Near eld broadband array design using a radially invariant modal expansion

This paper presents a new method of designing a beamformer having a desired broadband beampattern and a focusing capability to operate at any radial distance from the array origin The methodology uses the wave equation based representation of beampatterns to identify an orthogonal basis set of elementary beampatterns by which any arbitrary beampattern can be constructed A set of elementary beamformers are then designed for each elementary beam pattern and the desired beamformer is constructed by summing the elementary beamformers with frequency and source array distance dependent weights An important consequence of our result is that the beamformer can be factored in to three levels of ltering i beampattern independent elementary beamformers ii beampattern shape dependent lters and iii radial focusing lters where a single parameter can be adjusted to focus the array to a desired radial distance from the array origin As an illustration the method is applied to the problem of producing a practical array design that achieves a frequency invariant beampattern over the frequency range of which is suitable for speech acquisition using a microphone array and with the array focused either to far eld or near eld where at the lowest frequency the radial distance to the source is only three wavelengths PACS numbers GK Introduction Consider the problem of designing a microphone array for speech acquisition Not only does the array require a narrow main beam but it should operate uniformly over a large bandwidth and be able to cope with near eld sources Whilst there has been a deal of progress in designing broadband arrays having them operate well in the near eld still requires rather ad hoc solutions In this paper we will present a systematic way of designing near eld broadband sensor arrays In particular we will explicitly show how to parameterize the lter coe cients in order to focus the array to practically any operating radius from the array origin whilst maintaining a predetermined T D Abhayapala R A Kennedy R C Williamson Near eld broadband array design broadband angular beampattern Most of the array processing literature assumes a far eld source having only plane waves im pinging on the sensor array However in many practical situations such as microphone arrays in car environments the source is well within the near eld The use of far eld assumptions to design the beamformer in these situations may severely degrade the beampattern In many cases the approximate distance at which the far eld assumption begins to be valid is r L where r is the distance from an arbitrary array origin L is the largest array dimension and is the operating wavelength However we will show in section that this common rule of thumb for far eld approximation is not a necessary condition There appears to be little work in the literature on near eld beamforming In time delays were applied to compensate for di ering propagation delays due to spherical propagation However this is only an approximation and ignores the variation of the magnitude with distance and angle and assumes a point source In there was consideration initially for near eld theoretical development but this was ignored in the actual array design The few other related works we are aware of dealing with near eld arrays can be found in A novel technique useful in array design has been recently developed in It is based on writing the solution to the wave equation in terms of spherical harmonics and allowing a near eld beam pattern speci cation to be transformed to the far eld and the subsequent use of well understood far eld theory to design the near eld beamformer These near eld far eld transformations have been used for many years in the radio antenna community for reconstructing far eld antenna pat terns from near eld measurements though these works are computationally involved In the theory of near eld far eld transformation was used to establish a computationally simple design procedure that numerically implements the near eld far eld transformation Far eld broadband beamforming has been considered in and reviewed in J Acoust Soc Am In this paper a new method of beamforming is proposed in which an arbitrary desired beam pattern in both frequency and angle may be produced either in near eld or far eld We have used the wave equation based representation of beampatterns to identify the class of elementary beampatterns by which any arbitrary beampattern can be constructed These elementary beam patterns form an orthogonal basis set We use the concept of a theoretical continuous sensor to design elementary beamformers for each elementary beampattern Then the desired beamformer is constructed by summing the elementary beamformers with frequency and radial distance de pendent weights The proposed beamformer structure has three major processing blocks i a beampattern independent ltering block consisting of elementary beamformers ii a beampattern shape dependent ltering block and iii a radial focusing lter block where a single parameter can be adjusted to focus the array to di erent radial distances from the array origin The paper is organized as follows Section introduces the notion of elementary shape invariant beampatterns as building blocks of beampatterns The design of a general broadband theoretical continuous sensor as a sum of elementary continuous sensors is discussed in Section Section shows how to approximate the theoretical continuous sensor by a practical discrete array of sensors Guidelines for choosing the non uniformly spaced sensor locations and the consequence of spatial sampling are addressed in Section We conclude with an example broadband beamformer which can be focused to either near eld or far eld in Section Theory of Elementary Shape Invariant Beampatterns Beampattern Formulation In the near eld far eld transformation is obtained by solving the physical problem governed by the classical wave equation in the spherical co ordinate system Let r denote radial distance and be the azimuth and elevation angles as shown in Figure Then a general valid beampattern is T D Abhayapala R A Kennedy R C Williamson Near eld broadband array design constructed by combination of all possible modes of the form r H n kr P jmj n cos e jm where integers n and m jmj n index the modes k fc is the wave number f is the frequency of the wave and c is the speed of wave propagation The functions Pm n are associated Legendre functions and H n is the half odd integer order Hankel function of the rst kind which is de ned by H n Jn jYn where Jn and Yn are the half integer order Bessel functions of rst and second kind respectively A property we will rely on is that there is no nonnegative integer n and no real number r such that H n r This result follows from the fact that there are no common zeros for the functions Jn r and Yn r page The modes are associated with waves propagating towards the origin the half odd integer order Hankel functions of the second kind give the waves propagating away from the origin We assume that the propagation speed c is independent of frequency implying k is a constant multiple of frequency f Consequently throughout this paper we will often refer to k as frequency By combining modes for all possible n and m an arbitrary beampattern can be written as br k X n n X m n n k r H n kr P jmj n cos e jm where f n k g is a set of frequency dependent coe cients By introducing frequency dependence to the coe cients we can use to represent an arbitrary beampattern speci cation in both space angle and frequency To complete the beampattern transformation model it can be shown that the n k coe cients can be obtain from the analysis equation n k n r H n kr n m n m Z Z br k P jmj n cos e jm sin d d J Acoust Soc Am Since we can invert the representation via we conclude that the n k uniquely represent an arbitrary beampattern The equations and form an orthogonal transform pair This implies the following linearity property which will be used later if we have more than one beampattern at one xed radius then linearly adding them and transforming their sum to another radius is equivalent to the transformation of each beampattern separately to the second radius and then adding them together Elementary Beampatterns As we have seen by any physically realizable beampattern can be constructed by combining modes of Inversely an arbitrary beampattern can be decomposed into modes by These modes are also valid beampatterns Let us denote these beampatterns by