Abstract

Considering the influence of acceleration and the Gaussian envelope for a laser Doppler velocimeter (LDV), parameter estimation of a Doppler signal with a Gaussian envelope was investigated based on introducing acceleration. According to the theory of mathematics statistics, the Cramer–Rao lower bounds (CRLBs) of Doppler circular frequency and its first order rate were analyzed, formulas of CRLBs were given, and the power spectrum estimation with adjustment was discussed. The results of theory and the simulation show that the CRLBs are related to the data length, the signal-to-noise ratio (SNR), and the width of the Gaussian envelope, and they can be decreased by increasing the data length or improving the SNR; the larger the acceleration is and the narrower the Gaussian envelope is, the larger the CRLBs of Doppler circular frequency and its first order rate are; the gap between the variances of the measuring results and the CRLBs narrows when the SNR of the signal is improved, and is almost eliminated when the SNR is higher than 6dB. It is concluded that the model presented is much more suitable for a LDV than that acquired by Rife and Boorstyn [IEEE Trans. Inform. Theory 20, 591 (1974)].

© 2011 Optical Society of America

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References

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  1. N. Sun, “Autonomous navigation based on absolute velocity measuring instrument,” Aerospace Control 24, 27–30 (2006) (in Chinese).
  2. J. Zhou and X. Long, “Research on laser Doppler velocimeter for vehicle self-contained inertial navigation system,” Opt. Laser Technol. 42, 477–483 (2010).
    [CrossRef]
  3. D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,” IEEE Trans. Inform. Theory 20, 591–598 (1974).
    [CrossRef]
  4. G. Qi, “Cramer Rao bounds of real sinusoid parameter estimation from discrete-time observations,” J. Data Acquis. Process. 18, 151–155 (2003) (in Chinese).
    [CrossRef]
  5. R. Bamler, “Doppler frequency estimation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sens. 29, 385–390 (1991).
    [CrossRef]
  6. J. Wang, C. Zhang, Z. Ma, P. Ou, and X. Zhang, “Cramer-Rao lower bounds of parameter estimation from laser Doppler velocimetry,” Chin. J. Lasers 35, 1419–1422 (2008) (in Chinese).
    [CrossRef]
  7. W. Shu, “The Cramer-Rao bound for frequency estimation from LDA measurements,” Acta Metrologica Sinica 24, 36–39 (2003) (in Chinese).
  8. Y. Sun, “Design and adjustment of laser Doppler velocimeter,” in The Technology and Application of Laser Doppler Measurement (Shanghai Science and Technology Literature, 1995), pp. 132–155 (in Chinese).
  9. J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
    [CrossRef]
  10. X. Shen, Laser Doppler Velocimetry and Its Application (Tsinghua University Press, 2004) (in Chinese).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. D. Schwingshackl, T. Mayerdorfer, and D. Sträunigg, “Universal tone detection based on the Goertzel algorithm,” in Proceedings of the 2006 49th Midwest Symposium on Circuits and Systems, Vol. 1 (IEEE, 2006), 410–413.
    [CrossRef]
  15. H. Hu, “Research on signal processing of high accuracy in laser doppler velocimeter,” Thesis (University of Electronic Science and Technology of China, 2006), 23–25, 30–40 (in Chinese).
  16. K. Ding, M. Xie, B. Zhang, L. Zhao, and X. Zhang, “Principle and method of multiple modulation zoom spectrum analysis based on multiple analytical band pass filter,” J. Vibr. Eng. 14, 29–35 (2001) (in Chinese).
    [CrossRef]
  17. K. Ding, S. Zhong, and X. Zhu, “Phase difference correcting method for calibration of discrete spectrum,” J. Vibr. Shock 20, 52–55 (2001) (in Chinese).
    [CrossRef]
  18. K. Ding and L. Jiang, “Energy centrobaric correction method for discrete spectrum,” J. Vibr. Eng. 14, 354–358 (2001) (in Chinese).
    [CrossRef]
  19. X. Zhu and K. Ding, “The synthetical comparison of correcting methods on discrete spectrum,” Signal Process. 17, 91–97 (2001) (in Chinese).
    [CrossRef]

2010

J. Zhou and X. Long, “Research on laser Doppler velocimeter for vehicle self-contained inertial navigation system,” Opt. Laser Technol. 42, 477–483 (2010).
[CrossRef]

J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
[CrossRef]

2008

J. Wang, C. Zhang, Z. Ma, P. Ou, and X. Zhang, “Cramer-Rao lower bounds of parameter estimation from laser Doppler velocimetry,” Chin. J. Lasers 35, 1419–1422 (2008) (in Chinese).
[CrossRef]

2006

D. Schwingshackl, T. Mayerdorfer, and D. Sträunigg, “Universal tone detection based on the Goertzel algorithm,” in Proceedings of the 2006 49th Midwest Symposium on Circuits and Systems, Vol. 1 (IEEE, 2006), 410–413.
[CrossRef]

H. Hu, “Research on signal processing of high accuracy in laser doppler velocimeter,” Thesis (University of Electronic Science and Technology of China, 2006), 23–25, 30–40 (in Chinese).

N. Sun, “Autonomous navigation based on absolute velocity measuring instrument,” Aerospace Control 24, 27–30 (2006) (in Chinese).

2004

X. Shen, Laser Doppler Velocimetry and Its Application (Tsinghua University Press, 2004) (in Chinese).

2003

W. Shu, “The Cramer-Rao bound for frequency estimation from LDA measurements,” Acta Metrologica Sinica 24, 36–39 (2003) (in Chinese).

G. Qi, “Cramer Rao bounds of real sinusoid parameter estimation from discrete-time observations,” J. Data Acquis. Process. 18, 151–155 (2003) (in Chinese).
[CrossRef]

2001

K. Ding, M. Xie, B. Zhang, L. Zhao, and X. Zhang, “Principle and method of multiple modulation zoom spectrum analysis based on multiple analytical band pass filter,” J. Vibr. Eng. 14, 29–35 (2001) (in Chinese).
[CrossRef]

K. Ding, S. Zhong, and X. Zhu, “Phase difference correcting method for calibration of discrete spectrum,” J. Vibr. Shock 20, 52–55 (2001) (in Chinese).
[CrossRef]

K. Ding and L. Jiang, “Energy centrobaric correction method for discrete spectrum,” J. Vibr. Eng. 14, 354–358 (2001) (in Chinese).
[CrossRef]

X. Zhu and K. Ding, “The synthetical comparison of correcting methods on discrete spectrum,” Signal Process. 17, 91–97 (2001) (in Chinese).
[CrossRef]

1998

X. Shen, S. Zhang, P. Dong, and J. Yu, “A digital correlation signal processor used in LDV system,” J. Exper. Mech. 13, 294–301 (1998) (in Chinese).
[CrossRef]

1995

Y. Sun, “Design and adjustment of laser Doppler velocimeter,” in The Technology and Application of Laser Doppler Measurement (Shanghai Science and Technology Literature, 1995), pp. 132–155 (in Chinese).

1994

1991

R. Bamler, “Doppler frequency estimation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sens. 29, 385–390 (1991).
[CrossRef]

1984

1974

D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,” IEEE Trans. Inform. Theory 20, 591–598 (1974).
[CrossRef]

Bamler, R.

R. Bamler, “Doppler frequency estimation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sens. 29, 385–390 (1991).
[CrossRef]

Boorstyn, R. R.

D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,” IEEE Trans. Inform. Theory 20, 591–598 (1974).
[CrossRef]

Demarest, F. C.

Ding, K.

K. Ding, M. Xie, B. Zhang, L. Zhao, and X. Zhang, “Principle and method of multiple modulation zoom spectrum analysis based on multiple analytical band pass filter,” J. Vibr. Eng. 14, 29–35 (2001) (in Chinese).
[CrossRef]

K. Ding, S. Zhong, and X. Zhu, “Phase difference correcting method for calibration of discrete spectrum,” J. Vibr. Shock 20, 52–55 (2001) (in Chinese).
[CrossRef]

K. Ding and L. Jiang, “Energy centrobaric correction method for discrete spectrum,” J. Vibr. Eng. 14, 354–358 (2001) (in Chinese).
[CrossRef]

X. Zhu and K. Ding, “The synthetical comparison of correcting methods on discrete spectrum,” Signal Process. 17, 91–97 (2001) (in Chinese).
[CrossRef]

Dong, P.

X. Shen, S. Zhang, P. Dong, and J. Yu, “A digital correlation signal processor used in LDV system,” J. Exper. Mech. 13, 294–301 (1998) (in Chinese).
[CrossRef]

Feng, Q.

J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
[CrossRef]

Hu, H.

H. Hu, “Research on signal processing of high accuracy in laser doppler velocimeter,” Thesis (University of Electronic Science and Technology of China, 2006), 23–25, 30–40 (in Chinese).

Jiang, L.

K. Ding and L. Jiang, “Energy centrobaric correction method for discrete spectrum,” J. Vibr. Eng. 14, 354–358 (2001) (in Chinese).
[CrossRef]

Long, X.

J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
[CrossRef]

J. Zhou and X. Long, “Research on laser Doppler velocimeter for vehicle self-contained inertial navigation system,” Opt. Laser Technol. 42, 477–483 (2010).
[CrossRef]

Ma, S.

J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
[CrossRef]

Ma, Z.

J. Wang, C. Zhang, Z. Ma, P. Ou, and X. Zhang, “Cramer-Rao lower bounds of parameter estimation from laser Doppler velocimetry,” Chin. J. Lasers 35, 1419–1422 (2008) (in Chinese).
[CrossRef]

Mayerdorfer, T.

D. Schwingshackl, T. Mayerdorfer, and D. Sträunigg, “Universal tone detection based on the Goertzel algorithm,” in Proceedings of the 2006 49th Midwest Symposium on Circuits and Systems, Vol. 1 (IEEE, 2006), 410–413.
[CrossRef]

Otsuka, K.

Ou, P.

J. Wang, C. Zhang, Z. Ma, P. Ou, and X. Zhang, “Cramer-Rao lower bounds of parameter estimation from laser Doppler velocimetry,” Chin. J. Lasers 35, 1419–1422 (2008) (in Chinese).
[CrossRef]

Qi, G.

G. Qi, “Cramer Rao bounds of real sinusoid parameter estimation from discrete-time observations,” J. Data Acquis. Process. 18, 151–155 (2003) (in Chinese).
[CrossRef]

Rife, D. C.

D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,” IEEE Trans. Inform. Theory 20, 591–598 (1974).
[CrossRef]

Schwingshackl, D.

D. Schwingshackl, T. Mayerdorfer, and D. Sträunigg, “Universal tone detection based on the Goertzel algorithm,” in Proceedings of the 2006 49th Midwest Symposium on Circuits and Systems, Vol. 1 (IEEE, 2006), 410–413.
[CrossRef]

Shen, X.

X. Shen, Laser Doppler Velocimetry and Its Application (Tsinghua University Press, 2004) (in Chinese).

X. Shen, S. Zhang, P. Dong, and J. Yu, “A digital correlation signal processor used in LDV system,” J. Exper. Mech. 13, 294–301 (1998) (in Chinese).
[CrossRef]

Shu, W.

W. Shu, “The Cramer-Rao bound for frequency estimation from LDA measurements,” Acta Metrologica Sinica 24, 36–39 (2003) (in Chinese).

Sommargren, G. E.

Song, R.

J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
[CrossRef]

Sträunigg, D.

D. Schwingshackl, T. Mayerdorfer, and D. Sträunigg, “Universal tone detection based on the Goertzel algorithm,” in Proceedings of the 2006 49th Midwest Symposium on Circuits and Systems, Vol. 1 (IEEE, 2006), 410–413.
[CrossRef]

Sun, N.

N. Sun, “Autonomous navigation based on absolute velocity measuring instrument,” Aerospace Control 24, 27–30 (2006) (in Chinese).

Sun, Y.

Y. Sun, “Design and adjustment of laser Doppler velocimeter,” in The Technology and Application of Laser Doppler Measurement (Shanghai Science and Technology Literature, 1995), pp. 132–155 (in Chinese).

Truax, B. E.

Wang, J.

J. Wang, C. Zhang, Z. Ma, P. Ou, and X. Zhang, “Cramer-Rao lower bounds of parameter estimation from laser Doppler velocimetry,” Chin. J. Lasers 35, 1419–1422 (2008) (in Chinese).
[CrossRef]

Wei, G.

J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
[CrossRef]

Xie, M.

K. Ding, M. Xie, B. Zhang, L. Zhao, and X. Zhang, “Principle and method of multiple modulation zoom spectrum analysis based on multiple analytical band pass filter,” J. Vibr. Eng. 14, 29–35 (2001) (in Chinese).
[CrossRef]

Yu, J.

X. Shen, S. Zhang, P. Dong, and J. Yu, “A digital correlation signal processor used in LDV system,” J. Exper. Mech. 13, 294–301 (1998) (in Chinese).
[CrossRef]

Zhang, B.

K. Ding, M. Xie, B. Zhang, L. Zhao, and X. Zhang, “Principle and method of multiple modulation zoom spectrum analysis based on multiple analytical band pass filter,” J. Vibr. Eng. 14, 29–35 (2001) (in Chinese).
[CrossRef]

Zhang, C.

J. Wang, C. Zhang, Z. Ma, P. Ou, and X. Zhang, “Cramer-Rao lower bounds of parameter estimation from laser Doppler velocimetry,” Chin. J. Lasers 35, 1419–1422 (2008) (in Chinese).
[CrossRef]

Zhang, S.

X. Shen, S. Zhang, P. Dong, and J. Yu, “A digital correlation signal processor used in LDV system,” J. Exper. Mech. 13, 294–301 (1998) (in Chinese).
[CrossRef]

Zhang, X.

J. Wang, C. Zhang, Z. Ma, P. Ou, and X. Zhang, “Cramer-Rao lower bounds of parameter estimation from laser Doppler velocimetry,” Chin. J. Lasers 35, 1419–1422 (2008) (in Chinese).
[CrossRef]

K. Ding, M. Xie, B. Zhang, L. Zhao, and X. Zhang, “Principle and method of multiple modulation zoom spectrum analysis based on multiple analytical band pass filter,” J. Vibr. Eng. 14, 29–35 (2001) (in Chinese).
[CrossRef]

Zhao, L.

K. Ding, M. Xie, B. Zhang, L. Zhao, and X. Zhang, “Principle and method of multiple modulation zoom spectrum analysis based on multiple analytical band pass filter,” J. Vibr. Eng. 14, 29–35 (2001) (in Chinese).
[CrossRef]

Zhong, S.

K. Ding, S. Zhong, and X. Zhu, “Phase difference correcting method for calibration of discrete spectrum,” J. Vibr. Shock 20, 52–55 (2001) (in Chinese).
[CrossRef]

Zhou, J.

J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
[CrossRef]

J. Zhou and X. Long, “Research on laser Doppler velocimeter for vehicle self-contained inertial navigation system,” Opt. Laser Technol. 42, 477–483 (2010).
[CrossRef]

Zhu, X.

K. Ding, S. Zhong, and X. Zhu, “Phase difference correcting method for calibration of discrete spectrum,” J. Vibr. Shock 20, 52–55 (2001) (in Chinese).
[CrossRef]

X. Zhu and K. Ding, “The synthetical comparison of correcting methods on discrete spectrum,” Signal Process. 17, 91–97 (2001) (in Chinese).
[CrossRef]

Acta Metrologica Sinica

W. Shu, “The Cramer-Rao bound for frequency estimation from LDA measurements,” Acta Metrologica Sinica 24, 36–39 (2003) (in Chinese).

Aerospace Control

N. Sun, “Autonomous navigation based on absolute velocity measuring instrument,” Aerospace Control 24, 27–30 (2006) (in Chinese).

Appl. Opt.

Chin. J. Lasers

J. Wang, C. Zhang, Z. Ma, P. Ou, and X. Zhang, “Cramer-Rao lower bounds of parameter estimation from laser Doppler velocimetry,” Chin. J. Lasers 35, 1419–1422 (2008) (in Chinese).
[CrossRef]

High Power Laser Part. Beams

J. Zhou, Q. Feng, S. Ma, R. Song, G. Wei, and X. Long, “Error analysis of reference-beam laser Doppler velocimeter,” High Power Laser Part. Beams 22, 2581–2587 (2010) (in Chinese).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

R. Bamler, “Doppler frequency estimation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sens. 29, 385–390 (1991).
[CrossRef]

IEEE Trans. Inform. Theory

D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,” IEEE Trans. Inform. Theory 20, 591–598 (1974).
[CrossRef]

J. Data Acquis. Process.

G. Qi, “Cramer Rao bounds of real sinusoid parameter estimation from discrete-time observations,” J. Data Acquis. Process. 18, 151–155 (2003) (in Chinese).
[CrossRef]

J. Exper. Mech.

X. Shen, S. Zhang, P. Dong, and J. Yu, “A digital correlation signal processor used in LDV system,” J. Exper. Mech. 13, 294–301 (1998) (in Chinese).
[CrossRef]

J. Vibr. Eng.

K. Ding, M. Xie, B. Zhang, L. Zhao, and X. Zhang, “Principle and method of multiple modulation zoom spectrum analysis based on multiple analytical band pass filter,” J. Vibr. Eng. 14, 29–35 (2001) (in Chinese).
[CrossRef]

K. Ding and L. Jiang, “Energy centrobaric correction method for discrete spectrum,” J. Vibr. Eng. 14, 354–358 (2001) (in Chinese).
[CrossRef]

J. Vibr. Shock

K. Ding, S. Zhong, and X. Zhu, “Phase difference correcting method for calibration of discrete spectrum,” J. Vibr. Shock 20, 52–55 (2001) (in Chinese).
[CrossRef]

Opt. Laser Technol.

J. Zhou and X. Long, “Research on laser Doppler velocimeter for vehicle self-contained inertial navigation system,” Opt. Laser Technol. 42, 477–483 (2010).
[CrossRef]

Signal Process.

X. Zhu and K. Ding, “The synthetical comparison of correcting methods on discrete spectrum,” Signal Process. 17, 91–97 (2001) (in Chinese).
[CrossRef]

Other

Y. Sun, “Design and adjustment of laser Doppler velocimeter,” in The Technology and Application of Laser Doppler Measurement (Shanghai Science and Technology Literature, 1995), pp. 132–155 (in Chinese).

X. Shen, Laser Doppler Velocimetry and Its Application (Tsinghua University Press, 2004) (in Chinese).

D. Schwingshackl, T. Mayerdorfer, and D. Sträunigg, “Universal tone detection based on the Goertzel algorithm,” in Proceedings of the 2006 49th Midwest Symposium on Circuits and Systems, Vol. 1 (IEEE, 2006), 410–413.
[CrossRef]

H. Hu, “Research on signal processing of high accuracy in laser doppler velocimeter,” Thesis (University of Electronic Science and Technology of China, 2006), 23–25, 30–40 (in Chinese).

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Figures (10)

Fig. 1
Fig. 1

Relationship between the coefficient ε and the factor κ.

Fig. 2
Fig. 2

Signal processing block diagram.

Fig. 3
Fig. 3

Signal processing block diagram of the tracking filter.

Fig. 4
Fig. 4

Result comparison between the spectrum after Goertzel refinement and after direct FFT: (a) spectrum after Goertzel refinement when the frequency is 100 kHz , (b) spectrum after Goertzel refinement when the frequency is 200 kHz , (c) spectrum after Goertzel refinement when the frequency is 400 kHz , (d) result of FFT when the frequency is 100 kHz , (e) result of FFT when the frequency is 200 kHz , (f) result of FFT when the frequency is 400 kHz .

Fig. 5
Fig. 5

Relationship between the SNR and CRLBs of (a) the Doppler circular frequency and (b) its first order rate.

Fig. 6
Fig. 6

Comparison of four different results of CRLBs of Doppler circular frequency.

Fig. 7
Fig. 7

Comparison of two different results of CRLBs of the Doppler circular frequency’s first order rate.

Fig. 8
Fig. 8

CRLBs of (a) the Doppler circular frequency and (b) its first order rate at a different proportional factor.

Fig. 9
Fig. 9

Relationship between the variance of the Doppler frequency, as well as its first order rate, and the CRLBs and SNR when N = 128 and f = 200 kHz : (a) the Doppler circular frequency, (b) the first order rate of the Doppler circular frequency.

Fig. 10
Fig. 10

Result of each main block of the signal processing method: (a) the initial Doppler signal, (b) the spectrum of the initial Doppler signal, (c) the output of the tracking filter, (d) the spectrum of the output of the tracking filter, (e) the spectrum after Goertzel refinement, (f) the locally enlarged image of the spectrum after ratio correction.

Tables (2)

Tables Icon

Table 1 Configuration of Parameters in the Simulation

Tables Icon

Table 2 Comparison of Results between before Correction and after Correction

Equations (33)

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

Z = X + j Y .
X = [ X N , X N + 1 , X N ] T ,
Y = [ Y N , Y N + 1 , Y N ] T ,
X n = μ ( n ) + w ( n ) = A cos ( φ 0 + ω D t n + r D t n 2 ) exp ( t n 2 / T w 2 ) + w ( n ) ,
Y n = ν ( n ) + ( n ) = A sin ( φ 0 + ω D t n + r D t n 2 ) exp ( t n 2 / T w 2 ) + ( n ) .
SNR = A 2 / ( 2 σ 2 ) .
η = [ ω D r D A φ 0 T w ] T .
( η i ) CRLB ( J 1 ) i i ,
( ω D ) CRLB 2 σ 2 A 2 T 2 [ 2 π 4 K w 3 erf ( 2 N K w ) N K w 2 exp ( 2 N 2 K w 2 ) ] ,
( r D ) CRLB [ ( A 2 N 3 K w 2 T 4 2 σ 2 A 2 K w 4 T 4 N 8 σ 2 ) exp ( 2 N 2 K w 2 ) + 2 π A 2 K w 5 T 4 16 σ 2 erf ( 2 N K w ) A 2 T 4 N 2 K w 3 2 2 π σ 2 exp ( 4 N 2 K w 2 ) erf 1 ( 2 N K w ) ] 1 .
( ω ) CRLB 12 σ 2 A 2 T 2 K ( K 2 1 ) ,
( ω D ) CRLB γ σ 2 A 2 T 2 ( K 1 ) 3 ,
γ = 16 κ 3 ( 2 π / 4 ) erf ( 2 κ ) κ exp ( 2 κ 2 ) .
( r D ) CRLB { N 5 T 4 SNR [ ( 1 κ 2 1 4 κ 4 ) exp ( 2 κ 2 ) + 2 π 8 κ 2 erf ( 2 κ ) exp ( κ 2 ) 2 π κ 3 erf ( 2 κ ) ] } 1 .
J i j = 1 σ 2 n = N N ( μ n η i μ n η j + ν n η i ν n η j ) .
J 11 = A 2 σ 2 n = N N t n 2 exp ( 2 t n 2 / T w 2 ) , J 21 = J 12 = A 2 σ 2 n = N N t n 3 exp ( 2 t n 2 / T w 2 ) , J 31 = J 13 = 0 , J 41 = J 14 = A 2 σ 2 n = N N t n exp ( 2 t n 2 / T w 2 ) , J 51 = J 15 = 0 J 22 = A 2 σ 2 n = N N t n 4 exp ( 2 t n 2 / T w 2 ) , J 32 = J 23 = 0 , J 42 = J 24 = J 11 , J 52 = J 25 = 0 , J 33 = 1 σ 2 n = N N exp ( 2 t n 2 / T w 2 ) , J 43 = J 34 = 0 , J 53 = J 35 = 2 A σ 2 T w 3 n = N N t n 2 exp ( 2 t n 2 / T w 2 ) , J 44 = A 2 σ 2 n = N N exp ( 2 t n 2 / T w 2 ) , J 54 = J 45 = 0 , J 55 = 4 A 2 σ 2 T w 6 n = N N t n 4 exp ( 2 t n 2 / T w 2 ) .
J = ( J 11 0 0 0 0 0 J 22 0 J 11 0 0 0 J 33 0 J 35 0 J 11 0 J 44 0 0 0 J 35 0 J 55 ) ,
J 11 = A 2 T 2 2 σ 2 [ 2 π 4 K w 3 erf ( 2 N K w ) N K w 2 exp ( 2 N 2 K w 2 ) ] , J 22 = A 2 N 3 K w 2 T 4 2 σ 2 exp ( 2 N 2 K w 2 ) + 3 2 π A 2 K w 5 T 4 32 σ 2 erf ( 2 N K w ) 3 A 2 K w 4 T 4 N 8 σ 2 exp ( 2 N 2 K w 2 ) , J 33 = 2 π K w 2 σ 2 erf ( 2 N K w ) , J 44 = 2 π A 2 K w 2 σ 2 erf ( 2 N K w ) , J 55 = 3 2 π A 2 8 σ 2 K w T 2 erf ( 2 N K w ) 2 A 2 N 3 σ 2 K w 4 T 2 exp ( 2 N 2 K w 2 ) 3 A 2 N 2 σ 2 K w 2 T 2 exp ( 2 N 2 K w 2 ) , J 35 = A σ 2 K w 3 T [ 2 π 4 K w 3 erf ( 2 N K w ) N K w 2 exp ( 2 N 2 K w 2 ) ] .
( J 1 ) 11 = 1 J 11 = 2 σ 2 A 2 T 2 [ 2 π 4 K w 3 erf ( 2 N K w ) N K w 2 exp ( 2 N 2 K w 2 ) ] ,
( J 1 ) 22 = J 44 J 22 J 44 J 24 2 = [ ( A 2 N 3 K w 2 T 4 2 σ 2 A 2 K w 4 T 4 N 8 σ 2 ) exp ( 2 N 2 K w 2 ) + 2 π A 2 K w 5 T 4 16 σ 2 erf ( 2 N K w ) A 2 T 4 N 2 K w 3 2 2 π σ 2 exp ( 4 N 2 K w 2 ) erf 1 ( 2 N K w ) ] 1 .
( ω D ) CRLB 2 σ 2 A 2 T 2 [ 2 π 4 K w 3 erf ( 2 N K w ) N K w 2 exp ( 2 N 2 K w 2 ) ] ,
( r D ) CRLB [ ( A 2 N 3 K w 2 T 4 2 σ 2 A 2 K w 4 T 4 N 8 σ 2 ) exp ( 2 N 2 K w 2 ) + 2 π A 2 K w 5 T 4 16 σ 2 erf ( 2 N K w ) A 2 T 4 N 2 K w 3 2 2 π σ 2 exp ( 4 N 2 K w 2 ) erf 1 ( 2 N K w ) ] 1 .
X ( k ) = r = 0 N 1 x ( r ) e j 2 π N r k ,
X ( k ) = r = 0 N 1 x ( r ) W N r k .
W N k N = e ( j 2 π N ) ( k N ) = 1 ,
X ( k ) = r = 0 N 1 x ( r ) W N k r W N k N = r = 0 N 1 x ( r ) W N k ( N r ) .
y k ( n ) = m = x ( n ) W N k ( n m ) u ( n m ) ,
u ( n ) = { 1 ( n 0 ) 0 ( n < 0 ) .
X ( k ) = y k ( n ) | n = N .
H k ( z ) = 1 1 W N k z 1 = 1 W N k z 1 ( 1 W N k z 1 ) ( 1 W N k z 1 ) = 1 W N k z 1 1 2 cos ( 2 π k / N ) z 1 + z 2 .
g ( f ) = sin ( π f ) π f 1 2 ( 1 f 2 ) ,
g ( f ) / g ( f + 1 ) = y k / y k + 1 ,
f = ( y k 2 y k + 1 ) / ( y k + 1 y k ) .

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