贝叶斯压缩感知

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OLIVERI et al.: RELIABLE DIAGNOSIS OF LARGE LINEAR ARRAYS-A BAYESIAN COMPRESSIVE SENSING APPROACH 4629 recognizing the failure-free elements (m=0)and the BCs IV. NUMERICAL VALIDATION based approach reliably(Ym=86x 10)detects the faults This section is devoted to the numerical assessment of the n order to numerically solve(7), let us consider that ac- BCS-based fa Filure diagnosis approach and, as a by-product,to cording to Bayes theorem the conditional probability P(F) deduce suitable user's guidelines. For the numerical analysis is given by the following indexes have been adopted to quantify the error P(4[P(四fP(f in detecting the array failures(normalized diagnosis error, $) (9) P(E) 100 1∑ (16) where P(Flf) is the " likelihood and P() and P(E)are the priors over f and F, respectively as for the sparseness of f, it is enforced in a probabilistic and the measurement noise on the pattern samples((c lanner (i. e, privileging sparse distributions) by considering, noise ratio, SNR [dBD a"peak-Shaped"prior of f[17,[18. Towards this end,a Gaussian hierarchical prior for P(f) is invoked [17],[18],[201, W(() 2 [21] SNR (17) p(f)= lim P(fh)p(hdh (10) c1,2→0 respectively where A. Far-Field Regular Angular sampling P(团)-(2x)-N2∏ P (11) The first set of numerical experiments is aimed at analyzing the sensitivity of the proposed approach on its control parame- ters, namely n and K, when the radiation pattern is uniformly 1. .. N being the n-th independent hyperparam- sampled within the range uk E[-1,1 eter controlling the strength of the prior over /n[17]. Moreover, h;7=1 心…N} complies with the Gamma distribu k k=1 F (18) tion p(h)= Q e d K-1 a1 and a2 being the corresponding improper(ol, 02+0) Towards this end, a benchmark N=100 Taylor array(PSL sCa 30dB, t=6 [1] has been considered by setting =4%. Concerning the likelihood term, it is re-written as [20] Concerning the dependence on n(i.e, the initialization of the RVM solver for the computation of omp--see the Appendix) P(EIf=/P(ELL, o])p(o)da (12) Fig. 2(a)shows the behavior of the diagnosis error for different values of the SNR. As it can be observed, an unique 'optimal P(EII a] being computed as follows [20] alue does not exist since each curve (i.e, SNR value) presents a different minimum point for $. More specifically and similarly P(E)=20(-2mE-EP (13) to imaging problems [201, the smaller the SNR, the higher is the value of est SNR, being mbest ] SNR=arg[min[S(n, SNR) However, because of the unfeasibility/complexity of an accu- and a4 being the associated improper(a3, Paa se dq, a rate estimate of the SNR in practical scenarios, a trade-off/com while P(o)=a4(o-1)a3-le-aA0/ o Scale priors promise value nopt has been determined and successively used [17],[20] within the following numerical validation By substituting(10)and (12 )in(9), it turns out that SNRm (P(FIL, o)P(Ih)P(o)P(h) best」 SNRdSNR ≈2.07×10 (19) PF dhd SNRmaV - SNR P(E where SNRmin= 20 dB and snrmax =60 dB P(, h, o F)dhd (14) As for the angular sampling of the radiated pattern(i.e, the value of K), the plot of s versus vk/N for variOuS SNRs Finally, the solution of (7)is determined as the mode of P(fF) is reported in Fig. 2(b). As it can be noticed, the diagnosis ac- (see the appendix) curacy depends in a non-negligible way on v especially when the noise level decreases(e.g,与」B5d3-4×10-1v IoJMP Lamp t diag(h)Mp) e'e (15) S SNR=60 db=3x10-4 while Ero.0 dB=8x 10-1 vS EE SISNR120 dB=5x 10-1). Therefore and analogously to the where()'stands for the"transpose" operator. As far as \cimp Setting of nopt(19), a trade-off value has been chosen and thmp are concerned, they are numerically computed by NRma △ IsNR-:Dbest」sNR dsR applying the rvm to(32)as detailed in Appendix Dopt ≈1.11 (20) SNRmax- snr rnIn 4630 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 60. NO. 10 OCTOBER 201 101m160AN10.00.=%y= Taylor Array, N=100, PSL=30 dB, t-6, V=Vopt m=nopt T,Φ=2% Φ=4% I,Φ=4% ξ,d=8% 10910310710610510-410310210-110010 SNR-40 dB SNR=30 dB SNR=50 dB c SNR [dB l aylor Array, N=100, PSL=-30 dB, t=6, p=4%, n=nopt 02 Dolph Array, N-100, PSL--30 de, v-vcpt, n-nopt r,=2%-…13107 0 d=4% E98%一 r,Φ=8% 。9 10 10085 0.9 1.05 SNR=20 dE SNR=40dB=SNR=60d一 SNR=30 dB 二 SNR=50 dB SNR [dB] Fig. 2. Sensitivity Analysis(Taylor, N= 100, PSL =-30 dB,t= Dolph, N=100, PSL=-3C dB, D=4% d=4%)Behaviour of s versus(a)n and(b)u when SNR E 20, 60]dB 四E Failures In order to assess the robustness of the calibration setup [ i.e (19)and(20), the diagnosis of the same array has been car- ried out by considering different failure rates and noise levels [Fig. 3(a)]. The plots of the integral error versus the SNR show WMA DNWM NANAAN that the bcs-based approach reliably performs whatever for 10 SNR 2 50 dB($ <102). As expected, s reduces proportion f,-SNR=40 dB ally to the to the presence of the array failures. Moreover, the I, I -SNA=20 dB I-SNR=60 dB plots of the total confidence levelT (c) Fig 3. BCS Assessment(N 100, PSl=-30 dB)-Behaviour of s and ∑ (21) versus SnR for a) fao array (t a and b)Dolph aray. Diagnosis of a in Fig 3(a) point out a key feature and an advantage of the BCS diagnosis. By definition, such an output from the bcs With reference to the scenario snr=60 dB. the confidence data-processing a-priori quantifies the uncertainty (i. e, the de- level value is equal tor=6.78X10- [Fig 3(b) and the diag gree-of-reliability)of the failure estimation. More specifically, nosis error turns out 5=8.93 x 10[Fig 3(b)] with an excel the lower I, the more faithful to the actual array configuration lent prediction of the failure distribution [Fig 3(C)green line the bCs prediction. However, since the behaviour of s and r Otherwise(e.g, SNR=20 dB), a value of r= 1. 23X 10-2 he also extended to that of an a-priori performance index of the and a poor failure detection ig. 3(c) magenta linei g 3(b) always turn out to be almost coincident. the meaning of r can [Fig 3(b)] corresponds to an error 5=4. x 10[Fi diagnosis (i.e, the smaller the f value, the better is the bcs The following numerical experiments are devoted to a com- failure detection and vice-versa). These deductions do not de- parative assessment taking into account a reference state-of- end on the considered aut and they are not limited to a single the-art technique. The l1-norm based technique developed in [7 lest case. As a matter of fact, similar proofs have been yielded has been chosen and then applied to the scenarios at hand. I The also when dealing with a Dolph arrangement [e.g. ,n= 100. plots of the diagnosis error as a function of d for the /=100 PSL=-30 dB--Fig 3(b)] as confirmed by the diagnosis ex The Ll-Magic" Matlab package [22 has been used for the numerica ample summarized in Fig 3(c)and related to the case =4%c. simulations OLIVERI et al.: RELIABLE DIAGNOSIS OF LARGE LINEAR ARRAYS-A BAYESIAN COMPRESSIVE SENSING APPROACH 4631 Dolph Array, PSL=-30 dB Φ=4% Taylor Arrey, N=100, PSL=-30 dB,t-6. v=optn=nopt BCS N=100 1,N=100 BCS. N=200 11,N=200 BCs,N=1000 L1,N=1000 。 BCS·SNF=30dB L1-SNF=30 dB Bcs·sNR=40dB L1- SNF=40 dB BGS·SNF=50dB 105L L1- SNP=50 dB 8 10 214161820 Φ[%] Dolph Array, PSL=-30 dB, V=Voot, n=nopt, p=8Y Dolph Array, N=100, PSL=-30 dB, v=Vopt n=nopt BCS. N=100 L1,N=100 BCS. N=200 BCs.N=1000 L1,N=1000 BCs·SNR=30dB L1- sNR=30 dB eses BCS- SNR=40 L1- SNR=40 dB BCS-SNR=50 dB L1-SNR=50 dB SNR [dB] (b) Dolph Array, PSL=-30 dB, v=vcpt m=nlopt, =12% 102 BCS. N=100 L1,N=100 Fig 4. Comparative Assessment(N= 100, PSL =-30 dB)-Behaviour BCS. N=200 of s versus SNR:(a) Taylor array(t=6)and(b) Dolph array L1,N=200 BCS. N=100 L1,N=1000 PsL =-30 dB Taylor [Fig 4(a)]or Dolph[Fig 4(b)] arrange ments reveal that, besides providing enhanced functionalities (i. e, the estimation of the " confidence level"not available when exploiting the l1 solver), the bCs method gives a non-negligible performance improvement whatever the Snr and for most p values [e. g. Fig. 4(a)]. Unlike the l1-norm approach, the BCs 10o SN [dB] accuracy reduces as the failure rate increases, although similar results only arise when very large qp indexes are at hand [e. g Φ≈20%—Fig.4(a) Fig. 5. Comparative Assessment(Dolph, PSL =-30 dB)Behaviour off Similar conclusions hold true also versus SnR obtained by the BCS and the e1 approaches when(a)= 1% for wider and wider aper-(b)中=8%,and()中=12% tures. As representative example. the reconstruction error for a set of Dolph arrangements with different dimensions are re- ported in Fig. 5. With reference to a failure rate equal to = 4% N= 1000 array while the technique needs about 50 seconds [PSL =-30 dB--Fig 5(a)], it turns out that, besides the for solving the same problem.3 non-negligible failure-prediction improvement, the li-approach does not work when the snr is lower than 30 dB because of the numerical instabilities [Fig. 5(a)]. The numerical results con B. Alternative Sampling Strategies cerned with more and more failures [e.g p=8%--Fig 5(b) a=12%C--Fig 5(c)] still confirm previous outcomes. 2 In the next numerical experiments, the dependence of the bCs accuracy on the choice of the angular measurements has very efficient even for large apertures thanks to the statistical been investigated. Towards this end, a Dolph array 30 db =0.02 snR =60 dB)has been taken formulation and the exploitation of the fast rvm solver. Fo example, it requires less than 2 sec conds for the diagnosis of a into account as a reference benchmark either assuming regular samples("regular"Wk =-1+2((k-1/(K-1)),h 2It is worth observing that such outcomes actually confirm the results on e1 and BCS solvers by other authors in different applicative areas (e. g, see [23], Each simulation has been run on a laptop pc with a single CPu at 2. 16 GHz Figs. 5 and 6), as well by using a non-optimized matlab code 4632 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 60. NO. 10 OCTOBER 201 Dolph Array, N=200, PSL=30 de, p=2%, n SNR=60 dB Taylor Array, N=20, PSL=-20 dB, 1=6, p=10%, Non-Uniform Sampling(v=0. 25) W(u)2 0 104 BCS-Regular Samples L1-Regular Samples" 10 BCS.Rando L1· Random samples 00.1020.3040.5060.70.80911.11.21.3 1-080.6-0.4-0.20020.40.60.8 Fig. 6. Comparative Assessment [N= 200, PSl =-30 dB, =0.02 SNR=60 dB]Behaviour of s versus 1 obtained by BCS and Li approaches Taylor Array, N20, PSL=-2C dB, t=5, p=10%, Non-Uniform Sampling(v=0.25 for uniform and random samples 0B口 ◆Qa口口 1,.,K--Fig 6or random samples complying a uniform dis tribution within the whole visible range("random"P(u(k)) Diagnosed Faults 1/2 for u(k)E [-1, 1]-Fig. 6). As for these latter and to g Failures give a statistical relevance to the numerical analysis, 1000 dif- 802 ferent realization of the random process generating the u(k)set have been considered for each K value to compute the av erage s index. The plots of the accuracy as a function (Fig. 6)show that both BCS and li solvers achieve better per n■Nn|·ifnl- BCS x hint formances when a regular" displacement is adopted since ran domizing the angular samples usually yields to poorly-condi- Fig. 7. Non-Regular Sampling(Taylor, N=100, PSL=-30 dB, t=6 tioned E matrices with inaccurate or unreliable failure predic- SNR =60 dB). Power patterns (a) and array layouts (b). Reference array tions(≈40Fig.6) magenta Although a completely random sampling seems of no use, the same does not necessarily hold true for other non-regular arrangements. To investigate such a case, a strategy which more densely samples the pattern around its maxima(assumed at to rather than to M when(22)is adopted. Indeed, a value broadside, for simplicity) has heen analyzed. More specifically, v=0.3 allows a faithful failure detection up to =4%, while the following rule has been chosen v=0.5 is required when =8% [Fig 8(a)]. This is not an unexpected outcome, since from the CS theory [16] the number k-1 of measurements necessary to reliably retrieve a sparse vector l,k k=1,.,K,(22) must be proportional to the number of its non-zero entries[151 [16] B> 1 is a user-defined odd integer that establishes On the contrary, the same does not hold true for classical the sampling ' degree' within the angular range [6= 1 corre- linear schemes, such as those based on the truncated SVD sponds to(18)]. The plots of the sampled power pattern [B= (TSVD)[24 ]. Indeed, the plots of f as a function of v in 3--Fig. 7(a)] and the arising diagnosis [Fig. 7(b)] for the case Fig 9(Dolph array, N= 200, Psl =-30 dB 4%) in Fig. l(a), but assuming K=5, indicate that a quite accurate show that the tsvd method is not able to faithfully detect detection is yielded despite v l thanks to the non-uniform lo- the faulty arrangements whatever the snr when us 1 [e.g cation of the pattern samples. This suggests that( 22)can enable <TSVD g 1.2x 10-1 when SNR=60 dB, v=0.8(uniform a reduction of K well below n sampling)Fig. 9]. Unlike the BCs (red lines--Fig. 9), the To further assess such a BCs feature, the values of either application of (22) gives strongly unstable TSVD solutions as using(18)or(22)have been computed in correspondence with pointed out by the f svd behaviour for non-uniform samples a Taylor arrangement (N= 100, PSL 30 dB. SNR ith B=3 (blue lines--Fig. 9) 60 dB) and different failure rates [Fig. 8(a)]. As it can be no- Concerning the dependence of the bCs accuracy on B(22) ticed, the non-uniform sampling always outperforms the uni- choosing 6=0.3 usually gives the lowest s whatever the failure form approach in terms of accuracy [Fig. 8(a)] and it gives a rate and v [e.g., 3=0. 3=5.34x10-3 vs $)3=0.9=4. when reliable failure diagnosis($ 10)even when v< 1[e.g, v=0.5, =8%Fig 8(b)], while a strong oversampling of SN1.74X10-3(B=3)VS E A 4.06 when v=0.1,p=2%]. the mainlobe region, which takes place for large B [see(22) Moreover, it turns out that the trade-off w has to be proportional does not improve the diagnosis effectiveness OLIVERI et al.: RELIABLE DIAGNOSIS OF LARGE LINEAR ARRAYS-A BAYESIAN COMPRESSIVE SENSING APPROACH 4633 Taylor Array, N=100, PSL=-30 CB, t=6, SNR=60 dB Dolph Array, N=100, PSL=-3C dB: p =2% 000 10 5=10 界界是"=====界 mawaessena-es n0 0.30.405C60. 0.10.20.3040.50.60.708 v [normalized value y normalized value] Uniform. SNR=40 Uniform SNR=60 unom,Φ=2%— Uniforn,d=4%一 Uniform,=8% B=3,SNR=40 B=3.SNR=60 =3,①=2%- 阝=3,=4% 阝=3,①=8% Uniform Near Field. SNR=40 三 Uniform Near Field, SNR=60 ---- Fig. 10. Far-Field vs. Near- Field Sampling(Dolph, N= 100, PSL Taylor Array, N=100, PSL=-30 dB, t=6, SNR=60 dB, Non-Unifcrm Sampling 30 dB, op= 2%) Behaviour ofe versus v for ditferent sampling methods 02 and snr values 10x where p=- (0, 0, 2n) is the position of the n-th array element and p() is the k-th near-field probe position. As an example, let us consider a uniform near-field sampling (k) 0. R sin (k-1) (k-1) K ×一- K-1 K-1 (23) R being the distance in wavelengths between the array center p=4%,V=0.1-×-Φ=4%.V=0.3-米-=4%,V=0.5-米 Φ=8%,v=0.1-鲁-Φ=8%,v=0.3- =8%,v=0.5-鲁 and the measurement probes. The plots of s vS v in Fig. 10 ( Dolph array,N=100,PSL=-30dB,Φ-2%,R=75入) show that a near -field sampling yields strong improvements of Fig 8. Non-Regular Sampling (Tay lor, N=100,PSL=-30 dB,(=6). the diagnosis fidelity with respect to the far-field uniform mea Behaviour of E versus(a)v and(b)B obtained by BCs surement scheme and similar to that of (22) for high SNRs(e. g £F≈1.51×10-3vs.NF≈2.89×10-3for=0.1 SnR=60 dB--Fig. 10). As for this latter, better performances Dolph Array, N=200, PSL=-30 dB, p=4% are obtained in correspondence with more corrupted data(e.g H≈1.51×10-3vs.EM≈2.89×10-3forv=0.3, SNR=40 dB--Fig. 10). However, these improvements can be yielded at the expenses of the complexity of the measurement BCs,β=3,SNR=40dB process that requires the array to be accessible or in a controlled TSVD, Uniform, SNR=40 dB 10 TsvD.阝=3.SNR=40dB enviro BCS,阝-3SNR-60dB= TSVD, Uniform. SNR=Go dB 05 =10 TSVD,阝=3,SNR=60dB C. Detection of Partial Failures The final experiment is aimed at preliminarily assess the po tentials of the proposed technique in detecting"partial failures owards this end, let us model the partial failures according to 020.30 C.50.60708 the following equation instead of using(3) v normalized value 下 Un with probabilily dp 4%C) Behaviour of E versus v for BCS and TSVD approaches 30 dB, d Fig 9. Non-Regular Sampling(Dolph, N= 200, PSl= otherwise (24) k(k< 1) being the"failure factor. The same layout in Fig 10 has been diagnosed by assuming =4% and K E [0.0, 0.9 Besides(22), other sampling strategies can be exploited to SNR∈{40.50.60,70}dB-Fig.1l reduce v. As an example, near-field measurements can be also The plots of the normalized diagnosis error as a func- taken into account by just substituting(1)with tion of k show that the bcs technique reliably detects partial failures whatever the r index when Snr 50 dB (E 10Fig. 11). On the contrary, the diagnosis turns out to be less accurate if lower SNRs are at hand, especially when0.3(e.g,ξ≈7.,2×102 when k=0.7, Wn,exp i2T(p K).P), k=1,.,K SNR =40 dB--Fig. 11). As a matter of fact, the informative content of the difference field F(uk))tends to vanish as n-1 4634 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 60. NO. 10 OCTOBER 201 Dolph Array, N=100, PSL=-30 dB, c=4%, V=0. 4, B=3 v value [Fig. 2(b), while n must be set within the range 0.2≤≤2[Fig.2a)]; there is a trade-off between measurement setup complexity and the required samples K to reliably diagnose a layout of size N. Indeed v cannot be significantly below l if a regular far-field sampling is employed [i.e,(18)]. On the contrary, non-uniform far- field [(22) or near-field [(23) setups correspond to significantly smaller values ofv [e. g Fig. 8(a)l at the expenses of more complicated measure ment procedures 00.1020.30.4050.6070.80.9 Moreover, the numerical assessment has pointed out the fol- k [arbitrary unil] lowing outcomes SNH=40 dB SNH=50 dB- SNR=60 dB sNH=70dB一 as expected, the reliability of the BCs-based approach de Fig. 1l. Detection of partial failures(Dolph, N=100, PSL =-30 dB des as g increases(Fig 3) d=4%,U=0.4,8=0.3) Behaviour of E versus K the bcs positively compares with the state-of-the-art l1-based Cs diagnosis in terms of accuracy and robustness to noise whatever the array size [e. g because of /, 0 in(4). Accordingly, failure factors ap (5e1/BCs)≈6.5×101 when Snr=40dB,Φ=4%, proaching unity yield to more challenging diagnosis problems N=1000-Fig 5(a). moreover, the range of applica- when strongly corrupted data are at hand(Fig. 11) bility of the bcs technique extends also to heavier noise conditions (i.e SNR. 30 dB), where the l1-approach does not work [Fig. 5(a). Similar considerations still V. CONCLUSIONS hold true when comparing the bCs with linear inversion sche TSVD(Fig 9 The far field diagnosis of large linear arrays has been car- a"regular"angular sampling of the far-field radiation pat ried out through an innovative bCs technique able to provide tern [i.e, according to(18)or(22)] should be preferred the degree of reliability of the fault detection. Towards this pur- to ' random? measurements since this latter can cause the pose, a Bayesian formulation of the array diagnosis problem has degradation of the BCS prediction(e.g, N 1.0--Fig 6) been introduced and a fast rVm solver has been applied to de -. the proposed approach can be also extended to detect par- termine the array failures. A set of representative numerical ex tial failures although further investigations on its reliability amples has been illustrated to provide suitable insights on the are still needed performances, the potentialities, and the current limitations of Future works, out-of-the-scope of this paper, will be aimed at the Bcs-based diagnosis more extensively assessing the reliability of a Bcs-based ap- The main methodological innovations of this work rely in the proach for the diagnosis of linear arrays from near field mea- following key-features surements, as well. Moreover, the application to planar archi- a new formulation of the array diagnosis problem that en- tectures is at present under investigation ables the estimation of the local( 8)] and the total [(21)] APpeNDiX confidence-level/reliability-index of the diagnosis jointly with the retrieval of the element failures [(15)1 DERIVATION OF(15)AND OF (8) 2)a fast CS-based failure diagnosis technique suitable for By means of Bayes'theorem, the integrand in(14) can be (but not limited to) fair-field radiation measurements written as From the numerical analysis, the following guidelines for an optimal application of the BCS-based approach to the failure P(,12叫F)=P(fEF,,)P(b,叫F).(25) detection in linear arrays have been deduced without any a-priori knowledge on the level of the mea- The first term of (25 )can be expressed as surement noise or the scenario at hand, there exists a proper setup of the bcs calibration parameters [(19)and(20) P(|E,,) P(E[,])P( that enables faithful failure predictions in a wide range of P(四[h,]) working conditions (i. e, linear array architectures, SNR and its explicit form is equal to values, and failure rates) further accuracy improvements can be obtained when (f-m)S-1(f eXD some a-priori information on the noise level is avail- able (Section IV). More specifical y, optim perfor- P(fIlF, h, a)) (2x)+√S mances are yielded when 1 v 1.15 [Fig. 2(b) and 10-4 n) s 10[Fig. 2(a)] in the presence of since [17] low levels of noise(e.g. snr 2 30 dB) and if (18) is assumed. Otherwise(SNR< 20 dB), bCs detection exp[-2F'C-1E) accuracy is only marginally affected by the choice of the P(F[b,叫]) (2T)IVC (28) OLIVERI et al.: RELIABLE DIAGNOSIS OF LARGE LINEAR ARRAYS-A BAYESIAN COMPRESSIVE SENSING APPROACH 4635 C=o+EH-IE/, H= diag(h)] and the factors P(fh)and As far as the evaluation of h/MP and ioJMp P(FILf o])are reported in(11)and(13), respectively. In(27), is concerned, a RVM solver [17],[18],[20].[251 is applied to (32) starting from the initial setup OMP M P QuEF tho, being hn, 0= (29) 1exp(272()2/(Rr-m)( )where Rn= (k) Cxp(22T2n u(k) /∑ k=1OXp(2丌n and I=arimax[Rn]h EE 1 Finally, the knowledge of the variance ofP(fF) is required +H (30)[19] to determine the confidence level of the BCS estimate Since (36) models a multivariate Gaussian distribution, the are the mean and the covariance of the posterior(26) confidence level"y of f is the diagonal vector of the covari- As for the second term of(25), P(h,oE), a closed-form ance matrix S[19] as indicated in(8) computation is unfeasible and, usually, a"delta-function ap- proximation is employed (i.e, the unknown distribution is mod ACKNOWLEDgmENT elled with a pulse centered in its mode [17). Towards this end, The authors wish to thank Dr. s.Ji. Dr. y. Xue and prof. Since L Carin for sharing the bcs code online 25 moreover, the P(位,]F)∝P(E|,])P(bP(a)(31) authors are indebted to Prof, m.d. migliore for fruitful discus sions on the subject. and noticing that P(h)x const and P(o)c const (i.e they represent improper hyperpriors [17), it turns out that REFERENCES P([h:d]F)∝P(F:可]) [IR J Mailloux, Phased Array Antenna /handbook, 2nd ed. Norwood, MA: Artech house. 200 P(,以E)a∞上(E1E [2 G. Oliveri, M. Donelli, and A Massa, "Linear array thinning exploiting (2)sVC almost diffcrcncc scts, IEEE Trans. Antennas Propag., vol. 57, no 12 pp.3800-3812,Dec.2009 3 G. Oliveri, L. Manica, and A. Massa, ADS-based guidelines for [(28), whose mode is thinned planar arrays, IEEE Trans. Antennas Propag. vol. 58, no. 6, pp.1935-1948,Jun.2010 (h)MP, o]Mp)=arg max(h exp[-号(EC-1F) [4]TJ. Peters, A conjugate gradient-based algorithm to minimize the sidelobe level of planar arrays with element failures, IEEE Trans. An (2)√C tenna Propag., voL. 39, no. 10, pp. 1497-1504, Oct 199 arg max(h, g) C(h a)(32) [5]J. A. Rodriguez-Gonzalez, F. Ares-Pena, M. Fernandez-Delgado, R Iglesias, and S. Barro, " Rapid method for finding faulty elements in antenna arrays using far field pattern samples, IEEE Trans. Antennas re[17] l.57,no.6,pp.16791683,Jun.2 [6] M. Carlin, G. Oliveri, and A Massa, On the robustness to element fail c(h [NJg2丌+lgC|+P"C-](3 ures of linear ADS-thinned arrays, IEEE Trans. Antennas Propag vol.59,no.12,pp.48494853,Dc.2011 [7] M. D. Migliore, "A compressed sensing approach for is the logarithm of the marginal likelihood. Then, by introducing from a small set of near-field measurements. IEEE Trans Antennas the delta approximation [17], the second term of (25) can be Propag. vol. 59, no 6, pp. 2127 2133, Jun. 2011 [8]R. Iglesias, F. Ares, M. Delgado, J. Rodriguez, Bregains, and s now expressed as Barro, "Element Lailure detection in linear antenna arrays using case- based reasoning, IEEE Antennas Propag. Mag, vol. 50, no. 4, pp P(h,可]E)≈6(-{}MP,-{σ}MP).(34) 198-204,Aug.2008 [9J. A Rodriguez, F. Ares, E. Moreno, and G. Franceschetti, " Geneti algorithm procedure for linear array failure correction, "Electron. Lett. By substituting(34)and(27)in(25), we obtain vol.36,no.3,pp.196-198.Feb,2000. P(,,B)≈6(1-{MP,-{0}M) [10]JJ. Lee, E M. Ferrer, D. P. Woollen, and K. M. Lee, " Near-field probe used as a diagnostic tool to locate defective elements in an array an- (2丌 tenna, IEEE Trans. Antennas Propag., voL 36, no 3, pp. 884-889 Jun.1988 (-m)'s -(f [11]O.M. Bucci, A Capozzoli, and G. D'Elia, "Diagnosis of array faults x exp (35) from far-field amplitude-only data, IEEE TranS. Antennas Propag vol.48,no.5,Pp.647652,May2000 [120 M. Bucci, M.D. Migliore, G Panariello, and G. Sgambato,"Accu that when applied in(14)gives the following expression rale diagnosis of conformal arrays froin near-field data using the maLrix method, IEEE Trans. Antennas Propag., vol 53, no. 3, pp. 1114-1120, s Mar.2005 exp [13A Buonanno and M. D'Urso, On the diagnosis of arbitrary geom- P(|F) fully active arrays, pre (2x)N+v√/S (EuCAP), Barcelona, Apr. 12-16, 2010 [14 E.J. Candes and T. Tao, " Decoding by linear programming, ILLE h=hmP, o=omp Trans. Inf. Theory, vol. 51, no. 12, pp. 4203-4215, Dec. 2005 (36) [15]R GBaraniuk, " Compressive sampling, "IEEE Signal Process. Mag Since(36)is a Gaussian function, the mode ofP(f F)actually Pp.118-124,Jul.2007. coincides with its average value m. Therefore, (15)is yielded [16 E J Candes and M. B Wakin, "An introduction to compressive sam pling "IEEE Signal Process. Mag, vol. 25, no. 2, pp. 21-30, Mar by computing(29)with h= h mp and o =i 2008 4636 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 60. NO. 10 OCTOBER 201 [17] M.E. Tipping, "Sparse Bayesian learning and the relevance vector ma Paolo Rocca (M,08) received the M.S. degree chine, J. Machine learning Res, vol 1, pp. 211-244, 2001 (summa cum laude) in telecommunications engi [18 M. E. Tipping and A C Faul, Fast marginal likelihood maximization nccring and the Ph. D. dcgrcc in information and for sparse Bayesian models, "in Proc. 9th Int. Workshop Artificial In communication technologies from the university of telligence statistics, C. M. Bishop and B. J. Frey, Eds, 2003 [Online Trento, Italy, in 2005 and 2008, respectively. Available:http://citeseer.ist.psu.edu/611465.html He is currently an Assistant Professor at the De [19S.Ji, Y. Xue, and L. Carin, "Bayesian compressive sensing, IEEE partment of Information Engineering and Computer Trans. Signal Process., voL. 56, no 6, pp. 2346-2356, Jun 2008 Science, University of Trento and a member of the 20 G. Oliveri, P. Rocca, and A. Massa," A Bayesian compressive ELEDIA Research Center. He is the author coauthor pling-based inversion for imaging sparse scatterers, IEEE Tro of over 100 peer reviewed papers on international [21] G. Oliveri and A. Massa, "Bayesian compressive sampling for pat- dent at the Pennsyivani, ournals and conferences. He has been a visiting stu- Geosci Remote Sens, vol. 49, n0. 10, pp. 3993-4006, OcL. 2011 State University and at the University Mediterranea term synthesis with maximally sparse non-uniform linear arrays, " IEEE of Reggio Calabria. His main interests are in the framework of antenna array Trans. Antennas Propag vol 59, no. 2, pp. 467-481, Feb. 2011 synthesis and design, electromagnetic inverse scattering, and optimization tech [22] J. Romberg, " Ll-magic compressive sensing code, 2011 [Online]. niques for electromagnetics Available:http://users.ece.gatech.edu/justin/llmagic/ Dr rocca received the best ph d. thesis award from the ieee geoscience [23 J. Xu, Y Pi, and Z Cao, "Bayesian compressive sensing in synthetic and Remote Sensing Society Central Italy Chapter Ile serves as an Associate aperture radar imaging, IET Radar: Sonar Navigation, vol 6, no. 1, editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTER.S: ociate pp.2-8,Jan.2012 [24]W.H. Press, S. A. Tcukolsky, W. T. Vcttcrling, and B P. Flanncry York: Cambridge UniV Press, 200/7 tific Computing, 3rd ed. New Numerical Recipes The Art of scie Andrea Massa(M,96) received the"Laurea?"degree 125 SJi,Y Xue, and L Carin, "Bayesian compressive sensing code, 2009 in electronic engineering and Ph. D. degree in elec LoNline].Available:http:/ipeople.ee.duke.edu/lihan/cs/ tronics and computer science from the University of Genoa, Genoa, Italy, in 1992 and 1996, respectively From 1997 to 1999 he was an Assistant profe of clcctromagnctic ficlds at thc Dcpartment of Bio physical and Electronic Engineering, University of Genoa, teaching the university course of Electromag- tic Fields 1.F sociate Professor at the university of Trento. Since 2005, he has been a Full Professor ofelectroimagnelic fields at the University of Trento, where he currently teaches electromagnetic fields, inverse scattering techniques, antennas and wireless communications, Ind optimization techniques. At present, he is the Director of the ELEDIA Re Giacomo Oliveri (M,09)received the B.S. and M.s. search Center at the University of Trento and Deputy Dean of the Faculty of degrees(both summa cum laude)in telecommunica- Engineering. In addition, he is an Adjunct Professor at Penn State University tions engineering and the Ph. D. degree in space sci- and Visiting Professor at the Missouri University of Science and Technology ences and engineering from the University of Genoa, the Nagasaki University (Japan) and at the University of Paris Sud(france). His Italy, in 2003, 2005, and 2009, respectively research work since 1992 has been principally focused on electromagnetic di- Hc is currently an Assistant Professor at thc Dc- rcct and invcrsc scattering, microwave imaging, optimization tcchniqucs, wavc partment of Information Engineering and Computer propagation in presence of nonlinear media, wireless communications and ap Science, University of Trento and a member of the plications of electromagnetic fields to telecommunications, medicine and bi ELEDIA Research Center. He is the author/coauthor ology of over 100 peer reviewed papers on international Prof. Massa is a member ofthe Piers Technical Committee of the Inter - Uni jouRnals and conferences. He has been a visiting re- versity Research Center for Interactions Belween Electronagnetic Fields and searcher at the University of Paris Sud, France. His research work is mainly Biological Systems(ICEmB)and Italian representative in the general assembly focused on cognitive radio systems, electromagnetic direct and inverse prob- of the European Microwave Association(EuMA). He serves as an Associate lems, and antenna array design and synthesis Editor of the IeEe transactions on antennas and propagation

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