Abstract

An inverse-scattering problem concerning the determination of a time–frequency spreading function is addressed. Such a function characterizes a dense group of reflecting objects at different ranges and moving with different velocities. The problem, arising in radar and other remote-sensing techniques, is a classical inverse problem. The aim is to reconstruct a function of two variables by means of signals (of one variable) reflected from the environment being observed. The proposed approach is developed by recourse to the frame theory in order to provide a reconstruction formula that asymptotically converges to a unique spreading function. The realistic situation with respect to the transmission of a finite number of signals is further considered. In this case the reconstruction formula is shown to yield the orthogonal projection of the spreading function onto a subspace generated by the outgoing signals.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. J. Kelly, R. P. Wishner, “Matched-filter theory for high-velocity targets,” IEEE Trans. Mil. Electron. ME-9, 56–59 (1965).
    [CrossRef]
  2. G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Berlin, 1994).
  3. G. Kaiser, “Physical wavelets and radar—a variational approach to remote-sensing,” IEEE Antennas Propag. Mag. 38(1), 15–24 (1996).
    [CrossRef]
  4. C. H. Wilcox, “The synthesis problem for radar ambiguity functions,” (U.S. Army Mathematics Research Center, University of Wisconsin, Madison, Wisc., 1960).
  5. C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).
  6. H. Naparst, “Radar signal choice and processing for dense target environment,” Ph.D. dissertation (University of California, Berkeley, Calif., 1988).
  7. L. Rebollo-Neira, J. Fernandez-Rubio, “On the windowed Fourier transform,” IEEE Trans. Inf. Theory 45, 2668–2671 (1999).
    [CrossRef]
  8. M. Bernfeld, “Chirp Doppler radar,” Proc. IEEE 72, 540–541 (1984).
    [CrossRef]
  9. M. Bernfeld, “On the alternatives for imaging rotational targets,” in Radar and Sonar, Part II, F. A. Grünbaum, M. Bernfeld, R. E. Bluhat, eds. (Springer-Verlag, New York, 1992), pp. 37–44.
  10. H. Naparst, “Dense target signal processing,” IEEE Trans. Inf. Theory 37, 317–327 (1991).
    [CrossRef]
  11. R. J. Duffin, A. C. Shaffer, “A class of nonharmonic Fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
    [CrossRef]
  12. R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).
  13. J. R. Klauder, S. K. Skagerstam, Coherent States (World Scientific, Singapore, 1985).
  14. I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. (N.Y.) 27, 1271–1283 (1986).
    [CrossRef]
  15. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
  16. C. Heil, D. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).
    [CrossRef]
  17. S. T. Ali, J. P. Antoine, J. P. Gazeau, “Square integrability of group representation on homogeneous spaces. II. Coherent and quasi-coherent states. The case of the Poincaré group,” Ann. Inst. Henri Poincaré 55, 857–890 (1991).
  18. S. T. Ali, J. P. Antoine, J. P. Gazeau, “Continuous frames in Hilbert space,” Ann. Phys. (N.Y.) 222, 1–37 (1993).
    [CrossRef]
  19. I. Daubechies, “The Wavelets transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]

1999 (1)

L. Rebollo-Neira, J. Fernandez-Rubio, “On the windowed Fourier transform,” IEEE Trans. Inf. Theory 45, 2668–2671 (1999).
[CrossRef]

1996 (1)

G. Kaiser, “Physical wavelets and radar—a variational approach to remote-sensing,” IEEE Antennas Propag. Mag. 38(1), 15–24 (1996).
[CrossRef]

1993 (1)

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Continuous frames in Hilbert space,” Ann. Phys. (N.Y.) 222, 1–37 (1993).
[CrossRef]

1991 (2)

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Square integrability of group representation on homogeneous spaces. II. Coherent and quasi-coherent states. The case of the Poincaré group,” Ann. Inst. Henri Poincaré 55, 857–890 (1991).

H. Naparst, “Dense target signal processing,” IEEE Trans. Inf. Theory 37, 317–327 (1991).
[CrossRef]

1990 (1)

I. Daubechies, “The Wavelets transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

1989 (1)

C. Heil, D. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).
[CrossRef]

1986 (1)

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. (N.Y.) 27, 1271–1283 (1986).
[CrossRef]

1984 (1)

M. Bernfeld, “Chirp Doppler radar,” Proc. IEEE 72, 540–541 (1984).
[CrossRef]

1965 (1)

E. J. Kelly, R. P. Wishner, “Matched-filter theory for high-velocity targets,” IEEE Trans. Mil. Electron. ME-9, 56–59 (1965).
[CrossRef]

1952 (1)

R. J. Duffin, A. C. Shaffer, “A class of nonharmonic Fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
[CrossRef]

Ali, S. T.

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Continuous frames in Hilbert space,” Ann. Phys. (N.Y.) 222, 1–37 (1993).
[CrossRef]

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Square integrability of group representation on homogeneous spaces. II. Coherent and quasi-coherent states. The case of the Poincaré group,” Ann. Inst. Henri Poincaré 55, 857–890 (1991).

Antoine, J. P.

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Continuous frames in Hilbert space,” Ann. Phys. (N.Y.) 222, 1–37 (1993).
[CrossRef]

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Square integrability of group representation on homogeneous spaces. II. Coherent and quasi-coherent states. The case of the Poincaré group,” Ann. Inst. Henri Poincaré 55, 857–890 (1991).

Bernfeld, M.

M. Bernfeld, “Chirp Doppler radar,” Proc. IEEE 72, 540–541 (1984).
[CrossRef]

M. Bernfeld, “On the alternatives for imaging rotational targets,” in Radar and Sonar, Part II, F. A. Grünbaum, M. Bernfeld, R. E. Bluhat, eds. (Springer-Verlag, New York, 1992), pp. 37–44.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).

Cook, C. E.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).

Daubechies, I.

I. Daubechies, “The Wavelets transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. (N.Y.) 27, 1271–1283 (1986).
[CrossRef]

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Duffin, R. J.

R. J. Duffin, A. C. Shaffer, “A class of nonharmonic Fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
[CrossRef]

Fernandez-Rubio, J.

L. Rebollo-Neira, J. Fernandez-Rubio, “On the windowed Fourier transform,” IEEE Trans. Inf. Theory 45, 2668–2671 (1999).
[CrossRef]

Gazeau, J. P.

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Continuous frames in Hilbert space,” Ann. Phys. (N.Y.) 222, 1–37 (1993).
[CrossRef]

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Square integrability of group representation on homogeneous spaces. II. Coherent and quasi-coherent states. The case of the Poincaré group,” Ann. Inst. Henri Poincaré 55, 857–890 (1991).

Grossmann, A.

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. (N.Y.) 27, 1271–1283 (1986).
[CrossRef]

Heil, C.

C. Heil, D. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).
[CrossRef]

Kaiser, G.

G. Kaiser, “Physical wavelets and radar—a variational approach to remote-sensing,” IEEE Antennas Propag. Mag. 38(1), 15–24 (1996).
[CrossRef]

G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Berlin, 1994).

Kelly, E. J.

E. J. Kelly, R. P. Wishner, “Matched-filter theory for high-velocity targets,” IEEE Trans. Mil. Electron. ME-9, 56–59 (1965).
[CrossRef]

Klauder, J. R.

J. R. Klauder, S. K. Skagerstam, Coherent States (World Scientific, Singapore, 1985).

Meyer, Y.

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. (N.Y.) 27, 1271–1283 (1986).
[CrossRef]

Naparst, H.

H. Naparst, “Dense target signal processing,” IEEE Trans. Inf. Theory 37, 317–327 (1991).
[CrossRef]

H. Naparst, “Radar signal choice and processing for dense target environment,” Ph.D. dissertation (University of California, Berkeley, Calif., 1988).

Rebollo-Neira, L.

L. Rebollo-Neira, J. Fernandez-Rubio, “On the windowed Fourier transform,” IEEE Trans. Inf. Theory 45, 2668–2671 (1999).
[CrossRef]

Shaffer, A. C.

R. J. Duffin, A. C. Shaffer, “A class of nonharmonic Fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
[CrossRef]

Skagerstam, S. K.

J. R. Klauder, S. K. Skagerstam, Coherent States (World Scientific, Singapore, 1985).

Walnut, D.

C. Heil, D. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).
[CrossRef]

Wilcox, C. H.

C. H. Wilcox, “The synthesis problem for radar ambiguity functions,” (U.S. Army Mathematics Research Center, University of Wisconsin, Madison, Wisc., 1960).

Wishner, R. P.

E. J. Kelly, R. P. Wishner, “Matched-filter theory for high-velocity targets,” IEEE Trans. Mil. Electron. ME-9, 56–59 (1965).
[CrossRef]

Young, R. M.

R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).

Ann. Inst. Henri Poincaré (1)

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Square integrability of group representation on homogeneous spaces. II. Coherent and quasi-coherent states. The case of the Poincaré group,” Ann. Inst. Henri Poincaré 55, 857–890 (1991).

Ann. Phys. (N.Y.) (1)

S. T. Ali, J. P. Antoine, J. P. Gazeau, “Continuous frames in Hilbert space,” Ann. Phys. (N.Y.) 222, 1–37 (1993).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

G. Kaiser, “Physical wavelets and radar—a variational approach to remote-sensing,” IEEE Antennas Propag. Mag. 38(1), 15–24 (1996).
[CrossRef]

IEEE Trans. Inf. Theory (3)

L. Rebollo-Neira, J. Fernandez-Rubio, “On the windowed Fourier transform,” IEEE Trans. Inf. Theory 45, 2668–2671 (1999).
[CrossRef]

I. Daubechies, “The Wavelets transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

H. Naparst, “Dense target signal processing,” IEEE Trans. Inf. Theory 37, 317–327 (1991).
[CrossRef]

IEEE Trans. Mil. Electron. (1)

E. J. Kelly, R. P. Wishner, “Matched-filter theory for high-velocity targets,” IEEE Trans. Mil. Electron. ME-9, 56–59 (1965).
[CrossRef]

J. Math. Phys. (N.Y.) (1)

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. (N.Y.) 27, 1271–1283 (1986).
[CrossRef]

Proc. IEEE (1)

M. Bernfeld, “Chirp Doppler radar,” Proc. IEEE 72, 540–541 (1984).
[CrossRef]

SIAM Rev. (1)

C. Heil, D. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).
[CrossRef]

Trans. Am. Math. Soc. (1)

R. J. Duffin, A. C. Shaffer, “A class of nonharmonic Fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
[CrossRef]

Other (8)

R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).

J. R. Klauder, S. K. Skagerstam, Coherent States (World Scientific, Singapore, 1985).

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

M. Bernfeld, “On the alternatives for imaging rotational targets,” in Radar and Sonar, Part II, F. A. Grünbaum, M. Bernfeld, R. E. Bluhat, eds. (Springer-Verlag, New York, 1992), pp. 37–44.

G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Berlin, 1994).

C. H. Wilcox, “The synthesis problem for radar ambiguity functions,” (U.S. Army Mathematics Research Center, University of Wisconsin, Madison, Wisc., 1960).

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).

H. Naparst, “Radar signal choice and processing for dense target environment,” Ph.D. dissertation (University of California, Berkeley, Calif., 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

For every value of ϕ, the functions im(t, ϕ)=fm(t)exp(2πιϕt), m=1 ,, K, are the input for K filters, each of which is characterized by the corresponding impulse response g˜m(t)¯, m=1 ,, K. The orthogonal projection p˜(τ, ϕ) of the spreading function is obtained simply by adding together the K outputs om(τ, ϕ), m=1 ,, K.

Equations (83)

Equations on this page are rendered with MathJax. Learn more.

f(t)=αy[s(t-τ)],
s=c-vc+v,τ=2Rc-v,
f(t)=α exp(-2πiϕt)g(t-τ),
f(t)=--p(τ, ϕ)exp(-2πιϕt)g(t-τ)dτdϕ.
wgf(τ, ϕ)=-f(t)exp(2πιϕt)g(t-τ)¯dt,
f(t)=1 g2 --wgf(τ, ϕ)exp(-2πιϕt)g(t-τ)dtdϕ,
p(τ, ϕ)=wg f(τ, ϕ)g2+p(τ, ϕ),
c=mNemcm,cm=em,cl2.
g, f L2=-g(t)¯f(t)d t.
Af, f L2mN|gm, f L2|2Bf, f L2.
b=mNemgm, f L2,
bl22=b,bl2=mN|gm, f L2|2
Tˆ=mNemgm,L2.
T^=mNgmem,l2,
AI^L2GˆBI^L2,
Gˆ=mNgmgm,L2.
B-1I^L2G^-1A-1I^L2.
I^L2=mNgmgm,L2=mNgmgm,L2.
hm=gm+hm-kNhkgk, gmL2,mN,
I^L2=mNgmhm,L2=mNhmgm,L2.
f=mNgmhm, f L2=mNhmgm, f L2,
f=mNgmgm, f L2=mNgmgm, f L2,
f¯=mNgm¯f, gmL2=mNgm¯f, gmL2.
fm(t)=--p(τ, ϕ)exp(-2πιϕt)gm(t-τ)dτdϕ,
mN.
P(τ, t)=-p(τ, ϕ)exp(-2πιϕt)dϕ,
fm(t)=-P(τ, t)gm(t-τ)dτ,mN
fm(t)=-P(t-μ, t)gm(μ)dμP(t-μ, t)¯, gm(μ)L2,mN.
mNgm(t-τ)¯fm(t)
=mNgm(t-τ)¯-P(τ, t)gm(t-τ)dτ=mNgm(t-τ)¯P(t-μ, t)¯, gm(μ)L2=P(τ, t).
-mNgm(t-τ)¯fm(t)exp(2πιϕt)dt
=-P(τ, t)exp(2πιϕt)dt=p(τ, ϕ).
fm(t)=-P(τ, t)gm(t-τ)dτ
p(τ, ϕ)=-mNgm(t-τ)¯fm(t)exp(2πιϕt)dt.
p(τ, ϕ)=-mNgm(t-τ)¯exp(2πιϕt)×-P(τ, t)gm(t-τ)dτdt
=-mNgm(t-τ)¯exp(2πιϕt)×P(t-μ, t)¯, gm(μ)L2dt
=-P(τ, t)exp(2πιϕt)dtp(τ, ϕ).
p(τ, ϕ)=-mNhm(t-τ)¯exp(2πιϕt)fm(t)dt
=-mNhm(t-τ)¯exp(2πιϕt)×P(t-μ, t)¯, gm(μ)L2dt,
p(τ, ϕ)=-P(τ, t)exp(2πιϕt)d tp(τ, ϕ).
T^K=m=1Kemgm, L2,T^K=m=1Kgmem, l2,
G^K=T^KT^K=m=1Kgmgm, L2.
P^G˜=m=1Kg˜mgm, L2=m=1Kgmg˜m, L2.
m=1Kg˜m(t)¯-f(t)gm(t)d tm=1Kg˜m(t)¯f¯, gmL2
P^G˜u(t)=m=1Kg˜m(t)gm, uL2=m=1Kg˜m(t)-gm(t)¯u(t)d t
f(t)=m=1Kg˜m(t)-gm(t)¯f(t)dt.
f(t)=m=1Kg˜m(t)gm, f L2=P^G¯ f(t).
f(t)¯=m=1Kg˜m(t)¯-gm(t)f(t)¯d t=m=1Kg˜m(t)¯f,gmL2.
m=1Kg˜m(t-τ)¯fm(t)=m=1Kg˜m(t-τ)¯×P(t-μ,t)¯, gm(μ)L2.
m=1Kg˜m(t-τ)¯fm(t)=P˜(τ, t),
P(t-μ, t)¯=m=1Kcm(t)gm(μ),
P(τ, t)¯=m=1Kcm(t)gm(t-τ).
- m=1Kg˜m(t-τ)¯fm(t)exp(2πιϕt)dt=-P˜(τ,t)exp(2πιϕt)dt=p˜(τ, ϕ).
om(τ, ϕ)=-g˜m(t-τ)¯fm(t)exp(2πιϕt)dt.
fme(t)=-P˜e(τ, t)gm(t-τ)dτ,m=1 ,, K,
P˜e(τ, t)=k=1Kg˜k(t-τ)¯fko(t),
fto=m=1Kfmo(t)em,
ft=m=1Kfm(t)em.
ft, ftel2=ft, ftol2t.
fm(t)=-P(τ, t)gm(t-τ)dτ=-P(t-v, t)gm(v)dv=P(t-v, t)¯, gm(v)L2
fte=m=1Kfme(t)em,
fme(t)=-k=1Kfko(t)g˜k(t-τ)¯gm(t-τ)dτ
=k=1Kfko(t)-[G˜^K]-1gk(μ)¯gm(μ)dμ
=k=1Kfko(t)[G˜^K]-1gk, gmL2.
ft, ftel2=m=1Kfm(t)¯fme(t)=m=1K-P(t-v, t)gm(v)¯dv×k=1Kfko(t)[G˜^K]-1gk, gmL2
=m=1Kk=1Kfko(t)[G˜^K]-1gk, gmL2gm(v),
P(t-v, t)¯L2.
ft,ftel2=k=1Kfko(t)[G˜^K]-1gk(v), G^KP(t-v, t)¯L2.
ft, ftel2=k=1Kfko(t)G^K[G˜^K]-1gk(v), P(t-v, t)¯L2.
ft, ftel2=k=1Kfko(t)gk(v), P(t-v, t)¯L2
=k=1Kfko(t)fk(t)¯=ft, ft0l2.
fte, ftel2=fte, ftol2.
d2(t)=ft-ftol22=ft-fto, ft-ftol2.
d2(t)=ft-fte+fte-fto, ft-fte+fte-ftol2.
d2(t)=ft-ftel22+fte-ftol22,
T^KT^Kψk=λkψk,k=1 ,, K,
G˜^K=k=1rϕkλkϕk,L2.
[G˜^K]-1=k=1rϕk 1λk ϕk,L2,
g˜m=[G˜^K]-1gm=k=1rϕk 1λk ϕk, gmL2,
m=1 ,, K.
P^G˜=k=1rϕkϕk,L2.
P^G˜=k=1rϕkϕk,L2=k=1r G^Kϕkλk ϕk,L2=G^K[G˜^K]-1.
m=1Kg˜mgm,L2=m=1K[G˜^K]-1gmgm,L2=[G˜^K]-1G^K=P^G˜.

Metrics