IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1 Novel Compressed Sensing-based Channel...

1536-1276 (c) 2015 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.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TWC.2015.2505315, IEEETransactions on Wireless Communications

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1

Novel Compressed Sensing-based ChannelEstimation Algorithm and Near-Optimal Pilot

Placement SchemeYi Zhang, Ramachandran Venkatesan, Octavia A. Dobre, and Cheng Li

Faculty of Engineering and Applied ScienceMemorial University, NL, Canada A1B 3X5

Email: yz7384, venky, odobre, [email protected]

Abstract—This paper presents a novel recovery algorithmbased on sparsity adaptive matching pursuit (SaMP) and a newnear-optimal pilot placement scheme, for compressed sensing(CS)-based sparse channel estimation in orthogonal frequency di-vision multiplexing (OFDM) communication systems. Comparedwith other state-of-the-art recovery algorithms, the proposedalgorithm possesses the feature of SaMP of not requiring a prioriknowledge of the sparsity level, and moreover, adjusts the stepsize adaptively to approach the true sparsity level. Furthermore,we focus on the pilot pattern design in sparse channel estimation.Although a brute-force search guarantees the optimal pilotpattern, it is prohibitive to examine all possibilities due to highcomputational complexity. It is known that by minimizing themutual coherence of the measurement matrix when the signal issparse on the unitary discrete Fourier transform (DFT) matrix,the optimal set of pilot locations is a cyclic difference set(CDS). Based on this, we propose an efficient near-optimal pilotplacement scheme in cases where CDS does not exist. Simulationresults show that the proposed channel estimation algorithm, withthe new pilot placement scheme, offers a better trade-off betweenthe performance—in terms of mean squared error (MSE) andbit error rate (BER)—and complexity, when compared to otherestimation algorithms.

Index Terms—Sparse channel estimation, compressed sens-ing/compressive sensing, sparsity adaptive matching pursuit, pilotplacement, cyclic difference set.

I. INTRODUCTION

ORthogonal frequency division multiplexing (OFDM) hasbeen widely adopted in various wireless communication

standards, such as worldwide interoperability for microwaveaccess (WiMAX), long term evolution (LTE) [1], and highdefinition television (HDTV) broadcasting standards [2], dueto its high data rate, efficient spectral utilization and abil-ity to cope with multipath fading. In recent years, OFDMcommunication systems have also been exploited for under-water applications, e.g., pollution monitoring, offshore oil/gasexploration, environment surveillance [3], [4]. In coherentdigital wireless systems, obtaining accurate estimates of thechannel state information (CSI) is critical at the receiver [5].The data-aided channel estimation in OFDM communication

This work is supported in part by the Natural Science and EngineeringResearch Council (NSERC) of Canada and Research and Development Cor-poration Newfoundland and Labrador (RDC). Part of the work was presentedin the IEEE Wireless Communications and Networking Conference, March2015.

Manuscript received March 15, 2015; revised September 10, 2015.

systems can be performed by either inserting pilot tones intocertain subcarriers of each OFDM symbol, or by using allsubcarriers as pilots within a specific period [6]. Recently,studies have suggested that many multipath channels tendto exhibit a sparse structure in the sense that the majorityof the channel impulse response (CIR) taps end up beingeither zero or below the noise floor [7]. A few examplesinclude: a) in the North American HDTV broadcasting stan-dard, there are only a few significant echoes over a typ-ical delay spread [8], [9]; b) underwater acoustic (UWA)channels are characterized by a few dominant echoes overlarger time dispersion (in the order of hundreds of mil-liseconds) [4], [10]; c) channels of broadband wireless sys-tems in hilly environment also exhibit a sparse CIR [11],[12]. Conventional methods for CSI estimation, such as leastsquare (LS) and minimum mean-square error (MMSE) [13],cannot exploit the sparsity of the wireless channels and theyoften lead to excessive utilization of spectral and energyresources. On the other hand, channel estimation exploitingthe sparsity of channels reduces the required number ofpilots, and thus, effectively improves the spectral and energyefficiency [4], [7], [9], [14], [15].

More recently, advances in the new field of compressedsensing (CS) [16]–[18] have gained a fast-growing interest insignal processing and applied mathematics [19], [20]. It hasbeen shown in the literature that CS can be applied to sparsechannel estimation [4], [7], [15], [21]–[24]. Unlike traditionalchannel estimation methods, CS allows accurate reconstructionof the signal which is sparse on a certain basis, from asmall number of random linear projections/measurements [18].To ensure an accurate or even exact reconstruction of thetarget signal, a proper reconstruction algorithm and a properlydesigned measurement matrix are essential.

Existing algorithms to recover a target sparse signal aregenerally grouped in two categories: linear programming (LP)and dynamic programming (DP). The basis pursuit (BP)method in LP achieves a good MSE performance; however,its high computational complexity makes it less attractiveto real large-scale applications. Recently, the approximatemessage passing (AMP) algorithm [25] and its variants e.g.,generalized AMP [26] and expectation maximization (EM)-Bernoulli-Gaussian (BG)-AMP [27], were reported to achieve




the reconstruction performance1 almost identical to the LP-type methods while requiring less computational efforts [25].However, the performances of these algorithms were evaluatedfor certain statistics of the target signal, elements of mea-surement matrix, and noise. For example, the performanceof EM-BG-AMP relies on the BG distribution of the targetsignal. Alternatively, the orthogonal matching pursuit (OMP)algorithm [28] is the most popular algorithm in DP [15], [22].An improved OMP variant, referred to as the compressedsampling matching pursuit (CoSaMP) was proposed in [29],with the MSE performance close to that of the BP algorithm.However, CoSaMP requires knowledge of the channel sparsitylevel, which is often not available in practical applications.Although OMP can be further improved to work in the caseswhere the sparsity level is not known [30], the mutual inco-herence property (MIP) [31], which ensures the exact recoveryin the bounded noise cases, cannot be easily satisfied inpractice [30]. Recently, the sparsity adaptive matching pursuit(SaMP)2 algorithm was proposed to address this issue [32].While CoSaMP requires the level of sparsity as a prioriinformation to determine the number of iterations of thealgorithms, SaMP uses a stage-based approach to estimate thesparsity level, i.e., the estimated sparsity level is accumulatedwith a fixed preset step size stage by stage. The results showedthat SaMP can outperform the original OMP algorithm andits variants; however, its MSE performance and complexityare affected by the choice of the step size. Recently, a stage-wise algorithm which uses different step sizes for differentstages is proposed in [33]. However, it is not a truly adaptivealgorithm, as the change of step sizes depends on a specificrelation between the number of measurements and the sparsitylevel. In [34], we proposed a novel CS-based reconstructionalgorithm based on the SaMP algorithm, referred to as theadaptive step size SaMP (AS-SaMP), which can adaptivelyadjust the step size to achieve fast convergence. Simulationresults are provided for sparse channel estimation in UWA-OFDM systems to demonstrate the better MSE and BERperformance achieved by the proposed algorithm.

In this paper, we extend our previous work in [34] byproviding a symbolic computation complexity and theoreticalperformance analysis. Simulation results show that a goodperformance is attained by the AS-SaMP algorithm withoutincreasing the complexity significantly when compared withthe other mentioned recovery algorithms.

Furthermore, because different pilot placement choices willresult in different CS measurements, the result will directlyaffect the performance of the channel estimation algorithms.Equally spaced pilots are in general optimal for conventionalchannel estimation methods, which are, however, not truein CS-based methods [35]. In existing studies related toCS, randomly and deterministically placed pilot tones aremostly reported [35]–[39]. Although an exhaustive search ofall possible combinations of the pilot indices guarantees the

1The considered performance metric was the phase transition curve; fordetails, the reader is referred to [25].

2The name SAMP was originally used by the authors who proposed thealgorithm in [32]; we employ the abbreviation SaMP instead of SAMP in ordernot to be confused with the AMP (approximated message passing) algorithm.

optimal pilot pattern, the computational complexity increasesexponentially as the searching space expands. For example, if16 out of 256 subcarriers are used as pilots, there are

(25616

)≈

1025 different pilot patterns in the searching space. Thus, itis extremely computationally expensive for energy-efficientapplications, e.g., the power-constrained devices in underwateracoustic networks [40]. Moreover, provided a partial discreteFourier transform (DFT) measurement matrix, it is known thatif the pilot indices set is a cyclic difference set (CDS), themutual coherence of the measurement matrix is minimized[36], [38]. However, it is not guaranteed that a CDS will existfor every pilot size. In this paper, we investigate the problem ofpilot placement based on CDS. When CDS does not exist, wepropose a novel pilot pattern selection scheme which relieson the concatenated CDS with an iterative tail search (C-CDS with TS). Because the proposed design is deterministic,it is generally more computationally efficient than any othersearch-based methods. Simulation results demonstrate thatan improvement in the MSE and BER performance can beachieved using the proposed pilot placement scheme, whencompared to the randomly scattered pilots.

The following are the main contributions of this paper:• To extend our previous work in [34], we present a

comparative analysis of the performance of existing re-construction methods in terms of estimation accuracy andcomputational complexity.

• We propose a new efficient scheme for near-optimal pilotplacement to meet the requirement of the measurementmatrix for a satisfactory reconstruction.

The rest of this paper is organized as follows. In Section II,a review of CS fundamentals and the system model arepresented. In Section III, description of the new reconstructionalgorithm for sparse channel estimation is presented and acomparison with the existing reconstruction algorithms isperformed. Section IV introduces a novel pilot placementscheme based on C-CDS with TS. Section V presents thesimulation results and performance evaluation. Section VIconcludes the paper.

The following notation will be used for the rest of the paper.A bold symbol represents a set, a vector or a matrix, anda capital letter stands for frequency domain representation.XT denotes the transpose of X, X† denotes the Moore-Penrose pseudo-inverse matrix of X, which is defined as(XHX)−1XH , with XH as the Hermitian of X, and ‖ X ‖and ‖ X ‖1 denote the `2 and `1 norms of X, respectively.

II. CS FUNDAMENTALS AND SYSTEM MODEL

A. CS FundamentalsConsider a contaminated measurement vector of length M

obtained throughy = Φx + w, (1)

where x ∈ RN is a real-valued, one-dimensional, discrete-time signal vector, Φ is the sensing matrix, and w ∈ RN isa stochastic error term with bounded energy ‖ w ‖< ε [18].Assuming that x can be expanded in an orthonormal basisΨ as

x = Ψα, (2)




where α is the N × 1 coefficients vector, the signal x isK-sparse if and only if K coefficients (K N ) in αare non-zero, while the remaining coefficients are zero ornegligibly small.

Substituting (2) in (1), one obtains y = ΦΨα + w =Aα+ w, where A is referred to as the measurement matrix.Essentially, CS states that x can be recovered with high proba-bility, by solving the above under-determined problem. The re-liability of recovery depends on two constraints: 1) α is sparse;2) A satisfies the restrict isometry property (RIP) [16]–[18],which means that for an arbitrary level δ ∈ (0, 1) and anyindex set I ⊂ 0, 1, ..., N − 1 such that card(I) ≤ K,where card(·) denote the cardinality of the set, and forall α ∈ Rcard(I), the following relation holds: (1 − δ) ‖α ‖2≤‖ AIα ‖2≤ (1 + δ) ‖ α ‖2, where AI is the matrixcontaining the columns of which the indices are elements ofthe set I. An estimator of α can be achieved by solvinga convex optimization problem, which is formulated as [17],α = arg min ‖ α ‖1, subject to ‖ y − Aα ‖≤ ε, fora given ε > 0. If A satisfies RIP and α is sufficientlysparse, the norm of the reconstruction error is bounded by‖ α − α ‖≤ Cε, where C depends on the RIP relatedparameter δ of A rather than α [16], [17]. Particularly,if a measurement matrix is composed of random rows inan N × N DFT matrix, and if M > CδK logN , a K-sparse signal can be reconstructed with probability of at least1 − O(N−δ), where δ is the constant in the RIP and Cδ isapproximately linear with δ [16]. For more details, readers arereferred to articles on CS referenced in [20].

B. System Model

We consider an N -subcarrier OFDM system in which Psubcarriers are used as pilots. The symbols transmitted onthe kth subcarrier, X(k), 0 ≤ k ≤ N − 1, are assumed tobe independent and identically distributed random variablesdrawn from a phase-shift keying (PSK) or quadrature ampli-tude modulation (QAM) signal constellation. Assume that thediscrete multipath channel having the impulse response

h(n) =

Np−1∑p=0

ηp(n)δ(n− τp(n)), (3)

where Np is the number of paths, and ηp(n) and τp(n) arethe amplitude gain and the delay associated with the pth path,respectively. Although ηp(n) and τp(n) are time-varying, thecondition ηp(n) ≈ ηp can be true in one OFDM symbolperiod [15]. Moreover, by considering a relative stationarytransmitter and receiver, the assumption τp(n) ≈ τp can bemade [41]. Then the vector of received signal after DFT isexpressed as

Y = XH + W = XDh + W, (4)

where X is an N × N diagonal matrix with the elementsX(k), 0 ≤ k ≤ N − 1, on the main diagonal, Y =[Y (0), Y (1), ..., Y (N − 1)]T , H = [H(0), H(1), ...,H(N −1)]T , and W = [W (0),W (1), ...,W (N − 1)]T are the fre-quency response vectors of the received symbol, channel and

additive white Gaussian noise (AWGN), respectively, and h =[h(0), h(1), ..., h(L−1)]T , with L as the number of taps. The(m,n) element of D is given by [D]m,n = 1√

Ne−j2πmn/N ,

where 0 ≤ m ≤ N − 1 and 0 ≤ n ≤ L − 1. After extractingthe pilot subcarriers, there is

Yp = XpDph + Wp = Ah + Wp, (5)

where Yp = SY, Xp = SXST , Dp = SD, Wp = SW,and S is a P × N matrix for selected pilot subcarriers. Inaddition, A = XpDp is a P × L matrix, referred to as themeasurement matrix.

The goal of CS-based channel estimation is to estimate hfrom the received pilot Yp, given the measurement matrixA. In Eq (5), from which h needs to be estimated, the jthindex of h corresponds to the jth path delay τ(j), and thecorresponding value represents the gain of that path. The actualdelay of the channel may not coincide with the assumed delaypoints; this is known as the off-grid problem [42]3. Clearly,finer-grained delay points lead to a better approximation of thecontinuous delay, thus improving the estimation quality [22].In this paper, we use a sampling rate of N∆f , where ∆f isthe subcarrier spacing, and assume that τp is multiple integerof the sampling time, as it is commonly used in the literature[3], [4], [22].

III. THE AS-SAMP ALGORITHM WITH APPLICATION TOSPARSE CHANNEL ESTIMATION

A. Comparison of Reconstruction Algorithms in the Literature

This section focuses on the algorithms of the orthogonalmatching pursuit (OMP) family, that identify the support set4

of the target signal iteratively. At each iteration, one or morecolumns of the measurement matrix that are most correlatedwith the current residual are selected (this is referred to asthe maximum correlation test), and the residual is updated byprojecting the measurements onto the linear space spanned bythe selected columns. The reconstruction algorithms consid-ered here are the OMP, compressed sampling matching pursuit(CoSaMP), and sparsity adaptive matching pursuit (SaMP).Fig. 1 depicts the corresponding flow charts, where Ci, Fi, ri,and K denote the candidate support set, the final support set,the residual vector in the ith iteration, and the sparsity level ofthe target signal, respectively. As seen from Fig. 1, in the OMPalgorithm, Fi is expanded by adding coordinates successivelyand it uses only one maximum correlation test to add onecoordinate to Fi. On the other hand, the CoSaMP algorithmrefines a fixed-size Fi by selecting coordinates from a set ofcandidates Ci. It uses a preliminary correlation test and a finalcorrelation test, which are simply referred to as preliminarytest and final test, to add one or more coordinates to Fi. Thefinal test removes the wrong coordinates added in the prelim-inary test, which is referred to as backtracking, and thereforeimproves the accuracy of the estimation [29]. However, mostnatural signals are compressible rather than strictly sparse. The

3It is worth noting that the off-grid problem is common to all CS-basedalgorithms.

4For a vector β = [β1, β2, ..., βL] ∈ RL, the support set is defined asi|βi 6= 0 [28].




Initialization

Maximum

Correlation Test

Computing ri

Merging Final

Support Set

Exit

Yes

Halting condition?

No

Initialization

Preliminary Maximum

Correlation Test

Exit

Yes

Final Test

Ci( fixed size )

Fi( fixed size )

Halting condition?

No

Initialization

Preliminary Maximum

Correlation Test

Halting condition?

No

Exit

Yes

Final Test

Fi( adaptive size )

Updating Fiand r

i

Yes

No

Updating

Stage, Fi

and ri

Fi-1

Computing Signal

Estimation

Merging Final

Support Set

Fi-1

Computing Signal

Estimation and ri

Yes

Fi-1: Final support set from the previous iteration

Fi: Final support set at the current iteration

ri-1: Residual from the previous iteration

ri: Residual at the current iteration

Ci: Candidate support set at the current iteration

Computing Signal

Estimation and ri

Fi-1

Merging Final

Support Set

Ci( adaptive size )

OMP CoSaMP SaMP

<i i-1r r ?

Fig. 1. Flow charts of the OMP, CoSaMP, and SaMP algorithms.

sparsity level K for these signals could not be well-defined. Itis shown that the reconstruction accuracy can be significantlydegraded as we either underestimate or overestimate K [32].Unlike the CoSaMP algorithm, SaMP does not require a prioriknowledge of K. It adopts a stage-wise approach to identifyFi through the backtracking strategy. The size of Fi stays thesame among iterations in each stage; however, when it movesto the next stage, the size of Fi is increased by a fixed step sizes to search for more coordinates of the recovered signal whichcorrespond to the least residual. This process continues untilthe residual of the recovered signal falls below a predeterminedthreshold. Although SaMP guarantees exact recovery after afinite number of iterations (see proof in [32]), it leaves anopen question about the choice of the step size s to achieve thetrade-off between accuracy of estimation and complexity. Thismotivates us to address the problem of adaptively adjustingthe step size between consecutive stages. Recently, a variablestep size algorithm has been proposed in [33]. However, theincrement of the step size is based on a particular relationshipbetween the number of the measurements and the sparsitylevel, which is not always valid in applications. Therefore, wehave proposed a novel AS-SaMP algorithm, in our previouswork [34], which can adaptively adjust the step size; this willbe presented for completeness.

B. The AS-SaMP Algorithm

Since a smaller step size s in the SaMP algorithm leads toa better estimation accuracy while the complexity increases,and a larger s degrades the accuracy of the estimation whilethe complexity decreases, an adaptively adjusted s may leadto a better trade-off between the accuracy of estimation andthe complexity of the algorithm. Specifically, an adaptivelyadjusted s means that the change of s depends on how farthe current reconstruction state, e.g., current reconstructedsignal energy or its estimated sparsity level, is from thestate of the true signal. Because the sparse elements with

large values are reconstructed in the initial stages of thealgorithm, the energy difference of the reconstructed signalbetween consecutive stages is reduced at a declining rate asthe number of stages increases. In other words, the energy ofthe reconstructed signal tends to be stable when the estimatedsparsity level is close to the true sparsity level K. Followingthis property, the AS-SaMP algorithm begins with a largerstep size (the initial step size is denoted as sI ) to expeditethe convergence. Then the step size is adaptively decreasedto provide fine tuning in later stages, as the change rate ofthe reconstructed signal’s energy decreases. Consequently, anadditional threshold Γ is used to specify the beginning of thefine tuning. The pseudocode for the AS-SaMP algorithm isshown as Algorithm 1. The algorithm is stage-wise with avariable size of Fi in different stages. During a stage, it adoptstwo correlation tests iteratively, i.e., candidate and final tests,to search for a certain number of coordinates corresponding tothe largest correlation values between the signal residual andthe columns of the measurement matrix. Then, the algorithm

Algorithm 1 AS-SaMPInput: Received signal at pilot subcarriers Yp, measurement

matrix A, tolerance ε, threshold Γ, initial step size sI ;1: Initialize h = [0, 0..., 0]T , hold = [0, 0..., 0]T , rtemp =

[0, 0..., 0]T , indices set B0 = ∅, candidate support setC0 = ∅, residual r0 = Yp, size of final support setLF = s = sI , final support set F0 = ∅, iteration indexi = 1

2: while (‖ri−1‖ > ε) do3: Calculate signal SP = |AHri−1|4: Select indices set Bi in A corresponding to the LF

largest elements in SP Preliminary test5: Merge chosen indices and final support set from previ-

ous iteration into candidate support set Ci = Bi∪Fi−1

6: Refine candidate set to final set Fi by selecting indicescorresponding to the LF largest elements of |A†CiYp|Final test

7: Solve least-square problem h(Fi) = A†FiYp

8: Calculate current residual rtemp = Yp −AFiA†FiYp

9: if (‖rtemp‖ < ε) then10: ri = ri−1

11: Break12: else if (‖rtemp‖ ≥ ‖ri−1‖) then13: if (‖ h(Fi) ‖ − ‖ hold ‖< Γ) then14: s = ds/2e, LF = LF + s, hold = h(Fi), ri =

ri−1, i = i+ 1 Fine tuning15: else16: LF = LF + s, hold = h(Fi), ri = ri−1, i = i+ 1

Fast approaching17: end if18: else19: Fi−1 = Fi, ri = rtemp, i = i+ 120: end if21: end while22: return hOutput: Estimation of baseband channel impulse response h.




TABLE ITHE NUMBER OF ITERATIONS.

Methods OMP† CoSaMP SaMP‡ AS-SaMP‡

card(h) with |hmin| > 2b1−(2K−1)µ log(‖h‖/ε1) ≤ [−J log( |hmin|

‖h‖ ) −1log(CKs−SaMP

) + J]PL ≤ [−J log( |hmin|‖h‖ ) −1

log(CKs−AS−SaMP) + J]PL

† hmin is the non-zero element with the minimum magnitude in h.‡ h is the target signal and J is the total number of stages. CKs−SaMP and CKs−AS−SaMP are RIP-related parameters for SaMP and AS-SaMP, respectively (see Appendix

A for a detailed explanation).

moves to the next stage until the recovered signal with theleast residual is found. As opposed to SaMP, the proposedalgorithm incorporates two threshold values into the haltingcriterion: tolerance ε and Γ. Therefore, AS-SaMP halts whenthe residual’s norm is smaller than ε, in which ε is set tobe the noise energy. Meanwhile, s is decreased when theenergy difference of the reconstructed signal falls below Γ,whose value is chosen based on empirical observations. Theimpact of selecting Γ on performance is investigated in [34].Simulation results shows that the larger Γ is, the earlier finetuning starts, thus requiring a large number of iterations;however, a better reconstruction accuracy can be achieved [34].Starting with a sufficiently large initial step size (sI ≤ K),the algorithm quickly approaches the target signal. However,when the difference in the energy of the reconstructed signalsbecomes smaller than the preset Γ, the step size is reduced(by a factor of two) to avoid overestimation of the K-sparsetarget signal. This overestimation can significantly degradethe accuracy of the algorithm [32]. Furthermore, to extendour work in [34], we provide theoretical guarantee of exactrecovery of AS-SaMP, in both noiseless and noisy cases, withthe corresponding proofs in Appendix A.

C. Computational Complexity

In this section, the computational complexity of the existingalgorithms in the literature and the AS-SaMP algorithm iscompared in terms of the number of operations, which equalsthe product of the number of operations per iteration andthe number of iterations. Generally, each algorithm performssix major steps during each iteration5: forming signal proxy,identifying the largest indices, merging the set of indices,approximating the signal on the merged set of indices by least-squares, pruning to obtain next approximation, and updatingthe residual which is the part of the signal that has notbeen approximated [28], [29]. Assume that each step involvesstandard technique6; the dimension of the measurement matrixis P × L and the sparsity level is K. According to [29], theoperation counts per iteration for the corresponding steps areO(LP ), O(L), O(K), O(K2P ),O(K), and O(KP ), whereO(·) represents the big-O notation. Among the steps of eachalgorithm, the LS estimation dominates the contribution to

5The OMP algorithm involves five major steps during each iteration:forming signal proxy, identifying the largest indices, merging the set ofindices, approximating the signal on the merged set of indices by least-squares,and updating the residual [30].

6Fast implementation exists for each step of all the algorithms, e.g., in theLS estimation step, iterative methods such as Richardson’s iteration [29] orCholesky decompositions [43] can be used for efficiency. However, we adoptthe standard technique in each step for all the algorithms.

the complexity unless L is much larger than K2. In addition,the complexity of the algorithms also depends on the numberof iterations. Since the number of stages of SaMP is upperbounded by dK/se [32], and during each stage a portion ofcoordinates in the true support set are identified and refinedvia up to K iterations, an upper bound of the number ofiterations is dK/seK. On the other hand, due to the fastconvergence of the AS-SaMP algorithm, fewer stages arerequired to provide the same quality of estimates, and asthe computational complexity of each step is the same, theAS-SaMP algorithm is less complex when compared with theSaMP algorithm. Hence, a loose upper bound of the numberof iterations of AS-SaMP is also dK/seK. Note that the upperbound obtained for SaMP and AS-SaMP are quite loose, asthe number of iterations which varies from a stage to anotheris likely to be equal to or smaller than s( K) for most of thestages. Thus, we present an improved upper-bounded numberof iterations for the AS-SaMP algorithm in Table I (see Lemma2 in Appendix A for the proof) and we show that the upper-bounded number of iterations is smaller than that for the SaMPalgorithm in Corollary 1, Appendix A. Moreover, accordingto [30], the number of iterations which the OMP algorithminvolves is upper bounded by the number of the element inthe target signal with the amplitudes larger than 2b

1−(2K−1)µ ,where b is the upper bound of the `2 norm of the observationnoise and µ is the mutual coherence of the measurementmatrix (see the definition in Eq. (6), Section IV). For CoSaMP,the number of iterations is upper bounded by log(‖h‖/ε1)in which ε1 is a precision parameter of the reconstructedsignal [29]. A summary of the number of iterations of theconsidered algorithms is provided in Table I.

IV. PROPOSED PILOT PLACEMENT BASED ONCONCATENATED CDS

A. Problem Statement

According to the CS theory, an accurate recovery of a sparsesignal relies on the RIP of the measurement matrix. However,the RIP evaluation for a particular matrix is a non-deterministicpolynomial hard (NP) problem [18]. An alternative propertywhich evaluates if a measurement matrix can preserve wellthe information of the sparse signal in the measurements is themutual coherence of the measurement matrix [16]–[18]. In (5),the measurement matrix is the product of the transmitted pilotsand the DFT submatrix and is determined by both the symbolson the pilot subcarriers and the set of pilot location indices,which is also referred to as the pilot placement/arrangement.In this section, we focus on the pilot placement by assumingthat the same symbol is transmitted on all the pilot subcarriers.




The mutual coherence of a P × L measurement matrix Ais defined as the maximum absolute correlation between anytwo normalized columns, which is

µ(A) = max1≤i,j≤L, i6=j

| < aaai · aaaj > |‖aaai‖ · ‖aaaj‖

. (6)

Given the equal-power pilots, and substituting A with XpDp,(6) becomes

µ(A) = max1≤i,j≤L, i6=j

|X(kc)|2| < dpi · dpj > |‖dpi‖ · ‖dpj‖

, (7)

where c = 1, 2, ...P and dpi denotes the ith column ofDp with the mth element given by 1√

Ne−j2πikm/N , m =

1, 2, ..., P . We aim to find the set of pilot location indices Ω =k1, k2, ..., kP which minimizes µ(A). Although the optimalpilot pattern can be obtained through exhaustive searchingover all possible patterns, it is computationally prohibitiveto form the search space for a large number of subcarriersand pilots. Several methods have been suggested to search thesuitable solutions iteratively [35], [36], [38], [44]; however,the complexity of these methods is potentially high becausethe searching space grows rapidly (even exponentially) as thenumbers of subcarriers and pilots increase. In [39], the authorsidentify a collection of matrices formed by deterministicselection of rows of Fourier matrices which satisfy RIP. Inparticular, the selected indices of the rows correspond to theinteger outputs of certain polynomial functions [39].

In the following section, we propose a novel pilot placementscheme which aims to provide a near-optimal solution withoutsuffering from the fast-growing complexity.

B. Proposed Pilot Placement based on the Concatenated CDS

Suppose that the measurement matrix A is composed ofP rows of the N × L partial DFT matrix D in (4) and theindices set of the selected rows is Ω, and all the pilot symbolshave equal power. According to [36], [38], [44], the pilotarrangements based on CDS are optimal in minimizing themutual coherence of A. Recall the definition of a CDS [45],

Definition. Let G be a finite additive Abelian group of orderN . The P -element subset Q is called a (P,N, λ) CDS in G ifthe list of difference (q(i)− q(j)) moduloN , i, j = 1, 2, ...P ,and i 6= j, represents each nonidentity elements in G exactlyλ times.

For example, the (3, 7, 1) CDS is Q = 1, 2, 4, whichsatisfies that any integer between 1 and 6 will occur and repeatexactly once in the set q(i)− q(j) (modulo 7)|1 ≤ i 6= j ≤3. However, CDS exists only for some specific number ofsubcarriers and pilots. For the cases where there is no CDS,a pilot placement scheme based on the concatenated CDS isproposed. According to the definition of CDS, for an existing(P,N, λ) CDS of order P , assuming that G is the set of cyclicdifferences of any two elements of the CDS, every non-identityelement in the set G of order N has exactly the same numberof repetitions, λ [45]. In other words, if we denote the numberof repetitions of the different elements of G as λG = λg|g =1, 2, ...N−1, then λ1 = λ2 = ...λN−1 = λ which also meansthat the variance of λG is zero. Moreover, it is noticed that a

pilot pattern with a smaller variance of λG is likely to give asmaller mutual coherence of the resulting measurement matrix,and thus, more accurate estimates. Consider an OFDM systemwith N = 1024, in which P = 256 identical pilot symbolsare randomly scattered, and the number of taps of the sparseCIR is L = 256. In order to quantize the channel estimationerror, we adopt MSE, which is defined as

MSE = E[

N∑m=1

|H(m)− H(m)|2]. (8)

To show that as the variance of λG increases, it is likely thatso does the mutual coherence of A and the MSE of estimates,the Spearman’s rank correlation7 is adopted to measure thestrength of a monotonic relationship (i.e., values of elementsin a vector either increase or decrease with every increase inan associated vector) between paired vectors [46]. Table IIshows the Spearman’s rank correlation between any pair ofthe following four vectors: the variance of λG, the mutualcoherence µ(A), and the MSE for both the OMP and AS-SaMP algorithms, obtained based on 104 pilot patterns; 103

OFDM symbols and 10 dB signal-to-noise ratio (SNR) wereconsidered. From Table II, it can be seen that a smallervariance of λG tend to correspond smaller µ(A) and theaverage MSEs of OMP and AS-SaMP.

Algorithm 2 Pilot Placement Based on Concatenated CDSwith an Iterative Tail SearchInput: An existing (u, v, a) CDS C for concatenation, the

number of total subcarriers N , the number of pilot sub-carriers P , the partial DFT matrix D of which the (m,n)element is 1√

Ne−j2πmn/N , where 0 ≤ m ≤ N − 1,

0 ≤ n ≤ L − 1, and L is the number of taps of theCIR;

1: Initialize Ω0c = ∅, Ωtemp = ∅

2: for i from 1 to bNv c do3: Ωi

c = Ωi−1c

⋃[C + (i− 1)× v]

4: end for5: Pr = P − u× bNv c, Ω = Ωc

6: for j from 1 to Pr do7: Ωtemp = Ω8: Form all Pr−j+1 possible subsets of size j by adding

an element to Ωtemp:Ω = Ωtemp

⋃k ∈ Pr + 1, Pr + 2, ..., N\Ωtemp

9: Form the matrix A by selecting rows of D for each j-element sets generated from the previous step, and theindices set of the selected rows is Ω

10: For all (Pr − j + 1) of A matrices generated fromthe previous step, calculate the corresponding mutualcoherence, and choose the set with the minimum mutualcoherence

11: end for12: return ΩOutput: The pilot indices set Ω

7The Spearman’s rank correlation can take values from 1 to −1, with1 (−1) indicating that two vectors can be described using a monotonicincrease (decrease) function, and 0 meaning that there is no tendency forone vector to either increase (decrease) when the other increases [46].




TABLE IISPEARMAN’S RANK CORRELATIONS‡

var(λG)§ µ(A) The average MSE of OMP The average MSE of AS-SaMP

var(λG) 1 0.7475 0.7467 0.7527µ(A) 0.7475 1 0.7481 0.7534

The average MSE of OMP 0.7467 0.7481 1 —The average MSE of AS-SaMP 0.7527 0.7534 — 1‡ For two vectors of size V, A = [A(1), A(2), ..., A(V )] and B = [B(1), B(2), ..., B(V )], the Spearsman’s correlation is calculated from∑V

i=1(ai−a)(bi−b)√∑Vi=1(ai−a)2

∑Vi=1(bi−b)2

, where ai and bi are the positions in the ascending order (ranks) of A(i) and B(i), respectively. a and b are the

means of ai and bi, i = 1, 2, ..., V .§ The variance of λG is 1

N−1

∑N−1g=1 (λg)2 − ( 1

N−1

∑N−1g=1 λg)2.

It is noted that by concatenating a CDS the variance of thenumber of repetitions tends to be small. A concatenated CDSis shown in the following example. For the (3, 7, 1) CDS, aconcatenated CDS is obtained through 1, 2, 4, (1×7+1), (1×7+2), (1×7+4), (2×7+1), (2×7+2), (2×7+4), ..., (i×7+1), (i×7 + 2), (i×7 + 4), ..., where i ∈ Z+ and i ≥ 1. Fromthese observations, we propose a pilot placement scheme basedon the concatenated CDS for pairs of (P,N) where CDS doesnot exist. First, a CDS needs to be chosen for concatenationaccording to the ratio of the number of pilots to the number ofsubcarriers, i.e., P/N . Specifically, we select the existing CDSwith the parameters (u, v, a) in which u/v is the closest toP/N . For instance, to select indices for 256 pilots from 1024positions, the (133,33,8) CDS is used. After concatenating theselected CDS, we adopt an iterative procedure to find the restof pilot positions which minimize the mutual coherence ofthe resulting measurement matrix. We refer to this as to theiterative tail search; the pseudocode for the proposed schemeis shown as Algorithm 2. It is worth noting that the size of thesearch space is greatly reduced after concatenation, and hence,the proposed method converges much faster when comparedto the iterative methods in [36], [38], [39], [44].

V. SIMULATION RESULTS

A. Simulation Set-up

As UWA channels are inherently sparse, we consider UWAchannel estimation for a coded OFDM transmission withN = 1024 subcarriers and bandwidth of B = 9.8 kHz,leading to a subcarrier spacing of ∆f = 9.5 Hz. The cyclicprefix (CP) duration equals 26 ms, which corresponds tothe length of CP NCP = 256. Unless otherwise mentioned,the number of pilots is P = 256. The data symbols aredrawn independently from a 16-QAM constellation and arecoded using a (1024, 512) binary low-density parity-check(LDPC) code. We consider the channel model described in (3)with Np = 15 multipaths, in which the inter-arrival timesare exponentially distributed with a mean of 1 ms, i.e.,E[τj+1 − τj ] = 1 ms, j ∈ 0, 1, ..., Np − 1. The amplitudesare Rayleigh distributed with the average power decreasingexponentially with the delay, and the difference between theamplitudes at the beginning and the end of the CP is 20dB. These parameters are assumed to be constant within anOFDM symbol. In this paper, we consider the aforementionedOMP-like methods as well as an AMP-like method, namely

EM-BG-AMP [27]. The parameters for the considered OMP-like reconstruction algorithms are given in Table III. For EM-BG-AMP, the maximum number of EM8 iterations is 200and the convergence tolerance for EM is 10−5 [27]. ForSAMP and AS-SaMP, since there is a trade-off between theinitial step size s and the reconstruction speed, three choicesof the s value (s ≤ Np) are used, which correspond to asmall, medium and large step size. Also, the effect of variouschoices of Γ (0.01, 0.1, 1 and 10) on the MSE and CPUrunning time performance of AS-SaMP is evaluated in [34];the results showed that when Γ > 1, the MSE of the AS-SaMP algorithm starts to saturate. Therefore, the threshold Γis set to 1. Moreover, among the compared algorithms, onlyCoSaMP requires the sparsity level as a priori information,and the same stopping criterion is used for fairness, i.e., allalgorithms stop when the signal residual falls below ε. TheMSE and BER are used to measure the channel estimationaccuracy and the system performance, respectively. The CPUrunning time9 is used to provide a rough estimation of thechannel estimation computational complexity. Simulations areperformed in MATLAB R2014a using a 2.8 GHz Intel Corei7CPU with 8 GB of memory storage, and we use 104 Monte-Carlo trials to average the results. Performance of the AS-SaMP algorithm and proposed pilot placement scheme areshown next.

TABLE IIIPARAMETERS OF COMPARED ALGORITHMS.

Name MaxIter‡ Sparsity level K Step size s Tolerance ε Threshold Γ

OMP 20 not required not required norm(Noise)§ not requiredCoSaMP 20 15 not required norm(Noise) not requiredSaMP not required not required 1, 6, 8 norm(Noise) not requiredAS-SaMP not required not required initially 1, 6, 8 norm(Noise) 1‡ Maximum iterations.§ norm(V)=

√∑|V|2.

B. Performance of the AS-SaMP Algorithm

First, we compare the AS-SaMP algorithm with two classicalgorithms, namely least square (LS) and OMP using differentnumbers of randomly distributed pilots. Fig. 2 shows theMSE of these algorithms versus SNR. As the number of

8In EM-BG-AMP, the a priori distributions of the target signal and noiseare unknown and learned through the EM algorithm [27].

9The running time is used instead of the number of iterations for com-parison, as the complexity in each iteration varies for different algorithms.However, it is worth noting that different hardware configurations may resultin different running time measurements.




0 2 4 6 8 10 12 14 16 1810

−4

10−3

10−2

10−1

100

SNR (dB)

MSE

LS (P=64)

LS (P=128)

LS (P=256)

OMP (P = 64)

OMP (P = 128)

OMP (P = 256)

AS-SaMP (P = 64)

AS-SaMP (P = 128)

AS-SaMP (P = 256)

Fig. 2. MSE performance of the LS, OMP and AS-SaMP algorithms withvarious number of pilots.

pilots increases, MSE decreases for all algorithms. It is worthnoting that OMP has, in general, a better MSE performancethan LS for the same number of pilots. Similarly, AS-SaMPachieves a better MSE performance than the OMP algorithm.For example, at SNR = 15 dB and P = 64, the MSE for LS,OMP and AS-SaMP algorithms are 9× 10−2, 1.2× 10−2 and3.5×10−3, respectively. In other words, for the same level ofMSE performance, the AS-SaMP algorithm uses fewer pilotsthan the other two algorithms.

Next, Figs. 3, 4 and 5 plot the MSE, BER and the compu-tational complexity for all the algorithms, respectively. To getan idea of the potential MSE gain achieved by exploiting thesparsity into channel estimation, we compare the MSE perfor-mance of the considered algorithms with the MSE lower boundof an ideal channel estimator with known indices of the non-zero entries of h10 for the OMP-like algorithms. Accordingto results in Figs. 3 and 4, the CS-based channel estimatorsgive better MSE and BER performance than the conventionalLS estimator. In other words, the channel estimators based onthe SaMP and AS-SaMP algorithms outperform those basedon the OMP and CoSaMP algorithms in the sense that theformer algorithms offer the same performance even if fewerpilots were used. It can also be seen that, although EM-BG-AMP slightly outperforms the other algorithms for lowerSNRs (SNR < 8 dB), the proposed algorithm outperforms theother algorithms for higher SNRs (SNR ≥ 8 dB). Moreover,given the same sparsity level, the MSE of AS-SaMP is closerto the aforementioned lower bound than that of the otheralgorithms at higher SNRs. As shown in Fig. 5, the CPUrunning time of EM-BG-AMP is significantly larger thanthose of the other algorithms due to the iterative statisticalparameters learning process via EM, and at lower SNRs,EM-BG-AMP requires a larger number of iterations for EM.

10The ideal channel estimation is formed through h = AT†YP, where

AT is the submatrix obtained by extracting Np columns of A correspondingto the known indices set T. Thus, the lower bound of the MSE for thematching pursuit algorithms is Np

2σ2

trace(ATHAT)

[7], where σ2 is the noisepower and trace(x) is defined to be the sum of the elements on the maindiagonal of x.

0 2 4 6 8 10 12 14 16 1810

−4

10−3

10−2

10−1

100

SNR(dB)

MSE

LSOMPCoSaMPSaMP (s=6)

AS-SaMP (sI=6)

EM-BG-AMPKnown indices of non-zero entries

Fig. 3. MSE performance of the LS, OMP, CoSaMP, SaMP, AS-SaMP andEM-BG-AMP algorithms. The lower bound for known indices of the non-zeroentries of h is also included.

Furthermore, it is noted that the complexity of the CoSaMPalgorithm is higher than that of other algorithms; this canbe explained through Table IV in which the running time ofeach step per iteration for CoSaMP is higher than that forother algorithms. This can be further explained as the size ofthe support set for CoSaMP is larger (2K) than the supportset size for the other algorithms. Additionally, the complexityof the AS-SaMP algorithm is higher than that of the SaMP;this is because the step size is reduced during the fine tuningstages given the same initial step size in AS-SaMP. Moreover,from Table IV, it is noteworthy that the LS approximationstep dominates the contribution to the total running time periteration among the steps of all algorithms.

TABLE IVCPU RUNNING TIME (SEC) OF EACH STEP PER ITERATION FOR CS-BASED ALGORITHMS,

SNR = 10 DB AND P = 256.

Step OMP CoSaMP SaMP (s = 6) AS-SaMP (sI = 6)

Form proxy 4.21× 10−5 8.28× 10−5 5.11× 10−5 5.06× 10−5

Identification 7.23× 10−6 2.36× 10−5 1.80× 10−5 1.84× 10−5

Support merger 1.14× 10−5 1.80× 10−5 7.80× 10−6 8.03× 10−6

LS approximation 3.32× 10−4 4.4× 10−3 7.9× 10−4 8.46× 10−4

Pruning NA∗ 9.15× 10−5 7.99× 10−6 1.53× 10−5

Residual update 1.56× 10−5 4.80× 10−5 3.58× 10−5 5.50× 10−5

∗ NA: not applicable.

Figs. 6 and 7 depict the MSE performance and computa-tional complexity of the AS-SaMP and SaMP algorithms withdifferent step sizes, respectively. As can be seen, for a mediumor large initial step size (s = sI = 6 or s = sI = 8), AS-SaMP outperforms SaMP with a small increase in complexity,while for s = sI = 1, the same performance is achieved usinga slightly larger CPU running time for AS-SaMP. This canbe easily explained, as when s = sI = 1, AS-SaMP becomesequivalent to SaMP except an additional criterion for changingstages. Note that SaMP and AS-SaMP require s ≤ Np andsI ≤ Np, respectively, to avoid overestimation. In general,the AS-SaMP algorithm is more accurate without significantlyincreasing the complexity of the estimation.




0 2 4 6 8 10 12 14 16 1810

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR(dB)

BER

LSOMPCoSaMPSaMP (s=6)

AS-SaMP (sI=6)

EM-BG-AMP

Fig. 4. BER performance of the LS, OMP, CoSaMP, SaMP, AS-SaMP, andEM-BG-AMP algorithms.

0 2 4 6 8 10 12 14 16 180

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

SNR(dB)

Runningtime(sec)

CoSaMPOMPAS-SaMP (sI=6)

SaMP (s=6)

EM-BG-AMP

Fig. 5. Running time of the OMP, CoSaMP, SaMP, AS-SaMP, and EM-BG-AMP algorithms.

0 2 4 6 8 10 12 14 16 1810

−4

10−3

10−2

10−1

SNR (dB)

MSE

SaMP (s= 8)

SaMP (s = 6)

SaMP (s = 1)

AS-SaMP (sI = 8)

AS-SaMP (sI =6)

Fig. 6. MSE performance of the SaMP and AS-SaMP algorithms withdifferent step sizes.

C. Performance of the Proposed Pilot Placement Scheme

Here, we consider five pilot placement schemes, i.e., ran-dom, the procedure in [36], the stochastic sequential search

0 2 4 6 8 10 12 14 16 180

0.005

0.01

0.015

0.02

0.025

SNR (dB)

Runningtime(sec)

AS-SaMP (sI =1)

AS-SaMP (sI =6)

AS-SaMP (sI =8)

SaMP (s =1)

SaMP (s =6)

SaMP (s =8)

Fig. 7. Running time of the SaMP and AS-SaMP algorithms with differentstep sizes.

0 2 4 6 8 10 12 14 16 1810

−4

10−3

10−2

10−1

100

SNR (dB)

MSE

Random, OMP, µ = 0.31

C-CDS with TS, OMP, µ = 0.27

Random, AS-SaMP, µ = 0.31

C-CDS with TS, AS-SaMP, µ = 0.27

Fig. 8. MSE performance of the OMP and AS-SaMP algorithms with randomand the proposed pilot placement, for P = 64. Solid lines are used for OMPand dashed lines for AS-SaMP.

(SSS) in [38], the RIP-based scheme in [39], and our proposedscheme. Two polynomials are used to generate the pilot indicessets for the RIP-based scheme, i.e., f1(n) = 10n + n2 andf2(n) = 10n+ n2 + n3 [39]. Equal power pilots are assumedfor all scenarios. With the random scheme, the pilots areselected randomly among all subcarriers and 104 trials aregenerated for averaging the results. With our proposed scheme,the pilots are arranged based on the concatenated CDS withan iterative tail search, referred to as C-CDS with TS. Becausethe selection of the existing CDS depends on the ratio P/N ,the (273,17,1) and (133,33,8) CDS are chosen for the casesof P = 64 and P = 256, respectively.

Fig. 8 shows the MSE performance of the OMP and AS-SaMP algorithms with the random and the proposed pilotplacement scheme, given P = 64. For the randomly placedpilots, we use error bars to indicate the standard deviationsof the MSE; this was calculated based on 104 indices sets.It can be seen that the proposed method provides a superiorchannel estimation performance when compared to the randomplacements, as a reduced mutual coherence µ is obtained.




0 2 4 6 8 10 12 14 16 1810

−4

10−3

10−2

SNR (dB)

MSE

RIP-based in [39], f2(n) = 10n + n2 + n

3

RandomRIP-based in [39], f1(n) = 10n + n

2

SSS in [38](M 1 = 50, M 2 = 10)

Procedure in [36]

C-CDS with TS

Fig. 9. MSE performance of the AS-SaMP algorithm for different pilotplacements, for P = 256.

For instance, at SNR = 10 dB, the average MSEs of therandom scheme for OMP and AS-SaMP are approximately4× 10−2 and 1× 10−2 , respectively, while the MSEs of theproposed scheme for OMP and AS-SaMP are approximately3 × 10−2 and 8 × 10−3, respectively. It should be noted thatfor each SNR, the MSE with the proposed method is smallerthan the mean of the MSE minus the standard deviationwith the randomly placed pilots. More specifically, it equalsapproximately the mean minus twice the standard deviation; assuch, the proposed method provides a better MSE performancethan most of the random pilot arrangements. Also, AS-SaMPachieves a better MSE performance when compared with OMPfor both pilot placement schemes.

Fig. 9 shows the MSE of the AS-SaMP algorithm for thefive previously mentioned pilot placement schemes for P =256. Among them, the proposed C-CDS with TS has the bestperformance. Moreover, since the C-CDS pilot arrangement isdeterministic, and the iterative search is only conducted for thetail, the searching space is significantly reduced. Therefore,the number of iterations of the proposed method, which isproportional to its computational complexity, is significantlylower than that of the procedure in [36]. An example isprovided as follows. When N = 1024 and P = 256, theprocedure in [36] requires (2N − P + 1)P/2 = 229, 504iterations, and the SSS in [38] requires M1 × M2 × P =128, 000 iterations, where M1 and M2 are the number of theouter and inner loop, respectively. For our proposed scheme,if a (133,33,8) CDS is used for concatenation, there are1024−133×b 1024133 c = 93 subcarriers at the tail. To search therest of 25 (256 − 33 × b 1024133 c) pilot indices which minimizeµ, (2× 93− 25 + 1)× 25/2 = 2025 iterations are required.

Finally, we compare the BER performance for differentpilot placement schemes, with results shown in Figs. 10 and11. In Fig. 10, the AS-SaMP algorithm is considered forthe five pilot placement schemes. Clearly, AS-SaMP with theproposed C-CDS with TS is slightly better than the other pilotschemes. Fig. 11 compares the BER performance of the OMP,CoSaMP, SaMP and AS-SaMP algorithms with the randomand proposed pilot arrangements. In general, AS-SaMP with

0 2 4 6 8 10 12 14 16 1810

−7

10−6

10−5

10−4

10−3

10−2

10−1

SNR(dB)

BER

RIP-based in [39], f1(n) = 10n + n2 + n

3

RandomRIP-based in [39], f2(n) = 10n + n

2

SSS in [38] (M 1 = 10, M 2 = 50)

Procedure in [36]

C-CDS with TS

Fig. 10. BER performance of the AS-SaMP algorithm for different pilotplacements, for P = 256

0 2 4 6 8 10 12 14 16 1810

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR(dB)

BER

Random

C-CDS with TS

OMP

CoSaMP

AS-SaMP

SaMP

Fig. 11. BER performance of the OMP, CoSaMP, SaMP, and AS-SaMPalgorithms for random and the proposed pilot placement, for P = 256. Solidlines are used for OMP, dot lines for CoSaMP, dashed-dot lines for SaMP,and dashed lines for AS-SaMP.

the proposed pilot allocation scheme provides the best BERperformance among all the estimation algorithms with theconsidered pilot placement schemes.

VI. CONCLUSION

In this paper, we have presented an adaptive step sizeSaMP algorithm (AS-SaMP) and an efficient near-optimal pi-lot placement scheme for sparse channel estimation in OFDMsystems. The AS-SaMP algorithm features an adaptive stepsize adjustment strategy and possesses the advantage of not re-quiring a priori knowledge of the sparsity level of the channel.It is shown through performance analysis that the AS-SaMPalgorithm can significantly improve the estimation accuracywithout introducing significant additional complexity. In orderto ensure a satisfactory estimation, we have further proposeda near-optimal pilot placement scheme, which is based onthe concatenated cyclic difference set with an iterative tailsearch (C-CDS with TS). Because the searching space of theproposed method is significantly reduced, its complexity ismuch lower than the iterative procedures in the literature.




Simulation results show that the AS-SaMP algorithm with thenew pilot placement scheme provides a better MSE perfor-mance for the channel estimate, as well as the system BERwhen compared with existing schemes in the literature, withoutsignificantly increasing the computational complexity; thus, itoffers a better trade-off between complexity and performance.For future work, we plan to adapt the AS-SaMP algorithmand the proposed pilot placement scheme for sparse fast time-varying channel estimation.

APPENDIX ARECONSTRUCTION PERFORMANCE OF AS-SAMP

The recovery performance of the proposed AS-SaMP al-gorithm, is based on the theoretical performance of SaMPand subspace pursuit (SP) [47]; therefore, the proofs whichfollow the format in [32], [47] are developed for two cases:exact recovery from noiseless measurements and approximaterecovery from noisy measurements. Before stating Theorem 1for the exact recovery of the AS-SaMP algorithm, we needtwo results summarized in the lemmas below.

Lemma 1. Given an arbitrary K-sparse signal h and thecorresponding measurement Yp = Ah. Let the total numberof stages decided by AS-SaMP be J and si, i ∈ 1, 2, ..., Jbe the step size of the ith stage. If A satisfies the RIPwith parameter δ3KJ

< 0.06 [17], where KJ =∑Ji=1 si

is the estimated sparsity level, the last stage of AS-SaMP isequivalent to SaMP algorithm with estimated sparsity levelKJ , except possibly different contents in the final support setand the observation residual vector.

Proof: During the last stage of AS-SaMP, the final supportset has size KJ . Given the same size of the final support set,both algorithms use the same preliminary and final correlationtest, which returns the KJ indices corresponding to the largestabsolute values of |A†CjYp|, j denotes the iteration index. Theonly differences are in the content of the final support set andthe observation residual vector.

Lemma 2. AS-SaMP guarantees the convergence of the re-covery process. The upper-bounded number of iterations thatAS-SaMP involves is

− log(|hmin|‖ h ‖

)(−1

log(CK1)

+−1

log(CK2)

+ ...+−1

log(CKJ)) + J,

(9)where hmin is the non-zero element with the minimum magni-tude and CKi =

2δ3Ki(1+δ3Ki

)

(1−δ3Ki)3 , i = 1, 2, ..., J , δ3Ki is the RIP

parameter in the ith stage, and Ki is the size of final supportset in the ith stage.

Proof: Lemma 2.1 is introduced which serves as a foun-dation for the proof of Lemma 2.

Lemma 2.1. The energy difference between the signal cap-tured by the final support set from the current iterationand the final support set from the previous iteration, i.e.,‖ hFj ‖2 − ‖ hFj−1 ‖2, decreases as the number of itera-tions increases before the estimated sparsity level reaches thetrue sparsity level.

Proof of Lemma 2.1 is postponed to Appendix B. Similarto SaMP, AS-SaMP takes a finite number of iterations toapproach the sparse estimation. If the algorithm falls intoan infinite loop of a certain stage, the final support setwill repeat and this is in contradiction to the fact that theenergy difference decreases monotonically. Intuitively, AS-SaMP reaches the final estimation with the same estimatedsparsity level faster than SaMP because the most significantentries are reconstructed by selecting a larger number ofcoordinates into the support set during the initial stages. Letthe number of iterations required in the ith stage using the AS-SaMP algorithm be niti , i = 1, 2, ..., J . According to Theorem6 in [47], for each iteration in a particular stage both SaMPand AS-SaMP contain two correlation maximization tests andthe property below holds.

nit1 ≤− log( |hmin|

‖h‖ )

− log(CK1)+ 1,

nit2 ≤− log( |hmin|

‖h‖ )

− log(CK2)+ 1, ...,

nitJ ≤− log( |hmin|

‖h‖ )

− log(CKJ)

+ 1.

(10)

Let the total number of iterations required be ntotal, then

ntotal = nit1 + nit2 + ...nitJ ,

≤ − log(|hmin|‖ h ‖ )(

−1log(CK1)

+ ...+−1

log(CKJ )) + J.

(11)

Moreover, the upper-bounded number of iterations forAS-SaMP is compared with that for AS-SaMP in theCorollary below.

Corollary 1. Provided that A satisfies the RIP with param-eter δ3Ks−AS−SaMP

< 0.06 and δ3Ks−SaMP< 0.06, where

Ks−AS−SaMP and Ks−SaMP are the estimated sparsity levelfor AS-SaMP and SaMP, respectively, the upper-bounded num-ber of iterations for AS-SaMP is smaller than that for SaMP.

The proof of the Corollary 1 is deferred to in Appendix B.Now based on the lemmas above, a sufficient condition forexact reconstruction is drawn in the following theorem.

Theorem 1. (Exact recovery from noiseless measurements):Let Ks−AS−SaMP = sIJ , where sI = s1 is the initial stepsize and J is the total number of stages of the AS-SaMPalgorithm. If the sensing matrix A satisfies the RIP with theparameter δ3Ks−AS−SaMP

< 0.06, the AS-SaMP algorithm isguaranteed to exactly recover h from Yp via a finite numberof iterations.

Proof: Based on Lemma 1 and Lemma 2, when the RIPcondition is satisfied, because the last stage is equivalent toSaMP with estimated sparsity level Ks−AS−SaMP , the AS-SaMP algorithm guarantees exact recovery the target sig-nal after this stage, and it takes finite number of iterationsto reach Ks−AS−SaMP .

Remark: From the Lemma 1, a sufficient condition which isrequired for A to guarantee an exact recovery is δKJ

< 0.06,




where KJ =∑Ji=1 si and si is the step size in the ith stage.

As sI ≥ s2 ≥ · · · ≥ sJ , we have KJ ≤ sIJ . Therefore,a more restrictive requirement of the RIP parameter of Awill be δ3sIJ < 0.06 which is δ3Ks−AS−SaMP

< 0.06. Thesufficient condition for SaMP is more restrictive than SPalgorithm as the estimated sparsity level Ks−SaMP = sdK/sewhere s is the fixed step size in SaMP, is always largerthan the true sparsity level K [32]. Similarly, to compare therestrictiveness of the condition of the AS-SaMP, the values ofKs−SaMP , Ks−AS−SaMP and K needs to be compared. AsdK/Je ≤ sI ≤ sdK/se

J , so K ≤ sIJ ≤ sdK/se and thusK ≤ Ks−AS−SaMP ≤ Ks−SaMP .

Furthermore, because of monotonicity ofδ3K , δ3Ks−AS−SaMP

≤ δ3Ks−SaMP, as a result, if

δ3Ks−SaMP< 0.06, δ3Ks−AS−SaMP

< 0.06 is hold, whichmeans that the requirement of A for AS-SaMP is lessrestrictive than that for SaMP. Moreover, as A is a P × Npartial DFT matrix in our application, and indices of theP pilots are randomly chosen, A satisfies the RIP with anoverwhelming probability provided that

K ≤ C1P

(logN)6, (12)

where C1 depends only on the RIP parameter (by over-whelming probability, it means that the probability is at least1 − N−

1C1 ) [16] and K is the sparsity level of the target

CIR. In fact, (12) expresses the minimum number of pilots(P ≥ K(logN)6

C1) required such that a random subset of A

with average cardinality 3Ks−AS−SaMP satisfies the RIP withhigh probability. Specifically, for P ≥ 8K, the recovery rateis above 90% [16].

The second part is to investigate the approximate recoveryfrom inaccurate measurements of the AS-SaMP algorithm.Two types of inaccurate measurements are considered: oneis subject to noise perturbation and the other one is subject toapproximately sparse signal whose non-significant elementsare comparatively small (but not zero) and noise.

Theorem 2. (Approximate recovery from noisy measure-ments): Consider h ∈ RN as a K-sparse signal,Yp = Ah + Wp ∈ RP as the noisy measurement vectors andWp as a noise vector generated from a Gaussian distributionwith zero mean and variance σ2. If the measurement matrixA satisfies the RIP with parameter δ3Ks−AS−SaMP

< 0.03,the signal approximation h satisfies:

‖ h− h ‖ ≤1 + δ3Ks−AS−SaMP

δ3Ks−AS−SaMP (1− δ3Ks−AS−SaMP )‖Wp ‖

=1 + δ3Ks−AS−SaMP

δ3Ks−AS−SaMP (1− δ3Ks−AS−SaMP )σ

(13)

Corollary 2. (Approximate recovery from signal and noiseperturbations): Consider h ∈ RN as a compressible K-sparsesignal. Let hK represent the K most significant entries. Thesignal h is compressibly sparse if h−hK 6= 0. With the sameassumption of Theorem 2, if A satisfies the RIP with parameterδ6Ks−AS−SaMP

< 0.03, the reconstruction distortion of theAS-SaMP algorithm is written in Eq. (17).

The proofs of the Theorem 2 and Corollary 2 are sim-ilar to the corresponding theorem and corollary in [47] as

the AS-SaMP is equivalent to SaMP with the estimatedsparsity Ks−AS−SaMP at the last stage except for the dif-ferent contents of candidate and final support set whichdoes not affect stability of AS-SaMP under both signal andnoise perturbations.

APPENDIX BPROOF OF LEMMA 2.1 AND COROLLARY 1

A. Proof of Lemma 2.1

The proof is derived from Theorem 2 in [47] be-cause both the preliminary and final test are correlationmaximization tests.

Proof: Provided that the sensing matrix A satisfies theRIP with parameter δ3KJ

< 0.06.

‖ hF

t ‖2 ≤ C2Ki‖ h

Ft−1 ‖2,

≤ C2KJ‖ h

Ft−1 ‖2= ζ ‖ h

Ft−1 ‖2,

(18)

where hF

j is the reconstructed signal not captured by Fj afterthe jth iteration. (18) is based on 0 < CK1

≤ CK2≤ ... ≤

CKJ< 1, and therefore ζ = C2

KJ< 1. Thus, the following

derivation holds,

‖ hF

1 ‖2 −ζ ‖ h ‖2≤ 0 ≤‖ hF

2 ‖2,0 ≤‖ h

F1 ‖2 − ‖ h

F2 ‖2≤ ζ ‖ h ‖2,

(19)

where h = hF

0 . As ‖ hF

1 ‖2=‖ h ‖2 − ‖ hF1 ‖2 and‖ h

F2 ‖2=‖ h ‖2 − ‖ hF2 ‖2, (19) can be written as:

0 ≤ (‖ h ‖2 − ‖ hF1 ‖2)− (‖ h ‖2 − ‖ hF2 ‖2) ≤ ζ ‖ h ‖2,0 ≤‖ hF2 ‖2− ‖ hF1 ‖2≤ ζ ‖ h ‖2 .

(20)

Similarly, we have

0 ≤‖ hF3 ‖2 − ‖ hF2 ‖2 ≤ ζ2 ‖ h ‖2,0 ≤‖ hF4 ‖2 − ‖ hF3 ‖2 ≤ ζ3 ‖ h ‖2, · · ·

0 ≤‖ hFt ‖2 − ‖ hFt−1 ‖2 ≤ ζt−1 ‖ h ‖2 .(21)

As 1 > ζ > ζ2 > ζ3 > ζ4 · · · , the energy difference betweentwo consecutive iterations converges to a small positive valuewhich is related to the RIP parameter.

B. Proof of Corollary 1Proof: Since both the SaMP and AS-SaMP algorithms

use the preliminary and final tests, the upper-bounded numberof iterations in Lemma 2 can also be applied to the SaMPalgorithm. Consider the same target signal for both algorithmsand according to Lemma 2 we have

ntotal ≤ − log(|hmin|‖ h ‖ )(

−1log(CK1)

+ ...+−1

log(CKJ )) + J.

As 0 < δ3K1≤ δ3K2

... ≤ δ3KJ< 0.06, then 0 <

−1log(CK1

) ≤−1

log(CK2) ≤ ... ≤ −1

log(CKJ) . Thus we have

−1log(CK1

) + −1log(CK2

) + ...+ −1log(CKJ

) ≤−J

log(CKJ) , and therefore

ntotal ≤−J log(

|hmin|‖h‖ )

− log(CKJ) + J . With the same target signal and

the total number of stages, the upper bound only dependson CKJ

. According to Appendix A, 0 < δ3Ks−AS−SaMP≤

δ3Ks−SaMP< 0.06, and therefore, CKs−AS−SaMP

≤




‖ h− h ‖≤1 + δ6Ks−AS−SaMP

δ6Ks−AS−SaMP(1− δ6Ks−AS−SaMP

)(σ +

√1 + δ6Ks−AS−SaMP

K|h− hK |) (17)

CKs−SaMP. Clearly, 0 < −1

log(CKs−AS−SaMP) ≤

−1log(CKs−SaMP

)

and−J log(

|hmin|‖h‖ )

− log(CKs−AS−SaMP) + J ≤

−J log(|hmin|‖h‖ )

− log(CKs−SaMP) + J .

For example, given δ3Ks−AS−SaMP= 0.01 which leads to

log(CKs−AS−SaMP) = −5.59 and the total number of stages is

5. Suppose log( |hmin|‖h‖ ) = −7, the upper bound of the number

of iterations that the AS-SaMP algorithm involves is 11. Onthe other hand, with the same h and J , suppose δ3Ks−SaMP

=0.05, the upper bound of the number of iterations for the SaMPalgorithm is 16.

ACKNOWLEDGMENT

The authors would like to thank the Editor, Professor ShiJin, and anonymous reviewers for their valuable feedback.

REFERENCES

[1] L. Hanzo, Y. Akhtman, L. Wang, and M. Jiang, MIMO-OFDM forLTE,WIFI and WIMAX: coherent versus non-coherent and coopera-tive turbo-transceivers. John Wiley and IEEE Press, Dec. 2011.

[2] M. Sablatash, “Transmission of all-digital television: state of the art andfuture directions,” IEEE Trans. Broadcast., vol. 40, no. 2, pp. 102–121,Jun. 1994.

[3] B. Li, S. Zhou, M. Stojanovic, L. Freitag, and P. Willett, “Multicarriercommunication over underwater acoustic channels with nonuniformdoppler shifts,” IEEE J. Oceanic Eng., vol. 33, pp. 198–209, Apr. 2008.

[4] S. Zhou and Z. Wang, OFDM for Underwater Acoustic Communica-tions. John Wiley & Sons, Jun. 2014.

[5] H. Arslan and G. Bottomley, “Channel estimation in narrowband wire-less communication systems,” Wireless Commun. and Mobile Comput.,vol. 1, no. 2, pp. 201–219, Apr. 2001.

[6] S. Coleri, M. Ergen, A. Puri, and A. Bahai, “Channel estimationtechniques based on pilot arrangement in OFDM systems,” IEEE Trans.Broadcast., vol. 48, no. 3, pp. 223–229, Sep. 2002.

[7] W. Bajwa, J. Haupt, A. Sayeed, and R. Nowak, “Compressed channelsensing: a new approach to estimating sparse multipath channels,” Proc.IEEE, vol. 98, no. 6, pp. 1058–1076, Jun. 2010.

[8] I. Fevrier, S. Gelfand, and M. Fitz, “Reduced complexity decisionfeedback equalization for multipath channels with large delay spreads,”IEEE Trans. Commun., vol. 47, no. 6, pp. 927–937, Jun. 1999.

[9] S. Cotter and B. Rao, “Sparse channel estimation via matching pursuitwith application to equalization,” IEEE Trans. Commun., vol. 50, no. 3,pp. 374–377, Mar. 2002.

[10] M. Stojanovic, “Retrofocusing techniques for high rate acoustic com-munications,” J. Acoust. Soc. Am., vol. 117, no. 3, pp. 1173–1185, Mar.2005.

[11] S. Ariyavisitakul, N. Sollenberger, and L. Greenstein, “Tap-selectabledecision-feedback equalization,” IEEE Trans. Commun., vol. 45, no. 12,pp. 1497–1500, Dec. 1997.

[12] G. Gui, W. Peng, and F. Adachi, “Improved adaptive sparse channelestimation based on the least mean square algorithm,” in Proc. IEEE onWireless Commun. and Networking Conf. 2013 (WCNC’13), Shanghai,China, Apr. 2013, pp. 3105–3109.

[13] J. van de Beek, O. Edfors, M. Sandell, S. Wilson, and P. Borjesson,“On channel estimation in OFDM systems,” in Proc. IEEE 45th Veh.Technol. Conf., Chicago, IL, Jul. 1995, pp. 815–819.

[14] C. Carbonelli, S. Vedantam, and U. Mitra, “Sparse channel estimationwith zero tap detection,” IEEE Trans. Commun., vol. 6, no. 5, pp. 1743–1763, May 2007.

[15] C. Berger, S. Zhou, J. Preisig, and P. Willett, “Sparse channel estimationfor multicarrier underwater acoustic communication: from subspacemethods to compressed sensing,” IEEE Trans. Signal Process., vol. 58,no. 3, pp. 1708–1721, Mar. 2010.

[16] E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exactsignal reconstruction from highly incomplete frequency information,”IEEE Trans. Inf. Theory, vol. 52, pp. 489–509, Feb. 2006.

[17] E. Candes and T. Tao, “Near-optimal signal recovery from randomprojections: universal encoding strategies,” IEEE Trans. Inf. Theory,vol. 52, pp. 5406–5425, Dec. 2006.

[18] D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52,no. 4, pp. 1289–1306, Apr. 2006.

[19] R. Baraniuk, “Compresseive sensing,” IEEE Signal Process. Mag.,vol. 24, no. 4, pp. 118–121, Jul. 2007.

[20] “Compressive sensing resources,” Huston, TX,http://www.dsp.ece.rice.edu/cs/.

[21] G. Taubock, F. Hlawatsch, D. Eiwen, and H. Rauhut, “Compressiveestimation of doubly selective channels in multicarrier systems: leakageeffects and sparsity-enhancing processing,” IEEE J. Sel. Topics SignalProcess., vol. 4, no. 2, pp. 255–271, Apr. 2010.

[22] C. Berger, Z. Wang, J. Huang, and S. Zhou, “Application of compressivesensing to sparse channel estimation,” IEEE Commun. Mag., vol. 48,no. 11, pp. 164–174, Nov. 2010.

[23] H. Die, W. Xiaodong, and H. Lianghua, “A new sparse channel esti-mation and tracking method for time-varying OFDM systems,” IEEETrans. Veh. Technol., vol. 62, no. 9, pp. 4648–4653, Nov. 2013.

[24] R. Prasad, C. R. Murthy, and B. D. Rao, “Joint approximately sparsechannel estimation and data detection in OFDM systems using sparsebayesian learning,” IEEE Trans. Signal. Process., vol. 62, no. 14, pp.3591–3603, Jul. 2014.

[25] D. L. Donoho, A. Maleki, and A. Montanari, “Message passing algo-rithms for compressed sensing,” in Proc. Nat. Acad. Sci., vol. 106, Nov.2009, p. 1891418919.

[26] S. Rangan, “Generalized approximate message passing for estimationwith random linear mixing,” in Proc. IEEE Int. Symp. Inform. Thy.,Saint Petersburg, Russia, Aug. 2011, full version at arXiv:1010.5141.

[27] J. Vila and P. Schniter, “Expectation-maximization Bernoulli-Gaussianapproximate message passing,” in Proc. Asilomar Conf. Signals Syst.Comput., Pacific Grove, CA, Nov. 2011, pp. 799–803.

[28] J. Tropp and A. Gilbert, “Signal recovery from random measurementsvia orthogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53,no. 12, pp. 4655–4666, Dec. 2007.

[29] D. Needell and J. Tropp, “CoSaMP: iterative signal recovery from in-complete and inaccurate samples,” Commun. ACM: Research Highlightssection, vol. 53, no. 12, pp. 93–100, Dec. 2010.

[30] T. T. Cai and L. Wang, “Orthogonal matching pursuit for sparse signalrecovery with noise,” IEEE Trans. Inf. Theory, vol. 57, no. 7, pp. 4680–4688, Jul. 2011.

[31] D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomicdecomposition,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2845–2862,Jul. 2001.

[32] T. Do, G. Lu, N. Nam, and T. Tran, “Sparsity adaptive matching pursuitalgorithm for practical compressed sensing,” in Proc. the 42nd AsilomarConf. on Signals, Syst. and Comput., Pacific Grove, CA, Oct. 2008, pp.581–587.

[33] X. Bia, X. Chen, and Y. Zhang, “Variable step size stagewise adaptivematching pursuit algorithm for image compressed sensing,” in Proc.IEEE Int. Conf. on Signal Process., Commun. and Comput. (ICSPCC),Kunming, China, Aug. 2013, pp. 1–4.

[34] Y. Zhang, R. Venkatesan, O. A. Dobre, and C. Li, “An adaptive matchingpursuit algorithm for sparse channel estimation,” in Proc. IEEE onWireless Commun. and Networking Conf. (WCNC), New Orleans, LA,Mar. 2015, pp. 626–630.

[35] X. He, R. Song, and W. Zhu, “Optimal pilot pattern design for com-pressed sensing-based sparse channel estimation in OFDM systems,” J.Circuits, Syst., and Signal Process., vol. 31, no. 4, pp. 1379–1395, Aug.2012.

[36] P. Pakrooh, A. Amini, and F. Marvasti, “OFDM pilot allocation forsparse channel estimation,” EURASIP J. Adv. Signal Process., vol. 2012,no. 59, pp. 1–9, Mar. 2012.

[37] C. Qi and L. Wu, “Optimized pilot placement for sparse channelestimation in OFDM systems,” IEEE Signal Process. Lett., vol. 18,no. 12, pp. 749–752, Dec. 2011.

[38] C. Qi, G. Yue, L. Wu, Y. Huang, and A. Nallanathan, “Pilot designschemes for sparse channel estimation in OFDM systems,” IEEE Trans.Veh. Technol., no. 99, pp. 1–13, Jun. 2014.




[39] J. Haupt, L. Applebaum, and R. Nowak, “On the restricted isometry ofdeterministically subsampled fourier matrices,” in Proc. Conf. Inf. Sci.Syst. (CISS), Princeton, NJ, Mar. 2010, pp. 1–6.

[40] S. Climent, A. Sanchez, J. V. Capella, N. Meratnia, and J. J. Serrano,“Underwater acoustic wireless sensor networks: advances and futuretrends in physical, mac and routing layers,” Sensors, vol. 14, no. 1,pp. 795–833, 2014.

[41] P. Qarabaqia and M. Stojanovic, “Statistical modelling of a shallowwater acoustic communication channel,” in Proc. Underwater AcousticMeasurements Conf., Nafplion, Greece, Jun. 2009, pp. 1341–1350.

[42] G. Tang, N. Bhaskar, P. Shah, and B. Recht, “Compressed sensing offthe grid,” IEEE Trans. Inf. Theory, vol. 59, no. 11, pp. 7465–7490, Nov.2013.

[43] D. Donoho, V. Stodden, and Y. Tsaig, “About sparselab,” Mar. 2007.[Online]. Available: https://sparselab.stanford.edu/

[44] C. Qi and L. Wu, “A study of deterministic pilot allocation for sparsechannel estimation in OFDM systems,” IEEE Commun. Lett., vol. 16,no. 5, pp. 742–744, May 2012.

[45] C. J. Colbourn and J. H. Dinitz, “Other combinatorial designs,” inHandbook of Combinatorial Designs, second edition ed. CRC Press,2007, ch. 5, pp. 392–436.

[46] J. Myers and A. D. Well, in Research Design and Statistical Analysis,2nd ed. Lawrence Erlbaum, 2003.

[47] D. Wei and O. Milenkovic, “Subspace pursuit for compressive sensingsignal reconstruction,” IEEE Trans. Inf. Theory, vol. 55, no. 5, pp. 2230–2249, May 2009.

Yi Zhang received the B.Eng. degree in electronicinformation engineering from Xi’An University ofPost & Telecommunications, Xi’An, P. R. China,in 2008 and the M.ASc. degree in electrical andcomputer engineering from Memorial University,St.John’s, Canada, in 2010. Currently, she is workingtoward the Ph.D. degree at the Faculty of Engineer-ing and Applied Science at Memorial University.

Her research interests are in the areas of wire-less communications, digital signal processing, com-pressed sensing theory and their applications to

underwater acoustic communication systems.

Ramachandran Venkatesan is a Professor of com-puter engineering at Memorial University of New-foundland, Canada, where he has been workingsince 1987. He received his B. E. (Hons.) fromMadurai University and M.Sc.E. and Ph. D. degreesfrom University of New Brunswick, all in ElectricalEngineering. He worked in the industry as a weldingresearch engineer for several years. Dr. Venkatesanhas held several academic administrative positionsincluding Chair of Electrical and Computer Engi-neering, Associate Dean for Graduate Studies and

Research, Associate Dean for Undergraduate Studies, Acting Dean, and DeanPro Tempore of Engineering.

His current research interests include parallel processing architecturesand applications, error control coding, underwater communication, wirelesscommunications and optical communications. He is a registered professionalengineer and a senior member of IEEE.

Octavia A. Dobre is an Associate Professor withthe Faculty of Engineering and Applied Science atMemorial University, Canada. Previously she waswith New Jersey Institute of Technology, USA andPolytechnic Institute of Bucharest, Romania. Shewas the recipient of a Royal Society scholarshipat Westminster University, UK (2000), and held aFulbright fellowship at Stevens Institute of Technol-ogy, USA (2001). Octavia was a Visiting Professorwith Universit de Bretagne Occidentale, France,and Massachusetts Institute of Technology (MIT),

USA (2013).Her research interests comprise blind signal identification and parameter

estimation techniques, cognitive radio systems, spectrum sensing techniques,transceiver optimization algorithms, dynamic spectrum access, and coop-erative wireless communications, as well as optical communications. Shepublished +170 journal and conference papers in these areas, gave +40 invitedand keynote talks to industry and academia, and prepared +30 technicalreports for the industry partners. Octavia’s research has been supported bythe Natural Sciences and Engineering Research Council of Canada (NSERC),Mathematics of Information Technology and Complex Systems (MITACS),Canada Foundation for Innovation (CFI), Research and Development Corpo-ration (RDC), Atlantic Canada Opportunities Agency (ACOA), Defence andResearch Development Canada (DRDC), Communications Research Centre(CRC) Canada, Altera Corporation, DTA Systems, ThinkRF, Agilent Tech-nologies, and US Army Research Development and Engineering Command(RDECOM).

Octavia serves as Editor-in-Chief of the IEEE Communications Letters(2016-2017), as well as Editor for the IEEE Transactions on WirelessCommunications and IEEE Communications Surveys and Tutorials. She wasan Editor and Senior Editor for the IEEE Communications Letters, and GuestEditor for the IEEE Communications Magazine and IEEE Journal of SelectedTopics on Signal Processing. She has served as the General Chair of the IEEECWIT, and Technical Co-Chair of symposia at numerous conferences, such asIEEE VTC, IEEE GLOBECOM, CrownCom, IEEE WCNC, and IEEE ICC.She is the Chair of the Women in Communications Engineering ComSocstanding committee, and a registered Professional Engineer in the province ofNewfoundland and Labrador, Canada.

Cheng Li received the B.Eng. and M.Eng. de-grees from Harbin Institute of Technology, Harbin,P. R. China, in 1992 and 1995, respectively, andthe Ph.D. degree in Electrical and Computer Engi-neering from Memorial University, St.John’s, NL,Canada, in 2004.

He is currently a full professor at the Departmentof Electrical and Computer Engineering, Faculty ofEngineering and Applied Science, Memorial Univer-sity, St. John’s, NL, Canada. His research interestsinclude mobile ad hoc and wireless sensor networks,

wireless communications and mobile computing, switching and routing,and broadband communication networks. He was a recipient of the bestpaper award at the 2010 IEEE International Conference on Communications(ICC’10), Cape Town, June 2010. He is an editorial board member of WileyWireless Communications and Mobile Computing, an associate editor ofWiley Security and Communication Networks, and an editorial board memberof Journal of Networks, and KSII Transactions on Internet and InformationSystems. He has served a technical program committee (TPC) co-chair forthe ACM MSWIM’14, MSWIM’13, IEEE WiMob’11 and QBSC’10. He hasserved as a co-chair for various technical symposia of many internationalconferences, including the IEEE GLOBECOM, ICC, WCNC, and IWCMC.He has also served as the TPC member for many international conferences,including the IEEE ICC, GLOBECOM, and WCNC. Dr. Li is a registeredProfessional Engineer (P.Eng.) in Canada and is a Senior Member of theIEEE and a member of the IEEE Communication Society, Computer Society,Vehicular Technology Society, and Ocean Engineering Society.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1 Novel Compressed Sensing-based Channel...

Documents

Transcript of IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1 Novel Compressed Sensing-based Channel...