Abstract

We report the use of periodic, pseudonoise waveforms in a multifunction coherent ladar system. We exploit the Doppler sensitivity of these waveforms, as well as agile processing, to enable diverse ladar functions, including high range resolution imaging, macro-Doppler imaging, synthetic aperture ladar, and range-resolved micro-Doppler imaging. We present analytic expressions and simulations demonstrating the utility of pseudonoise waveforms for each of the ladar modes. We also discuss a laboratory pseudonoise ladar system that was developed to demonstrate range compression and range-resolved micro-Doppler imaging, as well as the phase recovery common to each of the coherent modes.

© 2010 Optical Society of America

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References

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  1. J. Abshire, J. Rall, and S. Manizade, “Altimetry and lidar using AlGaAs lasers modulated with pseudo-random codes,” NASA Conference Publication 3158 Part 2, presented at the 16th International Laser Radar Conference (ILRC), Cambridge, MA, 20-24 July 1992.
  2. M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.
  3. W. Stone, M. Juberts, N. Dagalakis, J. Stone and J. Gorman, “Performance analysis of next-generation LADAR for manufacturing, construction, and mobility,” NISTIR 7117, National Institute of Standards and Technology, Gaithersburg, MD, May 2004.
  4. Polytec, “Single point vibrometers,” http://www.polytec.com/usa/158_421.asp.
  5. R. R. Ebert and P. Lutzmann, “Vibration imagery of remote objects,” Proc. SPIE 4821, 1-10 (2002).
    [CrossRef]
  6. J. Czarske and O. Dölle, “Investigations on the frequency measuring error of laser Doppler velocimeters using the quadrature demodulation techniques,” Proc. SPIE 3749, 599-600(1999).
    [CrossRef]
  7. B. Redman, W. Ruff, and K. Aliberti, “Direct-detection laser vibrometry with an amplitude modulated ladar,” Proc. SPIE 5412, 218-228 (2004).
    [CrossRef]
  8. T. Green, S. Marcus, and B. Colella, “Synthetic-aperture-radar imaging with a solid-state laser,” Appl. Opt. 34, 6941-6949(1995).
    [CrossRef] [PubMed]
  9. W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
    [CrossRef]
  10. J. Ricklin, B. Schumm, and P. Tomlinson, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conference, Snowmass, CO (9-13 July 2007).
  11. J. Johnson, “Analysis of image forming systems,” in Image Intensifier Symposium, AD 220160 (Warfare Electrical Engineering Department, United States Army Research and Development Laboratories, 1958), pp. 244-273.
  12. N. Levanon and E. Mozeson, Radar Signals (Wiley InterScience, 2004).
    [CrossRef]
  13. M. Dierking, F. Heitkamp, and L. Barnes, “High temporal resolution laser radar tomography for long range target identification,” presented at the IEEE/OSA Signal Synthesis and Reconstruction Conference (Kona, Hawaii, June 1998).
  14. M. Richards, Fundamentals for Radar Signal Processing (McGraw-Hill, 2005).
  15. M. Skolnik, Radar Handbook (McGraw-Hill, 1990).
  16. M. Skolnik, Introduction to Radar Systems (McGraw-Hill, 2005).
  17. L. Cohen, Time Frequency Analysis: Theory and Applications, Signal Processing Series (Prentice Hall, 1995).
  18. M. Dierking and B. Duncan, “Genetically optimized periodic, pseudo-noise waveforms for multi-function coherent ladar,” presented at the 15th Coherent Laser Radar Conference, Toulouse, France, 22-26 June 2009.

2005 (1)

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

2004 (1)

B. Redman, W. Ruff, and K. Aliberti, “Direct-detection laser vibrometry with an amplitude modulated ladar,” Proc. SPIE 5412, 218-228 (2004).
[CrossRef]

2002 (1)

R. R. Ebert and P. Lutzmann, “Vibration imagery of remote objects,” Proc. SPIE 4821, 1-10 (2002).
[CrossRef]

1999 (1)

J. Czarske and O. Dölle, “Investigations on the frequency measuring error of laser Doppler velocimeters using the quadrature demodulation techniques,” Proc. SPIE 3749, 599-600(1999).
[CrossRef]

1995 (1)

Abshire, J.

J. Abshire, J. Rall, and S. Manizade, “Altimetry and lidar using AlGaAs lasers modulated with pseudo-random codes,” NASA Conference Publication 3158 Part 2, presented at the 16th International Laser Radar Conference (ILRC), Cambridge, MA, 20-24 July 1992.

Aliberti, K.

B. Redman, W. Ruff, and K. Aliberti, “Direct-detection laser vibrometry with an amplitude modulated ladar,” Proc. SPIE 5412, 218-228 (2004).
[CrossRef]

Barnes, L.

M. Dierking, F. Heitkamp, and L. Barnes, “High temporal resolution laser radar tomography for long range target identification,” presented at the IEEE/OSA Signal Synthesis and Reconstruction Conference (Kona, Hawaii, June 1998).

Beck, S.

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

Buck, J.

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

Buell, W.

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

Cisewski, M.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

Cohen, L.

L. Cohen, Time Frequency Analysis: Theory and Applications, Signal Processing Series (Prentice Hall, 1995).

Colella, B.

Czarske, J.

J. Czarske and O. Dölle, “Investigations on the frequency measuring error of laser Doppler velocimeters using the quadrature demodulation techniques,” Proc. SPIE 3749, 599-600(1999).
[CrossRef]

Dagalakis, N.

W. Stone, M. Juberts, N. Dagalakis, J. Stone and J. Gorman, “Performance analysis of next-generation LADAR for manufacturing, construction, and mobility,” NISTIR 7117, National Institute of Standards and Technology, Gaithersburg, MD, May 2004.

Dickinson, R.

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

Dierking, M.

M. Dierking and B. Duncan, “Genetically optimized periodic, pseudo-noise waveforms for multi-function coherent ladar,” presented at the 15th Coherent Laser Radar Conference, Toulouse, France, 22-26 June 2009.

M. Dierking, F. Heitkamp, and L. Barnes, “High temporal resolution laser radar tomography for long range target identification,” presented at the IEEE/OSA Signal Synthesis and Reconstruction Conference (Kona, Hawaii, June 1998).

Dobbs, M.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

Dölle, O.

J. Czarske and O. Dölle, “Investigations on the frequency measuring error of laser Doppler velocimeters using the quadrature demodulation techniques,” Proc. SPIE 3749, 599-600(1999).
[CrossRef]

Duncan, B.

M. Dierking and B. Duncan, “Genetically optimized periodic, pseudo-noise waveforms for multi-function coherent ladar,” presented at the 15th Coherent Laser Radar Conference, Toulouse, France, 22-26 June 2009.

Ebert, R. R.

R. R. Ebert and P. Lutzmann, “Vibration imagery of remote objects,” Proc. SPIE 4821, 1-10 (2002).
[CrossRef]

Gorman, J.

W. Stone, M. Juberts, N. Dagalakis, J. Stone and J. Gorman, “Performance analysis of next-generation LADAR for manufacturing, construction, and mobility,” NISTIR 7117, National Institute of Standards and Technology, Gaithersburg, MD, May 2004.

Green, T.

Harrison, F.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

Heitkamp, F.

M. Dierking, F. Heitkamp, and L. Barnes, “High temporal resolution laser radar tomography for long range target identification,” presented at the IEEE/OSA Signal Synthesis and Reconstruction Conference (Kona, Hawaii, June 1998).

Johnson, J.

J. Johnson, “Analysis of image forming systems,” in Image Intensifier Symposium, AD 220160 (Warfare Electrical Engineering Department, United States Army Research and Development Laboratories, 1958), pp. 244-273.

Juberts, M.

W. Stone, M. Juberts, N. Dagalakis, J. Stone and J. Gorman, “Performance analysis of next-generation LADAR for manufacturing, construction, and mobility,” NISTIR 7117, National Institute of Standards and Technology, Gaithersburg, MD, May 2004.

Kozlowski, D.

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

Krabill, W.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

Levanon, N.

N. Levanon and E. Mozeson, Radar Signals (Wiley InterScience, 2004).
[CrossRef]

Lutzmann, P.

R. R. Ebert and P. Lutzmann, “Vibration imagery of remote objects,” Proc. SPIE 4821, 1-10 (2002).
[CrossRef]

Manizade, S.

J. Abshire, J. Rall, and S. Manizade, “Altimetry and lidar using AlGaAs lasers modulated with pseudo-random codes,” NASA Conference Publication 3158 Part 2, presented at the 16th International Laser Radar Conference (ILRC), Cambridge, MA, 20-24 July 1992.

Marcus, S.

Marechal, N.

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

McGregor, D.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

Mozeson, E.

N. Levanon and E. Mozeson, Radar Signals (Wiley InterScience, 2004).
[CrossRef]

Neal, M.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

Rall, J.

J. Abshire, J. Rall, and S. Manizade, “Altimetry and lidar using AlGaAs lasers modulated with pseudo-random codes,” NASA Conference Publication 3158 Part 2, presented at the 16th International Laser Radar Conference (ILRC), Cambridge, MA, 20-24 July 1992.

Redman, B.

B. Redman, W. Ruff, and K. Aliberti, “Direct-detection laser vibrometry with an amplitude modulated ladar,” Proc. SPIE 5412, 218-228 (2004).
[CrossRef]

Richards, M.

M. Richards, Fundamentals for Radar Signal Processing (McGraw-Hill, 2005).

Ricklin, J.

J. Ricklin, B. Schumm, and P. Tomlinson, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conference, Snowmass, CO (9-13 July 2007).

Ruff, W.

B. Redman, W. Ruff, and K. Aliberti, “Direct-detection laser vibrometry with an amplitude modulated ladar,” Proc. SPIE 5412, 218-228 (2004).
[CrossRef]

Schumm, B.

J. Ricklin, B. Schumm, and P. Tomlinson, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conference, Snowmass, CO (9-13 July 2007).

Shum, C.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

Skolnik, M.

M. Skolnik, Radar Handbook (McGraw-Hill, 1990).

M. Skolnik, Introduction to Radar Systems (McGraw-Hill, 2005).

Stokes, S.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

Stone, J.

W. Stone, M. Juberts, N. Dagalakis, J. Stone and J. Gorman, “Performance analysis of next-generation LADAR for manufacturing, construction, and mobility,” NISTIR 7117, National Institute of Standards and Technology, Gaithersburg, MD, May 2004.

Stone, W.

W. Stone, M. Juberts, N. Dagalakis, J. Stone and J. Gorman, “Performance analysis of next-generation LADAR for manufacturing, construction, and mobility,” NISTIR 7117, National Institute of Standards and Technology, Gaithersburg, MD, May 2004.

Tomlinson, P.

J. Ricklin, B. Schumm, and P. Tomlinson, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conference, Snowmass, CO (9-13 July 2007).

Wright, T.

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

Appl. Opt. (1)

Proc. SPIE (4)

R. R. Ebert and P. Lutzmann, “Vibration imagery of remote objects,” Proc. SPIE 4821, 1-10 (2002).
[CrossRef]

J. Czarske and O. Dölle, “Investigations on the frequency measuring error of laser Doppler velocimeters using the quadrature demodulation techniques,” Proc. SPIE 3749, 599-600(1999).
[CrossRef]

B. Redman, W. Ruff, and K. Aliberti, “Direct-detection laser vibrometry with an amplitude modulated ladar,” Proc. SPIE 5412, 218-228 (2004).
[CrossRef]

W. Buell, N. Marechal, J. Buck, R. Dickinson, D. Kozlowski, T. Wright, and S. Beck, “Demonstrations of synthetic aperture imaging ladar,” Proc. SPIE 5791, 152-166 (2005).
[CrossRef]

Other (13)

J. Ricklin, B. Schumm, and P. Tomlinson, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conference, Snowmass, CO (9-13 July 2007).

J. Johnson, “Analysis of image forming systems,” in Image Intensifier Symposium, AD 220160 (Warfare Electrical Engineering Department, United States Army Research and Development Laboratories, 1958), pp. 244-273.

N. Levanon and E. Mozeson, Radar Signals (Wiley InterScience, 2004).
[CrossRef]

M. Dierking, F. Heitkamp, and L. Barnes, “High temporal resolution laser radar tomography for long range target identification,” presented at the IEEE/OSA Signal Synthesis and Reconstruction Conference (Kona, Hawaii, June 1998).

M. Richards, Fundamentals for Radar Signal Processing (McGraw-Hill, 2005).

M. Skolnik, Radar Handbook (McGraw-Hill, 1990).

M. Skolnik, Introduction to Radar Systems (McGraw-Hill, 2005).

L. Cohen, Time Frequency Analysis: Theory and Applications, Signal Processing Series (Prentice Hall, 1995).

M. Dierking and B. Duncan, “Genetically optimized periodic, pseudo-noise waveforms for multi-function coherent ladar,” presented at the 15th Coherent Laser Radar Conference, Toulouse, France, 22-26 June 2009.

J. Abshire, J. Rall, and S. Manizade, “Altimetry and lidar using AlGaAs lasers modulated with pseudo-random codes,” NASA Conference Publication 3158 Part 2, presented at the 16th International Laser Radar Conference (ILRC), Cambridge, MA, 20-24 July 1992.

M. Dobbs, W. Krabill, M. Cisewski, F. Harrison, C. Shum, D. McGregor, M. Neal, and S. Stokes, “A multi-functional fiber laser lidar for Earth science and exploration,” presented at the 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), Baltimore, MD, 2-4 June 2009.

W. Stone, M. Juberts, N. Dagalakis, J. Stone and J. Gorman, “Performance analysis of next-generation LADAR for manufacturing, construction, and mobility,” NISTIR 7117, National Institute of Standards and Technology, Gaithersburg, MD, May 2004.

Polytec, “Single point vibrometers,” http://www.polytec.com/usa/158_421.asp.

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Figures (15)

Fig. 1
Fig. 1

Three point targets encompassed within a diffraction limited beam footprint. Each target has an associated velocity and/or vibrational component. Individual returns and composite return are shown on the left.

Fig. 2
Fig. 2

Multifunction images for the three point target shown in Fig. 1 demonstrating: (a) range imaging, (b) range-Doppler imaging, (c) synthetic aperture imaging, and (d) micro-Doppler, or vibration, imaging.

Fig. 3
Fig. 3

Stripmap of SAL imaging geometry.

Fig. 4
Fig. 4

Summary of phase histories for range-Doppler (dashed-dottted curve), synthetic aperture (dashed curve), and micro- Doppler (solid curve) ladar imaging modes.

Fig. 5
Fig. 5

Effects of a residual sinusoidal Doppler phase modulation on PPN waveforms: (a) three identical noncontiguous transmit codes, (b) real component of the residual Doppler phase modulation for an assumed LOS velocity of 3.75 m / s , (c) transmitted waveforms after Doppler modulation, and (d) ideal matched filter output.

Fig. 6
Fig. 6

Peak matched filter output for a binary phase coded waveform is shown as a function of code length for LOS velocities of 1 m / s (dotted curve), 0.1 m / s (dashed curve), and 0.0375 m / s (solid curve). Calculated locations of the matched filter output nulls for these velocities are indicated by the diamond, triangle, and square markers, respectively.

Fig. 7
Fig. 7

Ideal IPR comparison for an LFM waveform (dotted curve), a PPN waveform (solid curve), and a 1 ns FWHM Gaussian pulse (dashed curve). Mainlobe resolution is nearly identical in each case, but significant differences are apparent in the sidelobes.

Fig. 8
Fig. 8

(a) Real component of the range-compressed macro- Doppler phase history for three point targets illuminated with a T c = 1 ns PPN waveform. (b) Fully compressed range-Doppler image correctly identifies the range and velocity of each target.

Fig. 9
Fig. 9

(a) Synthetic aperture phase history for the PPN waveform demonstrating the range-dependent phase migration for each of three point targets. (b) DBS fully compressed image demonstrating effective range cross-range localization of each of three targets.

Fig. 10
Fig. 10

Cross-range IPRs for the LFM (solid curve), PPN (dashed curve), and 1 ns FWHM Gaussian pulse waveforms (dotted curve). Cross-range phase modulation is the same for each waveform, producing nearly identical results in cross-range compression.

Fig. 11
Fig. 11

(a) Real component of the range-compressed micro- Doppler phase history for three point targets illuminated by a T c = 1 ns PPN waveform. (b) Micro-Doppler image of the point-target array.

Fig. 12
Fig. 12

Experimental PPN ladar setup. An AWG and phase modulator are used to generate PPN optical codes for transmission.

Fig. 13
Fig. 13

(a) Segment of the ideal PPN waveform (solid curve) has been temporally registered with the corresponding chips of the detected signal (dashed curve) returning from the vibrating target. (b) Ideal (solid curve) and measured (dashed curve) IPRs are in excellent agreement with one another.

Fig. 14
Fig. 14

Experimental micro-Doppler phase history for (a)  2 kHz vibrating target and (b) the corresponding normalized amplitude spectrum demonstrate excellent phase recovery when using PPN waveforms and matched filter processing.

Fig. 15
Fig. 15

Ideal (solid curve), range-compressed IR for a 1000 chip PPN waveform with T c = 2 ns . The dashed and dotted curves represent, respectively, the coherent addition of 10 and 100 sequentially measured, range-compressed IPRs. In (a) the experimental CNR is 0 dB , while in (b) the CNR is approximately 20 dB .

Equations (26)

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θ D = λ D ,
w ˜ ( t ) = w 0 ( t ) exp ( j θ ( t ) ) exp ( j 2 π f c t ) ,
s n ( t ) = A ( x n , y n , z n ) σ n w ( t γ n ) exp ( j 2 π f D n × ( t γ n ) ) = A ( x n , y n , z n ) σ n w ( t γ n ) exp ( j ϕ D n ( t γ n ) ) ,
v n - LOS ( t ) = v n o μ n sin ( ω n t ) ,
f D n = 2 v n - LOS ( t ) λ = 2 v n o λ 2 μ n sin ( ω n t ) λ .
ϕ D n ( t ) = 2 2 π λ v n o μ n sin ( ω n t ) d t = 2 2 π λ v n o t + 2 2 π λ μ n ω n cos ( ω n t ) ,
s ( t ) = n { A ( x n , y n , z n ) σ n w ( t 2 R n c ) × exp [ j 2 π 2 v n o λ ( t 2 R n c ) ] exp [ j 2 π λ 2 μ n ω n cos ( ω n ( t 2 R n c ) ) ] } .
q ( t ) = s ( τ ) w * ( τ t ) d τ = 1 { n S n ( ω ) · W * ( ω ) } ,
q R ( t ) = 1 { n A ( x n , y n , z n ) σ n exp ( j ω 2 R n c ) | W ( ω ) | 2 } .
q R ( t ) = n A ( x n , y n , z n ) σ n w ( t 2 R n c ) .
Δ R = c 2 B ,
q RD ( t ) = n A ( x n , y n , z n ) σ n exp ( j 2 π 2 v n o λ ( τ 2 R n c ) ) w ( τ 2 R n c ) w * ( τ t ) d τ ,
q RD ( t ) n A ( x n , y n , z n ) σ n × exp ( j 2 π 2 v n o λ T w t + T w 2 2 R n c T w ) T w T w w ( τ 2 R n c ) w * ( τ t ) d τ ,
q RD ( t ) n A ( x n , y n , z n ) σ n exp ( j 2 π 2 v n o λ T w t + T w 2 2 R n c T w ) w ( t 2 R n c ) ,
Δ v = λ 2 T W λ B 2 ,
R n ( t , u ) [ y n + u 2 2 y n u x n y n + x n 2 2 y n ] ,
s ( t , u ) = n A ( x n , y n , z n ) σ n w ( t 2 R n ( t , u ) c ) ,
q SA ( t , u ) = 1 { n A ( x n , y n , z n ) σ n exp ( j ω 2 R n ( t , u ) c ) | W ( ω ) | 2 } ,
q SA ( t , u ) = n A ( x n , y n , z n ) σ n w ( t 2 R n ( t , u ) c ) .
q μ D ( t ) = n A ( x n , y n , z n ) σ n × exp [ j 2 π λ 2 μ n ω n cos ( ω n ( τ 2 R n c ) ) ] w ( τ 2 R n c ) w * ( τ t ) d τ .
q μ D ( t ) n A ( x n , y n , z n ) σ n × exp [ j 2 π λ 2 μ n ω n cos ( ω n T w t + T w 2 2 R n c T w ) ] T w T w w ( τ 2 R n c ) w * ( τ t ) d τ .
q μ D ( t ) n A ( x n , y n , z n ) σ n exp [ j 2 π λ 2 μ n ω n cos ( ω n T w t + T w 2 2 R n c T w ) ] W ( t 2 R n c ) ,
w ( t ) = 1 N c T c m = 1 N c exp ( j π ϕ m ) rect ( t ( m 1 ) T c T c ) ,
N c T c = m 1 f d = m λ 2 v o ,
w LFM ( t ) = 1 T w rect ( 1 T w ) exp ( ± j π B T w t 2 ) ,
w G ( t ) = exp ( ln ( 2 ) t 2 B 2 ) ,

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