ieee transactions on communications, vol. 65,...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 6, JUNE 2017 2715

Compressive Channel Division Multiple Accessfor MTC Under Frequency-Selective Fading

Jinho Choi, Senior Member, IEEE, and Nam Yul Yu, Member, IEEE

Abstract— In this paper, we study random access formachine-type communications based on a multiple access schemethat exploits different channel state information (CSI) from activedevices under frequency-selective fading, which is called channeldivision multiple access (ChDMA). In ChDMA, in order to use alow-complexity compressive sensing (CS) algorithm for multiuserdetection (MUD) with a high transmission rate, we propose to useindex modulation. Since the CSI is used as a signature for MUD,the CSI estimation becomes crucial. For the CSI estimation,pilot transmission can be considered where each active devicerandomly chooses a pilot from a set of pre-determined pilots.Since the CSI estimation suffers from pilot collision in this case,we propose multiple pilot transmissions and show that a lowprobability of collision can be achieved. A CS-based approach isderived not only to estimate the CSI, but also to detect randomlyselected multiple pilot signals by exploiting the sparsity of activedevices over frequency-selective fading channels.

Index Terms— Random access, machine-type communications,compressive sensing.

I. INTRODUCTION

THERE has been a growing interest in machine-typecommunications (MTC) in order to support a number

of devices that are to be connected to a network [1]. Theapplications of MTC are diverse from health care to wirelesssensor networks and the Internet of Things (IoT). For MTC,random access is usually considered due to low signaling/control overhead in supporting a number of devices with alow probability of activity [1]–[3]. For example, in the longterm evolution-advanced (LTE-A) system, a random accessscheme, called random access (RACH) procedure, is proposedfor MTC [4]. In 5G systems, it is also expected to support up to30,000 devices per cell [5].

As in the RACH procedure, when multiple devicescan transmit randomly chosen preambles1 from a pool ofpreambles, an access point (AP) needs to detect them simulta-neously. If the AP is able to detect all transmitted preambles,

Manuscript received July 16, 2016; revised October 27, 2016,January 6, 2017, and March 5, 2017; accepted March 7, 2017. Date ofpublication March 13, 2017; date of current version June 14, 2017. This workwas supported by the “Climate Technology Development and Application”research project (K07732) through a grant provided by GIST in 2017. Thispaper was presented in part at the IEEE ICC, May 2016, and the IEEEICC, May 2017. The associate editor coordinating the review of this paperand approving it for publication was H. R. Bahrami. (Corresponding author:Jinho Choi.)

The authors are with the School of Electrical Engineering and ComputerScience, Gwangju Institute of Science and Technology, Gwangju 61005, SouthKorea (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TCOMM.2017.26816621Throughout the paper, we assume that preamble and pilot are interchange-

able. That is, a pilot sequence is also called a preamble sequence, vice versa.

those devices can successfully establish connections,which results in throughput improvement. Clearly, thisbecomes a key advantage of the RACH procedure oversingle-channel random access schemes (e.g., slotted ALOHA)that have a low throughput. In fact, the RACH procedurecan be seen as a multi-channel random access scheme withmultiple non-orthogonal channels where each channel ischaracterized by a different preamble.

In [6] and [7], it is suggested to support sporadic trafficsfrom devices in MTC separately from standard uplink pipesand to exploit the sparsity of active devices. For the casethat the activity of devices is low, the notion of compressiverandom access2 has been proposed in [8]–[13], where thesparse activity is exploited to derive computationally efficientmultiple signal detection schemes for a receiver at the APbased on compressive sensing (CS) algorithms [14]–[17].If the sparse activity is not exploited, the AP may use anexhaustive search to detect the set of active devices togetherwith a conventional multiuser detection (MUD) approach forjoint signal detection [18]. In this case, MUD with exhaustivesearch may suffer from a prohibitively high computationalcomplexity for a large number of devices unless multiplechannels are orthogonal. It is noteworthy that even if orthog-onal sequences are used to create multiple channels (as incode division multiple access (CDMA)), the orthogonality maynot be retained due to the distortion by channels as demon-strated in most multicarrier (MC)-CDMA systems [19]. Conse-quently, under a realistic channel environment (e.g., frequency-selective fading channels), compressive random access wouldbe promising for MTC.

In [20], compressive random access in conjunction with thechannel estimation is studied under a frequency-selective fad-ing channel environment with the channel estimation, whichis different from other approaches in [9], [10], and [12] wherethe channel state information (CSI) is assumed to be per-fectly known. Note that in [8], the channel estimation is alsoconsidered in compressive random access. In [21] and [22],compressive random access is studied for multicarrier systemswith the CS-based signal detection in the frequency-domain.Note that in [21], compressive random access is referred toas CS-based MUD in order to emphasize that the notionof CS is exploited for MUD. While CS-based MUD wouldbe a reasonable term, we prefer compressive random accessin this paper as in [8] in order to emphasize the nature of

2This term is used in [8] to refer to a random access scheme that allows areceiver to employ low-complexity CS algorithms for MUD.

0090-6778 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

2716 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 6, JUNE 2017

unscheduled sporadic transmissions for random access(in MTC) and the sparsity of activity that allows to derivelow-complexity CS-based MUD methods.

It is noteworthy that unlike the RACH procedure in [4],compressive random access is a “one-shot” transmissionscheme that can transmit a short message from an active devicewithout any handshaking process [20], [23]. That is, an activedevice transmits a packet that includes pilot and data blocks(as one-shot transmission) without any connection request andchannel allocation. While one-shot transmission is useful forvery short packets (of few bytes), it may not be practicalwhen the length of message is not short. Thus, in general,compressive random access would be suitable for MTC wheredevices transmit short messages of few bytes.

In this paper, we consider a compressive random accessscheme over frequency-selective fading channels that canmitigate some problems in conventional compressive randomaccess schemes. As in [8], [10], and [20], each device can havea unique spreading code. However, if the number of devicesis large, it may not be possible to assign a unique code foreach device due to various reasons (e.g., if orthogonal codesare used, the number of devices is limited by the length ofcodes). In the proposed compressive random access scheme,the CSI of active device is used as a signature for MUD.As a result, the proposed compressive random access schemecan support any number of devices as long as there are fewactive devices and any active device’s CSI is different fromeach other. For a higher transmission rate, we employ indexmodulation for data transmission. For the CSI estimation,we consider a conventional approach as the RACH proce-dure [4] with a pilot pool of finite size. In this case, unfortu-nately, the channel estimation suffers from pilot collision [22].To mitigate this, we consider multiple transmissions of pilotsin the proposed compressive random access scheme. As aresult, the proposed scheme can support a large number ofdevices of sparse activity with a negligible probability of pilotcollision. Note that this paper is based on [22] for the channelestimation, while the use of channels as signatures for multipleaccess and index modulation are not discussed in [22].

Since the size of the pilot pool is not necessarily large,there are also the following salient features in the proposedcompressive random access scheme: i) a good set of pilots orpilots (such as Zadoff-Chu (ZC) sequences [24]) can be used;ii) the complexity of CS algorithms for the CSI estimation canbe low.

The rest of the paper is organized as follows. In Section II,we present a system model for random access in MTC with anumber of devices with low activity. We propose a compres-sive random access scheme that uses the CSI as a signaturefor MUD in Section III. Since the proposed compressiverandom access scheme depends on the CSI, the CSI estimationis crucial, which is studied in Section IV. In this section,in particular, we study the CSI estimation with a commonpilot pool and propose the use of multiple pilot transmissionsto mitigate pilot collision. We present simulation results inSection V and conclude the paper with some remarks inSection VI. Note that for convenience, the list of symbolsused in the paper is given in Table 1.

TABLE I

TABLE OF SYMBOLS

Fig. 1. The structure of a packet consisting of multiple blocks for pilot anddata transmissions.

Notation: Matrices and vectors are denoted by upper-and lower-case boldface letters, respectively. The superscriptsT and H denote the transpose and complex conjugate, respec-tively. The p-norm of a vector a is denoted by ||a||p (If p = 2,the norm is denoted by ||a|| without the subscript). Thesuperscript † denotes the pseudo-inverse. For a vector a,diag(a) is the diagonal matrix with the diagonal elements froma. For a matrix X (a vector a), [X]n ([a]n) represents the nthcolumn (element, resp.). If n is a set of indices, [X]n is asubmatrix of X obtained by taking the corresponding columns.E[·] and Var(·) denote the statistical expectation and variance,respectively. CN (a, R) (N (a, R)) represents the distribution ofcircularly symmetric complex Gaussian (CSCG) (resp., real-valued Gaussian) random vectors with mean vector a andcovariance matrix R.

II. SYSTEM MODEL FOR RANDOM ACCESS

Suppose that there are a number of devices associatedwith an AP. The total number of devices is denoted by K .Throughout the paper, we assume that devices in MTC gen-erate sporadic traffics [6], [7]. In this case, although K canbe very large, only a fraction of them becomes active at atime to the AP. Thus, random access3 might be suitable forcommunications from devices of short packets to the AP dueto low signaling/control overhead.

Throughout the paper, we assume that each active devicetransmits a packet within a given frame and a packet consistsof (Td + Tp) blocks, where the first Tp blocks are pilot blocksand the last Td blocks are data blocks, which is illustratedin Fig. 1. For convenience, in this section, we assume thatTp = 1. A pilot block has N + Ncp symbols, where Ncp isthe length of cyclic prefix (CP) and N is the length of a pilotsequence. The structure of a data block is similar to that of apilot block. Throughout the paper, we assume that the channel

3This random access does not refer to the RACH procedure in LTE thatis used to establish a connection, but compressive random access that allowsone-shot transmission (such as the schemes in [10], [20], and [23]).

CHOI AND YU: COMPRESSIVE ChDMA FOR MTC UNDER FREQUENCY-SELECTIVE FADING 2717

is invariant within a frame (i.e., the coherence time is longerthan the length of frame).

For MTC, we consider sporadic transmissions from devicesas in [20]. Thus, in each frame, we may have a differentset of active devices. From this, we only consider signaltransmissions within one frame. In addition, throughout thepaper, we assume that the CSI is to be estimated at the APand active device is to randomly choose a pilot from a set ofpilots (or pilot pool).

A. Pilot Transmissions

For random access, each active device is to randomly chooseone out of L pre-determined pilots and transmit it in orderto allow the AP to estimate the CSI of active device. Theset of pilot sequences is given by C = {c1, . . . , cL}, wherecl ∈ C

N×1 denotes the lth pilot. We assume that all thedevices are synchronized and active devices can transmit theirrandomly selected pilots simultaneously. Denote by l(m) theindex of the pilot chosen by the mth active device and by Mthe number of active devices among K devices. The receivedsignal after removing CP when the pilot block is transmittedis given by

xq =M∑

m=1

Ncir−1∑

p=0

hm,pcl(m),(q−p)N + nq , q = 0, . . . , N − 1,

(1)

where hm,p is the channel impulse response (CIR) from themth active device to the AP, Ncir is the length of CIR, andnq ∼ CN (0, N0) is the background noise. Here, it is assumedthat Ncir ≤ Ncp and cl = [cl,0 . . . cl,N−1]T represents the lthpilot sequence in C and (x)N = x (mod N). For example,suppose that K = 64, L = 10, and M = 2. If the deviceindices of the first and second active devices are 2 and 54,respectively, we have k(1) = 2 and k(2) = 54. If the first andsecond active devices choose c4 and c10, respectively, we havel(1) = 4 and l(2) = 10.

Let x = [x0 . . . xN−1]T and n = [n0 . . . nN−1]T.In addition, define the discrete Fourier transform (DFT) matrixof size N × N as [F]n,q = e− j2πnq

N , where n, q ∈ {0, . . . ,N − 1}. The DFT of x becomes

y = Fx =M∑

m=1

HmFcl(m) + n, (2)

where n = Fn and Hk is the frequency-domain chan-nel matrix from device k to the AP, which is a diagonalmatrix and given by Hk = diag(Hk,0, . . . Hk,N−1). Here,

Hk,n = ∑Ncir−1p=0 hk,pe− j2πpn

N .

Unlike the approaches that use unique codes of devices forpilots [8], [20], the number of devices can be arbitrarily largewhen a pool of pilots is used. In other words, with a pilotpool of small size, it is possible to support a large numberof devices as long as the activity is low. Thus, regardless ofthe number of devices, K , a good set of pilots, C , in termsof auto- and cross-correlations (such as ZC sequences [24])or some other performance metrics associated CS algorithms

(e.g., the coherence [25], [26]) can be employed for given N inorder to achieve good CSI estimation performance. However,there is a drawback due to pilot collision that happens if thereare multiple active devices that choose the same pilot from C .Thus, it is desirable to have a low probability of pilot collision.For a given M , the probability of pilot collision, denoted byPpc, can be found as

Ppc = 1 −L!

(L−M)!L M

= 1 −M−1∏

m=1

(1 − m

L

)

≈ 1 −M−1∏

m=1

e− mL = 1 − e− M(M−1)

2L ≈ 1 − e− M22L . (3)

In (3), we consider the approximation that 1 − mL ≈ e− m

L

when m is small compared to L. For a given target Ppc, we

can also show that the required L would be L ≈ − M2

2 ln(1−Ppc).

For example, if Ppc = 0.01 is required when M = 5, we needL ≈ 1243. In general, for a lower Ppc, a larger L is required.Unfortunately, this results in the following two difficulties:i) as L increases, the computational complexity4 for the CSIestimation increases; ii) a low-complexity CS algorithm suchas the orthogonal matching pursuit (OMP) algorithm [27], [28]cannot provide a good performance when L is large.In particular, a practical approach is associated with thesecond difficulty, which results in a dilemma. That is, if L issmall, compressive random access suffers from pilot collision.On the other hand, if L is large, a low-complexity CSalgorithm cannot provide a good performance of the CSIestimation. In Section IV, we will propose a method tomitigate the above difficulty using multiple pilot transmissions.

B. Data Transmissions

After transmitting a randomly chosen pilot, an active devicetransmits T blocks of data symbols with CP. Denote by at;m =[at;m,0 . . . at;m,N−1]T the t th data block transmitted by themth active device, where at;m,q is the qth element of at;m .Then, after removing CP, the received signal becomes

xt;q =M∑

m=1

P−1∑

p=0

hm,pat;m,(q−p)N + nt;q,

where nt;q ∼ CN (0, N0) is the qth background noise ofdata block t . Let xt = [xt;0 . . . xt;N−1]T and nt =[nt;0 . . . nt;N−1]T. To perform MUD in the frequency-domain, we can consider the the DFT of xt , which is

yt = Fxt =M∑

m=1

HmFat;m + nt , (4)

where nt = Fnt .To allow MUD, each active device can use the same pilot

sequence that was chosen for pilot transmissions as a signature

4If an optimal estimation is considered for joint channel estimation andactivity detection, the complexity is prohibitively high, as it is proportionalto

( LM

).


when it transmits data symbols [20]–[22]. In this case, thesignal vector of the mth active device is given by

at;m = cl(m)dt;m, (5)

where dt;m is the data symbol of the mth active device in thetth data block. Then, yt in (4) becomes

yt =M∑

m=1

wmdt;m + nt = Wdt + nt , (6)

where wm = Hmcl(m), W = [w1 . . . wM ], and dt =[dt;k(1) . . . dt;k(M)]T.

III. COMPRESSIVE CHANNEL DIVISION

MULTIPLE ACCESS

In this section, we take into account the fact that eachactive device has a different CSI and exploit this for theproposed compressive random access scheme where CS-basedMUD can be used to detect signals from multiple activedevices with index modulation, which can achieve a highertransmission rate than that of conventional approach discussedin Subsection II-B. Throughout this section, we assume thatthe AP knows Hk for k ∈ A, where A denotes the set of activedevices, i.e., A = {k(1), . . . , k(M)}. The estimation of Hk willbe studied in Section IV with a common pilot pool.

A. Channel Division Multiple Access With Index Modulation

Noting that each active device can be characterized byits CSI in (4), we do not need to use additional signature(i.e., cl(m)) for MUD. In this subsection, we propose a multipleaccess based on CSI. While cl(m) is not used as additionalsignature, it will be used for index modulation to increase thetransmission rate.

Suppose that a device can choose Q positions in a data blockand transmit pulses in those positions as a generalization ofpulse position modulation (PPM). For example, if N = 4 andQ = 2, there are

(42

) = 6 possible combinations. This becomesindex modulation in the time domain and at;m becomes aQ-sparse vector, i.e., at;m ∈ �Q , where �Q denotes the setof Q-sparse vectors, i.e., �Q = {x | ||x||0 = Q}. Note thatthe resulting index modulation can be seen as a time-domainversion of the index modulation for orthogonal frequency divi-sion multiplexing (OFDM) [29]. For convenience, a Q-sparsevector, at;m, is referred to as an index modulated vector (IMV).In this case, the number of bits per block that can be trans-mitted by an active device becomes �log2

(NQ

)�. The receivedsignal during data block t at the AP is given by

yt =M∑

m=1

Hk(m)Fat;m + nt = �at + nt , (7)

where at = [aTt;k(1) . . . aT

t;k(M)]T and � is the followingmeasurement matrix:

� = [(Hk(1)F) . . . (Hk(M)F)] ∈ CN×N M .

This measurement matrix is determined by the CSI of activedevices and the channel matrices {Hm} are used as signa-tures of active devices for MUD. From this, the resulting

multiple access is referred to as channel division multipleaccess (ChDMA).

We have few remarks as follows.a: Since at ∈ �M Q , if N � M Q, at can be recovered from

yt in (7) using a CS algorithm in the frequency-domain. Wecan also exploit the structure of at for a better performance.Each subvector of at , at;m, is Q-sparse. A modified OMPalgorithm to take into account this property is presentedin [30].

b: We can consider the real-valued representation of at;m .Then, at;m becomes a real-valued vector of 2N elements. Inthis case, the number of bits per block becomes �log2

(2NQ

)�.That is, more bits5 can be transmitted using a total of Q sparsepulses in the real and imaginary domains. Note that comparedto the approaches in [20] and [21], the proposed approach cantransmit more information bits. For example, if N = 139 andQ = 2, each device can transmit 15 bits per data block inthe proposed approach. On the other hand, in the approachesin [20] and [21], which are based on (5), if quadrature phaseshift keying (QPSK) is used, each device can transmit 2 bitsper data block. Thus, the proposed approach can transmitmore information bits and have a lower overhead than thosein [20] and [21].

c: In ChDMA, as shown above, the AP can distinguish andrecover the signals from multiple active devices using theirdifferent channel matrices, {Hm}, which are used as signatures.Thus, the performance depends on the difference betweenHm’s or their cross-correlations. (e.g., if two channel matricesare highly correlated, the AP cannot differentiate them with ahigh probability). As an extreme case, if Ncir = 1 (i.e., flat-fading channels), the cross-correlations of Hm’s become 1 asthey are scalars, and ChDMA cannot be used in this case.From this, in general, we expect that ChDMA would providea poor performance if the length of CIR, Ncir , is too short. InSubsection III-B, we consider this issue in terms of recoveryguarantee (in general, a lower cross-correlation between Hm’simplies a better recovery performance).

B. Impact of Key Parameters on Recovery Guarantee

In this subsection, we consider the impact of key parameters(e.g., M , Q, and Ncir) on a recovery guarantee with thecoherence in ChDMA.

For convenience, we omit the time index t in this subsection.For a recovery guarantee in most CS algorithms, the propertiesof the measurement matrix, �, are important. For example,the coherence of � is often considered, which is defined as

μ(�) = maxl =m|φH

l φm |||φl || ||φm || , where φl is the lth column of �.

From [28] and [31], for a recovery guarantee (in the absence ofnoise), a sufficient condition is found as ||s||0 < 1

2

(1 + 1

μ(�)

).

This implies that

μ(�) <1

2M Q − 1. (8)

Thus, it is desirable to keep M Q constant for given �.In other words, the sparsity, Q, has to decrease as the

5In addition, pulse amplitude modulation (PAM) can be used for non-zeroelements of at;k to send more bits.


number of active devices increases to keep a certain recoveryperformance.

In order to see the impact of Ncir on a recovery guarantee,consider the following inner product of two columns of �:

νk,k′ ;l,m = ([HkF]l)H[Hk′F]m =

∑

n

Hk,n H ∗k′,ne− j 2π(l−m)n

N

= N∑

p

hk,ph∗k′,p+l−m . (9)

Since ||[HkF]l ||2 = ∑n |Hk,ne− j 2πln

N |2 = N∑

p |hk,p |2,from (9), we have

νk,k′ ;l,m = νk,k′ ;l,m||[HkF]l || ||[HkF]m ||

=∑

p hk,ph∗k′,p+l−m√∑

p |hk,p |2√∑

p |hk′,p|2, (10)

which is the normalized auto-correlation and cross-correlationfunctions of {hk,p} and {hk′,p}. Note that if |l − m| ≥ Ncir ,νk,k′ ;l,m becomes 0. From this, since the value of |νk,k′ ;l,m |tends to be large as |l − m| approaches 0, we are interestedin the value of |νk,k′ ;l,m | when l = m for the coherence of �,because the coherence of � is the largest value of |νk,k′ ;l,m |for (k, l) = (k ′, m).

In order to see the properties of νk,k′ ;l,l , we consider thefollowing assumption.A) The channel coefficients of the CIR are independent and

CSCG random variables as follows:

hk,p ∼ CN(

0,1

Ncir

). (11)

That is, the channels are assumed to be Rayleigh multi-path channels.

Under A), we can readily have E[νk,k′ ;l,l ] = 0, k = k ′.Property 1: Let X = |νk,k′ ;l,l |2, which a normalized inner

product of any two distinct columns of �. Note that X isindependent of l (i.e., the value of X is the same for all l).For a sufficiently large Ncir , the probability that X is less than

1(2M Q−1)2 is given by

Pr

(X ≤ 1

(2M Q − 1)2

)= 1 − exp

(− Ncir

(2M Q − 1)2

).

(12)

Proof: See Appendix A.From (8), to keep a certain recovery performance, it is

desirable to haveNcir

(2M Q − 1)2 ≈ Const. (13)

Thus, for a larger Ncir , we can have more active deviceswith a fixed Q or a large Q (i.e., a higher data rate) with afixed M .

In Fig. 2, a histogram of X is shown with the exponentialprobability density function (pdf) with parameter Ncir whenN ∈ {61, 139} and Ncir = 10. For each histogram, weuse 10,000 different realizations of {hk,p} with K = 2according to (11). We can see that the assumption that Xfollows the exponential pdf is reasonable and its distributionis independent of N .

Fig. 2. A histogram of X and the exponential pdf with parameter Ncir whenN ∈ {61, 139} and Ncir = 10. For histogram, the size of bin, denoted by δ,is set to 0.02.

IV. CHANNEL ESTIMATION WITH MULTIPLE

PILOT TRANSMISSIONS

For successful ChDMA, it is important to estimate Hk ,k ∈ A, as they are used as signatures in multiple access.To allow the AP to estimate the CSI, active devices need tosend pilots as mentioned in Section II. In this section, we firstderive a low-complexity CS-based approach for joint channelestimation and device activity with a set of pre-determinedpilot sequences, C . As mentioned earlier, when a commonpilot pool is used, pilot collision happens. To mitigate thisproblem, we then consider multiple pilot transmissions andderive an iterative method to resolve the uncertainty due topilot collision.

A. CS-Based Channel Estimation

For the channel estimation, we can either consider thefrequency-domain channel estimation [22] or the time-domainchannel estimation [20]. In this subsection, we derive a time-domain channel estimation approach. This approach could beeffective when ZC or Alltop sequences are employed for pilotdesign.

For convenience, let Cl denote the circulant matrix of cl .From (1), we have

x =M∑

m=1

Cl(m);Ncir hm + n, (14)

where Cl;Ncir represents the submatrix of Cl obtained by takingthe first Ncir columns and hk = [hk,0 . . . hk,Ncir−1]T. Let

� = [C1;Ncir . . . CL;Ncir ] ∈ CN×Ncir L . (15)

Then, (14) is rewritten as

x = �g + n, (16)

where g is a Ncir M-sparse vector of length Ncir L. In addition,let

gl =∑

k∈Al

hk ∈ CNcir×1, (17)


where Al represents the index set of the active devices thatchoose the lth pilot. Then, we have g = [gT

1 . . . gTL]T. Clearly,

we can see that g is block sparse as ||gl || is either 0 or apositive value. For convenience, we define the block sparsityfor g consisting of L subvectors of length Ncir as the number ofnon-zero subvectors, which is denoted by ||g||2,0. Accordingto [32], we can also define the block sparsity as follows:

||g||2,0 =L∑

l=1

�(||gl ||2 > 0),

where �(X) is the indicator function that is 1 if X is true and0 otherwise. Thus, the block sparsity of g is bounded by M ,i.e., ||g||2,0 ≤ M , as the number of active devices is M . If theblock sparsity is less than M , we can see that there is pilotcollision.

The maximum likelihood (ML) estimation of g for jointchannel estimation and activity detection can be formulatedas follows:

g = argmin||g||2,0≤M

||y − �g||2. (18)

From the support of g, we should be able to estimate thepilots used by active devices, and from the non-zero valuesof g, we are also able to estimate the CSI of active devices.However, since the computational complexity to solve (18)is prohibitively high, we need to exploit the sparsity of gfor the case that M L in order to derive low-complexityapproaches. In particular, we can use the block sparsity of g.In [32], the OMP algorithm is considered to exploit theblock sparsity and the resulting algorithm is called the blockOMP algorithm.

We now consider some specific sequences for a pilot poolor � in (15) to see the coherence.

Property 2: Suppose that N is prime. If the cl ’s are rootZC sequences [24], the coherence of � in (15) is 1√

Nfor

Ncir ≤ N, where L ≤ N − 1.Proof: See Appendix B.

According to Property 2, we can see that the root ZCsequences are attractive for pilot design. There is another setof sequences that can be used for C .

Property 3: For a prime number N ≥ 5, if cl ’s areorthogonal Alltop sequences [33], [34], i.e.,

[cl]n = 1√N

ej2πN (n3+(l−1)n), n =0, . . . , N − 1; l =1, . . . , L,

(19)

the coherence of � is 1√N

, where L ≤ N and Ncir ≤ N.Proof: The proof of this property is similar to that of

Property 2. Thus, we omit it.Note that the number of pilots in C can be larger than N −1

or N when the ZC or Alltop sequences are employed for C ,respectively, depending on Ncir . For example, if Ncir = 1,L can be N(N − 1) as all root ZC as well as cyclic shiftedversions are used for C . Thus, the maximum number of pilotsbecomes

Lmax = (N − 1)� N

Ncir� or N� N

Ncir� (20)

Fig. 3. An illustration of pilot selection with Tp = 2 and L = 7 when thereare M = 4 active devices. A shaded block represents a set of pilots that arechosen by an active device and a black block represents a pilot collision.

when the ZC or Alltop sequences are employed for C ,respectively.

B. Multiple Pilot Transmissions to Reduce Pilot Collisions

As mentioned in Subsection II-A, if L is small, the jointchannel estimation and activity detection suffers from a highprobability of pilot collision. On the other hand, its complexitybecomes prohibitively high for a large L (although a low-complexity CS algorithm such as the block OMP algorithm isused), while it is desirable to have a small size L for a goodCS recovery performance. Consequently, it might be desirableto have a pilot pool of a small L without suffering frompilot collision if possible. To this end, we consider multiplepilot transmissions with a small L, which can result in a lowprobability of pilot collision in this subsection.

Suppose that each active device can transmit B pilotsthrough the first B slots. For each pilot transmission, an activedevice can randomly and independently choose a pilot from C .Thus, if Tp = 1, the pilot transmission is identical to that inSubsection II-A. If Tp > 1, the event of pilot collision canbe seen in a higher-dimensional space. For example, whenTp = 2, we illustrate a pilot selection with L = 7 for 4 activedevices (i.e., M = 4) in Fig. 3, where the sets of selectedpilots’ indices by 4 active devices are {6, 2}, {2, 5}, {2, 5}and {6, 6}. Here, the first and second numbers in each bracerepresent the indices of the selected pilots in the first andsecond pilot transmissions by an active device, respectively.Although the 6th pilot is chosen by the 1st and 4th activedevices in the first pilot transmission, the CSI of them canbe separately estimated as they choose different pilots (i.e.,the 2nd and 6th pilots) in the second pilot transmission. Onthe other hand, the CSI of the 2nd and 3rd devices cannotbe estimated as they choose the same pilots in the first andsecond pilot transmissions. Thus, with Tp ≥ 1, the event ofpilot collision is the event that multiple active devices choosethe same set of pilots in consecutive pilot transmissions. Forconvenience, local pilot collision in each pilot transmission isreferred to as partial pilot collision (thus, if Tp = 1, partialpilot collision coincides with pilot collision). In the followingresult, we show that the probability of pilot event can rapidlydecrease with Tp.


Property 4: Suppose that Tp ≥ 1, i.e., there are multiplepilot blocks. In addition, assume that each active device is torandomly choose a pilot from C with equal probability for eachpilot block. Then, the probability of pilot collision is given by

Ppc ≈ 1 − e− M2

2LTp . (21)

Proof: See Appendix C.In order to see the impact of Tp on the probability of

pilot collision, we can consider an example with M = 10and L = 100. We have Ppc ≈ 0.393 with Tp = 1, whilePpc ≈ 0.005 with Tp = 2. This demonstrates that theprobability of pilot collision can be significantly lowered bymultiple pilot transmissions. For comparison purposes, we canconsider a longer pilot block with more pilot sequences in C .For example, suppose that the length of pilot block becomesdoubled (which is equivalent to Tp = 2 in terms of the totallength of pilot) and the number of pilots in C is also doubled.According to (3), the probability of pilot collision becomes

Ppc ≈ 1 − e− M24L . If M = 10 and L = 100, the resulting

probability of pilot collision is Ppc ≈ 0.221, which is not asignificant improvement compared to that by multiple pilottransmissions.

While the probability of pilot collision becomes signifi-cantly low by multiple pilot transmissions, the pilot overheadcan increase. To consider the pilot overhead, let

η = NTd

N(Tp + Td)= Td

Tp + Td,

which is the ratio of the total data length to the total packetlength (provided that the lengths of pilot and data sequencesare the same). As η approaches 1, we can assume that thepilot overhead is negligible. In MTC, for immobile devices, thecoherence time can be long so that Td can be sufficiently large.In this case, η is close to 1 as long as Tp Td. Consequently,with a sufficiently long coherence time, we prefer Tp = 2 toTp = 1 at the cost of negligible pilot overhead increase.

C. Approach to Resolve the AmbiguityDue to Partial Pilot Collision

In this subsection, we derive an approach to resolve theambiguity in the CSI estimation from g in the presence ofpartial pilot collision when Tp > 1. Note that although theprobability of pilot collision is sufficiently low when Tp > 1,the receiver still needs to resolve the ambiguity due to partialpilot collision since the received signal might be a super-position of the channels of some active devices as shownin (16) and (17), although the noise is negligible.

Denote by y[τ ] the received signal vector at the AP duringthe τ th pilot6 transmission. Then, we have

y[τ ] =M∑

m=1

Hmc[τ ],l(m) + n[τ ]

= �g[τ ] + n[τ ], τ = 1, . . . , Tp, (22)

6Note that we use square brackets, [·], in the subscript to differentiate thetime index for pilot transmissions from that for data transmissions, which isused without any brackets.

where c[τ ],l(m) is the selected pilot by the mth active deviceduring the τ th pilot transmission and g[τ ] and n[τ ] represent gand n from y[τ ], respectively. For convenience, let M[τ ] denotethe block sparsity of g[τ ]. If there is no pilot collision, we haveM[τ ] = M .

As shown above, from the estimate of g[τ ],l , we need tofind h. This would be straightforward if U is known. However,U is also unknown. In general, U can be expressed as

U =⎡

⎢⎣U(1)

...U(Tp)

⎤

⎥⎦ , U[τ ] ∈ RM[τ ]×M .

We can further characterize U[τ ] as follows: a) it is abinary matrix; b) each row vector has at least one 1; c) thetotal number of 1’s is M . Denote by UM the set of thebinary matrices that have the above three properties. Then,U[τ ] ∈ UM . Note that there is pilot collision for a row vector ofmore than one elements of 1. To estimate h from the estimatesof non-zero g[τ ],l’s that can be obtained from y[τ ] using a CSalgorithm, we can consider the following least squares (LS)approach:

minU[τ ]∈UM ,h

||g − (U ⊗ I)h||2, (23)

where g = [gT[1] . . . gT[Tp]]T and g[τ ] = [gT[τ ],(1) . . .

gT[τ ],(M[τ ])]T. Here, g[τ ],(m) denotes the estimate of the mth

non-zero subvector of g[τ ] from a CS algorithm.Since (23) is a combinatorial optimization problem, its com-

plexity can be prohibitively high. Thus, we need to consider alow-complexity iterative algorithm. The cost function in (23)can also be written as

Tp∑

τ=1

||G[τ ] − HUT[τ ]||2, (24)

where G[τ ] = [g[τ ],(1) . . . g[τ ],(M[τ ])]. Thus, for a given initialmatrix H, denoted by H(0), with i = 0 (here, i denotes theindex for iterations), we can find the U[τ ]’s to minimize thecost function as follows:

U(i)[τ ] = argmin

U[τ ]∈UM

||G[τ ] − H(i)UT[τ ]||2, τ = 1, . . . , Tp. (25)

Once the U(i)[τ ]’s are found, H can be updated as follows:

H(i+1) = argminH

Tp∑

τ=1

||G[τ ] − H(U(i)[τ ])

T||2. (26)

We have few remarks on the implementation of the aboveiterative method.

• In (25), to lower the computational complexity, wecan consider a relaxed constraint without imposingU[τ ] ∈ UM . For example, each row of U[τ ] can beassumed to be a binary vector of one element of 1 ortwo elements of 1. Since the probability that 3 devicescan choose the same pilot is very low (for a large N),we may ignore this case at the cost of slightly degradedperformance. For simulations in Section V, we will usethis simplification.


• Since the cost function in (24) has multiple minima, itis important to have a reasonable initial matrix H, H(0),in order to have a good performance. To decide a goodinitial matrix H, we use the non-zero g[τ ],(m)’s that havethe M smallest norms.

• From simulations, we find that two iterations are suffi-cient to converge in most cases when Tp = 2. Althoughthe probability of pilot collision decreases with Tp, wewill only consider the case of Tp = 2 as the pilot overheadincreases with Tp in simulations.

• In case of pilot collision (not partial pilot collision), it isnot possible to separately estimate the CSI of the devicesinvolved in pilot collision as the iterative method wouldprovide an estimate of the sum of the CSI. Althoughthe probability of pilot collision is low with Tp > 1,this event can happen. Thus, there should be additionalsteps to resolve this collision (e.g., we may use a randomback-off scheme).

V. SIMULATION RESULTS

In this section, we present simulation results when theCIR of each active device is independently generated underAssumption A). For CS-based MUD, the OMP algorithm isused, while the block OMP algorithm [32] is used for thechannel estimation to exploit the block sparsity. For a poolof pilots, we use orthogonal Alltop sequences in (19) withN = L. Note that L can be larger than N as shown in (20).

A. Simulation Results of Channel Estimation

As mentioned earlier, the main advantage of using a com-mon pool of pilots in compressive random access (over usinga unique pilot for each device) is to support any numberof devices at the cost of pilot collision. To compare theperformance of the channel estimation based on a commonpool of pilots with that based on unique pilots, we considerthe following two schemes:

• CRA-1: in this scheme, a common pilot pool is used withAlltop sequences of length N = 47;

• CRA-2: each device has a unique random pilot sequence(each element is one of {(±1± j)/

√2N}) and the number

of devices is up to K = {25N, 50N}, where N = 94 inthis scheme (i.e., the scheme in [8] and [20]).

In CRA-1, if Tp = 2, the total length of pilot sequencebecomes 47 × 2 = 94, which is the same as that ofCRA-2 where only one pilot sequence (per active device)is transmitted. For performance comparison purposes, weconsider the successful recovery rate that is the probabilitythat all the pilot sequences transmitted by active devices aredetected (or the probability of successful pilot detection).Thus, for CRA-1, any pilot collision results in unsuccessfulrecovery. In Fig. 4, the successful recovery rates of CRA-1 andCRA-2 are shown for various values of M when Ncir = 4 andsignal-to-noise ratio (SNR) = 20 dB. Here, the SNR is definedas SNR = ||cl ||2

N0= 1

N0. Note that in CRA-2, the block sparsity

is equal to M . On the other hand, in CRA-1, the block sparsityis less than or equal to M due to pilot collision. From Fig. 4,

Fig. 4. Successful recovery rate versus M when Ncir = 4 andSNR = 20 dB.

Fig. 5. Performance of pilot detection and channel estimation for differentvalues of Ncir when SNR = 20 dB: (a) Successful recovery rate; (b) NMSE(only for the signals that can be recovered by the block OMP algorithm).

we can see that the performance of CRA-1 can be similar tothat of CRA-2 although its N is a half of that of CRA-2 dueto good pilot sequences (Alltop sequences) and an effectivelysmaller block sparsity (than M). We can observe that althoughthe performances of CRA-1 and CRA-2 are similar to eachother when M is small (up to M = 7), the successful recoveryrate of CRA-1 decreases quickly and becomes much lowerthan that of CRA-2 for a large M (M ≥ 8) due to a higherprobability of pilot collision. However, fortunately, sincethe region of a high successful recovery rate is usually ofinterest for compressive random access, we can claim that theperformance of CRA-1 is comparable to that of CRA-2, whileit can support any number of devices. In the rest of simulationresults, we only consider CRA-1 unless stated otherwise.

Fig. 5 shows the performances of the pilot detection aswell as the channel estimation (for those active devices whosepilots are correctly detected) for different values of Ncir when


Fig. 6. Recovery rate of the iterative method with 3 iterations for variousvalues of M when L = 67 and Ncir = 5.

SNR = 20 dB. As Ncir increases, the variation of the signalpower (i.e., the squared norm of hk) becomes smaller as Ncir ,which can result in a better pilot detection performance.However, if Ncir becomes too large, the block OMP algorithmsuffers from a large block sparsity and cannot recover the pilotsignal well unless N = L is large.

For the active devices whose pilots are correctly detected,we carry out the channel estimation to estimate the CIRs ofthem. The normalized mean squared error (NMSE) of thechannel estimate, which is given by

NMSE = E[||gl − gl ||2]E[||gl ||2] ,

where gl is the estimated gl if gl is a non-zero vector, is shownin Fig. 5 (b). We can see that the NMSE increases with Ncir(as there are more channel coefficients to be estimated whenNcir increases).

B. Simulation Results of IterativeMethod to Resolve Ambiguity

In this subsection, we present simulation results of theiterative method to resolve ambiguity in Subsection IV-C whenTp = 2.

Fig. 6 shows the recovery error rate when the iterativemethod in Subsection IV-C is used to recover h from {g[τ ]},τ = 1, 2 when L = 67 and Ncir = 5 for various values ofM in the presence of pilot collision. The iterative methodcan provide nearly ideal performances when M is small(e.g., M = 2) as the recovery error rate is close to the collisionprobability with Tp = 2. However, as M increases, theperformance of the iterative method is degraded and the gapbetween the recovery error rate and the collision probabilitybecomes wider. We note that the recovery error rate of theiterative method with Tp = 2 can be lower than the probabilityof collision with Tp = 1, which demonstrates that the use ofmultiple pilot transmissions (with a low-complexity approachto resolve the ambiguity due to partial pilot collision) is aneffective means to mitigate pilot collision.

Fig. 7. Recovery rate of the iterative method with 3 iterations for variousvalues of L when M = 5 and Ncir = 5.

Fig. 8. Symbol error rate when L = 139, Q = 2, M = 4: (a) SER versusEbN0

(with Ncir = 10); (b) SER versus Ncir (with EbN0

= 10 dB).

In Fig. 7, we show the recovery error rate of the iterativemethod for various values of L when M = 5 and Ncir = 5.Clearly, as L increases, a better performance of resolving theambiguity due to partial pilot collision can be obtained.

C. Simulation Results of MUD in ChDMA

In this subsection, we show the symbol error rate (SER) ofChDMA when index modulation is used as a generalization ofPPM. Since the real-valued version of at;k is Q-sparse and itslength is 2N , we assume that each IMV can transmit log2

(2NQ

)

bits. Thus, the bit energy is given by Eb = A2

log2 (2NQ )

, where

A is the amplitude of non-zero elements of the real-valuedversion of at;k.

Fig. 8 (a) shows the SER for various values of EbN0

whenN = 139, Q = 2, M = 4, and Ncir = 10. We canobserve the error floor (i.e., the SER cannot be loweredalthough Eb

N0increases once Eb

N0is sufficiently high (in this

case, EbN0

≥ −5 dB)). This error floor is due to the recovery


Fig. 9. Throughput of the proposed approach: (a) throughput versus trafficintensity λ with Ncir = 8; (b) throughput versus Ncir with λ = 8. For datablocks, we assume that N = 139 and Eb

N0= 10 dB. For pilot blocks, we

consider two different lengths, N ∈ {139, 251} and SNR = 20 dB.

performance of the CS-based MUD method (i.e., the OMPalgorithm in this section). As discussed in Subsection III-B, abetter performance can be achieved if Ncir becomes larger orM Q becomes smaller.

According to (12), in ChDMA, the performance of signalrecovery depends on the product of M and Q. In addition, theperformance becomes better as Ncir increases for a fixed M Q.This prediction from (12) can be confirmed by Fig. 8 (b),where the SER for various values of Ncir with differentcombinations of M and Q is shown when L = 139 andEbN0

= 10 dB.

To see the overall performance, we consider the throughputof the proposed approach. The throughput is defined as theaverage number of the active devices that can successfullytransmit their data symbols. This throughput is similar to thatin ALOHA [35], where the throughput is the average numberof transmitting nodes that do not experience packet collision.To be a successful active device, the AP has to not only detectits pilot and estimate its CIR (from the pilot blocks), butalso correctly detect all data symbols that are transmitted byindex modulation through ChDMA (from data blocks). Forthe case of Tp = 2, we use the iterative method proposedin Subsection IV-C to resolve the ambiguity due to partialpilot collision with 3 iterations. We further assume that thenumber of active devices, M , is a Poisson random variablewith parameter λ. In this case, the average number of activedevices, E[M], is λ. Fig. 9 (a) shows the throughput for variousvalues of λ when Ncir = 8, Q = 2 and Td = 1. For each datablock, we assume N = 139 and Eb

N0= 10 dB. For pilot blocks,

we consider two different lengths, N ∈ {139, 251}, and assumeSNR = 20 dB. We can observe that the throughput or theaverage number of successful active devices increases with λand then decreases (due to more pilot collisions and degradedperformance of CS based MUD). For a large λ, the systembecomes overloaded and its throughput is degraded. To avoidthis problem, we need to consider the access control to decide

the access probability [35], which is beyond the scope of thepaper. We also see that a better performance is achieved for alonger pilot length or more pilot blocks as the probability ofpilot collision decreases.

In Fig. 9 (b), we show the throughput for various values ofNcir when Q = 2 and Td = 1. For data blocks, we assumeN = 139 and Eb

N0= 10 dB. For pilot blocks, we consider

N = 251 and SNR = 20 dB. While a better performance(in terms of the SER) is achieved for a longer Ncir in Fig. 8 (b)when no pilot collision is taken into account, we can observethat the throughput increases and then decreases with Ncirin Fig. 9 (b). Since the successful recovery rate increasesand then decreases with Ncir , which is shown in Fig. 5 (a),the throughput becomes also a ∩-shape function in Ncir .In particular, if Ncir becomes too large, the block OMPalgorithm cannot recover the signal due to a large blocksparsity and the resulting throughput decreases with Ncir . Wecan also confirm that more pilot blocks help to improve thethroughput by decreasing the probability of pilot collision.

VI. CONCLUDING REMARKS

We have proposed a compressive random access schemewith a relatively small number of good pilot sequences thatis suitable for MTC to support any number of devices overfrequency-selective fading channels. For a sufficiently lowprobability of pilot collision, we proposed to employ multiplepilot transmissions. Since the unique CSI of each active devicecan be exploited to allow the AP to differentiate data symbolsof one active device from those of other active devices, thereis no need to assign unique signature sequences to devices.Consequently, the proposed compressive random access wasable to support a large number of devices with sparse activityfor sporadic transmissions in MTC.

APPENDIX APROOF OF PROPERTY 1

For a sufficiently large Ncir , under A), from [36], we

have νk,k′ ;l,l ∼ CN(

0, 1Ncir

). From this, it follows that

X ∼ Exp (Ncir) = Ncire−Ncir x , k = k ′, where Exp(λ)represents the exponential distribution with parameter λ (themean is 1

λ ). Since FX (x) = 1 − exp (−Ncirx), (12) can beobtained.

APPENDIX BPROOF OF PROPERTY 2

For a given cl , consider Cl;Ncir . Each column vector ofCl;Ncir is a cyclic shifted vector of cl . That is, [Cl;Ncir ]n,p =[cl](n−p+1)N . Thus, if Ncir ≤ N , all column vectors of Cl;Ncir

are mutually orthogonal. In addition, for given two any root ZCsequences or any pair of distinct cl’s, we have |cH

l cm | = 1√N

,

l = m. Consequently, the coherence of � in (15) is 1√N

.

APPENDIX CPROOF OF PROPERTY 4

A pilot chosen by an active device for each pilot blockcan be seen as a random variable uniformly distributed


over {1, . . . , L}. For Tp ≥ 1, the set of the Tp pilots chosenby an active device is also seen as a random vector uni-formly distributed over {1, . . . , L}Tp . Thus, according to (3),

Ppc = 1−(LTp )!

(LTp −M)!LTpM ≈ 1−e

− M2

2LTp for LTp � M . This completesthe proof.

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Jinho Choi (SM’02) was born in Seoul,South Korea. He received the B.E. (magnacum laude) degree in electronics engineeringfrom Sogang University, Seoul, in 1989, andthe M.S.E. and Ph.D. degrees in electricalengineering from the Korea Advanced Instituteof Science and Technology, Daejeon, in 1991and 1994, respectively. He was with the Collegeof Engineering, Swansea University, U.K., as aProfessor/Chair in wireless. In 2013, he joined theGwangju Institute of Science and Technology as a

Professor. He authored two books published by Cambridge University Pressin 2006 and 2010, respectively. He received the 1999 Best Paper Award forSignal Processing from EURASIP and the 2009 Best Paper Award fromWPMC (Conference). He is currently an Editor of the IEEE TRANSACTIONSON COMMUNICATIONS.

Nam Yul Yu (M’07) received the B.S. degree inelectronics engineering from Seoul National Uni-versity, Seoul, South Korea, in 1995, the M.S.degree in electronic and electrical engineering fromthe Pohang University of Science and Technology,Pohang, South Korea, in 2000, and the Ph.D. degreein electrical and computer engineering from the Uni-versity of Waterloo, Waterloo, ON, Canada, in 2007.

From 2000 to 2003, he was with the Telecommuni-cation Research and Development Center, SamsungElectronics, South Korea, where he was involved in

channel coding schemes for wireless communication systems. In 2007, he wasa Senior Research Engineer with LG Electronics, South Korea, where hewas involved in the standardization of the 3GPP-LTE. From 2008 to 2014,he was an Assistant/Associate Professor with the Department of ElectricalEngineering, Lakehead University, Thunder Bay, ON, Canada. In 2014,he joined the Gwangju Institute of Science and Technology, Gwangju, SouthKorea, and is currently an Associate Professor with the School of ElectricalEngineering and Computer Science.

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