Waveform Design for Distributed Aperture using Gram-Schmidt Orthogonalization

Transcription

1 avefor Design for Distributed Aperture using Gra-Schidt Orthogonalization an Evren Yaran, rond Varslot, Birsen Yazıcı Deparent of Electrical, oputer Systes Engineering Rensselaer Polytechnic Institute roy, NY Eail: yaran, trond, Margaret heney Departent of Matheatical Sciences Rensselaer Polytechnic Institute roy, NY Eail: Abstract In this work, we consider a distributed aperture radar syste present a ethod for clutter rejecting wavefors reflectivity function reconstruction. his work generalizes the onostatic radar wavefor design ethod for range-doppler iaging, developed in 1, 2 to distributed aperture radar systes. he designed wavefors also lead to a filtered backprojection type reconstruction of the reflectivity function which can be efficiently ipleented in a parallel fashion. I. INRODUION In this work, we generalize the wavefor design strategy presented in 1, 2, which was developed for wideb range-doppler iaging using onostatic radar in the presence of clutter, to design wavefors develop a reconstruction ethod for distributed aperture which utilizes Gra-Schidt orthogonalization (GSO). II. SAERED FIELD e odel the antenna as a tie-varying current density j tr (t, x) over an aperture. his is appropriate for a wide variety of antennas 3, 4, 5. e assue that the electroagnetic waves eitted fro the antenna travels in a known background ignore polarization effects. Under these assuptions, the field eanating fro the antenna satisfies the scalar wave equation 0 (x) 2 t )u in (t, x) = j tr (t, x), (1) c 0 (x) is the speed of light in the background. Let g 0 (t, x; σ, y) be the Green s function of (2) 0 (x) 2 t )g 0 (t, x; σ; y) = δ(t σ)δ(x y). (2) hen the incident field is given by u in (t, x) = g 0 (t, x; σ, y)j tr (σ, y)dσ dy. (3) he odel we use for wave propagation, including the source, is (x) 2 t )u(t, x) = j tr (t, x). (4) c(x) is the speed of light in the distorted ediu. e write u = u in + u sc in (4) use (2) to obtain 0 (x) 2 t )u sc (t, x) = (x) 2 t u(t, x), (5) (x) = 1 c 2 (x) 1 c 2 (6) 0 (x). he reflectivity function contains all the inforation about how the scattering ediu differs fro given back ground. It is, or at least its discontinuities other singularities, that we want to recover. e ake the single-scattering approxiation to the scattered field u sc by replacing the full field u on the right side of (2) by the incident field u in. Solution of the resulting differential equation leads to u sc (t, x) g 0 (t, x; σ, z) (z) σu 2 in (σ, z)dσdz. (7) /07/$ IEEE avefor Diversity & Design

2 For the incident field (3), (7) becoes u sc (t, x) g 0 (t, x; σ, z) (z) ( ) σ 2 g 0 (σ, z; τ, y)j tr (τ, y)dτdy dσdz (8) III. MEASUREMEN MODEL For the rest of the paper, we will assue that the antenna is sall copared with the distance to the scatterers in the far filed focus to the odel j tr (t, x) = p(t)j(x)δ(x z j ), p is the transitted wavefor, referred to as the pulse, j(x) is the waveguide. For transitter location z j, j 0, receiver location x i, i 0, receiver antenna bea pattern j rc (t, x i ), we odel the easureents as e(t, x i, z j ) = u sc (., x i, z j ) t j rc (., x i )(t), (9) t denotes convolution in t. Let λ ij = (x i, z j ). Define κ(z; t; λ ij ; τ) = g 0 (τ, x; σ, z) σg 2 0 (σ, z; τ, z j ) j(z j )j rc (t τ, x i ) dσ dτ, (10) linear operators H G acting on p by H ( )(t, τ) = (z)κ(z; t; λ ij ; τ)dz = (. ), κ(. ; t; λ ij ; τ), (11) G p(t, z) = κ(z; t; λ ij ; τ)p(τ)dτ = κ(z; t; λ ij ;. ), p(. ), (12) respectively. hen, (9) defines a bilinear integral operator acting on p with kernel κ as follows: e(t, λ ij ) = (z)κ(z; t; λ ij ; τ)p(τ)dzdτ. (13) Using the notation H G, we rewrite (13) as e(t, λ ij ) = H ( )(t, τ)p(τ)dτ = H ( )(t,. ), p(. ), (14) e(t, λ ij ) = (z)g p(t, z)dz = G p(t,. ), (. ). (15) IV. LUER SUPPRESSION FILER In the presence of additive clutter, we odel easureents e by replacing the target by + in e: e (t, λ ij ) = H ( + )(t,. ), p(. ) (16) Let be a linear integral operator acting on the wavefor space with kernel η(t). e define the clutter suppression filter as the that iniizes = E for any wavefor p, i.e. H ( + )(t,. ), p(. ) H ( )(t,. ), p(. ) 2 dt p 2, (17) = in, p. (18) Let b n be an orthonoral basis for the wavefor space. Define P to be the operator whose atrix eleents are given by P = p n p p(t)p(t p n = p, b n / p, alias ) p = 2,n P b n (t)b (t ). Assuing are utually uncorrelated has zero ean, (18) can be equivalently expressed by ( = in tr ( I) K + K ) P ( I), P, (19) tr is the trace operator, K K are positive definite operators given by K = E H ( ) H ( ), (20) = E H () H (). (21) K /07/$ IEEE avefor Diversity & Design

3 Note that the cross correlation ters vanishes, EH () H ( ) = EH ( ) H () = 0, (22) since E (z)(z ) = 0. Let R (z, z ) = E (z) (z ) R (z, z ) = E(z) (z ) be the autocorrelation of, respectively. e can write K K in ters of R R as follows: K (t, τ) = κ R κ (t, τ) := κ(z; t; λ ij, τ)r (z, z )κ (z ; t; λ ij ; τ)dz dz, K (23) (t, τ) = κ R κ (t, τ) := κ(z; t; λ ij, τ)r (z, z )κ (z ; t; λ ij ; τ)dz dz. (24) e deterine by equating the variational derivative of (19) with respect to to 0: ( ) tr K + K K P = 0. (25) In order for (25) to hold for any P, ust be 1 = K + K (λ K ij) = I K + K 1 K (λ ij),(26) 1 denotes pseudo- or approxiate-inverse. Note that iniization of (19) with respect to all P is equivalent to iniizing E H( + ) H( ) 2 HS with soe additional assuptions (i.e. finite transit receive durations), A HS = tra A denotes the Hilbert Schidt nor, i.e. = in E H( + ) H( ) 2 HS = in tr ( I) K ( I) + K, which again leads to the clutter suppression filter in (26). V. DESIGNING HE RANSMIED AVEFORMS e want to design wavefor p s such that given the clutter suppression filter (z) = G p (., z), p n (. ) (27) for an orthonoral basis with respect to the inner product over z, i.e., (λ i j ) kl = δ k δ nl δ ii δ jj, (28) δ is the Kronecker delta function, which is equal to one when = n zero otherwise.hus the best approxiation to (z) will be (z) α,n,λ ij α (z), (29) =,. (30) For ease of exposition, we will first present the ain ideas of our discussion for a singe transitterreceiver pair case then generalize the ain ideas to distributed aperture. A. For a Single ransitter-receiver Pair Assuing that we have a single transitterreceiver pair, given an orthonoral set the best approxiation to (z) will be (z),n α Substituting (27) in (30), α = (z)g p (z). (31) (., z), p n (. )dz = e (., λ ij ), p n (. ), (32) e denotes the scattered field fro + due to transitted wavefor p. hus, by (32), (31) becoes (z),ne (., λ ij ), p n (. ) (z). (33) e can treat (33) as a filtered backprojection type reconstruction forula 4, 5, the filtering /07/$ IEEE avefor Diversity & Design

4 takes in two steps. First step is in transission, we filtering over the scene + by the transitting the wavefors p. Second step is atched filtering of the easureents with p n in receive. Finally, we backproject the atched filtered easureents with (z). 1) Miniu Mean Square Error avefors: Let {q } 0 be the eigenfunctions of the operator G, G q (t, z) = Q (z)q (t), (34) ordered such that corresponding eigenvalues are descending. hen G q (., z), q n (. ) = δ G q (., z), q (. ). (35) Define Q (z)=g q (., z), q (. ), (36) Q (z) = Q (z)/ Q. (37) If the transitted wavefors were p = q, then atching the echo with p eans projection of + onto Q. However { Q } 0 does not necessarily for an orthonoral set in the iage doain. In this regard, we want to design our wavefors fro {q } 0 such that the corresponding { (z)} 0 for an orthonoral set. onsequently, due to the linearity of the easureent odel, designing transitted wavefors is equivalent to for an orthonoral set { (z)} 0 fro { Q } 0, which we will present in the next section. Usage of eigenfunctions {q } siplifies the decoposition (31) as α (z) α (z), (38) =, = e (., λ ij ), p (. ). (39) If we were to transit a single pulse p 0, then the iniu ean square error (MMSE) E 2 2 is achieved when p 0 is the eigenfunction of the operator G corresponding to the largest eigenvalue, i.e. p 0 = q 0. Here f 2 2 = f(z) 2 dz. Siilarly, if N pulses were transitted then, the MMSE is achieved when p are chosen as the eigenfunctions of the operator G corresponding to the N largest eigenvalues, i.e. p i 2) onstruction of = q i for i = 0,..., N 1. (z) p (t): e will for { (z)} 0 by perforing the Gra- Schidt orthonoralization (GSO) process over { Q (z)} 0. Since the GSO depends on the choice of the order of { Q (z)} >0, in the light of the previous section, in order to obtain MMSE, we will choose { Q (z)} >0 with the increasing order in. Let 00 (z) = Q 0 (z). hen, by GSO, we for 1 (z) = k=0 β k kk (z) + β Q (z), (40) for soe constants β γ obtained by GSO. Due to the linearity of GSO, we can deterine the transitted wavefors along with the GSO of Q as follows. Let {p } 0 be the transitted wavefors. In order to obtain after atching the echo by p, by GSO of Q, p 0 (t)=q 0 (t)/ Q 0, (41) 1 q p k (t) γ k (t)= + q (t) β, k=0 Q k Q (42) 1 γ k = β γ nk. (43) n=k 3) In the Absence of the Eigenfunctions: It is not always possible to find the eigenfunctions of the operator G only a set of pulses s (t) can be transitted. hen one can still use the /07/$ IEEE avefor Diversity & Design

5 GSO to for a basis fro S (z) = G s (., z), s n (. ). (44) hus for each (z), one will obtain a pulse p (t). his will square the nuber of wavefors to be transitted. onsequently, the coputational burden is also square when copared the reconstruction using the wavefors deterined by the eigenfunctions of G. B. For Distributed Aperture In the case of distributed aperture, it is enough to generalize the GSO over {S (z)} for soe order on {(λ ij, )}. For exaple let the ordering on {(λ ij, )} be defined by (λ i j, ) < (λ ij, ) if i < i, or if i = i j < j, or λ i j = λ ij <. hen given {S (z)}, deterine (z) by GSO such that (z) is orthogonal to all (λ i j ) n (z), hence to all S (λ i j ) n (z), for (λ i j, ) < (λ00) (λ ij, ), with the initial condition 00 (z) = S (λ00) 00 (z)/ S (λ00) 00. VI. FROM GROUP HEOREI POIN OF VIE he proposed ethod is inspired fro statistical signal processing over groups developed in 1, 2. Under the group theoretic perspective (30) (31) corresponds to the Fourier inverse Fourier transfors. In this regard, Ŝ (z) = S (z)/ S can be treated as the atrix eleents of the irreducible unitary representations, denotes the atrix entry λ ij = (x i, z j ) is the corresponding irreducible subspace. In this setting GSO provides the square root of the linear operator known as the discrepancy operator for non-coutative groups, which fors the syetric Fourier basis { (z)} fro {Ŝ }. Finally corresponds to the iener filter. 1, 2 VII. SUMMARY he presented reconstruction ethod can be suarized by the following diagra: (z) (λ ij ) p (λ ij ) (t) 1 e(t, λ ij ),n,λ ij 4 2,p (λ ij ) α (z) 3 (λ Λ b ij ) (z) α he first step is transission of the filtered wavefors {p (t)}. he second step is the atching of the received echo with the transitted pulse. he third step is the backprojection of the atched signal. Finally, the iage is fored by suing over each backprojected iage. REMARK - ransission vs Receive: In stead of transitting the wavefors {p (t)} s one can still transit the wavefors {s (t)}. Since is linear Step 1 is assued to be linear, the easureents due to transitting {p (t)} can be obtained by weighted suation of the easureents ade by transitting {s (t)}. However, this induces ore coputation on the receiver side. he weighted suation can also be hled in Steps 2,3 4, due to the linearity of the steps. AKNOLEDGMEN e are grateful to Air Force Office of Scientific Research (AFOSR) the Defense Advanced Research Projects Agency (DARPA) for supporting this work under the agreeents FA , FA FA REFERENES 1 B. Yazıcı G. Xie, ideb extended range-doppler iaging wavefor design in the presence of clutter noise, IEEE rans. Inf. heory, vol. 52, pp , B. Yazici. Yaran, Deconvolution over Groups in Iage Reconstruction, 2006, vol. 141, pp M. heney B. Borden, Microlocal structure of inverse synthetic aperture radar data, Inverse Probles, vol. 19, pp , J. Nolan M. heney, Synthetic aperture inversion, Inverse Probles, vol. 18, pp , Nolan M. heney, Synthetic aperture inversion for arbitrary flight paths non-flat topography, IEEE ransactions on Iage Processing, vol. 12, pp , /07/$ IEEE avefor Diversity & Design