Design of wideband fixed beamformers using convex optimization and adaptive sensor calibration
We demonstrate optimization methods based on second-order cone programming (SOCP) for wideband beamforming arrays with applications to microphone arrays. All results (and much more) can be reproduced by downloading my MATLAB programs. Please note that these programs are open source and licensed under the GPL (GNU GENERAL PUBLIC LICENSE) version 3, 29 June 2007 (see file gpl-3.0.txt which is included with the MATLAB programs).
Two different design methods are investigated:
- linearly constrained minimum variance (LCMV) wideband beamformers with and without additional constraints
weighted least squares (WLS) wideband beamformers with and without additional constraints
The constrained optimization is performed with the CVX software package. You will need a working CVX installation to run my MATLAB programs. Details of the design algorithms can be found in G. Doblinger, “Optimization of wideband fixed beamformers with adaptive sensor calibration,” Proc. 18th European Signal Processing Conference EUSIPCO 2010, August 23-27, 2010, Aalborg, Denmark, paper number 1569291143.
Main design objectives:
- reduction of sidelobe levels using efficient convex optimization methods
- reduction of computational complexity by reducing the number of constraints
- robustness against sensor noise, and microphone tolerances
- adaptive sensor calibration before normal use of the array
Adaptive sensor calibration
Experimental results of an optimized LCMV beamformer
Experimental results of an optimized WLS beamformer
Wideband beamformers may be implemented by processing each sensor signal with optimized finite impulse response duration (FIR) filters as shown in Fig. 1.
The output signal power is needed in the optimization cost function, and will be calculated using a model of the spatio-spectral correlation matrix. This model contains contributions of uncorrelated sensor noise, diffuse noise field components, and jammers from certain directions. Alternatively, we can measure Sxx or Rxx. In the following, we summarize the optimization techniques to obtain the real-valued coefficients vector h of the FIR filters. The first part of the equation box shows the well-known LCMV problem which can be solved in closed form using the method of Lagrange multipliers. In the second part, we include additional constraints on the sidelobe levels, and on the norm of h. As shown in the experimental results, a better sidelobe behavior can be achieved as compared to the plain LCMV beamformer. The latter optimization problem (called LCMV+SOCP) can be solved by numerical optimization programs like the CVX software.
Beamformers based on an LCMV design offer a sharp mainlobe. This property is convenient, if there are no substantial movements of the source. In many situations, however, a broader mainlobe is desirable to avoid a significant performance loss is case of look direction mismatch. In such cases, the beamformer design may be based on the approximation of a desired beam pattern similar to the approximation of a spectral mask in an FIR filter design. Thus, the second approach we use in the design of wideband beamformers is based on a WLS criterion. The algorithm is summarized in the following equation box. The WLS optimization problem can also be solved in closed form. If we include additional constraints (WLS+SOCP problem), then the solution can be obtained with the convex optimization programs of the CVX software.
Adaptive sensor calibration
Optimized wideband beamformers are quite sensitive to a mismatch in sensor transfer functions. In case of microphone arrays, this means that we must use preselected sensors, or some calibration procedure. We use adaptive filters in the sensor paths (see Fig. 2) to compensate sensor tolerances by matching sensor transfer functions prior to normal use of the array. The system is connected between the sensor outputs and the beamformer inputs. It should be noted that we assume omnidirectional microphones with no angle-dependent errors in the polar patterns. Such errors cannot be compensated by the adaptive system we use in cascade to the beamformer.
In the calibration phase, the array is exposed to a wideband signal like speech or noise coming from broad side. The only requirement is an equal excitation of all sensors. After the calibration phase, the adaptive filter coefficients are frozen.
The reference path is a signal delay to ensure causal adaptive filters. After convergence, the filter coefficients are fixed, and the adaptive filter transfer function in each channel matches the transfer function of sensor 1. Our experiments show that in case of microphone arrays, sufficient convergence is obtained with adaptive FIR filters using the normalized least mean squares (NLMS) algorithm.
Experimental results of an optimized LCMV design
The results given in the sequel are for 1-dimensional, linear, uniform arrays with N = 8 sensors, and a look direction of 0° (endfire). Extensions to other array configurations are straight forward. We use a frequency band from 400 Hz up to 3200 Hz, with 8 kHz sampling frequency. Sensor spacing is 5 cm giving rise to an array size of 35 cm. Typically, the FIR filter length of both the beamformer, and the adaptive calibration filters is set to L = 50.
In order to investigate the influence of sensor errors, we model microphone frequency responses by 4th-order recursive filters approximating a typical magnitude response with a 6 dB decay below 700 Hz, and a 2 dB increase around 3 kHz. This response is randomly disturbed by small deviations to obtain maximum amplitude and phase errors of approximately 2 dB, and 10°, respectively.
Fig. 3 shows the 3-dimensional beam patterns with ideal sensors for both the plain LCMV design, and the improvement by SOCP.
Wideband beampatterns (i.e. beamformer response to a wideband source signal from different angles) are plotted in Fig. 4 (with calibrated sensors), and in Fig. 5 (non-calibrated sensors), respectively. The beam pattern with calibrated sensors are obtained after a calibration time period of 250 msec. using a wideband noise signal common to all sensors. In addition, sensor noise (SNR = 30 dB) is included to show the wide noise behavior of the beamformer. Ten different sets of sensors are used to obtain the beam patterns. The patterns of the arrays with calibrated sensors approximate the optimized pattern quite well. However, the desired spatial null at 90° is barely visible. We observe this property at other fixed beamformer designs as well.
Experimental results of an optimized WLS design
A design result based on the WLS criterion is shown in Fig. 6. Again, using SOCP improves the sidelobe behavior. In addition, a flatter mainlobe is achieved by SOCP. However, these optimzation methods results in a significantly higher sensitivity against sensor mismatch, and sensor noise as compared to the LCMV designs (see Fig. 7, and Fig. 8).
Optimized design methods for wideband beamformers based on SOCP offer a better beamforming behavior then standard design methods. By approximating the wideband beam pattern directly, our approach offers a lower computational complexity as compared to other methods. In order to obtain an improved sidelobe behavior with mismatched sensors, we must use an adaptive sensor calibration scheme. Experiments show that the resulting beam patterns are close to the optimized patterns of arrays with ideal sensors.
A list of references can be found in my EUSIPCO-2010 paper.