Blind array calibration for automotive radar

Abstract

A method of calibration for a radar array including: receiving a plurality of measurement vectors, each measurement vector associated with a range-Doppler (RD) cell of a RD map; determining from the plurality of measurement vectors a plurality of autocorrelation matrices, an autocorrelation matrix corresponding to each the measurement vector and RD cell; performing an iterative process to minimize a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming the plurality of autocorrelation matrices with a hypothesized calibration matrix; smoothing the plurality of transformed autocorrelation matrices using one or more subarray technique to provide a plurality of smoothed autocorrelation matrices; determining a discrepancy function for each of the smoothed autocorrelation matrices and for each of a plurality of hypothesized numbers of targets, to provide, for each smoothed autocorrelation matrix, a plurality of likelihood functions, one for each hypothesized number of targets; selecting, for each of the smoothed autocorrelation matrices, a minimal likelihood function corresponding to a single hypothesized number of targets; and summing the selected minimal likelihood functions to provide the cost function.

Description

[0001] BLIND ARRAY CALIBRATION FOR AUTOMOTIVE RADAR

[0002] TECHNOLOGICAL FIELD

[0003] The present disclosure, in some embodiments thereof, relates to methods for blind array calibration for multi-element radar and, more particularly, but not exclusively, to methods for blind array calibration for multi-element radar in presence of one or more of mutual coupling, phase, and gain errors, and even more particularly for blind array calibration for multi-element automotive radar.

[0004] BACKGROUND ART

[0005] The problem of high-resolution direction-of-arrival (DOA) estimation using an array of sensors has been investigated over the last four decades (see, e.g., H. Krim and M. Viberg, “Two decades of array signal processing research: The parametric approach,” IEEE Signal Process. Mag., vol. 13, pp. 67-97, 1996 ).

[0006] Radar array miscalibration is described, for example in A. Kisliansky, R. Shavit, and J. Tabrikian, “Direction of arrival estimation in the presence of noise coupling in antenna arrays,” IEEE Transactions on Antennas and Propagation, vol. 55, no. 7, pp. 1940- 1947, 2007).

[0007] Super-resolution methods are described, such as those based on polynomial (see, e.g. A. Barabell, “Improving the resolution performance of eigenstructure based direction-finding algorithms,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 8, 1983, pp. 336-339), spatial smoothing (see, e.g. T.-J. Shan, M. Wax, and T. Kailath, “On spatial smoothing for direction-of-arrival estimation of coherent signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 33, no. 4, pp. 806-811, 1985), and forward-backward averaging (see, e.g. S. Pillai and B. Kwon, “Forward / b ackward spatial smoothing techniques for coherent signal identification,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 1, pp. 8-15, 1989).

[0008] According to Allen and Diamond, “Mutual coupling in array antennas, ” MIT Lincoln Lab., Lexington, MA, USA, Tech. Rep., 1966, mutual coupling (MC), apparently, stems from several electromagnetic mechanisms, including, for example, one or more of direct electromagnetic radiation, indirect coupling caused by near-field scattering, and coupling inside the distribution network feed. A review of different interpretations of MC from both electromagnetic theory and signal processing perspectives is provided in B. Friedlander, “On the mutual coupling matrix in array signal processing,” 2020 54th Asilomar Conference on Signals, Systems, and Computers, pp. 1245-1249, 2020.

[0009] MC characterization has been investigated. Apparently, both MC and sensor phase and gain errors can be modeled by multiplying the ideal steering vector by an unknown complex square matrix, also herein termed a “calibration matrix” (see, e.g. Allen and Diamond, “Mutual coupling in array antennas,” MIT Lincoln Lab., Lexington, MA, USA, Tech. Rep., 1966).

[0010] Array calibration is a procedure that aims to correct errors in an initial array response model. Generally, calibration techniques can be classified into offline (see, e.g. B. Porat and B. Friedlander, “Accuracy requirements in off-line array calibration,” IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 2, 1997), and online methods (see, e.g. Sellone and Serra, “A novel online mutual coupling compensation algorithm for uniform and linear arrays,” IEEE Transactions on Signal Processing, vol. 55, no. 2, pp. 560-573, 2007).

[0011] Online self-calibration has been investigated, see e.g. F. Wang, R. Jiang, X. Tian, X. Liu, S. Jiang, B. Gu, and Y. Chen, “An efficient near-field model gain-phase selfcalibration method for uniform rectangular arrays,” IEEE Journal of Oceanic Engineering, pp. 1-14, 2022.

[0012] Simplified online calibration procedures are presented in T. Wei, B. Liao, H. Huang, and Z. Quan, “Online mutual coupling calibration using a signal source at unknown location,” in 2018 IEEE 10th Sensor Array and Multichannel Signal Processing Workshop (SAM), 2018, pp. 189-193 and Z. Ye, J. Dai, X. Xu, and X. Wu, “DOA estimation for uniform linear array with mutual coupling,” IEEE Transactions on Aerospace and Electronic Systems, vol. 45, no. 1, pp. 280-288, 2009.

[0013] A. Weiss and A. Yeredor, “Asymptotically optimal blind calibration of uniform linear sensor arrays for narrowband Gaussian signals,” IEEE Transactions on Signal Processing, vol. 68, pp. 5322-5333, 2020 apparently proposes a calibration algorithm for sensor phase and gain estimation in ULA (uniform linear array). This method is apparently based on the Toeplitz structure of the observations covariance matrix. In Z. Ye and C. Liu, “2-D DOA estimation in the presence of mutual coupling,”

[0014] IEEE Transactions on Antennas and Propagation, vol. 56, pp. 3150-3158, 2008, a 2-D DOA estimation method is proposed in the presence of MC for uniform rectangular arrays (URAs) based on the MUSIC algorithm.

[0015] Z. Zheng, K. Liu, W. Q. Wang, Y. Yang, and J. Yang, “Robust adaptive beamforming against mutual coupling based on mutual coupling coefficients estimation,” IEEE Transactions on Vehicular Technology, vol. 66, no. 10, pp. 9124- 9133, 2017 proposes an adaptive beamforming approach in the presence of MC, where the MCM is modeled with a banded Toeplitz symmetric structure.

[0016] In N. Petrov, O. Krasnov, and A. G. Yarovoy, “Auto-calibration of automotive radars in operational mode using simultaneous localization and mapping,” IEEE Transactions on Vehicular Technology, vol. 70, no. 3, pp. 2062-2075, 2021, calibration of gain and phase errors for automotive radar are apparently proposed.

[0017] Compressed sensing (CS) techniques (see, e.g. D. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289-1306, 2006) have apparently been applied to the problem of radar target localization.

[0018] A CS -based phase calibration method with targets in unknown directions was apparently proposed in N. Onhon and M. Cetin, “A sparsity -driven approach for joint SAR imaging and phase error correction,” IEEE Transactions on Image Processing, vol. 21, no. 4, pp. 2075-2088, 2012.

[0019] Online CS-based MC calibration was apparently proposed in A. M. Elbir and T. E. Tuncer, “Sparse support recovery for DOA estimation in the presence of mutual coupling,” in 2015 23rd European Signal Processing Conference (EUSIPCO), 2015, pp. 1366-1370.

[0020] The off-grid problem is apparently described in Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Transactions on Signal Processing, vol. 59, no. 5, pp. 2182-2195, 2011 when the actual sources do not fall on the discrete grid of angles used to form a basis.

[0021] The disclosures of all references mentioned throughout the present specification, as well as the disclosures of all references mentioned in those references, are hereby incorporated herein by reference. Acknowledgement of the above references herein is not to be inferred as meaning that these are in any way relevant to the patentability of the presently disclosed subject matter.

[0022] GENERAL DESCRIPTION

[0023] Following is a non-exclusive list of some exemplary embodiments of the disclosure. The present disclosure also includes embodiments which include fewer than all the features in an example and embodiments using features from multiple examples, even if not listed below.

[0024] Example 1. A method of calibration for a radar array comprising: receiving a plurality of measurement vectors, each measurement vector associated with a range-Doppler (RD) cell of a RD map; determining from said plurality of measurement vectors a plurality of autocorrelation matrices, an autocorrelation matrix corresponding to each said measurement vector and RD cell; performing an iterative process to minimize a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming said plurality of autocorrelation matrices with a hypothesized calibration matrix; smoothing said plurality of transformed autocorrelation matrices using one or more subarray technique to provide a plurality of smoothed autocorrelation matrices; determining a discrepancy function for each of said smoothed autocorrelation matrices and for each of a plurality of hypothesized numbers of targets, to provide, for each smoothed autocorrelation matrix, a plurality of likelihood functions, one for each hypothesized number of targets; selecting, for each of said smoothed autocorrelation matrices, a minimal likelihood function corresponding to a single hypothesized number of targets; and summing the selected minimal likelihood functions to provide said cost function.

[0025] Example 2. The method according to Example 1, wherein said discrepancy function is a likelihood function derived under an assumption that noise is Gaussian. Example 3. The method according to Example 2, wherein said receiving comprises receiving measurement snapshot array data vectors where a single measurement snapshot is used to determine each said RD cell.

[0026] Example 4. The method according to Example 1, wherein said smoothing is by applying one or both of spatial smoothing (SS) and forward-backward averaging (FBA) techniques.

[0027] Example 5. The method according to any one of Examples 1 to 4, wherein determining a likelihood function for each of said smoothed autocorrelation matrices for each of a plurality of hypothesized numbers of targets comprises calculating eigenvalues of the smoothed transformed autocorrelation matrix.

[0028] Example 6. The method according to Example 5, wherein said likelihood function includes a sum of said eigenvalues.

[0029] Example 7. The method according to Example 6, wherein said likelihood function comprises a penalty term selected to compensate for the specific number of hypothetical targets in the calculation.

[0030] Example 8. The method according to Example 7, wherein said penalty term is related to size and number of subarrays used during said smoothing.

[0031] Example 9. The method according to any one of Examples 1 to 8, wherein determining a likelihood function for each of said smoothed autocorrelation matrices for each of a plurality of hypothesized numbers of targets comprises determining noise variances for each hypothesized number of targets, the noise variance forming a negative term in said likelihood function.

[0032] Example 10. The method according to any one of Examples 1-9, wherein said performing is until reaching a stop condition.

[0033] Example 11. The method according to Example 10, wherein the stop condition is a number of iterations.

[0034] Example 12. The method according to any one of Examples 1-11, wherein said optimization is performed in a bounded parameter space.

[0035] Example 13. The method according to Example 12, wherein the bounded parameter space includes perturbations around the identity matrix.

[0036] Example 14. The method according to any one of Examples 10-13, wherein said bounded parameter space is based on physical understanding of mutual coupling behavior. Example 15. The method according to any one of Examples 1-14, comprising, preliminarily selecting a subset of said plurality of radar measurements.

[0037] Example 16. The method according to Example 15, wherein said selecting is of cells having a SNR above a predefined SNR threshold.

[0038] Example 17. The method according to any one of Examples 15-16, wherein said selecting is of cells is of those cells having range below a predefined range threshold.

[0039] Example 18. The method according to any one of Examples 1-17, wherein the plurality of measurement vectors are received from an automotive mounted radar array system; and wherein, upon finishing said performing, the optimized calibration matrix is used to correct subsequent radar measurements.

[0040] Example 19. The method according to Example 18, wherein said subsequent radar measurements are used in a navigation system of said automotive.

[0041] Example 20. The method according to any one of Examples 18-19, wherein said iterative process is performed online, during use of said automotive.

[0042] Example 21. The method according to any one of Examples 1-20, wherein said iterative process is re-performed periodically.

[0043] Example 22. The method according to any one of Examples 1-21, wherein said optimized calibration matrix is used to adjust a received calibration matrix.

[0044] Example 23. The method according to Example 22, wherein said received calibration matrix has been determined offline in a controlled calibration environment.

[0045] Example 24. A method of radar array calibration comprising: receiving a plurality of radar measurements; performing an iterative process minimizing a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming said plurality of radar measurements by applying a hypothesized calibration matrix to each of said plurality of radar measurements; smoothing each transformed radar measurement using one or more subarray smoothing technique; determining, for each smoothed transformed radar measurement, a minimal target likelihood function corresponding to a minimal number of targets, to provide a plurality of minimal target likelihood functions, one for each of said plurality of radar measurements, where calculation of the likelihood function is formulated to reflect separation between signal and noise subspaces; summing said plurality of likelihood functions to provide said cost function.

[0046] Example 25. The method according to Example 24, wherein said smoothing is by applying one or both of spatial smoothing (SS) and forward-backward averaging (FBA) techniques.

[0047] Example 26. The method according to any one of Examples 24-25, wherein said plurality of radar measurements include a plurality of autocorrelation matrices; wherein said receiving comprises receiving a plurality of measurement data vectors, each associated with a range-Doppler (RD) cell; and computing said autocorrelation matrices, an autocorrelation matrix computed for each measurement data vector and associated with a RD cell.

[0048] Example 27. The method according to any one of Examples 24-26, wherein said determining comprises, for each said smoothed transformed radar measurement: calculating a plurality of likelihood functions, one for each of a plurality of hypothesized number of targets; and selecting one of said plurality of said likelihood functions as said minimal target likelihood function.

[0049] Example 28. The method according to Example 27, wherein calculating said plurality of likelihood functions comprises calculating eigenvalues of the smoothed transformed autocorrelation matrix.

[0050] Example 29. The method according to Example 28, wherein said likelihood function includes a sum of said eigenvalues.

[0051] Example 30. The method according to Example 28, wherein said likelihood function comprises a penalty term selected to compensate for the specific number of hypothetical targets in the calculation.

[0052] Example 31. The method according to Example 30, wherein said penalty term is related to size and number of subarrays used during said smoothing.

[0053] Example 32. The method according to any one of Examples 27-31, wherein said determining comprises determining noise variances for each hypothesized number of targets, the noise variance forming a negative term in said likelihood function. Example 33. The method according to any one of Examples 24-32, wherein said performing is until reaching a stop condition.

[0054] Example 34. The method according to Example 33, wherein the stop condition is a number of iterations.

[0055] Example 35. The method according to any one of Examples 24-34, wherein said optimization is performed in a bounded parameter space.

[0056] Example 36. The method according to Example 35, wherein the bounded parameter space includes perturbations around the identify matrix.

[0057] Example 37. The method according to any one of Examples 35-36, wherein said bounded parameter space is based on physical understanding of mutual coupling behavior.

[0058] Example 38. The method according to any one of Examples 24-37, comprising, selecting a subset of said plurality of radar measurements, prior to said performing.

[0059] Example 39. The method according to Example 38, wherein said selecting is of cells having a SNR above a predefined SNR threshold.

[0060] Example 40. The method according to any one of Examples 38-39, wherein said selecting is of cells is of those cells having range below a predefined range threshold.

[0061] Example 41. The method according to any one of Examples 24-40, wherein said receiving is from an automotive mounted radar array system; and wherein, upon finishing said performing, the optimized calibration matrix is used to correct subsequent radar measurements.

[0062] Example 42. The method according to Example 41, wherein said subsequent radar measurements are used in a navigation system of said automotive.

[0063] Example 43. The method according to any one of Examples 41-42, wherein said performing is online, during use of said automotive.

[0064] Example 44. The method according to any one of Examples 24-43, wherein said performing is re -performed periodically.

[0065] Example 45. The method according to any one of Examples 24-44, wherein said optimized calibration matrix is used to adjust a received calibration matrix.

[0066] Example 46. The method according to Example 45, wherein said received calibration matrix has been determined offline in a controlled calibration environment. Example 47. A method of radar calibration comprising: receiving a plurality of radar measurements; processing said plurality of measurement vectors to identify noise and signal subspaces; and determining a calibration transformation configured to cancel radar array nonidealities by minimizing signal energy found in said noise subspace.

[0067] Example 48. A system for radar calibration comprising: a processor configured to: receive a plurality of measurement vectors, each measurement vector associated with a range-Doppler (RD) cell of a RD map; determine from said plurality of measurement vectors a plurality of autocorrelation matrices, an autocorrelation matrix corresponding to each said measurement vector and RD cell; perform an iterative process to minimize a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming said plurality of autocorrelation matrices with a hypothesized calibration matrix; smoothing said plurality of transformed autocorrelation matrices using one or more subarray technique to provide a plurality of smoothed autocorrelation matrices; determining a discrepancy function for each of said smoothed autocorrelation matrices and for each of a plurality of hypothesized numbers of targets, to provide, for each smoothed autocorrelation matrix, a plurality of likelihood functions, one for each hypothesized number of targets; selecting, for each of said smoothed autocorrelation matrices, a minimal likelihood function corresponding to a single hypothesized number of targets; and summing the selected minimal likelihood functions to provide said cost function.

[0068] Example 49. A radar navigation system comprising: - lo an automotive mounted radar system configured to provide a plurality of measurement vectors, each measurement vector associated with a range-Doppler (RD) cell of a RD map; a processor configured to: determine from said plurality of measurement vectors a plurality of autocorrelation matrices, an autocorrelation matrix corresponding to each said measurement vector and RD cell; perform an iterative process to minimize a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming said plurality of autocorrelation matrices with a hypothesized calibration matrix; smoothing said plurality of transformed autocorrelation matrices using one or more subarray technique to provide a plurality of smoothed autocorrelation matrices; determining a discrepancy function for each of said smoothed autocorrelation matrices and for each of a plurality of hypothesized numbers of targets, to provide, for each smoothed autocorrelation matrix, a plurality of likelihood functions, one for each hypothesized number of targets; selecting, for each of said smoothed autocorrelation matrices, a minimal likelihood function corresponding to a single hypothesized number of targets; and summing the selected minimal likelihood functions to provide said cost function.

[0069] Example 50. The radar navigation system according to Example 49, wherein said processor is configured to calibrate said radar system using said optimized calibration matrix.

[0070] Example 51. A subspace-based method for blind array calibration for radar, the method including: providing initial values by: Example 53. The method according to any one of Examples 51-52, wherein providing the noise variance value comprises setting

[0071] Example 54. The method according to any one of Examples 51-53, wherein providing the M measurement vectors {x1, ..., xm}, comprises providing M Range-Doppler (RD) measurement vectors {x1, ..., xm}.

[0072] Example 55. The method according to any one of Examples 51-54, wherein the minimizing the cost function C(ψ) comprises using a fmincon solver with the initial calibration matrix and a LB matrix and an UB matrix, provided as initialization values as part of the providing initial values, where LB and UB are pre-defined lower bound and upper bound matrices, respectively.

[0073] Example 56. The method according to any one of Examples 51-55 wherein the stop condition comprises performing no more than a specific number of iterations.

[0074] Unless otherwise defined, all technical and / or scientific terms used within this document have meaning as commonly understood by one of ordinary skill in the art / s to which the present disclosure pertains. Methods and / or materials similar or equivalent to those described herein can be used in the practice and / or testing of embodiments of the present disclosure, and exemplary methods and / or materials are described below. Regarding exemplary embodiments described below, the materials, methods, and examples are illustrative and are not intended to be necessarily limiting.

[0075] Some embodiments of the present disclosure are embodied as a system, method, or computer program product. For example, some embodiments of the present disclosure may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit,” “module” and / or “system.”

[0076] Implementation of the method and / or system of some embodiments of the present disclosure can involve performing and / or completing selected tasks manually, automatically, or a combination thereof. According to actual instrumentation and / or equipment of some embodiments of the method and / or system of the present disclosure, several selected tasks could be implemented by hardware, by software or by firmware and / or by a combination thereof, e.g., using an operating system. For example, hardware for performing selected tasks according to some embodiments of the present disclosure could be implemented as a chip or a circuit. As software, selected tasks according to some embodiments of the present disclosure could be implemented as a plurality of software instructions being executed by a computational device e.g., using any suitable operating system.

[0077] In some embodiments, one or more tasks according to some exemplary embodiments of method and / or system as described herein are performed by a data processor, such as a computing platform for executing a plurality of instructions. Optionally, the data processor includes a volatile memory for storing instructions and / or data and / or a non-volatile storage e.g., for storing instructions and / or data. Optionally, a network connection is provided as well. User interface / s e.g., display / s and / or user input device / s are optionally provided.

[0078] Some embodiments of the present disclosure may be described below with reference to flowchart illustrations and / or block diagrams. For example illustrating exemplary methods and / or apparatus (systems) and / or and computer program products according to embodiments of the present disclosure. It will be understood that each step of the flowchart illustrations and / or block of the block diagrams, and / or combinations of steps in the flowchart illustrations and / or blocks in the block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general-purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart steps and / or block diagram block or blocks.

[0079] These computer program instructions may also be stored in a computer readable medium that can direct a computer (e.g., in a memory, local and / or hosted at the cloud), other programmable data processing apparatus, or other devices to function in a particular manner, such that the instructions stored in the computer readable medium can be used to produce an article of manufacture including instructions which implement the function / act specified in the flowchart and / or block diagram block or blocks.

[0080] The computer program instructions may also be run by one or more computational device to cause a series of operational steps to be performed e.g., on the computational device, other programmable apparatus and / or other devices to produce a computer implemented process such that the instructions which execute provide processes for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0081] Some of the methods described herein are generally designed only for use by a computer, and may not be feasible and / or practical for performing purely manually, by a human expert. A human expert who wanted to manually perform similar tasks, might be expected to use different methods, e.g., making use of expert knowledge and / or the pattern recognition capabilities of the human brain, potentially more efficient than manually going through the steps of the methods described herein.

[0082] BRIEF DESCRIPTION OF THE DRAWING(S)

[0083] Some embodiments of the disclosure are herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of embodiments of the disclosure. In this regard, the description taken with the drawings makes apparent to those skilled in the art how embodiments of the disclosure may be practiced.

[0084] In the drawings:

[0085] FIG. 1A is a simplified schematic of a radar system, according to some embodiments of the disclosure;

[0086] FIG. IB is a simplified schematic of a radar system employed on a vehicle, according to some embodiments of the disclosure;

[0087] FIG. 2 is a flow chart of a method of radar calibration, according to some embodiments of the disclosure;

[0088] FIG. 3A presents eigenvalues of a smoothed autocorrelation matrix for two cases, with and without MC;

[0089] FIG. 3B presents eigenvalues of a smoothed autocorrelation matrix for two cases, with and without MC;

[0090] FIG. 4 illustrates MUSIC spatial spectra after SSFB (spatial smoothing (SS) and forward-backward averaging (FBA) techniques) for examples described above;

[0091] FIG. 5A illustrates RMSE as a function of SNR for the case of a single target at DOA of 30° for examples described above. FIG. 5 A shows a relation of DOA estimation RMSE, using MUSIC, in the presence of a single target at =30° with ULA, and unstructured MCM;

[0092] FIG. 5B illustrates RMSE as a function of SNR in presence of a single target at DOA of 30°for examples described above;

[0093] FIG. 6 illustrates RMSE as a function of SNR for the case of a single target at DOA of 30° for examples described above;

[0094] FIG. 7 illustrates the resolution performance of two targets with equal SNRs of 25dB;

[0095] FIGs. 8A-8D compare 2-D beam patterns of an array for the three cases of MC- free, MC with an uncalibrated array, and MC after calibration;

[0096] FIGs. 9A-9C compare beam patterns of an array for the three cases of MC-free, MC with an uncalibrated array, and MC after calibration of two point targets;

[0097] FIGs. 10A and 10B illustrate RMSE as a function of SNR for a case of a single target located at <) = 20°, 0 = 0° for examples described above; and

[0098] FIG. 11 is a simplified flow chart of a method for subspace-based blind calibration for multi-element radar, according to some embodiments of the disclosure.

[0099] DESCRIPTION OF SPECIFIC EMBODIMENTS

[0100] The present disclosure, in some embodiments thereof, relates to methods for blind array calibration for multi-element radar and, more particularly, but not exclusively, to methods for blind array calibration for multi-element radar in presence of one or more of mutual coupling, phase, and gain errors, and even more particularly for blind array calibration for multi-element automotive radar.

[0101] Overview

[0102] A broad aspect of some embodiments of the disclosure relates to calibrating multi-element radar (e.g., uniform planar arrays (UPA) or uniform linear arrays (UFA) using measurements acquired by the radar array, the measurements stemming from unknown targets, where calibrating is performed, without determining features of individual target / s appearing in the measurements.

[0103] A broad aspect of some embodiments of the disclosure relates to using an assumption that miscalibration of a radar array can violate spatial stationarity as measured by the array sensors. This violation may appear, for example, after application to measurement data of smoothing (e.g., by one or more of SS and FBA e.g., of measurement vector autocorrelation matrices) and subspace identification techniques, as leakage of signal energy into a noise subspace.

[0104] An aspect of some embodiments of the disclosure relates to exploiting this leakage effect to determine a calibration transformation, where the calibration transformation is optimized to minimize the leakage. In some embodiments, a calibration transformation (e.g., which may be applied to measurement data to cancel out array non-idealities) is optimized to minimize the signal energy appearing in the noise subspace. Where the calibration transformation may be understood to restore spatial stationarity in the smoothed autocorrelation matrices, by removing the leakage of signal subspace eigenvalues from the noise subspace.

[0105] Leakage of the signal subspace into the noise subspace, may be understood as decreasing a discrepancy function for nonparametric subspace estimation. In some embodiments, the discrepancy function is a likelihood function is derived under an assumption that the noise is Gaussian. However, in some embodiments, the likelihood function may be employed when noise is non-Gaussian. For example, it being theorized that the likelihood function will provide calibration as described herein under nonGaussian noise.

[0106] In some embodiments, a likelihood function is used (e.g., likelihood functions for different scenarios and data are calculated) in order to determine the calibration transformation. The likelihood function may be indicative of how likely the observed measurement / s result from the model parameters used in determining the likelihood function. For example, a likelihood function for a single measurement vector (e.g., corresponding to a single RD cell) may provide a measure of how likely a hypothesized calibration transformation and a hypothesized number of targets are to produce the observed measurements. It is theorized that the array non-idealities (leading to leakage of the signal into the noise subspace) reduce the likelihood function for the data involved.

[0107] Given both the calibration transformation and targets are unknown maximization of the likelihood function for a data set (measurement data vector) is, in some embodiments, performed to provide both. In some embodiments, nonparametric model order selection is used for target enumeration at the RD cells and / or for constructing a likelihood-based cost function. The likelihood-based cost function is optimized, optionally by minimizing leakage of energy from the signal subspace to noise subspace.

[0108] For example, in a two-stage process, where, for each measurement vector, a likelihood function corresponding to a minimal number of targets is determined, and then a calibration matrix is determined from the minimal target likelihood functions.

[0109] Based on an assumption that the same calibration transformation is applicable to all RD cells of the radar array, the cost function, in some embodiments, is a sum of the target determined likelihood functions for a plurality of cells is then optimized in an iterative process to provide a calibration transformation.

[0110] In some examples, a cost function with respect to (w.r.t.) a calibration matrix applied to the received data is optimized. In some examples, the cost function includes a superposition (e.g., sum) of a penalized negative log-likelihood at different range- Doppler (RD) cells, after minimization with respect to a number of targets.

[0111] In some embodiments, the calibration is considered to be “blind” where data used and processing performed, is for unknown target / s appearing in the measurement data. Where the targets, for each measurement (e.g., measurement vector) may have unknown number, intensity, range, and direction of arrival (DOA).

[0112] A potential benefit of blind calibration is the ability to perform the calibration “online”, e.g., while a radar system is in use e.g., without using a dedicated calibration setup.

[0113] In some embodiments, the calibration may be performed intermittently, the determined calibration transformation being used for a time period, before being redetermined. A potential advantage being that the processing requirements are intermittent and / or that the processing may be performed remotely (e.g., remote to a processor associated with the array system).

[0114] An aspect of some embodiments of the disclosure relates to, each measurement being a vector associated with a RD cell (having an unknown number of targets), from which a noise subspace (and a signal subspace) are determined as eigenvalues of a autocorrelation matrix of the vector. Where, in some embodiments, a calibration matrix is determined by performing an optimization process where, for calibrated measurements, leakage of the signal into the noise subspace is minimized where the optimization is performed using likelihood functions for a plurality of measurements. In some embodiments, the optimization includes minimizing a cost function which is a sum of likelihood functions, one function for each measurement. Where the summed likelihood functions are, for each measurement, the likelihood function providing a minimal number of targets.

[0115] In some embodiments, to select a likelihood function having a minimal number of targets, the likelihood functions are determined using a penalty function which depends on the hypothetical number of targets, the penalty function increasing with the number of hypothetical targets. Potentially, the penalty function compensates for decreasing likelihood with increasing number of hypothesized targets.

[0116] Generally, in this document, active radar arrays have been described where the method uses a plurality of data vectors, each one associated with an RD cell in the method. In some embodiments, however, the method may be used in passive radar arrays (e.g., where the radar array only receives signals and does not transmit radar). In some embodiments, for passive arrays, multiple time frames are used in the method in place of multiple RD cells.

[0117] Although description within this document is generally directed to SIMO radar, it should be understood that the general method and concepts behind the method may also be applied to MIMO (multiple-input multiple-output) radar systems e.g., after tailoring of the method to MIMO.

[0118] In some embodiments, the calibration methods described herein are employed in vehicular radar systems, e.g., automotive radar systems. Where, for example, a radar array mounted to a vehicle may be calibrated using the method / s described herein. In some embodiments, the calibrated radar data may be used, for example, in navigation of the vehicle (e.g., self-navigation).

[0119] Apparently, a limiting factor in the performance of multi-element radar systems is presence of mutual coupling (MC) between receive antenna elements and / or array imperfections, such as antenna phase and gain errors e.g., where different antennas have different phase and / or gain response. Data model / s may correspondingly be misspecified, potentially resulting in one or more of; high sidelobe levels in the beam pattern, low angular resolution, and biased angle estimation.

[0120] In this disclosure, a blind calibration scheme for uniform planar arrays is described. In some embodiments, methods may use multiple measurements of various scenarios, with an arbitrary and unknown number of targets-of-opportunity (also herein termed “targets”), having unknown directions-of-arrival (DOAs), and unknown intensities.

[0121] Exemplary methods use spatial smoothing (SS) and forward-backward averaging (FBA) techniques, in order to identify signal and noise subspaces. In the presence of MC and / or array imperfections, it is theorized that, the signal subspace leaks into the noise subspace. The example methods seek to find and compensate for model misspecification using a model-order selection criterion.

[0122] A potential benefit of the calibration method described herein, e.g., where model misspecification is compensated for, is improved DO A estimation performance in terms of accuracy and / or resolution. This may be of particular value in technologies having limited array aperture (e.g., automotive radar technology) which may affect DOA estimation performance.

[0123] Array miscalibration types include, for example; array geometry perturbation, sensor phase and gain errors, and mutual coupling (MC) between antenna elements. Although techniques herein may also be suitable for compensating for array geometry perturbation / s, associated with integrated circuity arrays having relatively static geometry, in general array non-idealities described herein are MC and sensor phase and / or gain errors.

[0124] A potential benefit of correction or compensation for model misspecification is increased or maintained performance of super-resolution method / s. This may be associated with reliance of super-resolution method / s on the data model, e.g., making them sensitive to modeling errors. This may be associated with, when estimation errors due to noise or interference are small, estimation errors due to model misspecification can become dominant e.g., compared to the noise and / or interference contribution / s. This may be associated with reliance of super-resolution method / s on specific array response structures.

[0125] Exemplary radar systems, such as automotive radars, may operate at millimeterwave (mmW) frequencies, such that small imperfections in a receiver chain may result in large phase errors e.g., due to the short wavelengths.

[0126] When an antenna array is designed on integrated circuits, the array geometry can be fixed and accurate. In such cases, mutual coupling, and / or phase errors, may provide sources of array mismatch or may be the dominating sources of array mismatch (see, e.g. B. Liao and S. C. Chan, “Adaptive beamforming for uniform linear arrays with unknown mutual coupling,” IEEE Antennas and Wireless Propagation Letters, vol. 11, pp. 464-467, 2012).

[0127] In some embodiments, the blind calibration methods herein are performed online e.g., in an uncontrolled environment with targets-of-opportunity whose locations and intensities are unknown.

[0128] Apparently, offline calibration is typically performed in a controlled environment, such as an anechoic chamber, where a transmitter with a strong power (e.g., a corner reflector) is placed at several known directions within a field-of-view of a radar, and a received signal by array elements of the radar is measured. Potential benefits of the online calibration methods described herein (e.g., with respect to offline calibration) are potentially reduced expense and / or time required for the process and / or increased ease of repetition of the calibration e.g., where calibration is repeated periodically e.g., to compensate for to thermal effects, aging of components, and / or changing environmental factors.

[0129] Description presented herein, provides techniques of blind array calibration in the presence of unstructured MC, and / or phase errors, and / or or gain errors for uniform linear and / or planar arrays.

[0130] The example method described herein is based on multiple independent measurements, where each measurement includes a single-snapshot array data vector, which is composed of the contribution of an arbitrary number of targets of opportunity with unknown DOAs and intensities, optionally in addition to additive noise. In some embodiments, the method described herein employs a single snapshot per RD cell. A potential benefit being, reduction of computational complexity e.g., in comparison to calibration methods which require multiple snapshots per RD cell.

[0131] A potential benefit of such assumptions is that they may match a problem of automotive radar, where independent measurements refer to measurements from Range- Doppler (RD) cells.

[0132] The example described herein uses an insight that, apparently, echo signals from far- field targets without multipath produce a spatially stationary field (see, e.g. S. Bose and A. O. Steinhardt, “Invariant tests for spatial stationarity using covariance structure,” Proc. IEEE ICASSP, vol. 44, no. 6, pp. 1523-1533, 1993).

[0133] Spatial stationarity allows the use of spatial smoothing (SS) and forwardbackward averaging (FBA) similar to that used in T.J. Shan, M. Wax, and T. Kailath, “On spatial smoothing for direction-of-arrival estimation of coherent signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 33, no. 4, pp. 806-811, 1985 in uniform linear or planar arrays.

[0134] Without wanting to be bound by theory, it is theorized that MC and / or array imperfections, such as phase, or gain errors, violate the spatial stationarity as measured by the array sensors.

[0135] This disclosure hereinbelow shows that applying SS and FBA (SSFB) on uniform planar arrays in the presence of such array misspecification results in leakage of the signal subspace into the noise subspace, and consequently, the log-likelihood for nonparametric subspace estimation is decreased.

[0136] In some examples, a cost function with respect to (w.r.t.) a calibration matrix applied to the received data is optimized. In some examples, the cost function is optionally composed of a superposition (e.g., sum) of a penalized negative loglikelihood at different range-Doppler (RD) cells, after minimization with respect to a number of targets. It should be noted that, within this document the terms “likelihood” and “log-likelihood” may be used interchangeably, where it may be understood that the log -likelihood may be used in calculations e.g., to simplify the calculations.

[0137] Such an approach, in some embodiments, does not require prior knowledge of the number of targets at each RD cell, nor does it require their DO As or intensities.

[0138] Before explaining at least one embodiment of the disclosure in detail, it is to be understood that the disclosure is not necessarily limited in its application to the details of construction and the arrangement of the components and / or methods set forth in the following description and / or illustrated in the drawings and / or the examples. The disclosure is capable of other embodiments or of being practiced or carried out in various ways.

[0139] Notation

[0140] Throughout the present application and claims, the following notation is adopted:

[0141] Italic letters are used for scalars, boldface lowercase letters for are used for column vectors, and boldface uppercase letters are used for matrices.

[0142] The superscripts (-)T, (•)*, (-)Hand (-)-1denote transpose, complex conjugate, conjugate transpose and inverse operators, respectively. E[-] denotes an expectation operator.

[0143] Evis an identity matrix of size N1 x N2, and are a column vector of size TV and a matrix of size N1 x N2, respectively, whose entries are equal to zero.

[0144] The Kronecker product is denoted by

[0145] Exemplary radar system

[0146] FIG. 1A is a simplified schematic of a radar array system 101, according to some embodiments of the disclosure.

[0147] In some embodiments, system 101 includes a radar array 103 which includes a plurality of antennas 107. In some embodiments, radar array is a uniform rectangular array (URA) having evenly spaced antenna elements, in a linear or planar (e.g., as illustrated in FIG. 1A) configuration.

[0148] Each antenna may be employed in one or both of transmission and reception of radar signals. System 101, in some embodiments, includes radar array circuitry 110 which includes receiver and / or transceiver circuitry. Radar array circuitry 110 may include, for example, one or more of power supply, amplification, analog-to-digital, and digital-to-analog, steering (e.g., phase shifter / s), beamforming, and local oscillator circuitries.

[0149] In some embodiments, system 101 is an active radar array where circuitry 110 is active, radar array 103 both transmitting and receiving radar signals.

[0150] Alternatively, in some embodiments, system 101 is a passive radar array where circuity 110 is active, radar array 103 only receiving radar signals.

[0151] In some embodiments, system 101 includes processing and memory circuitry (PMC) 113 which receives data from and sends control signals to array circuitry 111. Optionally, in some embodiments, a navigation system which includes system 101 may include one or more external PMC 119. Where processing functions may be divided between local and external PMCs 113, 119.

[0152] In some embodiments, system 101 may include one or more housing 108 where the housing / s 109 may be mounted to a vehicle e.g., an automotive vehicle (e.g., as illustrated in FIG. IB). In some embodiments, one or more of radar antenna array 103, radar array circuitry 111, and PMC 113 are hosted by housing 109 and / or are hosted by the vehicle (e.g., in multiple housings). In some embodiments, radar system 101 is part of a navigation system e.g., a vehicular navigation system. Where, in some embodiments, the navigation system includes one or more user interface 117. User interface / s 117 display data to user / s (e.g., radar measurement data e.g., navigation instruction / s). Where, in some embodiments, user interface / s 117 receive data from user / s e.g., where a user inputs one or more requirement of navigation system 100 e.g., a destination to be navigated to.

[0153] In some embodiments, the navigation system includes one or more vehicle control unit 115, which in some embodiments, controls movement of a vehicle e.g., to which radar system 101 is attached.

[0154] In some embodiments, processor / s 113, 119 generate data and / or control signal / s e.g. which are received vehicle control unit 115 for control of movement of the vehicle.

[0155] FIG. IB is a simplified schematic of a radar system employed on a vehicle 121, according to some embodiments of the disclosure.

[0156] FIG. IB illustrates an exemplary implementation of system 101 of FIG. 1A, where housing 109 is mounted to a vehicle 121, the housing hosting radar array 103 which may transmit and / or receive radar signals over a field of view (FOV) 123.

[0157] Exemplary method

[0158] FIG. 2 is a flowchart of a method of calibration, according to some embodiments of the disclosure.

[0159] An exemplary method is presented in this of the disclosure section. Further details (including optional details) regarding implementation of the method, nonbinding theory behind the method, and derivations associated with the method are presented after description of the method with respect to FIG. 2.

[0160] At 201, in some embodiments, radar measurements are received in a form of a plurality of data vectors. Where each data vector corresponds to a RD cell of a RD map.

[0161] In some embodiments, the receiving includes acquiring of radar measurements, which includes optionally (e.g., for active radar) emitting radar signals e.g., by a radar antenna array (e.g., radar array 102 FIG. 1A).

[0162] In some embodiments, acquiring of the measurements includes acquiring a plurality of data snapshots using the radar antenna array, where the snapshots include information regarding unknown targets. Where there may be an unknown number of targets each having, for example, unknown range (position with respect to position of measurement), DOA, and Doppler shift (radial velocity).

[0163] In some embodiments, each snapshot array data vector includes signal data received by elements (e.g., all elements) of a radar antenna array (e.g., array 103 FIG. 1A) at a moment in time, or corresponding to a single pulse of a transmitted radar signal (e.g., for active radar). Where a snapshot may be a complex vector of size N where the array has A receiver elements (e.g., antennas).

[0164] In some embodiments, a plurality of snapshots are processed to provide a RD map including a plurality of RD cells.

[0165] In some embodiments, the snapshots are processed to produce a Range-Doppler (RD) map of a plurality of RD cells, each RD cell corresponding to a specific combination of range (distance) and Doppler shift (relative velocity) of a target. Where each cell may be considered to have a range bin and a Doppler bin, where the range bin refers to values of range which are allocated to the cell, and the Doppler bin, likewise for Doppler values.

[0166] Where the RD map may be obtained, for example, by applying a matched filter (or pulse compression) for range estimation and a Fourier transform for Doppler estimation.

[0167] To create an RD map, data collected over multiple snapshots may be processed. Each RD cell in the map may be considered to be a combination of contributions from multiple snapshots that correspond to the same range and Doppler shift - the data from a single snapshot contributing to multiple RD cells. In some embodiments, the number of snapshots correspond to (e.g., are the same) the number of RD cells of the RD map. In some embodiments, a single snapshot is used to provide each RD cell.

[0168] In some embodiments, each RD cell value may be modelled (e.g., see section “exemplary data model”) as a data vector (different elements of the vector corresponding to different targets appearing within the RD cell) which is a sum of a signal vector (which is adjusted by a steering matrix and a mismatch matrix) and a noise vector (see equation (11) herein below).

[0169] At 205, in some embodiments, an initial calibration matrix is received. In some embodiments, the initial calibration matrix may simply be the identity matrix.

[0170] At 207, in some embodiments, an autocorrelation matrix is determined for each data vector (e.g., see step 1104 FIG. 11). At 209, in some embodiments, the autocorrelation matrices are transformed using the calibration matrix (e.g., see step 1108 FIG. 11). Where, for an initial iteration of the iterative process of steps 209 - 219, the initial calibration matrix is used to perform the transformation.

[0171] At 211, in some embodiments, the transformed autocorrelation matrices are smoothed e.g., by application of SS and / or FB techniques (e.g., see step 1110 FIG. 11).

[0172] At 213, in some embodiments, likelihood functions are determined for the smoothed transformed autocorrelation matrices (STAMs) - e.g., likelihood functions corresponding to the RD cells (e.g., see step 1116 FIG. 11). Where, in some embodiments, for each STAM corresponding to a RD cell, a plurality of likelihood functions are determined, one for each of a plurality of hypothesized number of targets appearing at the RD cell in question.

[0173] In some embodiments, the likelihood functions are penalized. Where a penalty term is incorporated into the likelihood function. The penalty term may be selected to compensate for the number of hypothetical targets, as, the likelihood function may increase with respect to the number of targets. The penalty term may also be related to size and number of subarrays used during smoothing.

[0174] At 215, in some embodiments, for each of the cells, a single likelihood function, corresponding to a minimal number of hypothesized targets is selected, e.g., see equation (29).

[0175] At 217, in some embodiments, a cost function is computed as a sum of these likelihood functions corresponding to minimal numbers of targets, one selected likelihood function per RD cell (e.g., see step 1118 FIG. 11).

[0176] At 219, an optimization process is performed to provide a calibration matrix which minimizes the cost function computed at step 217. Where, steps 209 to 217 may be repeated for different calibration matrices.

[0177] In some embodiments, optimization includes searching over a parameter space to find the optimal calibration matrix. In some embodiments, the parameter space may be confined. In an exemplary embodiment the parameter space is chosen to allow small perturbations around the identity matrix. For example, reflecting practical radar antenna situations where the mismatch matrix D is assumed to be close to the identity matrix.

[0178] In some embodiments, the parameter space is confined with upper and / or lower bounds. Where, in some embodiments, the bound / s may be defined based on assumptions regarding physical behavior of the radar system. For example, where it may be assumed that mutual coupling tends to decrease as the distances between sensor elements (e.g., antennas) increase.

[0179] It should be noted that, in some embodiments, the method is performed with subset of available data e.g., with data corresponding to a subset of RD cells. Where the cost function may be determined as a sum of likelihood functions associated with a subset of a total number of RD cells.

[0180] In some embodiments, the subset of RD cells is selected according to range (e.g., range bins) of the cells. Where, in some embodiments, short ranges (e.g., comparatively) may be used (e.g., only). A potential benefit of using short ranged RD cells is that the signal to noise ratio (SNR) of the data of the cells is potentially higher.

[0181] Alternatively or additionally, in some embodiments, the subset of RD cells is selected according to energy levels of the RD cells e.g., were those cells having low energy (e.g., comparatively) may be discarded.

[0182] Exemplary data model

[0183] In this section, a data model is first described, including model assumptions, to state a problem mathematically.

[0184] An example calibration method presented in this application can be used in signal processing for both passive and active arrays. Derivations are initially for active radar systems, and then that the exemplary method is applicable to passive radar systems (e.g., as well as to active systems).

[0185] Derivation (e.g., of a likelihood function) is under the assumption of Gaussian noise. In some embodiments, where noise is non-Gaussian, the derived likelihood function (termed a “discrepancy function”) may still be applied, to provide a calibration matrix. In this document, although description is generally regarding a likelihood function, it should be understood that the same mathematical function may be used to process radar data (e.g., in in the presence of non-Gaussian noise where it may be considered to be a discrepancy function.

[0186] In some embodiments, an exemplary radar array and associated circuitry (e.g., radar array 102 circuitry 111 FIG. 1A) is a single-input multiple-output (SIMO) radar, transmitting signals with a transmit baseband signal and a carrier frequency a>c. Referring back to FIG. 1A, an echo signal is received by URA 103 which has N = Nxx Nysensors with spacing the x-axis and y-axis, respectively, where the array response for a target 105 at azimuth Φ and elevation θ can be modeled as:

[0187] Where is wavelength and c is propagation speed e.g., of the radar source.

[0188] The complex function represents an antenna pattern, which is assumed, in the following derivations, to be identical for all the antenna elements 107.

[0189] A received data vector at the array after down-conversion to baseband can be modeled as:

[0190] Where Q is a total number of targets, and the q-th target complex amplitude, azimuth, elevation, range delay, and Doppler frequency shift.

[0191] A two-way range-delay and Doppler frequency shift are related to a corresponding target range, rq, and radial velocity vrqas follows:

[0192] The following assumptions are made to obtain this model:

[0193] Al) Far-field - this assumption allows removing a dependency of the steering vector on target range, and it will be used in the sequel.

[0194] A2) This assumption assures that the baseband signal does not significantly change along the observation time due to target motion and is used to avoid range migration. where Lqis an array aperture - This assumption is used to assure that the baseband signal is substantially identical along the array aperture, for signal bandwidth, W.

[0195] A detailed derivation of such a model under such assumptions can be found for example in J. Tabrikian, “Performance bound and techniques for target localization using MIMO radars ” in “MIMO Radar Signal Processing” . Hoboken, NJ: John Wiley & Sons, Inc., 2008. Conventional delay-Doppler processing methods employ matched- filtering on a received signal to produce a delay-Doppler map.

[0196] In the example method described, a maximum-likelihood (ML) estimator and a generalized likelihood ratio test in the presence of additive white Gaussian noise are implemented via matched-filtering, given by: where is a delay-Doppler ambiguity function, defined as

[0197] In cases where the transmit signal is a sequence of chirps e.g., Frequency- Modulated Continuous Wave (FMCW) chirps , a matched filtering operation in equation (5) may be performed using a two-dimensional fast-Fourier transform (FFT).

[0198] Let denote the m-th RD cell in the two-dimensional sampled delay- Doppler map (or RD map). Where each RD cell is a point in (or portion of) the map corresponding to a specific combination of range and Doppler shift (or ranges and Doppler shifts corresponding to bin sizes thereof), a value of the cell representing the measured signal strength or energy for that particular range and Doppler shift (or ranges and Doppler shifts).

[0199] Then, after sampling, equation (5) can be rewritten as:

[0200] At each RD cell, a small part of one or more targets may be present, since may be small for targets which are far away from a considered

[0201] RD cell. Let Amand (<pkm,0km) denote a number of targets and an azimuth and elevation of a k- th target at the zzz-th RD cell respectively.

[0202] A data vector collected by the array at the m-th RD cell can be modeled by:

[0203] Where data vector xmis a complex vector containing the received signals (actual measurements acquired by the array at the zzz-th RD cell) at each array element. This vector is a sum of the contributions from all the steering directions plus any noise in the system.

[0204] Where is a signal vector of complex amplitudes of the targets in the zzz-th RD cell.

[0205] The sequences (signal vector) and (noise vector) are assumed to be white and statistically independent, with distributions:

[0206] The matrix denotes a matrix of steering vectors (e.g., corresponding to directionality of the signals received at the radar array) in the zzz-th RD cell, whose columns are linearly independent:

[0207] Where, for K targets and m array elements, Am is an m x K matrix.

[0208] By observing equation (1) and equation (4), it can be verified that with no loss of generality, the antenna gain, can optionally be omitted, since it can be combined with aqwhich is assumed to be unknown.

[0209] Non-idealities of the array e.g., including one or more of MC, phase and gain mismatch for the different antenna elements, is now modeled as a DOA-independent linear transformation of steering vectors D. The data model in equation (9) is modified to:

[0210] Where the mismatch matrix includes deviations of measurements from the assumed model e.g., including one or more of the effect of MC, gain and / or phase errors compared to the assumed model.

[0211] This matrix, D referred to as MCM (mutual coupling matrix) “coupling matrix”, and “mismatch matrix”, is assumed to be unknown and is considered to be common among the different RD cells (e.g., for active arrays) and / or time frames (e.g., for passive arrays).

[0212] The model in (11) holds also for passive arrays, where the data sets can be collected over M time frames with varied scenarios and the m-th data set contains Kmnarrowband sources in the far-field.

[0213] In the above model, we considered point targets. In many cases, such as in automotive radars, one or more targets may be distributed in space. A distributed target can be represented by several discrete reflections, which may be spread in one or more of range, Doppler, and DOA. In radars having large bandwidth, the range bin of RD cells may be small, and thus, the distributed target can span multiple cells in the RD map. The small range bin reduces the possibility of target angular spread for each RD cell, but such cases may occur, especially in short ranges. Target distribution over range and Doppler increase the number of RD cells (range-Doppler bins), M, which contain targets. Target distribution in DOA results in increasing the number of targets per range- Doppler bin, Km. At each RD cell, distributed targets can be represented by multiple close point targets, and therefore, the model in (11) covers also the case of distributed targets, where the number of columns in A??? is increased. This point will be further discussed in the next section.

[0214] A goal of the described blind calibration method is to calibrate the array by finding a transformation, which cancels the effect of the mismatch matrix D by using multiple RD cell measurements. Each RD cell has unknown Km far-field targets with unknown directions and unknown powers (i.e. the matrices are unknown). Joint estimation of the target parameters over multiple RD cells and the mismatch matrix involves a multidimensional optimization problem. Herein, a method is presented that optionally avoids estimating target parameters at each RD cell. Optionally, the example method aims instead to find a calibration matrix that minimizes a cost function based on a nonparametric model order selection criterion.

[0215] Exemplary idea and motivation

[0216] In the present section, an idea behind a proposed example method is described. It is hypothesized that far-field stationary signals in a homogeneous propagation channel without multipath produce a spatially stationary field. Generally, having a spatially stationary field is exploited by the SS algorithm, which averages the autocorrelation matrices of subarrays with identical geometry and orientation to increase the rank of (number of signal sources also termed “targets” in) a received signal autocorrelation matrix (see, e.g., above-mentioned T.-J. Shan, M. Wax, and T. Kailath, “On spatial smoothing for direction-of-arrival estimation of coherent signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 33, no. 4, pp. 806-811, 1985).

[0217] In some embodiments, SS and / or FBA techniques are used as a preprocessing stage before using eigenstructure-based techniques that include identifying signal and noise subspaces.

[0218] Generally, SS and FBA techniques are be implemented for uniform planar arrays if the array elements have identical response without mutual coupling, i.e.,

[0219] For simplicity of notation, in the following, an objective cost function is derived based on a single RD cell and the cell index, m is omitted. The cost function will then be extended to multiple RD cells.

[0220] The autocorrelation matrix of the single RD cell received data x is given by

[0221] For SS, let xG = Gx, 7= denote the received data of the 7-th subarray of size Nsub, where G / denotes the sampling matrix from the full array to the 7-th subarray and L is the number of subarrays. For example, in the case of a and

[0222] It is noted that Nsub is a size of a subarray of a full radar array. The value of Nsub can optionally be any value from 2 up to a full size of the radar array (N elements).

[0223] The autocorrelation matrix of the 7-th subarray is given by:

[0224] The autocorrelation matrix after SS is given by:

[0225] Another technique for increasing the signal autocorrelation rank is FBA. The forward-backward averaged autocorrelation matrix is given by: where J / vis the exchange matrix of size N.

[0226] The SS and FBA techniques can be combined in order to further increase the rank of the signal autocorrelation matrix as follows:

[0227] Proposition 1. In the presence of K point targets, without MC, phase, or gain

[0228] Proof. By substitution of (12) into (16), without MC, phase, or gain errors, where D = I / v, we obtain: where we use the fact that jNsub and Gi are real matrices. It can be verified that GiA = GiAD, where I) is a diagonal matrix, that is, the columns of GiA do not rotate when moving from one subarray to another. Substitution of this relation into (17) yields

[0229] With a symmetric array, by choosing the array origin at the center of the first subarray, we obtain can be expressed as:

[0230] The “real” operation together with summation over L elements increases the rank of the signal component in the smoothed autocorrelation matrix by a factor of up to 2L, but it is limited by the size of the rank of the signal component Rs, which is the number of targets, K.

[0231] In the presence of MC or phase / gain errors, where D is not proportional to the identity matrix, the columns of do not stay co-linear when moving from one subarray to another, and therefore, the rank of the signal component of the autocorrelation matrix can increase to min In other words, the energy of the signal leaks into the noise subspace. In order to illustrate this phenomenon, consider a passive scenario with a linear array of N = 16 sensors with half a wavelength inter-element spacing. The sensor array measures single-snapshot narrowband signals from K = 2 far-field sources at directions -10°, 20°. Additive white Gaussian noise with unit variance was added to the measurements and the signal-to-noise ratio (SNR) for the two sources was set to subarrays of size Nsub = 9 was applied to evaluate the smoothed autocorrelation matrix The MCM is set similarly to the model presented in M. V. M. Lanne and A. Lundgren., Chapter 3: Calibration in Array Processing. Elsevier Inc., 2009, t. E. Tuncer and B. Friedlander, In Classical and Modern Direction-of-Arrival Estimation: where the elements of are generated as independent complex Gaussian random variables with Toeplitz statistics:

[0232] Where dpqdenotes the distance in half-a-wavelength units between the elements p and q of the array. The gain and phase errors of each sensor element is defined as:

[0233] Where and cp is a zero-mean Gaussian random variable with variance The gain and phase error standard deviations in the numerical simulations are defined as

[0234] Reference is now made to FIG. 3A, which presents eigenvalues of a smoothed autocorrelation matrix for two cases, with and without MC.

[0235] FIG. 3A is a graph with a qualitative x-axis 112 showing individual eigenvalues and a y-axis 114 showing a relative qualitative value of the eigenvalues. FIG. 3A shows eigenvalues for a case without mismatch 118 between an array calibration matrix and an array reception, in dark grey, on the left, and eigenvalues for a case with mismatch 120 between the array calibration matrix and the array reception, in light grey, on the right, where in both cases there are two targets present. The term mismatch is used herein to mean a mismatch between a true model which may include one or more of MC error and / or phase error and / or gain error, and an assumed computational model which does not include the above-mentioned errors. FIG. 3A shows eigenvalues of the autocorrelation matrix after SSFB with a single snapshot, for a ULA of N= 16 elements, and a subarray size of Nsub = 9 sensors in the presence of two far-field point sources at — 10 20°, and SNRi = SNR?. = 20dB.

[0236] FIG. 3A shows that in the absence of MC, where there is little or no mismatch, the noise subspace eigenvalues are approximately equal to or lower than the noise variance 116. The presence of MC, where there is mismatch 120, results in much higher eigenvalues of the noise subspace 116, reflecting spectral leakage from the signal subspace to the noise subspace. This phenomenon apparently stems from the effect of SSFB on the autocorrelation matrix, whose signal subspace is rotated due to the MC.

[0237] As mentioned in the previous section, the model in (11) can cover also a case of angular distributed targets, where the number of columns of A™ increases. The problem of distributed sources has been investigated in the literature (see, e.g. S. Valaee, B. Champagne, and P. Kabal, “Parametric localization of distributed sources,” IEEE Transactions on Signal Processing, vol. 43, no. 9, pp. 2144-2153, 1995). Distributed sources can be modeled by a large number of point sources, as modeled in the previous section, but naturally, it is impractical to estimate the DOA’s of these point sources.

[0238] However, it is possible to estimate the center and the width of each source. In D. Astely andB. Ottersten, “The effects of local scattering on direction of arrival estimation with music,” IEEE Transactions on Signal Processing, vol. 47, no. 12, pp. 3220-3234, 1999, it was shown that the spatial transfer function of distributed sources can be approximated by a linear combination of the point- source steering vector and its derivative, implying that it spans a 2-D subspace. In J. Tabrikian and H. Messer, “Robust localization of scattered sources,” in Proceedings of the Tenth IEEE Workshop on Statistical Signal and Array Processing (Cat. No.00TH8496), 2000, pp. 453-457 it was shown that angular spread causes extension of the signal subspace, where the size of the subspace depends on the source spread size.

[0239] We expect that, with distributed sources, the signal subspace size will increase compared to the case of point sources. Presence of MC is expected to further increase the signal subspace. However, in the case of MC, the signal subspace leaks into the original noise subspace.

[0240] In order to illustrate this issue, we consider a similar example as in FIG. 3A, but where each of the sources is composed of 5 point scatterers with 0.5° separation. Reference is now made to FIG. 3B, which presents eigenvalues of a smoothed autocorrelation matrix for two cases, with and without MC.

[0241] FIG. 3B is a graph with an x-axis 212 showing individual eigenvalues At, , and a y-axis 214 showing the relative qualitative value of the eigenvalues. FIG. 3B shows eigenvalues for a case without mismatch 218, in dark grey, on the left, and eigenvalues for a case with mismatch 220, in light grey, on the right, where in both cases there are two targets present.

[0242] FIG. 3B shows eigenvalues of the autocorrelation matrix after SSFB with a single snapshot, for a ULA of N= 16 elements, and a subarray size of Nsub = 9 sensors in the presence of two far- field distributed sources with centers at -10°, 20°, widths of Each distributed source includes 5 scatterers with 0.5° spacing.

[0243] It can be seen that the target distribution slightly enhances the signal subspace. For example, the third and fourth eigenvalues in the case of point target are around the noise level, while they are slightly above the noise level in the case of distributed targets.

[0244] The increase of the noise subspace eigenvalues implies that the assumed model does not match the data, resulting in the reduction of the likelihood function. Following this observation, we propose to transform the autocorrelation matrix in (12) by left and right multiplication by matrices respectively: where T is a hypothesized calibration matrix. For ψ which perfectly cancels the MC, i.e. V = D-1, potentially providing a MC-free autocorrelation matrix, where the signal subspace does not leak into the noise subspace.

[0245] From (23), it can be seen that since Rs is unknown, the normalization factor of ψ cannot be determined. Thus, with no loss of generality, we can normalize V, such that its determinant is equal to one, i.e. Thus, the transformation in (23) can be redefined as P4)

[0246] If is unknown, the transformation can be performed without the noise correction in (23). This solution will lead to spatial coloring of the noise, since the noise autocorrelation matrix after the transformation will be Noise coloring may cause a slight performance reduction of the proposed algorithm in the case of low SNR, where accuracy of identification of the noise subspace may be reduced. This is not a critical point, since at low SNRs the effect of array misspecification is smaller. Practically, for example, in calibration of automotive radar, in some embodiments, short ranges may be used (e.g., only), in which the SNR may be higher (e.g., in comparison to long ranges). Additionally or alternatively, RD cells with higher energy may be used (e.g., only).

[0247] In the next section, we present a model order selection based approach for MC cancellation, where it is proposed to find by minimizing the model order which is equivalent to maximizing the (evaluated) size of the noise subspace. Where maximizing the (evaluated) noise subspace may be considered equivalent to minimizing the evaluated signal subspace (or leakage of the evaluated signal subspace into the true noise subspace.)

[0248] It is noted that if ψ is not constrained, it may be orthogonal to the signal subspace or to part of it. Thus, V which minimizes the model order, may cancel the signal subspace. This can be solved by either setting some constraints on the rank of V or by considering measurements from multiple RD cells or time frames with different scenarios of targets, but with the same MCM. Such a setting can be helpful, since the same is not orthogonal to the steering vectors of multiple DO As. Hereinbelow this problem will be addressed using an iterative least square (iterative LS) approach.

[0249] Exemplary subspace-based blind calibration method

[0250] In automotive radar systems, many targets (or clutter echoes) may exist in different RD cells. It is assumed that the coupling matrix D is identical for all the RD cells. Based on this assumption, in some embodiments, multiple RD cells are used to determine the coupling matrix.

[0251] The likelihood function for estimating D with different RD cells can be stated based on the model in (11). However, generally, for such a purpose, the number of targets in each RD cell, is known. Even if the number of targets is known, the number of unknown parameters in this problem can be huge, since unknowns include all the target DO As at the considered RD cells in addition to the parameters in D. Accordingly, a nonparametric approach is adopted for the target spatial signature.

[0252] Starting with a single RD cell without MC and / or other non-idealities (i.e. D = ITV) and deriving the log -likelihood function after SSFB, where it is assumed that the samples used for estimating the autocorrelation matrix using SSFB are independent and identically distributed (i.i.d.) . In practice, the statistical independence assumption does not hold, which implies that we ignore information in the model due to the statistical dependency in the snapshots. In fact, such an information loss may result in a performance loss due to a smaller subarray aperture size.

[0253] Assume that under the k- th hypothesis, H / <, there are ^targets, where k= G,...,Nsub — 1. Let ULSSFB inc lude the concatenation of 2L i.i.d. measurements from the 2L subarrays of Nsub elements (including the backward subarrays), and let stand for the corresponding sample autocorrelation matrix. The matrixcanbe computed by substituting the matrix Rxin (16) with its sample estimation, which in the case of a single snapshot is given by

[0254] For a nonparametric target enumeration method, the list of unknown parameters for the case of k targets is given by M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoust., Speech, Signal Process., vol. 33, pp. 387-392, 1985 where are the eigenvalues and the corresponding eigenvectors of RXSSFB in(16), respectively. Let denote the eigenvalues

[0255] Then, the log -likelihood under hypothesis, Hk, after maximization w.r.t. the unknown parameters Qi. is given by above-mentioned M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoust., Speech, Signal Process., vol. 33, pp. 387-392, 1985: where the ML (maximum likelihood) estimate of the noise variance is given by and const is independent of the hypothesis index, k. If the noise variance is a- priori known, then it can be used in (27) instead of its estimate in (28).

[0256] For determining the number of targets, in some embodiments, a penalty term is added. Where the penalty can be chosen according to one or more feature of the criteria surveyed in P. Stoica and Y. Selen, “Model-order selection: a review of information criterion rules,” IEEE Signal Process. Mag., vol. 21, no. 4, pp. 36-47, 2004. The penalty P(k) may depend on the number of independent, unknown real parameters under each hypothesis.

[0257] In above-mentioned P. Stoica and Y. Selen, “Model-order selection: a review of information criterion rules,” IEEE Signal Process. Mag., vol. 21, no. 4, pp. 36-47, 2004 it was shown that the number of independent parameters in

[0258] Using the minimum description length (MDL) criterion, the penalty depends on the hypothesized number of targets, and the size and number of subarrays (e.g., in SSFB processing) is given by

[0259] By adding the penalty term to the negative of the maximized log-likelihood function in (27), the criterion for determining the number of targets is given by:

[0260] From (30) it can be seen that leakage of the eigenvalues from noise subspace to the signal subspace results in reduction of the maximized likelihood function w.r.t. k. Thus, in the proposed optimization framework, we search for a transformation ψ applied to the sample autocorrelation matrix according to (23), such that the corresponding log -likelihood after SSFB, is maximized. Specifically, an optimal calibration matrix, ψ is found as follows: where is obtained by substitution of in (25) with: As mentioned previously, such an optimization may also cancel the signal if the calibration matrix is orthogonal to the signal subspace. However, in the case of multiple RD cells with sufficiently large number of targets at different DOAs, the same (nonzero) V is not orthogonal to all the target steering vectors. Thus, in some embodiments, the calibration matrix is ψ optimized using multiple RD cells, by summing the contribution of the objective function (the likelihood function) from each RD cell after minimization w.r.t. the number of targets: where is a smoothed autocorrelation matrix in the m-th RD cell after applying the candidate calibration matrix, Y:

[0261] As mentioned previously, (for example, in the case of unknown noise variance) (35) can be computed by assuming o^ = Q.

[0262] A potential benefit of use of multiple RD cells is robustness of the solution and potential minimization of overfitting. The proposed optimization problem aims to find Ψ which restores the spatial stationarity in the smoothed spatial autocorrelation matrix, by removing the leakage of signal subspace eigenvalues into the noise subspace. This algorithm is based on targets-of-opportunity without any assumption on the prior knowledge of number of targets and their directions or intensities. Thus, the proposed algorithm can be considered as “blind”.

[0263] The optimization in (32) is not convex. Thus, when using conventional optimization methods, one may optionally confine the search over the parameter space Ωψ in order to avoid local minima. In exemplary practical cases (e.g., use-cases), the MCM, D, may be close to the identity matrix, and thus, Ωψ can be chosen to allow small perturbations around the identity matrix. Moreover, even though the MCM is random, a general physical understanding of its behavior may be assumed. For instance, in linear arrays, it may be assumed that the mutual coupling tends to decrease as the distances between sensor elements (e.g., antennas) increase. In some embodiments, is chosen based on such prior information. In general, Ωψ depends on lower and upper bounds as follows: where LB and UB are pre-defined complex lower bound and upper bound matrices, respectively, and the inequalities hold for real and imaginary parts, separately.

[0264] -based Blind Calibration Scheme M RD measurement vectors (11). calibration matrix.

[0265] - Optional - Lower and upper bound matrices, (LB, UB).

[0266] - Noise variance, o^ If o^ isunknown, set

[0267] - Compute the penalty function for k= G,...,Nsub— 1: t: Calibration matrix

[0268] Procedure:

[0269] 1: Compute

[0270] 2: Minimize the following function using an iterative algorithm, optionally an algorithm such as an fmincon solver with initialization and lower and upper bound matrices LB, UB: a) Compute for m = 1, M. b) Apply SSFB applying SSFB (spatial smoothing (SS) and forward-backward averaging (FBA) techniques) on c) Obtain the eigenvalues of and corresponding estimated noise variance for hypotheses d) Compute: e) Compute: end procedure

[0271] In some embodiments, computational complexity of exemplary algorithm 1 is mainly determined by steps 2.a)-2.c). These steps may be performed at each iteration of the optimization algorithm.

[0272] Steps 2. a and 2.b involve matrix multiplications while step 2.c involves M singular-value-decomposition of size Nsub. Accordingly, the computational complexity of the algorithm at each iteration is of the order 0(MN3sub). In some embodiments, the calibration process is not performed in real time. A potential benefit being that, in some embodiments the calibration process is performed, at least partially, by a remote processor (e.g., PCM 119 FIG. 1A) e.g., potentially reducing processor requirements for a mobile radar system 101. In some embodiments, the calibration process is performed intermittently, e.g., under an assumption that MCM variations may be slow, where the calibration can be performed periodically with long periods, e.g., potentially reducing impact of computation time of the method.

[0273] FIG. 11 is a simplified flow chart of a method for subspace-based blind calibration for multi-element radar, according to some embodiments of the disclosure.

[0274] The method of FIG. 11 includes: providing initial values (1100), by: providing M measurement vectors where D is a coupling matrix, Am is a matrix of steering vectors, smis a signal vector and vmis a noise vector; providing an initial calibration matrix providing a noise variance value and computing a penalty function where Nsub is a size of a subarray of the radar array being calibrated: computing M estimated matrices where is a conjugate transpose of minimizing a cost function C('P) by iteratively calculating a) to f) below until a stop condition is reached (1106): a) computing for to transform estimated autocorrelation matrices using a calibration matrix (e.g., using the initial calibration matrix during a first iteration where the calibration matrix may be adjusted between iterations); c) calculating eigenvalues d) obtaining corresponding estimated noise variances for hypotheses

[0275] Nsub - 1 by calculating: e) Calculating f) computing thereby obtaining a calibration matrix ψ (1120).

[0276] In some examples, the penalty function is where L is a number of subarrays.

[0277] In some examples, providing the noise variance value includes setting 0.

[0278] In some examples, providing the M measurement vectors includcs providing M Range-Doppler (RD) measurement vectors

[0279] In some examples, the minimizing the cost function C(ψ) optionally includes using a fmincon solver with the initial calibration matrix and a LB matrix and an UB matrix, optionally provided as initialization values as part of the providing initial values (1100).

[0280] In some examples the stop condition includes performing no more than a specific number of iterations.

[0281] Exemplary numerical demonstrations

[0282] In this section, simulations are used to demonstrate performance of the above- mentioned example blind calibration algorithm.

[0283] Performance of example methods is evaluated by simulations, in terms of DOA estimation accuracy and resolution. Results hereinbelow, apparently, demonstrate that the DOA estimation performance after calibration with the example methods is close to that of a perfectly calibrated array.

[0284] A case of ULA is first presented, where ULA may be considered to be is a special case of UPA. Then, results with UPA are presented.

[0285] In the ULA case, the proposed method is compared with the mismatch-free model and the uncalibrated model consisting of MC and sensor phase / gain errors. Apparently, unlike classical array calibration methods (see, e.g. Z. Zheng, K. Liu, W. Q. Wang, Y. Yang, and J. Yang, “Robust adaptive beamforming against mutual coupling based on mutual coupling coefficients estimation,” IEEE Transactions on Vehicular Technology, vol. 66, no. 10, pp. 9124-9133, 2017), which are usually based on multiple snapshots of a given set of targets, we consider an automotive radar case, which includes multiple RD cells with a single snapshot at each cell, and with different and unknown number of targets and power levels. To compare the proposed calibration method, we implemented an iterative algorithm for array calibration with similar attributes as in above-mentioned Z. Zheng, K. Liu, W. Q. Wang, Y. Yang, and J. Yang, “Robust adaptive beamforming against mutual coupling based on mutual coupling coefficients estimation,” IEEE Transactions on Vehicular Technology, vol. 66, no. 10, pp. 9124-9133, 2017.

[0286] The compared method is a coordinate-descent-like algorithm, described as follows: Estimate the DO As and the corresponding steering matrices and signal vectors at each RD cell, The MCM can be estimated by least- squares (LS), i.e. by minimizing is the Frobenius norm, and A = This objective function is minimized by D = The inverse of this matrix, is used to calibrate the data covariance matrix using (23), from which the target parameters can be estimated. This procedure can be performed iteratively till a convergence criterion is satisfied. A summary of the conventional method is presented in Algorithm 2.

[0287] Algorithm 2: Iterative LS Calibration

[0288] Input: - RD measurement vectors (11). Initialized solution.

[0289] Output: Calibration matrix ψ .

[0290] Procedure:

[0291] 1: Compute

[0292] 2: Compute for

[0293] 4: Estimate the DOAs for all RD cells using an eigen-structure-based technique, such as, by way of a non-limiting example, MUSIC (Multiple Signal Classification), and construct the estimated steering matrices, {Am}, for each RD cell using (10).

[0294] 5: Estimate the signal vector at each RD cell:

[0295] 6: Compute

[0296] 7: Repeat steps 2-6 till convergence or a stop condition is satisfied. end procedure

[0297] Consider a SIMO radar with ULA of N = 16 sensors with half a wavelength inter-element spacing. For the SS preprocessing, we set the subarray size to be Nsub = 9, resulting in L = N— Nsub+ 1 = 8 subarrays. Then, FBA doubles the number of sample measurements, i.e., 2L = 16. The numerical simulations described below include a data set for array calibration in multiple RD cells, referred to as training data, and a validation set for performance evaluation. In the calibration stage, a calibration matrix V is found, and in the validation stage, the calibration matrix is used by the DOA estimation algorithm. For the calibration process, we generate M = 300 RD cells, where at each cell, the following parameters were independently randomized with uniform distribution: number of targets NRs The data vectors were multiplied by the MCM, D, which was randomized based on a) unstructured, and b) Toeplitz symmetric models. In the unstructured case, the MCM was randomized according to the model described in (20)- (22), where the model parameters were set to: and Δ = 0.5. In the case of Toeplitz symmetric MCM, the first column of the MCM was set by taking the first column of the unstructured MCM matrix, with Δ = 0.9, and the main diagonal was set to one. The complex upper and lower bounds for estimating by the proposed algorithm were set as follows: the off-diagonal of both unstructured and Toeplitz symmetric MCM cases are:

[0298] The upper and lower bounds for the diagonal of Y were set to: while for the Toeplitz symmetric MCM, the diagonal elements are equal to one. DOA estimation performances of the MUSIC algorithm with SSFB preprocessing were evaluated and compared under the following cases: 1) proposed algorithm with and without prior knowledge of the noise variance 2) MC-free, 3) uncalibrated, and 4) calibrated using the iterative LS. The performances were evaluated in terms of root-mean-squared-error (RMSE) and probability of resolution, Pres, between two sources. The RMSE for the case of two targets is computed by the root of mean of the MSE for the two targets. The probability of resolution is defined as the probability of the event that the absolute estimation errors of both target DOAs are lower than e = 0.3 ° . The RMSE and Pres were evaluated using 2500 realizations produced with 25 MCMs, with 100 noise realizations per MCM. A. ULA with Unstructured MCM

[0299] In this subsection, we refer to the case of unstructured MCM, which is generated based on the model described in (20)-(22).

[0300] Reference is now made to FIG. 4, which illustrates MUSIC spatial spectra after SSFB (spatial smoothing (SS) and forward-backward averaging (FBA) techniques) for examples described above.

[0301] FIG. 4 is a graph with an x-axis 312 showing DOA in degrees and a y-axis 313 showing spatial spectrum in units of dB. FIG. 4 shows MUSIC spatial spectrum in the presence of two targets at with SNRs: 30dB and lOdB, respectively.

[0302] FIG. 4 illustrates spatial spectra lines for the following cases: MC free 314 (solid black line), uncalibrated 315 (solid red line), proposed example spectrum with unknown 316 (solid green line), and spectrum following iterative LS 317 (solid line).

[0303] FIG. 4 illustrates MUSIC spatial spectra after SSFB for the considered cases detailed above, in the presence of two targets located at -30° and -24°, with SNRs of 30dB and lOdB, respectively.

[0304] It is noted that there are two distinct peaks in the MC-free case 314. The localization performance is degraded in the presence of MC, which the other three cases represent. In the uncalibrated case 315, both peaks are lower and the peak corresponding to the target at -24° is obtained at a wrong location. It can be seen that the example proposed algorithm 316 is able to recover the two peaks, while the iterative LS algorithm 317 does not define the two peaks as well as the example proposed algorithm 316.

[0305] Reference is now made to FIG. 5A, which illustrates RMSE as a function of SNR for the case of a single target at DOA of 30°for examples described above. FIG. 5A shows a relation of DOA estimation RMSE, using MUSIC, in the presence of a single target at (p = 30 ° with ULA, and unstructured MCM.

[0306] FIG. 5A is a graph with an x-axis 412 showing SNR in units of dB and a y-axis 414 showing RMSE in units of degrees. FIG. 5 A illustrates a relation of error in detection (RMSE) to SNR for the following cases: MC free 416, uncalibrated 418, iterative LS 420 proposed example spectrum with unknown 422, and proposed example spectrum with a known 424. FIG. 5A presents RMSE as a function of SNR for the case of a single target at DOA of 30°. It is noted that in the uncalibrated case 418 the RMSE is stabilized at high SNRs, where the estimation errors due to miscalibration become dominant compared to the noise contribution, resulting in biased estimation. The proposed example methods 422, 424 are able to reduce the bias to 0.1°, but the bias does not vanish, which can be explained as follows: the calibrated array may result in some linear (small) phase errors on the elements. Since in the calibration stage the target DOAs are unknown, a small linear phase error, can be interpreted as approximately valid measurements of a calibrated array with DOAs which are slightly shifted, and thus, the proposed algorithm does not correct it. The linear phase error can result in a bias in DOA estimation. Another reason for the bias is explained as follows. During the calibration process, we rely on a finite number of RD cells, which results in some finite estimation error. In the DOA estimation stage with the calibrated array, this estimation error becomes dominant at high SNR’s, resulting in a biased estimation. In any case, such bias can optionally be removed by auto-calibration using a simple scheme as described in Section II-G in F. Engels, P. Heidenreich, M. Wintermantel, L. Stcker, M. Al Kadi, and A. M. Zoubir, “Automotive radar signal processing: Research directions and practical challenges,” IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 4, pp. 865-878, 2021, exploiting a geometric relation between measured angles and relative velocities w.r.t. the ego velocity.

[0307] In the next subsection, where we consider structured MCM, as commonly assumed in the literature, we see that the bias obtained by the calibrated array is much smaller. In addition, we can see in FIG. 5A that the performances of the proposed algorithm with (424) and without (422) prior knowledge of the noise variance are almost identical. The iterative LS method 420 provides poor performance, since it estimates the DOAs of the targets in the calibration stage based on an uncalibrated array, resulting in large errors in DOA estimation. In the iterative procedure, these large-error DOA estimates are used to calibrate the array, and thus this iterative approach does not provide a successful calibration algorithm.

[0308] Reference is now made to FIG. 5B, which illustrates RMSE as a function of SNR in presence of a single target at DOA of 30°for examples described above.

[0309] FIG. 5B shows two graphs with an x-axis 512 showing angular distance between two targets in units of degrees, a top graph having a y-axis 514 showing probability of resolution between the two targets at fixed SNR and a bottom graph having a y-axis 515 showing RMSE of DOA estimation in units of degrees. FIG. 5B illustrates probability of resolution and RMSE for the following cases: MC free 520, uncalibrated 522, iterative (least-squares) LS 524, proposed example method with unknown 526, and proposed example method with a known 528.

[0310] FIG. 5B shows Resolution performance of two targets with equal SNRs of 25dB, ULA and unstructured MCM. [a] - probability of resolution with resolution window of e = 0.3°. [b] - DOA estimation RMSE of the two targets.

[0311] FIG. 5B shows the resolution performance in terms of probability of resolution and RMSE of two targets as a function of their angular separation. The SNRs of the two targets are equal to 25dB. It can be seen that the MC severely degrades the probability of resolution. Even when the targets are far from each other, the probability of resolution is around 0.5. The proposed calibration approach is able to improve the resolution performance, although it is not fully recovered. We attribute this phenomenon to the bias term which is not fully removed, as explained in the case of single target. Cases where the estimation bias is greater than reduce the probability of resolution. The iterative LS calibration method exhibits lower performance compared to the proposed method.

[0312] B. ULA with Toeplitz Symmetric MCM

[0313] In this subsection, we evaluate the localization performance in the presence of MC, where the mismatch matrix, D, is Toeplitz symmetric, without sensor phase and gain errors. The justification for this model is that the coupling effect is a function of the distance between the sensors. In the case of a linear array, it is supposed to be a Toeplitz symmetric MCM B. Friedlander, “On the mutual coupling matrix in array signal processing,” 2020 54th Asilomar Conference on Signals, Systems, and Computers, pp. 1245-1249, 2020. The maximum number of iterations for the stop criterion of the algorithms (see algorithms 1 and 2) was set to 50.

[0314] Reference is now made to FIG. 6, which illustrates RMSE as a function of SNR for the case of a single target at DOA of 30° for examples described above.

[0315] FIG. 6 is a graph with an x-axis 612 showing SNR in units of dB and a y-axis 614 showing RMSE in units of degrees. FIG. 6 illustrates a relation of error in detection (RMSE) to SNR for the following cases: MC free 620, uncalibrated 622, iterative LS 624, proposed method with unknown 626, and proposed method with a known 628.

[0316] FIG. 6 shows a numerical study of a case where the MCM is structured.

[0317] FIG. 6 presents RMSE as a function of SNR for the case of a single target at DOA of 30°. FIG. 6 shows DOA estimation RMSE using MUSIC in the presence of a single target at =30° with UFA, and Toeplitz MCM.

[0318] Also, in the uncalibrated case 622, the RMSE is seen to stabilize at high SNRs, where the estimation error contains some bias due to miscalibration. It can be seen that the proposed method 626 628 reduces the bias significantly. Unlike the unstructured case shown in FIG. 5A, the proposed calibration method 626628 provides a much closer curve to the MC-free case 620. Here, we can see that the case of prior knowledge of the noise variance 628 slightly improves the performance of the proposed method. The improvement is evident in the small-error region. Back to the unstructured case shown in FIG. 5A, it can be seen that such a level of improvement due to the knowledge of noise variance is negligible compared to the bias. It is noted that in the unstructured MCM case shown in FIG. 5A, the prior knowledge of the noise variance did not improve the performance. Furthermore, we also see that the iterative ES approach 624 does not improve the performance compared to the uncalibrated case 622.

[0319] Reference is now made to FIG. 7, which illustrates the resolution performance of two targets with equal SNRs of 25dB.

[0320] FIG. 7 shows two graphs with an x-axis 712 showing angular distance between the two targets in units of degrees, a top graph having a y-axis 714 showing probability of resolution between the two targets and a bottom graph having a y-axis 715 showing RMSE of DOA estimation in units of degrees. FIG. 7 illustrates probability of resolution and RMSE for the following cases: MC free 720, uncalibrated 722, iterative (leastsquares) LS 724, proposed example method with unknown 726, and proposed example method with a known 728.

[0321] FIG. 7 shows resolution performance of two targets with equal SNRs of 25dB, ULA, and Toeplitz MCM. [a] - probability of resolution with resolution window of 0.3°. [b] - DOA estimation RMSE of the two targets. FIG. 7 illustrates resolution performance of two targets with equal SNRs of

[0322] 25dB. It can be seen that the MC case 720 also degrades the resolution performance here. Unlike the unstructured MCM case, shown in FIG. 5A, the proposed calibration method 726 728 achieves very close performance to the case of MC-free 720 in both close targets and distinct, further- apart targets. The iterative LS calibration approach 724 improves the performance but cannot obtain performance close to the MC-free case 720 or that of the proposed method 726728.

[0323] C. UP A with unstructured MCM

[0324] In this section, we show the calibration performance of the proposed method with UPA. In the previous two sections we found that prior knowledge of the noise variance does not significantly improve the performance of the algorithm. Accordingly, for simplicity of presentation, in this subsection we omit the case of known noise variance. In this test, we examine the performance of the algorithm in a more realistic scenario with distributed targets, similar to the scenario in S. Valaee, B. Champagne, and P. Kabal, “Parametric localization of distributed sources,” IEEE Transactions on Signal Processing, vol. 43, no. 9, pp. 2144-2153, 1995, and directional antenna elements, and a larger number of targets per range-Doppler cell.

[0325] Consider a SIMO radar with URA of Nx= Q,Ny= 4 sensors with half a wavelength inter-element spacing. For the SSFB preprocessing, we set the 2-D subarray size to be which yields subarrays. For the calibration process, cells were generated, where at each cell, the following parameters were independently randomized: number of targets:

[0326] It is assumed that the targets are distributed in azimuth such that the angular spread for each target was determined as follows: the target range and width were randomized according to respectively. Then, the target angular spread in azimuth was computed: The DOA’s of the target scatterers were determined uniformly over the target angles with a number of targets: The antenna patterns were set by

[0327] In this section, we do not present the results of the iterative LS -based method because it provided very poor performance. The reason for the poor performance is probably because here we use more targets at each RD cell (up to 10). Therefore, the probability of close targets is higher compared to the previous subsections in which up to 2 targets were assumed. Due to the limited resolution, in addition to the MC effect, the iterative LS algorithm provides poor results, which are used for the calibration stage.

[0328] Reference is now made to FIGs. 8A-8D, which compare 2-D beam patterns of an array for the three cases of MC-free, MC with an uncalibrated array, and MC after calibration.

[0329] FIG. 8A shows a zero-elevation cross-section through the 2D beam patterns.

[0330] FIG. 8A is a graph having an x-axis 812 of azimuth in degrees and a y-axis of antenna array gain of the beam pattern in dB, measured along a cross section through the 2D beam patterns at zero degrees elevation.

[0331] FIG. 8A shows relative power of the MC free case 816, the calibrated case 818, and the uncalibrated case 820.

[0332] FIGs. 8B, 8C and 8D show the 2D beam pattern using an x-axis 832 of azimuth in degrees, a y-axis 834, and color based on the color scale 836 shown on the right of FIGs. 8B, 8C and 8D.

[0333] FIG. 8B shows the beam pattern for the MC free case.

[0334] FIG. 8C shows the beam pattern for the uncalibrated case.

[0335] FIG. 8D shows the beam pattern for the calibrated case, calibrated by the proposed method described herein.

[0336] FIGs. 8A-8D show the 2D beam patterns in dB with a beam directed to =0°, Q =0° for MC free (FIG. 8B), MC without calibration (FIG. 8C), and MC with calibration (FIG. 8D). The lower images, Figs 8B-8D, show 2-D beam patterns, and the upper plot, FIG. 8A, presents the zero-elevation cut.

[0337] FIGs. 8A-8D compare the 2-D beam patterns of the array for the cases of MC- free (FIG. 8B), MC with uncalibrated array (FIG. 8C), and MC after calibration (FIG. 8D), where the upper plot, FIG. 8A, presents the zero-elevation cut.

[0338] It can be seen that the proposed calibration method is able to reconstruct the beam pattern with a good accuracy. Reference is now made to FIGs. 9A-9C, which compare beam patterns of an array for the three cases of MC-free, MC with an uncalibrated array, and MC after calibration of two point targets.

[0339] FIGs. 9A-9C show a 2D spatial spectra image using the MUSIC technique under URA configuration in presence of two targets with different SNRs.

[0340] FIGs. (A-9C compare the 2D DOA estimation image using MUSIC in the presence of two-point targets localized at -5, 5 deg, and with different SNRs.

[0341] FIGs. 9A, 9B and 9C show the 2D beam pattern using an x-axis 912 of azimuth in degrees, a y-axis 914, and color based on the color scale 916 shown on the right of FIGs. 9A, 9B and 9C.

[0342] FIG. 9A shows the beam pattern for the MC free case.

[0343] FIG. 9B shows the beam pattern for the uncalibrated case.

[0344] FIG. 9C shows the beam pattern for the calibrated case, calibrated by the proposed method described herein.

[0345] The DOA estimation performances using the three MC cases described above, were evaluated by examining the MUSIC spectrum in the presence of two point targets at azimuths: and zero elevation, with SNR’s: 20dB and 15dB. It can be seen that the presence of MC in the uncalibrated case (FIG. 9B) results in disappearance of the weaker target at while by using the proposed method (FIG. 9C), the peak is reconstructed. It should be noted that for the calibration stage, distributed targets were generated in a 1000 RD cell, however, for evaluating the calibration, we examine DOA estimation with a single RD cell and point targets.

[0346] Reference is now made to FIGs. 10A and 10B, which illustrate RMSE as a function of SNR for a case of a single target located at (p = 20 6 = 0 ° for examples described above.

[0347] FIGs. 10A and 10B illustrate DOA estimation RMSE with UPA, and unstructured MCM using MUSIC in the presence of a single target at 0.

[0348] FIG. 10A is a graph with an x-axis 1012 showing SNR in units of dB and a y- axis 1014 showing azimuth error, RMSE, in units of degrees.

[0349] FIG. 10B is a graph with an x-axis 1012 showing SNR in units of dB and a y- axis 1015 showing elevation error, RMSE, in units of degrees. FIGs. 10A and 1OB illustrate relations of error in detection (RMSE) to SNR for the following cases: MC free 1016, uncalibrated 1018, and calibrated by the proposed method 1020.

[0350] FIGs. 10A and 10B present RMSE as a function of SNR for a single target located at

[0351] It can be seen that the DOA estimation with the proposed calibration method 1020 significantly improves the performance compared to the uncalibrated case 1018. In elevation estimation, the RMSE stabilizes at high SNR’s due to the calibration estimation error, which implies biased estimation. A similar phenomenon occurs in azimuth estimation, but at higher SNR. The bias in such a case is smaller than the case of ULA, apparently since a larger number of RD cells were used in the calibration.

[0352] General

[0353] It is expected that during the life of a patent maturing from this application many relevant radar arrays will be developed and the scope of the term array is intended to include all such new technologies a priori.

[0354] The terms “comprising”, “including”, “having” and their conjugates mean “including but not limited to”.

[0355] As used herein, the singular form “a”, “an” and “the” include plural references unless the context clearly dictates otherwise. For example, the term “a unit” or “at least one unit” may include a plurality of units, including combinations thereof.

[0356] The words “example” and “exemplary” are used herein to mean “serving as an example, instance or illustration”. Any embodiment described as an “example or “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments and / or to exclude the incorporation of features from other embodiments.

[0357] The word “optionally” is used herein to mean “is provided in some embodiments and not provided in other embodiments”. Any particular embodiment of the disclosure may include a plurality of “optional” features unless such features conflict.

[0358] Unless otherwise indicated, numbers used herein and any number ranges based thereon are approximations within the accuracy of reasonable measurement and rounding errors as understood by persons skilled in the art. It is appreciated that certain features of the disclosure, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the disclosure, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable sub -combination or as suitable in any other described embodiment of the disclosure. Certain features described in the context of various embodiments are not to be considered essential features of those embodiments, unless the embodiment is inoperative without those elements.

[0359] Although the disclosure has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and broad scope of the appended claims.

[0360] Citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention. To the extent that section headings are used, they should not be construed as necessarily limiting. In addition, any priority document(s) of this application is / are hereby incorporated herein by reference in its / their entirety.

Claims

WHAT IS CLAIMED IS:

1. A method of calibration for a radar array comprising: receiving a plurality of measurement vectors, each measurement vector associated with a range-Doppler (RD) cell of a RD map; determining from said plurality of measurement vectors a plurality of autocorrelation matrices, an autocorrelation matrix corresponding to each said measurement vector and RD cell; performing an iterative process to minimize a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming said plurality of autocorrelation matrices with a hypothesized calibration matrix; smoothing said plurality of transformed autocorrelation matrices using one or more subarray technique to provide a plurality of smoothed autocorrelation matrices; determining a discrepancy function for each of said smoothed autocorrelation matrices and for each of a plurality of hypothesized numbers of targets, to provide, for each smoothed autocorrelation matrix, a plurality of likelihood functions, one for each hypothesized number of targets; selecting, for each of said smoothed autocorrelation matrices, a minimal likelihood function corresponding to a single hypothesized number of targets; and summing the selected minimal likelihood functions to provide said cost function.

2. The method according to claim 1, wherein said discrepancy function is a likelihood function derived under an assumption that noise is Gaussian.

3. The method according to claim 2, wherein said receiving comprises receiving measurement snapshot array data vectors where a single measurement snapshot is used to determine each said RD cell.

4. The method according to claim 1, wherein said smoothing is by applying one or both of spatial smoothing (SS) and forward-backward averaging (FBA) techniques.

5. The method according to any one of claims 1 to 4, wherein determining a likelihood function for each of said smoothed autocorrelation matrices for each of a plurality of hypothesized numbers of targets comprises calculating eigenvalues of the smoothed transformed autocorrelation matrix.

6. The method according to claim 5, wherein said likelihood function includes a sum of said eigenvalues.

7. The method according to claim 6, wherein said likelihood function comprises a penalty term selected to compensate for the specific number of hypothetical targets in the calculation.

8. The method according to claim 7, wherein said penalty term is related to size and number of subarrays used during said smoothing.

9. The method according to any one of claims 1 to 8, wherein determining a likelihood function for each of said smoothed autocorrelation matrices for each of a plurality of hypothesized numbers of targets comprises determining noise variances for each hypothesized number of targets, the noise variance forming a negative term in said likelihood function.

10. The method according to any one of claims 1-9, wherein said performing is until reaching a stop condition.

11. The method according to claim 10, wherein the stop condition is a number of iterations.

12. The method according to any one of claims 1-11, wherein said optimization is performed in a bounded parameter space.

13. The method according to claim 12, wherein the bounded parameter space includes perturbations around the identity matrix.

14. The method according to any one of claims 10-13, wherein said bounded parameter space is based on physical understanding of mutual coupling behavior.

15. The method according to any one of claims 1-14, comprising, preliminarily selecting a subset of said plurality of radar measurements.

16. The method according to claim 15, wherein said selecting is of cells having a SNR above a predefined SNR threshold.

17. The method according to any one of claims 15-16, wherein said selecting is of cells is of those cells having range below a predefined range threshold.

18. The method according to any one of claims 1-17, wherein the plurality of measurement vectors are received from an automotive mounted radar array system; and wherein, upon finishing said performing, the optimized calibration matrix is used to correct subsequent radar measurements.

19. The method according to claim 18, wherein said subsequent radar measurements are used in a navigation system of said automotive.

20. The method according to any one of claims 18-19, wherein said iterative process is performed online, during use of said automotive.

21. The method according to any one of claims 1-20, wherein said iterative process is re -performed periodically.

22. The method according to any one of claims 1-21, wherein said optimized calibration matrix is used to adjust a received calibration matrix.

23. The method according to claim 22, wherein said received calibration matrix has been determined offline in a controlled calibration environment.

24. A method of radar array calibration comprising: receiving a plurality of radar measurements; performing an iterative process minimizing a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming said plurality of radar measurements by applying a hypothesized calibration matrix to each of said plurality of radar measurements; smoothing each transformed radar measurement using one or more subarray smoothing technique; determining, for each smoothed transformed radar measurement, a minimal target likelihood function corresponding to a minimal number of targets, to provide a plurality of minimal target likelihood functions, one for each of said plurality of radar measurements, where calculation of the likelihood function is formulated to reflect separation between signal and noise subspaces; summing said plurality of likelihood functions to provide said cost function.

25. The method according to claim 24, wherein said smoothing is by applying one or both of spatial smoothing (SS) and forward-backward averaging (FBA) techniques.

26. The method according to any one of claims 24-25, wherein said plurality of radar measurements include a plurality of autocorrelation matrices; wherein said receiving comprises receiving a plurality of measurement data vectors, each associated with a range-Doppler (RD) cell; and computing said autocorrelation matrices, an autocorrelation matrix computed for each measurement data vector and associated with a RD cell.

27. The method according to any one of claims 24-26, wherein said determining comprises, for each said smoothed transformed radar measurement: calculating a plurality of likelihood functions, one for each of a plurality of hypothesized number of targets; and selecting one of said plurality of said likelihood functions as said minimal target likelihood function.

28. The method according to claim 27, wherein calculating said plurality of likelihood functions comprises calculating eigenvalues of the smoothed transformed autocorrelation matrix.

29. The method according to claim 28, wherein said likelihood function includes a sum of said eigenvalues.

30. The method according to claim 28, wherein said likelihood function comprises a penalty term selected to compensate for the specific number of hypothetical targets in the calculation.

31. The method according to claim 30, wherein said penalty term is related to size and number of subarrays used during said smoothing.

32. The method according to any one of claims 27-31, wherein said determining comprises determining noise variances for each hypothesized number of targets, the noise variance forming a negative term in said likelihood function.

33. The method according to any one of claims 24-32, wherein said performing is until reaching a stop condition.

34. The method according to claim 33, wherein the stop condition is a number of iterations.

35. The method according to any one of claims 24-34, wherein said optimization is performed in a bounded parameter space.

36. The method according to claim 35, wherein the bounded parameter space includes perturbations around the identify matrix.

37. The method according to any one of claims 35-36, wherein said bounded parameter space is based on physical understanding of mutual coupling behavior.

38. The method according to any one of claims 24-37, comprising, selecting a subset of said plurality of radar measurements, prior to said performing.

39. The method according to claim 38, wherein said selecting is of cells having a SNR above a predefined SNR threshold.

40. The method according to any one of claims 38-39, wherein said selecting is of cells is of those cells having range below a predefined range threshold.

41. The method according to any one of claims 24-40, wherein said receiving is from an automotive mounted radar array system; and wherein, upon finishing said performing, the optimized calibration matrix is used to correct subsequent radar measurements.

42. The method according to claim 41, wherein said subsequent radar measurements are used in a navigation system of said automotive.

43. The method according to any one of claims 41-42, wherein said performing is online, during use of said automotive.

44. The method according to any one of claims 24-43, wherein said performing is re-performed periodically.

45. The method according to any one of claims 24-44, wherein said optimized calibration matrix is used to adjust a received calibration matrix.

46. The method according to claim 45, wherein said received calibration matrix has been determined offline in a controlled calibration environment.

47. A method of radar calibration comprising: receiving a plurality of radar measurements; processing said plurality of measurement vectors to identify noise and signal subspaces; and determining a calibration transformation configured to cancel radar array nonidealities by minimizing signal energy found in said noise subspace.

48. A system for radar calibration comprising: a processor configured to: receive a plurality of measurement vectors, each measurement vector associated with a range-Doppler (RD) cell of a RD map; determine from said plurality of measurement vectors a plurality of autocorrelation matrices, an autocorrelation matrix corresponding to each said measurement vector and RD cell; perform an iterative process to minimize a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming said plurality of autocorrelation matrices with a hypothesized calibration matrix; smoothing said plurality of transformed autocorrelation matrices using one or more subarray technique to provide a plurality of smoothed autocorrelation matrices; determining a discrepancy function for each of said smoothed autocorrelation matrices and for each of a plurality of hypothesized numbers of targets, to provide, for each smoothed autocorrelation matrix, a plurality of likelihood functions, one for each hypothesized number of targets; selecting, for each of said smoothed autocorrelation matrices, a minimal likelihood function corresponding to a single hypothesized number of targets; andsumming the selected minimal likelihood functions to provide said cost function.

49. A radar navigation system comprising: an automotive mounted radar system configured to provide a plurality of measurement vectors, each measurement vector associated with a range-Doppler (RD) cell of a RD map; a processor configured to: determine from said plurality of measurement vectors a plurality of autocorrelation matrices, an autocorrelation matrix corresponding to each said measurement vector and RD cell; perform an iterative process to minimize a cost function to provide an optimized calibration matrix configured to cancel radar array non-idealities, the process comprising: transforming said plurality of autocorrelation matrices with a hypothesized calibration matrix; smoothing said plurality of transformed autocorrelation matrices using one or more subarray technique to provide a plurality of smoothed autocorrelation matrices; determining a discrepancy function for each of said smoothed autocorrelation matrices and for each of a plurality of hypothesized numbers of targets, to provide, for each smoothed autocorrelation matrix, a plurality of likelihood functions, one for each hypothesized number of targets; selecting, for each of said smoothed autocorrelation matrices, a minimal likelihood function corresponding to a single hypothesized number of targets; and summing the selected minimal likelihood functions to provide said cost function.

50. The radar navigation system according to claim 49, wherein said processor is configured to calibrate said radar system using said optimized calibration matrix.

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