compressive sensing based channel estimation for massive...

Research ArticleCompressive Sensing Based Channel Estimation forMassive MIMO Communication Systems

Athar Waseem 1 Aqdas Naveed1 Sardar Ali2 Muhammad Arshad3

Haris Anis1 and Ijaz Mansoor Qureshi4

1 International Islamic University Islamabad Pakistan2University of Buner Pakistan3University of Peshawar Pakistan4AIR University Islamabad Pakistan

Correspondence should be addressed to Athar Waseem atharwaseemiiuedupk

Received 9 November 2018 Revised 13 March 2019 Accepted 21 April 2019 Published 27 May 2019

Guest Editor Sungchang Lee

Copyright copy 2019 Athar Waseem et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Massive multiple-input multiple-output (MIMO) is believed to be a key technology to get 1000x data rates in wirelesscommunication systems Massive MIMO occupies a large number of antennas at the base station (BS) to serve multiple usersat the same time It has appeared as a promising technique to realize high-throughput green wireless communications MassiveMIMO exploits the higher degree of spatial freedom to extensively improve the capacity and energy efficiency of the systemThusmassive MIMO systems have been broadly accepted as an important enabling technology for 5th Generation (5G) systems Inmassive MIMO systems a precise acquisition of the channel state information (CSI) is needed for beamforming signal detectionresource allocation etc Yet having large antennas at the BS users have to estimate channels linked with hundreds of transmitantennas Consequently pilot overhead gets prohibitively high Hence realizing the correct channel estimation with the reasonablepilot overhead has become a challenging issue particularly for frequency division duplex (FDD) in massive MIMO systems In thispaper by taking advantage of spatial and temporal common sparsity of massive MIMO channels in delay domain nonorthogonalpilot design and channel estimation schemes are proposed under the frame work of structured compressive sensing (SCS) theorythat considerably reduces the pilot overheads formassiveMIMOFDD systemsTheproposed pilot design is fundamentally differentfrom conventional orthogonal pilot designs based onNyquist sampling theorem Finally simulations have been performed to verifythe performance of the proposed schemes Compared to its conventional counterparts with fewer pilots overhead the proposedschemes improve the performance of the system

1 Introduction

Multiple-input multiple-output (MIMO) systems have mul-tiple antennas at both the transmitter and receiver ends Byaddition of multiple antennas higher degree of freedom inwireless channels (in terms of time and frequency dimen-sions) can be obtained in order to achieve target of high datarates For this reason major performance progress can beattained in terms of system reliability and both spectral andenergy efficiency And also such higher degrees of freedomcan be exploited using beamforming given that channelstate information is available There are large number ofantenna elements (around tens or even hundreds) deployed

at both sides the transmitter and receiver It is importantto note that the transmit antennas may be distributed orcolocated in different applications Also the huge numberof receive antennas can be acquired by one device or dis-tributed to many devices [1 2] Additionally massive MIMOsystems help in minimizing the effects of noise and fastfading and also intracell interference can be reduced usingstraightforward detection and linear precoding methods Byappropriately implementing multiuser MIMO (MU-MIMO)in massive MIMO systems the design of medium accesscontrol (MAC) layer can be more simplified by getting ridof complicated scheduling algorithms [2] With MU-MIMOsignals to individual users can be easily separated using the

HindawiWireless Communications and Mobile ComputingVolume 2019 Article ID 6374764 15 pageshttpsdoiorg10115520196374764

2 Wireless Communications and Mobile Computing

same time-frequency resource at base station Thereforethese main advantages make it feasible to introduce themassive MIMO system as a promising candidate for 5Gwireless communication networks

One of the major issues in massive MIMO systems isthe accurate acquisition of the channel state information(CSI) for beamforming resource allocation signal detectionetc Due to large antennas placed at the BS the estimationof channels linked with hundreds of transmit antennas isrequired at users which results in high pilot overhead Hencethe precise channel estimation with the low pilot overhead isa challenging task [2 3]

There are several research challenges which are requiredto be solved beforemassiveMIMOcanbe fully integrated intofuture wireless systems [4 5]

(i) Beamforming will involve a huge amount of channelstate information which will be challenging for thedownlink

(ii) MIMO can be unfeasible for FDD systems but can beemployed in time division duplex (TDD) systems dueto having channel reciprocity

(iii) Conventional channel estimation approaches requirea large pilot and feedback overhead which typicallyscales proportionally with number of BS transmitantennas which results in unfeasible condition forlarge-scale FDDMIMO systems

(iv) In addition massive MIMO experiences pilot con-tamination from other cells if the transmit power ishigh otherwise it will be affected by thermal noiseMethods will be required to solve this in order todeliver the promised performance

(v) There is a need for channel models of massiveMIMO systems without which it will be difficult forresearchers to perfectly validate the algorithms andtechniques

(vi) For channel estimation TDD scenarios are onlyconsidered for massive MIMO due to the excessivecost of channel estimation and feedback Even forTDD toworkmassiveMIMOchannel calibration canprove to be a big achievement New methods andschemes will be needed for the purpose of channelestimation in massive MIMO systems

(vii) Extremely fast processing algorithms will be requiredfor processing the massive amount of data from theRF chains

The above-mentioned challenges have been addressed up tosome extent in the last few years To date many researchesonmassiveMIMO avoided the challenge of considering FDDsystems by simply assuming the TDD protocol The uplinkCSI can be more easily obtained at the BS due to the lessnumber of single-antenna users and the strong capabilityof processing at BS And then by leveraging the channelreciprocity property the CSI in the downlink can be directlytacked [6 7] However due to the fact that radio frequencychains suffer from calibration error and restricted coherence

time the CSI obtained in the uplink may not be correctfor the downlink [8 9] Additionally FDD systems have lowlatency as compared to TDD therefore the communicationis more efficient [10] Thus it is of significance to discoverthe challenging issue of channel estimation for FDD massiveMIMO systems which can assist massive MIMO to bebackward attuned with existing FDD dominated cellularnetworks There has been extensive analysis on channelestimation for traditional small-scale FDD MIMO systems[11 12] It has been established that the equally spaced andequally powered orthogonal pilots can be ideally suitableto estimate the noncorrelated Rayleigh MIMO channels forsingle OFDM symbol where the pilot overhead requiredincreases as the number of transmit antennas increases [11]The pilot overhead to estimate Rician MIMO channels canbe compacted by exploiting the spatial correlation of MIMOchannels [13 14] Furthermore by taking advantage of thetemporal channel correlation more reduced pilot overheadcan be attained to estimateMIMOchannels linked tomultipleOFDM symbols [15ndash17] Presently orthogonal pilots havebeen extensively introduced in the current MIMO systemsdue to small number of transmit antennas (ie up to eightantennas in LTE-Advanced system) they have reasonablepilot overhead [12 18] Though this is a critical issue inmassive MIMO systems due to massive figure of antennasat the BS (ie up to 128 antennas or even more at the BS[19]) An approach of exploiting the temporal correlationand the delay domain sparsity of channels for achievingthe reduced pilot overhead has been presented for FDDmassive MIMO systems in [20 21] but with large numberof transmit antennas the interference cancellation of pilotsequences of different transmit antennas will be difficult In[22ndash25] the spatial correlation in sparsity of delay domainMIMO channels is utilized to estimate channels achievingthe reduced pilot overhead but the assumption level of theknown channel sparsity to the user is impractical In [26ndash28] the compressive sensing (CS) based channel estimationschemes are presented by exploiting the spatial channel cor-relation however due to having nonideal antenna array theleveraged spatial correlation can be impaired [2 8] In [29] anovel doubly selective channel estimation scheme forMIMO-OFDM systems is proposed with both time and frequency-domain training (TFDT) based on structured compressivesensing (SCS) In [30] a new sparse channel model basedon dictionary learning is proposed which adapts to thecell characteristics and promotes a sparse representationThe learned dictionary is able to more robustly and effi-ciently represent the channel and improve downlink channelestimation accuracy Furthermore observing the identicalAOAAOD between the uplink and downlink transmissiona joint uplink and downlink dictionary learning and com-pressed channel estimation algorithm is proposed to performdownlink channel estimation utilizing information from thesimpler uplink training which further improves downlinkchannel estimation In [31] a channel estimator based onblock distributed compressive sensing (BDCS) is proposedfor the large-scale MIMO systems BDCS exploits structuredsparsity to reduce the pilot overhead In [32] a new way toestimate sparse channels by construction of beamforming

Wireless Communications and Mobile Computing 3

dictionary matrices is presented Continuous basis pursuit(CBP) algorithm that exploits the sparse nature of channelsto adaptively estimate the multipath mmWave channels isproposed In [33] an open-loop and a closed-loop channelestimation scheme for massive MIMO are presented but thechannel statistics cannot be perfectly known to the user inthe long term whereas in conventional broadband wirelesscommunication systems delay domain channels basicallyexhibit the sparse nature due to the large channel delayspread and having limited number of major scatterers in thepropagation environments [21 34]

In the meantime MIMO communication systems havesimilar scatterers in the transmission environment thereforeBS channels associated with one user and several transmitantennas experience similar type of path delays which showsthat these delay domain channels share common sparsityspecially in the case of not having very large aperture ofantenna array [2 35] Furthermore throughout the coher-ence time such sparsity is nearly unchanged which is dueto the fact that path delays fluctuate at very slow rate ascompared to path gains because of temporal correlationof channels [36] In [37] CS based block sparsity adaptivematching pursuit (BSAMP) algorithm is proposed whichtargets the estimation of channels of MIMO system withunknown number of channels paths The BSAMP exploitsthe joint sparse nature of massive MIMO channels In [38]CS based probability-weighted subspace pursuit (PWSP)algorithm is proposed which exploits the probability infor-mation attained from previously estimated CIRs to recoverthe uplink channels in massive MIMO scenario In [39]such properties of channels in MIMO are considered as thespatiotemporal common sparsity which is generally ignoredin current work Finally a general idea of the CS methodis presented in [40] in which fundamental setup recoverytechniques and guarantee of performance are discussedFurthermore different subproblems ofCS ie sparse approx-imation identification of support and sparse identificationare discussed with respect to some applications of wirelesssystems Design issues of wireless systems limitations andpotentials of CS algorithms useful tips and prior informationare also discussed to achieve maximum performance In [40]a valuable guidance is provided for researchers working in thearea of wireless communication systems Before getting intothe detail of spatiotemporal common sparsity we present anoverview of compressive sensing

In this paper we propose a CS based efficient nonorthogo-nal pilot scheme for massive MIMO communication systemsby exploring the temporal and spatial sparsity of massiveMIMO channels And then we propose a channel estimationscheme ie sparsity update CoSaMP (SUCoSaMP) whichexploits the temporal and spatial sparsity of massive MIMOchannelsThe proposed pilot scheme is significantly differentfrom the conventional schemes and substantially reduces thepilot overhead The proposed pilot scheme employs fullyidentical subcarriers for pilots of several transmit antennasin a specific antenna group The antennas placed at basestation (BS) are subdivided into groups based on the obser-vation that the coherence time of path gains is inverselyproportional to the system carrier frequency whereas the

variation in path delays is inversely proportional to the signalbandwidthTherefore the decision of making antenna groupsand determining the number of antennas to be included inone antenna group is taken according to the given systemparameters ie systems frequency system bandwidth andantenna spacing at BS Furthermore considering the antennaarray geometry of BS the proposed nonorthogonal pilotscheme is a space-time adaptive pilot scheme that adaptivelychanges its design according to the given system parametersThe proposedCS algorithmbased channel estimation schemeSUCoSaMP considers the initial sparsity level as 1 andthen regularly updates the sparsity level until the stoppingcriteria are met or a correct sparsity level is achieved in ascenario where sparsity level is unknown Compared withthe conventional CS algorithms SP and OMP and with otheravailable CS based algorithms the proposed CS based pilotand channel estimation scheme is tested through simulationson systems with different parameters It was verified thatthe proposed schemes provided improved channel estimationperformance

The organization of the paper is as follows Sections 2 and3 present the details about delay domain spatial and tem-poral sparsity of massive MIMO communication channelsrespectively Section 4 presents the proposed nonorthogonalpilot scheme based on CS theory Section 5 illustrates the CSbased massive MIMO channel estimation scheme Section 6provides the simulation results And in Section 7 we presentthe conclusion

Notation Uppercase and lowercase boldface letters representthe matrices and vectors respectively The matrix transposeinversion and conjugate transpose are denoted by ()119879 ()minus1and ()lowast respectivelyTheMoore-Penrose inversion ofmatrixis denoted by () The 1198972-norm operation and Frobenius-norm operation are denoted by 2 and 119865 respectivelyThe cardinality of a set and the j-th column vector of thematrix120595 are denoted by ||119862 and120595(119895) respectively Finally thecomplementary set of the set 119879 and the integer floor operatorare denoted by 119879119862 and lfloorrfloor respectively2 Delay Domain Spatial Sparsity

Considerable experimental researches have discovered thatin delay domain massive MIMO channels demonstratespatial sparsity This is due to the fact that the number ofsignificant scatterers is limited in wireless communication infading environments while in communication between basestation (BS) and users the transmission distance is very largeas compared to the distance between several antennas in anantenna array placed at BSThat is to say CIRs associatedwithseveral transmit antennas and one user exhibit similar pathdelays therefore they also share identical common supportof CIRs

Consider a massive MIMO-OFDM system where 119872transmit antennas are placed at BS The CIR between m-thtransmit antenna and one single-antenna user for z-thOFDMsymbol is expressed by

119889119911119898 = [119889119911119898 (1) 119889119911119898 (2) 119889119898119903 (119871)]119879 (1)


User 1 User 2

(a)

CIRs of 1st antenna group

Different tr

ansmit

antennas in

1st anten

na

groupDiffere

nt transm

it

antennas in

1st anten

na

group

Diff

eren

t Ant

enna

Gro

ups

Diff

eren

t Ant

enna

Gro

ups

Time(different OFDM symbols within coherence time for user 2)



Different tr

ansmit

antennas in

NＮＢ

antenna

groupDiffere

nt transm

it

antennas in

NＮＢ

antenna

group

CIRs of ＮＢ antenna groupCIRs of

ＮＢ antenna group

(b)

Figure 1 Delay domain spatial and temporal sparsity of massive MIMO channels (a) limited number of scatterers and common scatterersin wireless communication scenario (b) spatial and temporal sparsity of massive MIMO channels in delay domain between two users andcolocated antenna array

where 1 le 119898 le 119872 and 119871 is the channel length equivalentto the maximum delay spread Let 119878119911119898 be the sparsity levelof CIR between one transmit-receive antenna pair ie thenumber of nonzero elements in 119889119911119898 the support of 119889119911119898 canbe expressed as119875119911119898 = supp 119889119911119898 = 119897 1003816100381610038161003816119889119911119898 [119897]1003816100381610038161003816 gt 0

with 1 le 119897 le 119871 (2)

where 119878119911119898 = |119875119911119898|119862 fulfilling 119878119911119898 ≪ 119871 And due to spatialsparsity we have the following

1198751199111 = 1198751199112 = sdot sdot sdot = 119875119911119872 (3)The delay domain spatial sparsity with specific system param-eters will be detailed in Section 5

Figure 1 shows the common sparse pattern of CIRs fordifferent transmit-receive antenna pairs

3 Delay Domain Temporal Sparsity

In [39] the authors have discovered that fast time-varyingchannels illustrate temporal correlation That is to say thevariations in path gains are significant whereas the pathdelays are almost invariant for various consecutive OFDMsymbols The reason is that path delays variation durationover time-varying channels is inversely proportional to thesignal bandwidth while the coherence time of path gainsis inversely proportional to carrier frequency of the system[28]


Consider a system with signal bandwidth B = 10MHzand carrier frequency of system 119891119888 = 2GHz the path delaysvary much slower than the path gains [11] Therefore due tothe nearly invariant path delays the CIRs share the commonsparsity for R adjacent OFDM symbols over the coherencetime of path delays The supports of CIRs associated with Rsuccessive OFDM symbols signify the following

1198751119898 = 119875119911119898 = sdot sdot sdot = 119875119877119898 1 le 119898 le 119872 (4)

Figure 1 demonstrates the temporal sparsity of massiveMIMO channels

4 Proposed Nonorthogonal Pilot Scheme

In this section we shall propose a nonorthogonal pilotscheme based on the above-mentioned two observationsFirst the basic idea to divide the antennas placed at BS intosubgroups is proposed The proposed scheme of dividingthe antennas into subgroups is based on system parametersand the temporal and spatial sparsity of massive MIMOchannels Then after creating the antenna groups a specificnonorthogonal pilot scheme is proposed

41 Proposed Scheme for Creating the Antenna Groups Therewill be the uniform distribution of antennas to be includedin subgroups Consider a system with signal bandwidth Bsystem carrier frequency 119891119888 total number of antennas placedat the BS in a uniform linear antenna array M number ofsubgroups 119873119892 and number of antennas in one group 119872119899119892Themaximum resolvable distance119863119898119886119909must be smaller thanc10B [38] that is to say to have two channel taps resolvablethe time interval of arrival must be less than 110B where c isthe speed of light [38]The two successive antennas are spacedby the distance 119889119898 = 1205822 therefore the maximum distancebetween two antennas can be 119889119879 = 119889119898(119872 minus 1) and themaximum distance between two antennas in a single groupcan be 119889119872119899119892 = 119889119898(119872119899119892 minus 1) In order to guarantee the spatialsparsity and based on the condition that119863119898119886119909119888 lt 110119861 the119889119872119899119892 must satisfy 119889119872119899119892 lt 119863119898119886119909

And the formula for number of antennas 119872119899119892 in eachsubgroup119873119892 can be derived as follows

119889119898 (119872119899119892 minus 1) le 11988810119861 (5)

1205822 (119872119899119892 minus 1) le

11988810119861 (6)

119872119899119892 le 1198885119861 (119888119891119888) + 1 (7)

which clearly shows that 119872119899119892 is a function of 119861 and 119891119888Therefore the119873119892 can be given by the following

119873119892 = lfloor 119872119872119899119892rfloor + 120572 (8)

The constant 120572 represents an integer to make sure theuniform distribution of antennas in each subgroup ie

1-D Antenna Array at the BS

The 1st antenna group

1ＭＮ TA ndash Mngf TA M ndash Mngf+1 TA -MTA

The ＮＢ antenna group

Figure 2 1D antenna array at the BS

119872 is completely divisible by 119873119892 Similarly actual numberof antennas 119872119899119892119891 in each group can be calculated by thefollowing

119872119899119892119891 = 119872119873119892 (9)

The formula for 119873119892 will ensure the spatial sparsity for anymassive MIMO system with large 1D antenna array placedat BS Moreover since the 119872119899119892 is a function of systemparameters B and 119891119888 if the B and 119891119888 are changed the119873119892 stillensures the spatial sparsity for antennas And also 119873119892 is theminimumnumber of subgroups a systemmust have to ensurespatial sparsity moreover by increasing119873119892 spatial sparsity ofsystemwill remain preserved For example consider a systemwith the following specifications

M = 128 1D antenna array 119891119888 = 2GHz B = 20MHz119872119899119892 given in (7) can be calculated as 21 119873119892 will be equalto 8 with 120572 = 2 and 119872119899119892119891 will be equal to 16 Based onthese calculations the initial condition is satisfied that is119889119872119899119892 lt 119863119898119886119909 hence the spatial sparsity is preserved for thesystem

The119873119892 antenna groups are shown in Figure 2 where TAdenotes the transmit antenna

In Section 42 first we will derive a massive MIMOsystemmodel And then specific pilot designwill be proposedin Section 43

42 Massive MIMO System Model Consider a massiveMIMO-OFDM system with M transmit antennas placed atBS communicating with one user 120585119899 represents the index setof pilot subcarriers for n-th antenna group the choice of 120585119899will be detailed in Section 43) There are total119873 subcarriersin one OFDM symbol out of which119873119901 corresponds to pilotsubcarriers and the pilot sequence of m-th transmit antennain n-th antenna group is denoted by smn isin C119873119901x1 yzn isinC119873119901x1 is the received vector of the pilot sequence of z-thOFDM symbol of n-th antenna group at user for119873119892 antennagroups yzn can be expressed as

yzn =119872119899119892119891sum119898=1

diag smn R|120585119899n [ dmzn0(119873minus119871)times1

] + wzn (10)

with 1 lt n lt 119873119892yzn =

119872119899119892119891sum119898=1

Smn RL1003816100381610038161003816120585119899n dmzn + wzn


= 119872119899119892119891sum119898=1

120595mndmzn + wzn

(11)

where R isin C119873x119873 is the Discrete Fourier Transform matrix(DFT) yzn is expressed after exclusion of guard intervaldmzn isin C119871x1 is the CIR vector of mth transmit antenna ofn-th antenna group for zth OFDM symbol R|120585119899n isin C119873119901x119873

is a submatrix comprised of 119873119901 rows selected according to120585119899 RL|120585119899n isin C119873119901x119871 contains the first L columns of R|120585119899 n isinC119873119901x119873 Smn = diagsmn isin C119873119901x119873119901 120595mn = SmnRL|120585119899n isinC119873119901x119871 andwzn represents theAdditiveWhiteGaussianNoiseof n-th antenna group for z-th OFDM symbol

The equation can be rearranged as

yzn = 120595ndzn + wzn (12)

where 120595n = [1205951n1205952n 120595119872119899119892119891 n] isin C119873119901times119872119899119892119891119871 and theaggregate CIR vector of n-th antenna group is given by dzn =[d1zn d2zn d119872119899119892119891 zn]T isin C119872119899119892119891119871times1 As explained earlierthe CIR vector dmzn exhibits spatial and temporal sparsitytherefore dzn is also a sparse signalThe system in equation isan underdetermined system due to the fact that119873119901 lt 119872119899119892119891119871and cannot be reliably solved using the traditional channelestimation methods [39]

Moreover the equation can be rearranged into structuredsparse form according to [39] as

yzn = Anhzn + wzn (13)

where hzn = [hT1zn hT2zn dTLzn]T isin C119872119899119892119891119871times1 is a struc-tured sparse equivalent CIR vector hlzn = [1198891zn[119897] 1198892zn[119897] 119889119872119899119892 zn[119897]]T for 1 le 119897 le 119871 and after reformulation 120595n canbe converted into An where An can be written as

An= [A1nA2n ALn] isin C119873119901times119872119899119892119891119871 (14)

where A119897119899 = [120595(119897)1119899120595(119897)2119899 120595(119897)119872119899119892119891 119899] = [1198601119899119897 1198602119899119897 A119872119899119892119891 119899119897] isin C119873119901times119872119899119892119891

After converting to structured form we will have 119873119892equations to be solved simultaneously corresponding to eachsubgroupThe received pilot vectors of zth OFDM symbol for119873119892 subgroups can be expressed as follows

yz1 = A1hz1 + wz1

yz2 = A2hz2 + wz2

yzn = Anhzn + wzn

yz119873119892 = A119873119892 hz119873119892 + wz119873119892

(15)

The system equation of each subgroup exhibits structuredsparsity which provides the motivation to apply CS theoryto recover the high dimension structured sparse signal hz119873119892from the low dimension received pilot vector yz119873119892 TheCS based sparse signal recoverychannel estimation will beexplained in Section 5

43 Proposed Pilot Design Mostly wireless communicationsystems detect the data with the help of pilot signals Specif-ically the purpose of pilot signals is to assist the receiverin estimating the wireless channels and then the receivercoherently detects the data on the basis of estimated channel[41]

In massive MIMO communication systems due to largenumber of transmit antennas at BS the number of channelsgets prohibitively high and thus results in high pilot overheadfor estimation of these channels Therefore there is a need formethods to reduce the high pilot overhead inmassiveMIMOcommunication systems to achieve the target of high datarate

In this research paper CS theory is applied to reduce thehigh pilot overhead based on the fact that the wireless chan-nels undergo spatial and temporal sparsity CS based pilotdesign significantly reduces the pilot overhead as compareto conventional pilot design Figure 3 demonstrated the pro-posed uniformly distributed and identical pilot subcarrierfor multiple antennas while Figure 4 shows the conventionpilot design (ie orthogonal pilots for different antennas)The proposed pilot design allows the system to provide moreresources to data

The CS based pilot design permits the antennas of eachsubgroup to occupy identical subcarriers for pilots within agroup while pilot subcarriers for each subgroup are com-pletely different from each other as shown in Figure 3 Thedesign is based on the choice of 120585119899 The guard interval can becalculated by119873119866 = 119873119861x(maximum channel delay spread)Consider a set119874 = 1 2 119873minus119873119866 The 120585119899 is a subset of119874represents the index of pilot subcarriers for n-th subgroupand is identical for all the antennas within that group theformula for creating the 120585119899 can be given by

120585119899 = 119899 + 119904119901119886119888119894119899119892 lowast 119902 lfloor 119872119872119899119892rfloor minus 120572 le 119873119892le 119904119901119886119888119894119899119892 0 lt 119902 lt 119873119901 minus 1 and 119904119886119901119888119894119899119892= lfloor119873 minus119873119866119873119901 rfloor ge lfloor 119872119872119899119892rfloor minus 120572

(16)

where119873119901 gt 119871 and by solving the inequality lfloor(119873minus119873119866)119873119901rfloor gelfloor119872119872119899119892rfloor minus 120572 we have the following

119872119899119892119891 le 119873119901 le (119873 minus 119873119866)(119872 minus 120572119872119899119892)119872119899119892 (17)

There are 119873119866 unique sets associated with 120585119899 ie 1205851 120585119899 120585119873119892 There will be total 119873119892119873119901 subcarriers occupied


The 1st antenna group The 2nd antenna group

T x 1 T x 2

T x 1 Pilot

T x 2 Pilot

Null Pilot

Data

Freq

uenc

y

T x Mngf T x MT x M-Mngf + 2T x M-Mngf + 1T x 2 MngfT x Mngf + 2T x Mngf + 1


T x Mngf Pilot

T x Mngf +1 Pilot

T x Mngf + 2 Pilot

T x Mngf Pilot

T x M - Mngf + 1 Pilot


T x M Pilot

Figure 3 Proposed pilot design

for pilot vectors for 119872 transmit antennas at BS Figure 3demonstrates the proposed pilot design119873119901 = |120585119899|119862 is the number of subcarriers for pilot vector inOFDM symbol of nth subgroup

Pilot subcarriers of nth group are the null pilots for all theother119873119892 minus 1 subgroups whereas119873minus119873119866 minus119873119892119873119901 subcarriersare available for data

Pilot overhead ratio is defined by 120573119901 = 119873119892119873119901119873Figure 4 shows the conventional pilots in which different

pilots are allocated to different antennas resulting in very highpilot overhead

5 Massive MIMO Channel Estimation Basedon Compressive Sensing

The basic idea presented by CS theory is to recover a signalwhich is sparse in somedomain fromextremely small amount

of nonadaptive linear measurements by applying convexoptimization In a different opinion it relates the preciserecovery of a sparse vector of high dimension by reducingits dimension From another point of view the problem canbe considered as calculation of a signalrsquos sparse coefficientwith respect to an overcomplete systemThe concept of com-pressed sensing was primarily applied for random sensingmatrices which allow for a reduced amount of nonadaptivelinear measurements These days the idea of compressedsensing has been generally replaced by sparse recovery

In (13) it can be seen that hzn demonstrates structuredsparsity in delay domainwhere hzn is 119878119911119898119899-sparse vector dueto

119875119911119898119899 = supp hzn = 119897 10038161003816100381610038161003816ℎ119911119899 [119897]10038161003816100381610038161003816 gt 0with 1 le 119897 le 119871 (18)


T x 1 T x 2 T x M

T x 1 Pilot

T x 2 Pilot

T x M

Null Pilot

Data

Freq

uenc

y

Figure 4 Conventional orthogonal pilot design

where 119878119911119898119899 = |119875119911119898119899|119862 fulfilling 119878119911119898119899 ≪ 119871 And due tospatial sparsity we have the following

1198751199111119899 = 1198751199112119899 = sdot sdot sdot = 119875119911119872119899 (19)

The desired small correlation ofAn according to CS theory isachieved which is described in [39] therefore reliable sparserecovery is guaranteed It is further shown in [39] that anytwo columns of An attain excellent cross correlation betweenthem according to RMT since pilot design proposed in [39]is the special case of proposed pilot design Therefore theproposed pilot design is simple to employ and also supportivein terms of compatibility with current wireless networks[39]

For 119870 adjacent OFDM symbols having identical patternof pilots we have

Yn = AnHn +Wn (20)

where Yn = [yzn yz+1n yz+119870minus1n]C119873119901times119870

The An derived in massive MIMO model satisfies theStructure Restricted Isometric Property (SRIP) conditionSpecifically SRIP can be given by the following

radic1 minus 120575 100381710038171003817100381710038171003817A119873119892100381710038171003817100381710038171003817119865 le 10038171003817100381710038171003817Anhzn10038171003817100381710038171003817119865radic1 minus 120575 100381710038171003817100381710038171003817A119873119892100381710038171003817100381710038171003817119865 (21)

The definition and justification for SRIP are discussed indetail in [39] where 120575 isin [0 1)

Our aim is to recover hzn given yzn Under the frame-work of CS theory the massive MIMO channels hzn areestimated by means of the following problem

h = arg min 10038171003817100381710038171003817hzn100381710038171003817100381710038171st 10038171003817100381710038171003817yzn minus Anhzn

100381710038171003817100381710038172 lt 120598(22)

where 120598 is the noise variance There are many algorithmsthat can solve the problem For example projected gradientmethods or interior point methods can be used for applyingconvex optimization A famous greedy algorithm is orthogo-nal matching pursuit (OMP)


Input Sensing matrix A119899 and noisy measurement vector yznOutput An sparse estimation h of channelsdmzn119898=119872119899119892119891 119899=119873119892119898=1 119899=1

Step 1 (Initialization)1 h0 larr997888 0Trivial initial approximation2 119907larr997888 yznCurrent samples = input samples3 119896 larr997888 0Iterative index4 119904 larr997888 1Initial sparsity levelStep 2 Solve the structure sparse vector hzn to (15)Repeat1 119896 larr997888 119896 + 12 119910larr997888 AlowastnkMake the signal proxy3 Ω larr997888 supp(y2s)Identify large components4 T larr997888 Ω cup supp(h119896minus1)Merge supports5 ℎ|119879 larr997888 119860119899119879yznSignal estimation by least squares6 ℎ|119879119862 larr997888 07 h119896 larr997888 ℎ119904Prune to get next approximation8 119907larr997888 yzn minus A119899h119896Update the current samplesif 1199071198962 lt 119907119896+129 Iteration with fixed sparsity levelelse10 Update sparsity level h119904 larr997888 h119896 119907119904 larr997888 119907119896 119904 larr997888 119904 + 1end ifUntil stopping criterion trueStep 3Obtain channels ℎ = h119904minus1 and obtain estimation of channels dmzn119898=119872119899119892119891 119899=119873119892119898=1 119899=1 according to (11)-(15)

Algorithm 1 Proposed SUCoSaMPrecovery algorithm

For channel estimation purpose we propose theSUCoSaMP algorithm derived from basic CoSaMP asdescribed in Algorithm 1 There will be 119873119892 similar parallelprocessing required for estimating the massive MIMOchannels with 119873119892 sub-antenna groups ie the samealgorithm will be working simultaneously with user toestimate channels of119873119892 subgroups

There are many natural approaches of stopping the algo-rithmWe follow the following stopping criterion if 119907119896+12 gt119907119904minus12 the iteration is stopped [39] The information ofcorrect sparsity level 119878119911119898119899 is usually not available and alsoit is practically not possible to have prior knowledge ofcorrect sparsity level whereas information about sparsitylevel plays a significant role in compressive sensing problemof solving underdetermined system and it is also requiredas prior information by most of the CS based algorithmsThe proposed SUCoSaMP algorithm does not require priorinformation of sparsity level because it adaptively acquires thesparsity level and avoids the unrealistic assumption of havingprior information of correct sparsity level

In steps 21 to 29 the target of SUCoSaMP algorithm isto obtain the solution hzn to (15) with fixed sparsity levels similar to conventional CoSaMP The condition 1199071198962 le119907119896+12 shows that the solution hzn to (15) has been acquiredand then the new iteration is started with updated sparsitylevel 119904 + 1 This process is repeated until the stopping criteriaare true and then the iteration is stopped We get the solutionto (15) with updated sparsity level and obtain the channelsie ℎ = h119904minus1

Note that the proposed SUCoSaMP algorithm worksin the same way as the CoSaMP algorithm The CoSaMPalgorithm is basically based on basic OMP The SUCoSaMPalgorithm has incorporated some other ideas from the liter-ature to present strong guarantees that OMP and CoSaMPcannot provide and to speed up the algorithm as comparedto OMP

The major advantage of SUCoSaMP over OMP CoSaMPand basic subspace pursuit (SP) algorithms is that it doesnot require prior knowledge of sparsity level which is an


Table 1 Major steps involved in CoSaMP and SUCoSaMP

CoSaMP SUCoSaMP

1 Classification The algorithm creates a replacement of the residual from the existing samples andplaces the biggest components of the replacement

Follows the same step ofCoSaMP

2 Support union The set of recently recognized components is joined with the set of components thatemerge in the present approximation


3 Estimation The algorithm finds the solution of a least-squares problem to estimate the objectivesignal on the combined set of components


4 Pruning The algorithm generates a fresh estimation by keeping only the biggest entries inthis least-squares approximation of signal


5 Sample Update Finally the algorithm updates the samples such that they reflect the residual theun-approximated elements of the signal


6 Stopping Criterion Until stopping criterion is true Until first stoppingcriterion is true

7 Sparsity levelUpdate None The algorithm updates the

sparsity level

8 Stopping Criterion None Until second stoppingcriterion is true

Table 2 CoSaMP and SUCoSaMP hypothesis

CoSaMP SUCoSaMP

1 The sparsity level s is fixed (ie initialization with fixed value ofsparsity level)

The sparsity level s is not fixed (ie initialization with sparsitylevel equals 1) It adaptively acquires the correct sparsity level

2 The sensing matrix A119899 has restricted isometry constant1205754119904 le 01A119899 satisfies the Structure Restricted Isometric Property (SRIP)

condition according to [39]

3 The signal hzn isin C119872119899119892119891119871times1 is random except where prominent The signal hzn isin C

119872119899119892119891119871times1 is a structured sparse equivalent CIRvector

4 wzn represents arbitrary noise vectorwzn represents the Additive White Gaussian Noise of nth

antenna group for zth OFDM symbol

unrealistic assumption specifically in wireless communica-tion scenario

There are two differences between SUCoSaMP andCoSaMP Firstly SUCoSaMP estimates the channels ieit recovers the high dimensional sparse vector by utilizingstructured sparsity of massive MIMO channels from onevector of low dimension Secondly SUCoSaMP adaptivelyacquires the sparsity level In contrast the CoSaMP recoversthe sparse vector without exploiting the structured sparsityand it requires prior information of correct sparsity level

The proposed SUCoSaMP algorithm is exactly the sameas CoSaMP algorithm up to step 29The proposed algorithmstops the iteration with fixed sparsity level (ie the currentvalue of s) if 1199071198962 gt 119907119896+12 and then it performs anadditional step that is step 210 to update the sparsitylevel by adding 1 to the current value of sparsity level (ie119904 larr997888 119904 + 1)

There are two different types of iterations in SUCoSaMPone on k and one on s and finally the iterations on s arestopped if 119907119896+12 gt 119907119904minus12 Table 1 elaborates some furtherdetails of major steps involved in CoSaMP and SUCoSaMPalgorithms And Table 2 demonstrates the hypothesis ofCoSaMP and SUCoSaMP algorithms

Table 3 System parameters

Parameter TypeValueTotal number of transmit antennas 128Number of transmit antennas in one sub-group 16Number of antennas groups (119873119892) 8Modulation 16QAMGuard Interval 16Number of pilot subcarriers (119873119901) 32 64System bandwidth 20MHzDFT size 2048

6 Simulation Results

Simulations have been performed in MATLAB in orderto verify the effectiveness of the proposed methods Meansquare error performance of proposed scheme is comparedwith the conventional OMP CoSaMP Structured SubspacePursuit (SSP) and Adaptive Structured Subspace Pursuit(ASSP) algorithms Simulation parameters are mentionedin Table 3 for the proposed system The BS has 1D 1x128antenna array (119872 = 128) The system bandwidth and


0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

Figure 5 Comparison of MSE of each algorithm under different SNRs (a) with119873119901 = 32 (b) with119873119901 = 64

carrier frequency are set to 119861 = 20MHz and 119891119888 = 2GHzrespectively There are 119873119892 = 8 sub-antenna groups with16 transmit antennas in each group to ensure the spatialchannel sparsity within group The OFDM subcarriers areset as 119873 = 2048 guard interval is 119873119866 = 16 which couldfight the delay spread up to 64 120583119904 and 16QAM modulationis used The numbers of pilot subcarrier 119873119901 in OFDMsymbol transmitted by each antenna in each antenna groupand channel length 119871 are varied over a reasonable range toverify the performance of the proposed system The pilotpositions are uniformly distributed according to (16) andare identical for the entire antennas within one group Thenumber of multipath channels is randomly chosen and thechannel multipath amplitudes and positions follow Rayleighand random distribution respectively

Figure 5 shows the MSE performance of the proposedSUCoSaMP algorithm with different values of 119873119901 versussignal to noise ratio (SNR) The performance of SUCoSaMPis compared with the conventional algorithms OMP andCoSaMP and with SSP and ASSPThe sparsity level for OMPCoSaMP and SSP is known in simulations whereas the ASSPand proposed SUCoSaMP adaptively acquire the correctsparsity level It is observed that the channel estimationperformance of all the algorithms is improved by increasingpilot subcarriers 119873119901 And overall the proposed SUCoSaMPoutperforms all the other algorithms This is because theSUCoSaMP takes full advantage of structured sparsity ofmassive MIMO channels Since the prior information ofsparsity level of massive MIMO wireless channel is not arealistic assumption the proposed SUCoSaMP algorithmhas a clear advantage over the conventional algorithmsFurthermore theMSE gap between the proposed SUCoSaMPand the conventional algorithm remains almost constant asthe SNR increases which makes SUCoSaMP superior in bothlow and high SNR wireless communication scenarios

In Figures 6(a)ndash6(h) individual MSE performance versusSNRof different sub antenna groups (ie from antenna group1198731 to 1198738) is presented for 119873119901 = 64 and is compared withconventional algorithms OMP and CoSaMP and with SSPand ASSP It can be seen in the figures that SUCoSaMP isevidently superior to conventional algorithmsThis is becausethe provided sparsity level for conventional algorithms can-not be the actual sparsity ofmassiveMIMOwireless channelsHowever the adaptive process in ASSP and SUCoSaMPis realized by fixed increment in sparsity level thereforethe sparsity level estimation in these algorithms is slightlyoverestimated or underestimated It is again observed thatwith high number of pilot subcarriers 119873119901 the performanceof SUCoSaMP is improved along with the other algorithmsFurthermore it is observed that theMSE gap between the pro-posed SUCoSaMP and the conventional algorithm slightlyincreases or remains almost constant with the increase inSNR resulting in SUCoSaMPbeing better in different wirelesscommunication environments with respect to SNR

In Figures 7(a)ndash7(h) MSE performance versus SNR ofindividual antenna groups (ie from antenna group 1198731 to1198738) is presented with 119873119901 = 32 and comparison is providedwith conventional algorithms OMP and CoSaMP and withSSP and ASSP It can be seen from the figures that theSUCoSaMP algorithm for each antenna group can achievehuge MSE gains over conventional algorithms However it isobserved that the MSE performance of all algorithms slightlydegraded with decrease in number of pilot subcarriers 119873119901Results show that SUCoSaMP efficiently considers the effectof SNR and performs better in both high and low SNRscenarios

7 Conclusion

This paper proposes a nonorthogonal pilot design and aCS based algorithm SUCoSaMP to estimate the massive


0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)

Figure 6 Comparison of MSE of each algorithm under different SNRs for individual antenna groups with119873119901 = 64 (a) MSE performanceversus SNR of 1st antenna group (1198731) (b) MSE performance versus SNR of 2nd antenna group (1198732) (c) MSE performance versus SNR of 3rdantenna group (1198733) (d) MSE performance versus SNR of 4th antenna group (1198734) (e) MSE performance versus SNR of 5th antenna group(1198735) (f) MSE performance versus SNR of 6th antenna group (1198736) (g) MSE performance versus SNR of 7th antenna group (1198737) (h) MSEperformance versus SNR of 8th antenna group (1198738)


0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



MIMO channels The reduction of high pilot overhead inmassive MIMO systems and the recovery ability when thesparsity level of massive MIMO channels is unknown are themain focus of this research By taking advantage of spatialand temporal common sparsity of massive MIMO channelsin delay domain the proposed nonorthogonal pilot designand channel estimation scheme under the frame work ofCS theory significantly reduce the pilot overheads for mas-sive MIMO systems and also outperform the conventionalalgorithms in performance This research may be extendedby incorporating the proposed schemes in the multicellscenarios

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C-X Wang F Haider X Gao et al ldquoCellular architecture andkey technologies for 5G wireless communication networksrdquoIEEE Communications Magazine vol 52 no 2 pp 122ndash1302014

[2] F Rusek D Persson and B K Lau ldquoScaling up MIMOopportunities and challenges with very large arraysrdquo IEEESignal 2013

[3] H Yang Y Fan D Liu and Z Zheng ldquoCompressive sensingand prior support based adaptive channel estimation inmassiveMIMOrdquo in Proceedings of the 2nd IEEE International Conferenceon Computer and Communications ICCC rsquo16 IEEE 2016

[4] J G Andrews S Buzzi and W Choi ldquoWhat will 5G berdquo IEEEJournal on Selected Areas in Communications vol 32 no 6 pp1065ndash1082 2014

[5] W H Chin Z Fan and R Haines ldquoEmerging technologies andresearch challenges for 5G wireless networksrdquo IEEE WirelessCommunications Magazine vol 21 no 2 pp 106ndash112 2014

[6] W Xu T Shen Y Tian and Y Wang ldquoCompressive channelestimation exploiting block sparsity in multi-user massiveMIMO systemsrdquo in Proceedings of the 2017 IEEE WirelessCommunications and Networking Conference WCNC rsquo17 IEEE2017

[7] J Zhang B Zhang S Chen and X Mu ldquoPilot contaminationelimination for large-scale multiple-antenna aided OFDM sys-temsrdquo IEEE Journal 2014

[8] E Bjonson J Hoydis and M Kountouris ldquoMassive MIMOsystems with non-ideal hardware energy efficiency estimationand capacity limitsrdquo IEEE Transactions 2014

[9] Y S Cho J KimW Yang and C KangMIMO-OFDMWirelessCommunications with MATLAB 2010

[10] Y Xu G Yue and SMao ldquoUser grouping formassiveMIMO inFDD systems new design methods and analysisrdquo IEEE Accessvol 2 pp 947ndash959 2013

[11] B Hassibi and B M Hochwald ldquoHow much training is neededin multiple-antenna wireless linksrdquo IEEE Transactions onInformation Theory vol 49 no 4 pp 951ndash963 2003

[12] L CorreiaMobile BroadbandMultimediaNetworks TechniquesModels and Tools for 4G 2010

[13] Z Zhou X Chen and D Guo ldquoSparse channel estimationfor massive MIMO with 1-bit feedback per dimensionrdquo in Pro-ceedings of the IEEE Wireless Communications and NetworkingConference WCNC rsquo17 2017

[14] E Bjornson and B Ottersten ldquoA framework for training-basedestimation in arbitrarily correlated RicianMIMOchannels withRician disturbancerdquo IEEE Transactions on Signal Processing2010

[15] X Ma F Yang S Liu and J Song ldquoTraining sequence designand optimization for structured compressive sensing basedchannel estimation in massive MIMO systemsrdquo in Proceedingsof the GlobecomWork (GC rsquo16) 2016

[16] I Barhumi G Leus andM Moonen ldquoOptimal training designfor MIMO OFDM systems in mobile wireless channelsrdquo IEEETransactions on Signal Processing vol 51 no 6 pp 1615ndash16242003

[17] H Minn and N Al-Dhahir ldquoOptimal training signals forMIMO OFDM channel estimationrdquo IEEE Transactions onWireless Communications 2006

[18] Y-H Nam Y Akimoto Y Kim M-I Lee K Bhattad and AEkpenyong ldquoEvolution of reference signals for LTE-advancedsystemsrdquo IEEE Communications Magazine vol 50 no 2 pp132ndash138 2012

[19] L Lu G Y Li and A L Swindlehurst ldquoAn overview of massiveMIMO benefits and challengesrdquo IEEE Journal 2014

[20] N Gonzalez-Prelcic K T Truongt C Rusu and R W HeathldquoCompressive channel estimation in FDD multi-cell massiveMIMO systems with arbitrary arraysrdquo in Proceedings of the 2016IEEE GlobecomWorkshops GC Wkshps rsquo16 2016

[21] L Dai Z Wang and Z Yang ldquoSpectrally efficient time-frequency training OFDM for mobile large-scale MIMO sys-temsrdquo IEEE Journal on Selected Areas in Communications vol31 no 2 pp 251ndash263 2013

[22] M S Sim J Park and C-B Chae ldquoCompressed channelfeedback for correlated massive MIMOrdquo Commun 2016

[23] Z Gao L Dai and Z Wang ldquoStructured compressive sens-ing based superimposed pilot design in downlink large-scaleMIMO systemsrdquo IEEE Electronics Letters vol 50 no 12 pp896ndash898 2014

[24] CQi andLWu ldquoUplink channel estimation formassiveMIMOsystems exploring joint channel sparsityrdquo IEEE ElectronicsLetters vol 50 no 23 pp 1770ndash1772 2014

[25] Z Gao L Dai Z Lu and C Yuen ldquoSuper-resolution sparseMIMO-OFDM channel estimation based on spatial and tem-poral correlationsrdquo IEEE Communications Letters vol 18 no 7pp 1266ndash1269 2014

[26] S L H Nguyen and A Ghrayeb ldquoCompressive sensing-basedchannel estimation for massive multiuser MIMO systemsrdquo inProceedings of the 2013 IEEE Wireless Communications andNetworking Conference WCNC rsquo13 2013

[27] Z Gao L Dai Z Wang and S Chen ldquoSpatially commonsparsity based adaptive channel estimation and feedback forFDD massive MIMOrdquo IEEE Transactions on Signal Processingvol 63 no 23 pp 6169ndash6183 2015

[28] W Shen L Dai B Shim and S Mumtaz ldquoJoint CSIT acqui-sition based on low-rank matrix completion for FDD massive


MIMO systemsrdquo IEEE Communications Letters vol 19 no 12pp 2178ndash2181 2015

[29] X Ma F Yang S Liu and J Song ldquoDoubly selective channelestimation forMIMO systems based on structured compressivesensingrdquo in Proceedings of the 2017 13th International WirelessCommunications and Mobile Computing Conference (IWCMCrsquo17) 2017

[30] Y Ding and B D Rao ldquoDictionary learning based sparsechannel representation and estimation for FDDmassiveMIMOsystemsrdquo 2016

[31] B Gong L Gui Q Qin and X Ren ldquoBlock distributed com-pressive sensing-based doubly selective channel estimation andpilot design for large-scale MIMO systemsrdquo IEEE Transactionson Vehicular Technology vol 66 no 10 pp 9149ndash9161 2017

[32] S Sun and T S Rappaport ldquoMillimeter Wave MIMO channelestimation based on adaptive compressed sensingrdquo in Proceed-ings of the 2017 IEEE International Conference on Communica-tions Workshops (ICC Workshops rsquo17) 2017

[33] J Choi D J Love and P Bidigare ldquoDownlink trainingtechniques for FDD massive MIMO systems Open-loop andclosed-loop training with memoryrdquo IEEE Journal of SelectedTopics in Signal Processing vol 8 no 5 pp 802ndash814 2014

[34] D Hu X Wang and L He ldquoA new sparse channel estimationand tracking method for time-varying OFDM systemsrdquo IEEETransactions on Vehicular Technology vol 62 no 9 pp 4648ndash4653 2013

[35] T Santos J Karedal and P Almers ldquoModeling the ultra-wideband outdoor channel measurements and parameterextraction methodrdquo IEEE Transactions on Wireless Communi-cations vol 9 no 1 pp 282ndash290 2010

[36] I E Telatar and D N Tse ldquoCapacity andmutual information ofwideband multipath fading channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol46 no 4 pp 1384ndash1400 2000

[37] I Khan M Singh and D Singh ldquoCompressive sensing-basedsparsity adaptive channel estimation for 5G massive MIMOsystemsrdquo Applied Sciences vol 8 no 5 p 754 2018

[38] Y Nan L Zhang and X Sun ldquoWeighted compressive sensingbased uplink channel estimation for TDD massive MIMOsytemsrdquo IET Communications vol 11 no 3 pp 355ndash361 2017

[39] Z Gao L Dai W Dai B Shim and Z Wang ldquoStruc-tured compressive sensing-based spatio-temporal joint channelestimation for FDD massive MIMOrdquo IEEE Transactions onCommunications vol 64 no 2 pp 601ndash617 2016

[40] J Choi B Shim Y Ding B Rao and D Kim ldquoCompressedsensing for wireless communications useful tips and tricksrdquoIEEE Communications Survey and Tutorials vol 19 no 3 pp1527ndash1550 2017

[41] A Lozano and N Jindal ldquoOptimum pilot overhead in wirelesscommunication a unified treatment of continuous and block-fading channelsrdquo in Proceedings of the 2010 European WirelessConference (EW rsquo10) 2010

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same time-frequency resource at base station Thereforethese main advantages make it feasible to introduce themassive MIMO system as a promising candidate for 5Gwireless communication networks

One of the major issues in massive MIMO systems isthe accurate acquisition of the channel state information(CSI) for beamforming resource allocation signal detectionetc Due to large antennas placed at the BS the estimationof channels linked with hundreds of transmit antennas isrequired at users which results in high pilot overhead Hencethe precise channel estimation with the low pilot overhead isa challenging task [2 3]

There are several research challenges which are requiredto be solved beforemassiveMIMOcanbe fully integrated intofuture wireless systems [4 5]

(i) Beamforming will involve a huge amount of channelstate information which will be challenging for thedownlink

(ii) MIMO can be unfeasible for FDD systems but can beemployed in time division duplex (TDD) systems dueto having channel reciprocity

(iii) Conventional channel estimation approaches requirea large pilot and feedback overhead which typicallyscales proportionally with number of BS transmitantennas which results in unfeasible condition forlarge-scale FDDMIMO systems

(iv) In addition massive MIMO experiences pilot con-tamination from other cells if the transmit power ishigh otherwise it will be affected by thermal noiseMethods will be required to solve this in order todeliver the promised performance

(v) There is a need for channel models of massiveMIMO systems without which it will be difficult forresearchers to perfectly validate the algorithms andtechniques

(vi) For channel estimation TDD scenarios are onlyconsidered for massive MIMO due to the excessivecost of channel estimation and feedback Even forTDD toworkmassiveMIMOchannel calibration canprove to be a big achievement New methods andschemes will be needed for the purpose of channelestimation in massive MIMO systems

(vii) Extremely fast processing algorithms will be requiredfor processing the massive amount of data from theRF chains

The above-mentioned challenges have been addressed up tosome extent in the last few years To date many researchesonmassiveMIMO avoided the challenge of considering FDDsystems by simply assuming the TDD protocol The uplinkCSI can be more easily obtained at the BS due to the lessnumber of single-antenna users and the strong capabilityof processing at BS And then by leveraging the channelreciprocity property the CSI in the downlink can be directlytacked [6 7] However due to the fact that radio frequencychains suffer from calibration error and restricted coherence

time the CSI obtained in the uplink may not be correctfor the downlink [8 9] Additionally FDD systems have lowlatency as compared to TDD therefore the communicationis more efficient [10] Thus it is of significance to discoverthe challenging issue of channel estimation for FDD massiveMIMO systems which can assist massive MIMO to bebackward attuned with existing FDD dominated cellularnetworks There has been extensive analysis on channelestimation for traditional small-scale FDD MIMO systems[11 12] It has been established that the equally spaced andequally powered orthogonal pilots can be ideally suitableto estimate the noncorrelated Rayleigh MIMO channels forsingle OFDM symbol where the pilot overhead requiredincreases as the number of transmit antennas increases [11]The pilot overhead to estimate Rician MIMO channels canbe compacted by exploiting the spatial correlation of MIMOchannels [13 14] Furthermore by taking advantage of thetemporal channel correlation more reduced pilot overheadcan be attained to estimateMIMOchannels linked tomultipleOFDM symbols [15ndash17] Presently orthogonal pilots havebeen extensively introduced in the current MIMO systemsdue to small number of transmit antennas (ie up to eightantennas in LTE-Advanced system) they have reasonablepilot overhead [12 18] Though this is a critical issue inmassive MIMO systems due to massive figure of antennasat the BS (ie up to 128 antennas or even more at the BS[19]) An approach of exploiting the temporal correlationand the delay domain sparsity of channels for achievingthe reduced pilot overhead has been presented for FDDmassive MIMO systems in [20 21] but with large numberof transmit antennas the interference cancellation of pilotsequences of different transmit antennas will be difficult In[22ndash25] the spatial correlation in sparsity of delay domainMIMO channels is utilized to estimate channels achievingthe reduced pilot overhead but the assumption level of theknown channel sparsity to the user is impractical In [26ndash28] the compressive sensing (CS) based channel estimationschemes are presented by exploiting the spatial channel cor-relation however due to having nonideal antenna array theleveraged spatial correlation can be impaired [2 8] In [29] anovel doubly selective channel estimation scheme forMIMO-OFDM systems is proposed with both time and frequency-domain training (TFDT) based on structured compressivesensing (SCS) In [30] a new sparse channel model basedon dictionary learning is proposed which adapts to thecell characteristics and promotes a sparse representationThe learned dictionary is able to more robustly and effi-ciently represent the channel and improve downlink channelestimation accuracy Furthermore observing the identicalAOAAOD between the uplink and downlink transmissiona joint uplink and downlink dictionary learning and com-pressed channel estimation algorithm is proposed to performdownlink channel estimation utilizing information from thesimpler uplink training which further improves downlinkchannel estimation In [31] a channel estimator based onblock distributed compressive sensing (BDCS) is proposedfor the large-scale MIMO systems BDCS exploits structuredsparsity to reduce the pilot overhead In [32] a new way toestimate sparse channels by construction of beamforming










119889119911119898 = [119889119911119898 (1) 119889119911119898 (2) 119889119898119903 (119871)]119879 (1)


User 1 User 2

(a)


Different tr

ansmit

antennas in

1st anten

na

groupDiffere

nt transm

it

antennas in

1st anten

na

group

Diff

eren

t Ant

enna

Gro

ups

Diff

eren

t Ant

enna

Gro

ups




Different tr

ansmit

antennas in

NＮＢ

antenna

groupDiffere

nt transm

it

antennas in

NＮＢ

antenna

group



(b)



with 1 le 119897 le 119871 (2)














119889119898 (119872119899119892 minus 1) le 11988810119861 (5)

1205822 (119872119899119892 minus 1) le

11988810119861 (6)

119872119899119892 le 1198885119861 (119888119891119888) + 1 (7)


119873119892 = lfloor 119872119872119899119892rfloor + 120572 (8)








119872119899119892119891 = 119872119873119892 (9)






yzn =119872119899119892119891sum119898=1


] + wzn (10)

with 1 lt n lt 119873119892yzn =

119872119899119892119891sum119898=1

Smn RL1003816100381610038161003816120585119899n dmzn + wzn


= 119872119899119892119891sum119898=1

120595mndmzn + wzn

(11)




yzn = 120595ndzn + wzn (12)








yz1 = A1hz1 + wz1

yz2 = A2hz2 + wz2

yzn = Anhzn + wzn

yz119873119892 = A119873119892 hz119873119892 + wz119873119892

(15)







(16)


119872119899119892119891 le 119873119901 le (119873 minus 119873119866)(119872 minus 120572119872119899119892)119872119899119892 (17)




T x 1 T x 2

T x 1 Pilot

T x 2 Pilot

Null Pilot

Data

Freq

uenc

y



T x Mngf Pilot

T x Mngf +1 Pilot

T x Mngf + 2 Pilot

T x Mngf Pilot



T x M Pilot












T x 1 T x 2 T x M

T x 1 Pilot

T x 2 Pilot

T x M

Null Pilot

Data

Freq

uenc

y



1198751199111119899 = 1198751199112119899 = sdot sdot sdot = 119875119911119872119899 (19)



Yn = AnHn +Wn (20)







100381710038171003817100381710038172 lt 120598(22)













CoSaMP SUCoSaMP













sparsity level



CoSaMP SUCoSaMP


















0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











119889119911119898 = [119889119911119898 (1) 119889119911119898 (2) 119889119898119903 (119871)]119879 (1)


User 1 User 2

(a)


Different tr

ansmit

antennas in

1st anten

na

groupDiffere

nt transm

it

antennas in

1st anten

na

group

Diff

eren

t Ant

enna

Gro

ups

Diff

eren

t Ant

enna

Gro

ups




Different tr

ansmit

antennas in

NＮＢ

antenna

groupDiffere

nt transm

it

antennas in

NＮＢ

antenna

group



(b)



with 1 le 119897 le 119871 (2)














119889119898 (119872119899119892 minus 1) le 11988810119861 (5)

1205822 (119872119899119892 minus 1) le

11988810119861 (6)

119872119899119892 le 1198885119861 (119888119891119888) + 1 (7)


119873119892 = lfloor 119872119872119899119892rfloor + 120572 (8)








119872119899119892119891 = 119872119873119892 (9)






yzn =119872119899119892119891sum119898=1


] + wzn (10)

with 1 lt n lt 119873119892yzn =

119872119899119892119891sum119898=1

Smn RL1003816100381610038161003816120585119899n dmzn + wzn


= 119872119899119892119891sum119898=1

120595mndmzn + wzn

(11)




yzn = 120595ndzn + wzn (12)








yz1 = A1hz1 + wz1

yz2 = A2hz2 + wz2

yzn = Anhzn + wzn

yz119873119892 = A119873119892 hz119873119892 + wz119873119892

(15)







(16)


119872119899119892119891 le 119873119901 le (119873 minus 119873119866)(119872 minus 120572119872119899119892)119872119899119892 (17)




T x 1 T x 2

T x 1 Pilot

T x 2 Pilot

Null Pilot

Data

Freq

uenc

y



T x Mngf Pilot

T x Mngf +1 Pilot

T x Mngf + 2 Pilot

T x Mngf Pilot



T x M Pilot












T x 1 T x 2 T x M

T x 1 Pilot

T x 2 Pilot

T x M

Null Pilot

Data

Freq

uenc

y



1198751199111119899 = 1198751199112119899 = sdot sdot sdot = 119875119911119872119899 (19)



Yn = AnHn +Wn (20)







100381710038171003817100381710038172 lt 120598(22)













CoSaMP SUCoSaMP













sparsity level



CoSaMP SUCoSaMP


















0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



User 1 User 2

(a)


Different tr

ansmit

antennas in

1st anten

na

groupDiffere

nt transm

it

antennas in

1st anten

na

group

Diff

eren

t Ant

enna

Gro

ups

Diff

eren

t Ant

enna

Gro

ups




Different tr

ansmit

antennas in

NＮＢ

antenna

groupDiffere

nt transm

it

antennas in

NＮＢ

antenna

group



(b)



with 1 le 119897 le 119871 (2)














119889119898 (119872119899119892 minus 1) le 11988810119861 (5)

1205822 (119872119899119892 minus 1) le

11988810119861 (6)

119872119899119892 le 1198885119861 (119888119891119888) + 1 (7)


119873119892 = lfloor 119872119872119899119892rfloor + 120572 (8)








119872119899119892119891 = 119872119873119892 (9)






yzn =119872119899119892119891sum119898=1


] + wzn (10)

with 1 lt n lt 119873119892yzn =

119872119899119892119891sum119898=1

Smn RL1003816100381610038161003816120585119899n dmzn + wzn


= 119872119899119892119891sum119898=1

120595mndmzn + wzn

(11)




yzn = 120595ndzn + wzn (12)








yz1 = A1hz1 + wz1

yz2 = A2hz2 + wz2

yzn = Anhzn + wzn

yz119873119892 = A119873119892 hz119873119892 + wz119873119892

(15)







(16)


119872119899119892119891 le 119873119901 le (119873 minus 119873119866)(119872 minus 120572119872119899119892)119872119899119892 (17)




T x 1 T x 2

T x 1 Pilot

T x 2 Pilot

Null Pilot

Data

Freq

uenc

y



T x Mngf Pilot

T x Mngf +1 Pilot

T x Mngf + 2 Pilot

T x Mngf Pilot



T x M Pilot












T x 1 T x 2 T x M

T x 1 Pilot

T x 2 Pilot

T x M

Null Pilot

Data

Freq

uenc

y



1198751199111119899 = 1198751199112119899 = sdot sdot sdot = 119875119911119872119899 (19)



Yn = AnHn +Wn (20)







100381710038171003817100381710038172 lt 120598(22)













CoSaMP SUCoSaMP













sparsity level



CoSaMP SUCoSaMP


















0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










119889119898 (119872119899119892 minus 1) le 11988810119861 (5)

1205822 (119872119899119892 minus 1) le

11988810119861 (6)

119872119899119892 le 1198885119861 (119888119891119888) + 1 (7)


119873119892 = lfloor 119872119872119899119892rfloor + 120572 (8)








119872119899119892119891 = 119872119873119892 (9)






yzn =119872119899119892119891sum119898=1


] + wzn (10)

with 1 lt n lt 119873119892yzn =

119872119899119892119891sum119898=1

Smn RL1003816100381610038161003816120585119899n dmzn + wzn


= 119872119899119892119891sum119898=1

120595mndmzn + wzn

(11)




yzn = 120595ndzn + wzn (12)








yz1 = A1hz1 + wz1

yz2 = A2hz2 + wz2

yzn = Anhzn + wzn

yz119873119892 = A119873119892 hz119873119892 + wz119873119892

(15)







(16)


119872119899119892119891 le 119873119901 le (119873 minus 119873119866)(119872 minus 120572119872119899119892)119872119899119892 (17)




T x 1 T x 2

T x 1 Pilot

T x 2 Pilot

Null Pilot

Data

Freq

uenc

y



T x Mngf Pilot

T x Mngf +1 Pilot

T x Mngf + 2 Pilot

T x Mngf Pilot



T x M Pilot












T x 1 T x 2 T x M

T x 1 Pilot

T x 2 Pilot

T x M

Null Pilot

Data

Freq

uenc

y



1198751199111119899 = 1198751199112119899 = sdot sdot sdot = 119875119911119872119899 (19)



Yn = AnHn +Wn (20)







100381710038171003817100381710038172 lt 120598(22)













CoSaMP SUCoSaMP













sparsity level



CoSaMP SUCoSaMP


















0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



= 119872119899119892119891sum119898=1

120595mndmzn + wzn

(11)




yzn = 120595ndzn + wzn (12)








yz1 = A1hz1 + wz1

yz2 = A2hz2 + wz2

yzn = Anhzn + wzn

yz119873119892 = A119873119892 hz119873119892 + wz119873119892

(15)







(16)


119872119899119892119891 le 119873119901 le (119873 minus 119873119866)(119872 minus 120572119872119899119892)119872119899119892 (17)




T x 1 T x 2

T x 1 Pilot

T x 2 Pilot

Null Pilot

Data

Freq

uenc

y



T x Mngf Pilot

T x Mngf +1 Pilot

T x Mngf + 2 Pilot

T x Mngf Pilot



T x M Pilot












T x 1 T x 2 T x M

T x 1 Pilot

T x 2 Pilot

T x M

Null Pilot

Data

Freq

uenc

y



1198751199111119899 = 1198751199112119899 = sdot sdot sdot = 119875119911119872119899 (19)



Yn = AnHn +Wn (20)







100381710038171003817100381710038172 lt 120598(22)













CoSaMP SUCoSaMP













sparsity level



CoSaMP SUCoSaMP


















0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




T x 1 T x 2

T x 1 Pilot

T x 2 Pilot

Null Pilot

Data

Freq

uenc

y



T x Mngf Pilot

T x Mngf +1 Pilot

T x Mngf + 2 Pilot

T x Mngf Pilot



T x M Pilot












T x 1 T x 2 T x M

T x 1 Pilot

T x 2 Pilot

T x M

Null Pilot

Data

Freq

uenc

y



1198751199111119899 = 1198751199112119899 = sdot sdot sdot = 119875119911119872119899 (19)



Yn = AnHn +Wn (20)







100381710038171003817100381710038172 lt 120598(22)













CoSaMP SUCoSaMP













sparsity level



CoSaMP SUCoSaMP


















0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



T x 1 T x 2 T x M

T x 1 Pilot

T x 2 Pilot

T x M

Null Pilot

Data

Freq

uenc

y



1198751199111119899 = 1198751199112119899 = sdot sdot sdot = 119875119911119872119899 (19)



Yn = AnHn +Wn (20)







100381710038171003817100381710038172 lt 120598(22)













CoSaMP SUCoSaMP













sparsity level



CoSaMP SUCoSaMP


















0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













CoSaMP SUCoSaMP













sparsity level



CoSaMP SUCoSaMP


















0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

005

01

015

02

025

03N

MSE

2616 18 20 28 302214 2412SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

0

005

01

015

NM

SE

16 18 20 26 28 3022 2414SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)






7 Conclusion



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

005

01

015

02

025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

0

005

01

015

02

025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(a)

14 16 18 20 22 24 26 28 3012SNR

000501

01502

02503

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(b)

0005

01015

02025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(c)

0005

01015

02025

03

NM

SE14 16 18 20 22 24 26 28 3012

SNRCoSaMPOMPSSP

ASSPSUCoSaMP

(d)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(e)

14 16 18 20 22 24 26 28 3012SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(f)

0

005

01

015

02

025

03

NM

SE

14 16 18 20 22 24 26 28 3012SNR

CoSaMPOMPSSP

ASSPSUCoSaMP

(g)

12 14 16 18 20 22 24 26 28 30SNR

0005

01015

02025

03

NM

SE

CoSaMPOMPSSP

ASSPSUCoSaMP

(h)




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Data Availability




References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


compressive sensing based channel estimation for massive...

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