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

Relationship between Drive Signal and Stability in MIDS Modulator
Shigeto TAKEOKA1
1Shizuoka Institute of Science and Technology, JapanABSTRACT
We propose a matrix driving method using a multi-input delta-sigma (MIDS) modulator to drive a large
number of speakers. The MIDS modulator shares a single quantizer among multiple ΔΣ modulators and isable to drive the speaker array using the multiplication output individually and simultaneously even though
the wiring is common. In other words, N × Nch speaker arrays can be driven individually with 2 × N wirings,
instead of using time division as in imaging devices.
In this paper, we discuss the relationship between the drive method and stability for these MIDS
modulators. In general, it is known that input signal amplitude of the ΔΣ modulator is limited by the peakvalue of the transfer function for quantization noise. Meanwhile, in the MIDS modulator can be used as acommon quantizer for multiple signals, and the quantization noise from each input signal can be related tothe other input signals. Therefore, simulations are performed to confirm the limit values of the input signals
at the time of driving by the MIDS modulator. In the simulation, we compare the method combined withthe random signal and the method of quantizing the modulator alternately by the X- and Y- axes.
Consequently, in the proposed method, the stable input range is determined by the sum of the energies of
the input signals regardless of the drive method, while the stable pole location differs depending on thedrive method.
Keywords: Loudspeaker array, Multi-input delta-sigma modulator, MIDS, Super-multichannel processing, 1bit
signal processing

1. INTRODUCTION
Sound waves are physical phenomena by which three-dimensional vibrations propagate; however,to reproduce an arbitrary wavefront strictly based on the Huygens principle, it is necessary to set
control points, such as speakers, every half-wavelength distance for the target frequency band. We
have constructed a recording/reproducing system, such as a 1024 channel microphone array, thatsatisfies the sampling theorem in space by using high-speed 1-bit signal processing [1]. However, it isdifficult to construct a large-scale system by individually wiring each element; therefore, a method
capable of controlling multiple points with a simpler structure is desired.
In a normal speaker, because the voltage difference between terminals is proportional to theacoustic output, it is inevitable that a strong correlation exists between the outputs when multiple
speaker elements are driven using a common wiring. However, these problems may be solved if thespeaker output is proportional to the product of the voltages between the terminals. Therefore, wepropose a multichannel drive method that introduces multi-input delta–sigma (MIDS) modulation toencode signals input to a speaker array drive system using the multiplication output transducer array[2-4]. In the proposed method, a free structure wherein the inputs are combined with random signals oralternately quantized in the X- and Y-axes is possible. Therefore, we compare the stabilities of theinput signals and pole locations between the two drive methods in this study.

2. MULTI-INPUT DELTA-SIGMA MODULATOR
2.1 Driving by common wiring
To construct a speaker array of extremely high density and a large number of channels, theindividual-wiring method shown in Fig. 1 poses an implementation limit. Alternatively, in a speaker
1takeoka.shigeto@sist.ac.jp3113

Control Signal Y

array in which the wiring is common, the outputs are proportional to the difference signals and alwaysexhibit a correlation. In an imaging device, a multichannel drive with common wiring is realiz ed forscanning; however, a hardware limitation exists in the drive of the sound signal, thus requiring highsampling and dynamic ranges compared with those for images. Further, it is difficult to increase thescale of the structure in which an IC is installed for each element. Therefore, in this research, we use aspeaker that radiates an output proportional to the product of two input control signals, and consider anarray to connect those speakers with common wiring. The scheme for this is shown in Fig. 2. By using
speakers that output the multiplication result, it is possible to drive individually although it is commonwiring. For example, considering an array in which 1-bit signals (1, -1) are output, as shown in Fig. 3,
it can be easily noted that each speaker output is determined only by the combination of the respective
input lines. With such a structure, individual control of the N × N channel can be performed with 2 ×
N control signals as shown in Fig.2. As array control using a multiplication output speaker andcommon wiring, a structure using an electrostrictive element is proposed[5]. and a method using theSemi Discrete Decomposition for generating drive signals is proposed[6-7]. However, themultiplication output structure is weak to noise between steps on hardware when the drive signal iscomposed of multibit, and the effective sampling frequency is limited in time division driving .
Therefore, the purpose of this research is to realize a speaker array that can be driven passively andindividually while having common wiring, not being driven by time division, and additionally can bedriven with 1-bit signal. Hence, we propose the MIDS modulator as a method of generating signals.

NormalSpeaker䠇 䠉 䠇 䠉䠇 䠉 䠇 䠉䠇 䠉 䠇 䠉Control SignalFig 1 – Individually wired arrayControl Signal X

0XOWLSOLFDWLRQRXWSXW㽢㽢䠝䠄-1,1)㽢㽢B䠄-1,1)C䠄-1,1)D䠄-1,1)

Fig 2 – Array using common wiring  Fig3 – Combination ofwith multiplication output        1-bit signal

2.2 Delta Sigma modulator
ǼȊ modulation is widely known as a quantization method for continuous s ignals such as soundsignals[8]. It is a method of controlling the frequency characteristic of quantization noise by feedingback the quantizer output and obtaining a dynamic range in a desired band even with low bitquantization. The output of the first-order ΔΣ modulator, which is the basic structure, is shown in Eq.
(1); the block diagram is shown in Fig. 4, and the spectrum of the simulation result of the output
signal when a sine wave is input is shown in Fig. 5. As shown, the dynamic range of the audible
range is obtained while the number of quantization bits is 1.
(1)ܸ ൌ ܷ ൅ ሺͳ െ ܼ ିଵሻܰ‫ݍ‬V : Quantized signal

U : Input signal

Nq : Quantization noise for Qin㻺㼝㼁㻗㻽㼕㼚㻙㻽㼂㻙㼆㻙㻝㻗Figure 4 – First-order ΔΣ modulator3114Figure 5 – Spectrum of output by 1st -order ΔΣ modulator

2.3 MIDS modulator
As a method of generating a signal for driving a multiplication-output-type speaker array describedin Section 2.1, we proposed the MIDS. A block diagram of the proposed method with four
inputs/outputs and one quantization output is shown in Fig. 6. Unlike the normal ΔΣ modulator, thequantizer is common to multiple ΔΣ modulators and the product of the quantizer output Y and theindividual signals X 1 to X 4 is the output of each channel. The basic idea of the proposed method is that
although the control signal is common, the errors generated at each point are individually fed back andnoise shaping is applied. In this method, X 1 to X4 must be instantaneously uncorrelated with each otherto output individual signals. The common quantizer Q ' produces an output that minimizes the sum ofthe error energy, which becomes Eq. (2) when driving with a 1-bit signal.
ேܻ ൌ  ൥෍ ܺ௞ ൈ ܳ݅݊௞ ൩(2)௞ୀଵAs an example, consider a system in which four points are controlled by four M -sequence signals(random signals) and one control line, as shown in Fig. 7. The spectrum of each output (the productof the common quantizer output and each random signal) by the modulator is shown in Fig. 8.
Different sine waves were used for the input signal. As shown, the input signals were output while
maintaining their independence, and noise shaping was applied.
㼄㻝㻿㻻㼁㻺㻰㻌㻰㻭㼀㻭㻝㻗㼄㻝㽢㼅㻽㼕㼚㻝

㻙
㼆㻙㻝

㻗㼄㻝㽢㼅

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㻙㻗㻽䇻

Figure 7 – Combination with random signal㻻㼁㼀㻼㼁㼀㻟㻙㻙㻗㼄㻟㽢㼅㼄㻠㻗㼄㻟㽢㼅㻽㼕㼚㻟㼆㻙㻝㻿㻻㼁㻺㻰㻌㻰㻭㼀㻭㻠

Random Signal X1䡚X4(M-sequence)㼄㻞㽢㼅㼄㻟㻗㻻㼁㼀㻼㼁㼀㻞ControlSignalY㻙㼆㻙㻝㻿㻻㼁㻺㻰㻌㻰㻭㼀㻭㻟㻻㼁㼀㻼㼁㼀㻝㻙㼄㻠㽢㼅㻽㼕㼚㻠㻻㼁㼀㻼㼁㼀㻠㻙㻙㼆㻙㻝㻗㼄㻠㽢㼅㼅

Figure 6 – Block diagram of the proposed method
Figure 8 – Simulation results for four channels(first-order modulator)
Furthermore, when viewed from one channel, as shown in Fig. 9, the quantizer input of the otherchannel can be considered to be added as a whitened dither signal by being multiplied by the individual
signal X. Assuming that these assumptions hold, ξܰ times whiteness dither can be added to Qin in thechannel units. That is, if the level of Nq is considered to be constant, it can be regard ed as equivalent
to having a gain ݇ ൌ ξܰ in front of the quantizer, as shown in Fig. 10. Furthermore, the transfer
function of this equivalent block diagram is Eq. (3).
㼄㻝䡚㼄㻠

㻽㼕㼚㻞㻌

㽢㻽㼕㼚㻟㻌

㽢

㻽㼕㼚㻠㻌

㽢

㻽㼕㼚㻝㻌

㽢

㻽䇻

㻺㼝

㻗
㻺㼝
㻗

㻽

㼁  
Figure 9 – Quantizer seen from one channel㻸㼛㼛㼜㻌㻲㼕㼘㼠㼑㼞

㼗㻽㼂

Figure 10 – Channel unit equivalent block diagram3115ଵି௭ షభ௞ܸ ൌ ଵାሺ௞ିଵሻ௭షభ ܷ ൅ଵାሺ௞ିଵሻ௭ షభNq(3)We have confirmed by simulation that the equivalent block diagram of Fig. 10 matches
approximately the quantization noise spectrum of the proposed method (however, the gain k is 1/N
only in the first-order system). In addition, the suppression characteristics of the quantization noise
can be controlled by changing the order of the loop filter, as in the case of a normal ΔΣ modulator. In
the proposed method, the system may become unstable in the second- or higher-order system, but thestability can be maintained by controlling the pole location of the loop filter as in the normal modulator.
At that time, the transfer function shown from each channel is expressed by the following: ௞ήௌ்ி ሺ௭ሻே்ி ሺ௭ሻభభܸ ൌ ௞ାሺଵି௞ሻே்ிܷ ൅ ௞ାሺଵି௞ሻே்ிNqሺ௭ሻሺ௭ሻభ

(4)

భ

STF1 and NTF1 are transfer functions of the input signal and quantization noise when k = 1.
2.4 Drive Signal Combination
The proposed method can be used even if one side of the signal is a pseudorandom signal. Hence,the control signal that is changed according to the sound source is only one side, and advantages suchas simplification of hardware and prevention of limit cycles can be obtained. Therefore, in this study,
we examined the stability of the two drive methods, as described below. Method 1 is a method in which
the X-axis direction is a random signal of M-sequence and the Y-axis direction is a control signal, as
shown in 11(a). Method 2 is a method in which both the X- and Y-axes are used as control signals, andthey are alternately quantized, as shown in Fig. 11 (b). For convenience, these control methods areherein named as one-dimensional and two-dimensional control, respectively. In these methods, theeffective sampling frequency and information amount of the control signal are the same, and when
the same loop filter is used, the spectrum after quantization coincides. Fig. 12 shows the spectrum ofa third-order modulator in which the pole of NTF1 is 0.93 in each method. The results in theconfigurations of (a) and (b) of Fig. 11 are indicated by “a” and “b” , respectively, in Figs. 12 and 13.
The characteristics are highly consistent. Meanwhile, differences are observed in the range of polelocations that operate stably. Fig. 13 shows the spectrum when the pole location is 0.9. As shown, thetwo-dimensional control operates stably even at the pole location that becomes unstable whenquantized by a one-dimensional control signal. Consequently, a better S/N can be obtained in thetwo-dimensional control. Therefore, although the transfer function as a filter is the same, the rangeof stable operation differs depending on the drive type.

X ܴଵ ܴଶܴଷܴସܴହ

ܴ଺ X ܵଵିଵY ܵଵ ܵଶܵଷ

ܵସ

ܵହ

ܵ଺

Figure 11(a) – Combination with random signal

Figure 12 – Comparison of quantization results
within a stable range (pole location = 0.93)

3116Yܵଵିଶܵଶିଵܵଵିଷܵଶିଶ

ܵଶିଷ

Figure 11(b) – Alternate quantization method

Figure 13 – Result of moving the pole
(pole location = 0.9)

3. COMPARISON OF STABILITY BY DRIVE METHOD
3.1 Stability to input signals
In a normal ΔΣ modulator, the level of an input signal that operates stably is limited according to
the peak of the frequency characteristic in the loop filter. Meanwhile, the proposed method ischaracterized by that the quantizer of the input signals are common, and that the stable operating input
range will be determined by the input level for each channel and the entire system. Therefore, in theproposed method composed of four M-sequence signals and one control signal shown in Figs. 7 and 11(a), the relationship between the location of the unstable pole and the input signal was confirmed bysimulation. Fig.14 shows the levels of the input signals for which the simulation was performed. The
numerical value of the level is normalized by ͳȀξʹܰ to the quantizer output -1,1. Moreover,considering the limit cycle, the notation “0” is approximated by a minute input of 1/1000. The
frequencies of the input signals are bin numbers 9 to 13 for the window length 65536, and the signals
are different sine waves. The number of quantization bits is one. The poles of NTF 1 were slid and thevalues at which the system became unstable were compared. The results for driving points 4 to 64 areshown in Fig. 15. The value of the pole is shown as a normalized value between the value when all
the inputs are 1 and that when all the inputs are 0. Although the input signals are different from eachother, the unstable pole values show the same tendency for the group in which the sum of squares of
the input signals is 2 (A,B,C) and the group in which they are 1(D,E). As described in Section 2.3, inthe proposed method, each quantizer input can be regarded as whitened and added to each other. The
results in Fig. 15 suggest that the stable input range is also determined by th e sum of the input signalenergy.

A૚Ȁ ૛૚Ȁ ૛૚Ȁ ૛૚Ȁ ૛B૙૙૚૚C૙૚Ȁ ૛૚Ȁ ૛૚D૙Ǥ ૞૙Ǥ ૞૙Ǥ ૞૙Ǥ ૞E૙૙૙૚ Figure 14 – Combination of input signal levels(one-dimensional control)Figure 15 – Stable pole position with respect
to driving points (one-dimensional control)

Meanwhile, simulations were performed in the case of alternate quantizing; as shown in Fig. 11 (b) ,and is simulated with a 4 × 4 structure. The input channel is divided by 2 × 2, and similar to Fig. 14, asimulation is performed by inputting a sine wave with the amplitude value of the pattern shown inFig. 16. The obtained minimum pole location for a stable operation is shown in Fig. 17. Even in thetwo-dimensional control, the pairs (A,B) and (C,D) in which the energy sums of the quantizer
input signals coincide are well matched. It was suggested that the stable range was determined by theenergy sum of the input signal in the two-dimensional control.

3117ͳȀ ʹͳȀ ʹͳͳȀ ʹͳȀ ʹͳȀ ʹͳȀ ʹͲ

AB

ͲǤͷ

ͲǤͷ

ͳ

ͲͲǤͷ

ͲǤͷ

Ͳ

Ͳ

C

D

Figure 16 – Combination of input signal levels(two-dimensional control)

Figure 17 – Stable pole position with respect
to driving points (two-dimensional control)

3.2 Differences in stability due to driving methods compared at quantizer input
The design of a normal ΔΣ modulator is performed by calculating the peak of the transfer
function of quantization noise. Meanwhile, in the proposed method, as shown in Fig. 9 and Section
3.1, the quantization noise itself changes in according to the input of all channels. In addition to that,as shown in Section 2.4, although the transfer function of the system is the same, the range of stable
operation differs depending on the drive type. Thus, in order to discuss stability, it is necessary toconsider parameters including these terms that occur non-linearly. Therefore, we considered that theamount of feedback energy from the quantizer was different depending on the driving method, andcompared the RMS value of the quantizer input and the stable operation range.
In the simulations, the input signal is a sine wave of amplitude ͳȀξʹܰ with different frequencies
in each channel. The locations of the poles of NTF 1 were reduced and the RMS values of thequantizer input at the limit of stable operation were plotted. The simulation to 4 to 64 ch wasperformed and an approximate curve was added. The results for the one- and two-dimensional controls
are shown in Figs. 18(a) and 18(b), respectively. The approximate curve in the figure is shown below.
Third-order modulator in one-dimensional control: ͲǤͷͷ ൈ ܰଵȀସFifth-order modulator in one-dimensional control: ͲǤ͵ ൈ ܰଵȀସ
Third-order modulator in one-dimensional control: 1.2ൈ ܰଵȀଷFifth-order modulator in one-dimensional control: 0Ǥ ͹ͺ ൈ ܰଵȀଷThe two-dimensional control can operate stably with a large quantizer input, thus implying thatthe stable operation range is wide. In addition, the results fit well to the order of ܰଵȀସ inone-dimensional control and to the order of ܰଵȀଷ in two-dimensional control. Although thetheoretical background of these values is a future subject, we could obtain an index indicating thedifference in stability in each drive system. The transfer function to the quantizer input can be foundby subtracting Nq from NTF in Eq. (4). Therefore, by obtaining an approximate curve, an appropriate
pole location can be back calculated for the number of driving points.

(a)

3118

one-dimensional control(b) two-dimensional control
Figure 18 – Driving point vs. RMS value of quantizer input

4. CONCLUSIONSWe herein discussed the relationship between driving method and stability in our proposed MIDSmodulator. We examined two driving methods: a driving method that quantized one dimensionally incombination with a random signal, and a two-dimensional driving method that quantized the X- andY-axes alternately. As a result of simulation by the combination of multiple input levels, it wasconfirmed that the sum of energy of the input signal was related to the range of pole location for astable operation in each driving method. In addition, by analyzing the RMS value of the quantizer
input with respect to the number of driving points, the RMS value was approximated to the power
curve specific to the driving method. By obtaining these approximate curves, the pole location forstable operation could be calculated with respect to the number of driving points, that is, thedifference in stability between them was demonstrated.
In the proposed method, although the speaker required a multiplication output structure, thespeaker array could be driven individually while using common wiring. Moreover, a passive
structure could be constructed using a diode, a capacitor speaker, etc., and a “super multi-channel”
speaker array could be created. The findings in this paper suggested that stability increased in theorder of the power as the number of combinations of control signals increased. In the future, wewould like to consider a method in which the two-dimensional quantization method is not performed
alternately on the X- and Y-axes, but instead all points are collectively performed.

ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) (17K12732)

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and Delta-sigma Modulator with Multiple inputs. IEICE Technical Report, vol. 117, no. 328,
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Multiplication Output Loudspeaker Array Using Multiple Inputs Delta-sigma Modulator. IEICE
Technical Report, vol. 118, no. 234, EA2018-56. October 2018. p. 13-19.
5. Naoki Yatame, Yasutaka Tamura et al. Edge-Connected 2 Dimensional Array Composed of Nonlinear
Acoustic Devices. Japanese Journal of Applied Physics, vol. 41, 2002. p.3348-3353.
6. TAMARA G.KOLDA et al. Computation and Uses of the Semidiscrete Matrix Decomposition ,ACM
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