Applications of the SRV constraint in broadband pattern synthesis

doi:10.1016/j.sigpro.2007.11.001

Signal Processing

Volume 88, Issue 4, April 2008, Pages 1035-1045

https://doi.org/10.1016/j.sigpro.2007.11.001 Get rights and content

Abstract

In this paper, the spatial response variation (SRV) is defined to measure the fluctuation of the array spatial response within the desired frequency band. Applying the SRV constraint in the optimization formulations to produce the frequency invariant beampattern (FIBP) is investigated. An efficient way to formulate the FIBP synthesis problems is proposed. Examples are demonstrated to show the effect of applying the SRV constraint in broadband pattern synthesis.

Introduction

In applications using sensor arrays to process broadband signals, such as audio teleconferencing using microphone and loudspeaker arrays to process the speech signals, the array beampattern with a constant beamwidth over the frequency band of interest is desired to avoid the effect of lowpass filtering [1], [2]. Some applications even require the array spatial response in the whole field of view to remain constant with respect to frequency. A typical example is the multi-beam beamforming network in the beamspace adaptive array [3], [4]. As indicated in [3], the frequency invariant property is required in the whole field of view of each beam to prevent distortion of interferences arriving from any direction. The frequency-dependent response will damage the replica of the interference and hence degrade the performance of adaptive beamforming severely. Both the above two types of beampatterns are referred to as the frequency invariant beampattern (FIBP) in literatures.

In order to get a constant beamwidth over the frequency range of interest, the harmonic nesting approach [1], [5] uses several subarrays with properly chosen apertures and geometries and combines their outputs in a frequency-dependent way. To further improve the frequency invariance of the beampattern, the filter-and-sum beamformer is used in each subarray in [6]. In [7], the asymptotic theory of unequally spaced arrays is used to derive the relationship between the beampattern characteristics and the functional requirements on sensor spacings and weightings. The theory for the far-field frequency invariant beamformer (FIB) presented in [8] offers deep insights into the issue of the FIBP synthesis. Multi-rate and single-rate methods combining the finite impulse response (FIR) or infinite impulse response (IIR) filters are studied in [9], [10]. In Refs. [3], [4], [11], the fan filter is used to produce the FIBP for the uniformly spaced arrays. The design of the FIBP for planar arrays like the concentric ring array and the rectangular array can be found in [12], [13], [14]. The superdirective array with the FIBP is studied in [15], [16].

Existing designs synthesize the beampattern with the frequency invariant feature in either the whole spatial region [3], [8], [9] or only the mainlobe directions [1], [2], [5]. It is desirable to have a method that allows us to specify arbitrary angles at which the spatial response is frequency invariant. Moreover, there is no criterion to measure the extent of the frequency invariance of the synthesized beampattern quantitatively.

In this paper, we propose a spatial response variation (SRV) constrained pattern synthesis method to produce the FIBP. The SRV is defined to measure the fluctuation of the array spatial response over the desired frequency band. By constraining the average SRV in the specified angles to be smaller than a threshold, a desirable tradeoff between the frequency invariance and other characteristics of the beampattern can be obtained. By changing the spatial region specified in the SRV constraint, the frequency invariant feature can be obtained for any direction of interest. Therefore, relative to the existing approaches, the SRV constrained pattern synthesis method shows greater flexibility. Although the SRV is defined in a form which looks like the mean square error, it is completely different from the minimum mean square error method. This will be explained in detail in Section 3 where the definition of the SRV is presented. Besides, an efficient way to formulate the FIBP synthesis problems is investigated. The number of constraints and hence the complexity of the optimization formulations are significantly reduced.

This paper is organized as follows: In Section 2, the mathematical formulations of the array model and the definition of the convex optimization problem are introduced. In Section 3, the definition of the SRV constraint and its applications in broadband pattern synthesis are presented. In Section 4, examples of synthesizing the FIBP are demonstrated. Brief conclusions are drawn in Section 5.

Section snippets

Background

For a broadband array with L sensors and J taps connected after each element, the beamforming output is the sum of all the LJ weighted tap signals. The $LJ \times 1$ real-valued weight vector $w$ for the broadband array is defined as follows: $w = [w_{1, 1} \dots w_{L, 1} \dots w_{1, J} \dots w_{L, J}]^{T},$ where the superscript T denotes the transpose. The two-dimensional spatial response is expressed by $H (f, θ) = w^{T} s (f, θ) .$ The beampattern is defined as the norm square of the spatial response. $s (f, θ)$ is the $LJ \times 1$ steering vector which is written as $s (f,$

Definition of the SRV constraint

The SRV in the direction $θ$ is defined as follows: $SRV (θ) = \frac{1}{B_{Ω}} \int_{f \in Ω} | w^{T} s (f, θ) - w^{T} s (f_{0}, θ) |^{2} d f .$ It represents the variation of the spatial response within the frequency range $Ω$ . $B_{Ω}$ denotes the bandwidth of $Ω$ . $f_{0}$ is the reference frequency selected within the frequency range. In definition (6), the template for the desired response is not fixed but a function of the weight vector which is to be optimized. Therefore, the value of the SRV reflects the true fluctuation of the synthesized beampattern at

Example 1: synthesis of the FIBP with the frequency invariance in either the mainlobe region or the whole field of view

This example is designed to show that the proposed method can synthesize the beampattern with the frequency invariant feature in either the mainlobe region or the whole field of view. The influence of various SRV constraints on the characteristics of the synthesized beampattern can be observed.

The simulation on a uniform linear array with 15 sensors and 15 taps per element is performed. The normalized operating frequency band of the array is assumed to be $[0.2, 0.4]$ . The sensor spacing is

Conclusions

In this paper, the SRV constraint is defined to restrict the fluctuation of the spatial response over the desired frequency band for the broadband array. The optimization formulation using the SRV constraint in the efficient way to synthesize the FIBP is studied. Examples are demonstrated to show the effect of applying the SRV constraint in broadband pattern synthesis.

Acknowledgement

The authors would like to acknowledge the anonymous reviewers for their valuable comments and suggestions that helped to improve the quality of this paper.

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View full text

Applications of the SRV constraint in broadband pattern synthesis

Abstract

Introduction

Section snippets

Background

Definition of the SRV constraint

Example 1: synthesis of the FIBP with the frequency invariance in either the mainlobe region or the whole field of view

Conclusions

Acknowledgement

Appl. Acoust.

Signal Process.

Wideband beamspace adaptive array utilizing FIR fan filters for multibeam forming

IEEE Trans. Signal Process.

A generalized sidelobe canceller employing two-dimensional frequency invariant filters

IEEE Trans. Antennas Propag.

Wide-bandwidth constant beamwidth acoustic array

J. Acoust. Soc. Amer.

Theory and design of broadband sensor arrays with frequency invariant far-field beam patterns

J. Acoust. Soc. Amer.

FIR filter design for frequency invariant beamformers

IEEE Signal Process. Lett.