Cramer-Rao bounds for estimating range, velocity, and direction with an active array

We derive Cramer-Rao bound (CRB) expressions for the range (time delay), velocity (Doppler shift), and direction of a point target using an active radar or sonar array. First, general CRB expressions are derived for a narrowband signal and array model and a space-time separable noise model that allows both spatial and temporal correlation. We discuss the relationship between the CRB and ambiguity function for this model. Then, we specialize our CRB results to the case of temporally white noise and the practically important signal shape of a linear frequency modulated (chirp) pulse sequence. We compute the CRB for a three-dimensional (3-D) array with isotropic sensors in spatially white noise and show that it is a function of the array geometry only through the "moments of inertia" of the array. The volume of the confidence region for the target's location is proposed as a measure of accuracy. For this measure, we show that the highest (and lowest) target location accuracy is achieved if the target lies along one of the principal axes of inertia of the array. Finally, we compare the location accuracies of several array geometries.

[1]  Arye Nehorai,et al.  Effects of sensor placement on acoustic vector-sensor array performance , 1999 .

[2]  J. Moura,et al.  Passive systems theory with narrow-band and linear constraints: Part II - Temporal diversity , 1979, IEEE Journal of Oceanic Engineering.

[3]  Helga Bunke,et al.  Identifiability and Estimability , 1974 .

[4]  J. Moura,et al.  Passive systems theory with narrow-band and linear constraints: Part I - Spatial diversity , 1978, IEEE Journal of Oceanic Engineering.

[5]  Charles E. Cook,et al.  Radar Signals: An Introduction to Theory and Application , 1967 .

[6]  J. Litva,et al.  Radar Array Processing , 1993 .

[7]  P. Stoica,et al.  Maximum likelihood methods in radar array signal processing , 1998, Proc. IEEE.

[8]  Aleksandar Dogandzic,et al.  Space-time fading channel estimation and symbol detection in unknown spatially correlated noise , 2002, IEEE Trans. Signal Process..

[9]  N. L. Johnson,et al.  Linear Statistical Inference and Its Applications , 1966 .

[10]  Pertti Vainikainen,et al.  Real-time 3-D spatial-temporal dual-polarized measurement of wideband radio channel at mobile station , 2000, IEEE Trans. Instrum. Meas..

[11]  Martin Haardt,et al.  Smart antennas for combined DOA and joint channel estimation in time-slotted CDMA mobile radio systems with joint detection , 2000, IEEE Trans. Veh. Technol..

[12]  A. W. Rihaczek.  Principles of high-resolution radar , 1969 .

[13]  Khaled H. Hamed,et al.  Time-frequency analysis , 2003 .

[14]  Arye Nehorai,et al.  Polarimetric modeling and parameter estimation with applications to remote sensing , 1995, IEEE Trans. Signal Process..

[15]  E. Vonesh,et al.  Linear and Nonlinear Models for the Analysis of Repeated Measurements , 1996 .

[16]  I. Ueda,et al.  Tactical cylindrical active phased array radar , 1996, Proceedings of International Symposium on Phased Array Systems and Technology.

[17]  E. J. Kelly,et al.  The Radar Measurement of Range, Velocity and Acceleration , 1961, IRE Transactions on Military Electronics.

[18]  James Ward,et al.  Space-time adaptive processing for airborne radar , 1994, 1995 International Conference on Acoustics, Speech, and Signal Processing.

[19]  José M. F. Moura,et al.  Ambiguity in radar and sonar , 1998, IEEE Trans. Signal Process..

[20]  J. Moura,et al.  Passive systems theory with narrow-band and linear constraints: Part III - Spatial/Temporal diversity , 1979, IEEE Journal of Oceanic Engineering.

[21]  S. Kay.  Fundamentals of statistical signal processing: estimation theory , 1993 .

[22]  S. Pasupathy,et al.  Optimum Active Array Processing Structure and Space-Time Factorability , 1974, IEEE Transactions on Aerospace and Electronic Systems.

[23]  Aleksandar Dogandzic,et al.  Estimating range, velocity, and direction with a radar array , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[24]  Don H. Johnson,et al.  Array Signal Processing: Concepts and Techniques , 1993 .

[25]  Aleksandar Dogandzic,et al.  Cramer-Rao bounds for estimating range, velocity, and direction with a sensor array , 2000, Proceedings of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop. SAM 2000 (Cat. No.00EX410).

[26]  S. P. Stapleton,et al.  A mobile base station phased array antenna , 1992, 1992 IEEE International Conference on Selected Topics in Wireless Communications.

[27]  James P. Reilly,et al.  Optimal design of linear array of sensors , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[28]  W. Wiesbeck,et al.  Design and measurement of conformal antennas on cylindrical and spherical geometries , 1999, 1999 IEEE Africon. 5th Africon Conference in Africa (Cat. No.99CH36342).

[29]  R. O. Nielsen.  Azimuth and elevation angle estimation with a three-dimensional array , 1994 .

[30]  J. Ward,et al.  Cramer-Rao bounds for target angle and Doppler estimation with space-time adaptive processing radar , 1995, Conference Record of The Twenty-Ninth Asilomar Conference on Signals, Systems and Computers.

[31]  Andrew P. Sage,et al.  On Sonar Signal Analysis , 1970, IEEE Transactions on Aerospace and Electronic Systems.

[32]  A. Gualtierotti.  H. L. Van Trees, Detection, Estimation, and Modulation Theory, , 1976 .

[33]  Akira Ishimaru,et al.  Equally Spaced Spherical Arrays , 1968 .

[34]  E. Cowley,et al.  Adaptive Radar Detection and Estimation , 1993 .

[35]  A. Wald.  Tests of statistical hypotheses concerning several parameters when the number of observations is large , 1943 .

[36]  B. Mahafza.  Introduction to radar analysis , 1998 .