Books. B. Borden, Radar Imaging of Airborne Targets, Institute of Physics, 1999.

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1 Books B. Borden, Radar Imaging of Airborne Targets, Institute of Physics, C. Elachi, Spaceborne Radar Remote Sensing: Applications and Techniques, IEEE Press, New York, W. C. Carrara, R. G. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms, Artech House, Boston, G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing, CRC Press, New York, L.J. Cutrona, Synthetic Aperture Radar, in Radar Handbook, second edition, ed. M. Skolnik, McGraw-Hill, New York, C.V. Jakowatz, D.E. Wahl, P.H. Eichel, D.C. Ghiglia, and P.A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach, Kluwer, Boston, I.G. Cumming and F.H. Wong, Digital Processing of SAR Data: Algorithms and Implementation, Artech House, 2005

2 Outline 1. introduction, history, frequency bands, db, real-aperture imaging 2. radar systems: stepped-frequency systems, I/Q demodulation 3. 1D scattering by perfect conductor 4. receiver design, matched filtering 5. ambiguity function & its properties 6. range-doppler (unfocused) imaging 7. introduction to 3D scattering 8. ISAR 9. antenna theory 10. spotlight SAR 11. stripmap SAR 12. time permitting: deramp processing, Doppler from successive pulses

3 3D Mathematical Model We should use Maxwell s equations; but instead we use ( 2 1 c 2 (x) 2 t ) E(t, x) = j(t, x) }{{} source Scattering is due to a perturbation in the wave speed c: 1 c 2 (x) = 1 c 2 0 V (x) }{{} For a moving target, use V (x, t). reflectivity function

4 Basic facts about the wave equation fundamental solution g ( 2 c 2 ) 0 2 t g(t, x) = δ(t)δ(x) g(t, x) = δ(t x /c 0) 4π x = e iω(t x /c 0 ) 8π 2 dω x g(t, x) = field at (t, x) due to a source at the origin at time 0 Solution of is ( 2 c t ) u(t, x) = j(t, x), u(t, x) = g(t t, x y)j(t, y)dt dy frequency domain: k = ω/c 0 ( 2 + k 2 )G = δ G(ω, x) = eik x 4π x

5 Introduction to scattering theory ( 2 c 2 (x) t 2 ) E(t, x) = j(t, x) ( 2 c t )E in (t, x) = j(t, x) write E = E in + E sc, c 2 (x) = c 2 0 V (x), subtract: ( 2 2 t ) E sc (t, x) = V (x) 2 t E(t, x) use fundamental solution E sc (t, x) = g(t τ, x z)v (z) τ 2 E(τ, z)dτdz. Lippman-Schwinger integral equation frequency domain Lippman-Schwinger equation: E sc (ω, x) = G(ω, x z)v (z)ω 2 E(ω, z)dz

6 single-scattering or Born approximation E sc (t, x) EB sc := g(t τ, x z)v (z) τ 2 E in (τ, z)dτdz useful: makes inverse problem linear not necessarily a good approximation! In the frequency domain, E sc B (ω, x) = e ik x z 4π x z V (z)ω2 E in (ω, z)dz

7 E in is obtained by solving ( 2 + k 2 )E in = J For now, suppose J(ω, x) = P (ω)δ(x x 0 ) E in (ω, x) = G(ω, x y)p (ω)δ(y x 0 )dt dy = P (ω) eik x x0 4π x x 0 Then the scattered field back at x 0 is EB sc (ω, x 0 ) = P (ω) ω 2 e 2ik x0 z (4π) 2 x 0 z 2 V (z)dz In the time domain this is E sc B (t, x 0 ) = e iω(t 2 x 0 z /c) 2π(4π x 0 z ) 2 k2 P (ω)v (z)dωdz Note 1/R 2 geometrical decay power decays like 1/R 4

8 η(t, x 0 ) = Matched filtering p (t t)eb sc (t, x 0 )dt ( ) 1 e iω (t t) P (ω )dω 2π e iω(t 2 x 0 z /c) 2π(4π x 0 z ) 2 k2 P (ω)v (z)dωdz dt. do t and ω integrations = e iω(t 2 x 0 z /c) 2π(4π x 0 z ) 2 k2 P (ω) 2 V (z)dωdz Effect of matched filter is to replace P (ω) by P (ω) 2.

9 Far-field expansion z << x 0 x 0 z = x 0 x 0 z + O( x 0 1 ) where x = x/ x Proof: x 0 z = (x 0 z) (x 0 z) = x 0 2 2x 0 z + z 2 = x x0 z x 0 + z 2 2 x 0 use 1 a = 1 a 2 2 ( + = x 0 1 x ) 0 z x 0 + O( x 0 2 ) ( ) = x 0 x z 0 z + O x 0 Note: Whether z << x 0 depends on location of origin of coordinates. e ik x0 z x 0 z = eik x x 0 0 e ikc x 0 z ( 1 + O ( )) ( z x O ( )) k z 2 x 0

10 Far-field approximation to data applies to smallish target (Born-approximated) output of correlation receiver is: η B (t, x 0 ) = e iω(t 2 x 0 z /c) 2π(4π x 0 z ) 2 k2 P (ω) 2 V (z)dωdz put origin of coordinates in target, use far-field expansion η B (t, x 0 1 ) 32π 2 x 0 2 e iω(t 2 x0 /c+2x c0 z/c) k 2 P (ω) 2 V (z)dωdz in the frequency domain: D B (ω, x 0 ) e2ik x0 32π 2 x 0 2 k2 P (ω) 2 e 2ikc x 0 z V (z)dz }{{} F[V ](2k c x 0 )!

11 Outline 1. introduction, history, frequency bands, db, real-aperture imaging 2. radar systems: stepped-frequency systems, I/Q demodulation 3. 1D scattering by perfect conductor 4. receiver design, matched filtering 5. ambiguity function & its properties 6. range-doppler (unfocused) imaging 7. introduction to 3D scattering 8. ISAR 9. antenna theory 10. spotlight SAR 11. stripmap SAR 12. time permitting: deramp processing, Doppler from successive pulses

12 Airborne targets

13 Inverse Synthetic Aperture Radar (ISAR) fixed radar, moving target (usually airplane or ship) transmit pulse train typical pulse length 10 4 sec, rotation rate Ω = 10 /sec 1/6 R/sec, target radius 10m distance traveled during pulse 10 4 m start-stop approximation for airborne target, measurements from nth pulse contain no reflections from earlier pulses take out translational motion via tracking and range alignment, leaving only rotational motion

14 Range alignment

15 ISAR vs SAR

16 Mathematics of ISAR at start of nth pulse, V (x) = q(o(θ n )x) where O is an orthogonal matrix For example: if radar is in plane to axis of rotation ( turntable geometry ), cos θ sin θ 0 O(θ) = sin θ cos θ D B (ω, θ n ) = e 2ik x0 32π 2 x 0 2 k2 P (ω) 2 e 2ikc x 0 z q(o(θ n )z)dz }{{} y ( use x 0 O 1 (θ n )y = O(θ n )x 0 y) e 2ik x0 32π 2 x 0 2 k2 P (ω) 2 e 2ik O(θ n) x c0 y V ( y)dy }{{} F[q](2kO(θ n ) c x 0 )!

17 Region in Fourier space where we have data The backhoe data dome Ω (turntable geometry)

18 Polar Format Algorithm (PFA) applies to turntable geometry, frequency-domain data 1. interpolate to rectangular grid 2. use 2D Fast Fourier Transform

19 An ISAR image of a Boeing 727 taking off from LAX

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23 Assume turntable geometry ISAR Resolution Notation Take x 0 = (1, 0, 0), write ê θ = O(θ)x 0. D(k, θ) = e 2ik(y 1 cos θ+y 2 sin θ) q(y 1, y 2, y 3 )dy 1 dy 2 dy 3 e 2ik(y 1 cos θ+y 2 sin θ) q(y 1, y 2, y 3 )dy 3 dy 1 dy 2 } {{ } q(y 1,y 2 ) get 2D image of q = target projected onto plane containing radar

24 ISAR Resolution, continued D(k, θ) = χ Ω (kê θ )F[q](kê θ ) to form image, take 2D inverse Fourier transform I(x) = e ix kê θ D(k, θ)kdkdθ = e ix kê θ e iy kêθ q(y)dykdkdθ Ω = e i(x y) kê θ kdkdθ q(y)d 2 y Ω } {{ } K(x y) point spread function (PSF), imaging kernel, ambiguity function

25 ISAR Resolution: analysis of PSF K(z) = Ω e iz kê θ kdkdθ Take ê θ = (cos θ, sin θ) (1, θ) (small-angle approx.) z = r(cos ψ, sin ψ) b = k 2 k 1 k c = ω c /c Down-range resolution K(r, 0) 2Φ i d dr [e ik cr b2 sinc(br/2) ] K(r, 0) bω c Φ e ik cr sinc br 2 down-range resolution is 4π/b note oscillatory factor

26 Cross-range Resolution K(0, r) 2bk c Φ sinc ( brφ 2 ) sinc(k c rφ) k c b K(r, 0) 2bk c Φ sinc(k c rφ) cross-range resolution is 2π/(k c Φ)

27 D B (ω, θ n ) η B (t, θ n ) ISAR in the time domain e 2ik x0 32π 2 x 0 2 k2 P (ω) 2 Fourier transform e 2ik O(θ n) x c0 y q( y)dy }{{} F[q](2kO(θ n ) c x 0 ) e iω(t 2 x0 /c+2o(θ n ) c x 0 y/c) ω 2 P (ω) 2 dω q(y)dy η B ( t + 2 x 0 c, θ n ) = e iω(t+ τ {}}{ 2O(θ n )x 0 y/c) ω 2 P (ω) 2 dω }{{} R δ(s τ) e iω(t s) ω 2 P (ω) 2 dω } {{ } χ(t s) ds q(y)dy

28 ( 2 x 0 ) η B t +, θ n c where R[q](s, µ) = = ( ) χ(t s) δ s 2O(θ n) x 0 y q(y)dy c }{{} R[q] ( ) 2O(θ n ) x = χ R[q] 0 c δ(s µ x)q(x)dx s, 2O(θ n) d x 0 c «is the Radon transform ds For ISAR, use waveform so that χ δ (high range-resolution waveform)

29 Radon transform R bµ [q](s) = R[q](s, µ) = = Properties Projection-slice theorem: δ(s µ x)q(x)dx = x bµ q(s µ + x )d n 1 x s=bµ x q(x)d n x F s σ (R bµ [f])(σ) = (2π) (n 1)/2 F n [f](σ µ) 1D Fourier transform nd Fourier transform here Fourier transforms have symmetric 1/ 2π convention

30 Filtered backprojection (FBP): (R # h) f = R # (h R[f]) backprojection operator filter where R # operates on h(s, µ) via (R # h)(x) = h(x µ, µ)d µ S n 1 R # integrates over part of h corresponding to all lines through x R # is the formal adjoint of R, i.e., Rf, h L2 (R S n 1 ) = f, R # h L2 (R n )

31 Radon transform

32 1 view 4 views 8 views 16 views 60 views

33 ISAR summary use high range-resolution waveform (broad frequency band) time-domain formulation gives rise to Radon transform invert via FBP or... frequency-domain formulation gives Fourier transform of target on polar grid interpolate to cartesian grid, use FFT can make images from an aperture of a few degrees because of high carrier frequency ISAR is sometimes called range-doppler imaging Doppler shift can be inferred from phase of successive measurements

34 Where more work is needed on ISAR model; Born approximation leaves out: multiple scattering shadowing creeping waves find target/sensor motion 3D imaging template library look-up fast algorithms

35 Outline 1. introduction, history, frequency bands, db, real-aperture imaging 2. radar systems: stepped-frequency systems, I/Q demodulation 3. 1D scattering by perfect conductor 4. receiver design, matched filtering 5. ambiguity function & its properties 6. range-doppler (unfocused) imaging 7. introduction to 3D scattering 8. ISAR 9. antenna theory 10. spotlight SAR 11. stripmap SAR 12. time permitting: deramp processing, Doppler from successive pulses