The present invention relates generally to the field of loudspeakers and more specifically to means of controlling the directional characteristics of loudspeakers. Still more specifically, the present invention relates to the application of acoustic beamforming for controlling the directional characteristics of a loudspeaker unit comprising a plurality of individual loudspeaker drivers distributed over a surface.
The directivity of loudspeakers has been subject to extensive consideration among loudspeaker designers over the years. The general consensus appears to be that investigation of the correlation between loudspeaker directivity and various perceptual aspects may be of great importance in the development of future innovative sound systems. Recently, a literature study of the topic was presented by Evans et al. Based on a literature review concerning the optimal directivity for stereo reproduction, loudspeaker directivity control and previous investigations into the influence of directivity upon listeners, the inventors found that results from previous studies do not provide definite evidence for relationships between directivity type and the perceptual attribute under investigation. This is caused by the multiple kinds of loudspeakers used in the experiments, as sources with different directivity. It is apparent that this will introduce the risk of judging other parameters than directivity due to inherent differences in the individual loudspeaker types. It would consequently be advantageous to have access to a single loudspeaker or loudspeaker unit, the directional characteristics of which can be varied without affecting other parameters of the loudspeaker unit.
According to the invention there is provided a loudspeaker unit which offers an extended range of loudspeaker directivities. The loudspeaker unit according to the present invention implements controllable directivity, thereby providing a foundation for achieving supportive listening test data in future experimental investigations.
According to the invention there is provided a loudspeaker unit comprising a uniform circular array of loudspeaker drivers for broadband audio reproduction by means of acoustic beamforming. The loudspeaker unit according to the invention complies with a series of specifications and requirements valid for free field conditions: The beam pattern must be steerable to a certain focus direction in the horizontal plane (0-360°) and the beam width should be variable from an omni-directional to a narrow beam characteristic. Due to the fact that ideal conditions will not be ideally met in practice, side lobes (or secondary lobes) might be formed outside the main lobe direction. From a practical point of view, a side lobe level of −20 dB relative to the main lobe may be acceptable, but other—also more stringent—requirements may also be specified. Furthermore, the physical dimensions should be minimized in order to reduce room interaction. In the detailed description of the invention, a given target function will be implemented that satisfies frequency invariance in the frequency range 500 Hz to 4 kHz. The detailed description comprises both simulated directivity patterns obtained according to the teachings of the present invention and measured results from a real prototype loudspeaker unit, measured in an anechoic room.
Within the theory of beamforming, two concepts are often considered regarding the beamformer-weighting of the signals for each array element of the beamformer, when controlling a circular array: (1) Phase compensation and amplitude tapering, and (2) the concept of phase modes. The first method concerns optimizing beam pattern characteristics (e.g. half-power bandwidth of the main lobe and minimizing side lobe levels), while advantage is taken of the inherent circular periodicity using method (2).
In this specification, the directivity is defined as the ratio of the position dependent frequency response to the frequency response of a reference position. The directivity is evaluated only in the horizontal plane. The orientation is expressed in cylindrical coordinates and the directivity is given by the expression:
where (rref, φref) is the direction of the beam pattern focus, or equivalently, the maximum, and G is the frequency response.
The synthesis of the desired directivity or beam pattern is based on a spatial Fourier analysis. The procedure for determining the beamformer-weight for each array element (loudspeaker driver) is (1) the desired pattern is determined based on the specific directivity target function; (2) a spatial Fourier analysis of the directivity pattern is applied, and; (3) the weights are determined by the resulting Fourier coefficients and the sound field transfer function (from each element to a given observation point).
Theory
The following paragraphs briefly present the most important theory relevant for the present invention. The derivation of the sound field generated from a line source located on an infinitely long cylinder is outlined. The results of this derivation are later utilized as the transfer function applied in the present invention and for simulations of the resulting directivity characteristics. The presented theory winds up with a description of the method of the directivity pattern synthesis applied in the present invention: the concept of phase modes.
Line Source on a Cylinder
Referring to
The meaning of the symbols used in
Of special interest is the corresponding radial particle velocity of the surface:
where the coefficients Em are expressed by:
With ε0=2 and εm=1 for m>0. In the case where only a single line element of the cylinder is vibrating (see
with u0 being the line source velocity and dα→0. Thus, an infinitely long and thin line source is defined on the surface of a rigid cylinder of infinite extent. The Fourier-series expansion of this function is:
The boundary condition on the surface is given as ur(a, φ)=ua(φ), which implies that the coefficients Am must satisfy:
Some examples of the sound pressure normalized with the maximum value are shown in
Simulation results are based on a uniform circular array of line sources on an infinitely long cylinder. Expansion of the solution in order to simulate the sound field generated by an array can be performed. Here, the solution for N equidistant line sources is introduced by the superposition principle. A phase difference φ−φn in equation (2) is included in the cosine term, where
Each source contributes to the sound field, and hence, summing over N contributions in (2) is required in order to obtain the complete solution.
In general, this solution introduces a number of ideal conditions, which cannot be satisfied in practice. The length of the cylinder and line sources must obviously be truncated in a practical implementation. This implies that simulations for frequencies with wavelengths comparable or larger than the truncated cylinder length may not give proper results. On the other hand, this somewhat ideal solution accounts for near field terms and allows acoustic parameters to be determined analytically at any distance from the cylinder surface.
Phase Modes
In the following, a loudspeaker producing a specific directivity pattern, H(φ, f), is considered. A specific directivity pattern, H(φ, f), is considered. This target directivity can be approximated with an array consisting of N elements, by adjusting the amplitude and phase of the individual elements with specific element weight, wn(f)
where g(φ, r, φn, f) is the transfer function from the n'th array element, with angular position φn, to an observation point (φ, r) as shown in
Assuming a uniform circular array, consisting of arbitrary elements mounted on an arbitrary baffle geometry, the radiated directivity can be controlled using the concept of phase modes. Using this method, specific element weights are determined to adjust the array response. Making use of the circular periodicity inherent in the array configuration, the target directivity can be expanded into circular harmonics using a Fourier series representation,
where M is the number of harmonics truncating the general summation from −∞ to ∞. The complex coefficient of the Fourier series expansion ap can be numerically determined using the discrete Fourier transform as
where p is an integer and N is the total number of array elements.
In accordance with the placement of the elements in the array, the weights must be 2π periodic. Hence, the weights can be decomposed in circular harmonics:
Here, âp(f) denotes the Fourier coefficients of the expanded element weights (not to be confused with the corresponding coefficients of the target directivity ap(f)). Each harmonic of the target directivity can be determined through summation across the weighted array elements. The elements are described by the acoustic transfer function g(φ, r, φn, f) and weighted by the p'th harmonic of the element weights â(f)ejpφn
In case of the infinite cylinder with line sources, the transfer function g(φ, r, φn, f) is given by (2) and (7). Besides, analytical derivations g(φ, r, φn, f) may also be determined through FEM simulations or measurements.
Moving the constant âp(f) out of the summation and applying (10) it is possible to write (13) as
Rearranging the terms, it is possible to determine the unknown p'th harmonic of the specific element weights from the given harmonic of the target directivity and the obtained acoustic transfer function
The element weights are calculated from a summation of the M harmonics at the angle of the element angular position φn
Substituting the expression for âp into (16) the equation can be written as
Here, it is seen that the term ejpφ is a scaling factor, which does not change between the calculations of the specific element weights. Hence, it can be calculated for an arbitrary angle (e.g. φ=0 for simplicity).
Weighting the array elements with the calculated values results in the directivity Ĥ(φ, f), which ideally is equal to the target directivity H(φ, f)
According to a first aspect of the invention there is provided a method for controlling the directivity of a sound-emitting device, the method comprising:
According to a second aspect of the present invention, there is provided a circular loudspeaker array with controllable directivity comprising a plurality of sound sources distributed over the surface of a body, each of the sound sources being driven by a separate power amplifier, the input terminal of which is provided with the output signal from a corresponding filter, such that the frequency response of each individual sound source can be controlled, where each filter is provided with an input signal corresponding to a plurality of input channels Ch1, Ch2 . . . ChN.
According to an embodiment of the invention, the sound-emitting device comprises a cylindrical body provided with end pieces at either longitudinal end.
According to an embodiment of the invention, the sound sources are uniformly distributed over a circular path on the surface of the body, specifically (but not limited hereto) over a circular path substantially in parallel with the end pieces. According to a specific embodiment of the invention, the surface of the body is substantially rigid.
According to an embodiment of the invention, each of the filters has filter characteristics that are determined according to the method defined above. Other methods of determining suitable filter characteristics may however be applied.
The invention will be better understood by reading the following detailed description of an embodiment of the invention and with reference to the figures of the drawing, wherein:
a) shows a schematic representation of an embodiment of the invention comprising six loudspeakers:
b) shows a photo of the measurement setup for the experimental study of a uniform circular array with six 2″ loudspeaker drivers in an anechoic room;
In the following, both simulated results of the application of the beamforming method of the invention and actual measurements using a simple six-loudspeaker embodiment are shown.
Simulations
The simulation results presented below were made for an infinite cylinder with equidistantly spaced line sources as outlined above. The chosen array configuration consists of 24 line sources positioned on an infinite cylinder with a=0.15 m. This combination of elements and array radius allows beamforming up to a frequency of approximately 4.3 kHz, according to the sampling criterion. The directivity pattern of the array is obtained under free field conditions, which removes otherwise disturbing reflections from the simulation. To avoid influence of near field components, the simulated directivity is determined from the sound pressure at a radial distance of r=3.5 m. Determining the directivity at a specific radial distance is possible due to the chosen derivation as presented above. The main lobe of the target directivity is oriented towards 0° and the corresponding weights applied to the array elements are calculated following the procedure described in the above section on phase modes.
Target Function
Applying the concept of phase modes to control the array makes it possible to form a large variety of directivity patterns. The chosen target directivity pattern for the simulation has the smallest beam width desired for the psychoacoustic experiments mentioned in the background of the invention. This corresponds to a beam width of 23° at 3 dB pressure attenuation (being equal to half power bandwidth assuming far field conditions). Due to the narrow beam width, this pattern is especially demanding to realize, as the transition from main lobe with high amplitude to reduced level occurs across a small angular variation. This steep slope necessitates accurate control of the array elements to facilitate such destructive interference. It is assumed that if this narrow pattern can be realized, the remaining less demanding and broader directivities can also be realized using this configuration.
a) shows a contour plot of the target directivity pattern across frequency. The target is shown for comparison purposes, and according to the used directivity definition (1), the calculated pressure is normalized with respect to the pressure in the focus direction. The straight contour lines of the target response reflect frequency invariance of the target directivity. In relation to the goals defined previously, frequency invariance is desired in the specified frequency interval of concern.
Target Function Realization
Through a simulation study it has been found that the target directivity could be formed with side lobe level below −20 dB, within the frequency range of concern, using 24 elements in the circular array. Side lobe level is here defined as the dB difference between the maximal amplitude of the main lobe and the amplitude of the side lobes.
Increasing the number of elements reduces the separation distance between the individual elements, which increases both the upper and lower frequency limit. A low frequency limit is effectively introduced due to an increased sensitivity to errors in the element weights at low frequencies. This effect is caused by the inter-element wavelength distance being very small, hence the phase change in the sound pressure emitted from one source at the position of neighboring sources is accordingly small. Hereby, a large amount of destructive interference is needed to form the correct directivity pattern. The amount of destructive interference is illustrated with
Introducing Errors in the Simulation
Constructing a circular array on a cylindrical baffle introduces errors in the system due to production tolerances. The main error sources arise from non-uniform angular placement of the elements and deviations in the baffle geometry compared with the ideal case. The significance of such errors can be illustrated by inclusion in the simulation of the array. The non-uniform placement of array elements is modeled by adding uniform distributed random variation to the angular position. Deviations in the baffle structure and finite approximations of the infinite cylinder and line source are modeled as a uniformly distributed random variation in the transfer function (symbolized with g(φ, r, φn, f) in (13)) used in the design procedure of the element weights.
a) shows the effect of a random variation in angular element placement of ±1°. It is seen that the angular variation highly affects the realized directivity pattern, especially at low frequencies, as indicated by reference numeral 3 in
b) shows the effects of adding random variation of 0.5 dB amplitude to the transfer function, which the element weights are based upon. Again the effect (reference numeral 4) of the variation is seen mainly at low frequencies where the concept is most sensitive.
In
Experimental Results
The concept of phase modes described above has been examined experimentally using a uniform circular array consisting of six equidistant loudspeaker drivers. The objective of the experiments presented in this section has been to verify the applicability of the concept of phase modes as a beamforming method. A small scale model was implemented primarily in order to verify the theory and simulations. In order to be able to reproduce the range of directivity patterns described previously, a larger number of loudspeakers should be used, for instance four times as many loudspeakers as in the small scale model described in this section. However, in order to examine the aspect of frequency invariance, a single predefined directivity pattern was utilized as target. The directivity target is shown in
As shown in
Pure tone signals were processed by off-line filtering and played back using a PC. The specific directivity pattern, used as reference in the experiments, was determined at five frequencies 500, 700, 1000, 1400 and 2000 Hz. Two different approaches for determining the weights of each array element were applied: In the first approach the sound field transfer function g(φ, r, φn, f) was simulated using the theory presented above, while in the second approach the transfer function was measured in a large anechoic room for each array element and applied. These two cases will be referred to in the following as gsim and gmeas, respectively. The experimental results are shown in
Discussion of Results Obtained with the Small Scale Embodiment of the Invention
Comparing the measured results reveals significant differences between measurements obtained with gsim and gmeas, respectively. It is apparent that better performance is obtained when the utilized element weights are determined by the measured transfer function for the physical array structure under test, rather than an ideal simulation. When the weights are determined using the ideal simulated transfer function, this evidently implies that deviations from the actual physical sound field will occur. This corresponds to an error imposed on g(φ, r, φn, f) and the performance is getting worse, as expected. However, this is mainly found for the side and back lobes, whereas the main lobe is generally well reproduced, at least above 500 Hz.
Evaluating frequency invariant reproduction of a specific beam pattern is a rather difficult task considering only six elements. However, the results presented in
Inaccuracies in the practical implementation are inevitable and are seen to affect the experimental results. This is further aggravated when a hand-made model is under consideration. In the above section on the introduction of errors in the simulation it was shown that severe side lobes occur at low frequencies when introducing a random error in the angular placement of the array elements. The measurements below 1400 Hz seem to reveal a consistent error around 90° to 180° where the side lobes are not resolved properly. Presumably, these errors are associated with tolerances in the handcrafted small scale model with respect to the actual positioning of elements comprising a circular array. Individual differences in the resulting directivities measured using gsim and gmeas reveal whether the errors may be attributed to deviations from the physical sound field. However, the errors are found also when applying the measured transfer function, which must contribute with a somewhat more realistic representation of the sound field. This suggests that the requirement of a cylindrical array of equidistant elements is not perfectly met. In order to support this hypothesis, a simulation of a similar setup was performed with a random angular error imposed on all elements except at focus direction. The errors imposed on the angular position is within the range of ±4° and maintained over frequency. The results are shown in
The weights determined for each of the six elements are calculated for focus direction equal to the angular orientation of an element. It is apparent that the resolution of the focus direction heavily depends on the number of elements implemented. The directivity patterns presented in
It is important to apply a control scheme for the number of active elements in the beamforming process that can maintain performance at low frequencies. The high sensitivity found here, is due to the interaction between low frequency excitation and a large density of array elements (relative to the wavelength of sound at low frequencies). A solution would be to reduce the number of loudspeakers contributing to the beamforming, for instance by a factor of two, below a specific frequency limit.
Conclusions
From the simulated results it has been demonstrated that it is possible to realize the desired directivity patterns across a frequency range from 500 to 4000 Hz applying 24 sources on a 0.15 m radius uniform circular array. However, it was also seen that using phase modes to control the beamforming introduces high sensitivity towards element weight errors at low frequencies.
A small scale practical embodiment of the invention comprising six 2″ “full-range” loudspeaker drivers mounted in a 0.1 m radius circular array has been implemented as described above. Even though it was not possible to realize the target directivities with six sources, the measurements obtained using the small scale embodiment showed very good agreement between measurements and the expected results from the simulations at 1000 to 2000 Hz. Significant deviations in the low frequency range 500 to 700 Hz might be attributed to production inaccuracies.
From these results it is concluded that the target directivity patterns can be realized with a practical 24 element array mounted on a 0.15 m radius circular array in the high range of the frequency range of interest. By plotting the ratio of the maximal sound pressure at the main lobe of the array to the on-axis pressure from a corresponding single source, a very large reduction in sound pressure was seen at low frequencies. From these results, problems due to high weight error sensitivity are expected below approximately 1700 Hz in the practical 24 element array.
Both the desired directivity 15 and the directivity 16 of the specific source are decomposed into p harmonics in the respective steps 20 and 21. Here, (10), describing target directivity decomposed in circular harmonics, and (11), describing target harmonic strength calculated using DFT, may be used.
For the directivity 16 of the source, a summation of array element contributions to the p'th harmonic of the unweighted array is determined in step 22, after which harmonic weights are calculated as the ratio of desired harmonic strength and the harmonic strength of the unweighted array (step 23 where weightp=(desired harmonic strength/source harmonic strength). Here (12), describing that a weighting function may be regarded as a 2pi periodic function which can be decomposed into weight harmonics, (13), which describes target harmonic determined from transfer function and weight harmonic, and (15), describing weight harmonic strength determined from target harmonic strength and transfer function harmonic strength, may be used.
After this the sourceweight(n) is calculated through a summation of harmonic weights at source positions (step 24 where sourceweight(n)=sum over harmonics of weightp(φn)). Here (16), describing source weights determined by summation of weight harmonics at angular source position, may be used.
These calculations result in the weights for each source (loudspeaker) 26 in the array 25 at the specific frequency where the shown source directivity 16 applies. Once the weights have been calculated for each frequency of interest, the complete frequency response of each individual filter 10 implementing the source weights as a function of frequency can be constructed.
These expressions have already been given in the preceding paragraphs.
Alternative methods for determining the filter characteristics may also be used without departing from the scope of the invention. In practice it would also be possible to avoid measuring the sound source directivities and to decompose these in circular harmonics. Instead it is possible to measure or calculate the transfer function g(φ, r, φn, f) between each individual source and a single observation point. For each circular harmonic, this transfer function is provided with a phase change corresponding to the position of the particular sound source in the array for the specific circular harmonic. This determines how much the individual source contributes to the circular harmonic in the observation point. Summing these contributions from each source results in an indication of how much the unweighted array excites each individual circular harmonic. This result is subsequently used to determine the weights based on how the array itself excites the circular harmonics and how they have to be excited in order to obtain the target directivity of the array.
It should be noted that the control method for providing the weights for each sound source according to the invention only requires that the sources be placed uniformly in a circle. The design of the baffle has no consequence as it is only needed to know the transfer function of the sources in order to be able to control the array. The number of sources will depend on the radius of the array, which precision is desired and within which frequency interval the desired directivity is to be obtained.
Number | Date | Country | Kind |
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PA 2010 00446 | May 2010 | DK | national |
Filing Document | Filing Date | Country | Kind | 371c Date |
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PCT/EP11/57532 | 5/10/2011 | WO | 00 | 11/9/2012 |