Abstract

We investigate the ability of detection techniques based on the likelihood ratio to discriminate between heterodyne lidar Doppler estimates at low signal levels using examples generated by simulation. The distinction between estimates that are regarded as acceptable and as spurious is based on the Cramer–Rao lower bound. The conditional false alarm probability, which ordinarily describes recording detection of a signal when none is present, is then found to be an approximate upper bound on the probability of selection of a spurious estimate. The method is superior theoretically to similar techniques based on detection functions other than the likelihood ratio. The likelihood ratio also provides a basis for reprocessing rejected data in the light of contextual information provided by those estimates that are accepted.

© 1997 Optical Society of America

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References

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  1. B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. I. Incoherent spectral accumulation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
    [CrossRef]
  2. B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. II. Incoherent correlogram accumulation,” IEEE Trans. Geosci. Remote Sensing 31, 28–35 (1993).
    [CrossRef]
  3. P. T. May, R. G. Strauch, “An examination of some algorithms for spectral moment estimation,” J. Atmos. Ocean. Technol. 6, 731–735 (1989).
    [CrossRef]
  4. D. A. Merritt, “A statistical averaging method for wind profiler Doppler spectra,” J. Atmos. Ocean. Technol. 12, 985–995 (1995).
    [CrossRef]
  5. B. J. Rye, R. M. Hardesty, “Cramer-Rao lower bound-limited Doppler estimation using discrimination,” in Proceedings of the Seventh Conference on Coherent Laser Radar Applications and Technology, (Commission National for Education and Science, Paris, 1993), pp. 217–220.
  6. B. J. Rye, R. M. Hardesty, “Limits on Doppler lidar detectabilityand precision,” presented at the European Space Agency Doppler Wind Lidar Workshop, Noordwijk, Netherlands, September, 1995.
  7. E. S. Chornoboy, “Optimal mean velocity estimation for Doppler weather radars,” IEEE Trans. Geosci. Remote Sensing 31, 575–586 (1993).
    [CrossRef]
  8. D. S. Zrnic, “Estimation of spectral moments for weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
    [CrossRef]
  9. R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. 11, 1217–1230 (1994).
    [CrossRef]
  10. B. J. Rye, R. M. Hardesty, “Spectral matched filters in coherent laser radar,” J. Mod. Opt. 41, 2131–2144 (1994).
    [CrossRef]
  11. R. G. Frehlich, “Effects of wind turbulence on coherent Doppler lidar performance,” J. Atmos. Ocean. Technol. 14, 54–75 (1997).
    [CrossRef]
  12. H. L. van Trees, Detection, Estimation, and Modulation Theory. Part I: Detection, Estimation, and Linear Modulation Theory (Wiley, New York, 1968).
  13. M. J. Levin, “Power spectrum parameter estimation,” IEEE Trans. Inf. Theory IT-11, 100–107 (1965).
    [CrossRef]
  14. B. J. Rye, “Return power estimation for targets spread in range,” in Coherent Laser Radar: Technology and Applications, Vol. 19 of 1995 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1995), pp. 202–205.
  15. J. R. Anderson, “High performance velocity estimators for coherent laser radars,” in Coherent Laser Radar: Applications and Technology, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1991).
  16. B. J. Rye, R. M. Hardesty, “Time series identification and Kalman filtering techniques for Doppler lidar velocity estimation,” Appl. Opt. 28, 879–891 (1989).
    [CrossRef] [PubMed]
  17. F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, “Infrared 10.6-micron heterodyne detection with gigahertz IF capability,” IEEE J. Quantum Electron. QE-3, 484–492 (1967).
    [CrossRef]
  18. A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
    [CrossRef]
  19. B. J. Rye, “Comparative precision of distributed-backscatter Doppler lidars,” Appl. Opt. 34, 8341–8344 (1995).
    [CrossRef] [PubMed]
  20. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  21. R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar: corrigenda,” J. Atmos. Ocean. Technol. 12, 445–446 (1995).
    [CrossRef]

1997 (1)

R. G. Frehlich, “Effects of wind turbulence on coherent Doppler lidar performance,” J. Atmos. Ocean. Technol. 14, 54–75 (1997).
[CrossRef]

1995 (3)

D. A. Merritt, “A statistical averaging method for wind profiler Doppler spectra,” J. Atmos. Ocean. Technol. 12, 985–995 (1995).
[CrossRef]

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar: corrigenda,” J. Atmos. Ocean. Technol. 12, 445–446 (1995).
[CrossRef]

B. J. Rye, “Comparative precision of distributed-backscatter Doppler lidars,” Appl. Opt. 34, 8341–8344 (1995).
[CrossRef] [PubMed]

1994 (2)

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. 11, 1217–1230 (1994).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Spectral matched filters in coherent laser radar,” J. Mod. Opt. 41, 2131–2144 (1994).
[CrossRef]

1993 (3)

E. S. Chornoboy, “Optimal mean velocity estimation for Doppler weather radars,” IEEE Trans. Geosci. Remote Sensing 31, 575–586 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. I. Incoherent spectral accumulation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. II. Incoherent correlogram accumulation,” IEEE Trans. Geosci. Remote Sensing 31, 28–35 (1993).
[CrossRef]

1989 (2)

P. T. May, R. G. Strauch, “An examination of some algorithms for spectral moment estimation,” J. Atmos. Ocean. Technol. 6, 731–735 (1989).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Time series identification and Kalman filtering techniques for Doppler lidar velocity estimation,” Appl. Opt. 28, 879–891 (1989).
[CrossRef] [PubMed]

1979 (1)

D. S. Zrnic, “Estimation of spectral moments for weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
[CrossRef]

1971 (1)

A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
[CrossRef]

1967 (1)

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, “Infrared 10.6-micron heterodyne detection with gigahertz IF capability,” IEEE J. Quantum Electron. QE-3, 484–492 (1967).
[CrossRef]

1965 (1)

M. J. Levin, “Power spectrum parameter estimation,” IEEE Trans. Inf. Theory IT-11, 100–107 (1965).
[CrossRef]

Anderson, J. R.

J. R. Anderson, “High performance velocity estimators for coherent laser radars,” in Coherent Laser Radar: Applications and Technology, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1991).

Arams, F. R.

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, “Infrared 10.6-micron heterodyne detection with gigahertz IF capability,” IEEE J. Quantum Electron. QE-3, 484–492 (1967).
[CrossRef]

Arcese, A.

A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
[CrossRef]

Chornoboy, E. S.

E. S. Chornoboy, “Optimal mean velocity estimation for Doppler weather radars,” IEEE Trans. Geosci. Remote Sensing 31, 575–586 (1993).
[CrossRef]

Frehlich, R. G.

R. G. Frehlich, “Effects of wind turbulence on coherent Doppler lidar performance,” J. Atmos. Ocean. Technol. 14, 54–75 (1997).
[CrossRef]

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar: corrigenda,” J. Atmos. Ocean. Technol. 12, 445–446 (1995).
[CrossRef]

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. 11, 1217–1230 (1994).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Hardesty, R. M.

B. J. Rye, R. M. Hardesty, “Spectral matched filters in coherent laser radar,” J. Mod. Opt. 41, 2131–2144 (1994).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. I. Incoherent spectral accumulation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. II. Incoherent correlogram accumulation,” IEEE Trans. Geosci. Remote Sensing 31, 28–35 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Time series identification and Kalman filtering techniques for Doppler lidar velocity estimation,” Appl. Opt. 28, 879–891 (1989).
[CrossRef] [PubMed]

B. J. Rye, R. M. Hardesty, “Cramer-Rao lower bound-limited Doppler estimation using discrimination,” in Proceedings of the Seventh Conference on Coherent Laser Radar Applications and Technology, (Commission National for Education and Science, Paris, 1993), pp. 217–220.

B. J. Rye, R. M. Hardesty, “Limits on Doppler lidar detectabilityand precision,” presented at the European Space Agency Doppler Wind Lidar Workshop, Noordwijk, Netherlands, September, 1995.

Levin, M. J.

M. J. Levin, “Power spectrum parameter estimation,” IEEE Trans. Inf. Theory IT-11, 100–107 (1965).
[CrossRef]

May, P. T.

P. T. May, R. G. Strauch, “An examination of some algorithms for spectral moment estimation,” J. Atmos. Ocean. Technol. 6, 731–735 (1989).
[CrossRef]

Merritt, D. A.

D. A. Merritt, “A statistical averaging method for wind profiler Doppler spectra,” J. Atmos. Ocean. Technol. 12, 985–995 (1995).
[CrossRef]

Pace, F. P.

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, “Infrared 10.6-micron heterodyne detection with gigahertz IF capability,” IEEE J. Quantum Electron. QE-3, 484–492 (1967).
[CrossRef]

Peyton, B. J.

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, “Infrared 10.6-micron heterodyne detection with gigahertz IF capability,” IEEE J. Quantum Electron. QE-3, 484–492 (1967).
[CrossRef]

Rye, B. J.

B. J. Rye, “Comparative precision of distributed-backscatter Doppler lidars,” Appl. Opt. 34, 8341–8344 (1995).
[CrossRef] [PubMed]

B. J. Rye, R. M. Hardesty, “Spectral matched filters in coherent laser radar,” J. Mod. Opt. 41, 2131–2144 (1994).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. I. Incoherent spectral accumulation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. II. Incoherent correlogram accumulation,” IEEE Trans. Geosci. Remote Sensing 31, 28–35 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Time series identification and Kalman filtering techniques for Doppler lidar velocity estimation,” Appl. Opt. 28, 879–891 (1989).
[CrossRef] [PubMed]

B. J. Rye, “Return power estimation for targets spread in range,” in Coherent Laser Radar: Technology and Applications, Vol. 19 of 1995 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1995), pp. 202–205.

B. J. Rye, R. M. Hardesty, “Cramer-Rao lower bound-limited Doppler estimation using discrimination,” in Proceedings of the Seventh Conference on Coherent Laser Radar Applications and Technology, (Commission National for Education and Science, Paris, 1993), pp. 217–220.

B. J. Rye, R. M. Hardesty, “Limits on Doppler lidar detectabilityand precision,” presented at the European Space Agency Doppler Wind Lidar Workshop, Noordwijk, Netherlands, September, 1995.

Sard, E. W.

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, “Infrared 10.6-micron heterodyne detection with gigahertz IF capability,” IEEE J. Quantum Electron. QE-3, 484–492 (1967).
[CrossRef]

Strauch, R. G.

P. T. May, R. G. Strauch, “An examination of some algorithms for spectral moment estimation,” J. Atmos. Ocean. Technol. 6, 731–735 (1989).
[CrossRef]

Trombini, E. W.

A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
[CrossRef]

van Trees, H. L.

H. L. van Trees, Detection, Estimation, and Modulation Theory. Part I: Detection, Estimation, and Linear Modulation Theory (Wiley, New York, 1968).

Yadlowsky, M. J.

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar: corrigenda,” J. Atmos. Ocean. Technol. 12, 445–446 (1995).
[CrossRef]

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. 11, 1217–1230 (1994).
[CrossRef]

Zrnic, D. S.

D. S. Zrnic, “Estimation of spectral moments for weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, “Infrared 10.6-micron heterodyne detection with gigahertz IF capability,” IEEE J. Quantum Electron. QE-3, 484–492 (1967).
[CrossRef]

IEEE Trans. Geosci. Electron. (1)

D. S. Zrnic, “Estimation of spectral moments for weather echoes,” IEEE Trans. Geosci. Electron. GE-17, 113–128 (1979).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (3)

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. I. Incoherent spectral accumulation and the Cramer-Rao bound,” IEEE Trans. Geosci. Remote Sensing 31, 16–27 (1993).
[CrossRef]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in Doppler lidar. II. Incoherent correlogram accumulation,” IEEE Trans. Geosci. Remote Sensing 31, 28–35 (1993).
[CrossRef]

E. S. Chornoboy, “Optimal mean velocity estimation for Doppler weather radars,” IEEE Trans. Geosci. Remote Sensing 31, 575–586 (1993).
[CrossRef]

IEEE Trans. Inf. Theory (2)

A. Arcese, E. W. Trombini, “Variances of spectral parameters with a Gaussian shape,” IEEE Trans. Inf. Theory IT-17, 200–201 (1971).
[CrossRef]

M. J. Levin, “Power spectrum parameter estimation,” IEEE Trans. Inf. Theory IT-11, 100–107 (1965).
[CrossRef]

J. Atmos. Ocean. Technol. (5)

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar: corrigenda,” J. Atmos. Ocean. Technol. 12, 445–446 (1995).
[CrossRef]

P. T. May, R. G. Strauch, “An examination of some algorithms for spectral moment estimation,” J. Atmos. Ocean. Technol. 6, 731–735 (1989).
[CrossRef]

D. A. Merritt, “A statistical averaging method for wind profiler Doppler spectra,” J. Atmos. Ocean. Technol. 12, 985–995 (1995).
[CrossRef]

R. G. Frehlich, M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar,” J. Atmos. Ocean. Technol. 11, 1217–1230 (1994).
[CrossRef]

R. G. Frehlich, “Effects of wind turbulence on coherent Doppler lidar performance,” J. Atmos. Ocean. Technol. 14, 54–75 (1997).
[CrossRef]

J. Mod. Opt. (1)

B. J. Rye, R. M. Hardesty, “Spectral matched filters in coherent laser radar,” J. Mod. Opt. 41, 2131–2144 (1994).
[CrossRef]

Other (6)

H. L. van Trees, Detection, Estimation, and Modulation Theory. Part I: Detection, Estimation, and Linear Modulation Theory (Wiley, New York, 1968).

B. J. Rye, R. M. Hardesty, “Cramer-Rao lower bound-limited Doppler estimation using discrimination,” in Proceedings of the Seventh Conference on Coherent Laser Radar Applications and Technology, (Commission National for Education and Science, Paris, 1993), pp. 217–220.

B. J. Rye, R. M. Hardesty, “Limits on Doppler lidar detectabilityand precision,” presented at the European Space Agency Doppler Wind Lidar Workshop, Noordwijk, Netherlands, September, 1995.

B. J. Rye, “Return power estimation for targets spread in range,” in Coherent Laser Radar: Technology and Applications, Vol. 19 of 1995 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1995), pp. 202–205.

J. R. Anderson, “High performance velocity estimators for coherent laser radars,” in Coherent Laser Radar: Applications and Technology, Vol. 12 of 1991 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1991).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (8)

Fig. 1
Fig. 1

Scatterplot of 180 pairs of values of the maximum log likelihood ln[p(x|F)]max and the frequency estimate F 1*, the latter plotted relative to the receiver bandwidth f1* = F 1*/F S (see Appendix A). The mean spectrum is described by Eqs. (2) and (16), and data samples are processed to obtain F 1* by correlation with the modified Levin filter function [Eq. (7) ]. The sample size is M = 64, the relative frequency of the signal shift and bandwidth are, respectively, f 1 = 0.2 and f 2 = F 2/F S = 0.02, and the wideband signal-to-noise ratio δ = -10 dB (α1 ≈ 1.2, N pc ≈ 10).

Fig. 2
Fig. 2

(a) Histogram (squares) of estimates obtained from simulations for parameters M = 64, f 1 = 0.0, f 2 = 0.02, δ = - 10 dB, and a fit (solid curve) to these values obtained from Eq. (1). The dashed curve is a Gaussian curve with a standard deviation equal to the Cramer–Rao lower bound σCR (Appendix A). (b) Parameters of curves fitted to histograms of the type exemplified in (a). The parameters are the same as for (a) except that the effect of increase in the sample size is illustrated by an accumulation of returns from n pulses at constant M. The squares give the ratio σ g CR and the curve is the fitted value of a [Eq. (1)].

Fig. 3
Fig. 3

Example of a probability density function PDF0 (dotted curve) and a complementary distribution function CDF0 (solid curve) of the log-likelihood difference ln(Λg′) for noise-only data with M = 64 obtained from simulations. The signal parameters that are needed in the processing algorithm [Eqs. (7) and (16)] are f 2 = 0.05 and α1 = 2.5 (δ ≈ -5 dB, N pc ≈ 20).

Fig. 4
Fig. 4

Log-likelihood difference (gLLD) values taken from CDF curves (as in Fig. 3) for noise-only data at different P FA [Eq. (9)]. The range gate and relative signal bandwidth are constant (M = 64, f 2 = 0.05), and the gLLD is shown as a function of degeneracy. The P FA are 30% for the lowest (heavy solid) curve and decrease for higher curves through 3%, 0.3%, 0.03%, and 0.003%.

Fig. 5
Fig. 5

Conditional detection probability P D (for with-signal data) at thresholds set to correspond to a false alarm probability (with noise-only data) of P FA = 1%. The parameters for the simulation were M = 64, f 2 = 0.05. The P D, shown as a function of signal strength (N pc), compare the use of the gLLD with the different detection functions defined in Eq. (11). These functions are the gLLD (solid curve, open circles), r0* (dashed curve, open squares), m1* (solid curve), r1* (dashed curve, solid squares), ρ* (dashed curve, open triangles), and ε* (dashed curve, crosses).

Fig. 6
Fig. 6

(a) The fraction of frequency shift estimates F1* falling within ±Δ F S of the true value F 1 as Δ is increased relative to a constant value of σCR. Only estimates with gLLD ln Λ g′ greater than a threshold value, set to make P FA = 1%, are considered. Wins (estimates within ±ΔF S of F 1) are shown as a fine solid curve and sells as a heavy curve. The standard deviation of the wins is shown as a dashed curve (right-hand ordinate). Note that this standard deviation is generally less than σCR, which is possible because it characterizes only selected estimates. The parameters used in the simulations are M = 64, f 1 = 0.0, f 2 = 0.02, δ = -10 dB (α1 ≈ 0.5, N pc ≈ 10). (b) Example of complementary distribution functions CDF0 (ln Λ g′) for noise-only data (heavy solid curve) and CDF1 (ln Λ g′) for with-signal data (fine solid curve) obtained from simulations with M = 64, f 2 = 0.05, and degeneracy α1 = 2.5 (δ ≈ -5 dB, N pc ≈ 20). The dotted curves are the total winner and bluff probabilities p W and p B obtained with signal by our multiplying CDF1(ln Λ g′) with the a priori ratio R 1 and (1 - R 1), respectively [Eqs. (13) and (15) ]. In general, at a given value of the log-likelihood difference, ln Λ g′, P FA [from CDF0(ln Λ g′), Eq. (9)] is greater than p B, in agreement with the inequality in Eq. (13), and P D is greater than p W . However, closer examination of the P FA and p B curves at low false alarm probability values as shown on the right-hand ordinate, indicates that this inequality no longer applies when P FA < 0.5%.

Fig. 7
Fig. 7

(a) Contour plot of the total win probability p W as a function of effective photocount N pc and photocount degeneracy α1 for a signal of bandwidth f 2 = 0.05 and P FA = 1%. The dashed curves correspond to constant values of the sample size M = N pc/[√(2π) f 2 α1] equal to 10 (upper left), 100, and 1000 (lower right). (b) Graph of total probability of a bluff p B as a function of N pc for various α1, corresponding to different sections through (a) (i.e., 10 < M < 1000). The line markers indicate α1 = 0.1 (solid squares), 0.3 (plus signs), 1.0 (open squares), 3.0 (crosses), and 10.0 (solid triangles). (c) Graph of the probability p S of unrecognized acceptable estimates as a function of N pc for the same values of α1 as in (b).

Fig. 8
Fig. 8

(a) Comparison of the probabilities of wins, bluffs, and sells at a false alarm probability P FA of 1% obtained with (curves) and without (points) prior knowledge of signal strength. The parameters are f 2 = 0.05 and M = 64. Wins are shown as solid squares and a heavy curve, bluffs as open squares and a thin curve, and sells as crosses and a dashed curve. The signal strength is parameterized by N pc and is the true value. In general, there is little difference between results obtained with the two methods, i.e., little is gained from prior knowledge. The main exception to this is at low signal levels where p B > P FA. (b) Standard deviation (referred to receiver bandwidth) of the winners for various false alarm probabilities for the same parameters as (a). The points refer to values obtained without prior knowledge of signal strength, open squares for P FA = 0.1%, solid squares for P FA = 1.0%, and crosses for P FA = 10%. Somewhat surprisingly, these results all have a standard deviation that approximates the value expected from the CRLB corresponding to the true signal parameters (heavy curve). This is not the case for the results obtained with prior knowledge (solid curve for P FA = 0.1% and dashed curve for P FA = 10%), where the standard deviation is much less than the CRLB for low signal levels (the technique selects only the peak of the scatter plot of Fig. 1) and greater than the CRLB at high signal levels, especially for the higher P FA (= 10%) when estimates within the wings on the estimate distribution are included among the wins.

Equations (25)

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

gF1*=1-a+a2πσg exp-F1*-F122σg2,
ϕi=ϕiS+ϕiN.
Λχ=p1x|H1maxp0x|H0max,
p0x|H0=i=0M-1nϕNnxin-1 exp-nxi/ϕNΓn,
lnp0x|H0=qx+rx; ϕN, qx=Mn lnn-lnΓn+n-1i=0M-1 lnxi, rx; ϕN=-n i=0M-1lnϕN+xi/ϕN.
lnp1x|H1=qx-ni=0M-1lnϕi+xi/ϕi,  =qx+rx; ϕN+lnΛx, ϕN, ϕS,
lnΛx, ϕN, ϕS=ni=0M-1xiϕNϕiSϕi-lnϕiϕN.
lnΛgx, ϕN, ϕS*=ni=0M-1xiϕNϕiS*ϕi*-lnϕi*ϕN.
PFAln Λg=CDF0ln Λg=1--lnΛgPDF0lnΛgxdlnΛgx.
PDln Λg=CDF1ln Λg=1--lnΛgPDF1lnΛgxdlnΛgx,
r0*=1/Mi=1Mr0i,  r1*=1/M-1i=1M-1Rer1i, ρ*=r1*/r0*,  m0*=1/M-1i=1M-1r02i  m1*=1/M-1i=1M-1r1i21/2, μ*=m1*/m0*.
PWln Λg=CDF11ln Λg,  PSln Λg=1-PWln Λg,  PBln Λg=CDF10ln Λg.
pBln Λg=1-R1PBln ΛgPFAln Λg
pWln Λg=R1PWln Λg  PDln Λg.
pSln Λg=R1PSln Λg=R1-pWln Λg.
ϕiSϕN=α1 exp-Fi-F122F22,
Npc=i=0M-1ϕiSQFhν=δM.
σCR=F2Npc1g1α1,  g1α1=α12π-x2 exp-x2dx1+α1 exp-x2/22.
m1=Mk=02M-1μkS=M2 Rek=0M-1μkS-1,
α1=δk=02M-1 μkS=δk=02M-1rkSr0S=2Npcϕ0Si=02M-1ϕiS=ϕ0SϕN.
lnpZ, R=-lnR-Z+R-1Z,
lnp1Z|H1=-M lnN-lnDGδ+IDcc-N-1Z+DGδ+IDcc-1Z.
Z+DGδ+IDcc-1Z=Z+DΓDccZ=2Rem=0M-1i=0M-m-1ziccγi,i+mzi+m×exp-jωm-i=0M-1zi2γi,i,
lnΛ=N-1Z+DΨDccZ-lnGδ+I=N-12Rem=0M-1i=0M-m-1ziccΨi,i+mzi+mexp-jωm-i=0M-1zi2Ψi,i-ln Gδ+I,
lnΛ=1/NZ+DGδΓDccZ-lnGδ+I.

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