In a multipath environment, to obtain accurate received times for multiple speakers using a single receiver, it is necessary to use broadband transmission signals and avoid interference between the signals. For such a case, we propose a transmission/reception method to improve the precision while maintaining the accuracy. In our proposed method, all speakers simultaneously transmit phase-shifted pulses multiple times and the overlapped received signals are separated without interference. Through a real environmental experiment, we confirmed the efectiveness of this method.
With the spread of mobile devices, such as smartphones, various location-based services are being developed. In indoor environments, global navigation satellite system (GNSS) could have significant errors, owing to the shielding of radio waves. Therefore, indoor positioning methods using sensors embedded in mobile devices have been widely studied [1].
Almost all mobile devices have a built-in microphone for calls and voice recognition. With multiple speakers installed in an indoor environment, the positions of microphones of multiple mobile devices can be estimated simultaneously. In such a method, the microphone receives signals transmitted from each speaker, and their received times are estimated. The microphone’s position is then calculated using the diferences between these received times.
waves are received with a delay, relative to the direct wave, and could cause systematic errors by interfering it. To avoid this error, the time resolution must be improved. This can be realized by increasing the bandwidth allocated to each speaker. In practice, however, the available bandwidth is limited. Some techniques, such as code-division multiplexing (CDM), assign a broadband signal to each speaker. However, this could cause interference between signals and result in systematic errors.
To avoid these problems, time-division multiplexing (TDM) is commonly used. It allocates the entire bandwidth signal to each speaker and transmits the signals in sequence, as shown in Figure 1. In TDM transmission schemes, if signals are transmitted, even in empty slots, the signal-to-noise ratio (SNR) can be improved by increasing the signal energy. However, interference between signals could reduce the accuracy. Therefore, it is necessary to be able to separate the signals for each speaker without interference.
In this paper, we propose a method that simultaneously transmits signals from speakers times and separates them without interference after reception. This method can increase the SNR by times while maintaining accuracy as high as TDM. The proposed method improves the SNR by times because signals are added for each speaker in the separation process. The proposed method uses phase-shift keying modulated chirp signals. By setting the appropriate phase value for each signal, the received signal can be separated into signals from each speaker without interference.
• We devised a phase-division multiplexing (PDM) transmission/reception method that improves the SNR by times while maintaining accuracy, using the same number of slots as TDM. • Through real-environment experiments, we confirmed the efectiveness of the proposed method.
as TDM.
This paper is organized as follows: Section 2 shows the related work for this paper. In Section 3, the details of the proposed method are described. In Section 4, the efectiveness of the proposed method is shown through comparative experiments with the conventional method (TDM) in a real environment. In Section 5, we discuss the experimental results and limitations. We present the conclusion in Section 6.
In a positioning method called time diference of arrival (TDoA), acoustic signals are transmitted from multiple speakers, and the microphone position is determined using the diference between the received times of these signals. When there are three speakers, let 1, 2, and 3 be the transmission times of the acoustic signals and 1, 2, and 3 be the received times of each signal. The relationship between these times and the positions of the speakers and microphone is represented as follows: ∥2 − ∥ − ∥1 − ∥ = ( (2 − 2) − (1 − 1)) ∥3 − ∥ − ∥2 − ∥ = ( (3 − 3) − (2 − 2)) where the 2D positions of the three speakers are 1, 2, and 3. indicates the 2D position of the microphone. ∥·∥ is the Euclidean norm. Solving this for gives the position of the microphone.
Next, we describe the transmission/reception method to obtain the received times. There are two
main types of transmission schemes: simultaneous transmissions and sequential transmissions from
each speaker. First, we explain the case of simultaneous transmission (1 = 2 = 3 = 0). This method
requires only one slot to obtain all the received times. Therefore, the positioning time is short. Let
() (0 ≤ ≤ ) be the signal from speaker and ℎ () be the impulse response representing the
propagation characteristics between speaker and the microphone. The received signal () is
where ∗ indicates a convolution. () is noise following a white Gaussian distribution.
(
Let be the discretized form of () and 1, 2, and 3 be the vector of discretized analytic signals corresponding to transmission signals 1 (), 2 (), and 3 (). These are convolved into to enhance the signal.
= [1 , 2 , . . . ],
= ∑︁ =1
+−1 where and are the th elements of and , respectively. is the number of samples when () is discretized. This process is known as a matched filter [ 2]. The received time is obtained as the time when the absolute value of is the maximum.
For positioning, 1 (), 2 (), and 3 () must be diferent to identify which speaker transmitted it. When calculating , could respond to () ( ≠ ). This causes an error in the estimated . In particular, when the microphone position is close to speaker and far from speaker , the response of () during the convolution of is larger. This is because the received signal of speaker is larger, causing a systematic error. This is called the near–far problem.
Various signal-multiplexing methods have been proposed to reduce such interference, such as frequency-division multiplexing (FDM) [3, 4, 5], orthogonal frequency-division multiplexing (OFDM) [6] and orthogonal chirp-based method [7]. However, when pulse signals are superimposed, the interference between these signals cannot be reduced to zero at an arbitrary position, in principle1.
When reverberation is present, as in Equation (
By transmitting signals sequentially from each speaker, the interference between the signals of each speaker can be zero. This method is known as TDM. TDM enables us to assign the entire bandwidth to all speakers, thus maximizing the suppression of reverberation. Therefore, TDM is suitable for situations where accurate and precise positioning is required, such as robot arm control [15, 16]. However, because the transmissions from each speaker are sequential, the time required for positioning increases compared to that of simultaneous transmissions.
In TDM, transmitting signals in empty slots can improve the precision because the signal energy increases. However, this can cause interference, which reduces accuracy. Therefore, the signals must be separated without interference to improve the SNR while maintaining accuracy.
In this section, we propose a transmission/reception method that improves the SNR by times while maintaining the accuracy of TDM. This is realized by transmitting signals times from speakers simultaneously and separating them without interference at the receiver.
In the proposed method, the following chirp signal is used for the transmitted signal () of speaker ( = 1, 2, . . . , ).
() = ∑︁ =1 (), () = {︄ (, , ) ( − 1) ≤ ≤ ( − 1) + , 0
( − 1) + < < (︃ (︃ 2)︃ (, , ) = () sin 2 0 + 2
)︃
+ ,
1The convolution of pulses can be expressed as the product of a matrix and a vector. Because the upper part of this matrix is
a lower triangular matrix, the columns of this matrix are linearly independent. Therefore, the only vector that makes the
result of this product a zero vector is the zero vector. This indicates that pulses that do not interfere with each other cannot
be constructed.
(
As can be seen from these equations, the transmitted signals are phase-shift modulated chirp signals. Because the proposed method uses this phase-shift modulation for multiplexing, we call the proposed method PDM. Figure 2 shows the transmission scheme.
H ( ()) = (ℎ ∗ ) ( − )
2The − 1 sample points are required to bring the matched-filter output to points.
(
∑︁ ∑︁ =1 =1 (), is the th element of the discretized complex chirp signal: ∑︁ =1 =
(+−1 + +−1) .
(︃ (︃ 2)︃ )︃ () = () exp 2 0 + 2 , 0 ≤ ≤ .
Let be the signal component of and ′ be the noise component of . Let be the component of transmitted by speaker ; then, these relations are = ∑︁ =1
.
= [1 , 2 , . . . , ]
= [1 , exp( 2 )1 , . . . , exp( )1 ],
as in Equation (
() =
0 {︄ ( ) exp( (2 ( 0+ 2
2)+)) 2 ( − 1) ≤ ≤ ( − 1) + ( − 1) + < < .
(︃ (︃ (︃ 2)︃ exp 2 0 + 2 , exp( 1 ) = 1.
= [1, exp( 2 )1, . . . , exp( )1, 0−1].
We apply the matched filter to the Hilbert transform output + . Letting denote this output, its th element can be represented as follows: = ′ + ′
= ⎜⎜⎛⎜... ⎜ ⎝ ′ = (11, 21, · · ·, 1 )∗,
· · · 2/ · · · 2 ( −1)/
... . . . ... 2 ( −1)/ · · · 2 ( −1) ( −1)/ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ′ = + , , ∼ N (0, 2).
−1 = 1
∗
(
By multiplying by −1, we obtain ′ as follows: 1 −1 = ′ + −1′ = ′ + ∗′. (25)
The proposed method assumes that indoor acoustic-wave propagation is a linear time-invariant system,
such that Equation (
For the transmission signal, we set 0 = 15 kHz, 1 = 23 kHz, and = 10 ms. These signals can be measured with microphones embedded in mobile devices such as smartphones [3, 10]. We set the transmission interval to 210 ms. This value is suficient to remove the reverberation [ 3, 10]. The sampling rate was set to 48 kHz. In estimating the received time, we upsampled 1 10 times.
The audio interface used in this experiment had a systematic time error of 2.0417 ms between channel three and the other channels. Therefore, in this experiment, we computed the positions using the received times calibrated by this value. 4.1.2. Position calculation
() = ( ∥2 − ∥ − ∥1 − ∥ − (2 − 1))2 + ( ∥3 − ∥ − ∥2 − ∥ − (3 − 2))2 (26) 3Because ′ is the matched-filter output, its th element is correlated with the previous and next samples. Thus, when < , the variance is not 2/.
= 21 − 0.521, = 32 + 0.532 21 and 32 are the gradients of the asymptotes of each hyperbola and are obtained as follows: 21 = √︁∥2 − 1 ∥2 − 2 (1 − 2)2 , 32 = √︁∥2 − 3 ∥2 − 2 (2 − 3)2
(1 − 2) (2 − 3) where is the speed of sound.
For Equation (26), we conducted a grid search in 1-cm increments in the range ±1 m centered at
this intersection point. Then, another grid search was conducted in 1-mm increments within ±10 cm
centered at the position of the first grid search result.
4.2. Measurement results
We show 2D plots of the TDM and PDM positioning results and the cumulative distribution function
(CDF) in Figures 7, 8, and 9, respectively. Table 1 shows the ratio of variances between TDM and PDM
on each axis and systematic errors of each method in order. Only at position (
Figures 7, 8 and 9, and Table 1 show that PDM improved the precision and its systematic error was comparable to that of TDM. From Table 1, we can see that the variance on each axis is approximately one-third compared to that of TDM. These indicate the efectiveness of the proposed method.
For both TDM and PDM, the spread of the positioning results increases as the distance from the coordinate origin increases. This depends on the relationship between the speakers and microphone positions. (27) (28)
5.1. Outlier 5.2. Computational complexity Compared to TDM, the additional calculation in PDM is Equation (25). Because there are only nonzero components in each column of −1 and the unit-matrix part of −1 does not require multiplication, the required number of multiplications is ( − 1) × . For Equation (25), as 1 contains a direct-wave component in its front, the only range requiring computation is the front of each 1 . This can further reduce the number of product operations. 5.3. Microphone movement In our experiments, measurements were conducted with a stationary microphone. If the microphone moves during the measurements, the reverberation will change. In this case, there is a diference between the observed data and the observation model (Equation (20)), and an error may occur in the iflter output ′ after separation. We would like to address this point in future work.
In this paper, we proposed a method to improve the precision without degrading the accuracy of TDM by simultaneously transmitting pulse sequences with the same bandwidth and diferent initial phases and separating them at the receiver side. Through real-world measurements, we confirmed that our method can improve the positioning precision without worsening the accuracy compared to the conventional method (TDM).
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