Design of wideband beamformers with sparse arrays
Background information on beamformer design:
The standard design method is based on superdirective arrays (MVDR beamformers). In order to show the problems involved in the design of these arrays, we consider the simple case of a uniform array exposed to a singlefrequency wave field (Fig. 1).
In case of a narrowband beamformer operating at frequency f, the complexweighted sensor signals are summed up to obtain the beamformer output signal. The beam pattern P(u)² represents the output signal power versus azimuth variable u at frequency f.
Example: Superdirective (MVDR) Beamformer
A minimumvariancedistortionlessresponse (MVDR) beamformer minimizes the output signal power but maintains signals from the desired direction. Figure 2 shows the (narrowband) beam patterns of an MVDR beamformer at approximately f = 5 kHz (above) and f = 500 Hz (below). The visible regions (blue) contain the azimuth range from 0° to 180° and are normally plotted using polar coordinates (diagrams at right hand side of Fig. 2).
Problem: MVDR beamformers exhibit large bumps in the invisible region at low frequencies. We can say that a superdirective beamformer shifts energy from the visible to the invisible region. High energy in the invisible region is an indicator of high sensitivity to sensor noise.
Example: Sparse Array Beamformer
In a sparse array not all positions of the sensor grid are occupied. In the example in Fig.3, we use a onedimensional sparse array where only 16 of 29 uniform sensor positions are populated with sensors. The other positions are empty resulting in a nonuniform array. As shown in Fig. 3, the sparse array exhibits nearly the same mainlobe behavior as in Fig. 2. The sidelobe levels are increased, as compared to the MVDR design. However, there is much less energy in the invisible region at low frequencies. Thus, sparse array beamformers are in general much more robust than comparable MVDR beamformers.
Design of sparse array beamformers:
Simulated annealing (SA) is a standard discrete optimization technique that can be applied to the design of sparse arrays. In our approach, we use SA only to find the optimum positions of N sensors located at M > N possible locations. The beamformer weights are not computed by SA because we want to reduce the degrees of freedom in the SA optimization algorithm. Otherwise SA would be very timeconsuming. We want to achieve a comparable wideband beampattern as a full array with all M positions occupied. The design flow of our SA design is sketched in Fig.4.
By selecting a proper starting temperature T the cool down phase of the SA algorithms is slow enough to hopefully achieve a global minimum. The sensitivity to local minima is reduced by allowing the selection of solutions with higher cost functions. The probability for such a selection depends on T and is high at the beginning of the iteration loop. Permutations of the sparse array are obtained by a random selection of two positions and exchanging their states (state = 1, if there is a sensor, 0 if not). Details of our sparse array design with SA are published in
G. Doblinger, “Optimized design of interpolated array and sparse array wideband beamformers,” Proc. 16^{th} European Signal Processing Conference EUSIPCO 2008, Lausanne, Switzerland, Aug. 2529, 2008.
A MATLAB implementation of the proposed algorithm can be found here. Please note that the programs are published under the GNU GENERAL PUBLIC LICENSE (see file attached to the programs).
Some experimental results on sparse array beamforming:
Figure 5 shows a typical design result of a sparse array beamformers with 60° look direction, and desired spatial nulls at 20° and 160°. The beampatterns are measured with a white noise source signal bandlimited to 0.3 kHz – 5.5 kHz. Compared to the beam pattern of a uniform array 8 sensors (blue curve) the sparse array (red curve) is clearly superior. The sharper mainlobe of sparse arrays is due to the better beamforming at low frequencies (larger array length than in case of the uniform array).
Influence of sensor noise:
Figure 6 shows the influence of sensor noise (SNR = 25 dB). Compared to Fig. 5, sidelobe levels are only slightly increased. However, the sharpness of spatial nulls is reduced.
Influence of room reverberations:
In order to test the beamformer design in a room environment, we show the influence of simulated room acoustics on the beam patterns in Fig. 7.
Room size is L x W x H = 6 m x 5 m x 3 m with a reverberation time T60 = 0.13 seconds. Moderate room reverberations increase sidelobe levels by approximately 2 dB, and affect spatial nulls. However, we observe only a minor influence on the mainlobe performance.
In the following animation, we illustrate the wideband beam pattern measurement. You can observe the room impulse response in addition to the successive beam pattern measurement at 10° azimuth steps.
Conclusions:
Comparing sparse arrays with uniform MVDR arrays, we notice the following items:

sparse arrays have sharper mainlobes

sparse arrays are less prone to sensor noise and room acoustics

sparse arrays show higher sidelobe levels

sparse arrays have significantly larger sizes

sparse array layouts change, if design parameters (look direction, spatial nulls) are altered.