the spreading factor,lead to under-determined equation systems whose solutionrequires sparsity, i.e., sporadic activity. Overall, CDMA isvery attractive for M2M scenarios, due to its adaptive andflexible support of different data rates as well as Qualitiesof Service, flexible and scalable number of supporteddevices, and low rate channel coding.

1.2. Compressive Sensing

The application of CS theory is a growing research fieldfor many research areas, e.g., distributed signal processing[8] and radar [9]. In communications it is a very recentapproach with much potential for further research. Inliterature, the focus has been on the application of CStheory to communication problems, where some amountof sparsity can be assumed, e.g., in channel estimation [10,11], for ultra-wideband communication [12] and sensornetworks [13, 14, 15, 7]. This section provides a briefoverview of CS assumptions, reconstruction approachesand algorithms. For a thorough overview of CS see, e.g.,[16].

Generally speaking, CS is concerned with efficientsampling assuming the original data is compressible.Formally, this can be described in the following way: Let zbe a compressible signal with respect to the basis Φ, suchthat z = Φx and x is sparse, i.e., only few entries are non-zero. Note that for a particular family of signals the basis Φmay not be unique, e.g., natural pictures are compressiblewith respect to many bases, such as Fourier or wavelets,see [17]. However, the basis Φ is always determined by

the physical constraints of the signal. In the following, werefer to Φ as the sparsity basis.

The data acquisition process of z is modeled bydetermining the correlation of z to a system Ψ. Further,we incorporate the possibility of inexact measurementsinto the data acquisition model. Hence, the availableinformation of the physical signal z is given by

y = Ψz + n , (1)

where n is a noise vector with known stochastic properties.Therefore, the length of y is determined by the dimensionsof Ψ.

The usual task of Compressed Sensing is to find agood measurement matrix Ψ for a given sparsity basisΦ. Here, the objective is not only to guarantee goodrecovery properties for the sparse signal, but to ensurethose properties with as few measurements as possible.This allows to use less measurements than Nyquistsampling leading to under-determined equation systemswith less equations than variables. The minimum amountof measurements required for reliable reconstructiondepends on the sparsity of x and the dimensions of Φ [18].

Reconstruction for CS is based on finding the sparsedescription x of the physical signal z, i.e., a sparsesolution x that fulfills (1). To obtain such a sparsesolutions, multiple approaches are discussed in literature,which are principally based on the formulation of wellsuited optimization problems. The basic CS reconstructionproblem is

minx∈RN

‖x‖0 subject to y = ΨΦx , (2)

where ‖·‖0 denotes the l0-pseudo norm that counts thenumber of non-zero entries in the vector. Problem (2)aims to find a suitable solution x with as few non-zeroentries as possible. Note, that (2) is known to be non-convex and NP-hard to solve, which motivates relaxationapproaches. A common relaxation of (2) in the presenceof noise is the so-called Basis Pursuit De-Noising (BPDN)[19], where the non-convex l0-pseudo norm is relaxed tothe convex l1-norm. Another approach, stemming from theregression point of view, is the so-called Least AbsoluteShrinkage and Selection Operator (LASSO) [20], wherethe l1-norm is used as a penalty term to encourage sparsesolutions for the Least Squares (LS) estimation problem.Furthermore, regularization of the LASSO problem byan additional weighted l2-norm to enhance convergencefor badly conditioned matrices A is then termed Elastic-Net [21]. For the relaxations of (2) (BPDN/LASSO,Elastic-Net), there are many different approaches andalgorithms, which aim to efficiently determine the solutionof the general CS reconstruction problem. A thoroughoverview of the different algorithmic approaches, however,is beyond the scope of this paper.

In the area of sensor networks, two directions can bedifferentiated: the application of known CS reconstructionalgorithms for continuous signals and the extensions

2 Eur. Trans. Telecomms. 2013; 00:1–12 c© 2013 John Wiley & Sons, Ltd.DOI: 10.1002/ett

Prepared using ettauth.cls

C. Bockelmann, H. F. Schepker, A. Dekorsy CS-based Multi-User Detection for M2M

to finite, discrete alphabets. The first investigates theapplicability of CS detectors in a communications context[7], and the second aims at new formulations for knowndetectors, like sparsity-aware CS MAP detection [13].Such CS MAP detectors extend known approaches, like thesphere detector, to detect the transmitted symbols and non-active nodes. Furthermore, block-detection strategies havebeen devised to exploit the fact that nodes usually transmitmultiple symbols, which implies activity over defined timeintervals. Such structure helps to detect node activity moreaccurately [13, 15, 22].

For the scenario discussed in the following, the CSMAP approach is not applicable as it is numerically toodemanding. Here, we focus on efficient CS algorithmsthat supply continuous estimates for further processing.Therefore, the greedy Group Orthogonal Matching Pursuit(GOMP) detector, which exploits block-information, willbe detailed in Section 3.

1.3. Paper Focus and Contribution

In this paper, we provide a detailed introduction to CSbased joint Multi-User data and activity detection forCDMA uplink transmission transferring the ideas fromCS theory to M2M communication. Special attention ison the application and influence of channel codes insuch CS detected CDMA systems. Based on the CDMAchip-rate Multi-User description [23], we will explainhow CS algorithms can be applied to jointly detectthe activity of sensor nodes and their data reducingrequired control signaling overhead. Furthermore, for thelong term practical application of CS-based Multi-Userdetection schemes, a wider system context has to beconsidered, which focuses on the impact of CS detectorson other system components. Thus, we provide insightabout the behavior of the activity detection, which stronglyinfluences the impact of CS detection. To this end, weextend the analysis of CS detectors beyond the SymbolError Rate (SER), which has been the only qualityindicator in previous publications [7, 13]. In contrast, wealso illustrate different activity error events. Specifically,we compare the combination of a computationally efficientgreedy CS algorithm to a baseline system employingenergy detection and LS filtering that implements separateactivity and data detection. Additionally, on a framelevel channel coding influences the performance of bothapproaches and offers another opportunity for enhancedactivity detection. To this end, we propose to augmentthe channel decoding for enhanced activity detection toachieve activity-aware decoding.

1.4. Organization and Notation

The remainder of this paper is organized as follows:Section 2.1 introduces the general M2M scenario that isconsidered, and which is mathematically summarized inSection 2.2. Section 3 discusses strategies for separateas well as joint activity and data detection. Furthermore,channel coding in combination with the presented CS

Multi-User detectors is the topic of Section 4. Then,numerical results are discussed in detail in Section 5 andfinally a conclusion is given.

Throughout this paper, the following notation isemployed: G is a set of indices of groups and G is thecomplementary set. Further, Γ(G) specifies the indicesof those columns of a matrix A contained in the groupsindexed by G. Furthermore, AΓ(G) specifies the sub-matrix that only contains those columns with indices inΓ(G), and likewise xΓ(G) contains only those elementsof x with indices in Γ(G). Additionally, x`, A` and G`

specify the respective variable during the `th iteration.Herein, A† is the Moore-Penrose pseudoinverse of A, andAH the Hermitian matrix of A.

2. SYSTEM DESCRIPTION

2.1. Machine-to-Machine Scenario

Figure 1. Sensor network - star topology

In this paper, we consider a M2M scenario, whereK sensor nodes communicate with a central aggregationnode, as shown in Fig. 1, typically denoted as startopology. All sensor nodes are devices of low complexity,while the central node allows for advanced signalprocessing, such as sophisticated detection and messageaggregation for forwarding. In principle, the sensor nodesare only active on occasion, i.e., measurements have tobe transmitted regularly or event driven, which leads tosporadic communication. To model sensor node activity,we adopt a statistical approach, where each sensor nodeis active for a short time period, with a given activityprobability pa. This activity probability, which we assumeto be identical for all sensor nodes, determines the numberof active nodes in a statistical sense. In the following, thebase assumption is that out of the K sensor nodes in thenetwork the majority are silent, i.e., the activity probabilityis rather small with pa � 1. Given a large number of nodesK, this is a valid assumption for practical applications.

To facilitate sporadic and simultaneous medium access,CDMA with Pseudo random Noise (PN) sequences isused instead of the CSMA/CA scheme employed in IEEEstandards. Here, we choose PN sequences due to their well

Eur. Trans. Telecomms. 2013; 00:1–12 c© 2013 John Wiley & Sons, Ltd. 3DOI: 10.1002/ettPrepared using ettauth.cls


known correlation properties for uplink communication.CDMA offers a number of attractive properties for M2Mcommunication as noted before. It allows for a veryflexible non-orthogonal medium access with some degreeof asynchronicity on the chip level between sensor nodes,which is well suited for the low overhead communicationthat is the goal of efficient M2M communication schemes.

2.2. CDMA Chip-rate System Model

Enc Π PNbk ck dk

Figure 2. Sensor node transmitter model of sensor k

Based on the previous section, we formulate a mathe-matical model of the sporadic sensor node communicationfor further analysis. The basic transmitter setup of thesensor nodes is depicted in Fig. 2. We assume that an activenode ka transmits a data frame of Nb information bitsbka ∈ {0, 1}Nb . The information is encoded by a channelcode of code rate RC for error protection, yielding alength Nc code word vector cka ∈ {0, 1}Nc . After randominterleaving, the code bits are mapped to symbols yieldinga symbol vector dka ∈ ANd that contains the symbols of aframe, where A denotes the alphabet. In the following, wewill restrict the description to BPSK and, thus, the overallsystem description is real-valued. An extension to QAMmodulation alphabets is straightforward, e.g., by widelylinear processing. Note that the activity probability padetermines activity on a frame level, i.e., a sensor nodeis active or inactive for multiples of Nd = Nc symbols.Consequently, the symbol vector of an inactive node ki ismodeled as all-zeros, dki = 0 . Then the so-called aug-mented alphabet A0 = {A ∪ 0}, which is the modulationalphabet A augmented and extended by the zero symbolthat indicates inactivity or “no data”, describes all possibletransmit symbols.

With CDMA as medium access scheme, each sensornode k uses a real-valued PN sequence sk ∈ {±1}NS ,to spread each of its information symbols dk,i with i =1, . . . , Nc to NS chips. Without loss of generality, weassume that the spreading factor NS, i.e., the number ofchips per information symbol, is identical for all sensornodes. Therefore, the amount of symbols in a fixed framelength will be identical for all nodes.

We assume that sensor nodes switch activity on thesame symbol time basis, i.e., they are either active for thenext Nc symbols or inactive (frame synchronous), whichis illustrated in Fig. 3. In a practical system this has tobe ensured, e.g., by a synchronization signal from theaggregation node, which is beyond the scope consideredhere. However, it is still possible that the chips transmittedat a synchronous symbol time are not received at thesame chip time in this uplink scenario, thus yielding anasynchronicity of a few chips at the aggregation node.Such typical CDMA uplink considerations can be easily

0 NC 2NC 3NC

Node 1

Node 2

Node 3

Figure 3. Example for synchronous frame activity from thesensor node perspective

incorporated in the model by delayed versions of thespreading sequences similarly to [24]. In order to simplifythe notation, we assume that all frames are receivedsynchronously at the aggregation node. Furthermore, thespreaded symbols of each node are distorted by a sensornode specific frequency selective channel hk ∈ RLh oflength Lh which is constant for a whole frame the receivedvector at the aggregation node is

y =

K∑k=1

HkSkdk + n , (3)

where

Sk =

sk 0 00 sk 0

0 0. . .

(4)

denotes the spreading matrix of user k and

Hk =

hk,0 0 . . .hk,1 hk,0

......

. . .hk,Lh-1 hk,Lh-2

0 hk,Lh-1

0 0. . .

(5)

is the convolution matrix of the channel of userk and additive white Gaussian noise n ∼ N

(0, σ2

n)

superimposes the signal at the receiver. The chip-ratemodel (3) includes Inter-Symbol-Interference (ISI) withinone frame, due to the frequency selective channel.For simplicity, we do not capture frame start and endprocessing in this model. Thus, we omit the last Lh − 1rows of Hk and set the matrix to be Hk ∈ RNSNc×NSNc .

Based on [23], equation (3) can be written as a Multi-user chip-rate model as

y = Ax + n , (6)

where the matrix A ∈ RNSNc×KNc models the summationin (3), combining spreading and channel influences. Thevector x ∈ RKNc contains the data from all nodes, i.e., itis the stacked vector of all node frames dk

x = [dT1 , . . . ,dTK ]T . (7)




3. ACTIVITY AND DATA DETECTION

3.1. Compressive Sensing Detection Model

To clarify the connection of Compressed Sensing and theCDMA model developed in the previous section, recallthe CS model (1). In terms of the CDMA model, thesparsity basis Φ is given by the set of PN sequences,which are used to separate the sensor nodes, and thechannel influence can be interpreted as the measurementmatrix Ψ. However, in contrast to the typical CS problem,the measurement matrix Ψ cannot be chosen arbitrarily,but is given physically, and the sparsity basis Φ is notconstrained by physical properties of a signal, but can bechosen freely. Instead of viewing sparsity basis Φ andmeasurement matrix Ψ as separate parts of the systemdescription, it is also widely accepted to combine the twomatrices to an overall matrix A that characterizes thetransition from the sparse domain to the observation [16],which is exactly the CDMA system equation (6). Thismeans that with (6) as the system model by solving (2)we jointly detect data and activity of sensor nodes withoutrequiring activity control signaling.

Unfortunately, the computational complexity of thisapproach quickly gets prohibitive. As the size of the Multi-user vector x is given by KNc a restricted model fordetection purposes is required for reasonable code wordsizes Nc. Therefore, (6) is divided into sub-problems,each consideringL consecutive transmit symbols per node.Thus, each of the Nc/L sub-problems is determined bythe system matrix Aν ∈ RNSL×KL, which is the same forall ν = 1, . . . , Nc/L, as both PN sequences and channelsare assumed to be constant for an entire frame. In order tosimplify the detection model, we neglect the ISI betweensub-problems in the detection. Depending on the channellength Lh the choice of the sub-problem size L willdetermine the performance loss due to remaining ISIwith negligible losses for L/Nc → 1 and more severelosses for L/Nc → 0. The transmit vector xν ∈ AKL0 thensummarizes L out of Nc transmit symbols for each of theK nodes in the ν th sub-problem as

xν = [d1,L(ν−1)+1, . . . , d1,Lν , . . .

, dK,L(ν−1)+1, . . . , dK,Lν ]T . (8)

Note, that the properties of A in a typical CDMAsetup are different from the usual assumptions in CSliterature. In CS theory, the columns of A are usuallyassumed to be normalized to length 1, i.e., columns withidentical `2-norm, which may have a severe impact on thereconstruction quality. In the previously defined CDMAsystem, both the channel and the PN sequences will onaverage have unit norm. However, due to the randombehavior of these two influences, the column norms mayvary strongly, and thus statistical reconstruction qualitywill deviate from the ideal CS behavior. Therefore, weintroduce an effective channel Hν , which is a diagonal

matrix of the column norms of Aν such that

Aν = AνHν . (9)

Using this modification, CS detection is applied to aunit norm matrix Aν which has two advantages: i)the performance of the greedy CS algorithm discussedin Section 3 is enhanced [25] and ii) either soft-valuecalculation or hard quantization after CS detection canexploit Hν as the effective channel.

In summary, the following simple system equationfully characterizes the communication for each of theν = 1, . . . , Nc/L equivalent CS detection models

yν = AνHνxν + nν , (10)

where nν ∈ RNSL is real-valued AWGN noise N(0, σ2

n)

and yν ∈ RNSL denotes the received vector at theaggregation node.

3.2. Detection Strategies and Assumptions

Energy

Detection

LS

FilterDEC

GOMPmajoriy

decisionDEC

GOMPactivity-aware

DEC

a)

b)

c)

Figure 4. Detection strategies for data and activity detection

Besides the CS based detection, which is the main focusof this paper, other activity detection strategies can beapplied. Fig. 4 summarizes the three strategies that willbe compared here. Fig. 4a shows separate activity anddata detection, where energy detection (ED) is employedto identify active nodes and data estimation is thenachieved by LS filtering, hard/soft decision and subsequentdecoding. Accordingly, ED-LS will be the baseline systemto compare with and is briefly described below. Fig. 4bshows the CS-MUD approach using the Group OrthogonalMatching Pursuit (GOMP) detailed below followed by amajority decision to jointly detect activity and data. Due tothe fact that the GOMP will be applied per sub-problemas described in the previous section node activity is onlydecided for groups of L symbols requiring an additionalframe level decision. The majority decision is the optimalhard activity decision in this context, because frameactivity can be interpreted as a repetition code of code rate1/NC regarding activity. Thus, frame activity knowledgeis exploited even though CS-MUD is applied only per sub-problem. Again, hard/soft decision and standard decodingcan be applied to recover the data. Finally, Fig. 4c depicts acombined detection strategy of CS-MUD and an extended



decoder that incorporates the decision which nodes areactive into decoding based on the knowledge of the codestructure. More details about this “activity-aware” decoderwill be discussed in Section 4. Regarding complexity, ED-LS is obviously the simplest. The GOMP adds additionalcomplexity due to the iterative approach, where the maincomplexity stems from repeated LS filtering and finally themost complex scheme employs the activity-aware Viterbi,which has to be used for nearly all nodes to decide activityinstead of only the active ones. Subsequently, the energydetector and GOMP will be discussed in more detail.

The base assumption for all detection strategies inthis paper is perfect CSI knowledge at the aggregationnode, i.e., A and σ2

n are known. In a practical setup, thechannel knowledge has to be acquired, e.g., by semi-blindmethods exploiting the structure imposed by spreadingsimilar to [26]. Furthermore, for slowly changing channelsestimation could be facilitated by a regular estimationphase with channel tracking in active phases of the sensornodes. A detailed scheme for CSI acquisition, though, isbeyond the scope of this paper.

3.3. Energy Detection

The baseline activity detector for CDMA systems employsa matched filter to correlate the columns of A withthe received signal y and sums this correlation for allNC symbols of a frame to determine active users. Thecorrelation of the kth user is then given by

rk =∑j∈Γ(k)

∣∣∣AHj y∣∣∣ , (11)

where Γ (k) denotes the column indices of user k.Principally, LS and MMSE filtering could also be usedhere, but are omitted due to complexity reasons.

Usually, active users are identified based on acorrelation threshold given by the assumed noise model[27]. However, to provide a fair comparison to the GOMPand to avoid threshold dependencies we assume that theenergy detector has perfect knowledge of the instantaneousnumber of active users Ka. Thus, the Ka nodes with thehighest correlations rk are collected in the set of activenodesG. Subsequently, Least Squares filtering with A†Γ(G)

followed by a hard or soft decision can be applied toestimate the transmitted data of the active users. Note thatthe computational complexity of the pseudoinverse quicklygets prohibitive for large frame sizes NC. Therefore,energy detection exploits the full frame structure, but LSfiltering will be applied to the sub-problems (9) ignoringsome amount of ISI.

3.4. Group Orthogonal Matching Pursuit

The Group Orthogonal Matching Pursuit algorithm [28] isa block-sparsity exploiting member of a family of greedyCS algorithms. The main property of these algorithms islow complexity. However, the drawback of any greedyapproach is error propagation, since previous choices

Algorithm 1 Group Orthogonal Matching Pursuit(GOMP)

G0 = ∅, ` = 0, r0 = yrepeat` = `+ 1kmax = arg max

k

∑j∈Γ(k)

∣∣AHj r`−1

∣∣ with k ∈ G`−1

G` = G`−1 ∪ kmax

x`Γ(G`) = A†

Γ(G`)y and x`

Γ(G`)

= 0

r` = y −Ax`

until ` = Ka

for symbol activity are not re-evaluated. Even thoughthe GOMP will be applied to sub-problems (10), thedescription in this section omits the index ν for the sakeof notational clarity.

The OMP [29, 30], from which the GOMP is derived, isbased on two observations: the sparse solution in generalcontains only a few non-zero values and the measurementsin the absence of noise are in a space supported by asub-matrix of A determined by the position of these non-zero elements. Therefore, the OMP iteratively builds thesupport of x, by building the set of columns Γ of matrix Athat define the space for the measurements y. The GOMPextends this idea by using knowledge about block-sparsity.Due to the fact that L symbols are stemming from a singlenode, this group of symbols has to be either active or non-active. Therefore, the support has to be build by addingindices of groups to a group index set G instead.

During each iteration, the GOMP adds the group kmax,which has the highest correlation to the previous residualr`−1 = y −Ax`−1 to the setG`−1. This step implementsthe activity detection based on previous activity anddata decisions. After this activity detection, the valuesof the non-zero elements in x are determined by LeastSquares estimation using the sub-matrix AΓ(G`) of all

columns in Γ(G`). Afterwards, the current residual r`

is calculated. As the subsequent iteration depends onthe previous residual r`−1, the activity detection of theGOMP is influenced by errors in the data estimation. Theseiterations are continued up to an appropriate stoppingcriterion, which is discussed in detail in [31]. Withoutloss of generality, we use the ideal stopping criterion ofterminating the algorithm after a number of iterations equalto a known sparsity Ka, i.e., terminating at ` = Ka.

In contrast to the baseline system using energy detectionand LS filtering, here the correlation with respect tothe combination of spreading sequence and channel isdone iteratively and on the level of the sub-probleminstead of the whole frame. Furthermore, LS estimationand activity detection are done jointly such that the LSestimation enhances the activity detection step by reducingthe residuum. This suppresses highly correlated nodes thatare likely chosen by ED, if both exhibit an overall highcorrelation rk.




S/P

CS-MUD

CS-MUD

DEC

DEC

Node Sorting

Node perspectiveDetection model perspective

n

y

y1

yNC

L

x1

xNC

L

d1

dK

c1

cK

b1

bK

Figure 5. Detection structure at the aggregation node for CS based joint activity and data detection

4. CS DETECTION AND DECODING

4.1. General Aspects

According to the system model and the GOMP algorithmdiscussed in the previous section, decisions are madeon a sub-problem level, neglecting the frame activity ofsensor nodes. The received vector y is separated formultiple CS-MUD detectors per (10), which is illustratedin Fig. 5 and the resulting symbol estimates xν have tobe sorted to reconstruct the node specific frames. Thus, theestimated frame dk of sensor node k is build as the stackedvector of L estimates from ν = 1, . . . , Nc/L detector runs(cf. Algorithm 1)

dk = [x1,L(k−1)+1, . . . , x1,Lk, . . . ,

xNc/L,L(k−1)+1, . . . , xNc/L,Lk] . (12)

Thereby, each frame contains continuous non-zero entries,where data has been estimated, and zero entries, wherethe sensor has been estimated as inactive. In other wordsthe effective channel after CS-MUD behaves similar to anerasure channel. As a result, it is not obvious which sensornodes were active on a frame level and how decoding canbe accomplished.

After frame reconstruction, either a quantizer orLLR calculation is required to demodulate the non-zerooutput of the CS-MUD using the effective channelsHν . Originally, the transmit frame of active nodesonly contained symbols from the alphabet ANc , whichposes the question how to handle zero symbols in theestimated frame. The optimal processing should exploitsoft-information to preserve reliability information fordecoding. Due to the hard decided activity by the GOMP,however, soft-information is not available for zeros andonly in the sense of the effective channel coefficients afterLS filtering for non-zero entries. Thus, soft-values for non-zeros can be estimated easily similar to soft-processing forlinear filters, often employed in communication systems.

From the channel coding point of view each zero isequivalent to an erased bit and in terms of LLR valuesconveys no decision in either direction. Such, any channelcoding scheme that allows to handle erased bits in thedecoder should be suitable. In the following, however, wewill focus on convolutional codes as a specific channelcoding example, as erased code bits pose no difficulty for

a Viterbi decoder. In comparison to optimal symbol-by-symbol decoding through a BCJR decoder, the (Soft-in)Viterbi decoder offers the advantage to determine optimalcodewords, which can be exploited directly for activitydetection as will be shown in the next section. Still,depending on the percentage of erased bits, decoding errorsmight be very likely, due to a lack of information. As a ruleof thumb, for Viterbi decoding, a decision after 5LC statetransitions does not influence the remainder of the codeword, where LC is the constraint length. So, if a wholeblock of bits in the range of 5LC has been erased, this partof the code word is entirely random.

Independent of the number of symbols in a framethat have been estimated as active, the Viterbi decoderestimates the most likely code word, and thus generates anestimate of the transmitted information bits bk. Consideran extreme example: an inactive node k is detected suchthat its estimated frame dk only contains a single non-zero entry dk,n. It seems obvious that this sensor has beeninactive, however, the channel decoder cannot estimateinactivity as is. Hence, the two options already discussedas Fig. 4b, majority decision plus CS-MUD (M-CS-MUD)and activity-aware Viterbi decoding plus CS-MUD (V-CS-MUD) seem viable. In the following, we focus onthe second approach to exploit the code redundancy foractivity detection.

4.2. Activity-aware Viterbi Decoding

ξinactive

ξminViterbi best hypothesis

Erasure hypothesis

Figure 6. Activity-aware Viterbi extension by erasure path

To incorporate the knowledge that a node could havebeen inactive instead of sending a code word intothe channel decoder, the Viterbi decoder needs to beaugmented. Viterbi decoding using the Euclidean distancemetric determines the state transition costs using a BPSK



notation of the code bits. This means that an erasedsymbol, i.e., a zero, simply acts as a “don’t care” inthe metric, which is independent of hard or soft decisioninputs. If the majority of the bits of a code word areerasures, the accumulated path metric will no longerprovide any meaningful decision as most parts of theestimated minimum metric path will be arbitrarily chosen.Therefore, it is necessary to calculate the probability ofan all-erasure word in addition to Viterbi decoding, whichis illustrated in Fig. 6. The all-erasure path at the topof the figure indicates the necessary augmentation toconsider inactivity of a sensor node and has a Euclideandistance metric ξinactive,k, which simply is the l2-norm ofthe quantized code word vector ck in case of hard decisions

ξinactive,k = ‖ck‖2 . (13)

Alternatively, the metric can be calculated in the samesense for soft-values. Comparing the best code wordhypothesis determined by Viterbi decoding with thiserasure hypothesis is a generalized log-likelihood ratiotest comparing the two hypotheses “active” and “inactive”[32]. Using the path metrics the problem can be cast asfollows

ξinactive,k − ξmin,kActive

≷Inactive

0 . (14)

Thus, if ξinactive,k is smaller than or equal to the minimumsum path metric ξmin,k after Viterbi decoding, the kth

sensor node is estimated as inactive by the decoder. Insummary, the channel decoder is used as a decision devicefor sensor node activity on a frame level.

5. NUMERICAL EVALUATION

5.1. Parameters

In the following, we discuss numerical evaluations of thedetectors described in Section 3. More specifically, thefocus is on a detailed analysis of the activity detection andthe influence of the channel code.

An interesting setup for CS detection is given byoverloaded CDMA systems, which result in under-determined equation systems for CS detection. Especiallyin such under-determined systems, CS algorithms offerpotential gains over conventional MUD schemes if sensornode activity is unknown [7]. We focus on two scenariosto demonstrate the influence of different problem sizeson the performance. The channel is modeled by Lh =6 i.i.d. Rayleigh distributed taps with an exponentialdecaying power delay profile for both cases. Due to theconstant channel per frame, other channel statistics wouldmainly influence the amount of ISI and to a smaller degreethe correlation, but do not change the overall behavior.

The first scenario uses K = 128 sensor nodes, whereeach node transmits using a PN sequence of length NS =32. Hence, the CDMA system is under-determined bya factor of four. Furthermore, the frame length is Nc =

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−6

10−5

10−4

10−3

10−2

10−1

100

Es/N0

SER

ED + LS

OMP, L8

GOMP, L2

GOMP, L4

GOMP, L8

oracle LS

Figure 7. Symbol Error Rate for the augmented alphabetafter activity and data detection, before decoding; Parameters:

K = 128 nodes and activity probability pa = 2%

104 symbols using a standard terminated non-systematicnon-recursive [5; 7]8 convolutional code (code rate RC =1/2, constraint length LC = 3). Consequently, Nb = 50information bits are transmitted by every active node.

The second scenario with K = 32 sensor nodes anda PN sequence length of NS = 8 is used to demonstratethe effect of channel coding and the problem size onperformance. Furthermore, two different codes are utilizedto demonstrate the achievable gains. On the on hand, theterminated [5; 7]8 convolutional code is used as a “weak”code and the [133; 145; 175]8 convolutional code (coderate RC = 1/3, constraint length LC = 7) as a “strong”code. The frame lengths are fixed to Nc = 104 in bothcases. Note, that all presented results are restricted to harddecisions after symbol detection. An extension to soft-information is straightforward as discussed in Section 4and effects all presented results in the same way.

5.2. Results without Channel Coding

Fig. 7 shows the symbol error rate (SER) over theaugmented alphabet A0 before channel decoding. Thus,the SER contains both errors due to incorrect activitydetection, and errors due to incorrect data estimationequally weighted. If not noted otherwise, the sub-problemsize L = 8 and activity probability pa = 0.02 are used.In addition to the approaches presented in Fig. 4a andc, i.e., ED-LS and GOMP with activity-aware Viterbidecoding, we show simulation results for the OMP. This isa GOMP with index group size 1 in terms of Algorithm 1and demonstrates the gains by exploiting block-sparsity.Furthermore, the oracle LS is a LS estimation for knownactivity. Effectively, the SER of the oracle LS is onlydetermined by data errors of the LS estimation andcontains no support errors. First, we note that the GOMPof any sub-problem size L has much lower SER thanthe OMP in this scenario, which is obviously due to




0 0.02 0.05 0.110

−4

10−3

10−2

10−1

100

pa

SER

ED + LS

GOMP

oracle LS

Figure 8. Symbol Error Rate vs. activity probability beforedecoding; Parameters: K = 128 nodes, ES/N0 = 12 dB and

sub-problem size L = 8

the exploitation of block-sparsity. The larger the sub-problem size, the larger the gains. Applying the GOMPto the whole problem, i.e., L = Nc, can be expectedto outperform energy detection over the whole SNRrange, but is computationally prohibitive. Secondly, theED-LS scheme shows superior performance compared tothe CS schemes at low SNR, but quickly hits an errorfloor. The energy detector exploits the full frame activityknowledge, which leads to a good performance for highnoise levels due to the averaging. However, systematicactivity estimation errors caused by highly correlatedcolumns of A quickly dominate the performance and leadto an error floor. In overloaded CDMA systems, spreadingsequences are highly correlated with a high probability andthe OMP/GOMP approach explicitly suppresses nodes,whose columns are highly correlated with a node alreadydetected as active. Lastly, all schemes have a much higherSER than the oracle LS. Therefore, we can conclude thattheir SER is mainly determined by errors in the activitydetection.

To demonstrate the effect of different activity probabil-ities pa, Fig. 8 shows the SER at Es/N0 = 12 dB. As canbe seen, the performance of both approaches deteriorateswith increased pa and the difference between GOMP andED gets smaller. The more nodes are active, the less errorsare mainly caused by highly correlated nodes. In terms ofCS theory, a higher activity probability corresponds to anoverall higher mean sparsity and increased probability ofhigh sparsity values. Sparsity and the degree of underdeter-mination are tightly connected and determine, if a systemis reliably solvable by CS methods. Furthermore, the sizeof the system, i.e., the dimensions of A, influences theperformance, which is illustrated by Fig. 9.

Here, the second scenario with K = 32 nodes exhibitsthe same degree of sparsity and underdetermination as thefirst scenario when normalized to the problem dimensions.The overall performance of the CS schemes is visiblydegraded in comparison to Fig. 7, whereas the energy

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−5

10−4

10−3

10−2

10−1

100

Es/N0

SER

ED + LS

OMP

GOMP

oracle LS

Figure 9. Symbol Error Rate before decoding; Parameters:K = 32 nodes, activity probability pa = 2% and sub problem

size L = 8

detector shows the same behavior. This indicates that CSdetection performs better for larger system sizes, eventhough sparsity and underdetermination are comparable. InCS theory, phase transitions [33] are used for the noise freecase to describe the parameter sets of underdeterminationand sparsity for which perfect recovery is still reliable.Furthermore, both OMP and GOMP lead to an error floorat high SNR, which is due to systematic errors caused bycorrelations. As data detection errors can be improved bynon-linear data estimation and by the subsequent channeldecoding, we focus the investigation on activity errors.

5.3. Results with Channel Coding

In the following, we investigate numerical resultsafter decoding. Specifically, activity errors of an entirecode word/frame are considered, where we differentiatebetween frame missed detection (FMDR), i.e., an activenode is detected as inactive, and frame false alarm rates(FFAR), i.e., the opposite case.

Fig. 10 shows the FFAR (dashed) and FMDR (solid) forthe first scenario with K = 128 nodes and pa = 2%. Forsimplicity, only the ED-LS and GOMP results for L = 8are shown. It is quite apparent, that the activity detectionperformance of the energy detector is the determiningfactor for the overall SER performance. Especially, themissed detection rate dominates in comparison to the falsealarm rate. Similarly, the GOMP shows a missed detectionrate behavior analogous to the SER results presented inFig. 7. In contrast, no false alarms up to a FFAR of 10−5

could be observed in our evaluation, which allows theconclusion that the FFAR is much lower than the FMDR inthis case. These results are especially interesting in termsof the overall CDMA system. On the one hand, higherlayer processing does not need to determine “garbage”data estimated for truly inactive users, if false alarms aremostly negligible. On the other hand, most higher layerprocessing includes some amount of error detection likeCRC codes, which implicitly allow to validate the activity



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10−4

10−3

10−2

10−1

100

Es/N0

fram

eactivityerrors

ED + LS (MDR)

ED + LS (FAR)

GOMP (MDR)

Figure 10. Frame missed detection (solid) and false alarm rates(dashed) after decoding; Parameters: K = 128 nodes, activity

probability pa = 2% and sub-problem size L = 8

decision of the physical layer schemes. From this point ofview, a heightened false alarm rate would not harm systemperformance, but an improved missed detection rate couldtremendously improve the detection of truly active nodes.Thus, new or modified detection algorithms are requiredthat offer more flexibility to trade-off false alarms andmissed detections.

Up to this point, the specific improvements achieved bychannel coding have not been discussed. The presentedresults were focused on Fig. 4a (ED-LS) and Fig. 4c (V-CS-MUD). To illustrate the differences between Fig. 4b(M-CS-MUD) and Fig. 4c (V-CS-MUD), i.e., majoritydecision before Viterbi decoding and activity-awareViterbi decoding, Fig. 11 shows the FFAR and FMDR forthe second scenario with K = 32 nodes, a sub-problemsize of L = 8 and activity probability pa = 0.02. Resultsfor both approaches are indicated by M-CS-MUD and V-CS-MUD in the figure. Furthermore, two different channelcodes have been applied to illustrate the outcome ofdifferent coding strengths.

Once again, energy detection exhibits the sameperformance as in the first scenario. The missed detectionrate (solid lines) for the CS-MUD approaches, however,surprisingly indicates no relevant performance differencescomparing majority decision and activity-aware decoding,even comparing the two codes. A simple explanation forthis behavior lies in the structure of the activity-awareViterbi detection. The additional erasure path only helps toavoid false alarms, but does not improve missed detectionsin any way. Consequently, the code strength does notinfluence this behavior. Furthermore, majority decision isalready optimal for hard decisions if the code structurecannot be exploited as in the case of false alarms.

Considering the false alarm rate (dashed lines), activity-aware Viterbi decoding actually shows slight performancegains over majority decision, whose performance obvi-ously does not depend on the coding scheme. The perfor-mance differences of activity-aware decoding comparing

0 5 10 15 2010

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Es/N0

fram

eactivityerrors

ED + LS

M-CS-MUD, R 12 , LC3

M-CS-MUD, R 13 , LC7

V-CS-MUD, R 12 , LC3

V-CS-MUD, R 13 , LC7

Figure 11. Frame missed detection (solid) and false alarm rates(dashed) after decoding;Parameters: K = 32 nodes, activity

probability pa = 2% and sub-problem size L = 8

the “weak” and the “strong” code, though, are negligibleas the main performance factor is the coding gain. Theassumed block fading channel offers no diversity.

Interestingly, the false alarm rate increases with theEs/N0 instead of decreasing and the performance gaincompared to the majority decision gets smaller. Thereare two opposing influences that cause the observedbehavior: in the under-determined scenario it is possiblethat the columns Ak1 and Ak2 of two nodes k1 andk2, which represent their spreading sequence and channelinfluences, are highly correlated. Thus, whenever one ofthese nodes, say k1, is active we are likely to estimate theincorrect, highly correlated node k2 as active instead. Asthe correlation between nodes is the same for the entireframe, this type of error likely results in a frame error.More importantly, stronger channel coding only slightlyhelps to cope with it, as the received frame is a validcodeword with high probability due to the systematicnature of this error.

The second influence on the frame false alarm rate isthe presence of noise. While the influence of noise leadsto more errors on a symbol level, it eases frame errordetection. The main reason for this is, that correlationbetween nodes is constant for an entire frame, while thenoise is different for each transmitted chip. Thus, for eachtransmitted sub-frame, a different node may be incorrectlydetected as being active. This increases the likelihood thatno node is incorrectly decided as being active for themajority of symbols. We can conclude that the noise helpsto disperse the symbol-wise activity errors over nodes,thereby making frame-wise activity errors less likely.

6. CONCLUSION

In this paper, we have presented a scheme for applicationof CS algorithms to a channel coded CDMA systememployed for sensor node communication. To this end,




a CDMA chip-rate model has been formulated anddecomposed into smaller sub-problems to lower thecomputational complexity at the cost of sub optimality.The major advantage of CS-MUD is that activity and datadetection can be jointly facilitated on the physical layer,which may be exploited to lower the required signalingoverhead for M2M systems. Here, we concentrated on twoaspects: i) the detailed behavior of the GOMP detectorwith respect to activity detection and associated errorsto provide meaningful performance indicators and ii) theapplication of channel coding in combination with suchCS-MUD schemes to enhance the overall performance ofthe system, especially the activity detection.

Regarding the first point we have shown that the twopossible activity errors (false alarms, missed detections)have to be differentiated to fully understand the impactof CS detectors on communication systems. Especially,missed detections determine the overall performance,whereas false alarms are less likely and can be furtherreduced using channel coding and adapted decoders.To achieve the second point, the erasure-like overallbehavior of transmitter, channel and CS-MUD has beenincorporated in the decoding process by a slightlyenhanced Viterbi decoding scheme. The numerical resultspresented in this paper clearly indicate that the applicationof channel coding in this context enables an enhancedactivity detection. However, this improvement is limitedto the case of false alarms, which, in principle, is the lesscritical error. On the one hand, far less false alarms areoccurring, and on the other hand, further processing onhigher layers might ensure detection of misinformation.Still, activity errors are tightly coupled and enhancing thefalse alarm error rate may still prove useful, if the decodingresults are fed back to the CS-MUD in an iterative fashion.

Based on our results, a variety of future research topicsshould be addressed. On the one hand, theoretical analysisregarding the properties of the system matrix A in aCS context is urgently required to provide insights onthe fundamental bounds of different algorithms. On theother hand, the activity detection clearly needs to beenhanced with respect to the missed detections. To thisend, possible approaches are iterative schemes, whichexploit the channel coding for enhanced false alarm ratesor enhanced CS-MUD that primarily focuses on misseddetection rates. The main conclusion we draw from Fig. 10and 11 is that known CS algorithms are not well suitedfor systems, where the severity of activity estimationerrors is not symmetrical. As already mentioned, misseddetections are the dominating effect here and may alsohave an overall higher impact in a systems context, whichleads to asymmetrical error importance. The usual goalof CS, however, is to find solutions that are as sparse aspossible independent of this aspect, which is only idealin terms of reconstruction error or the SER over theaugmented alphabet. Therefore, for future applications,modified criteria have to be adopted to formulate optimaland suboptimal detection algorithms.

ACKNOWLEDGEMENTS

This work was supported in part by the German ResearchFoundation (DFG) under grant DE 1858/1-1.

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AUTHORS’ BIOGRAPHIES

Prof. Armin Dekorsy holds the chair of the Departmentof Communications Engineering, University of Bremen,since April 2010. He received his Dipl.-Ing. (FH)(B.Sc.) degree from Fachhochschule Konstanz, Germany;Dipl.-Ing. (M.Sc.) degree from University of Paderborn,Germany; PhD degree from the University of Bremen,Germany, all in communications engineering. From 2000to 2007 he worked as research engineer at DeutscheTelekom AG and as Distinguished Member of TechnicalStaff (DMTS) at Bell Labs Europe, Lucent Technologies.In 2007 he joined Qualcomm GmbH as European ResearchCoordinator conducting Qualcomms’ internal and externalEuropean research projects like ARTIST4G, BeFemto,and WINNER+. His current research interests includeresource management, transceiver design and digitalsignal processing for wireless communications systems inhealth care, automation and mobile communications. Prof.Dekorsy is member of ITG expert committee ”Informationand System Theory”, VDE and IEEE communications andsignal processing society.

Dr.-Ing. Carsten Bockelmannn received his Dipl.-Ing.(M.Sc.) degree in electrical engineering in 2006 andhis PhD degree 2012 both in electrical engineering andfrom the University of Bremen, Germany. Since 2012he is working as a post doctoral researcher at theUniversity of Bremen coordinating research activitiesregarding the application of compressive sensing/samplingto communication problems. His current research interestsinclude compressive sensing and its application incommunications contexts, as well as channel coding andtransceiver design.

Dipl.-Ing. Henning F. Schepker graduated from theRWTH Aachen University in Aachen, Germany, in 2009.He started working as a PhD candidate at the Departmentof Communications Engineering at the University ofBremen in Germany in 2010. His research is focused onCompressive Sensing in communication technology.



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