ARRAY CALIBRATION WITH MODIFIED ITERATIVE HOS-SOS (MIHOSS) ALGORITHM

Transcription

1 19th European Sgnal Processng Conference EUSIPCO 2011 Barcelona, Span, August 29 - September 2, 2011 ARRAY CALIBRATION WITH MODIFIED ITERATIVE HOS-SOS ALGORITHM Metn Aktaş, and T. Engn Tuncer Electrcal and Electroncs Engneerng Department, Mddle East Techncal Unversty, Ankara, TURKEY maktas@eee.metu.edu.tr, etuncer@metu.edu.tr sam/ndex.html ABSTRACT Jont drecton-of-arrval DOA and sensor poston estmaton for randomly deployed sensors s ntroduced n Iteratve HOS-SOS IHOSS algorthm [1]. IHOSS algorthm explots the advantages of both hgher-order-statstcs HOS and second-order-statstcs SOS wth an teratve algorthm usng two reference sensors. The teratve algorthm s guaranteed to converge. IHOSS algorthm solves the poston ambguty by usng source sgnals observed at multple frequences and hence t s applcable for wdeband sgnals. In ths paper, we propose Modfed-IHOSS algorthm to solve the same problem for narrowband sgnals. In, t s assumed that the nomnal sensor postons are known. It s shown that ambguty problem s solved effectvely wthout any assumpton on the poston perturbatons. The upper bound of perturbatons for unambguous sensor poston estmaton s presented. The performance of approaches to the Cramér-Rao bound for both DOA and poston estmaton. 1. INTRODUCTION The devaton of array parameters from the assumed model generates errors for the array processng applcatons ncludng DOA estmaton and beamformng. Gan/phase msmatch of antennas, mutual couplng and antenna poston errors are some examples of array modelng errors. Array calbraton s the task of estmatng the errors n array model as well as the DOA and source parameters. In ths paper, we focus on array calbraton for the sensor poston errors. Sensor poston errors are mportant n practcal applcatons where the sensors are dstrbuted n a wde area or there are sensor dsplacements due to the platform as n the case of sensors on the wng tps of a plane. In ths paper, jont DOA and sensor poston estmaton s done n a settng where the nomnal sensor postons are known. In ths case, t s assumed that there are two reference sensors whose postons are known perfectly. The rest of the sensors are dstrbuted randomly n a large area. Whle the nomnal postons of the dstrbuted sensors are known, there s no assumpton on the perturbaton for the sensor postons. Note that the problem defned above s dfferent than the partally calbrated arrays PCA [3] snce the number of sources s not restrcted to be less than the number of reference sensors. Furthermore, the sensor postons are estmated n our case as opposed to [3]. In the lterature, array calbraton problem for the sensor poston errors s nvestgated n two settngs, namely small error [4] and large error approxmatons [5]. In small error approxmaton, the perturbatons are assumed to be small and array calbraton s performed by usng a frst order approxmaton. The frst order approxmaton s not applcable as the perturbatons are ncreased. Large error approxmaton [5] s proposed to crcumvent the lmtatons of the small error approxmaton. However the DOA estmaton problem s consdered for a unform crcular array and for some fxed source DOAs. One of the man problems n sensor poston estmaton from the knowledge of source observed data s the ambguty n sensor postons. Ambguty arses due to the wrap around n array steerng matrx phase terms. Prevously IHOSS algorthm [1], [6] s presented, whch jontly uses HOS and SOS approaches teratvely for the estmaton of both source DOAs and sensor postons. The teratve process s guaranteed to converge. In IHOSS algorthm, except the two reference sensors there s no a pror nformaton about the sensor postons. The postons of the two reference sensors are assumed to be known. IHOSS algorthm consders the ambguty problem n sensor poston estmaton and solves the problem by usng the source sgnals observed at multple frequences. Hence t s applcable for wdeband sgnals. In ths paper, IHOSS algorthm s modfed for the narrowband sgnals and the new algorthm s called as. Snce for the narrowband case, source sgnals can only be observed at sngle frequency, requres to know the nomnal sensor postons to solve the ambguty problem. It s proved that the ambguty problem can be solved f the perturbatons are bounded. The upper bound for the perturbatons s also presented. Both IHOSS and algorthms can effectvely be used n the array calbraton problem for the sensor poston errors for dfferent applcatons. Snce dfferent assumptons are used for IHOSS and algorthms, the comparson between them s not far. 2. PROBLEM STATEMENT It s assumed that the array s composed of randomly deployed M sensors and there are L far-feld sources. Two sensors are selected as the reference sensors. The sensor postons are randomly perturbed from ther nomnal postons except the reference sensors. The postons of the reference sensors are assumed to be known and the dstance between them s less than or equal to λ/2, where λ s the wavelength of the ncomng source sgnals. Under these assumptons, the receved sgnal vector for the sensor array can be wrtten as, xt=aθ,p 0 + Pst+vt, t = 1,2,...,N 1 where, N s the number of snapshots, st=[s 1 t,...,s L t] T s the L 1 vector of L sources, vt s the M 1 vector of Gaussan nose. Source sgnals are assumed to be EURASIP, ISSN

2 non-gaussan and they can be correlated but not coherent. Nose s assumed to be statstcally ndepent wth the source sgnals. Θ =[θ 1,...,θ L ] s the source DOA vector, P 0 =[p 0 1T,...,p 0 T M ] T and P =[ p T 1,..., pt M ]T are the nomnal sensor postons and the perturbatons n postons, respectvely. AΘ,P s the M L array steerng matrx, composed of, [ ] { AΘ,P 0 + P = exp j 2π [ p 0 m λ m,x + p m,x cosθ + p 0 m,y + p m,y snθ ]} 2 where, θ s the drecton-of-arrval of th source n azmuth, p 0 m =[p 0 m,x, p 0 m,y] and p m =[ p m,x, p m,y ] are the 2D nomnal poston of the m th sensor and the 2D perturbaton of the m th sensor poston, respectvely. Snce the postons of the two reference sensors are known, ther perturbatons are zero,.e., p m = 0, m = 1,2.. T s the transpose operator. The goal n ths paper s to estmate both DOAs of L sources and the perturbaton parameters of M 2 sensors. 3. ALGORITHM In ths secton, algorthm s ntroduced for a soluton to the problem descrbed n Secton 2. algorthm s based on the IHOSS algorthm [1], whch uses the HOS and SOS approaches jontly. The basc dfference between the IHOSS and s ther soluton of the ambguty n sensor postons. IHOSS algorthm requres observatons at multple frequences. On the other hand, MI- HOSS uses the nomnal sensor postons to solve the ambguty problem and can be appled for narrowband sgnals. 3.1 HOS Based Blnd DOA Estmaton In [7], t s shown that HOS approach can be used to fnd the DOA and array steerng matrx estmates for random sensor geometres wthout knowng the sensor postons except the two reference sensors. In ths respect cumulant matrx composed of fourth-order cumulants are used together wth the ESPRIT algorthm. On the other hand, as explaned n [1], ths approach can be employed for DOA estmaton as long as the source sgnals are ndepent. IHOSS algorthm [1] overcomes ths lmtaton by proposng a new cumulant matrx estmaton technque, whch s more robust to the depency between source sgnals,.e., L C = A 1 C s A H 1 A 1 C s A H 2 =1 A 2 C s A H 1 A 1 C s A H 3 1 where  s the estmate of the array steerng matrx, A, and A j = Q A q j A, j {1,2}. q j s the complex conjugate of the j th row of the matrx Q = I ÂZ Â.. s the Moore-Penrose pseudonverse operator. Z s the L L dagonal matrx whose dagonal elements are one except the th element. The th element s set to zero. C s s the L 2 L 2 source cumulant matrx n the form of, C s, j=cums k t,s l t,s mt,s nt 4 = Lm 1+l, 1 m,l L j = Ln 1+k, 1 n,k L Note that the cumulant matrx estmate n 3 s a generalzed cumulant matrx estmate whch mproves the parameter estmates depng on the accuracy of the array steerng matrx estmaton. In [1], t s shown that f the actual array steerng matrx s known,.e.,  = A, the cumulant matrx estmate n 3 smplfes to, [ AR HOS C = s A H AR HOS s DA H ] AD H R HOS s A H AR HOS s A H 5 L L dagonal matrces R HOS s R HOS and D are defned as, s = dagγ 1,γ 2,...,γ L 6 D = dag e j 2π λ Δcosθ1,...,e j 2π λ Δcosθ L 7 where γ = Cums t,s t,s t,s t and Δ λ/2 s the dstance between the two reference sensors. The reference sensors are assumed to be located at 0,0 and Δ,0 on the coordnate system for smplcty. Note that, the cumulant matrx n 5 s n the same form of the correlaton matrx n the ESPRIT algorthm. The only dfference s the source correlaton matrx defned for SOS s replaced by R HOS s. Snce the cumulant matrx n 5 s not avalable n practce, ts estmate n 3 s used for the parameter estmaton. The DOA and array steerng matrx estmates are found from the egenvalue decomposton of C,.e., CS = SΛ s as n the ESPRIT algorthm. Λ s s the dagonal matrx composed of the L largest egenvalues of the matrx C and 2M L matrx S =[ S T 1 S T 2 ]T s obtaned from the egenvectors correspondng to these egenvalues. S 1 and S 2 are M L matrces. The DOA and the array steerng matrx estmates are found by applyng the ESPRIT algorthm,.e., ˆθ = cos 1 Φ, 2πΔ λ 8 A = S 1 Ψ 9 where Φ, s the phase term of the th dagonal element of the matrx Φ. L L dagonal matrx, Φ, and L L matrx, Ψ, are related as, S 1 S 2Ψ = ΨΦ 10 Note that knowng the dstance and the drecton between the two reference sensors are suffcent for the DOA estmaton as n 8. However, t s not the case for the array steerng matrx estmaton. In the ESPRIT algorthm the array steerng matrx estmaton s found up to an unknown scale factor as n 9. To fnd the scale factor, n addton to the dstance and the drecton between the two reference sensors, t s requred to know one of the reference sensor poston. Snce t s assumed that the frst reference sensor s located at 0,0, the frst row of the array steerng matrx has to be consst of all ones. Then, the actual array steerng matrx can be found from 9,.e.,  = AH 1 11 where H = daga 11,a 12,...,a 1L and a j s the th row and j th column of matrx A. 615

3 3.2 Unambguous Sensor Localzaton Once the DOA and array steerng matrx estmatons are found, sensor locatons can be estmated usng 2. Due to 2π ambguty, the elements of the array steerng matrx n 2, correspondng to m th sensor and th source can be rewrtten n the followng form, a m, = e j 2π λ [p 0 m+ p muθ λk m,] 12 where, k m, s an nteger specfed for the m th sensor and the th source. uθ =[ cosθ, snθ ] T s the unt drecton vector of the th ncomng source. When all the ncomng sources are consdered, the followng relaton can be wrtten, p 0 m + p m U ˆΘ= λ 2π ˆΞ m + λk m, 1 m M 13 where ˆΞ m = [ â m,1 â m,2... â m,l ] 14 k m = [ k m,1 k m,2... k m,l ] 15 U ˆΘ = [ u ˆθ 1... u ˆθ L ] 16 ˆx stands for the estmaton of x and â m, s the phase term of the array steerng matrx element estmate n 12. The poston perturbaton of the m th sensor can easly be found from 13 n the least squares sense as, λ ˆ p m k m = 2π ˆΞ m + λk m U ˆΘ p 0 m, 1 m M 17 Note that the poston perturbaton estmate n 17 takes dfferent values for dfferent k m values. Therefore, ˆ p m k m values are consdered as the ambguous poston perturbaton estmates of the m th sensor. If the poston perturbaton s lmted, the ambguty problem can be solved by selectng the sensor poston perturbaton estmate wth mnmum norm,.e., ˆ p m = argmn k m ˆ p m k m 18 The upper bound of perturbatons for unambguous sensor poston estmatons s gven n Lemma-1. Lemma-1: Let d m be the mnmum dstance between actual and estmated poston perturbaton of the m th sensor,.e., ˆ p m k m p m d m. Then, the ambguty problem n sensor poston estmaton s solved f the followng condton s satsfed for, 1 m M, k 1 m k 2 m U ˆΘ max p m < λ mn 2 d m k 1 m k 2 m m k m 19 The proof of Lemma-1 s not gven due to space lmtatons. 3.3 SOS-Based Algorthm Sensor poston matrx estmate, ˆP = P 0 + ˆ P, s constructed usng 18 wth the nomnal sensor postons and used n the algorthm to generate the pseudospectrum,.e., Γθ = a H θ, ˆPGG H aθ, ˆP 1 where G s the M M L matrx whose columns are composed of the egenvectors correspondng to M L smallest egenvalues of the correlaton matrx obtaned n the SOS approach. The DOA and the array steerng matrx estmates for the SOS approach are obtaned by fndng the L largest peaks of the pseudospectrum,.e., { } L ˆθ = argmaxγθ 20 =1 θ Â = [ a ˆθ 1, ˆP,a ˆθ 2, ˆP,...,a ˆθ L, ˆP ] The Cost Functon and The Algorthmc Steps algorthm teratvely updates the DOA and array steerng matrx estmates usng the HOS and SOS approaches sequentally as summarzed n Table 1. The cost functon used at each teraton to select the best array steerng vector estmates for each source s defned by the pseudospectrum,.e., Γâ = â H GG H â 1 22 where â are the array steerng vector estmate for the th source. Note that the cost functon, Γâ s non-negatve. At each teraton, n, wehaveγâ n Γâ n 1 0. Therefore, the proposed algorthm s guaranteed to converge to a certan value, Γ, at the of the teratons. However, the convergence to ths value does not mean that the global optmum s reached as t s the general dsadvantage of all teratve algorthms [8]. 4. PERFORMANCE RESULTS algorthm s compared wth the [2] and small error approxmaton [4], llustrated as n the fgures, for DOA and sensor poston estmatons. [1] s also evaluated for both DOA and sensor poston estmaton. Whle and algorthms are teratve methods, algorthm s non teratve one. As stated n Table 1, starts wth the SOS algorthm and terates HOS and SOS approaches to update both DOA and sensor poston estmatons. Also [4] algorthm starts wth algorthm and teratvely updates both DOA and sensor poston estmatons usng SOS approach. Therefore, comparng algorthm wth MI- HOSS and algorthms shows the effectveness of the teraton processes. Note that for a far comparson, the sensor poston estmaton algorthm descrbed n Secton 3.2 s also appled for the algorthm. It s assumed that there are two far-feld sources and M = 10 sensors. Each sensor poston except the two reference sensors s randomly selected from a unform dstrbuton n the deployment area of 2λ 2λ. The reference sensors are placed at 0,0 and λ/2,0. The postons of the sensors other than the reference sensors are arbtrarly perturbed. The perturbaton values are randomly selected wth a unform dstrbuton. For the parameter estmaton, N = 1000 snapshots are collected. The performance results are the average of 100 trals. At each tral, source sgnals, nose, the sensor postons except the reference sensors, the perturbatons and the DOA angles of source sgnals are changed randomly. The dfference between the DOA angles of the source 616

4 Table 1: Pseudocode for algorthm n = 0. Fnd the ntal values of the array steerng vector for each source, â 0, wth SOS approach as n 21 usng the nomnal sensor postons.; Termnaton = true. Estmate the proposed cumulant matrx from the array output and â n as n 3. Then, fnd the DOA estmates, ˆθ HOS usng 8 and the array steerng matrx  HOS, usng 11, for 1 L; Fnd the sensor poston estmates, ˆP = P 0 + ˆ P,asn 18 usng 16 and 14 wth ˆθ HOS and  HOS, for 1 L; Fnd ˆθ SOS usng ˆP as n 20. Then, fnd â SOS usng ˆP and ˆθ SOS as n 21; for = 1 to L do f Γâ SOS Γâ n then else â n+1 Γâ n+1 â n+1 = â SOS, ˆθ n+1 = ˆθ SOS =Γâ SOS = â n, ˆθ n+1 = ˆθ n ;,, Termnaton = false; f Termnaton = falsethen n = n + 1, Go to Step 2; else Fnd the fnal estmate of sensor postons usng ˆθ n and â n, 1 L; sgnals s set to 40 degrees. The source sgnals have a unform dstrbuton and the nose s addtve whte Gaussan and uncorrelated wth the source sgnals. The performance results for the DOA and sensor poston estmatons at dfferent SNR values are llustrated n Fg. 1. The sensor poston perturbaton s lmted to 0.1λ. Its seen that both and small error approach algorthm have a floorng effect for both DOA and sensor poston estmatons. As t s seen n Fg. 1, algorthm slghtly mproves the performance. It s also seen that after approxmately SNR = 7dB algorthm sgnfcantly outperforms and closely follows for both DOA and sensor poston estmatons. In Fg. 2, the performance of the algorthms s presented for dfferent poston perturbatons. SNR s set to 30 db. As t s seen n Fg. 2, the parameter estmaton performance of algorthm s not affected from the value of perturbatons and closely follows. It s also observed n Fg. 2-b that, algorthm effectvely solves the ambguty problem up to a perturbaton value of 0.42λ. The condton presented n Lemma-1 s not satsfed for further ncrease n perturbatons and sensor postons can not be found unambguously. Note that DOA estmaton s accurate and s not affected by the sensor poston ambguty as shown n Fg. 2-a. Ths s due to the fact that array steerng matrx estmate s accurate whle the postons are ambguous. The performance of both and algorthm degrade sgnfcantly for the large perturbaton values. DOA RMSE, DEGREES NORMALIZED SENSOR POSITION RMSE, xλ SNR, db 10 3 a SNR, db b Fgure 1: a DOA and b poston estmaton RMSE values for dfferent SNR values and sensor poston perturbaton of 0.1λ. algorthm slghtly outperforms algorthm only for very small perturbatons less than 0.01λ. For the perturbatons less than λ outperforms both and SmallEror algorthms as well as. The reason for ths fact s that teratve processes n and algorthms decrease the estmaton performances for the extremely small perturbatons. As shown n 17, pseudonverse operator s used for sensor poston estmaton, whch s not an exact soluton. Iteratvely updatng sensor postons may result worse poston estmaton than the nomnal sensor postons when the perturbaton s extremely small. The smlar explanaton s also vald for the algorthm. Whle does not specfy any algorthm for sensor poston estmaton, t uses perturbatons as unknown parameters and tres fnd the mnmum varance for both DOA and sensor poston estmatons jontly. Hence, assumes that there are always errors n sensor postons even f there s no. 617

5 DOA RMSE, DEGREES NORMALIZED SENSOR POSITION RMSE, xλ On the other hand algorthm fnds the DOA and sensor poston estmatons n a sngle step. It does not assume that there are errors n sensor postons and does not update the estmatons teratvely POSITION PERTURBATION, xλ 10 3 a REFERENCES [1] M. Aktaş, and T. E. Tuncer, Iteratve HOS-SOS IHOSS Algorthm for Drecton-of-Arrval Estmaton and Sensor Localzaton, IEEE Trans. on Sgnal Proc., vol. 58, pp , Dec [2] R. Schmdt, Multple Emtter Locaton and Sgnal Parameter Estmaton, Proc. RADC Spectrum Estmaton Workshop, 1979, pp [3] M. L and Y. Lu, Source Bearng and Steerng-Vector Estmaton usng Partally Calbrated Arrays, IEEE Trans. on Aerospace and Electronc Systems, Vol. 45, No. 4, pp , October 2009 [4] A. J. Wess and B. Fredlander, Array shape calbraton usng egenstructure methods, Twenty-Thrd Aslomar Conference on Sgnals, Systems and Computers, Vol. 2, pp , 1989 [5] B. P. Flanagan and K. L. Bell, Array self-calbraton wth large sensor poston errors, Elsever Scence Sgnal Processng pp , [6] M. Aktaş and T. E. Tuncer, Iteratve HOS-SOS IHOSS Based Sensor Localzaton and Drecton-Of- Arrval Estmaton, IEEE Internatonal Conference on Acoustcs, Speech, and Sgnal Processng, ICASSP- 2010, March [7] M. C. Dogan and J. M. Mel, Applcatons of Cumulants to Array Processng - Part I: Aperture Extenson and Array Calbraton, IEEE Trans. on Sgnal Proc., Vol. 43, No. 5, pp , May [8] A. J. Wess and B. Fredlander, DOA and Steerng Vector Estmaton Usng a Partally Calbrated Array, IEEE Trans. on Aerospace and Electronc Systems, Vol. 32, pp , POSITION PERTURBATION, xλ b Fgure 2: a DOA and b poston estmaton RMSE values for dfferent sensor poston perturbatons and SNR = 30 db. 5. CONCLUSION A new method for jont DOA and sensor poston estmaton s presented when the sensors are randomly deployed and arbtrarly perturbed from ther nomnal postons. It s assumed that the dstance and the drecton between two reference sensors are known. HOS and SOS approaches are employed jontly n an teratve manner. The teratve method s guaranteed to converge. Several smulatons are done and t s shown that the proposed method mproves the performance of DOA and sensor poston estmaton sgnfcantly and approaches to the. 618