Astrid Lundgren

Calibration in Array Processing

High-resolution direction-or-arrival estimation has been an active area of research since late 1970s. The methods find a wide range of applications, including passive listening arrays, radar and sonar, and spatial (or space-time) characterization of wireless communication channels. Conventional beamforming-based techniques for direction estimation are limited by the aperture (or physical size) of the array. In contrast, parametric methods promise an unlimited resolution in theory. These methods take advantage of a precise mathematical model of the received array data, for example due to incoming plane waves. In practice, the resolution and estimation accuracy is limited by noise as well as errors in the assumed data model. The focus of this chapter is on modeling errors, and in particular calibration techniques to mitigate such errors. Perhaps the most natural and common approach is to measure the response of the array in an anechoic chamber. These calibration measurements are then used to update the data model, either in the form of explicit unknown parameters or in a non-parametric way. Under favorable conditions it is also possible to estimate the response model together with the unknown directions, so-called auto-calibration. The purpose of this chapter is to give an overview of existing techniques and discuss their respective pros and cons. We will also elaborate on how the methods can be extended to more general situations, for example including frequency and polarization dependence. It should be mentioned that practical calibration also involves hardware adjustments, to compensate for temperature drift etc. The methods considered here can be classified as software calibration, where errors are handled by adjusting the assumed data model rather than correcting for it.

Calibrating an Array with Scan Dependent Errors Using a Sparse Grid

Today most arrays have a very high mechanical and electrical accuracy. Our aim is to allow for larger errors (low manufacturing cost), while keeping the calibration grid sparse (low calibration cost). To be able to handle the imperfections causing scan dependent errors (like position errors), we suggest a new array receive calibration method based on local models. The results show that our method performs much better than linear interpolation or global (direction independent) calibration regarding direction of arrival estimation using MUSIC for arrays with large position errors. In beamforming for e.g. communication, however, the global calibration performs better (lower side lobes) since it is optimized for the whole calibration region, while the local calibration is optimized for one direction.

Array interpolation based on Local Polynomial Approximation with application to DOA estimation using weighted MUSIC

The problem of Direction-of-Arrival (DOA) estimation using an array of sensors has received much attention for more than 3 decades. This is due to a rich interest from application areas such as radar, sonar and wireless communication channel characterization. However, high resolution DOA estimation requires an accurate model of the array response. This is usually achieved by measuring the response using sources at known positions (calibration). This paper considers interpolation of the calibration measurements using knowledge of a nominal response model. Standard linear interpolation is compared to an approach based on Local Polynomial Approximation (LPA). We also derive a weighted MUSIC estimator, which is applied using error estimates from the interpolation. Both LPA interpolation and weighted MUSIC are found to improve the performance, but not uniformly in all scenarios.

Optimized Beamforming Calibration in the Presence of Array Imperfections

For arrays with position and channel errors the calibration becomes very crucial. In the traditional calibration methods one can choose between an optimal SNR and a beam pattern with low side lobes. In this paper we formulate a beam pattern synthesis method which optimize the trade-off between the two criteria. A classical problem with position errors in an array, is that it is difficult to get low side lobes over the whole side lobe region, since the position errors give rise to direction dependent errors. In this paper this problem is solved by using local (direction dependent) correction matrices in the beam pattern optimization. The new way of using local correction matrices leads to the lowest possible uniform side lobe level, for the chosen SNR, beamwidth and beam pointing direction.

Gaussian processes framework for validation of linear and nonlinear models

Abstract A statistical nonlinear model validation method is suggested based on the Gaussian processes framework. Instead of testing for correlation between the residuals and certain test variables, as in traditional statistical tests, a parameterized model of the correlation is proposed and the significance for this model is tested. The test makes it possible to validate against nonlinear models without making detailed assumptions on the structure of the nonlinearities.

Adaptive beamforming using calibration vectors with unknown gain and phase

The importance of taking the direction dependent amplitude and phase of the received signals (due to the mutual coupling) into account in the calibration is demonstrated using adaptive beamforming (Capons method). The method is compared to a direction dependent (local) calibration method, which is shown to be the preferred method if there are position errors in the array. In reality, robust versions of Capons method is always used. The calibration is considered as a complement to these robust methods, to reduce the array steering vector errors and thereby further improve the performance of the robust Capon methods. The calibration also allows for larger errors in the array, which translates to a potential to decrease the manufacturing cost.

Two-Step Esprit with Compensation for Modelling Errors using a Sparse Calibration Grid

In arrays with scan dependent errors, such as large position errors, a dense calibration grid can become necessary. Calibration time is, however, very expensive and keeping the measured calibration grid as sparse as possible is important. In this paper it is shown how interpolation using local models can be used to make the calibration grid more dense without increasing the number of measurements. Furthermore, it is shown how the performance of the DOA estimation with ESPRIT using arrays with large position errors can be improved by a second step including weighted calibration.

Nonlinear instrument variable methods based on local linear models

The aim of the present study is to derive nonlinear instrument variable methods by using local linear models. Two algorithms to estimate consistent local ARX-models of the system order are presented. A local ARX-model with a regressor of higher order than the system is simulated to estimate an approximately noise-free data set. In the first algorithm this approximately noise-free data is used as estimation data to a local ARX-model of the system order. The second algorithm uses the simulated data as instrument in a local instrument variable method. The algorithms are demonstrated on both simulated and laboratory data.