COPRIME ARRAYS AND SAMPLERS FOR SPACE-TIME ADAPTIVE PROCESSING

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1 COPRIE ARRAYS AND SAPLERS FOR SPACE-IE ADAPIVE PROCESSING Chun-Lin Liu 1 and P. P. Vaidyanathan 2 Dept. of Electrical Engineering, California Intitute of echnology, Paadena, CA 91125, USA cl.liu@caltech.edu 1 and ppvnath@ytem.caltech.edu 2 ABSRAC hi paper extend the ue of coprime array and ampler for the cae of moving ource. Space-time adaptive proceing (SAP play an important role in etimating direction-of-arrival (DOA and radial velocitie of emitting ource. owever, the detection performance i fundamentally limited by the array geometry and the temporal ampler at each enor. Coprime array and coprime ampler offer an enhanced degree of freedom of O(N uing only O( + N phyical enor or ample. In thi paper, we propoe coprime joint angle-doppler etimation (coprime JADE, which incorporate both coprime array and coprime ampler with the SAP framework. Nonuniform time ample at different enor can be ued to generate a ampled autocorrelation matrix, from which we compute a patial moothed matrix. It will be proved that patial moothed matrice can be ued in the USIC algorithm for parameter etimation. With ufficient naphot, coprime JADE ditinguihe O( 1N 1 2N 2 independent ource if it correpond to coprime array and coprime ampler with coprime integer ( 1, N 1 and ( 2, N 2, repectively. It i verified through imulation that coprime JADE reolve the angle-doppler information better compared to other conventional algorithm. Index erm Coprime array, coprime ampler, joint angle- Doppler etimation, the USIC algorithm. 1. INRODUCION Space-time adaptive proceing (SAP i a tandardized technique in airbone I radar ignal proceing [1, 2]. From time ample at different enor, adaptive weighting are applied to thee data to derive a 2D beamformer for direction-of-arrival (DOA and Doppler frequencie. Etimator baed on power pectrum denity (PSD [1], minimum variance ditortionle repone (VDR [3], USIC algorithm [4], ESPRI algorithm [5], time-pace-time USIC [6], and compreive joint angular-frequency power pectrum etimation [7] are alo popular in extracting the angle-doppler information. Coprime array have received attention in DOA etimation ince they can reolve more ource than the number of enor [8]. Coniting of two uniform linear array (ULA whoe interelement pacing are d and Nd, coprime array provide an enhanced degree of freedom of O(N uing O( + N enor, auming and N are coprime integer. hi property facilitate uperreolution application in DOA etimation [9, 10], pectrum ening [8], and other application. In thi paper, we propoe a novel joint angle-doppler etimation framework uing coprime array for enor and coprime ampler at hi work wa upported in part by the ONR grant N , and the California Intitute of echnology. the output of each enor. hi will be called coprime JADE in thi paper. Deigning coprime array with coprime integer 1 and N 1 in the patial domain and coprime ampler with coprime integer 2 and N 2 in the time domain enable uperreolution on the angle- Doppler plane. By uing coprime array and ampler, it will be proved that coprime JADE can identify O( 1N 1 2N 2 ditinct ource. hi paper i organized a follow. In Section 2, the data model i briefly reviewed. In Section 3, coprime JADE i introduced in detail. In Section 4, we propoe identifiability guarantee for coprime JADE, which are verified by imulation in Section 5 before Section 6 conclude thi paper. 2. E DAA ODEL Conider D ource emitting electromagnetic wave in the pace. Our array receive the emitted ignal at enor location nd in pace, and then take ample at time intant m in time, where d = λ/2 denote the minimum pacing between enor and i the minimum ampling interval. Each ource i characterized by it DOA i and it radial velocity v i. he received ignal X (n, m can be modelled a a random variable defined by X (n, m = A ie j2π i n e j2π f i m + N (n, m, (1 where {A i} D are zero-mean random variable with the uncorrelated property E [A i A j] = σi 2 δ i,j. ere i and f i are normalized DOA and normalized Doppler frequencie defined by i = d in i, fi λ = λ vi, where i, f i [ 1/2, 1/2. N(n, m tand for additive white noie with E [N (n, m N (n, m ] = σ 2 δ n,n δ m,m. Noie i aumed uncorrelated with the ource {A i} D. Eq. (1 can be reformulated uing matrice. Defining a random matrix X = [X(n, m], Eq. (1 become X = A iv ( i v ( t fi + N, (2 where v ( i denote patial teering vector correponding to i, ( and v t fi are temporal teering vector with repect to normalized Doppler frequencie f i. he upercript tand for matrix tranpoe operation and N = [N(n, m]. Vectorizing (2 lead to the following expreion x = A iv,t i + n, (3

2 Coprime array 1 d N 1 d v Source Coprime ampler Contruct data matrix X = [X(n, m] Fig. 1. A ytem diagram of coprime JADE. where x = vec (X, n = vec (N, and v,t i = vt v ( i repreent pace-time teering vector related to the angle-doppler frequency pair i. ere i the Kronecker product between matrice. In practice, multiple naphot are taken to etimate econd order tatitic if ource are tationary within thee naphot. Aume thee multiple meaurement are written a x (k, where k = 1, 2,..., K. he autocorrelation matrix of x can be etimated a ˆR x = 1 K x(kx (k, (5 K k=1 where the upercript tand for tranpoe-conjugate of matrice. he etimated autocorrelation matrix can then be applied to beamformer, USIC algorithm, or coprime array/ampler. 3. JOIN ANGLE-DOPPLER ESIAION In thi ection, we preent a novel JADE etting uing coprime array and coprime ampler to achieve much higher degree of freedom. From the coprime ample, the autocorrelation matrix i firt etimated. Baed on the difference et pertaining to coprime array and coprime ampler, the patial moothing method i applied to the ampled autocorrelation matrix, yielding a larger-ize patial moothed matrix. hen the USIC algorithm i applied to etimate the angle-doppler information. Fig. 1 depict the ytem diagram of coprime JADE. he coprime array i compoed of two uniform linear array with interelement pacing 1d and N 1d, where 1 and N 1 are coprime integer. he enor location are explicitly defined over the grid nd, where n {0, 1,..., (N 1 1 1, N 1, 2N 1,..., (2 1 1N 1}. here are N enor in total. At each enor, two ampler operate at ampling rate 2 and N 2 in parallel to obtain coprime ample, where 2 and N 2 are alo coprime integer. hee time-domain ample are defined at time intant m, where m {0, 2,..., (N 2 1 2, N 2, 2N 2,..., (2 2 1N 2}. Each ampler take N ample per enor in one naphot. Next, the autocorrelation matrix of x i tudied. From (3, the ideal autocorrelation matrix of x i found to be [ E xx ] = σi 2 v,t i v,t i + σ 2 I. (4 1N N 1 2N 2... q 0 p... 2N 2 Z Z p,q Z 0,0 Fig. 2. Relationhip between Z and it ubmatrix Z p,q. he term v,t i v,t i can be rewritten a follow ( ( v t fi v ( ( t fi v ( i v ( i, which i becaue of (4 and the property (A B (C D = (AC (BD. Note that the entrie in v ( i v ( i are in the form of e j2π i ñ, where the range et of ñ contain all integer from 1N 1 to 1N 1 [9]. ence, there exit an arrangement that convert v ( i v ( i to a new teering vector w [ ] ( i = e j2π i ñ, where ñ = 1N 1,..., 1N 1. w ( i can be alo regarded a teering vector correponding to a (2 1N element ULA in coarray domain. he ame argument ( hold true in the time domain, where a new teering vector w t fi i viewed a the collection of 22N uniform ample. According to the equivalence of (2 and (3, a matrix Z C (2 1N 1 +1 (2 2 N 2 +1 i contructed from E [ xx ] uch that Z = σi 2 w ( i w ( t fi + σ 2 e 1e 2, (6 where e i = [δ l,i N i +1] 2 in i +1 l=1 for i = 1, 2. Eq. (6 can be interpreted a a um of D coherent ource in 2D. hen, for coherent ource, patial moothing [11 13] i a technique for etimating the autocorrelation matrix of Z, from which the USIC algorithm [4] can reolve the DOA and the Doppler frequency information. o apply patial moothing, the ubmatrix of Z i defined a Z p,q C ( 1N 1 +1 ( 2 N 2 +1, which i depicted in Fig. 2, Z p,q = σi 2 e j2π( i p+ f i q w ( i w ( t fi + σ 2 e 1,pe 2,q,

3 where w ( i C 1 N 1 +1 and w t C 2 N 2 +1 are pace and time teering vector of an ULA and uniform ampling tarting with 0, repectively. e 1,p i a ubvector contructed from the ( 1N pth entry to the (2 1N pth entry of e 1. ere p {0, 1,..., 1N 1}, q {0, 1,..., 2N 2}. We then define the patial moothed matrix R a R = 1 ( 1N 1 + 1( 2N N 2 N z p,qz p,q, (7 where z p,q = vec (Z p,q. It follow from (7 that R i a ermitian, poitive emidefinite matrix of ize ( 1N 1 + 1( 2N by-( 1N 1 + 1( 2N heorem 1. he patial moothed matrix R can be expreed a ( D 2 R = C 2 σi 2 w,t i w,t i + σ I 2, (8 where w,t i = wt w ( i are pace-time teering vector, and C = (( 1N 1 + 1( 2N /2. Proof. he proof i ketched a follow. Analogou to (2 and (3, z p,q = vec (Z p,q can be written in term of w,t i. hen, putting z p,q into (7 lead to an expreion in term of ummation over p, q, i 1, i 2, where i 1 and i 2 are indice which run over all ource. he cro term in R are replaced with teering vector, w w t 1 N ( i1 w ( i2 = ( ( 2 N fi1 wt fi2 = ( 1 N w t fi2 w ( i2 = e j2π i1 p e j2π i2 p, e j2π f i1 q e j2π f i2 q, 2 N hen R /C 2 can be expreed a i 1 =1 i 2 =1 e j2π( i2 p+ f i2 q (e2,q e 1,p. ( [σ 2 i1 σ 2 ( i2 w t fi1 w ( ( t fi1 wt fi2 w ( t fi2 ( w ( i1 w ( i1 w ( i2 w ] ( i2 + 2 σi 2 σ 2 w,t i w,t i + σ 4 I. (9 Rearranging (9 prove thi theorem. Note that the theorem imilar to heorem 1 have already been propoed in the context of 1D coprime array [9], 1D neted array [13], and 2D neted array [14]. It i alo oberved that heorem 1 reemble heorem 3 of [14] becaue coprime JADE follow the ame formulation a coprime array in two dimenion [15] and neted array in two dimenion [14]. In other word, our work bridge the gap between SAP and two dimenional array proceing. he former etimate the DOA and the Doppler frequency while the latter eek the azimuth and the elevation in two dimenion. owever, coprime JADE and coprime array in 2D are related to different deign problem. In coprime JADE, enor location are fixed and the ame ampling pattern i applied to each enor, yielding eparable ample in the pace-time domain. In coprime array in 2D, enor can be arbitrarily placed over noneparable ampling lattice. Coprime JADE tranform eparable ample to the angle-doppler plane, which i noneparable ince i can be placed anywhere on the angle-doppler plane. Coprime array in 2D convert their noneparable input to the azimuth-elevation domain, which i alo noneparable. heorem 1 ugget that R can be utilized in USIC algorithm to etimate i. It i becaue R hare the ame eigenvector a the term between the parenthee in (8. Aume the ignal ubpace of R i repreented a U, and let U n denote the noie ubpace of R. Following the definition of USIC pectra [4], the USIC pectrum for coprime JADE i then defined a 1 P USIC (, f = U n w,t (, f 2, 2 where, f [ 1/2, 1/2 and 2 denote Euclidean norm of vector. hen, the DOA and the Doppler frequencie can be etimated by detecting the peak in P USIC (, f. 4. OW ANY SOURCES CAN COPRIE JADE IDENIFY? he number of identifiable ource for coprime JADE will be dicued in thi ection. Under ome condition, the USIC pectrum P USIC (, f can identify up to 1N 1 2N 2 correct angle- Doppler pair. It wa hown in Section 3 that coprime JADE i cloely related to neted array or coprime array in two dimenion. In 2D, identifiability iue become difficult and can be categorized into almoture identifiability [16] and exact identifiabilty [14]. In thi paper, we propoe the following theorem on exact identifiability, baed on the theoretical reult on identifiability propertie given in [14], heorem 2. Conider ditinct ource S = { i } D where D 1N 1 2N 2. Aume that a the et { i } D take at mot 1N 1 ditinct value and b { } D fi contain at mot 2N2 ditinct value. hen, P USIC (, f ha a ingularity if and only if (, f S. Proof. According to heorem 4 in [14], letting = 1N and N = 2N 2 + 1, we obtain that P USIC (, f ha a ingularity if and only if w,t (, f = w,t i for ome 1 i D. Provided with w,t (, f, w ( i the firt 1N entrie of w,t (, f and wt ( f correpond to the (1 + (1N 1 + 1lth entrie of w,t (, f, l = 0, 1,..., 2N 2. ence w,t (, f = w,t i i equivalent to w ( = w ( i and wt ( f = w t for ome 1 i D. Since w ( and wt ( f correpond to ULA and uniform ampling, repectively, applying heorem 1 in [17] twice implie that w ( and wt ( f are invertible. ence, for ome 1 i D, = i and f = f i, which complete the proof. heorem 2 indicate the maximum number of identifiable ource i 1N 1 2N 2. hi reult can be interpreted a a combination of coprime array and coprime ampler. heorie in coprime array proved that up to 1N 1 DOA can be detected [9]. Coprime ampler have up to 2N 2 degree of freedom [9]. If the D ource exhibit 1N 1 ditinct value projected onto and 2N 2 ditinct value onto f i, then thee value are ditinguihable uing coprime array and coprime ampler.

4 Ideal PSD VDR ULA USIC coprime JADE (a (b (c Fig. 3. Angle-Doppler pattern uing different method and different ource. (a 121 uniform ource, (b 400 uniform ource, and (c 20 random ource. x axe tand for normalized DOA while y axe repreent normalized Doppler frequencie f. he color i revered (black how larger value to enhance the contrat. 5. NUERICAL RESULS he performance of different algorithm i compared in thi ection. We compare Fourier-baed power pectrum denity (PSD [1], inimum variance ditortionle repone (VDR [3], USIC algorithm with ULA [4], and coprime JADE. he former three method ue ULA plu uniform time ample at each enor. In thee imulation, 1 = 2 = 4 and N 1 = N 2 = 5 o that the number of enor and the number of ample per enor are exactly 12 in all method. According to (5, the autocorrelation matrix of x i etimated from 1000 naphot. he input ignal i mixed with additive white Gauian noie with 0 db SNR. Simulation reult are hown in Fig. 3. In the firt imulation, 11 2 = 121 ource are placed uniformly on the angle-doppler plane. hi example hit the limit of ULA and uniform ampling ince there are only 12 enor and 12 ample at each enor. PSD and VDR fail to eparate all 121 ource but USIC-baed method how imilar pattern a the ideal cae. In the econd imulation, the number of ource i increaed to 20 2 = 400, which i the maximum number of ditinguihable ource for coprime JADE. he firt three method cannot detect all 400 ource. ULA USIC method cannot reolve the pectrum due to it rank deficiency. owever, coprime JADE till work in thi extreme cae. Our third example elect 20 random ource. PSD and VDR till cannot reolve thee ource while ULA USIC and coprime JADE work. owever, a een from the circled part in thee figure, ULA USIC produce more blurry pectrum than coprime JADE, for cloely paced ource. herefore, coprime JADE ha higher reolution and i uitable for cloely paced ource. 6. CONCLUDING REARKS In thi paper, we have formulated coprime JADE a a novel approach to angle-doppler etimation, inpired by the concept of coprime array and coprime ampler. From the input ample, it wa realized by contructing a matrix a a um of 2D complex inuoid, from which a patial moothed matrix wa etablihed. heorem 1 on the patial moothed matrice allowed a definition of the USIC pectrum over the angle-doppler plane. Coprime JADE exhibited unique identifiability up to 1N 1 2N 2 ditinct ource a long a thee ource atify ome condition. hee reult were verified through imulation and coprime JADE reolved cloely paced ource much better than ome conventional approache. Further reearch will be directed toward the connection between coprime JADE and coprime array in high dimenion [14, 15]. Coprime JADE take the ame number of ample at each enor, which follow the ame formulation a eparable coprime array in two dimenion. It could be poible to deign different ampling cheme at different enor uch that the number of ampler i reduced but the performance i unchanged.

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