1

Downlink Synchronization Techniques forHeterogeneous Cellular Networks

Neelakandan Rajamohan∗, Arun Pachai Kannu†

Department of Electrical EngineeringIndian Institute of Technology Madras

Chennai - 600036, IndiaEmail: ∗ee10d011@ee.iitm.ac.in, †arunpachai@ee.iitm.ac.in

Abstract—Base stations (BS) in a cellular network send uniquesynchronization signals based on their physical layer cell-identity,in every frame. In order to synchronize with a BS, a mobileterminal (MT) needs to identify the following synchronizationparameters - starting time of the frame, relative frequency offsetand the cell-identity of the corresponding BS. In a heteroge-neous cellular network, due to the increased density of low-powered base stations, the MT receives signals from multipleBS in its vicinity. We consider the downlink synchronizationproblem where the MT tries to recover the synchronizationparameters of all the BS in its neighborhood, using which, MTcan subsequently choose the suitable BS for its connection. Inany given scenario, the number of BS in the neighborhood ofMT is relatively small compared to the total number of BS inthe entire network. Exploiting this sparseness, we present a two-stage synchronization approach where the first stage identifies theframe timing using an approximation to the maximum likelihooddetector and the second stage recovers the cell identities and thecorresponding frequency offsets of all the neighborhood BS usingblock sparse signal recovery framework of compressive sensing.We analytically and numerically study the recovery guaranteesof our proposed techniques and establish their superior per-formance over existing successive interference cancellation andmatched filtering approaches.

Keywords— timing detection, frequency offset estimation, cellsearch, compressive sensing, block sparse signal recovery

I. INTRODUCTION

A cellular network comprises of base stations (BS) servingusers in their respective coverage areas. Each BS in a cellularnetwork sends its data in a sequence of frames, with eachframe having a synchronization signal unique to the physicallayer cell-identity of the corresponding BS. Before a mobileterminal (MT) can exchange information with a BS, MTneeds to acquire the synchronization parameters of that BSwhich includes finding the starting time of frames from thatBS, estimation of relative frequency offset and recovering thephysical layer identity of that BS. Recovering the synchro-nization parameters by the MT in a cellular network [1]–[5] is referred as downlink synchronization or cell searchand is typically done during the initial acquisition phase (thestart-up phase when the MT is switched ON). Conventionaldownlink synchronization procedure done by the MT findsthe synchronization parameters of the BS corresponding tothe strongest received signal power.

This work is sponsored by Dept. of Science and Technology, India.A patent has been filed based on this work.

In order to address the growing demand for data communi-cation among mobile phone users, a heterogeneous cellularnetwork (HetNet) with many low powered BS within thecoverage area of conventional BS is being envisioned. In aHetNet, there are macro cells (conventional BS with coverageradius of few kilometers and transmit power levels of 46 dBm),pico cells (operator deployed BS with coverage radius of fewhundreds of meters and transmit power levels of 20 to 23 dBm)and femto cells (user deployed BS to provide coverage insidehome with open or restricted access) [6], [7].

Due to the increased density of BS in a HetNet, the MTreceives superposition of signals from multiple BS in itsvicinity. In several scenarios in HetNets, a mobile terminalmay need to identify and connect to a BS which does notcorrespond to the strongest received power [8]. For instance,

• MT may try to connect to the closest macro BS butthe strongest signal may come from a near-by (restrictedaccess) femto BS

• MT may need to connect to a pico-BS (in order to permitcell splitting and enhance overall network throughput[9]) while receiving the strongest signal from the nearestmacro BS or near-by femto BS

Hence, we consider the HetNet downlink synchronizationproblem in the initial acquisition phase, in which the MTtries to acquire the synchronization parameters of all theBS in its neighborhood, thereby enabling the MT to subse-quently choose a BS suitable for its connection. The notionof neighbor BS encompasses the base stations with receivedpower sufficiently higher than the noise floor at the MT toenable further communication. Further, acquiring the neighborBS parameters are essential in several advanced techniquesincluding positioning of the MT [10], inter cell interferencecoordination [11] and coordinated multi-point transmission[12].

In any given scenario, the number of BS in the neighbor-hood of MT is relatively small compared to the total numberof BS in the entire network. Exploiting this sparseness, wepresent a two-stage synchronization approach where the firststage identifies the frame timing using an approximation tothe maximum likelihood (ML) detector and the second stagerecovers the cell identities and the corresponding frequencyoffsets of all the neighborhood BS using block sparse signalrecovery framework of compressive sensing (CS) [13], [14].

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The closely related works in [15]–[17] consider the recoveryof neighbor BS parameters with the perfect knowledge offrame timing and absence of relative frequency offsets, whilewe consider the initial acquisition phase where frame timinginformation is not available and the residual frequency offsetsare present (details are provided in Section II).

In our work, the proposed timing detection algorithm basedon joint correlator detector presented in Section III is noveland is shown to be equivalent to ML detector under someidealistic assumptions. In the application of block sparse signalrecovery framework in our HetNet synchronization problempresented in Section IV, due to the presence of frequencyoffsets, the structure of the sensing matrix in our correspondingCS model and its block sparse signal recovery propertiesdiffer significantly from those of the sensing matrix consideredin [17]. More details on these differences are explained inthe Section IV-A. Also, in addition to the standard blocksparse signal recovery algorithms in the literature, we alsostudy several new single-step recovery algorithms for our CSmodel and analytically characterize their recovery guarantees,in Section IV-B. In Section V, we present the simulationstudies of our proposed techniques and establish their superiorperformance over the existing matched filtering (MF) [5],[18] and successive interference cancellation (SIC) [16] basedapproaches.

Notation: matrices/vectors are denoted by bold upper-case/lowercase letters, lp norm by ‖ · ‖p, inner product by 〈·〉,determinant by det(·), transpose by (·)t, hermitian by (·)∗,sets by mathcal font A, cardinality of the set by |A|, and setminus operation by A \ B and δ(·) denotes Kronecker deltafunction.

II. SYSTEM MODEL AND PROBLEM STATEMENT

We consider the downlink scenario where the mobile ter-minal receives signals transmitted by the base stations in itsvicinity. Let I = {1, · · · , C} denotes the set of cell identitiesavailable in a cellular system, where C denotes the totalnumber of unique cell identities in the network. For example,in 3GPP-LTE system, the value of C is 504. Let B ⊂ Idenotes the set of BS in the vicinity of MT from which itreceives signals. We call the set B as the neighbor set of MTand |B| = S denotes the number of neighbor base stations.Without loss of generality, we assume that the cell identitiesof all neighbor BS are distinct. In the discrete time base bandmodel, the received observation sequence {y(n)} in a givenMT is

y(n) =∑q∈B

Lq−1∑`=0

√Pqhq(n; `)tq(n− `)ej2πεqn + w(n), (1)

where w(n) denotes the additive noise, {tq(n)} denotes thetransmit signal with power level Pq from BS with identityq, εq denotes the normalized relative frequency offset and,{hq(n; `)} and Lq denote the channel impulse response (CIR)at time n between MT and BS with identity q and thecorresponding delay spread respectively. The frequency offsetεq lies in the interval [−εmax, εmax] with εmax being themaximum possible frequency offset, which depends on the

Fig. 1. Alignment of SS from different BS in the received frame. (Not toscale)

oscillator specifications in the cellular standards. The channeldelay spread Lq ≤ Lmax,∀q ∈ B, where the maximumdelay spread Lmax depends on the transmit power levels ofthe macro cell. The number of neighbor base stations in thevicinity of the mobile terminal is bounded as S ≤ Smax whereSmax depends on the density of low powered BS.

Base stations in a cellular system transmit information in asequence of frames whose structure is specified in the cellularstandards. In every frame (of length F ), at a predeterminedposition (in time), each base station transmits a trainingsequence (of length N ) which is specific to its identity. Thisis illustrated in Fig. 1. Focusing on a specific frame, let{xq(n), n = 0, · · · , N − 1} denotes the known/fixed train-ing/synchronization signal (SS) of BS with identity q and Tqrepresents the starting location of the training signal receivedfrom BS with identity q. Hence, tq(n + Tq) = xq(n), n ∈{0, · · · , N − 1}. We assume that the starting time of theframes from all the neighbor BS in the set B are time aligned(within a degree of accuracy accounting for the propagationdelays). Frame time alignment is a feasible assumption andcan be accomplished if the low powered BS within a macrocoverage area align their frames with that of their macro BS,by monitoring the macro timing using synchronization signaltransmitted by the macro BS. This frame time alignment istypical in the study of HetNets [7], [19]. With the transmissionof the synchronization signals from the neighbor set B beingtime aligned, we can assume that there is a time window[T, T + Lmax] such that [Tq, Tq + Lq] ⊂ [T, T + Lmax] forall q ∈ B (as shown in the Fig. 1). This assumption is madein the neighbor cell search problem addressed in [16] as well.We consider cyclic prefixed OFDM transmission (as in LTE)and assume that the cyclic prefix duration is larger than Lmax.

We are interested in developing downlink synchronizationalgorithms for the mobile terminal in our HetNet model (1).Synchronizing with a BS involves finding the starting time ofits frame, its cell identity and its relative frequency offset.However, as previously discussed, synchronizing with thestrongest BS may not be sufficient in HetNets. So, we areinterested in recovering the identities of all the neighbors in B,

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along with their residual frequency offsets {εq} and their frametiming. Note that, once the MT recovers the synchronizationparameters of all its neighbor BS, it can subsequently connectto a suitable BS for exchanging its data/information.

III. FRAME TIMING DETECTION

Since frames are sent in a continuous stream, and sinceeach frame has a synchronization signal at a fixed location,it follows that, any contiguous collection of (F + N − 1)1

received samples {y(n), n = 0, · · · , F + N − 1} will havea synchronization signal present somewhere in that collec-tion. By identifying the location of the training signal inthe collected samples, the location of the start of the framecan be inferred. To proceed further, we assume that channelcoherence time is larger than the duration of the trainingsignal N as done in [16], which is satisfied in typical mobilityenvironments. For convenience, let us denote hq(`) = hq(n; `)for n ∈ [T, · · · , T + N − 1]. Let us define the vectorsyk = [y(k), · · · , y(k+N − 1)]t for k ∈ {0, · · · , F − 1}. Theobservation vector yT := y = [y(T ), · · · , y(T + N − 1)]t

is the received vector2 corresponding to the superpositionof synchronization signals from the neighbor BS in B. Forconvenience, we use the common length L = Lmax for theCIR from all the base stations, by suitably padding zeros.Specifically, for ` = 0, · · · , L − 1, we define zq(`) =√Pqhq(`)e

j2πεq(T+`)U [T − Tq + `] with U [n] being the unitstep sequence (U [n] = 1 for n ≥ 0 and 0 else) and with the no-tion that hq(`) = 0,∀` ≥ Lq . We vertically concatenate zq(`)to form L×1 vector zq , which corresponds to the CIR vectorfor BS with identity q. Defining the matrix Xq,ε ∈ CN×L withits `th column being {xq(n−`)ej2πε(n−`), n = 0 to N−1, ` =0 to L− 1}, we have

y =∑q∈B

Xq,εqzq + w, (2)

where w = [w(T ), · · · , w(T + N − 1)]t is the noise vector.The relative delays between the arrival times of the SS fromdifferent BS are taken into account in the construction ofvectors zq . Since the time window [T, T + N − 1]3 capturesthe training sequences from all the BS at the MT (includingall the multipaths), we can identify the common frame timingof all the BS in the set B by recovering the value of T .

If we assume additive white Gaussian noise (AWGN) withvariance σ2

w, zq is zero-mean Gaussian with covariance Cq

and CIRs of different BS are uncorrelated, it follows that y iszero-mean Gaussian with covariance

CB,εq =∑q∈B

Xq,εqCqX∗q,εq + σ2

wI, (3)

given the neighbor set B and frequency offsets εq . Givenk = T , the neighbor set is B and the corresponding frequency

1Additional N − 1 samples are necessary to ensure that always we havecontiguous collection of N samples of SS.

2For the value of T such that Tq ∈ [T, T + Lmax] ∀q ∈ B, the vector ycaptures SS from the all the BS in the set B.

3Due to the presence of cyclic prefix, the positioning of this window hassome flexibility (up to the difference between cyclic prefix duration andLmax) [20].

offsets are εq , the conditional pdf of yk is zero mean Gaussianwith covariance CB,εq and to find the maximum likelihoodestimates, we need to maximize this conditional pdf over k,B and εq . Since εq is a continuous parameter, we divide theinterval [−εmax, εmax] into M frequency bins (we consider Mto be odd for convenience) and perform a discrete search asproposed in [21]. Specifically, defining the set correspondingto the center frequencies of bins as E = {ε1, · · · , εM} withεi = εmax( 2i−1

M − 1), we jointly estimate the frame timing(T ), set of neighbor BS (B) and the corresponding frequencyoffsets (εq) as

(T , B, εq)ML = arg maxk∈{0,··· ,F−1}

B⊂I such that |B|≤Smax

q∈B and εq∈E

exp(−y∗kC

−1

B,εqyk

)det(CB,εq )

,

(4)

where exp(·) denotes the exponential function. This opti-mization process is computationally expensive, since it needsto compute the determinant and inverse of CB,εq for allk ∈ {0, · · · , F − 1}, for all possible sets of neighbor BS (B)and their corresponding frequency offset hypotheses from E .Also, it requires the knowledge of noise variance σ2

w, transmitpower levels Pq and the covariance of CIR for all BS in B,which may not be typically available during the initial cellsearch. Due to these reasons, the above ML search procedureis not feasible.

We propose a simplified metric, referred as joint correlatordetector (JCD), which identifies the timing as,

(T , B, εq)JCD = arg maxk∈{0,··· ,F−1},

εi∈E

∑q∈B

∥∥X∗q,εiyk∥∥2

2(5)

where B = {q :∥∥X∗q,εiyk∥∥2

2≥ τjcd} ⊂ I and |B| ≤ Smax.

Above JCD expression is equivalent to computing the corre-lation metric

∥∥X∗q,εiyk∥∥2

2for all {k, i, q}, and summing the

top (at-most) Smax metrics (provided they cross threshold τjcd)for each timing hypothesis k, and finding where the maximumsum occurs. The threshold is incorporated since the MT doesnot know the exact number of neighbor BS S and the choiceof threshold gives a trade off between the successful detection(of identities in B) versus false alarms (wrongly detectingidentities /∈ B).

Lemma 1. With known value of S and threshold set as τjcd =0, JCD detection in (5) becomes identical to the ML detectionin (4) when all the following conditions hold true.

1) There is no frequency offset, i.e., εq = 0, ∀q ∈ B.2) Covariance matrix Cq = ρI,∀q ∈ B for a constant ρ.3) The training sequences have ideal correlation proper-

ties, i.e.,∑N−1n=0 xi(n)x∗j (n + `) = δ(i − j)δ(`) for

i, j ∈ I and ` ∈ {0, · · · , L− 1}.4) ‖yk‖2 is constant for all k.

Proof: Note that, if all the channel taps in CIR from allneighbor BS are i.i.d. Gaussian and the power levels of allneighbor BS are identical then the covariance Cq = ρI whereρ denotes the received signal power from each neighbor BS.

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Using the first two conditions in lemma, the covariance of thetraining signal observations (3) is given by

CB,0 = ρ∑q∈B

Xq,0X∗q,0 + σ2

wI. (6)

From the ideal correlation properties of training signals, itfollows that X∗q,0Xp,0 = δ(p−q)I . Using this, it follows thatdet(CB,0) = σ

2(N−SL)w (σ2

w+ρ)SL. Note that the determinantdoes not depend on the set of interferers B. From the matrixinversion lemma, we also have

C−1B,0 =

1

σ2w

I − ρ

σ2w(ρ+ σ2

w)

∑q∈B

Xq,0X∗q,0. (7)

Now the ML metric given in (4) can be rewritten as,

(T , B) = arg maxk∈{0,··· ,F−1}

B⊂I such that |B|=S

∑q∈B

ρ∥∥X∗q,0yk∥∥2

2

σ2w(ρ+ σ2

w)− 1

σ2w

y∗kyk.

(8)From the final assumption in the lemma, above expressionsimplifies to (5) in the absence of frequency offsets.

Though JCD is identical to ML only under some idealisticassumptions, it is computationally far less intensive (no ma-trix inversions) and does not require any apriori informationon signal and noise powers. This makes JCD attractive forpractical implementations.

Conventionally, the frame timing is detected using matchedfilter (MF) based approach [5]

(T , q, εq)MF = arg maxk,q,m

∣∣∣∣∣N−1∑n=0

y(k + n)x∗q(n)e−j2πεmn

∣∣∣∣∣ ,q ∈ I, εm ∈ E . (9)

MF performs moving window correlation of received sampleswith the training signals (with appropriate rotating phasors)from every BS and picks the timing hypothesis, cell-identityand frequency offset bin-index corresponding to the maximumcorrelation. If there is only a single BS in the neighborhoodof MT with flat fading channel (i.e., S = 1 and L = 1),the MF approach is identical to JCD (with threshold τjcd =0) and hence will be ML under the idealistic assumptions ofLemma 1. However, in the presence of interfering BS (S > 1),MF is sub-optimal and it typically identifies the timing basedon the BS with the strongest received signal power.

The complexities in terms of number of multiplications,to compute the metrics for ML4, JCD and MF approachesare given in Table I. For ML, the complexity is dominatedby the matrix inversion in (4), which typically takes O(N3)operations.

Since all the neighbor BS share the same frame timing(within a window of duration Lmax), JCD works well inidentifying their common frame timing. But the performanceof JCD in recovering the identities of all neighbor BS andtheir frequency offsets is shown to be poor (analyticallyand numerically) in the subsequent sections. Hence, in thefollowing section, we develop new techniques for recovering

4ML complexity includes both timing detection and neighbor BS identifi-cation since ML detects the timing and neighbor BS jointly.

Algorithm No. of MultiplicationsML F

(MCS

)O(N3)

JCD FMCNLMF FMCN

TABLE ICOMPLEXITY FOR TIMING DETECTION.

the cell identities in the neighbor set B and their correspondingfrequency offsets.

IV. NEIGHBOR SET IDENTIFICATION

A. Observation Model in CS FrameworkIn this section, we present how the parameters of the

neighbor base stations (cell identities and frequency offsetvalues) can be recovered using the compressive sensing (CS)framework [22], [23], after their common frame timing isfound. For simplicity in notations, we present the details ofour approach assuming that the detected timing T from (5)is correct such that [Tq, Tq + Lq] ⊂ [T , T + Lmax],∀q ∈ B.Based on the timing detected from JCD, we can collect thesynchronization signal observations as in (2) and subsequentlyrecover the neighbor set B along with their frequency offsets.From (2), we see that only S out of a total of C base stationscontribute to the observation vector y. Typically S is smallcompared to the value of C since the number of base stationsin the vicinity of the MT is very small compared to thetotal number of base stations in the entire cellular network.This sparseness of the neighbor BS facilitate the applicationof compressive sensing based algorithms for the recoveryof synchronization parameters of neighbor set B. The exactdetails of our approach are explained below.

Since frequency offset of each BS (εq) in (2) is alsounknown, we define the bin frequency in E which is closestto the εq as,

εq = minε∈E|ε− εq|, ∀q ∈ B. (10)

To proceed further, we approximate Xq,εq using Xq,εq .Specifically, we rewrite (2) as

y =∑q∈B

Xq,εqzq + w, (11)

where the effective noise w = w+∑q∈B(Xq,εq −Xq,εq )zq ,

is the sum of additive noise and the approximation error.Note that, as the number of bins M increases, the differencebetween εq and εq decreases, leading to the reduction in the ap-proximation error. Now, let us construct the super-block train-ing matrix X of size N × LMC by horizontally concatenat-

ing as X =

X1,ε1 , · · · ,X1,εM︸ ︷︷ ︸M blocks with q=1

, · · · ,XC,ε1 , · · · ,XC,εM︸ ︷︷ ︸M blocks with q=C

.

Also, defining ziq =

{zq q ∈ B & εi = εq0L×1 else

, we define

the super-block channel vector z of size LMC × 1 by ver-tically concatenating {z1

1, · · · , zM1 , · · · , z1C , · · · , zMC }. Now,

(11) can be rewritten as,

y = X z + w. (12)

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From its construction, it is clear that the only S out of CMblocks ziq in z are non-zero (corresponding to q ∈ B & εi =εq). Hence z is S-block sparse signal [14], [17], [24]. Thesensing matrix X is a block matrix with concatenation of CMtoeplitz blocks {Xq,εi} and only S out of them are active inproducing y, due to the block sparse structure of z. Notethat recovering the support of z (i.e., location of non-zeroblocks) will lead to identification of the neighbor BS B andthe corresponding closest bin frequencies εq given in (10).Hence, we focus on block-sparse signal recovery algorithmsto recover z from y and establish conditions for successfulsupport recovery.

Block restricted isometry property [14], [23], [25], mutualsubspace incoherence [26] and mutual incoherence [27] ofthe sensing matrix X play a crucial role on the recoveryguarantees of our algorithms. First, we characterize theseproperties for our sensing matrix with randomly generatedtraining sequences {xq(n)}, which are constant modulus com-plex sequences with uniform phase (CMCSUP). The matrix Xis said to satisfy block restricted isometry property (RIP) oforder S with a RIP constant δ ∈ [0, 1), if

(1− δ) ‖z‖22 ≤ ‖X z‖22 ≤ (1 + δ) ‖z‖22for all S-block sparse vectors z. Mutual subspace incoherence(µ) of X characterizes the smallest angle between columnspaces of any two blocks Xp,εi and Xq,εm , whereas, mutualincoherence (γ) of X characterizes the smallest angle betweenany two columns of X . These can be defined as,

µ = maxp,q∈I & εi,εm∈E

(p 6=q OR i 6=m)

maxaip,a

mq

|〈aip,amq 〉|‖aip‖2‖amq ‖2

(13)

where aip and amq belong to column spaces of Xp,εi andXq,εm respectively, and,

γ = maxi 6=m

|〈ai,am〉|‖ai‖2‖am‖2

(14)

where ai,am are any two columns of X . These propertiesfor our sensing matrix X in (12) are characterized as follows.

Theorem 1. For the randomly generated training sequencesxq(n) = 1√

Nejθ(q,n) , with θ(q,n) being i.i.d. uniformly dis-

tributed in [−π, π], if the length of the sequence N satisfies

N ≥12(SL)2 log

(48L2CM

β

)(δ − (M − 1)α)2

(15)

then X satisfies block RIP of the order S with RIP constantδ ∈ ((M − 1)α, 1) with probability at least 1− β, where

α = maxi,m∈{1,··· ,M}, i 6=m

1

N

∣∣∣∣∣N−1∑n=0

ej2π(εi−εm)n

∣∣∣∣∣ (16)

Proof : See Appendix B.

Theorem 2. For the random training signal described in The-orem 1 with length N satisfying (15) with δ ∈ ((M − 1)α, 1),the mutual subspace incoherence µ of X satisfies,

α ≤ µ ≤Lmax

{δ−(M−1)α

SL , α}

1− δ−(M−1)αS

(17)

and the mutual incoherence γ of X satisfies,

α ≤ γ ≤ max

(δ − (M − 1)α

SL, α

)(18)

with probability at least 1− β, where α is given by (16).

Proof : See Appendix C.Previously, the block RIP and µ of the sensing matrix in theabsence of frequency offset (εq = 0) have been characterizedin [17]. Due to the presence of the frequency offsets, thestructure of our sensing matrix X in (12) is different fromthe sensing matrix for the model considered in [17]. TheRademacher distributed training sequences considered in [17]can not be used to study our sensing matrix (more detailsare given in Appendix A), instead, we need to use complexsequences with uniform phase. Our novel results in Theorem 1and Theorem 2 illustrate the impact of the number of fre-quency bins M on the block-RIP, µ and γ of the sensingmatrix through the bound in (15) and through the value of αin (16). If the bin frequencies in E are not orthogonal (α 6= 0)over the duration of N , then γ, µ and block RIP constant δ ofX are bounded away from zero, unlike the results from [17].We will see in the following section that smaller values of µ,γ and RIP constant δ, better the recovery performance.

The condition on the length N (15) of the (randomlygenerated) training sequences required for satisfying the CSrelated properties (RIP, µ, γ) in Theorem 1 and Theorem 2is sufficient (but not necessary). In general, the sufficientconditions on the length N are loose, i.e., the actual value ofN needed to guarantee the desired recovery properties maybe smaller than those values obtained analytically. Due tothe structure present in the sensing matrix X , getting tightbounds on length N is a hard problem. Discussions on whythe theoretical bounds are typically loose for structured sensingmatrices can be found in [14], [28], [29].

B. Detection Algorithms

In this subsection, we apply existing as well as new blocksparse signal recovery algorithms to recover the neighborBS parameters (cell-identities and frequency offsets) fromour model in (12) and theoretically establish their detectionperformance.

1) Mixed Norm Minimization (MNM): The super-blockchannel vector z is recovered by the solution to the followingconvex optimization which minimizes the l2/l1 mixed normas [14], [17],

minz

M∑i=1

C∑q=1

‖ziq‖2 subject to ‖X z − y‖2 ≤ λ (19)

where λ is a threshold of our choice. Let z denote the recon-structed super-block channel vector (solution to (19)). Fromblock-RIP characterization in Theorem 1 and [17, Theorem3], we have the following recovery guarantee of MNM.

Theorem 3. For the model in (12), if the effective noise isbounded as ‖w‖2 ≤ λ and if X satisfies block-RIP of order2S with block-RIP constant δ < 2

√2−17 ≈ 0.2612, then the

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solution of optimization problem (19) satisfies the followinginequality,

‖z − z‖2 ≤4√

1 + δ

1− (1 + 2√

2)δλ. (20)

It follows that MNM recovers z perfectly in the absenceof effective noise (λ = 0) when X satisfies the sufficientblock RIP condition given in the above theorem. For thenoisy case, we recover the neighbor BS parameters as follows.Note that the reconstructed vector z has MC blocks {ziq},corresponding to M frequency bins and C cell identities. Foreach cell identity q ∈ I, we estimate the best frequency binεq as

εq = εi, where i = arg maxm∈{1,··· ,M}

‖zmq ‖1, (21)

and set the corresponding recovered channel vector for thatcell identity as zq = ziq . We find the set of neighbor BS Bwith |B| ≤ Smax such that ∀q ∈ B, we have ‖zq‖1 ≥ τmnmand ‖zq‖1 ≥ ‖zp‖1 for any p /∈ B. This simply amounts tochoosing the top at-most Smax cells in terms of the l1 norm fortheir reconstructed channel (provided they are above a chosenthreshold τmnm). The value of threshold gives a trade offbetween successful detection of all neighbor BS versus falselyidentifying wrong cell identities.

2) Subspace Correlation Pursuit (SCP): SCP, describedbelow, is a greedy block sparse recovery algorithm [14], [27]and an extension of orthogonal matching pursuit to exploit theblock sparse structure.

Step 1. Initialize i = 1, B = ∅, Bε = ∅, r0 = y, X = ∅.Step 2. Find the index (qi, εi) such that

(qi, εi) = arg maxp∈I\B,ε∈E

‖X∗p,εri−1‖2

Step 3. B ← B ∪ {qi}, Bε ← Bε ∪ {(qi, εi)}Step 4. Concatenate: X = [X X qi,εi ]Step 5. z = arg minz ‖y − Xz‖2Step 6. ri ← y − Xz and i← i+ 1Step 7. If i ≤ Smax and ‖ri‖2 ≥ τscp, go to Step 2, else setz ← z and stop.

When the algorithm stops, Bε gives the set of detected BSalong with their corresponding closest frequency offset binindex. The reconstructed CIR of detected cells are available inz. SCP chooses a block Xq,ε which gives the maximum cor-relation with the residual in each iteration. The residual aftereach iteration corresponds to the projection of the observationvector onto the orthogonal complement of the column space ofall the previously chosen blocks. The metric in Step 2 of SCPis similar to sufficient signal metric derived in [16] under theassumptions that the CIRs corresponding to all the neighborBS are uncorrelated and have the same covariance matrix withhigh SNR. We establish the following recovery guarantee ofthe SCP algorithm for our synchronization problem.

Theorem 4. For the observation model given in (12) with‖w‖2 ≤ λ, the parameters (cell identity and the closestfrequency offset bin) recovered by SCP at the ith iteration are

identical to the true values for all i ≤ S (i.e. , qi ⊂ B, εi =εqi), if

S <1

2L+

1

Lγ

(1

2−√Lλ

d

)(22)

and the residual after S iterations satisfies

‖rS‖2 ≤ λ

(1 +

1√1− µ(S − 1)

)(23)

where d = minq∈B‖zq‖2.

Proof : See Appendix D.For the chosen threshold τscp, if the algorithm runs for exactlyS iterations, then SCP perfectly identifies the set B with nofalse detections. It should be noted that if the algorithm is setto run for Smax iterations without any threshold (τscp = 0),then the MT will perfectly detect all the BS in neighbor set Bif (22) is satisfied but there will be (Smax−S) falsely detectedcell identities.

3) Subspace Matching Pursuit (SMP): SMP is an anothergreedy algorithm for block sparse signal recovery in CS frame-work [26], [30], [31]. With Πq,ε denoting the the projectionmatrix onto the column space of block Xq,ε, at ith iteration,SMP recovers parameters of a neighbor BS as

(qi, εi) = arg minp∈I\B,ε∈E

‖ri−1 −Πp,εri−1‖2 (24)

which replaces the step 2 in SCP algorithm. The remainingsteps are same as SCP. A block Xq,ε is chosen by SMP, if theerror corresponding to the projection of the residual onto thecolumn space of that block has minimal norm compared tothe other blocks. SMP involves more computation comparedto SCP due to the projection of residual onto the column spaceof each block. Applying the recovery guarantee for the SMPalgorithm from [17], [26] to our model, we have the followingresult.

Theorem 5. For the observation model given in (12) with‖w‖2 ≤ λ, the parameters (cell identity and the closestfrequency offset bin) recovered by SMP at the ith iterationare identical to the true values for all i ≤ S (i.e. , qi ⊂B, εi = εqi), if

S <1

2+

1

µ

(1

2− λ

c

)(25)

and the residual after S iterations satisfies (23), where c =minq∈B‖Xq,εqzq‖2.

4) Single-step Subspace Matching Pursuit (SSMP): InSSMP, we avoid the iterative procedure of SMP and recoverthe parameters of all the neighbor BS in a single step, basedon the projection norm as follows.

Step 1: Compute projection norm Gi,m = ‖Πi,εmy‖2 fori ∈ I, εm ∈ E .Step 2: For each cell identity q ∈ I, find the detected binindex corresponding to the maximum projection norm asmq = arg maxm∈{1,··· ,M}Gq,m.

7

Step 3: Choose neighbor set B with |B| ≤ Smax such that∀q ∈ B, we have Gq,mq > τssmp and Gq,mq > Gp,mp forany p /∈ B.

Essentially, SSMP chooses the top (at-most Smax) distinct cellidentities in terms of their projection norm values (providedthey are above a chosen threshold).

Theorem 6. For the observation model in (12) with ‖w‖2 ≤λ, the parameters (cell identities and closest frequency offsetbins) corresponding to the top S largest values of ‖Πi,εmy‖2are identical to the true values, if

S <1

2

(1 +

1

µ

c2c1

)− λ

µc1(26)

where c1 = maxq∈B‖Xq,εqzq‖2, c2 = minq∈B‖Xq,εqzq‖2.

Proof : See Appendix E.For the chosen threshold, if |B| = S, SSMP correctly identifiesthe neighbor set B with no false alarms and when |B| 6= S,the parameters corresponding to min

(S, |B|

)largest values

of ‖Πi,εmy‖2 are the true values if (26) is satisfied. For λ = 0,SSMP recovery condition is equivalent to amplification of µby a factor of c1

c2when compared with the SMP recovery

condition. Hence, complexity reduction in SSMP by avoidingthe iterative process of SMP results in a poorer detection per-formance. Essentially the performance of single shot SMP isapproximately identical to performance of SMP with increasedmutual incoherence parameter µ by a factor of c1

c2, which

depends on range of received signal strengths from the BSpresent in the neighbor set B.

5) Joint Correlation Detector (JCD): When the timingwindow detected by JCD is correct, the performance of JCD(5) in recovering the neighbor BS parameters is characterizedas follows.

Theorem 7. For the observation model in (12) with ‖w‖2 ≤λ, the parameters (cell identities and closest frequency offsetbins) corresponding to the top S largest values of

∥∥X∗i,εmy∥∥2

2are identical to the true values, if

S <1

2+d2

d1

[1

2L

(1 +

1

γ

)−(

1

2+

λ√Lγd2

)](27)

where d1 = maxq∈B‖zq‖2, d2 = minq∈B‖zq‖2.

Given the detected timing, JCD chooses the blocks in asingle step based on the largest correlation metric values∥∥X∗i,εmy∥∥2

2. Above theorem can be proved along the similar

principles of SSMP proof, using the bounds of the correlationmetrics obtained in the proof of SCP. When compared toother recovery algorithms, JCD poses stringent condition forrecovery of the parameters of all the BS in the neighbor setB.

6) Successive Interference Cancellation: A successive in-terference cancellation (SIC) based approach was developedin [16] for the neighbor cell detection problem, with theassumption that the residual frequency offsets are negligible.We can also employ this SIC method for our neighbor setdetection problem, by adapting it suitably to handle the non-zero residual frequency offsets. SIC essentially is a greedy

Algorithm No. of MultiplicationsSMP KMCN2 +N

(K3L2 +K2L

)+O(L3)

SCP KMCNL+N(K3L2 +K2L

)+O(L3)

SSMP MCN2

JCD MCNLSIC [16] KMCNL+KN2

TABLE IICOMPUTATIONAL COMPLEXITY COMPARISON.

algorithm which detects the cells based on correlation metric,same as the Step 2 of SCP. However, SIC computes theLS channel estimates for each detected cell separately. Thecomputation of residual at the (i + 1)th iteration is done asri+1 ← ri−Πqi,εiri, where qi and εi denote the cell identityand the frequency bin index detected at the ith iterationrespectively.

7) Remarks on Recovery Algorithms: MNM, SMP and SCPalgorithms exist in CS literature while the single-step varia-tions SSMP and JCD are novel block sparse signal recoverytechniques. Also, the recovery guarantees of SCP, SSMP andJCD are novel results in the CS framework. MNM is a convex-programming algorithm which can be solved using secondorder cone programming in polynomial time complexity ofO(√MC) [14], [32]. For the other algorithms, we measure the

real-time complexity in terms of the number of multiplicationsperformed in running the algorithm. The iterative SCP, SICand SMP algorithms incur complexity during the computationof metric for detection (correlation or projection) and in com-puting the residual involving the LS estimation of the channelcoefficients. The projection matrices Πq,ε corresponding toevery sub-block of X required to compute the projection errorin (24) can be pre-computed and stored in the MT. For Kiterations, the estimate of the number of multiplications neededfor SCP, SIC and SMP are given in Table II. SSMP and JCDdetects the parameters of all the neighbor cells in a single stepand their complexities are lower than the iterative techniques.

V. SIMULATION RESULTS

A. Simulation Parameters

We study the synchronization performance of the algorithmsusing Monte-Carlo simulations. We use the following param-eters; synchronization signal bandwidth of 1 MHz, carrier fre-quency of 2 GHz, frequency offsets are (uniformly distributed)within the range of ±7ppm (closely aligned with the LTEspecifications). We consider AWGN noise and use extendedtypical urban model and extended pedestrian A model [33] forthe multipath channels from different BS along with Dopplercorresponding to a mobility of 30 Kmph. The number ofneighbor BS is set as S = 6, and is unknown at MT, whichsets the value of Smax = 9. For the neighbor BS powerlevels, we consider two scenarios, 1) equal - the averagereceived power levels of all the neighbor BS are equal and2) unequal - the difference between strongest and weakestis set as 10 dB and the remaining power levels are chosenuniformly in between. Successful recovery of parameters of a

8

neighbor BS is declared if its identity, frame timing and itsclosest frequency offset bin index are all recovered correctly.In order to fairly compare all the algorithms, we choose theirthresholds to satisfy the constraint that the average number offalsely detected cells is less than 1, i.e., E{|B \ B|} < 1. Webenchmark the performance of our algorithms with that of theideal ML detector described in (4), which in addition knowsthe exact value of S.

B. Detection Performance ComparisonIn Figure 2, we plot the detection performance versus signal

to noise ratio (SNR), with CMCSUP training sequences de-scribed in Theorem 1. Signal power is computed by summingthe received powers from all the neighbor BS. In Figure 2(a), we plot the probability of correct timing detection of ML,MF and JCD. Detected frame timing T is declared correct ifthe arrival times of all the neighbor BS falls within a narrowwindow [T − L

2 , T + L2 ] around the detected timing. It can

be seen that JCD performs close to ML and is significantlybetter than MF for both equal (Eq) and unequal (Ueq) powerlevels. Performance of ML in both equal and unequal powersare almost identical.

In Figure 2 (b), we plot the probability of successfully iden-tifying the synchronization parameters of all the base stationsin the neighbor set B. Here, the frame timing is obtainedusing JCD algorithm and the neighbor set parameters areobtained using corresponding detection algorithms describedin Section IV-B. For comparison, we also consider the typicalapproach which detects the strongest BS first by using MFtechnique (9) and then subsequently detects the other neighborcells’ parameters one by one using SIC [16] technique. Ourproposed approach of timing detection using JCD and subse-quent cell identity detection and frequency offset estimationusing CS algorithms perform significantly better than theconventional MF+SIC technique. In fact, the timing detectionwith JCD with neighbor set parameter recovery using MNMgives the best performance, next to the ML detection. Also,the two stage approach performs better than the single stageJCD which recovers all the synchronization parameters as per(5). Our results show that, with some additional complexity,the detection performance can be significantly improved overthe conventional MF+SIC approach.

Figure 3 shows the effect of number bins M on the detectionperformance of the algorithms. As M increases, the binapproximation of the frequency offset gets better and improvesthe variance of the residual noise w in (12). On the otherhand, increase of M makes the angle between column spacesof constituent blocks in X smaller, and results in the increaseof α in (16). Since the training sequence length N remainsfixed, increasing M beyond some value breaks the conditionin (15) (due to the increase in α) and the theoretical recoveryguarantees fail. Due to these two contrasting impacts of M ,the performances of the algorithms improves up to some valueof M after which degradation occurs.

C. Detection with LTE sequencesWe also study the successful synchronization parameters

recovery of all the neighbor cells in B, for the LTE frame

−10 −5 0 5 10

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

Pro

babi

lity

of T

imin

g D

etec

tion

ML

JCD−Eq

JCD−Ueq

MF−Eq

MF−Ueq

−10 −5 0 5 10

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

Pro

b. o

f Suc

cess

ful R

ecov

ery

of a

ll N

eigh

bor

BS

Par

amet

ers

Unequal Power Levels

ML

MNM

SMP

SCP

SSMP

SIC

MF+SIC

JCD

Fig. 2. CMCSUP sequences, N = 128, C = 500, M = 3, F = 144×102.

structure with LTE synchronization signals [1], [4] and isshown in Fig. 4. In LTE, there are 504 unique cell identities.Synchronization signals are present in two locations in eachLTE frame, referred as preamble and midamble. Synchro-nization signal of a BS comprises of primary and secondarysynchronization sequence (PSS and SSS) transmitted consecu-tively. PSS remains the same in both preamble and midamblewhile SSS differs. When PSS and SSS are concatenated, thereare a total of 504 × 2 sequences (of total length 124) whichconvey the cell-identity and preamble/midamble information.Our algorithms use this concatenated PSS and SSS sequencesto recover the synchronization parameters. We choose the restof the simulation parameters as discussed in Section V-A.

From Fig. 4, in both equal and unequal power levels ofneighbor BS, our proposed methods (JCD timing detectionfollowed by CS based parameter recovery) perform betterthan the conventional MF+SIC method. SIC performance isworse in the equal power case, in agreement with the commonwisdom that SIC works well in the strong or weak interferenceregimes. On the other hand, MNM performs better in the equalpower case since it recovers the parameters of all the neighborBS jointly.

In Fig. 5, we show the performance of our approaches indetecting the synchronization parameters for the BS with theweakest received power. For this study, we set the averagereceived power levels from all the interfering neighbor BS(except the weakest one) at 0 dB and the noise power at -10dB.The detection performance plotted versus signal to interferenceplus noise ratio (SINR) of the weakest base station (treatingother neighbor base stations as interference) shows that ourproposed methods work well even at very low SINR levels.

In Figure 6, we show the performance of detecting syn-chronization parameters of all the BS present in B versus thenumber of neighbor BS S. For this study, we set the powerlevels of all the neighbor BS at 0 dB and the Smax is set asS + 3. When S is increased beyond N/L (i.e., SL > N ),the concatenated matrix X in step 5 of the algorithms (SMPand SCP) will have non trivial null space (thereby failingto find the least squares channel estimates) and hence theywill not be able to recover B. MNM recovers more than the

9

1 2 3 4 5 6 7 8 9 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Bins (M)

Pro

b. of S

ucc

ess

ful R

eco

very

of all

Neig

hbor

BS

Para

mete

rs

MNM

SMP

SCP

SSMP

SIC

JCD

Fig. 3. CMCSUP Seq., N = 128, F = 14400, C = 500, SNR = 10 dB.

−10 −5 0 5 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

Pro

b. o

f Suc

cess

ful R

ecov

ery

of a

ll N

eigh

bor

BS

Par

amet

ers

Equal Power Levels

ML

MNM

SMP

SCP

SSMP

SIC

MF+SIC

JCD

−10 −5 0 5 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

Pro

b. o

f Suc

cess

ful R

ecov

ery

of a

ll N

eigh

bor

BS

Par

amet

ers

Unequal Power Levels

Fig. 4. LTE Synchronization Performance; M = 3.

−16 −15 −14 −13 −12 −11 −10 −9 −80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR [dB]

Pro

b. of S

ucc

ess

ful R

eco

very

of W

eake

st B

S P

ara

mete

rs

ML

MNM

SMP

SCP

SSMP

SIC

MF+SIC

JCD

Fig. 5. LTE Synchronization with the weakest power BS; M = 3.

aforementioned limit, however, the reconstructed channel zwill be unreliable.

2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Neighbor BS (S)

Pro

b. of S

ucc

ess

ful R

eco

very

of all

Neig

hbor

BS

Para

mete

rs

MNM

SMP

SCP

SSMP

SIC

MF+SIC

JCD

Fig. 6. LTE Training Sequences; M = 3, SNR = 12 dB.

D. Comparison of Theoretical and Simulation performances

In this section, we compare the simulation performanceof the greedy (SCP, SMP) and the single-step (SSMP, JCD)algorithms shown in Figure 6 with the performance from theirtheoretical recovery conditions. Towards that, we keep the JCDbased timing detection common and the subsequent neighborset parameters recovery performance is obtained using theoryas described below.

First, we consider the theoretical performance of SCPalgorithm. From our model, it is clear that the set of activeblocks in X , which contribute to the observation y is givenby {Xq,εq , q ∈ B}. While Theorem 4 gives the SCP recoveryguarantee for any set of active blocks (of size S), the followingresult (which can be proved similar to Theorem 4) discussesthe recovery of a specific set of active blocks.

Lemma 2. Given a specific set of active blocks {Xq,εq , q ∈B}, SCP recovers the active set parameters perfectly in thefirst S iterations, if

S <1

2L+

1

LγB

(1

2−√Lλ

d

), (28)

where d = minq∈B ‖zq‖2 and γB = maxai 6=ak|〈ai,ak〉|‖ai‖2‖ak‖2

with ai being any column of active blocks {Xq,εq , q ∈ B}and ak being any column of X (from active or non-activeblocks but not identical to ai).

The above sufficient recovery condition is identical to thatof Theorem 4, except that γB is replaced by γ in (22).Clearly, γ is the maximum of γB, taken over all the possiblesets of active blocks (by varying B and {εq}). Intuitively,this is satisfying because Theorem 4 gives sufficient recoverycondition for any possible set of active blocks. From the matrixX constructed using the LTE synchronization sequences, wecan compute the values of γB for all the possible sets of activeblocks of size S. When the set of active blocks is chosenrandomly with uniform distribution (as done in simulations), a

10

corresponding probability distribution is induced for γB. Usingthis distribution of γB, we can compute the probability thatthe condition (28) is satisfied for any specific value of S, i.e.,P{γB < 1

2LS−1

(1− 2

√Lλd

)}. Here, we set the value of λ

as 4 times the standard deviation of the noise, which ensuresthat the norm of the noise is bounded within λ with very highprobability. The value of L is set based on the delay spread ofthe multipath channel model. We also set the value of d in (28)as the smallest value of ‖zq‖2 corresponding to the successfulrecovery of all the neighbor BS parameters by SCP in thesimulations. This probability computed from the analyticalrecovery condition for various values of S is plotted along withthe detection performance from the simulations in Figure 7.For the chosen simulation parameters, the value 2

√Lλd � 1

and hence the theoretical probability of successful recovery ofa randomly chosen set of active blocks is approximately equalto P{γB < 1

2LS−1}.In a similar manner, by rearranging the terms in (27),

the theoretical probability of successful recovery of JCD iscomputed as P{γB < d2−2λ

√L

d1L(2S−1)+d2(L−1)} where d1 andd2 are set as the maximum and minimum value of ‖zq‖2respectively, corresponding to the successful recovery of allthe neighbor BS parameters by JCD in the simulations. In thesame fashion, theoretical performance of SMP and SSMP areobtained by replacing µ in their respective recovery condi-tions with µB = maxXa 6=Xb

{maxu,w∈CL

|〈Xau,Xbw〉|‖Xau‖2‖Xbw‖2

},

with Xa ∈ CN×L being an active block from the set{Xq,εq , q ∈ B} and Xb ∈ CN×L being any block fromX (which is not identical to block Xa). We note that thetheoretical recovery results are reasonably good in estimatingthe recovery performance of these algorithms and their relativeperformances. Nevertheless, the simulation performance isslightly better than that of the theory, which is a commontrend in the compressive sensing literature [28].

2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Neighbor BS (S)

Pro

b. of S

ucc

ess

ful R

eco

very

of all

Neig

hbor

BS

Para

mete

rs

SMP−Sim

SMP−Theory

SCP−Sim

SCP−Theory

SSMP−Sim

SSMP−Theory

JCD−Sim

JCD−Theory

Fig. 7. LTE Training Sequences; M = 3, SNR = 12 dB.

VI. CONCLUSIONS

We considered the downlink synchronization problem inHetNets to identify the synchronization parameters of all theBS in the vicinity of the mobile terminal. We proposed a

two stage approach involving JCD based timing detection andCS based recovery of cell-identities and frequency offsets.We studied the performance of the proposed techniques bothanalytically and numerically. Our proposed methods, performclose to ML upper bound and are superior to the existingSIC and MF algorithms. Future work involves extensionof the proposed ideas to utilize observations from multipleframes/antennas using the multiple measurement models ofcompressive sensing.

APPENDIX

A. Bound on off diagonal entries in Gram matrix of X

In this section, we bound the absolute values of the offdiagonal entries of full gram matrix G = X∗X when thetraining signals are randomly generated as given in Theorem 1.This bound will be subsequently used in characterizing blockRIP constant δ, mutual incoherence γ and mutual subspaceincoherence µ of the sensing matrix X . The absolute valuesof the entries in the matrix G, which are the inner productsbetween the columns of X , are represented in general formas Γ(p, q, k, `, i,m) =∣∣∣∣∣

N−1∑n=0

xp(n− k)x∗q(n− `)ej2π[εi(n−k)−εm(n−`)]

∣∣∣∣∣ , (29)

for p, q ∈ I, i,m ∈ {1, · · · ,M} = M, k, ` ∈ {0, · · · , L −1} = L.

Let us collect the entries in G in a set as Φ ={Γ(p, q, k, `, i,m) : p, q ∈ I, `, k ∈ L, i,m ∈ M}. We writeΦ = ΦD ∪ ΦR where ΦD = {Γ(p, q, `, k, i,m) : p = q, ` =k, i = m} ∪ {Γ(p, q, `, k, i,m) : p = q, ` = k, i 6= m} andΦR = Φ \ ΦD. ΦD contains the deterministic entries in Gwhile ΦR contains the random entries of G. ΦD containsdiagonal entries in G which are equal to one. The off diagonalentries of G present in ΦD can be written as,∣∣∣∣∣

N−1∑n=0

xp(n− `)x∗p(n− `)ej2π(εi−εm)n

∣∣∣∣∣ (30)

where p ∈ I, i,m ∈ M, i 6= m, ` ∈ L. The maximum valueof the above deterministic term is α as given in (16) andthere exists (at least) one pair (i 6= m) for which the abovesummation is equal to α.

The random entries in ΦR can be written as ΦR ={Γ(p, q, k, `, i,m) : p = q, k 6= `, i,m)} ∪ {Γ(p, q, k, `, i,m) :p 6= q, `, k, i,m)}. These two disjoint cases are dealt sepa-rately. Let us first consider the case p = q, k 6= ` in (29) andthese terms can be represented as,

Γ(p, p, k, `, i,m) =

∣∣∣∣∣N−1∑n=0

xp(n− k)x∗p(n− `)ej2π(εi−εm)n

∣∣∣∣∣(31)

for p ∈ I, i,m ∈ M, k, ` ∈ L. We use complex Hoeffding’sinequality to bound the summation in (31) which can be statedas follows. For i.i.d. θk uniformly distributed in [−π, π], andany vector v ∈ CK , we have

P

(∣∣∣∣∣K−1∑k=0

vkejθk

∣∣∣∣∣ ≥ t)≤ 4 exp

(− t2

4 ‖v‖22

). (32)

11

Note that multiplying xq(n) with any (deterministic) complexexponential results again in a phase uniformly distributed in[−π, π] and thus complex Hoeffding’s inequality (32) can beapplied5 to bound the sum in (31). Using Haznal-Sezemereditheorem on equitable coloring as in [34], the terms in thesummation (31) can be grouped into three sets S1,S2,S3,in such a way that entries in different sets are independentwith S1 ∪ S2 ∪ S3 = {xp(n − k)x∗p(n − `)ej2π(εi−εm)n} forn ∈ {0, 1, · · · , N − 1} and bN3 c ≤ |Sw| ≤ d

N3 e, for w =

1, 2, 3. Denoting the entries in Sw by {sw,u}, u = 1, · · · , |Sw|,we can write the sum in (31) as Γ(p, p, k, `, i,m) =∣∣∣∑3

w=1

∑|Sw|u=1 sw,u

∣∣∣. For each Sw, applying Hoeffding’s in-equality (conveniently assuming N is divisible by 3), we havefor any value of t, P

{∣∣∣∑|Sw|u=1 sw,u

∣∣∣ ≥ t3

}≤ 4 exp

(−t2N

12

)and the union bound on the sum of three such groups becomes,P{∣∣∣∑3

w=1

∑|Sw|u=1 sw,u

∣∣∣ ≥ t} ≤ 12 exp(− t

2N12

). Applying

union bound over C(L2 − L)2M unique off diagonal entriesof the case p = q, k 6= `,

P(

Γ(p, p, k, `, i,m) ≥ t)≤ 24ML2C exp

(− t

2N

12

)(33)

∀p ∈ I, k, ` ∈ L, i,m ∈M and k 6= `.Let us consider the other case of random sums, i.e., p 6= q

in (29) and these terms can be represented as,

Γ(p, q, k, `, i,m) =

∣∣∣∣∣N−1∑n=0

xp(n− k)x∗q(n− `)ej2π(εi−εm)n

∣∣∣∣∣p, q ∈ I, p 6= q. Since the training sequences are indepen-dent, Hoeffding inequality presented in (32) can be directlyapplied to bound the above term. Applying union bound overthe 2ML2(C2 − C) unique entries belong to this case, theprobability that any random sum in ΦR belong to the casep 6= q in (29) can be bounded as,

P(

Γ(p, q, k, `, i,m) ≥ t)≤ 8ML2C2 exp

(− t

2N

4

)(34)

∀p, q ∈ I, `, k ∈ L, i,m ∈ M and p 6= q. Applying unionbound over (33) and (34) for both the cases, probability thatabsolute of any random sum φr ∈ ΦR, exceeding the value t,P (|φr| ≥ t)

≤ 8L2CM

{C exp

(− t

2N

4

)+ 3 exp

(− t

2N

12

)}≤ 16L2CM max

{C exp

(− t

2N

4

), 3 exp

(− t

2N

12

)}.

(35)

With β = 48L2CMexp(− t

2N12

), if N ≥

12log(

48L2CMβ

)t2 , in

the above bound, second term inside the max dominates andwe have P (|φr| ≥ t) ≤ β. These results are summarized asfollows.

Lemma 3. For random training signals considered in Theo-rem 1, the off diagonal entries of full gram matrix G = X∗Xare bounded as follows. Any deterministic off diagonal entry

5This is not possible if the training sequences take values ± 1√N

as inscaled Rademacher distribution.

(in ΦD) is upper bounded by α given in (16). For any randomentry φr ∈ ΦR, we have, P (|φr| ≤ t) ≥ 1− β for any t > 0,

if N satisfies N ≥12log

(48L2CM

β

)t2 .

B. Block RIP characterization

We construct the matrix XP (of size N ×SL) by concate-nating S blocks out of CM blocks {Xp,εi , p ∈ I, εi ∈ E}and let P be a subset of the Cartesian product I × E with|P| = S, which denotes the actual pair of indices used in thecomposition of XP . From the Rayleigh-Ritz theorem [35], Xsatisfies block RIP of order S with RIP constant δ if the eigenvalues of the gram matrix GP = X∗PXP lie in the interval[1− δ, 1 + δ], for all possible choices of P .

Since all the diagonal entries of GP is equal to 1, fromthe Gershgorin disc theorem [35], it is sufficient to show thatthe sum of absolute values of the off diagonal entries in anycolumn of GP is less than δ. Towards that, let us use thebound on absolute values of off diagonal entries of full grammatrix G = X∗X given in Lemma 3 in Appendix A. If sumof any SL off diagonal entries in any column of the full grammatrix G = X∗X is less than δ, then the eigen values of GPwill lie in the interval [1− δ, 1 + δ] for any choice of P .

We need to show that the worst case sum of SL off diagonalentries in any column of G is less than δ. Towards that,we assume S > M since we are interested in finding themaximum value for the sum. Note that any column of G willhave M−1 deterministic off-diagonal entries whose maximumvalue is α (from Lemma 3). Hence we have the bound onthe RIP constant δ ≥ (M − 1)α. Now, for δ > (M − 1)α,setting t = δ−(M−1)α

SL in Lemma 3 in Appendix A, we areguaranteed that sum of any SL entries (including deterministicand random entries) in any column of G does not exceed δwith probability 1− β, if N satisfies (15).

C. Mutual Subspace Incoherence and Mutual Incoherence

The expression for mutual subspace incoherence can bealternatively written as,

µ = maxp,q∈I & εi,εm∈E

(p 6=q OR i 6=m)

{max

u,w∈CL

|〈Xp,εiu,Xq,εmw〉|‖Xp,εiu‖2‖Xq,εmw‖2

}.

(36)

Let xkp,i and xkq,m denote kth column of Xp,εi and Xq,εm

respectively. From Lemma 3 in Appendix A, we have,|〈xkp,i,x`q,m〉| ≤ max

{δ−(M−1)α

SL , α}

with probability at least1− β if N satisfies (15). Now, expanding the numerator termin (36),

|〈Xp,εiu,Xq,εmw〉| =L∑k=1

u∗k

L∑`=1

〈xkp,i,x`q,m〉w`

≤ max

{δ − (M − 1)α

SL, α

} L∑k=1

|uk|L∑`=1

|w`|

≤ Lmax

{δ − (M − 1)α

SL, α

}‖u‖2‖w‖2 (37)

12

where the last step follows from the equivalence of l1 and l2norms. Using (37) in the expression for µ, we have

µ ≤ max

Lmax{δ−(M−1)α

SL , α}‖u‖2‖w‖2

‖Xp,εiu‖2‖Xq,εmw‖2

≤

Lmax{δ−(M−1)α

SL , α}

min{‖Xp,εiu‖2

‖u‖2‖Xq,εmw‖2‖w‖2

} . (38)

Considering the denominator term in the above equation,

‖Xp,εiu‖2‖u‖2

=

√u∗X∗p,εiXp,εiu

u∗u≥√νmin

(X∗p,εiXp,εi

)(39)

where νmin is the minimum eigen value by Rayleigh-Ritztheorem [35]. It is evident that the off diagonal entries ofX∗p,εiXp,εi are the subset of random sums ΦR explainedin Appendix A and are bounded above by δ−(M−1)α

SL withprobability at least 1 − β if N satisfies (15). We can boundthe sum of L such entries by δ−(M−1)α

S and from Gersh-gorin theorem, eigen values of X∗p,εiXp,εi lie in the interval[1− δ−(M−1)α

S , 1+ δ−(M−1)αS ]. Taking the minimum of eigen

value in (39) and substituting in (38) we get the desired upperbound in (17).

Also, for some i 6= m, the absolute value of inner productsbetween the corresponding (unit-norm) columns from theblocks Xp,εi and Xp,εm , given by the expression (30), willbe equal to α. This fact, along the definition of µ in (13),establishes that µ ≥ α. Based on the same reasoning, we havethe mutual incoherence defined in (14) is bounded as γ ≥ α.Also, noting that γ = maxi6=mGi,m where Gi,m is the (i,m)th

entry in the full gram matrix G = X∗X , the upper bound (18)follows from Lemma 3.

D. Recovery Guarantee for SCP

Without loss of generality, we consider the model (12) asy =

∑Si=1 Xi,ε1zi + w with ‖z1‖2 ≥ ‖z2‖2 ≥ · · · ≥ ‖zS‖2,

i.e., active blocks correspond to the first S cell-identities withthe first frequency bin index. SCP chooses one of the correctblocks in the first iteration if,∥∥X∗1,ε1y∥∥2

>∥∥X∗t,εmy∥∥2

(40)

for all {t > S, εm ∈ E} and {t ≤ S, εm 6= ε1} (which are theinactive blocks). Defining y =

∑Si=1 Xi,ε1zi, we have,∥∥X∗1,ε1y∥∥2

≥∥∥X∗1,ε1 y∥∥2

−∥∥X∗1,ε1w∥∥2

, (41)∥∥X∗1,ε1 y∥∥2≥∥∥X∗1,ε1X1,ε1z1

∥∥2−

S∑i=2

∥∥X∗1,ε1Xi,ε1zi∥∥

2.

Note that∥∥X∗1,ε1X1,ε1z1

∥∥2≥∣∣νmin

(X∗1,ε1X1,ε1

)∣∣ ‖z1‖2and

∥∥X∗1,ε1Xi,ε1zi∥∥

2≤

∣∣νmax(X∗1,ε1Xi,ε1

)∣∣ ‖zi‖2. Themaximum (νmax) and the minimum (νmin) eigen valuesare bounded using Lemma 3 in Appendix A and Gersh-gorin disc theorem as

∣∣νmin(X∗1,ε1X1,ε1

)∣∣ ≥ (1 − (L −

1)γ) and∣∣νmax

(X∗1,ε1Xi,ε1

)∣∣ ≤ Lγ. Therefore

∥∥X∗1,ε1 y∥∥2≥ (1− (L− 1)γ)‖z1‖2 −

S∑i=2

Lγ‖zi‖2

≥ (1− SLγ + γ)‖z1‖2.

Similarly, for all {t > S, εm ∈ E} and {t ≤ S, εm 6= ε1}, wehave ∥∥X∗t,εm y∥∥2

≤S∑i=1

‖X∗t,εmXi,ε1zi‖2

≤S∑i=1

Lγ‖zi‖2 ≤ SLγ‖z1‖2.

The correlation of noise with any block can be boundedas∥∥X∗q,εmw∥∥2

≤√Lλ. Using these bounds in (40), SCP

chooses the correct subspace in the first iteration if,

S <1

2L+

1

2Lγ− λ√

Lγ‖z1‖2. (42)

After the first iteration, the residual r1 is orthogonal to thecolumns of the detected block. The above argument carriesforward for each iteration and hence SCP correctly recoversthe parameters of the neighbor BS in the first S iterationsif (22) is satisfied. Further, with this successful recovery, theresidual after S iterations can be written as,

‖rS‖2 =

∥∥∥∥∥∥y −∑q∈B

Xqzq

∥∥∥∥∥∥2

=

∥∥∥∥∥∥∥∑q∈B

Xq (zq − zq)︸ ︷︷ ︸eq

+w

∥∥∥∥∥∥∥2

≤∑q∈B‖Xqeq‖2 + λ (43)

and by directly applying [26, Theorem 3], the above bound(43) is equivalent to (23).

E. Recovery Guarantee of Single-step SMP

Without loss of generality, assume that B = {1, 2, · · · , S}and εi = ε1 ∀i ∈ B. Let us write the observation in (12) asy = y + w =

∑Si=1 aivi + w, with |ai| = ‖Xi,ε1zi‖2,vi ∈

column space of (Xi,ε1) and ‖vi‖2 = 1. Assume that thecoefficients satisfy |a1| ≥ · · · ≥ |aS |. Top S largest values ofthe projection Πi,εk(y) will correctly recover all the neighborset parameters if, for {i ≤ S}, we have

‖Πi,ε1(y)‖2 > ‖Πt,εm(y)‖2 (44)

for all {t ≤ S, εm 6= ε1} and {t > S, εm ∈ E}. For i ≤ S,we have ‖Πi,ε1(y)‖2 ≥ ‖Πi,ε1(y)‖2 − ‖Πi,ε1(w)‖2 and‖Πi,ε1(y)‖2 ≥ ‖Πi,ε1(aivi)‖2 −

∥∥∥∑Sk=1,k 6=i Πi,ε1(akvk)

∥∥∥2.

It follows that∥∥Πi,ε1(aivi)

∥∥2

= |ai| and‖∑Sk=1,k 6=i Πi,ε1(akvk)‖2 ≤ |a1|

∑Sk=1,k 6=i

∥∥Πi,ε1(vk)∥∥

2≤

|a1|(S − 1)µ since blocks of X have a mutual subspaceincoherence of µ. The projection of effective noise on anysubspace satisfies ‖Πq,εm(w)‖2 ≤ λ. Putting these together,for all i ≤ S,

‖Πi,ε1(y)‖2 ≥ |aS | − (S − 1)µ|a1| − λ. (45)

13

Similarly, for all {t > S, εm ∈ E} and {t ≤ S, εm 6= ε1}

‖Πt,εm(y)‖2 ≤ |a1|Sµ+ λ. (46)

Putting (45) and (46) together, recovery condition in (44) willbe satisfied if |aS | − (S − 1)µ|a1| − λ > Sµ|a1| + λ, whichis same as (26) since |a1| = maxq∈B ‖Xq,εqzq‖2 and |aS | =minq∈B ‖Xq,εqzq‖2.

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