Abstract

A high-resolution random-modulation continuous wave lidar for surface detection using a semiconductor laser diode is presented. The laser diode is intensity modulated with the pseudorandom binary sequence. Its enhanced resolution is achieved via interpolation and a novel front-end analog technique, lowering the requirement of the analog-to-digital converter sampling rate and the associated circuitry. Its mathematical model is presented, including the derivation of the signal-to-noise ratio and the distance standard deviation. Analytical and experimental results demonstrate its capability to achieve distance accuracy of less than 2cm within 2.6ms acquisition time, over distances ranging from 1 to 12m. The laser diode emits 1.4mW of optical power at a wavelength of 635nm.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. N. Takeuchi, H. Baba, K. Sakurai, and T. Ueno, “Diode-laser random-modulation cw lidar,” Appl. Opt. 25, 63–67 (1986).
    [CrossRef] [PubMed]
  4. Y. Emery and C. Flesia “Use of the A1-and the A2-sequences to modulate continuous-wave pseudorandom noise lidar,” Appl. Opt. 37, 2238–2241 (1998).
    [CrossRef]
  5. R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. 43, 557–571 (2005).
    [CrossRef]
  6. V. Mitev, R. Matthey, J. Pereira do Carmo, and G. Ulbrich, “Signal-to-noise ratio of pseudo-random noise continuous wave backscatter lidar with analog detection,” Proc. SPIE 5984, 598404 (2005).
    [CrossRef]
  7. H. S. Lee and R. Ramaswami, “Study of pseudo-noise cw diode laser for ranging applications,” Proc. SPIE 1829, 36–45 (1992).
    [CrossRef]
  8. J. J. Esteban, I. Bykov, A. Marin, G. Heinzel, and K. Danzmann, “Optical ranging and data transfer development for LISA,” in Journal of Physics: Conference Series (Institute of Physics, 2009), Vol.  154, pp. 012025.
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. B. Buttgen, M’H. A. El Mechat, F. Lustenberger, and P. Seitz, “Pseudonoise optical modulation for real-time 3-D imaging with minimum interference,” IEEE Trans. Circuits Syst. 54, 2109–2119 (2007).
    [CrossRef]
  12. R. Rasshofer and K. Gresser, “Automotive radar and lidar systems for next generation driver assistance functions,” Advances in Radio Science 3, 205–209 (2005).
    [CrossRef]
  13. Y. Wang, Y. Wang, and H. Le, “Multi-spectral mid-infrared laser stand-off imaging,” Appl. Opt. 41, 1063–1070 (2002).
    [CrossRef]
  14. J. Kalisz, “Review of methods for time interval measurements with picosecond resolution,” Metrologia 41, 17–32(2004).
    [CrossRef]
  15. D. Luenberger and Y. Ye, “Chapter 8 basic descent methods,” in Linear and Nonlinear Programming (Springer, 2008), pp. 215–262.
    [CrossRef]
  16. R. Davenport, “FFT processing of direct sequence spreading codes using modern DSP microprocessors,” in Proceedings of the IEEE 1991 National Aerospace and Electronics Conference (NAECON) 1991 (IEEE, 2002), pp. 98–105.
  17. B. Bundschuh, D. Schneider, and M. Grindel, “Feasibility study of a compact low cost correlation LIDAR using a pseudo noise modulated diode laser and an APD in the current mode,” in International Geoscience and Remote Sensing Symposium, 1996. IGARSS ’96.’Remote Sensing for a Sustainable Future (IEEE, 2002), Vol.  2, pp. 999–1001.
    [CrossRef]

2010

2009

J. J. Esteban, I. Bykov, A. Marin, G. Heinzel, and K. Danzmann, “Optical ranging and data transfer development for LISA,” in Journal of Physics: Conference Series (Institute of Physics, 2009), Vol.  154, pp. 012025.
[CrossRef]

2008

D. Luenberger and Y. Ye, “Chapter 8 basic descent methods,” in Linear and Nonlinear Programming (Springer, 2008), pp. 215–262.
[CrossRef]

2007

B. Buttgen, M’H. A. El Mechat, F. Lustenberger, and P. Seitz, “Pseudonoise optical modulation for real-time 3-D imaging with minimum interference,” IEEE Trans. Circuits Syst. 54, 2109–2119 (2007).
[CrossRef]

2005

R. Rasshofer and K. Gresser, “Automotive radar and lidar systems for next generation driver assistance functions,” Advances in Radio Science 3, 205–209 (2005).
[CrossRef]

R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. 43, 557–571 (2005).
[CrossRef]

V. Mitev, R. Matthey, J. Pereira do Carmo, and G. Ulbrich, “Signal-to-noise ratio of pseudo-random noise continuous wave backscatter lidar with analog detection,” Proc. SPIE 5984, 598404 (2005).
[CrossRef]

2004

F. Lin and M. L. Jia, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
[CrossRef]

J. Kalisz, “Review of methods for time interval measurements with picosecond resolution,” Metrologia 41, 17–32(2004).
[CrossRef]

2002

Y. Wang, Y. Wang, and H. Le, “Multi-spectral mid-infrared laser stand-off imaging,” Appl. Opt. 41, 1063–1070 (2002).
[CrossRef]

R. Davenport, “FFT processing of direct sequence spreading codes using modern DSP microprocessors,” in Proceedings of the IEEE 1991 National Aerospace and Electronics Conference (NAECON) 1991 (IEEE, 2002), pp. 98–105.

B. Bundschuh, D. Schneider, and M. Grindel, “Feasibility study of a compact low cost correlation LIDAR using a pseudo noise modulated diode laser and an APD in the current mode,” in International Geoscience and Remote Sensing Symposium, 1996. IGARSS ’96.’Remote Sensing for a Sustainable Future (IEEE, 2002), Vol.  2, pp. 999–1001.
[CrossRef]

2001

M. Amann, T. Bosch, M. Lescure, R. Myllyl, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

1998

1992

H. S. Lee and R. Ramaswami, “Study of pseudo-noise cw diode laser for ranging applications,” Proc. SPIE 1829, 36–45 (1992).
[CrossRef]

1986

1983

Amann, M.

M. Amann, T. Bosch, M. Lescure, R. Myllyl, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Baba, H.

Bosch, T.

M. Amann, T. Bosch, M. Lescure, R. Myllyl, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Buller, G.

Bundschuh, B.

B. Bundschuh, D. Schneider, and M. Grindel, “Feasibility study of a compact low cost correlation LIDAR using a pseudo noise modulated diode laser and an APD in the current mode,” in International Geoscience and Remote Sensing Symposium, 1996. IGARSS ’96.’Remote Sensing for a Sustainable Future (IEEE, 2002), Vol.  2, pp. 999–1001.
[CrossRef]

Buttgen, B.

B. Buttgen, M’H. A. El Mechat, F. Lustenberger, and P. Seitz, “Pseudonoise optical modulation for real-time 3-D imaging with minimum interference,” IEEE Trans. Circuits Syst. 54, 2109–2119 (2007).
[CrossRef]

Bykov, I.

J. J. Esteban, I. Bykov, A. Marin, G. Heinzel, and K. Danzmann, “Optical ranging and data transfer development for LISA,” in Journal of Physics: Conference Series (Institute of Physics, 2009), Vol.  154, pp. 012025.
[CrossRef]

Danzmann, K.

J. J. Esteban, I. Bykov, A. Marin, G. Heinzel, and K. Danzmann, “Optical ranging and data transfer development for LISA,” in Journal of Physics: Conference Series (Institute of Physics, 2009), Vol.  154, pp. 012025.
[CrossRef]

Davenport, R.

R. Davenport, “FFT processing of direct sequence spreading codes using modern DSP microprocessors,” in Proceedings of the IEEE 1991 National Aerospace and Electronics Conference (NAECON) 1991 (IEEE, 2002), pp. 98–105.

El Mechat, M’H. A.

B. Buttgen, M’H. A. El Mechat, F. Lustenberger, and P. Seitz, “Pseudonoise optical modulation for real-time 3-D imaging with minimum interference,” IEEE Trans. Circuits Syst. 54, 2109–2119 (2007).
[CrossRef]

Emery, Y.

Esteban, J. J.

J. J. Esteban, I. Bykov, A. Marin, G. Heinzel, and K. Danzmann, “Optical ranging and data transfer development for LISA,” in Journal of Physics: Conference Series (Institute of Physics, 2009), Vol.  154, pp. 012025.
[CrossRef]

Flesia, C.

Gresser, K.

R. Rasshofer and K. Gresser, “Automotive radar and lidar systems for next generation driver assistance functions,” Advances in Radio Science 3, 205–209 (2005).
[CrossRef]

Grindel, M.

B. Bundschuh, D. Schneider, and M. Grindel, “Feasibility study of a compact low cost correlation LIDAR using a pseudo noise modulated diode laser and an APD in the current mode,” in International Geoscience and Remote Sensing Symposium, 1996. IGARSS ’96.’Remote Sensing for a Sustainable Future (IEEE, 2002), Vol.  2, pp. 999–1001.
[CrossRef]

Heinzel, G.

J. J. Esteban, I. Bykov, A. Marin, G. Heinzel, and K. Danzmann, “Optical ranging and data transfer development for LISA,” in Journal of Physics: Conference Series (Institute of Physics, 2009), Vol.  154, pp. 012025.
[CrossRef]

Jia, M. L.

F. Lin and M. L. Jia, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
[CrossRef]

Kalisz, J.

J. Kalisz, “Review of methods for time interval measurements with picosecond resolution,” Metrologia 41, 17–32(2004).
[CrossRef]

Krichel, N.

Le, H.

Lee, H. S.

H. S. Lee and R. Ramaswami, “Study of pseudo-noise cw diode laser for ranging applications,” Proc. SPIE 1829, 36–45 (1992).
[CrossRef]

Lescure, M.

M. Amann, T. Bosch, M. Lescure, R. Myllyl, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Lin, F.

F. Lin and M. L. Jia, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
[CrossRef]

Luenberger, D.

D. Luenberger and Y. Ye, “Chapter 8 basic descent methods,” in Linear and Nonlinear Programming (Springer, 2008), pp. 215–262.
[CrossRef]

Lustenberger, F.

B. Buttgen, M’H. A. El Mechat, F. Lustenberger, and P. Seitz, “Pseudonoise optical modulation for real-time 3-D imaging with minimum interference,” IEEE Trans. Circuits Syst. 54, 2109–2119 (2007).
[CrossRef]

Marin, A.

J. J. Esteban, I. Bykov, A. Marin, G. Heinzel, and K. Danzmann, “Optical ranging and data transfer development for LISA,” in Journal of Physics: Conference Series (Institute of Physics, 2009), Vol.  154, pp. 012025.
[CrossRef]

Matthey, R.

V. Mitev, R. Matthey, J. Pereira do Carmo, and G. Ulbrich, “Signal-to-noise ratio of pseudo-random noise continuous wave backscatter lidar with analog detection,” Proc. SPIE 5984, 598404 (2005).
[CrossRef]

R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. 43, 557–571 (2005).
[CrossRef]

McCarthy, A.

Mitev, V.

R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. 43, 557–571 (2005).
[CrossRef]

V. Mitev, R. Matthey, J. Pereira do Carmo, and G. Ulbrich, “Signal-to-noise ratio of pseudo-random noise continuous wave backscatter lidar with analog detection,” Proc. SPIE 5984, 598404 (2005).
[CrossRef]

Myllyl, R.

M. Amann, T. Bosch, M. Lescure, R. Myllyl, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Pereira do Carmo, J.

V. Mitev, R. Matthey, J. Pereira do Carmo, and G. Ulbrich, “Signal-to-noise ratio of pseudo-random noise continuous wave backscatter lidar with analog detection,” Proc. SPIE 5984, 598404 (2005).
[CrossRef]

Ramaswami, R.

H. S. Lee and R. Ramaswami, “Study of pseudo-noise cw diode laser for ranging applications,” Proc. SPIE 1829, 36–45 (1992).
[CrossRef]

Rasshofer, R.

R. Rasshofer and K. Gresser, “Automotive radar and lidar systems for next generation driver assistance functions,” Advances in Radio Science 3, 205–209 (2005).
[CrossRef]

Rioux, M.

M. Amann, T. Bosch, M. Lescure, R. Myllyl, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Sakurai, K.

Schneider, D.

B. Bundschuh, D. Schneider, and M. Grindel, “Feasibility study of a compact low cost correlation LIDAR using a pseudo noise modulated diode laser and an APD in the current mode,” in International Geoscience and Remote Sensing Symposium, 1996. IGARSS ’96.’Remote Sensing for a Sustainable Future (IEEE, 2002), Vol.  2, pp. 999–1001.
[CrossRef]

Seitz, P.

B. Buttgen, M’H. A. El Mechat, F. Lustenberger, and P. Seitz, “Pseudonoise optical modulation for real-time 3-D imaging with minimum interference,” IEEE Trans. Circuits Syst. 54, 2109–2119 (2007).
[CrossRef]

Sugimoto, N.

Takeuchi, N.

Ueno, T.

Ulbrich, G.

V. Mitev, R. Matthey, J. Pereira do Carmo, and G. Ulbrich, “Signal-to-noise ratio of pseudo-random noise continuous wave backscatter lidar with analog detection,” Proc. SPIE 5984, 598404 (2005).
[CrossRef]

Wang, Y.

Ye, Y.

D. Luenberger and Y. Ye, “Chapter 8 basic descent methods,” in Linear and Nonlinear Programming (Springer, 2008), pp. 215–262.
[CrossRef]

Advances in Radio Science

R. Rasshofer and K. Gresser, “Automotive radar and lidar systems for next generation driver assistance functions,” Advances in Radio Science 3, 205–209 (2005).
[CrossRef]

Appl. Opt.

IEEE J. Sel. Top. Quantum Electron.

F. Lin and M. L. Jia, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
[CrossRef]

IEEE Trans. Circuits Syst.

B. Buttgen, M’H. A. El Mechat, F. Lustenberger, and P. Seitz, “Pseudonoise optical modulation for real-time 3-D imaging with minimum interference,” IEEE Trans. Circuits Syst. 54, 2109–2119 (2007).
[CrossRef]

Metrologia

J. Kalisz, “Review of methods for time interval measurements with picosecond resolution,” Metrologia 41, 17–32(2004).
[CrossRef]

Opt. Eng.

M. Amann, T. Bosch, M. Lescure, R. Myllyl, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. 43, 557–571 (2005).
[CrossRef]

Proc. SPIE

V. Mitev, R. Matthey, J. Pereira do Carmo, and G. Ulbrich, “Signal-to-noise ratio of pseudo-random noise continuous wave backscatter lidar with analog detection,” Proc. SPIE 5984, 598404 (2005).
[CrossRef]

H. S. Lee and R. Ramaswami, “Study of pseudo-noise cw diode laser for ranging applications,” Proc. SPIE 1829, 36–45 (1992).
[CrossRef]

Other

J. J. Esteban, I. Bykov, A. Marin, G. Heinzel, and K. Danzmann, “Optical ranging and data transfer development for LISA,” in Journal of Physics: Conference Series (Institute of Physics, 2009), Vol.  154, pp. 012025.
[CrossRef]

D. Luenberger and Y. Ye, “Chapter 8 basic descent methods,” in Linear and Nonlinear Programming (Springer, 2008), pp. 215–262.
[CrossRef]

R. Davenport, “FFT processing of direct sequence spreading codes using modern DSP microprocessors,” in Proceedings of the IEEE 1991 National Aerospace and Electronics Conference (NAECON) 1991 (IEEE, 2002), pp. 98–105.

B. Bundschuh, D. Schneider, and M. Grindel, “Feasibility study of a compact low cost correlation LIDAR using a pseudo noise modulated diode laser and an APD in the current mode,” in International Geoscience and Remote Sensing Symposium, 1996. IGARSS ’96.’Remote Sensing for a Sustainable Future (IEEE, 2002), Vol.  2, pp. 999–1001.
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Principle of traditional RM-CW systems and the proposed MRMCW technique. Accuracy of the RM-CW system is given by the period of the system clock, and MRMCW exhibits subclock period accuracy via a process of interpolation on the digital CCF.

Fig. 2
Fig. 2

Schematic of the MRMCW lidar principle. See text in first paragraph of Section 2 for component denomination.

Fig. 3
Fig. 3

15   bit M-sequence. T c denotes the chip time.

Fig. 4
Fig. 4

15   bit M-sequence ACF. N T c represents the total integration time; the offset is T c , negligible only for long PRBS.

Fig. 5
Fig. 5

CCF. m represents the sample where the digital CCF is maximal, τ denotes the round-trip time delay, C m 1 and C m + are samples around the maximum, C max = C ( t 0 = τ ) will be referenced in Appendix B.

Fig. 6
Fig. 6

Digital CCF of two spaced targets. The dark dashed trace represents the first surface samples; the light dashed trace represents the second surface samples.

Fig. 7
Fig. 7

Digital CCF of two nearby targets. The dark dashed trace represents the first surface samples; the light dashed trace represents the second surface samples.

Fig. 8
Fig. 8

Gradient descent technique. The minimum of the cost function Ω is approached iteratively, in this example, initially x 0 = [ 2 , 0 ] T , at the 100th iteration x 100 = [ 1 , 4.5 ] T , results in a minimized cost function with an error of 2 × 10 6 .

Fig. 9
Fig. 9

Integrator output voltage range with a mixer applied prior to the ADC, shown as the solid trace which exhibits a small voltage range. Without the mixer is shown as the dashed trace which exhibits large voltage range. The mixer stops potential integrator saturation and reduces the dynamic range requirement of the ADC.

Fig. 10
Fig. 10

Block diagram of the XCORR and the M-sequence generator (LFSR). Seven taps with the digital CCF are illustrated.

Fig. 11
Fig. 11

Assembly of the MRMCW demonstrator, showing the (1) APD module, (2) LNA, (3) mixer, (4) integrator, (5) FPGA board with an (6) ADC board underneath the FPGA, (7) laser drivers, (8, 9) laser modules, and (10) lens tube RX.

Fig. 12
Fig. 12

Block diagram of the demonstrator system. See text in second and third paragraphs of Section 5 for component denomination.

Fig. 13
Fig. 13

MRMCW distance measurements are compared with an industrial laser range finder (BOSCH DLE 70 with 1 mm distance accuracy). The white diffusive screen is placed in various locations, distances are measured with a 0 ° incidence angle, laboratory fluorescent lighting conditions, and an acquisition time of 2.6 ms .

Fig. 14
Fig. 14

Distance standard deviation with mutual interference is shown as the light dashed trace. Without mutual interference is shown as the dark dashed trace. Theoretical shot noise limited standard deviation evaluated by Eq. (18) with and without the consideration of the overlap function (obtained experimentally) are shown as the solid trace and dashed trace.

Fig. 15
Fig. 15

CCF with respect to the ADC trigger delay. The measured result is shown as the solid trace which is obtained by delaying the ADC sampling trigger by 0 130 ns in 1 ns steps. The theoretical triangle is shown as the dashed trace obtained by Eq. (11).

Fig. 16
Fig. 16

Distance nonlinearity due to distortion in the CCF.

Fig. 17
Fig. 17

Preliminary experimental result for the gradient descent technique. The measured digital CCF shown as the solid trace are estimated as circles, which are a combination of the dark dashed trace representing the first surface samples and the light dashed trace representing the second surface samples. The targets are placed at 369 cm and 1292 cm away from the lidar, triangles are separated with calculated results of 385 cm and 1280 cm , respectively.

Tables (1)

Tables Icon

Table 1 Specification of the Demonstrator

Equations (37)

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a ( t ) = i = 1 N a i ψ ( t i T c ) ,
ψ ( t ) = { 1 0 t < T c 0 otherwise ,
R ( t 0 ) = 0 N T c a ( t ) a ( t + t 0 ) d t ,
{ N T c | 1 t 0 T c | L < t 0 < L + 0 otherwise ,
P s ( t ) = G P t ( t τ ) .
G = η t η r α cos θ Y r A r π D 2 ,
P b = η r α E λ ω tan 2 β A r ,
I i = i T s ( i + 1 ) T s P r ( t ) d t .
C n = i = 1 r c s N I i A i n E s R ^ ( n T s τ ) ,
E s = G P 0 N T c ,
C ( t 0 ) = E s R ^ ( t 0 τ ) ,
D = 1 2 c τ .
SNR max = ξ E s [ μ e ξ ( E s + E b ) ] 1 / 2 = ( ξ μ e ) 1 2 E s ( E s + E b ) 1 2 ,
ξ = ( h f ) 1 ,
E b = P b N T c ,
τ = ( m 1 ) T s + C m + 1 C m 1 + C m + 1 T c ,
σ D 2 4 c T c SNR max .
σ D [ π h c 2 D 2 8 α f μ e T c cos θ η r η t A r N P 0 ( 1 + π E λ ω tan 2 β D 2 cos θ η t P 0 ) ] 1 / 2 .
Ω = n = 1 N [ C n ( E s , τ ) S n ] 2 .
x k + 1 = x k Γ Ω ( x k ) .
C n = E s 0 R ^ ( n T s + τ 0 ) + E s 1 R ^ ( n T s + τ 1 ) .
I i = i T s ( i + 1 ) T s P r ( t ) d t ,
I i = 0 r s c N T s P r ( t ) ψ ( t i T s ) d t .
C n = i = 1 r s c N I i A i n .
C n = i = 1 r c s N ( 0 r c s N T s P r ( t ) ψ ( t i T s ) d t · A i n ) ,
C n = 0 r c s N T s P r ( t ) i = 1 r c s N ψ ( t i T s ) A i n d t ,
C n = 0 N T c P r ( t ) a ( t n T s ) d t = 0 N T c ( P s ( t ) + P b ) a ( t n T s ) d t = 0 N T c P s ( t ) a ( t n T s ) d t + 0 N T c P b a ( t n T s ) d t .
C n G P 0 0 N T c a ( t τ ) a ( t n T s ) d t ,
C n E s R ^ ( τ n T s ) ,
C n E s R ^ ( n T s τ ) .
τ = ( m 1 ) T s + C m + 1 C m 1 + C m + 1 T c ,
σ τ = [ ( δ C m 1 τ ) 2 σ C m 1 2 + ( δ C m + 1 τ ) 2 σ C m + 1 2 ] 1 / 2 ,
δ C m 1 τ = C m + 1 T c ( C m 1 + C m + 1 ) 2 ,
δ C m + 1 τ = T c C m 1 + C m + 1 C m + 1 T c ( C m 1 + C m + 1 ) 2 .
σ τ 2 2 T c σ C max C max 2 2 T c SNR max ,
σ D 2 4 c T c SNR max .
σ D c T c 2 SNR max .

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