PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Three-dimensional resonance control based on spatial wave
synthesis with parametric array loudspeaker
Shiori SAYAMA1; Masato NAKAYAMA2, 3; Takanobu NISHIURA4
1

Graduate School of Information Science and Engineering, Ritsumeikan University, Japan
2
3

College of Design Technology, Osaka Sangyo University, Japan

Research Organization of Science and Technology, Ritsumeikan University, Japan
4

College of Information Science and Engineering, Ritsumeikan University, Japan

ABSTRACT
A parametric array loudspeaker (PAL) realizes a sharper directivity utilizing a demodulation of an intense
amplitude modulated (AM) wave. The demodulation is modeled as spatial sound sources generated with thedemodulated audible sound in the acoustic beam. Spatial sound sources form resonance and anti-resonancepositions because of the interference between spatial sound sources demodulated with the non-linear
interaction in the air. Thus, the sound pressure level (SPL) of the PAL is reduced at anti-resonance positions.
However, it suggests that resonance positions might be reformed by spatial wave synthesis with phased arrayprocessing. Therefore, we propose three-dimensional resonance control based on spatial wave synthesis with
the PAL. In this paper, we employ the PAL consists of line-type PALs corresponding to column. The PAL canapply different delay filters to each column. The proposed method designs inverse filters with a multichannel
adaptive algorithm as delay filters for line-type PALs under the constraint condition that the SPL is
maximized by spatial wave synthesis at the target position. The proposed method enables to influence thenon-linear interaction and improve the SPL at the target position by applying designed filters to each column.
Finally, we confirmed the effectiveness of the proposed method through evaluation experiments.
Keywords: Parametric array loudspeaker, Resonance control, Spatial wave synthesis

1. INTRODUCTION
Recently, an electrodynamic loudspeaker is widely used to emit audible sound such as music andvoice. The electrodynamic loudspeaker has a wider directivity and can transmit the audible sound to awide area. However, it is difficult for the electrodynamic loudspeaker to emit the audible sound to an
only target listener who requires the audible sound. Therefore, the audible sound emitted by theelectrodynamic loudspeaker becomes a noise for the non-listener [1]. To solve this problem, aparametric array loudspeaker (PAL) [2, 3] which has a sharper directivity has recently been focused on.
The PAL realizes the sharper directivity by utilizing an ultrasonic wave. The PAL emits an intense
amplitude modulated (AM) wave which is designed by modulating the ultrasonic wave (carrier wave)
with an audible sound. The emitted intense AM wave from the PAL is gradually demodulated into theoriginal audible sound by the non-linear interaction in the process of propagation in the air [4]. In theprocess of the demodulation, the PAL generates spatial sound sources with the demodulated audible
sound and constructs a vertical array in the acoustic beam with the non-linear interaction. Therefore,
the PAL can realize the sharper directivity and emits the audible sound in a particular area [4, 5, 6].
The sound pressure level (SPL) of the audible sound emitted from the electrodynamic loudspeakerdecay by distance right after it is emitted. On the other hand, the SPL of the demodulated audible soundgradually increases with the occurring demodulation. The demodulated audible sound has the
maximum SPL at the distance which ceases the demodulation. After ceasing the demodulation, the
SPL of the demodulated audible sound decays by distance. Furthermore, spatial sound sources form
1
2, 3
4

is0309hk@ed.ritsumei.ac.jp
nakayama@ise.osaka-sandai.ac.jp
nishiura@is.ritsumei.ac.jp1053

resonance and anti-resonance positions of the demodulated audible sound in the acoustic beam
because of the interference between spatial sound sources demodulated with the non-linear interaction.
Thus, the SPL of the demodulated audible sound is attenuated at anti-resonance positions. To control
the resonance positions, we propose the method to reform them by spatial wave synthesis with phasedarray processing. In the preliminary experiment, we confirmed that resonance positions can bereformed by emitting AM waves imparted different delays from each line-type PAL constituting the
PAL. It is considered that imparting delays to each column influences the composite waves
demodulated by the non-linear interaction in the air. In addition, we confirmed that the SPL of thedemodulated audible sound increases in the preliminary experiment. The proposed method designs
inverse filters with a multichannel adaptive algorithm as delay filters for line-type PALs to improve
the SPL of the demodulated audible sound at the target position. We design filters under constraint
condition that the SPL of the demodulated audible sound is maximized by spatial wave synthesis. By
applying designed filters which control phases to each column, we change the sound pressure
distribution of resonance positions and improve the SPL at the target position. As a result of theevaluation experiment, we confirm the effectiveness of the proposed method.

2. PARAMETRIC ARRAY LOUDSPEAKER
2.1 Principle of Parametric Array Loudspeaker
The PAL realizes a sharper directivity by utilizing an ultrasonic wave and a parametric array effect.
Generally, the high frequency wave has the straightness than the low frequency wave. For this reason,
the PAL utilizing the ultrasonic wave, which has high frequency, can transmit the sound straightly. In
addition, the PAL utilizes the parametric array effect by consisting of multiple small ultrasonictransducers with close-packed structure. When AM waves are emitted from each ultrasonic transducer
simultaneously, they are synthesized spatially and realize the directivity. The PAL emits an intense AM
wave which is designed by modulating the carrier wave with an audible sound. Figure 1 shows theoverview of the demodulation of the emitted AM wave in time domain. The carrier wave 𝑣" (𝑡) and the
audible sound 𝑣& (𝑡) are indicated as follows:𝑣' (𝑡) = 𝐴' cos(2𝜋𝑓' 𝑡),(1)𝑣0 (𝑡) = 𝐴0 cos(2𝜋𝑓0 𝑡),(2)where 𝑡 represents a time index, 𝐴" and 𝐴& represent the maximum amplitudes of the carrier wave
and the audible sound, 𝑓" and 𝑓& represent the frequencies of the carrier wave and the audible sound.
The AM wave 𝑣12 (𝑡) is designed by modulating the carrier wave 𝑣" (𝑡) with the audible sound 𝑣& (𝑡).
From Eq. (1) and (2), the AM wave 𝑣12 (𝑡) is indicated as follows:
𝑣12 (𝑡) = {1 + 𝑚 ∙ 𝑣0 (𝑡)}𝑣' (𝑡),(3)𝑚 = 𝐴0 ⁄𝐴' ,(4)where 𝑚 (0 < 𝑚 ≤ 1) is an amplitude modulation index. Equation (3) can be transformed as follows
by trigonometric functions.
𝑣12 (𝑡) = {1 + 𝑚 ∙ 𝑣0 (𝑡)}𝑣' (𝑡)= {1 + 𝑚𝐴0 cos(2𝜋𝑓0 𝑡)}𝐴' cos(2𝜋𝑓' 𝑡)= 𝐴' cos(2𝜋𝑓' 𝑡) + 𝑚𝐴0 𝐴' cos(2𝜋𝑓0 𝑡) cos(2𝜋𝑓' 𝑡)= 𝐴' cos(2𝜋𝑓' 𝑡) + 𝑚𝐴0 𝐴' cos{2𝜋(𝑓' + 𝑓0 )𝑡}⁄2 + 𝑚𝐴0 𝐴' cos{2𝜋(𝑓' − 𝑓0 )𝑡}⁄2.

(5)

From Eq. (5), we can confirm that the AM wave consists of the carrier wave (frequency: 𝑓' ), thehigher sideband wave (frequency: 𝑓' + 𝑓0 ) and the lower sideband wave (frequency: 𝑓' − 𝑓0 ). Whenthe intense AM wave as a primary wave is emitted from the PAL, the sum and difference frequency
waves of the carrier wave and sideband waves are secondarily demodulated along the primary beam in
the process of propagation in the air due to the non-linear interaction. Generally, the high frequency
wave decay more rapidly than the low frequency wave because the sound absorption increases withfrequency. Therefore, only the difference frequency wave, which has low frequency and is perceived
by humans, can be transmitted an appreciable distance and remain sufficient amplitudes than the highfrequency wave such as the carrier wave.

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Figure 1 – Overview of the demodulation of the emitted AM wave (Time domain).
2.2 Generation of Virtual Sound Source
Figure 2 shows the generation model of a parametric array [1]. As mentioned in the previous
section, the intense AM wave as a primary beam emitted from the PAL is demodulated in the process ofthe propagation in the air due to the non-linear interaction. In the primary beam, secondary spatial
sound sources of the difference frequency wave are virtually created and distribute along a narrow
beam. This generation model is referred as the parametric array. In the parametric array, each spatial
sound source operates as a point sound source. Therefore, they form resonance and anti-resonancepositions in the acoustic beam because of the interference between spatial sound sources demodulatedwith the non-linear interaction in the air. Figure 3 shows the relationship between the distance and the
SPL of the demodulated audible sound. In the Fig. 3, a horizontal axis shows the distance from the PAL
and a vertical axis shows the power of the demodulated audible sound. From Fig. 3, we can confirm
that the several resonance and anti-resonance positions are generated and the power of thedemodulated audible sound mottles in the audible area. Hence, the listener who stands at theanti-resonance position may obtain the demodulated audible sound with insufficient power. In thispaper, we propose the method to control the generation process of resonance positions and improve the
SPL of the demodulated audible sound at the target position.

Figure 2 – Generation model of parametric array.

Figure 3 – Relationship between the distance and the SPL of the demodulated audible sound.

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3. PROPOSAL OF THREE-DIMENSIONAL RESONANCE CONTROL BASED ON
SPATIAL WAVE SYNTHESIS WITH PARAMETRIC ARRAY LOUDSPEAKER
3.1 Overview of Proposed Method
In this paper, we propose the method to control three-dimensional resonance positions of thedemodulated wave. We reform resonance positions by spatial wave synthesis with phased arrayprocessing with the PAL. Figure 4 shows the overview of the proposed method. In Fig. 4, 𝑥(𝑡) shows
the input signal and 𝑔B (𝑡)(𝑖 = 1, 2, ⋯ , 𝑀) shows the transfer function, where 𝑖 represents a column
index and M is a number of line-type PALs. ℎB (𝑡)(𝑖 = 1, 2, ⋯ , 𝑀) shows the inverse filter. Figure 5
shows the PAL which we use in this paper. In Fig. 4 and 5, 𝑆B (𝑖 = 1, 2, ⋯ , 𝑀) shows the line-type PALof 𝑖th column. The PAL is consisted of some line-type PALs corresponding to column and enables
array processing. In the proposed method, we design some delay filters for each column to improve the
SPL of the demodulated audible sound at the target position. By imparting delays to each column, it ispossible to influence the composite waves demodulated by non-linear interaction in the air. As delay
filters, the proposed method designs inverse filters for line-type PALs under the constraint condition
that the SPL is maximized by spatial wave synthesis at the target position. The inverse filter has theinverse characteristics of a system. In this paper, we utilize inverse filters in order to correct transfer
functions between each line-type PAL and the target position. It is possible to change the phase of thesound emitted from some line-type PALs by designing inverse filters in consideration of phases.
In the proposed method, we first measure transfer functions 𝑔B (𝑡) between each line-type PAL andthe target position. Transfer functions can be measured using the TSP method [8]. Next, in the Step2,
we design inverse filters ℎB (𝑡) with a multichannel adaptive algorithm in order to add different delays
to each line-type PAL. Details on calculation of the inverse filter coefficient are described in the next
section. The filtered input signal 𝑣0I (𝑡) is indicated as follows:
𝑣K B (𝑡) = 𝑥(𝑡) ∗ ℎB (𝑡),(12)where ∗ means convolution operation. In the proposed method, we modulate the carrier wave 𝑣12I (𝑡)with the filtered input signal 𝑣0I (𝑡) and emit the modulated wave 𝑣12I (𝑡) from each line-type PAL.
The modulated wave 𝑣12I (𝑡) is indicated as follows:
𝑣LM B (𝑡) = {1 + 𝑚 ∙ 𝑣K B (𝑡)}𝑣' (𝑡).

(13)

By emitting the AM wave applying designed filters which control phases to each column, we enable toinfluence the non-linear interaction and change the process of the spatial wave synthesis. Therefore, it
becomes to change the sound pressure distribution of resonance positions and improve the SPL at thetarget position.

Figure 4 – Overview of the proposed method.

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Figure 5 – PAL consisted of some line-type PALs.
3.2 Calculation of inverse filter coefficient
The proposed method utilizes the NLMS for the adaptive algorithm [9]. A filter design procedure isfollowing:Calculate the input signal 𝑟B (𝑡)(𝑖 = 1, 2, ⋯ , 𝑀) of the adaptive algorithm as follows:𝑟B (𝑡) = 𝑥(𝑡) ∗ 𝑔B (𝑡).

i.

ii.

Set initial coefficient ℎB (0) as time 𝑡 = 0.

iii.

Calculate the output signal 𝑦(𝑡) and error signal 𝑒(𝑡) as follows:
T𝑦(𝑡) ≈ ∑M
BVW 𝒉B (𝑡) 𝒓B (𝑡),e(𝑡) = 𝑑(𝑡) − 𝑦(𝑡),

(6)

(7)
(8)

where 𝒓B (𝑡) and 𝒉B (𝑡) are given as follows:𝒓B (𝑡) = [𝑟B (𝑡), 𝑟B (𝑡 − 1), ⋯ , 𝑟B (𝑡 − 𝐿 + 1)]T ,

(9)

T

(10)
𝒉B (𝑡) = ]ℎB,W (𝑡), ℎB,^ (𝑡), ⋯ , ℎB,_ (𝑡)` ,where L represents the length of the filter coefficient. In the proposed method, 𝑒(𝑡) is
calculated by the output single 𝑦(𝑡) and the desired signal 𝑑(𝑡). The output signal 𝑦(𝑡) in Eq.
(9) is the approximation of the output signal emitted from the PAL.
iv.

Estimate the coefficient 𝒉B (𝑡 + 1) as follows:𝒉B (𝑡 + 1) = 𝒉B (𝑡) + 𝛼𝑒(𝑡)𝑟B (𝑡)⁄(𝒓B (𝑡)T 𝒓B (𝑡) + 𝛽),(11)where 𝛼 represents the step size which controls the size of the coefficient correction, 𝛽represents the small positive constant which prevents denominator in Eq. (11) from becoming
zero.
v.

Increment a variable by 1 and repeat procedures ⅲ and ⅴ.

4. EVALUATION EXPERIMENT
4.1 Experimental Condition
We carried out the objective evaluation experiment to confirm the effectiveness of the proposed
method. The proposed method designs some filters for each column to improve the SPL of thedemodulated audible sound at the target position. We compared the SPL of the demodulated audible
sound in the case of the conventional method which emits AM waves which have coordinate phase andthe case of the proposed method. Table 1 shows the experimental conditions in the evaluation

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experiment, Table 2 shows the experimental equipment and Fig. 7 shows the experimental
arrangement. Table 3 shows the conditions for inverse filter design and Fig. 6 shows the experimental
arrangement for inverse filter design. In this experiment, we set the desired position at the 0.8 m spotfrom the PAL as shown in Fig. 6. We utilize the NLMS for adaptive algorithm, add the delay to thedesired signal to satisfy causal relation, and set each parameter experimentally. In addition, we utilize
the PAL as shown in Fig. 5. The PAL is consisted of fifteen line-type PALs corresponding to column
and each of them has ten small ultrasonic transducers. However, the propose method designs inverse
filters for the five channel PAL in consideration of easy convergence. Figure 8 shows the five channel
PAL utilized in the evaluation experiment. We realize the five channel PAL by dividing the PAL into
five as shown in Fig. 8.
We expect that the proposed method changes the sound pressure distribution of resonance positions
and improves the SPL at the target position. The SPL of the demodulated audible sound 𝑃def iscalculated as follows:st𝑃def = q r 𝑃(𝑓)vw(𝑓h − 𝑓g )

(14)

sVsu

where 𝑓g and 𝑓h represent the maximum and minimum frequency of the evaluation frequency bandand 𝑃(𝑓) is the SPL of the frequency 𝑓. Moreover, the evaluated frequency band is 1 ± 0.01 kHz
because we evaluate the pure tone.
Table 1 – Experimental conditions for measuring the sound pressure distribution.
Sampling frequency

96 kHzQuantization

16 bits

Carrier frequency

40 kHzAmbient noise level

𝐿1 = 19.2 dB

Environmental

Soundproof room(𝑇no = 0.15 s)

Sound source

White noise (0~8 kHz)

Evaluation distance

0.30〜1.30 m (0.02 m spacing)

Table 2 – Experimental equipment.
Ultrasonic transducer

SPL (Hong Kong) Limited, UT1007-Z325R

Power amplifier

VICTOR, PS-A2002

A/D, D/A converter

RME, FIREFACE UFX

Microphone amplifier

THINKNET, MA-2016C

Microphone

SONY, ECM-88B

Table 3 – Experimental conditions for designing inverse filters.

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Adaptive algorithm

NLMS

Filter length3840 points

Step size

𝛼 = 0.10000

Parameter

𝛽 = 0.00001

Desired signal

White noise (0〜8 kHz)Figure 6 – Experimental arrangement for designing inverse filter.

Figure 7 – Experimental arrangement for measuring the sound pressure distribution.

Figure 8 – Channel segment of the PAL in the evaluation experiment.

Figure 9 – SPLs of the demodulated audible sound.
4.2 Experimental Result
Figure 9 shows the experimental result. In Fig. 9, a horizontal axis shows the distance from the PAL
and a vertical axis shows the power of the demodulated audible sound (1kHz).
As shown in Fig. 9, the SPL of the demodulated audible sound at the target position (0.8 m spotfrom the PAL) is improved about 4.1 dB compared to the conventional method. This is because inverse
filters are designed to correct transfer functions between each line-type PAL and the target position.
In Fig. 9, it is also confirmed that the sound pressure distribution of the resonance positions has1059

changed compared to the conventional method. At the 0.65 m and 0.85 m spots from the PAL, theanti-resonance position of the demodulated wave, which existed in the conventional method, becomesthe resonance positions in the proposed method and the SPL of the demodulated audible sound is
improved. It is considered that the process of spatial wave synthesis has changed by applying theinverse filters to each line-type PAL and resonance positions have reformed. From the evaluation
experimental result, we confirmed the effectiveness of the proposed method.

5. CONCLUSIONSThe PAL realizes a sharper directivity and can transmit the audible sound to only the particular area
by utilizing AM wave. However, in demodulation process, some spatial sound sources are generated in
the acoustic beam and form some resonance and anti-resonance positions of the demodulated wave
because of the interference between spatial sound sources. Thus, the SPL of the demodulated audible
sound is attenuated at anti-resonance positions. In this paper, we propose the method to control
three-dimensional resonance positions of the demodulated wave. The proposed method reforms
resonance positions by spatial wave synthesis with phased array processing. We design inverse filterswith a multichannel adaptive algorithm as delay filters for line-type PALs to improve the SPL of thedemodulated audible sound at the target position. We conducted the evaluation experiment to confirm
the effectiveness of the proposed method. From the result of the evaluation experiment, we confirmedthat the proposed method can improve the SPL of the demodulated audible sound at the target positionand change the sound pressure distribution of the resonance positions. In the future work, we intend toreconsider the filter design method to change only phase characteristic of inverse filters.

ACKNOWLEDGEMENTS
This work was partly supported by JST COI and JSPS KAKENHI Grant Numbers JP18K19829,
JP18K11365, and JP19H04142.

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