mt-bcs-baseddoaandbandwidthestimationofunknown...

Research ArticleMT-BCS-Based DoA and Bandwidth Estimation of UnknownSignals through Multiple Snapshots Data

Shi Hui Zhang Qing He Zhang Li Ping Shi Chao Yi and Guang Xu Liu

e College of Computer and Information China ree Gorges University Yichang China

Correspondence should be addressed to Qing He Zhang zqhctgueducn

Received 18 October 2019 Accepted 10 December 2019 Published 6 January 2020

Academic Editor Chien-Jen Wang

Copyright copy 2020 Shi Hui Zhang et al-is 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

-e Direction-of-Arrival (DoA) and bandwidth (BW) estimation strategy impinging on a linear array using multiple snapshotsdata is addressed within the multitask Bayesian Compressive Sensing (MT-BCS) -e DoA estimation is used as the recon-struction of sparse signal constrained by the Laplace prior through multitask Bayesian Compressive Sensing Receiving widebandsignal data through linear array the space is divided into I parts according to the equal interval-e data of interest are assumed tobe represented as I-dimensional vector and the wideband signal can be reconstructed accurately using only a small number M-e receiving antenna operates in the frequency range [fmin fmax] Starting from the voltages measured at the output of the arrayelements at a multiple time instants at fp fmin + Δf p 1 P the retrieval of the DoAs is addressed by means of acustomized strategy based on MT-BCS in order to correlate the solutions obtained over different frequency samples -ebandwidth of the signals is obtained as a byproduct by identifying at which frequencies the MT-BCS estimations include a signalalong the ith (i 1 I) sampling direction From the outputs of different frequencies we can know the DoA and BW of signalsA preliminary numerical result is reported to show the behavior of the proposed approach in multiple snapshots data

1 Introduction

Direction-of-Arrival (DoA) estimation has important ap-plications in many traditional fields such as smart antennaarray radar signal processing geophysical or seismicsensing acoustics multiple-input multiple-output (MIMO)and other applications related to finding the direction of theincoming signals or sources [1ndash4] In fact the DoAsknowledge of receiving signal can be properly used to locatethe corresponding source and the adaptive beamforming ofthe receiving antenna pattern can improve the sensitivity ofthe system to the desired signal direction or suppress un-necessary interference

In some scientific literature several effective method-ologies have been proposed about DoA estimation such asthe multiple signal classification (MUSIC) [5] and the signalestimation parameter via rotational invariance technique(ESPRIT) [6] For the limitations that the need of a prioriknowing the number of incoming signals and the calcu-lation of the correlation matrix are usually computationally

expensive and their standard implementations are rarelyavailable especially nowadays with the huge proliferation ofwireless devicesservices and the presence of non-collaborative users [7] More recently DoA estimationstrategies based on the Compressive Sensing (CS) [8 9] areproposed and have shown promising features and results CSis an enabling paradigm for many applications where there isthe need of overcoming the Shannonrsquos limit in data ac-quisition and to recover sparse signals from far fewermeasurements [10] -anks to these features that thecomputational efficiency the accuracy and the robustness tothe noise CS-based strategies have already been applied to avariety of applications in electromagnetic engineering[11 12] But for guaranteeing reliable estimations thesampling matrix must satisfy the restricted isometry prop-erty (RIP) [13] when applying CS Unfortunately such acondition cannot be easily verified because of it results arecomputationally demanding [14] But innovative ap-proaches using the Bayesian compressive sensing (BCS)[15 16] have been proposed and they have improved this

HindawiInternational Journal of Antennas and PropagationVolume 2020 Article ID 6179280 7 pageshttpsdoiorg10115520206179280

problem However most of the works about DoA estimationare mainly focused on narrowband signals In [17] thenarrowband algorithm is simply extended to broadband Asfor the DoAs estimation of wideband signals because of theguiding vector of broadband signal frequency the signalspace of different sub-bands is inconsistent which makes itdifficult to separate the signal space from noise spaceGeneral subspace-based estimation techniques [18] basedon the root-MUSIC [19] and maximum likelihood [20] areused to estimate the DoAs of the wideband signal but theyalso have limitations -is problem has been shown in[21 22] where theMT-BCS [23] has been customized to dealwith wideband signals while exploiting the correlationamong different frequency samples taken from singlesnapshot data and multiple snapshots respectively

In this paper the DoAs and BW estimation problem ofwideband signals are formulated within the MT-BCSframework based on Laplace priors Starting from the keyobservation that the wideband signal affecting the antennaarray is essentially sparse in spatial domain Because thesignal is a wideband signal we sample the signal with acertain range of frequencies in order to estimate thebandwidth of the signal Set the sampling frequency range to[fmin fmax] the DoAs and BW are estimated at the sametime by which the range of frequencies has measured data

-e rest of the paper is organized as follows -eBayesian modeling is mathematically formulated in Section2 where the wideband signal model of MT-BCS usingLaplace priors is described A set of representative numericalresults is then reported and discussed in Section 3 wherereference DoAs and bandwidth estimation methods areperformed Finally some conclusions are drawn in Section 4

2 Bayesian Modeling

CS theory can cover certain signals from far fewer samples ormeasurements than traditional methods and it is a veryimportant step to restore the signal we want In some worknarrowband signals are recovered by MT-BCS [24 25] Inthis section a simple MT-BCS model about widebandsignals based on Laplace priors is shown

21 Model for BCS Let us consider a set of K widebandsignals impinging on a planar distribution of N sensors islocated -e DoAs of the signals are denoted asθk k 1 K and operating in the frequency range[fmin fmax] -e voltages collected from the output of thearray elements at P different frequency samples fp fmin +

Δf(2p minus 1)2 p 1 P being Δf (fmax minus fmin)

(P minus 1) -e input data of the problem are the voltagesv

(p)n n 1 N measured by each sensor expressed as

v(p)n 1113944

K

k1sk 1113954y middot he

j 2πλ(p)( )xn sin θk + η(p)n n 1 N

(1)

where sk k 1 K is the amplitude of the impingingsignals h is the antenna effective length λ(p) being the free-

space length at pth frequency and η(p)n n 1 N the

contribution of the noise at each sensor-en in order to employ the proposed MTBCS-based

methodology the problem is reformulated by sampling theangular domain of interest over a very fine grid of IgtgtKangular location being θi minus (π2) + π(i minus 1)(I minus 1)

i 1 2 I Under this assumption and considering themultiple-snapshots case (1) can be rewritten in a matrixform as

v(p)

l Φ(p)(θ)s(p)

l + ηp

l l 1 L (2)

where L is the number of snapshots s(p)

l is the vector ofestimated signals Φ(p)(θ) ΨA(p)(θ) Ψ isin RMtimesN is themeasurement matrix we will use the Gaussian matrix in thispaper andM (Mltlt I) is the number of samples [16] A(p) isthe matrix of array manifold

A(p)(θ)

ej 2πλ(p)( )x1 sin θ1 ej 2πλ(p)( )x1 sin θ2


middot middot middot ej 2πλ(p)( )xI sin θ1

middot middot middot ej 2πλ(p)( )x2 sin θI

⋮ ⋮

ej 2πλ(p)( )xN sin θ1 ej 2πλ(p)( )xN sin θ2

⋱ ⋮

middot middot middot ej 2πλ(p)( )xN sin θI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)

where xn (n minus 1)d n 1 N is the distance betweenthe ith array element and the reference array element and d isthe array element spacing

22 Model for MT-BCS Based on Laplace Priors In [15] wecan learn about the general model of MT-BCS based onLaplace priors So this section of the model simply recountsthe expressions of the wideband signals we want to useSpecify sparse signal s(p)

l as a Laplace priors about parameterλ and it is defined as

p s(p)

l

11138681113868111386811138681113868 λ1113874 1113875 λ2exp minus

λ2s(p)

l

11113888 1113889 (4)

And the mean and covariance of signals are given by

μ βΣΦTv(p)

l (5)

Σ βΦTΦ + Λ1113960 1113961minus 1

(6)

where Λ diag(1ci) i 1 I ci is the signal hyper-parameter and estimated as

ci

minus si si + 2λ( 1113857 + si

si + 2λ( 11138572

minus 4λ si minus q2i + λ( 1113857

1113969

2λs2i if q2i minus si gt λ

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

with

si Si

1 minus ciSi

(8)

qi Qi

1 minus ciSi

(9)

2 International Journal of Antennas and Propagation

Si βϕTi ϕi minus β2ϕT

i ΦΣΦTϕi (10)

Qi βϕTi v

(p)

l minus β2ϕTi ΦΣΦ

Tv(p)

l (11)

where ϕi i 1 I expresses the ith column ofΦ β is thenoise hyperparameter and Gamma prior obeying parame-ters a and b Parameter λ realizes the following Gammahyperprior

p(λ | ]) Gamma(λ | ]2 ]2)(10) (12)

-e parameters estimation of λ β and υ is expressed asfollowing

λ N minus 1 + ]2

1113936Ni1 ci2( 1113857 + ]2

(13)

β LN2 + a

1113936Ll1 lang v

(p)

l minus Φs(p)

l

2rang21113874 1113875 + b

(14)

1 InputΦ (v(p)1 v(p)

2 v(p)

L )

2 Outputc [c1 c2 cI]T λ ] β

3 Initialize all c 0 λ 04 While convergence criterion not met do5 Choose a ci i isin [1 I]

6 If q2i gt λ and ci 0then add ci to the model

7 Else if q2i gt λ and ci gt 0then re-estimate ci using (7)

8 Else if q2i lt λthen prune ith column of Φ from the model and set ci 0

End if9 Update μ LΣ using (5) and (6)10 Update si qiusing (8)sim(11)11 Update λ ] β using (12)sim(15)12 end while

ALGORITHM 1 Algorithm of MT-BCS

ndash80 ndash60 ndash40 ndash20 0 20 40 60 80Angular direction

0

02

04

06

08

1

12

14

16

18

2

Sign

al en

ergy

Actualf1 = 06GHzf2 = 07GHzf3 = 08GHz

f4 = 09GHzf5 = 10GHzf6 = 11GHz

Figure 1 Actual and estimated DoAs for multiple snapshots

Table 1 Estimated DoAs and signal energy for differentfrequencies

1113957θ 1113957s

f1 (minus 9 minus 5 60) (08029 05749 03021)f2 (minus 9 minus 5 60) (10005 09356 08111)f3 (minus 9 minus 5 60) (08117 09323 09916)f4 (minus 5 44 60) (05714 minus 00070 07933)f5 (13 39 60) (00212 minus 00048 02998)f6 (minus 9 minus 5 60) (00044 minus 00021 00015)

1 2 3Signal index (k)

0

1

2

3

4

5

6Fr

eque

ncy

inde

x (p

)

Actual BWEstimated BW

Figure 2 Actual and estimated signals bandwidth

International Journal of Antennas and Propagation 3

ln(υ2) + 1 minus ψ(υ2) + ln λ minus λ 0 (15)

From the above formula we can see that it is known thatc λ β and ] can find μ and Σ using (5) and (6) It is knownthat μ and Σ can also find c λ β and ] using (7)sim(15) -eprocedure is summarized in Algorithm 1

3 Numerical Results

In this section we present experimental results thatdemonstrate the performance of MT-BCS based on

Laplace priors to recovery wideband signals Differentfrom narrow narrowband signals the wideband signalexists only when sampling frequency fp is within therange of bandwidth

We use the following default setup in the experimentalresults reported in this section Signal matrix of length I isgenerated where K coefficients are located at signal ran-domly and the rest (I-K) of the coefficients are set equal tozero As the measurement matrix Ψ we chose a Gaussianmatrix where the columns ψi are Gaussian distributed onthe sphere RI Moreover we present results with noisy

10 12 14 16 18 20L

0

1

2

3

4

5

6

7RM

SE


f4 = 09GHzf5 = 10GHz

0 2 4 6 8

(a)

10 12 14 16 18 20L

0

05

1

15

2

25

3

RMSE

0 2 4 6 8

(b)



ndash5 10SNR

0

2

4

6

8

10

12

14

RMSE

50

(c)

ndash5 50 10SNR

0

1

2

3

4

5

6

7

8

RMSE

(d)

Figure 3 Root mean square error (RMSE) versus LSNR (a) Behavior of RMSE versus L for different frequencies of fp p 1 5 (b)Behavior of average RMSE versus SNR set SNR 10 (c) Behavior of RMSE versus SNR for different frequencies of fp p 1 5 (d)Behavior of average RMSE versus SNR set L 20


acquisitions where for the noisy observations we add whiteGaussian noise

In the first set of experiments let us consider K 3signals arriving on a linear array of N 30 elements halfwavelength spaced at f0 (fmin + fmax)2 and designed towork in the frequency range [fmin fmax] [06 11]GHz-e actual signals are characterized by a sinusoidal wavewith amplitude sk + 1 k 1 K DoAs equal toθ1 minus 9 deg θ2 minus 5 deg and θ3 60 deg and bandwidthsb1 [06 08]GHz b2 [06 09]GHz andb3 [06 10]GHz In order to estimate the DoAs and thesignal bandwidth by means of the MT-BCS the measureddata are considered characterized by a signal-to-noise ratioequal to SNR 10 dB and are sampled at P 6 frequencies(f1 f2 f3 f4 f5 f6) (06 07 08 09 10 11)GHzbeing Δf 01GHz Set snapshots L 10 and the resulteddata of pth frequency in multiple snapshots is calculated by

1113958spavg 1113944

L

l1

1113957sp

l L (16)

where 1113957s

p

l are the estimated signal energy in lth snapshot atpth frequency sampling Moreover the angular grid has

been discretized with I 181 samples in order to obtain aresolution of Δθ 1 deg

-e actual DoAs and those estimated values by theproposed spectral correlation MT-BCS based strategy areshown in Figure 1 and Table 1 As it can be observed inTable 1 there are some values that are small enough to beignored -is is mainly caused by noise We ignore thesevalues below 01 It can be observed that DoA θ1 minus 9 degand bandwidth b1 [f1 f3] for s1 θ2 minus 5 deg and b2

[f1 f4] for s2 and θ3 60 deg and b3 [f1 f5] for s3which is shown Figure 1 and Table 1

-e actual and estimated bandwidths of the K 3 im-pinging signals are shown in Figure 2 It is possible toobserve that both the directions of the incoming signals arecorrectly retrieved (Figure 1) as well as their band widths

-en we will see reconstruction error by RMSE -eequation of RMSE is expressed as follows

RMSE 1113944P

p11113944

K

k1

1113944

T

t1

1L

1113944

L

l1

1113954θfpktl minus θk⎛⎝ ⎞⎠

211139741113972

PK (17)

where T is number of experiments and 1113954θfpktl is the esti-mated value of kth signal at pth frequency

In the second set of experiments let us consider K 2signals designed to work in the frequency range[fmin fmax] [06 10]GHz DoAs equal to θ1 minus 55 degand θ2 minus 32 deg and in order to facilitate the use of RMSEcalculation error we set the two signals which have the samebandwidths [06 10] GHz -e measured data are sampledat P 5 frequencies (f1 f2 f3 f4 f5) (06 07 08 09

10)GHz being Δf 01GHz Average reconstruction er-rors of T 500 runs are shown for the case in Figure 3 for alltypes of signals

It is noted that RMSE results in lower values than settingsnapshots minor from Figure 3(b) and the frequency close tof0 show better performance than others from Figure 3(a)


0

02

04

06

08

1

12

14

16

18

2

Sign

al en

ergy

ActualL = 1L = 5

L = 10L = 20

Figure 4 Actual and estimated DoAs for average by (18)

Table 2 -e estimated DoAs for frequencies verse snapshots1113957θ

L 1 L 5 L 10 L 20

f1 (minus 55 minus 32 10) (minus 55 minus 32 12) (minus 55 minus 32 12) (minus 55 minus 49minus 32)

f2(minus 55 minus 32

minus 23)(minus 56 minus 55

minus 32) (minus 55 minus 32) (minus 55 minus 32)

f3 (minus 55 minus 32 26) (minus 55 minus 32) (minus 55 minus 32) (minus 55 minus 32)f4 (minus 55 minus 32) (minus 55 minus 32) (minus 55 minus 32) (minus 55 minus 32)




minus 32) (minus 55 minus 32 48)

Table 3 -e estimated signal energy

1113957sL 1 L 5 L 10 L 20

f1(02934 03126

minus 0016)

(0293503044

minus 00134)

(0293503144

minus 00133)

(02920minus 0008703136)

f2(08098 08065

minus 0205)

(minus 000870801408189)

(0809008349)

(0802907948)

f3(09969 09979

00296)(0993809857)

(0997810013)

(0987909796)

f4 (08068 08045) (0801408012)

(0796907886)

(0808108057)

f5(03197 03047

minus 00317)

(029650289700098)

(000930305903050)

(0301803106

minus 00033)

1113958savg

(06453 064524minus 0041 00063

minus 00032 00059)

(0001705379063990002000027)

(00019064060648800027)

(06385000170640900007)


-e results of the same experimental setup with differentSNR are shown in Figures 3(c) and 3(d) Similar per-formance decrease by the increase of SNR can be ob-served -ese suggest that the estimating DoA results canget lower reconstruction error in multiple snapshots andhigher SNR For wideband signals different samplingfrequencies can obtain different reconstruction results-is will have an impact on the accuracy of DoA esti-mation results

In order to obtain better reconstruction performanceswe will add signal energy 1113958s

pavg in (16) at different frequencies

of the same signal and average it and then get 1113958savg (16) canbe rewritten as follows

1113958savg 1113944P

p11113944

L

l1

1113958s

(p)

l

LP (18)

In Figure 4 the reconstruction DoA results using (18) atdifferent snapshots are shown We have also given esti-mated DoAs and values at pth frequency as shown inTable 2 1113958savg is the average signal energy of all estimatedangles from low to high -e signal energy that shows theunrelated angle is much lower than what we get in the firstset experiments-is shows that using (18) to estimate DoAcan get better reconstruction results than using (16) Wealso can note that as the number of snapshots increasesunrelated angles no longer exist Frequency away from f0still has poor reconstruction results but this is an im-provement after the signal energy of all frequencies isaveraged that is the signal energy of the unrelated anglebecomes smaller

We ignore values less than 01 as in the first set of ex-periments It can be concluded that the DoAs and band-widths of signals form the result shown in Figure 4 andTables 2 and 3 θ1 minus 55 deg and b1 [f1 f5] for s1 andθ2 minus 32deg and b2 [f1 f5] for s2-e estimated result ofsignal bandwidths is shown in Figure 5

4 Conclusion

In this paper we formulated the DoAs and BW-estimatedproblem of the wideband signal using MT-BCS based onLaplace priors and presented a framework for multiplesnapshots data Using this framework we first get the actualand estimated DoAs for multiple snapshots at differentfrequencies At the same time the signal BW is obtained byobserving at which frequency sampling point there is a signalenergy after ignoring values that are small enough -en weuse the RMSE to evaluate the reconstruction results indifferent snapshots and SNRs finding with the increasing ofsnapshots or SNR the reconstruction results are better

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China under Grant no 61771008

References

[1] P Stoica P Babu and J Li ldquoSPICE a sparse covariance-basedestimation method for array processingrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 629ndash638 2011

[2] F Liu J Wang C Sun and R Du ldquoSpatial differencingmethod for DOA estimation under the coexistence of bothuncorrelated and coherent signalsrdquo IEEE Transactions onAntennas and Propagation vol 60 no 4 pp 2052ndash2062 2012

[3] C H Niow and H T Hui ldquoImproved noise modeling withmutual coupling in receiving antenna arrays for direction-of-arrival estimationrdquo IEEE Transactions on Wireless Commu-nications vol 11 no 4 pp 1616ndash1621 Apr 2012

[4] F Wen J Shi and Z Zhang ldquoJoint 2D-DOD 2D-DOA andpolarization angles estimation for bistatic EMVS-MIMOradar via PARAFAC analysisrdquo IEEE Transactions on Vehic-ular Technology 2019

[5] R Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[6] R Roy and T Kailath ldquoESPRIT-Estimation of signal pa-rameters via rotational invariance techniquesrdquo IEEE Trans-actions on Acoustics Speech and Signal Processing vol 37no 7 pp 984ndash995 1989

[7] M Carlin P Rocca G Oliveri F Viani and A MassaldquoDirections-of-arrival estimation through bayesian com-pressive sensing strategiesrdquo IEEE Transactions on Antennasand Propagation vol 61 no 7 pp 3828ndash3838 2013

[8] E J Candes and M B Wakin ldquoAn introduction to com-pressive samplingrdquo IEEE Signal Processing Magazine vol 25no 2 pp 21ndash30 2008

[9] S Ji Y Xue and L Carin ldquoBayesian compressive sensingrdquoIEEE Transactions on Signal Processing vol 56 no 6pp 2346ndash2356 2008

21Signal index (k)

0

1

2

3

4

5

6

Freq

uenc

y in

dex

(p)


Figure 5 Actual and estimated signal bandwidth


[10] R G Baraniuk ldquoMore is less signal processing and the datadelugerdquo Science vol 331 no 6018 pp 717ndash719 2011

[11] W L Chan M L Moravec R G Baraniuk andD M Mittleman ldquoTerahertz imaging with compressedsensing and phase retrievalrdquo Optics Letters vol 33 no 9pp 974ndash976 2008

[12] L C Potter E Ertin J T Parker and M Cetin ldquoSparsity andcompressed sensing in radar imagingrdquo Proceedings of theIEEE vol 98 no 6 pp 1006ndash1020 2010

[13] R Baraniuk ldquoCompressive sensing [lecture notes]rdquo IEEESignal Processing Magazine vol 24 no 4 pp 118ndash121 2007

[14] A Massa P Rocca and G Oliveri ldquoCompressive sensing inelectromagneticsmdasha reviewrdquo IEEE Antennas and PropagationMagazine vol 57 no 1 pp 224ndash238 2015

[15] M Lustig D L Donoho J M Santos and J M PaulyldquoCompressed sensingMRIrdquo IEEE Signal ProcessingMagazinevol 25 no 2 pp 72ndash82 2008

[16] S D Babacan R Molina and A K Katsaggelos ldquoBayesiancompressive sensing using Laplace Priorsrdquo IEEE Transactionson Image Processing vol 19 no 1 pp 53ndash63 2010

[17] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation eory vol 52 no 4 pp 1289ndash1306 2006

[18] Z D Lei X K Huang and S J Zhang ldquoA fast algorithm fordirection of arrival estimation of multiple wide-band sour-cesrdquo Journal of the China Railway Society vol 19 no 4pp 46ndash50 1997

[19] B Ottersten and T Kailath ldquoDirection-of-arrival estimationfor wide-band signals using the ESPRIT algorithmrdquo IEEETransactions on Acoustics Speech and Signal Processingvol 38 no 2 pp 317ndash327 1990

[20] A B Gershman and M G Amin ldquoCoherent wideband DOAestimation of multiple FM signals using spatial time-fre-quency distributionsrdquo in Proceedings of the 2000 IEEE In-ternational Conference on Acoustics Speech and SignalProcessing pp 3065ndash3068 Istanbul Turkey June 2000

[21] M A Hannan P Rocca and A Massa ldquoRobust BCS-basedDirection-of-Arrival and bandwidth estimation of unknownsignals for cognitive radarrdquo in Proceedings of the 2018 IEEEInternational Symposium on Antennas and Propagation ampUSNCURSI National Radio Science Meeting Boston MAUSA July 2018

[22] S Ji D Dunson and L Carin ldquoMultitask compressivesensingrdquo IEEE Transactions on Signal Processing vol 57 no 1pp 92ndash106 2009

[23] M A Hannan N Anselmi G Oliveri and P Rocca ldquoJointDoA and bandwidth estimation of unknown signals throughsingle snapshot data and MT-BCS approachrdquo in Proceedingsof the 2017 IEEE International Symposium on Antennas andPropagation amp USNCURSI National Radio Science MeetingSan Diego CA USA July 2017

[24] M Carlin P Rocca G Oliveri and A Massa ldquoBayesiancompressive sensing as applied to directions-of-arrival esti-mation in planar arraysrdquo Journal of Electrical and ComputerEngineering vol 2013 Article ID 245867 12 pages 2013

[25] A Massa M Bertolli G Gottardi et al ldquoCompressive sensingas applied to antenna arrays synthesis diagnosis and pro-cessingrdquo in Proceedings of the IEEE International Symposiumon Circuits amp Systems Meeting Florence Italy May 2018


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problem However most of the works about DoA estimationare mainly focused on narrowband signals In [17] thenarrowband algorithm is simply extended to broadband Asfor the DoAs estimation of wideband signals because of theguiding vector of broadband signal frequency the signalspace of different sub-bands is inconsistent which makes itdifficult to separate the signal space from noise spaceGeneral subspace-based estimation techniques [18] basedon the root-MUSIC [19] and maximum likelihood [20] areused to estimate the DoAs of the wideband signal but theyalso have limitations -is problem has been shown in[21 22] where theMT-BCS [23] has been customized to dealwith wideband signals while exploiting the correlationamong different frequency samples taken from singlesnapshot data and multiple snapshots respectively

In this paper the DoAs and BW estimation problem ofwideband signals are formulated within the MT-BCSframework based on Laplace priors Starting from the keyobservation that the wideband signal affecting the antennaarray is essentially sparse in spatial domain Because thesignal is a wideband signal we sample the signal with acertain range of frequencies in order to estimate thebandwidth of the signal Set the sampling frequency range to[fmin fmax] the DoAs and BW are estimated at the sametime by which the range of frequencies has measured data

-e rest of the paper is organized as follows -eBayesian modeling is mathematically formulated in Section2 where the wideband signal model of MT-BCS usingLaplace priors is described A set of representative numericalresults is then reported and discussed in Section 3 wherereference DoAs and bandwidth estimation methods areperformed Finally some conclusions are drawn in Section 4

2 Bayesian Modeling

CS theory can cover certain signals from far fewer samples ormeasurements than traditional methods and it is a veryimportant step to restore the signal we want In some worknarrowband signals are recovered by MT-BCS [24 25] Inthis section a simple MT-BCS model about widebandsignals based on Laplace priors is shown

21 Model for BCS Let us consider a set of K widebandsignals impinging on a planar distribution of N sensors islocated -e DoAs of the signals are denoted asθk k 1 K and operating in the frequency range[fmin fmax] -e voltages collected from the output of thearray elements at P different frequency samples fp fmin +

Δf(2p minus 1)2 p 1 P being Δf (fmax minus fmin)

(P minus 1) -e input data of the problem are the voltagesv

(p)n n 1 N measured by each sensor expressed as

v(p)n 1113944

K

k1sk 1113954y middot he

j 2πλ(p)( )xn sin θk + η(p)n n 1 N

(1)

where sk k 1 K is the amplitude of the impingingsignals h is the antenna effective length λ(p) being the free-

space length at pth frequency and η(p)n n 1 N the

contribution of the noise at each sensor-en in order to employ the proposed MTBCS-based

methodology the problem is reformulated by sampling theangular domain of interest over a very fine grid of IgtgtKangular location being θi minus (π2) + π(i minus 1)(I minus 1)

i 1 2 I Under this assumption and considering themultiple-snapshots case (1) can be rewritten in a matrixform as

v(p)

l Φ(p)(θ)s(p)

l + ηp

l l 1 L (2)

where L is the number of snapshots s(p)

l is the vector ofestimated signals Φ(p)(θ) ΨA(p)(θ) Ψ isin RMtimesN is themeasurement matrix we will use the Gaussian matrix in thispaper andM (Mltlt I) is the number of samples [16] A(p) isthe matrix of array manifold

A(p)(θ)



middot middot middot ej 2πλ(p)( )xI sin θ1

middot middot middot ej 2πλ(p)( )x2 sin θI

⋮ ⋮

ej 2πλ(p)( )xN sin θ1 ej 2πλ(p)( )xN sin θ2

⋱ ⋮

middot middot middot ej 2πλ(p)( )xN sin θI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)

where xn (n minus 1)d n 1 N is the distance betweenthe ith array element and the reference array element and d isthe array element spacing

22 Model for MT-BCS Based on Laplace Priors In [15] wecan learn about the general model of MT-BCS based onLaplace priors So this section of the model simply recountsthe expressions of the wideband signals we want to useSpecify sparse signal s(p)

l as a Laplace priors about parameterλ and it is defined as

p s(p)

l

11138681113868111386811138681113868 λ1113874 1113875 λ2exp minus

λ2s(p)

l

11113888 1113889 (4)

And the mean and covariance of signals are given by

μ βΣΦTv(p)

l (5)

Σ βΦTΦ + Λ1113960 1113961minus 1

(6)

where Λ diag(1ci) i 1 I ci is the signal hyper-parameter and estimated as

ci

minus si si + 2λ( 1113857 + si

si + 2λ( 11138572

minus 4λ si minus q2i + λ( 1113857

1113969

2λs2i if q2i minus si gt λ

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

with

si Si

1 minus ciSi

(8)

qi Qi

1 minus ciSi

(9)



i ΦΣΦTϕi (10)

Qi βϕTi v

(p)


Tv(p)

l (11)


p(λ | ]) Gamma(λ | ]2 ]2)(10) (12)


λ N minus 1 + ]2

1113936Ni1 ci2( 1113857 + ]2

(13)

β LN2 + a

1113936Ll1 lang v

(p)

l minus Φs(p)

l

2rang21113874 1113875 + b

(14)


2 v(p)

L )









0

02

04

06

08

1

12

14

16

18

2

Sign

al en

ergy





1113957θ 1113957s



0

1

2

3

4

5

6Fr

eque

ncy

inde

x (p

)






3 Numerical Results




10 12 14 16 18 20L

0

1

2

3

4

5

6

7RM

SE



0 2 4 6 8

(a)

10 12 14 16 18 20L

0

05

1

15

2

25

3

RMSE

0 2 4 6 8

(b)



ndash5 10SNR

0

2

4

6

8

10

12

14

RMSE

50

(c)

ndash5 50 10SNR

0

1

2

3

4

5

6

7

8

RMSE

(d)





1113958spavg 1113944

L

l1

1113957sp

l L (16)

where 1113957s

p







RMSE 1113944P

p11113944

K

k1

1113944

T

t1

1L

1113944

L

l1


211139741113972

PK (17)






0

02

04

06

08

1

12

14

16

18

2

Sign

al en

ergy

ActualL = 1L = 5

L = 10L = 20



L 1 L 5 L 10 L 20











1113957sL 1 L 5 L 10 L 20

f1(02934 03126

minus 0016)

(0293503044

minus 00134)

(0293503144

minus 00133)

(02920minus 0008703136)

f2(08098 08065

minus 0205)

(minus 000870801408189)

(0809008349)

(0802907948)

f3(09969 09979

00296)(0993809857)

(0997810013)

(0987909796)

f4 (08068 08045) (0801408012)

(0796907886)

(0808108057)

f5(03197 03047

minus 00317)

(029650289700098)

(000930305903050)

(0301803106

minus 00033)

1113958savg

(06453 064524minus 0041 00063

minus 00032 00059)

(0001705379063990002000027)

(00019064060648800027)

(06385000170640900007)






1113958savg 1113944P

p11113944

L

l1

1113958s

(p)

l

LP (18)



4 Conclusion


Data Availability




Acknowledgments


References










21Signal index (k)

0

1

2

3

4

5

6

Freq

uenc

y in

dex

(p)























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



i ΦΣΦTϕi (10)

Qi βϕTi v

(p)


Tv(p)

l (11)


p(λ | ]) Gamma(λ | ]2 ]2)(10) (12)


λ N minus 1 + ]2

1113936Ni1 ci2( 1113857 + ]2

(13)

β LN2 + a

1113936Ll1 lang v

(p)

l minus Φs(p)

l

2rang21113874 1113875 + b

(14)


2 v(p)

L )









0

02

04

06

08

1

12

14

16

18

2

Sign

al en

ergy





1113957θ 1113957s



0

1

2

3

4

5

6Fr

eque

ncy

inde

x (p

)






3 Numerical Results




10 12 14 16 18 20L

0

1

2

3

4

5

6

7RM

SE



0 2 4 6 8

(a)

10 12 14 16 18 20L

0

05

1

15

2

25

3

RMSE

0 2 4 6 8

(b)



ndash5 10SNR

0

2

4

6

8

10

12

14

RMSE

50

(c)

ndash5 50 10SNR

0

1

2

3

4

5

6

7

8

RMSE

(d)





1113958spavg 1113944

L

l1

1113957sp

l L (16)

where 1113957s

p







RMSE 1113944P

p11113944

K

k1

1113944

T

t1

1L

1113944

L

l1


211139741113972

PK (17)






0

02

04

06

08

1

12

14

16

18

2

Sign

al en

ergy

ActualL = 1L = 5

L = 10L = 20



L 1 L 5 L 10 L 20











1113957sL 1 L 5 L 10 L 20

f1(02934 03126

minus 0016)

(0293503044

minus 00134)

(0293503144

minus 00133)

(02920minus 0008703136)

f2(08098 08065

minus 0205)

(minus 000870801408189)

(0809008349)

(0802907948)

f3(09969 09979

00296)(0993809857)

(0997810013)

(0987909796)

f4 (08068 08045) (0801408012)

(0796907886)

(0808108057)

f5(03197 03047

minus 00317)

(029650289700098)

(000930305903050)

(0301803106

minus 00033)

1113958savg

(06453 064524minus 0041 00063

minus 00032 00059)

(0001705379063990002000027)

(00019064060648800027)

(06385000170640900007)






1113958savg 1113944P

p11113944

L

l1

1113958s

(p)

l

LP (18)



4 Conclusion


Data Availability




Acknowledgments


References










21Signal index (k)

0

1

2

3

4

5

6

Freq

uenc

y in

dex

(p)























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




3 Numerical Results




10 12 14 16 18 20L

0

1

2

3

4

5

6

7RM

SE



0 2 4 6 8

(a)

10 12 14 16 18 20L

0

05

1

15

2

25

3

RMSE

0 2 4 6 8

(b)



ndash5 10SNR

0

2

4

6

8

10

12

14

RMSE

50

(c)

ndash5 50 10SNR

0

1

2

3

4

5

6

7

8

RMSE

(d)





1113958spavg 1113944

L

l1

1113957sp

l L (16)

where 1113957s

p







RMSE 1113944P

p11113944

K

k1

1113944

T

t1

1L

1113944

L

l1


211139741113972

PK (17)






0

02

04

06

08

1

12

14

16

18

2

Sign

al en

ergy

ActualL = 1L = 5

L = 10L = 20



L 1 L 5 L 10 L 20











1113957sL 1 L 5 L 10 L 20

f1(02934 03126

minus 0016)

(0293503044

minus 00134)

(0293503144

minus 00133)

(02920minus 0008703136)

f2(08098 08065

minus 0205)

(minus 000870801408189)

(0809008349)

(0802907948)

f3(09969 09979

00296)(0993809857)

(0997810013)

(0987909796)

f4 (08068 08045) (0801408012)

(0796907886)

(0808108057)

f5(03197 03047

minus 00317)

(029650289700098)

(000930305903050)

(0301803106

minus 00033)

1113958savg

(06453 064524minus 0041 00063

minus 00032 00059)

(0001705379063990002000027)

(00019064060648800027)

(06385000170640900007)






1113958savg 1113944P

p11113944

L

l1

1113958s

(p)

l

LP (18)



4 Conclusion


Data Availability




Acknowledgments


References










21Signal index (k)

0

1

2

3

4

5

6

Freq

uenc

y in

dex

(p)























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




1113958spavg 1113944

L

l1

1113957sp

l L (16)

where 1113957s

p







RMSE 1113944P

p11113944

K

k1

1113944

T

t1

1L

1113944

L

l1


211139741113972

PK (17)






0

02

04

06

08

1

12

14

16

18

2

Sign

al en

ergy

ActualL = 1L = 5

L = 10L = 20



L 1 L 5 L 10 L 20











1113957sL 1 L 5 L 10 L 20

f1(02934 03126

minus 0016)

(0293503044

minus 00134)

(0293503144

minus 00133)

(02920minus 0008703136)

f2(08098 08065

minus 0205)

(minus 000870801408189)

(0809008349)

(0802907948)

f3(09969 09979

00296)(0993809857)

(0997810013)

(0987909796)

f4 (08068 08045) (0801408012)

(0796907886)

(0808108057)

f5(03197 03047

minus 00317)

(029650289700098)

(000930305903050)

(0301803106

minus 00033)

1113958savg

(06453 064524minus 0041 00063

minus 00032 00059)

(0001705379063990002000027)

(00019064060648800027)

(06385000170640900007)






1113958savg 1113944P

p11113944

L

l1

1113958s

(p)

l

LP (18)



4 Conclusion


Data Availability




Acknowledgments


References










21Signal index (k)

0

1

2

3

4

5

6

Freq

uenc

y in

dex

(p)























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






1113958savg 1113944P

p11113944

L

l1

1113958s

(p)

l

LP (18)



4 Conclusion


Data Availability




Acknowledgments


References










21Signal index (k)

0

1

2

3

4

5

6

Freq

uenc

y in

dex

(p)























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


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