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Antenna array calibration methods and narrowband direction finding (DF) techniques will be outlined and compared for a uniform circular array. DF is stated as an inverse problem, which solution requires a parametric model of the array itself. Because real arrays suffer from mechanical and electrical imperfections, analytic array models are per se not applicable. Mitigation of such disturbances by a global calibration matrix will be addressed, and methods to estimate this calibration matrix will be recapped from literature. Also, a novel method will be presented, which circumvents the problem of a changed noise statistic due to calibration. Furthermore, local calibration, where array calibration measurements are incorporated in the DF algorithm, is considered as well. Common DF algorithms will be outlined, their assumptions regarding array properties will be addressed, and required preprocessing steps such as the beam-space transformation will be presented. Also, two novel DF techniques will be proposed, based on the Capon beamformer, but with reduced computational effort and higher resolution for bearing estimation. Simulations are used to exemplary compare calibration and DF methods in conjunction with each other. Furthermore, measurements with a single and two coherent sources are considered. It turns out that global calibration enables computational efficient DF algorithms but causes biased estimates. Furthermore, resolution of two coherent sources necessitates array calibration.

Direction finding (DF) is a task which occurs in several applications of surveillance, reconnaissance, radar, or sonar. Basically, DF can be defined as estimation of the bearing of one or multiple signal sources with respect to (w.r.t.) a reference point in space. Typically, an array of spatially distributed sensors is placed at this reference point and the array output is exploited for DF. Hence, DF estimation is an inverse problem. Solving the inverse problem requires a parametric model of the array output in terms of the parameters of interest: azimuth of arrival (AoA)

In order to derive a parametric model of the array output, a model of the sensor array itself is necessary. The array model highly depends on the array geometry and the characteristic of each sensor. Theoretical array models typically assume omnidirectional sensors and an ideal array geometry, which cannot be assured for real arrays. Apart from these assumptions, real arrays suffer from disturbances as, e.g., mutual coupling between the sensors or the support structure of the array, unknown sensor gain, and phase or mechanical imperfections [

All investigations are subject to a uniform circular array (UCA). UCAs feature a very attractive geometry, because their aperture covers the whole azimuth range and hence ambiguous free AoA estimates are ensured, on the contrary to, e.g., uniform linear array (ULA). Also, UCAs can be employed to estimate elevation too, but generally not unambiguous. For simplification, only AoA estimation and copolarised sources w.r.t. the array sensors are considered. For the conducted investigations it is not necessary to consider elevation and arbitrarily polarised sources. However, neglecting source polarisation and assuming fix elevation may result in biased estimates in real DF applications [

The goal of this paper is to jointly investigate array calibration methods and narrowband DF techniques. Global calibration, where a direction independent calibration matrix is used, will be considered. Methods to estimate the global calibration matrix from array calibration measurements are reviewed and a new method is proposed, which accounts for the change of the noise statistic due to the application of the calibration matrix. Also, local calibration, where the array calibration data are incorporated in the DF method, is considered. Known DF techniques will be outlined and two novel DF techniques based on the Capon beamformer will be proposed, featuring a reduced computational effort and better resolution in case of multiple sources. Some of the considered DF techniques take advantage of special array structures, which are not provided by UCAs. Hence, beam-space transformation will be briefly recapped. Simulations and measurements are employed for the investigations. Measurements with two coherent sources, hence sources with a fix phase relation, will be considered. Resolution of coherent sources is crucial in DF [

The reminder of the paper is organised as follows: a parametric model of the array output is derived in Section

Mathematical notation is as follows: scalars are italic letters. Vectors are in column format and written as boldface, lower-case, italic letters. Matrices correspond to boldface, upper-case letters. The matrix operations

DoA estimation requires a parametric model of the measurement data in terms of the DoAs. Consider an array of

Spherical coordinate system and angle definition.

The objective of direction finding is to estimate the AoAs from the array observations

Consider an uniform circular array with equiangular spaced omnidirectional sensors, placed on a circumference of radius

Several DF techniques are known from literature; see [

Considered narrowband DF techniques, the number of identifiable sources, and the considered structure of the steering vectors.

DF Methods | Reference | No. sources | Steering vector |
---|---|---|---|

CML | [ | | arbitrary |

UML | [ | | arbitrary |

MUSIC-1 | [ | | arbitrary |

root-MUSIC | [ | | Vandermonde |

MUSIC-2 | [ | | arbitrary |

ESPRIT | [ | | Vandermonde |

MODE | [ | | arbitrary |

IQML | [ | | Vandermonde |

Capon-1 | [ | | arbitrary |

Bartlett | [ | | arbitrary |

root-Capon | This work | | Vandermonde |

Capon-2 | This work | | arbitrary |

Generally, the Capon beamformer attempts to minimise the power contribution from interferer directions, while maintaining the gain in the direction of interest. The estimator is given by maximising a 1D spatial spectrum [

Restating cost function (

Another DF estimator based on the Capon beamformer is derived by exploiting array manifold separation [

Considering the cost function (

The ESPRIT, IQML, root-Capon, and root-MUSIC algorithm are naturally applicable for DF with ULAs. Hence, they necessitate steering vectors with Vandermonde structure. According to (

The beam-space transformation assumes an UCA, which follows the element-space model (

In global calibration, the disturbed array output is mapped onto a reference array output, whereas the disturbances are assumed as independent on the direction of impingement. In the following, a linear relationship between disturbed and reference array output is assumed. Then, global array calibration is done by a calibration matrix

The calibration matrix is derived from array calibration measurements, which are conducted on a test range or in an anechoic chamber. These measurements are a set of array outputs for known directions of impingement and comprise the array characteristics as well as the disturbances. Introduce the matrices

In local calibration, the array disturbances are considered as depending on the direction of impingement. Consideration of direction dependent disturbances is accomplished by using the array calibration measurements

Because the calibration measurements describe the array for discrete angles only, whereas DF algorithms require a continuous description, interpolation is required. Here, the EADF [

Table

Considered methods to estimate calibration matrix

Method | Reference | Estimation space |
---|---|---|

Wax | [ | element- and beam-space |

Sommerkorn | [ | element- and beam-space |

Kortke | [ | element- and beam-space |

Friedlander | [ | element-space |

Ng | [ | element-space |

See | [ | element- and beam-space |

Haefner | This work | element- and beam-space |

As pointed out, the second-order statistic of the noise is changed by applying the calibration matrix. This can deteriorate the performance of some DF methods, because they assume the noise covariance matrix to be diagonal. Introducing the constrain

In order to investigate the effect of calibration on the array characteristic, a real array will be considered. The array under consideration is the Poynting DF-A0046 UCA (see Figure

Poynting DF-A0046 UCA (dashed box) featuring 5 vertically polarised dipoles and 5 horizontally polarised monopoles. Only the dipoles will be considered in this paper.

First, the influence of global calibration on the array geometry is investigated. As stated previously, practical arrays suffer from mechanical imperfections, such that the assumed circular geometry is not assured. The estimated and assumed sensor positions are shown in Figure

Normalised sensor positions in the x-y-plane (array top view) as assumed by the model, and before and after global calibration. The sensor positions are normalised to the centre wavelength

Furthermore, the effect of calibration on the sensor characteristics is investigated. Magnitude and phase of the model and the sensor response before and after calibration are shown in Figures

Measured response of the first sensor of the Poynting DF-A0046 UCA, the corresponding response of the element-space model and the sensor response after calibration with the calibration matrix estimated by the method of Wax. The plots show the (a) magnitude and (b) phase of the response.

In order to compare the various calibration matrix estimators and the DF techniques, Monte-Carlo simulations with varying signal to noise ratio (SNR) are carried out. Data are generated according to (

Parameters of the simulations.

Parameter | Value |
---|---|

Transmit signal | zero-mean, circularly normal distributed |

Centre frequency | 305 MHz |

Receive array | Poynting DF-A0046 |

SNR | −10 dB to 40 dB |

No. snapshots | 50 |

AoAs | uniformly distributed |

The estimated global calibration matrices are applied to the generated array outputs and DF is conducted afterwards. The ESPRIT estimator and the MUSIC-1 estimator are exemplary utilised for AoA estimation in beam-space and element-space, respectively. For the ESPRIT estimator, calibration according to the array model in beam- and element-space with subsequent beam-space transformation is applied. The resulting RMSEs are shown in Figures

RMSE of AoA estimates with the ESPRIT algorithm and calibration w.r.t. (a) the element-space model and subsequent beam-space transformation and (b) the beam-space model. The CRLB is plotted as red dashed line.

RMSE of AoA estimates with the MUSIC-1 algorithm and calibration w.r.t. the element-space model. The CRLB is plotted as red dashed line.

Obviously, the RMSE curves for all calibration methods converge to a certain value, indicating biasedness. Furthermore, the ESPRIT estimates never attain the CRLB, because the beam-space transformation introduces errors resulting in an increased variance of the estimates. Furthermore, the RMSEs of the ESPRIT estimates indicate that calibration w.r.t. the array model in element-space slightly outperforms the calibration w.r.t. the array model in beam-space. An explanation can be given by the global calibration matrix itself. Basically, the matrix describes disturbances due to coupling or electrical and mechanical imperfections. Therefore, the calibration matrix has a clear physical meaning in the element-space. Calibration matrix estimation w.r.t. to the beam-space model assumes a virtual array, such that the calibration matrix has no longer a clear physical meaning. Hence, the disturbances are not described properly and the calibration becomes less powerful. The proposed estimation method performs comparably worse, which can be related to the constraint of a Hermitian calibration matrix causing a less powerful calibration. Hence, variation of the noise statistic is not as a crucial for the bearing estimation as remaining calibration errors. Comparison of the RMSEs of the ESPRIT and MUSIC-1 estimator indicates that the MUSIC-1 estimator attains the CRLB for SNRs around 0 dB, but the ESPRIT estimator slightly outperforms the MUSIC-1 estimator in terms of minimal achievable RMSE. Overall, the estimation method by Sommerkorn in conjunction with the ESPRIT performs best.

First, the considered DF techniques in conjunction with global calibration will be compared. The global calibration matrix is estimated in element-space by the method of Sommerkorn. The calculated RMSEs are shown in Figure

RMSE of AoA estimation with several DF techniques for (a) global calibration using the method of Sommerkorn to estimate the calibration matrix and (b) local calibration using the EADF as array model. The CRLB is plotted as red dashed line.

In order to verify the explanation of biased estimates due to global calibration, the RMSE for local calibration will be investigated. Note that the beam-space estimators are excluded, because they cannot be applied under local calibration. The calculated RMSEs are depicted in Figure

Calibration and experimental measurements were performed on a test range in Paardefontein, South Africa, using the 5 vertically polarised dipoles of the Poynting DF-A0046 UCA as receive array. First, a single source has been placed at

Setup for the test measurements.

Parameter | Value |
---|---|

Transmit signal | multi–sine |

Centre frequency | 305 MHz |

Bandwidth | 10 MHz |

Transmit antennas | logarithmic periodic dipole antenna |

Transmit polarisation | vertical |

SNR | approx. 25 dB |

Receive array | uniform circular array with 5 dipols |

Receive polarisation | vertical |

AoAs | 25^{∘}; |

To show the difference between global and local calibration, the spectrum of the Bartlett, Capon-1, and MUSIC-1 method is calculated for both calibration schemes. The respective spectra for the measurement with the single source are shown in Figure

Spectra of Bartlett beamformer, Capon-1 beamformer, and MUSIC-1 for global and local calibration. The global calibration matrix has been calculated by the method of Sommerkorn. Measurement data with a single source at 25^{∘} are used to calculate the spectra.

First, the estimators are applied to the measurements without previous calibration. The estimation results for the single and dual source measurement are shown in Table

Estimated AoAs without array calibration. The setup is a single source (25°) and a coherent dual source (−30°, 25°) scenario.

| | | |
---|---|---|---|

| | | |

ESPRIT | | | |

root-MUSIC | | | |

IQML | | | |

root-Capon | | | |

Capon-1 | | | |

Bartlett | | | |

MUSIC-1 | | | |

CML | | | |

UML | | | |

MODE | | | |

Capon-2 | | | |

MUSIC-2 | | | |

The method by Sommerkorn is utilised to estimate the global calibration matrix in element-space. Estimated AoAs for the single and dual source case are shown in Table

Estimated AoAs under global and local array calibration. The setup is a single source (25°) and a coherent dual source (−30°, 25°) scenario.

| | | | |
---|---|---|---|---|

| | | ||

ESPRIT | global | | | |

root-MUSIC | global | | | |

IQML | global | | | |

root-Capon | global | | | |

Capon-1 | global | | | |

Bartlett | global | | | |

MUSIC-1 | global | | | |

CML | global | | | |

UML | global | | | |

MODE | global | | | |

Capon-2 | global | | | |

MUSIC-2 | global | | | |

Capon-1 | local | | | |

Bartlett | local | | | |

MUSIC-1 | local | | | |

CML | local | | | |

UML | local | | | |

MODE | local | | | |

Capon-2 | local | | | |

MUSIC-2 | local | | | |

Last, the scenario with the closer located sources is considered. The estimation results are shown in Table

Estimated AoAs under global and local array calibration. The setup is a coherent dual source (−10°, 8°) scenario.

| | | |
---|---|---|---|

| | ||

ESPRIT | global | | |

root-MUSIC | global | | |

IQML | global | | |

root-Capon | global | | |

Capon-1 | global | | |

Bartlett | global | | |

MUSIC-1 | global | | |

CML | global | | |

UML | global | | |

MODE | global | | |

Capon-2 | global | | |

MUSIC-2 | global | | |

Capon-1 | local | | |

Bartlett | local | | |

MUSIC-1 | local | | |

CML | local | | |

UML | local | | |

MODE | local | | |

Capon-2 | local | | |

MUSIC-2 | local | | |

In summary, calibration is not necessary in the single source case, because some estimators can tackle the array disturbances. However, calibration is required in case of two coherent sources. Furthermore, beam-space methods are quite sensitive due to the beam-space transformation and the few number of sensors, such that these methods are not able to resolve two closely spaced sources. Overall, DF methods in conjunction with local calibration show the best estimation accuracy.

Calibration of and direction finding with uniform circular arrays has been investigates in this paper. Several assumptions have been drawn for the conducted investigations. First, source signals are assumed to imping in the azimuth plane. Second, cross-polar sensor characteristics have been neglected, because the sources are assumed to be copolar.

Global and local calibration of the UCA have been investigated. As shown by simulations, DF in conjunction with global calibration results in biased estimates due to remaining model errors, which are especially severe for high SNRs. On the contrary, DF in conjunction with local calibration results in unbiased estimates. Also, it was shown that local calibration is superior to global calibration in terms of the achievable root mean-square error for SNR above 10 dB. Furthermore, test measurements with a single and two coherent sources have been considered. In case of a single source, array calibration is not necessary for several estimation methods. However, calibration was found to be required in order to resolve coherent sources. Also, local calibration was found to outperform global calibration in case of closely spaced coherent sources. Comparison of the DF techniques based on simulations and test measurements indicates that maximum-likelihood estimators (CML, UML, and MODE) and MUSIC type estimators (MUSIC-1 and MUSIC-2) show better estimation accuracies than beam-space estimators (ESPRIT, IQML, and root-MUSIC).

In summary, choosing the appropriate DF technique and calibration method is an application specific trade-off between required estimation accuracy, computational complexity, and also calibration measurement effort.

The used data have not been made available due to confidentiality agreements with research collaborators.

The authors declare that they have no conflicts of interest.

This work was partially funded by the Bundesministerium für Bildung und Forschung [grant number 16BN1203]. Stephan Häfner is funded by the Deutsche Forschungsgemeinschaft [grant number 317632307]. The authors acknowledge support for the Article Processing Charge by the German Research Foundation (DFG) and the Open Access Publication Fund of the Technische Universität Ilmenau. The authors like to thank Dipl.-Ing Uwe Trautwein for supporting the calibration and test measurements.