Method and apparatus for determining the directional frequency response of an arrangement of transducer elements

Abstract

Techniques for determining the directional frequency response of an arrangement of transducer elements are disclosed. In an example, a method includes providing a simulation of locations of the transducer elements, in the spatial domain; providing a beamforming direction and a frequency range; converting the simulation of locations from the spatial domain into corresponding frequency response values in a spatial frequency domain, such that, for each frequency of a plurality of frequencies in the frequency range, a spatial frequency contour is defined, each of the spatial frequency contours intersecting at the origin; determining the frequency response by applying a transformation to the frequency response values for the provided beamforming direction and frequency range and translating the spatial frequency domain into a modified frequency domain; and outputting the frequency response.

Claims

1. A computer-implemented method for determining a frequency response as a function of direction of an arrangement of transducer elements, the method comprising: (i) providing a simulation of locations of the arrangement of the transducer elements, in a spatial domain; (ii) providing a beamforming direction and a frequency range; (iii) converting the simulation of locations from the spatial domain into corresponding frequency response values in a spatial frequency domain, the frequency response values having spatial frequency coordinates, such that: for each frequency of a plurality of frequencies in the frequency range, a spatial frequency contour, defined by a locus of points in the spatial frequency domain corresponding to the location of the frequency response values associated with the frequency response, as a function of direction, of the arrangement of the transducer elements with respect to that frequency and the beamforming direction, passes through an origin of the spatial frequency domain, wherein each of the spatial frequency contours for each of the plurality of frequencies intersect at the origin; (iv) determining the frequency response, as a function of direction, of the arrangement of transducer elements by applying a transformation to the frequency response values for the provided beamforming direction and frequency range, wherein applying the transformation comprises translating the spatial frequency domain into a modified frequency domain, such that the contours avoid intersecting; and (v) outputting the frequency response values in the modified frequency domain, as the frequency response, as a function of direction, of the arrangement of the transducer elements with respect to the beamforming direction.

2. The computer-implemented method of claim 1, wherein applying the transformation is performed such that the contours are substantially parallel.

3. The computer-implemented method of claim 1, wherein applying the transformation further comprises translating the spatial frequency domain, having coordinates k.sub.x, k.sub.y, and optionally k.sub.z, into the modified frequency domain, having coordinates g.sub.x, g.sub.y and optionally g.sub.z, using the transformation k.sub.x=g.sub.x, and k.sub.y=g.sub.y, and optionally k.sub.z=g.sub.z, wherein the beamforming direction is x.

4. The computer-implemented method of claim 1, wherein applying the transformation further comprises translating the spatial frequency domain, having coordinates k.sub.x and k.sub.y, into the frequency domain, having coordinates , f, using the transformation k.sub.x=f(cos 1); and k.sub.y=f sin , wherein the beamforming direction is x.

5. The computer-implemented method of claim 1, wherein converting the simulation of locations comprises applying a Fourier Transform to the simulation of the locations of the arrangement of the transducer elements.

6. The computer-implemented method of claim 1, wherein providing the simulation of locations comprises providing a spatial function defining the arrangement of the transducer elements.

7. The computer-implemented method of claim 1, wherein providing the simulation of locations comprises allocating a weighting value to one or more of the transducer elements.

8. The computer-implemented method of claim 7 wherein the weighting value is a measure of a gain of an associated transducer element.

9. The computer-implemented method of claim 1 wherein the frequency response of the arrangement of the transducer elements is determined for transducer elements arranged to transmit acoustic signals or electromagnetic signals.

10. The computer-implemented method of claim 1 wherein the frequency response of the arrangement of the transducer elements is determined for transducer elements arranged to receive acoustic signals or electromagnetic signals.

11. The computer-implemented method of claim 1 wherein the frequency range comprises the range from about 20 Hz to 20 kHz.

12. The computer-implemented method of claim 1 wherein the method further comprises controlling a computer processor to generate an evaluation metric of a performance of the arrangement.

13. The computer-implemented method of claim 12 wherein the evaluation metric is a directivity index.

14. A data processing device comprising a processor that s adapted to perform the method of claim 1.

15. A non-transitory computer-readable medium comprising program instructions which, when executed by a processor, cause the processor to perform operations comprising: (i) providing a simulation of locations of an arrangement of transducer elements, in a spatial domain; (ii) providing a beamforming direction and a frequency range; (iii) converting the simulation of locations from the spatial domain into corresponding frequency response values in a spatial frequency domain, the frequency response values having spatial frequency coordinates, such that: for each frequency of a plurality of frequencies in the frequency range, a spatial frequency contour, defined by a locus of points in the spatial frequency domain corresponding to the location of the frequency response values associated with a frequency response, as a function of direction, of the arrangement of the transducer elements with respect to that frequency and the beamforming direction, passes through an origin of the spatial frequency domain, wherein each of the spatial frequency contours for each of the plurality of frequencies intersect at the origin; (iv) determining the frequency response, as a function of direction, of the arrangement of transducer elements by applying a transformation to the frequency response values for the provided beamforming direction and frequency range, wherein applying the transformation comprises translating the spatial frequency domain into a modified frequency domain, such that the contours avoid intersecting; and (v) outputting the frequency response values in the modified frequency domain, as the frequency response, as a function of direction, of the arrangement of the transducer elements with respect to the beamforming direction.

16. A system comprising: a non-transitory computer-readable medium storing computer-executable program instructions; and a processing device communicatively coupled to the non-transitory computer-readable medium for executing the computer-executable program instructions, wherein executing the computer-executable program instructions configures the processing device to perform operations comprising: (i) providing a simulation of locations of an arrangement of transducer elements, in a spatial domain; (ii) providing a beamforming direction and a frequency range; (iii) converting the simulation of locations from the spatial domain into corresponding frequency response values in a spatial frequency domain, having spatial frequency coordinates, such that: for each frequency of a plurality of frequencies in the frequency range, a spatial frequency contour, defined by a locus of points in the spatial frequency domain corresponding to the location of the frequency response values associated with a frequency response, as a function of direction, of the arrangement of the transducer elements with respect to that frequency and the beamforming direction, passes through an origin of the spatial frequency domain, wherein each of the spatial frequency contours for each of the plurality of frequencies intersect at the origin; (iv) determining the frequency response, as a function of direction, of the arrangement of transducer elements by applying a transformation to the frequency response values for the provided beamforming direction and frequency range, wherein applying the transformation comprises translating the spatial frequency domain into a modified frequency domain, such that the contours avoid intersecting; and (v) outputting the frequency response values in the modified frequency domain, as the frequency response, as a function of direction, of the arrangement of the transducer elements with respect to the beamforming direction.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will now be described, purely by way of example, with reference to the accompanying drawings, in which;

(2) FIG. 1 shows a flow diagram of a method according to a first aspect of the invention.

(3) FIG. 2 shows a perspective view of an arrangement of transducer elements according to a first aspect of the invention.

(4) FIG. 3 shows a two-dimensional representation of the simulation of locations of the arrangement of transducer elements of FIG. 2 when converted into the spatial frequency domain according to a first aspect of the invention.

(5) FIG. 4 shows a two-dimensional representation of the directional frequency response of the arrangement of transducer elements of FIG. 2, as a polar plot, according to a first aspect of the invention.

(6) FIGS. 5a and 5b show a three-dimensional representation of the directional frequency response of the arrangement of transducer elements shown in FIG. 2, as a spherical polar plot, according to a first aspect of the invention.

(7) FIG. 6 shows a two-dimensional representation of the directional frequency response of the arrangement of transducer elements shown in FIG. 2, as a Cartesian plot, according to a first aspect of the invention.

(8) FIG. 7a shows a perspective view of a further arrangement of transducer elements according to a first aspect of the invention.

(9) FIG. 7b shows a two-dimensional representation of the directional frequency response of the further arrangement of transducer elements shown in FIG. 7a, as a polar plot, according to a first aspect of the invention.

(10) FIG. 8 shows a perspective view of yet a further arrangement of transducer elements according to a first aspect of the invention.

(11) FIG. 9a shows a two-dimensional representation of the determined directional frequency response of the arrangement of transducer elements shown in FIG. 8, as a polar plot, according to a first aspect of the invention.

(12) FIG. 9b shows a two-dimensional representation of the measured directional frequency response of the arrangement of transducer elements shown in FIG. 8, as a polar plot.

(13) FIG. 10a shows a two-dimensional representation of the determined directional frequency response of the arrangement of transducer elements shown in FIG. 8, as a Cartesian plot, according to a first aspect of the invention.

(14) FIG. 10b shows a two-dimensional representation of the measured directional frequency response of the arrangement of transducer elements shown in FIG. 8, as a Cartesian plot.

(15) The drawings are for illustrative purposes only and are not to scale.

DETAILED DESCRIPTION

(16) With reference to FIG. 1, in a first embodiment of the method of determining the directional frequency response of an arrangement of transducer elements, a simulation of the locations of an arrangement of transducer elements is provided 101 as a periodic spatial function. The transducer elements, having a two-dimensional arrangement, are located in an xy plane of the spatial domain, having spatial coordinates x,y. The transducer elements could equally have a one-dimensional or a three-dimensional arrangement. The periodic spatial function is determined by sampling the sensor space with an infinite grid of lattice points. A sin c filter is used as an anti-aliasing filter to determine appropriate gain values at each of the lattice points due to the proximity, or otherwise, of any transducer elements.

(17) A beamforming direction and frequency range are provided appropriate to the intended application 102, being to determine the directional frequency response of an arrangement of microphones for detecting audio signals. The beamforming direction is selected in the positive x direction and the frequency range is selected as 0 Hz to 24 kHz. A two-dimensional spatial Fast Fourier Transform is applied to the periodic spatial function to convert the periodic spatial function from the spatial domain to the spatial frequency domain 103. The spatial frequencies are converted into equivalent temporal frequencies by multiplication by the speed of propagation of sound in air.

(18) The step of determining the directional frequency response 104 is achieved by applying a transformation to the frequency response values for the selected beamforming direction, and frequency range.

(19) For each of the frequencies in the frequency range 0 Hz to 24 kHz, there is a locus of points, in the spatial frequency domain, having coordinates k.sub.x, k.sub.y, k.sub.z. Each point has a respective frequency response value corresponding to the magnitude, in decibels, of the directional response of the arrangement of transducer elements at this frequency.

(20) The locus of points for each of the frequencies defines a three-dimensional spherical spatial frequency contour passing through the origin k=0 of the spatial frequency domain.

(21) The spatial frequency domain is translated into the modified frequency domain, by applying the transformation:
k.sub.x=g.sub.x{square root over (g.sub.x.sup.2+g.sub.y.sup.2+g.sub.z.sup.2)},k.sub.y=g.sub.y, and k.sub.z=g.sub.z

(22) where g.sub.x, g.sub.y, g.sub.z are the modified frequency coordinates of the resulting modified frequency domain. Each of the spatial frequency contours is translated, such that, when mapped into the coordinates in the modified frequency domain, the modified frequency contours are arranged as a nested family of spherical contours, each being centred on the origin.

(23) The directional frequency response of the arrangement of transducer elements is outputted in three-dimensional polar coordinates derived from the Cartesian coordinates g.sub.x, g.sub.y, g.sub.z.

(24) With reference to FIG. 2, in a second embodiment of the method of determining the frequency response of an arrangement of transducer elements, 48 microphone transducer elements 206 are arranged in two dimensions in the xy plane of the spatial domain 207. Each of the 48 microphones is configured to receive sound in the selected frequency range of 0 Hz to 24 kHz. The beamforming direction is specified relative to the arrangement of transducer elements, in the positive x direction, to detect signals travelling in the negative x direction as indicated by arrow A. The frequency range of 0 Hz to 24 kHz is selected. Equally, other beamforming directions and frequency ranges may be selected.

(25) The 48 microphones are equidistantly spaced at 36 mm in the x direction and 39 mm in they direction, and, for the purposes of determining the simulation of their locations, are defined within a sensor space having dimensions of 3.6 m3.6 m. The resulting periodic spatial function of the arrangement of 48 microphones 206 is determined by application of a sin c anti-aliasing filter.

(26) With reference to FIG. 3, by means of the application of a spatial Fast Fourier Transform to the periodic spatial function, the simulation of locations of the arrangement of microphones 206 are converted from the spatial domain 207 into corresponding frequency response values in the spatial frequency domain 319, having coordinates k.sub.x, k.sub.y, k.sub.z. The spatial frequencies are converted to equivalent temporal frequencies in Hertz. The locus of points for each of the frequencies defines a spherical spatial frequency contour. For example, the sets of points corresponding to the frequencies 24 kHz, 18 kHz, 12 kHz, 6 kHz, define frequency contours 313, 314, 315, 316 respectively. All of the frequency contours 313, 314, 315, 316 pass through the origin k=0.

(27) Each point within the spatial frequency domain 319 has a respective frequency response value in decibels, as indicated by the degree of shading in FIG. 3. The straight lines B, C, D and E correspond to the angles 45, 90, 135 and 180 with respect to the beamforming direction.

(28) With reference to FIG. 4, the step of determining the directional frequency response is achieved by translating the spatial frequency domain 319, and the associated spatial frequency contours, into a modified frequency domain 419 according to the transformation:
k.sub.x=g.sub.x{square root over (g.sub.x.sup.2+g.sub.y.sup.2+g.sub.z.sup.2)},k.sub.y=g.sub.y, and k.sub.z=g.sub.z

(29) where y.sub.x, g.sub.y, g.sub.z are the coordinates of the resulting modified frequency domain 419.

(30) Respective frequency response values, associated with the spatial frequency contours 313, 314, 315, 316 in the spatial frequency domain 319, are translated, such that the resulting contours 413, 414, 415, 416 for each of the plurality of frequencies in the modified frequency domain 419, are arranged as a nested family of spherical contours, each centred on the origin. Frequency contours 413, 414, 415, 416 are the translation of the spatial frequency contours 313, 314, 315, 316 respectively. The straight lines G, H, I and J correspond to the directional frequency response at angles 45, 90, 135 and 180 with respect to the beamforming direction.

(31) Following the application of the transformation, the directional frequency response of the arrangement of microphones 206 is outputted in Cartesian coordinates (g.sub.x, g.sub.y), but could equally be outputted in polar coordinates (, f), as illustrated by the circular gridlines.

(32) With reference to FIGS. 5a and 5b, the translated frequency contours 513, 515, of the modified frequency domain 520, are illustrated in three-dimensional Cartesian coordinates (g.sub.x, g.sub.y, g.sub.z). Frequency contours 513 and 515 correspond to frequency contours 413 and 415 as illustrated in FIG. 4 in two dimensions.

(33) Thereby, the directional frequency response of the arrangement of microphones is outputted, as the modified frequency domain 419, as illustrated in two dimensions in FIG. 4, and in three dimensions in FIGS. 5a and 5b.

(34) With reference to FIG. 6, in a third embodiment of the method of determining the directional frequency response of an arrangement of transducer elements the modified frequency domain 622 is outputted, as the determined directional frequency response. The third embodiment is similar to the second embodiment, but the step of determining the directional frequency response is achieved by translating the spatial frequency domain 319, and the associated spatial frequency contours 313, 314, 315, 316, into a modified frequency domain 622 according to the transformation:
k.sub.x=f(cos 1); and k.sub.y=f sin ,

(35) where , f are the coordinates of the modified frequency domain 622, wherein is the angle, with respect to beamforming direction, and f is the frequency, of the resulting directional frequency response.

(36) The locus of points, and the respective frequency response values, associated with the spatial frequency contours 313, 314, 315, 316, are translated such that the frequency contours for each of the plurality of frequencies are arranged as respective parallel linear contours 613, 614, 615, 616. The straight lines K, L, M and N correspond to the angles 45, 90, 135 and 180 with respect to the beamforming direction.

(37) With reference to FIGS. 7a and 7b, in a fourth embodiment of the method of determining the frequency response of an arrangement of transducer elements, seventeen microphone transducer elements 706 are arranged in two-dimensional concentric rings in the xy plane of the spatial domain 707. Each of the seventeen microphones is configured to receive sound in the selected frequency range of 0 Hz to 12 kHz. The beamforming direction is specified relative to the arrangement of transducer elements 706, in the positive x direction, to detect signals travelling in the negative x direction as indicated by arrow Z. The frequency range of 0 Hz to 12 kHz is selected. Equally, other beamforming directions and frequency ranges may be selected.

(38) The seventeen microphones 706 are arranged in three concentric rings, and, for the purposes of determining the simulation of their locations, are defined within a sensor space having dimensions of 7.2 m7.2 m. The resulting periodic spatial function of the arrangement of seventeen microphones 706 is determined by application of a sin c anti-aliasing filter.

(39) The simulation of locations of the arrangement of microphones are converted from the spatial domain 707, into corresponding frequency response values in the spatial frequency domain using a spatial Fourier Transform. The spatial frequencies are converted to equivalent temporal frequencies in Hertz. The locus of points for each of the frequencies defines a frequency contour.

(40) The step of determining the directional frequency response, is achieved by application of the following transformation to the spatial frequency domain:
k.sub.x=g.sub.x{square root over (g.sub.x.sup.2+g.sub.y.sup.2+g.sub.z.sup.2)},k.sub.y=g.sub.y, and k.sub.z=g.sub.z

(41) where g.sub.x, g.sub.y, g.sub.z are the coordinates of the resulting modified frequency domain 719.

(42) Respective frequency response values, associated with the spatial frequency contours, are translated, such that, the spatial frequency contours for each of the plurality of frequencies are arranged as a nested family of spherical contours, each centred on the origin.

(43) Following the application of the transformation to each of the sets of points and the respective frequency response values, the frequency response of the arrangement of microphones 706 is outputted.

(44) The directional frequency response of the arrangement of transducer elements is outputted in three-dimensional polar coordinates , , f derived from the Cartesian coordinates g.sub.x, g.sub.y, g.sub.z according to the transformation:
g.sub.x=f cos cos
g.sub.y=f sin cos
g.sub.z=f sin

(45) With reference to FIG. 8, in a fifth embodiment of the method of determining the directional frequency response of an arrangement of transducer elements an arrangement of 16 microphone transducer elements 806 are arranged in two dimensions in the xy plane of the spatial domain 807. Each of the 16 microphones is configured to receive sound in the selected frequency range of 0 Hz to 24 kHz. The beamforming direction is specified relative to the arrangement of the transducer elements 806, in the positive x direction, to detect signals travelling in the negative x direction. The frequency range of 0 Hz to 24 kHz is selected.

(46) The simulation of the locations of the arrangement of 16 microphones 806 is provided by a periodic spatial function.

(47) By means of the application of a spatial Fast Fourier Transform to the periodic spatial function, the simulation of locations of the arrangement of microphones 806 are converted from the spatial domain 807 into corresponding frequency response values in the spatial frequency domain, having coordinates k.sub.x, k.sub.y, k.sub.z.

(48) With reference to FIG. 9a, the step of determining the directional frequency response is achieved by translating the spatial frequency domain, and the associated frequency contours, into a modified frequency domain 919 according to the transformation:
k.sub.x=g.sub.x{square root over (g.sub.x.sup.2+g.sub.y.sup.2+g.sub.z.sup.2)},k.sub.y=g.sub.y, and k.sub.z=g.sub.z

(49) where g.sub.x, g.sub.y, g.sub.z are the coordinates of the resulting modified frequency domain 919. FIG. 9a shows a representation of the resulting directional frequency response in polar coordinates , , f, which are derived from the Cartesian coordinates g.sub.x, g.sub.y, g.sub.z according to the transformation:
g.sub.x=f cos cos
g.sub.y=f sin cos
g.sub.z=f sin

(50) FIG. 9b shows the measured directional frequency response of the arrangement of microphones 806. The array of microphones was rotated on a turntable in an anechoic chamber in presence of test signals from a loudspeaker, from which the directional frequency response was calculated.

(51) In both FIG. 9a and FIG. 9b, the degree of shading represents the magnitude of the response of the arrangement of microphones 806. As can be seen, there is a good correlation between the magnitude of the directional frequency response in both FIGS. 9a and 9b. For example, the main lobe 923, 922 is clearly discernible and shows good degree of similarity in both the measured and the determined frequency response.

(52) With reference to FIGS. 10a and 10b, in a fifth embodiment of the method of determining the directional frequency response of an arrangement of transducer elements a measured directional frequency response and a determined directional frequency response of the arrangement of transducer elements 806 is outputted.

(53) The fifth embodiment is similar to the fourth embodiment, but the step of determining the directional frequency response is achieved by translating the spatial frequency domain into a modified frequency domain according to the transformation:
k.sub.x=f(cos 1); and k.sub.y=f sin ,

(54) where , f are the coordinates of the modified frequency domain, wherein sp is the angle, with respect to beamforming direction, and f is the frequency, of the resulting directional frequency response. FIG. 10a shows a representation of the resulting directional frequency response as a Cartesian plot.

(55) FIG. 10b shows a Cartesian representation of the measured directional frequency response of the arrangement of microphones 807, measured according to the method of embodiment 4 as used to produce figure the response illustrated in FIG. 9b.

(56) In both FIG. 10a and FIG. 10b, the degree of shading represents the magnitude of the response of the arrangement of microphones 807. As can be seen, there is a good correlation between the magnitude of the directional frequency response in both FIGS. 10a and 10b. For example, the main lobe 1023, 1022 is clearly discernible and shows good degree of similarity in both the measured and the determined directional frequency response.