PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Design of a constant beamwidth beamformer for the parametric
array loudspeaker
Chuang SHI; Ruyu BAI
University of Electronic Science and Technology of China, Chengdu, China
Corresponding email: shichuang@uestc.edu.cn

ABSTRACTThe parametric array loudspeaker (PAL) is an unconventional type of directional sound source that transmits
inaudible ultrasonic frequencies in air to generate the audible difference frequency based on the parametric
array effect. The convolution model describes the beampattern of the PAL by convolving the productbeampattern with the Westervelt beampattern. As a result, the beamwidth of the PAL becomes narrower when
the difference frequency increases. A change of timbre is therefore perceived when people stand at an off-axis
angle of the PAL. Hence, a constant beamwidth beamformer is required. This paper works on the beamformer
design to result in the product beampattern that becomes broader when the difference frequency increases, inorder for the Westervelt beampattern to be compensated. A constant half-power-beamwidth (HPBW) is
successfully achieved for the typical frequency range of the PAL.
Keywords: Parametric array loudspeaker, Beamformer design, Constant beamwidth

1. INTRODUCTION
The principle of the PAL is explained by the parametric array effect (1, 2). When two ultrasonicfrequencies are transmitted together, the acoustic energy of the ultrasonic frequencies is transferred
into new frequencies by the nonlinearity of air. Among those new frequencies, the difference
frequency can be made audible to human beings (3, 4). The difference frequency inherits the narrow
beam of ultrasonic frequencies. Therefore, the PAL is known to be one of the most effective means tocreate a directional sound beam (5, 6). A typical design of the PAL consists of a driver circuit and anarray of ultrasonic transducers. One of the two ultrasonic frequencies is set as the carrier frequency andthe other is given by the sideband frequency.
PALs can be readily deployed in applications of sound field control, whereby tailored enhancement
and attenuation of acoustic components are required (7, 8). With phased array technique, PALs possess
controllable beampatterns. They are also known as steerable PALs (9, 10). There have been several
attempts to develop sophisticated array signal processing methods for the steerable PAL (11, 12).
However, the theoretical works are far from practical implementations, because of a lack of effective
directivity models. Since 2015, the convolution model has been established, providing an accurate
directivity model for the PAL (13). The convolution model has opened an opportunity in thebeampattern design of the PAL. The desire for a constant beamwidth originates from the timbre
variation perceived by people standing at an off-axis angle of the PAL. Therefore, this paper presents
a constant HPBW beamformer specially designed for the PAL consisting of 100 ultrasonic transducers.

2. CONVOLUTION MODEL OF PALS
The well-known Westervelt’s directivity ‫ܦ‬ௐ ሺߠሻ, or called the Westervelt beampattern in this paper,is expressed asͳ‫ܦ‬ௐ ሺߠሻ ൌ(1)߱ଶ tanସ ߠඨͳ ൅ ଶ ௗͶܿ଴ ሺߙଵ ൅ ߙଶ ሻଶ
where ߱ௗ is the difference frequency; ܿ଴ is the speed of sound; ߙଵ and ߙଶ are the absorption
coefficients of the two ultrasonic frequencies. Sometimes, ߙଵ ൅ ߙଶ ൌ ʹߙ଴ is approximated, where3077

ߙ଴ is the absorption coefficient of the center frequency. The Westervelt beamwidth gets narrower
when ߱ௗ increases or ߙ଴ decreases. Figure 1 shows the Westervelt beampattern and HPBW withrespect to the difference frequency when the carrier frequency is 40 kHz.

Figure 1 – Westervelt beampattern and beamwidth for a typical PAL
The convolution model explains that the difference frequency beampattern is estimated by theconvolution between the Westervelt beampattern and the product beampattern of the two ultrasonicfrequencies, i.e.
‫ܦ‬ௗ ሺߠሻ ൌ ሾ‫ܦ‬ଵ ሺߠሻ ൈ ‫ܦ‬ଶ ሺߠሻሿ ٘ ‫ܦ‬ௐ ሺߠሻ,(2)where ‫ܦ‬ଵ ሺߠሻ and ‫ܦ‬ଶ ሺߠሻ are the beampatterns of the ultrasonic transducer array at the two ultrasonicfrequencies; and ٘ denotes the convolution operator. Considering the beampatterns of individual
ultrasonic transducers, equation (2) is further extended to‫ܦ‬ௗ ሺߠሻ ൌ ሾ‫ܣ‬ଵ ሺߠሻ ൈ ‫ܦ‬ଵ ሺߠሻ ൈ ‫ܣ‬ଶ ሺߠሻ ൈ ‫ܦ‬ଶ ሺߠሻሿ ٘ ‫ܦ‬ௐ ሺߠሻ,(3)
where ‫ܣ‬ଵ ሺߠሻ and ‫ܣ‬ଶ ሺߠሻ are the beampattern of the ultrasonic transducers at the two ultrasonicfrequencies. The ultrasonic transducers can be modeled as piston sources, whereby a larger radius
leads to a narrower beampattern.

3. CONSTANT BEAMWIDTH BEAMFORMER DESIGN
Table 1 lists the parameters of the PAL considered in the constant HPBW beamformer design.
Table 1 – Parameters in the constant HPBW beamformer design
Diameter of ultrasonic transducers

0.08 meter

Spacing between ultrasonic transducers

0.10 meter

Number of ultrasonic transducers

100Carrier frequency

40000 HzIn order to compensate for the Westervelt beampattern, the constant HPBW beamformer has to also
include an adjustable parameter with a large dynamic range. Therefore, the Dolph-Chebyshev weights
are adopted. The product beampattern of the two ultrasonic frequencies can be adjusted largely by thesidelobe attenuation.
Firstly, the sidelobe attenuation of the carrier frequency is fixed at 6 dB. The lower and upper single
sideband modulation methods are considered. They have only one sideband. The sidelobe attenuationof the sideband frequency varies from 6 dB to 50 dB. The contour plot of the resultant HPBW of thedifference frequency based on the convolution model is shown in Figure 2. With exception for the line
representing 25 degrees of HPBW, no other line is able to cover the typical frequency range of the PAL,

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which is from 500 Hz to 8000 Hz. Moreover, the difference between the contour plots of the lower andupper single sideband modulation methods is trivial. This suggests that 25 degrees of HPBW is also
applicable to the double sideband modulation method.

Figure 2 – Difference frequency HPBW with respect to the sidelobe attenuation of the sideband frequency
Secondly, the sidelobe attenuations resulting in 25 degrees of HPBW are extracted for differencefrequencies. After which, they are used to obtain the constant beamwidth beampatterns. The results arealmost the same for the single and double sideband modulation methods. To maintain presentation
clarity, only the result of the double sideband modulation method is shown in Figure 3. To illustrate theconstant beamwidth, the magnitude above -3dB are colored with yellow in the subplot below.

Figure 3 – Constant beamwidth beampattern of the double sideband modulation method
Furthermore, the second harmonic beampattern is obtained by the convolution model, whereby theWestervelt beampattern is computed at the second harmonic frequency and beampatterns of the two
sideband frequencies in the double sideband modulation method are multiplied. The second harmonic
distortion is calculated by the ratio of the difference frequency beampattern and the second harmonicbeampattern. The results are shown in Figure 4. The HPBW of the second harmonic beampattern isnarrower than that of the difference frequency beampattern. This results in a rapid decay of the second
harmonic distortion at off-axis angles of the PAL, especially for relatively low difference frequencies.

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Figure 4 – Second harmonic beampattern of the double sideband modulation method

4. CONCLUSIONSIn this paper, a constant HPBW beamformer is designed for the PAL, which has been constructedusing 100 ultrasonic transducers. A constant HPBW is achieved from 500 Hz to 8000 Hz, when thespacing between the ultrasonic transducers is 0.01 meter. The Dolph-Chebyshev weights are adopted.
An initial testing shows that only 25 degrees of HPBW can be designed to cover the typical frequency
range of the PAL. It has been observed that because of the Westervelt beampattern, several strong
constraints have been incurred in the constant HPBW beamformer design. Therefore, a systematicapproach is yet to develop. Furthermore, when the double sideband modulation method is applied, theconstant HPBW beamformer is achieved for the first time. Since the second harmonic beampattern isnarrower, it is advantageous to listen to the PAL on a slightly off -axis angle for reduced second
harmonic distortion.

ACKNOWLEDGEMENTS
This work is jointly supported by the National Science Foundation of China (Grant No. 61701090)
and Sichuan Science and Technology Program (Project No. 2019YJ0182).

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