Passive Imaging of Moving Targets with Sparsely Distributed Receivers

Ling Wang and Birsen Yazıcı∗∗ Department of Electrical, Computer and Systems Engineering,Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180 USA Email: [email protected]

Abstract—We develop a novel passive image formation method speed c can be expressed as for moving targets using measurements from a sparse array of −2 −2 2 receivers that rely on illumination sources of opportunity. We c (t, x)= c0 [1 + n (x − vt)] (2) use a physics-based approach to model the wave propagation and develop a passive measurement model that expresses the where c0 denotes the speed in the background medium and the measurement at each receiver in terms of the measurement at a second term in (2) represents the perturbation due to deviation different receiver. This model eliminates the need for knowledge about the locations and waveforms in the proposed from the background reflectivity that is caused by the moving image formation method. We formulate the image formation scatterer. In this work, we focus specially on radar imaging problem as a Generalized Likelihood Ratio Test (GLRT) for and assume that c0 is equal to the speed of light in vacuum. unknown target location and velocity using the proposed passive −2 2 Let qv(x − vt)= c0 n (x − vt). The moving scatterers in measurement model. We form the image in spatial and velocity the spatial volume d3x centered at x give rise to c−2(t, x)= space using the space- and velocity-resolved test-statistics. −2 3 c0 + qv(x − vt)d v [8]. I.INTRODUCTION TheR propagation medium is characterized by the Green's function. For free-space, the Green's function is given by With the rapid growth of broadcasting stations, base stations, communication and navigation satellites, δ(t − |x − z|/c0) g(x, z,t)= . (3) as well as relatively low cost and rapid deployment of re- 4π|x − z| ceivers, passive radar imaging using of opportu- We assume that the electromagnetic waves decay rapidly as nity has emerged as an active area of research in recent years they penetrate the ground. We then write q (x − vt) in terms [1]–[7]. v of 2D location and 2D velocity as in In this paper, we develop a passive image formation method for moving targets using static distributed receivers. The qv(x − vt)= qv(x − vt)δ(x3 − h(x))δ(v3 − D · v) (4) method is based on a new passive measurement model and 2 2 a space- and velocity-resolved binary hypothesis testing. Our where x = (x, x3), x ∈ R and v = (v, v3), v ∈ R , h method requires no information on the location of transmitters represents the ground topography and D = [ ∂h ∂h ]. ∂x1 ∂x2 or the transmitted waveforms. Additionally, it does not rely on For the slow-mover case where the speed of the target is receivers with high directivity. much slower than the speed of light c0, the scattered field measurement at the receiver located at x0 due to a transmitter II. PASSIVE MEASUREMENT MODEL located at z is expressed as [8] For a single transmitter with isotropic located at p¨[αy,v,x0,z (t − |y − x0|/c0) − |y − z|/c0 + Tz] z transmitting the waveform p(t) starting at time t = −T , m(t)= z Z (4π)2|y − x ||y − z| the propagation of electromagnetic waves in a medium can be 0 y y v (5) described using the scalar wave equation, × qv( )d d + n(t)

2 −2 2 where [∇ − c (t, x)∂ ]E(t, x, z)= δ(x − z)p(t + Tz) (1) t 1 − y\− z · v/c α = 0 (6) y,v,x0,z \ where c is the wave speed in the medium and E is the electric 1+ y − x0 · v/c0 field. Note that this model can be extended to realistic antenna models and multiple transmitters in a straightforward manner. is the Doppler-scale-factor [8] and n(t) is the additive thermal A single scatterer moving at velocity v corresponds to an noise at the receiver. Note that y = (y,h(y)) and v = (v,D · index-of-refraction distribution n2(x − vt). Thus, the wave v). In Fourier domain, (5) becomes 2 iφ −ω e y,v,x0 ω This work was supported by the Air Force Office of Scientific Research mˆ (ω)= 2 3 pˆ (AFOSR) under the agreements FA9550-07-1-0363 and FA9550-09-1-0013 (4π) Z |y − x0||y − z|αy,v,x0,z αy,v,x0,z  and National Science Foundation (NSF) under Grant No. CCF-08030672. ** Corresponding author × qv(y)dydv +ˆn(ω). (7) ′ where where Ws(y , y) is a spatial windowing function of unit ω |y − z| |y − x | amplitude centered at a hypothetical target location y and φ = T − − ω 0 . (8) z gˆ(xi, y,ω) represents the Fourier transform of (3). αy,v,x ,z  c0  c0 0 We define the back-propagation operator as the inverse of In the analysis that follows, we consider a sparse distribution −1 Py,v,i and denote it by Py,v,i. Using (11), we obtain of N receivers located at xi ,i = 1, · · · ,N. We assume −1 −1 −1 that there is a single transmitter located at z0 transmitting Py,v,i[ˆmi](ω)= Sv,i Gy,i [ˆmi](ω) (15) waveform p starting at time t = −T . This assumption allows 0 0 where −1 is the inverse of , −1 is the inverse of us to simplify the analysis and distill the important aspects of Gy,i Gy,i Sv,i Sv,i th and mˆ i represents the measurement at the i receiver. Note our imaging theory. The results obtained in this paper can −1 easily be extended to multiple-transmitter case. that if Py,v,i does not exist, it may be replaced with its Passive measurement model cannot depend on the trans- pseudoinverse. th mitter location or transmitted waveforms. We develop an We now express the measurement at the i receiver, mˆ i th alternative measurement model that expresses measurements in terms of the measurement at the j receiver, mˆ j , by at each receiver in terms of the measurements at a different back-propagating mˆ j measured at xj to a hypothetical target receiver, which involves a back-propagation operation and a location with a hypothetical velocity via the back-propagation operator and then forward-propagating the resulting field to forward-propagation operation. j ˜ xi via the forward-propagation operator. Let mˆ (ω) denote Let u = −q (y)ω2Eˆin(y, v,ω) where i v the ith measurement expressed in terms of the jth measure- ω |y−z0| th i T0− µ˜y,v,z0 “ c0 ” ment. We define the passive measurement model for the j ˜in e ω Eˆ (y, v,ω)= pˆ0 (9) measurement as follows: 4π|y − z0|µ˜y,v,z0 µ˜y,v,z0  j −1 mˆ (ω)= Py,v,iP mˆ j (ω)+ˆni(ω) (16) \ ˆ˜in i y,v,j with µ˜y,v,z0 =1−y − z0 ·v/c0. Note that E can be viewed th as the incident field observed by a moving target. The scale where nˆi(ω) is the additive thermal noise at the i receiver. factor, µ˜, accounts for the Doppler scaling effect induced by Taking the jth receiver as a reference, we form the following the movement of the target on the field due to the transmitted measurement vector for N receivers: waveform. j j j T m = mˆ 1 mˆ 2 · · · mˆ N (17) Let mˆ i denote the Fourier transform of the measurement th j   at the i receiver. Using (7), we define the linear operator, where mˆ i , i =1, · · · ,N and i 6= j, denotes the measurement th Py,v,i, that propagates the field u to the i measurements, mˆ i modeled in terms of the reference measurement mˆ j . i.e., Similarly, we vectorize the “reference measurements”, mˆ j ’s, and the noise: Py,v,i[u](ω) =ˆmj (ω) (10) T m = mˆ mˆ · · · mˆ (18) as the forward-propagation operator with respect to the ith r j j j  T receiver. Using (7), we express Py,v,i as follows: n = nˆ1 nˆ2 · · · nˆN . (19)   Py,v,i[u](ω) = Gy,iSv,i[u](ω). (11) Note that m, mr and n are all (N − 1) vectors. We represent the composition of the back-propagation and where S is a scaling operator that accounts for the Doppler- v,i forward-propagation operators as a diagonal matrix as follows: scale factor observed by the ith receiver due to the movement −1 −1 of the target at the hypothetical velocity v; and Gy,i in (11) is Py,v = diag Py,v,1Py,v,j ··· Py,v,N Py,v,j (20) a propagation operator that accounts for the time delay from   where i 6= j and P is (N − 1) × (N − 1). the target to the ith receiver in free-space. y,v Using (16), (17)-(20), we form the vectorized passive mea- S is given by v,i surement model as follows: ′ ′ Sv,i[u](ω)= Wv(v , v)µy,v′,i u(µy,v′,i ω)dv (12) m(ω)= P m (ω)+ n(ω) (21) Z y,v r ′ where Wv(v , v) is a windowing function of unit amplitude for some range of ω. in velocity space centered at the hypothetical velocity v and Note that in (21) all the operations are understood to be \ elementwise. µy,v,i =1+ y − xi · v/c0, (13) III.IMAGE FORMATION IN SPATIAL AND VELOCITY SPACE which accounts for the Doppler-scale-factor observed by the We formulate the image formation problem as a test of ith receiver due to the movement of the target at the hypo- thetical velocity v. binary hypothesis, which has its roots in the Generalized Likelihood Ratio Test (GLRT) [9]. We extract a space- and Gy in (11) is given by ,i velocity-resolved test-statistic as opposed to reconstructing the ′ ′ ′ unknown quantities of interest themselves due to the limited Gy,i[u](ω)= Ws(y , y)ˆg(xi, y,ω)u(y , v,ω)dy (14) Z number of measurements available. The image is then formed in (y, v) domain by the space- and velocity-resolved test- and analyze how moving point targets are resolved in the four- statistic where the location and the velocity can be identified dimensional image λ(y, v), y, v ∈ R2. by thresholding the image. Let K(y, y0; v, v0) denote the PSF of the four-dimensional We define the space- and velocity-resolved binary hypoth- imaging operator. We define K as the expected value of the esis test as follows: image of a moving point target represented by (29), i.e., H : m = n K(y, y0; v, v0) := E [λ(y, v)]. 0 (22) H1 : m = Py,vmr + n Without loss of generality, we first assume that there is a single pair of receivers present in the scene and then extend the where Py,v, mr, m and n are as defined in (17)-(21). The null results to the case where there are multiple pairs of receivers. hypothesis states that the measurement is due to noise whereas For a deterministic moving point target model given by (29), the alternative hypothesis states that the measurement is due we have to a target located at y moving with velocity v. 2 iφ Using (16), (18), (19) and (22), we obtain ω T e y0,v0,j E[ˆmj (ω)] = 2 3 (4π) |y0 − xj ||y0 − z0|α E [m|H0] = 0 (23) y0,v0,xj ,z0 m R R (24) ω Cov [ |H0] = n =: 0 × pˆ0 (30) αy0,v0,xj ,z0  E [m|H1] = Py,vE [mr|H1]= Py,vmr (25) H Cov [m|H1] = Py,vRnr Py,v + Rn =: R1 (26) where αy0,v0,xj ,z0 is as described by (6) and where E [·] denotes the expectation operator and Cov [·] de- ω |y − z | |y − x | H 0 0 0 j notes the covariance operator, mr denotes E [mr|H1], Py,v φy0,v0,j = Tz1 − − ω . αy0,v0,xj ,z0  c0  c0 denotes the Hermitian transpose of Py,v, Rn denotes the auto- (31) ′ H ′ covariance of the vector n, i.e., Rn(ω,ω )=E n(ω)n (ω ) For notational simplicity, we drop z0 from the subscript of α R n and nr is the autocovariance of the noise vector r = for the rest of our paper. ′ H ′ [ˆnj , nˆj , · · · , nˆj ], i.e., Rnr (ω,ω )=E nr(ω)nr (ω ) . We assume that there are two receivers located at x1 and We determine the test-statistic by maximizing the signal- x2 and take the measurement at the x1 as the reference. Thus, to-noise ratio (SNR) of the test-statistic while constraining Using(27), (28), (11), (15), (12), (14) and (29), we have the associated discriminant functional to be linear. The linear discriminant functional involved in our problem has the form |y − x1| −1 K(y, y0; v, v0)= γy,v,21 S2 (ω)× |y − x2| Z m w wH m ∗ j λ = h , i := dω = wi (ω)ˆmi (ω)dω (27) −ik (|y−x2|−γy,v,21|y−x1|) ∗ Z Z e E[ˆm1(γy,v,21 ω)]E[ˆm (ω)] dω Xi6=j 2 (32) where λ denote the the output of the discriminant functional, which we call the test-statistic and w is a template given by where T \ w = [w1 w2 · · · wN ] . 1+ y − x2 · v/c0 Under the assumption that the noise at different receivers γy,v,21 = \ . (33) 1+ y − x1 · v/c0 are wide sense stationary and mutually uncorrelated, the maximization of the SNR of λ [9] results in an optimal linear Using (30), (32) becomes template given by −1 4 −ikr21 ⋆ −1 K(y, y0; v, v0)= β S (ω)ω e w = S (ω)Py,v(ω)mr(ω) (28) Z 2 γy,v,21 1 −1 ik − (cTz −|y0−z0|) where S is the inverse of S defined by S(ω) = × e h“ αy0,v0,x1 αy0,v0,x2 ” 0 i R ′ R ′ ′. Using (24) and (26), We 1/2( 1(ω,ω )+ 0(ω,ω ))dω γy v ω −1 , ,21 ∗ Rcan approximate S by a diagonal matrix. (See [10] for × pˆ0 ω pˆ0 dω (34) −1 −1 αy0,v0,x1  αy0,v0,x2  details.) We denote diagonal elements of S by Si (ω),i = 1, · · · ,N and i 6= j, which is a function of the power spectral where β is a s scaling term due to geometric spreading factors density function of noise at the ith receiver and the kernel of and Doppler-scale-factors and Py,v. r21 = |y−x2|−γy,v,21|y−x1|−(|y0−x2|−γy,v,21|y0 −x1|) . IV. RESOLUTION ANALYSIS (35)

In this section, we assume that the surface topography is flat, Note that if Tz0 is chosen to be equal to |y0 − z0|/c0, the i.e., h(y)= h, for some y ∈ R2 and therefore set y = [y,h], second exponential term disappears. v = [v, 0]. We focus our analysis on the deterministic moving Examining (34), we see that if point target model given by γy v 1 , ,21 = , (36) y y y v v qv( )= Tδ( − 0)δ( − 0) (29) αy0,v0,x1 αy0,v0,x2 (34) becomes Based on the analysis above, we conclude that PSF peaks 2 at the intersection of the passive-iso-Doppler manifold defined T |y − x1| K(y, y ; v, v )= by (39), and the passive-iso-range manifold defined by (43). 0 0 (4π)4|y − z |2|y − x ||y − x |α6 0 0 0 1 2 y0,v0,x2 Note that (37) can be interpreted as a generalized auto- 2 −1 4 −i kr21 ω of the transmitted waveform p1. The spread × S2 (ω)ω e pˆ0 dω (37) Z αy v x  of the passive-iso-Doppler manifold and the passive-iso-range 0, 0, 2 manifold are both related to the shape of this generalized auto- −1/2 2 ω which defines the correlation of S2 (ω)ω pˆ1 α ambiguity function. Hence, the resolution of the reconstructed  y0,v0,x2  with itself in time domain. Clearly, this correlation peaks when image in (y, v) is determined by the overlapping area between r21 =0, i.e., the passive-iso-Doppler and passive-iso-range manifolds. For N > 2 receivers, it can be shown that multiple pairs |y−x |−γ |y−x | = |y −x |−γ |y −x | . (38) 2 y,v,21 1 0 2 y,v,21 0 1 of receivers generate multiple four-dimensional passive-iso- The analysis above shows that the PSF of the imaging oper- Doppler and passive-iso-range manifolds. These manifolds ator for two receivers and one transmitter becomes maximum intersect at the correct target location and correct velocity in under the two conditions given by (36) and (38). (y, v) space and contribute to the reconstruction of the target Substituting (6) into (36), we have image. The test-statistic value at the the correct target location and correct velocity increases by roughly a factor of N − 1 as 1+ y\− x · v /c γ = 0 2 0 0 = γ . (39) compared to the two-receiver scenario as described in (34). y,v,21 \ y0,v0,21 1+ y0 − x1 · v0/c0 V. CONCLUSION Note that y0 = (y0,h) and v0 = (v0, 0). Under the slower- In this work, we proposed a new passive radar imaging mover assumption, γy,v,ij can be approximated as follows: method for moving targets using sparsely distributed receivers. 1+ y\− x · v/c i 0 y\x y\x v We performed extensive numerical simulations to verify our γy,v,ij = \ ≈ 1 + ( − i − − j ) · /c0 . 1+ y − xj · v/c0 method. Due to page limitations, numerical simulations will (40) be presented at the conference. 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