The detection of the leading edge of a signal using a threshold value can be used as a method for estimating the received time, which is not easily afected by reverberation or the sensor characteristics of the speaker and microphone. When the noise level is relatively high compared with the signal, the threshold must be set high to prevent false alarms, causing a systematic error in the received time estimation. The noise can be suppressed by transmitting the signal multiple times and averaging the results; however, it is necessary to wait for the reverberation to disappear after each transmission. Therefore, this measurement is time consuming. In this study, we propose a noise suppression method that can be applied to cases in which signals are transmitted multiple times without waiting for the disappearance of reverberation. For the proposed method, an analysis of noise suppression performance was conducted, and the limits of performance were provided under the given conditions. In addition, numerical experiments on the performance of the received time estimation in a reverberant environment were conducted to compare the proposed method with the conventional method and confirm the efectiveness of the proposed method.
With the recent spread of mobile devices, indoor positioning techniques for these devices have been extensively studied[1]. In this study, we discuss a positioning method that uses the propagation time of acoustic signals. When using the propagation time, it is important to estimate the received time of the acoustic signal. In indoor environments, reverberation and the physical characteristics of speakers and microphones can cause large systematic errors in the estimation of received time.
The detection of the leading edge of the signal with a threshold value can be used as a method that is less sensitive to reverberation and sensor characteristics. In detection using thresholds, it is necessary to set a threshold based on the noise level to avoid false noise detection. When the signal-to-noise ratio is low, the threshold must be set high, the parts of the leading edge of the signal are delayed, and a systematic error is caused in the estimation of the received time. Therefore, to reduce systematic errors with the desired probability of a false alarm, the signal-to-noise ratio must be improved.
If the noise follows a white normal distribution, it can be suppressed by transmitting the signal multiple times and averaging them. However, this method requires waiting for the disappearance of the reverberation after each transmission. Therefore, it takes a long time to send all the signals. This could be a problem, especially when positioning is required at many locations.
In this study, we propose a noise suppression method that can be applied even when the transmission time is set shorter than the reverberation time to reduce the time required for signal transmission. By applying an orthogonal projection matrix that preserves the transmitted signal component of the received signal, only the noise component is suppressed.
The contributions of this study are as follows: • We devised a noise suppression method that can be applied when the transmission period is set to less than the reverberation time. • The relationship between the transmission period and the number of transmissions and the noise suppression performance of the proposed method is discussed. The limitations of the noise suppression performance of the proposed method under given conditions are also provided. • We conducted numerical experiments on the performance of received time estimation under multiple reverberation conditions, and demonstrated the efectiveness of the proposed method over the conventional method. In addition, we conducted numerical experiments regarding the performance of received time estimation when the required transmission time is changed, and discussed their relationship.
The remainder of this study is structured as follows. In Section 2, we describe conventional methods for estimating the received time. In Section 3, we describe the proposed method and discuss the relationship between the transmission period, number of transmissions, and noise suppression performance. In Section 4, we discuss the results of the experiments comparing the proposed method with conventional methods. Finally, in Section 5, we present our conclusions.
First, we described the observational model discussed in this study. The start times of the
transmission and the transmission signal are set to 0 and (), respectively. The signal length
was . A signal transmitted through a speaker is nonlinearly distorted, depending on the
physical properties of the speaker. This is expressed as follows:
() = Θ(())
(
In indoor environments, reverberation is caused by multiple delayed waves reflected on walls, ceilings, and other surfaces. When this signal is observed using a microphone, nonlinear distortion occurs at the microphone. Let ℎ() and Θ(· ) be the indoor impulse response and microphone characteristics that cause distortions, respectively. The reverberation signal can be expressed as follows:
() = Θ((ℎ * )())
The matched filter is a well-known method for estimating the received time in noise[ 2]. In the following, we explain the matched filter in the case where the transmission signal () is a sinusoidal wave, as an example.
() = ( − Δ − ) + () represents the clock ofset between the transmitter and receiver. () is white normal noise.
If is known, the propagation time Δ can be estimated from the received time. The distance can be estimated by multiplying Δ by the propagation velocity. Even if is not known, can be eliminated if the transmission times of multiple speakers are known. Therefore, the distance between the speaker and the microphone can be estimated. * is a convolution.
Let Δ be the time required for propagation between the speaker and microphone. The
received signal can be expressed as:
(
( )( − ), () = {︃exp(2 ) 0 ≤ ≤
0 ℎ is frequency, which is a constant multiple of 1/ . The matched filter output is expressed as where () denotes the reference signal. For simplicity, is assumed as zero in this study. If there is no influence of reverberation or sensor characteristics, the absolute value of () reaches its maximum value, where = Δ. Therefore, in this case, the received time must be the time when the absolute value of () is largest. In this paper, this method is referred to as the maximum peak method. The maximum peak method is optimal when there is only white normal noise and no reverberation or sensor characteristic influence[ 2], and it is used in various acoustic positioning methods[3, 4, 5, 6, 7, 8, 9].
In a reverberant environment, as expressed in Eq. (
The maximum peak method assumes that the direct waves of the received and reference
signals are coincident. However, in practice, because distortions occur depending on the sensor
(Eq. (
One way to avoid both the reverberation and distortion of the received signal is to measure () in advance and use this signal as a reference. However, because the delayed waves that constitute reverberation and the sensor characteristics vary depending on the location of the speaker and microphone, a large amount of premeasurement is required.
To avoid the efects of reverberation and sensor characteristics, detection of the leading edge of
the signal can be used to estimate the received time. Specifically, a threshold was set for the
absolute value of the matched filter output, and the first point at which it exceeded the threshold
was detected. From the matched filter equation (Eq. (
In this section, we discuss the probability of false noise detection. This is called the false-alarm probability. When a matched filter is applied to white normal noise, the probability density distribution of the absolute value of its output is a Rayleigh distribution. We describe the Rayleigh distribution when a discretized matched filter is used because the process with discrete data is discussed in Section 3 and later. Let denote the signal length converted to the number of samples. Let denote the sampling period, and let be a consist multiple of . In this case, = /. Let be the threshold; then, the false alarm probability is calculated as follows: = ∫︁ ∞
︂(
2 exp − 22
2 )︂
(
As can be observed from these equations, setting a large threshold reduces the false-alarm probability. However, a larger threshold value caused a greater delay in the estimated received time. Therefore, to reduce the systematic error in the received time with a suficiently small false-alarm probability, the signal-to-noise ratio of the received signal must be improved.
Let 1, 2, . . . , be samples that follow a white normal distribution with mean 0 and variance 2, and the variance of the average ∑︀
=1 / is 2/. Therefore, if the noise of the received signal follows a white normal distribution, it can be suppressed by averaging multiple transmitted signals. Hereafter, this method will be referred to as the averaging method.
In reverberant environments, if the averaging method is used with the transmission period set to less than the reverberation time, the transmitted signal components will be corrupted during averaging by the delay waves of reverberation. In this case, the received time cannot be properly estimated. Therefore, to improve the signal-to-noise ratio, the signal transmission period should be longer than the reverberation time. However, this requires a significant amount of time. In the next section, we propose a noise suppression method that can improve the signal-to-noise ratio even when the transmission period is set below the reverberation time to reduce the required transmission time.
In the proposed method, () is transmitted times in period . Let denote the signal transmission period converted to the number of samples. For simplicity, let be constant multiples of . In this case, = /.
Let be the discrete data of the signal that is received and sampled and let be its length. Let be suficiently large. Subsequently, can be expressed as follows:
⎡ ⎢ = ∑︁ ⎢
⎢ =1 ⎣⎢ Δ− 1 (− 1)
(− )
− − Δ+1− (− 1)
⎤
⎥
⎥⎥ + .
⎥
⎦
(
The noise suppression method detailed in Section 3.3 assumes that 1 is contained in the first elements of the vector to which the orthogonal projection matrix is applied. To satisfy this assumption, the signal was cut out of . Let denote the signal. Hereafter, the range from the ifrst to the -th sample of is called the decision interval.
The signal cutout procedure is described in detail. First, a discrete matched filter is applied to ,+− 1/√ . the received signal to determine the starting point of the signal cutout as = Here, ,+− 1 is the cutout of from the -th to the ( + − 1)-th.
Let Δ(
⎡′(− 1)
= ∑︁ ⎢⎣ ′
=1 ′(− )
⎤
⎥
⎦
= [︀ 01
denotes a vector with zeros before and after . ′ denotes the length of . ′ is the zero matrix
whose size is × ′. 1, 2 are − 1 + Δ(
Here, we discuss how to set the parameter ′. The length of contained in is determined
by Θ(· ), Θ(· ), and ℎ(), and varies depending on the sensor characteristics and environment.
Therefore, is assumed to be unknown. In addition, because the starting point of the cutout is
Δ(
In the proposed method, the orthogonal projection matrix
= ( )− 1 was applied to . To show that ( )− 1 exists, it is suficient to show that has full rank, which is proved in the Appendix. The result ′ can be expressed as follows: ′ = = ( )− 1 + = + This equation shows that signal component can be retained without corruption. For the noise component, let be ′, then, the -th element of ′ can be expressed as ′ = where refers to the -th columns of . Because ′ is a linear sum of white normal noise, ′ follows a normal distribution, and its variance ( ′)2 is ( ′)2 = 2. This implies that represents the noise suppression performance; if this value is less than one, a smaller threshold value can be set with the same false alarm probability. In this case, as discussed in section 2.3.1, the systematic error in the received time estimation can be reduced. The detailed process is as follows: The threshold value (2) is calculated from a given false-alarm probability (2) and ( ′)2. The absolute value of the discrete matched filter output ′ = (′,+− 1) /√ is calculated, and detection is performed using the threshold (2). ′,+− 1 is the cutout of ′ from the -th to the ( + − 1)-th.
Let Δ(
(9) (10) (11) (12) (13) 3.3. Relationship between transmission period, number of transmissions, and noise suppression performance In this section, we first show that to study the noise suppression performance of in the decision interval, it is suficient to investigate the diagonal components of ( )− 1. Next, we show that the noise suppression performance can be expressed in a simple form using , , and ′ under the given conditions, and we discuss the relationship between each parameter and the noise suppression performance. In addition, the performance limits of the proposed method under these conditions are discussed. 3.3.1. Analysis of is the ( + 1)-th and the later rows of . Therefore, can be written as: (1, representing the noise suppression performance, is the -th diagonal component of . Here, is a projective matrix and is symmetric. Hence, = = . Therefore, is identical to the -th diagonal component of . Here, can be expressed as = 0 ) where 1, is the first to -th rows of the identity matrix of degree ′, and 0 0 = ︂( ( )−1,1 0( )− 1
︂) = ︂( 1, ︂) ( )− 1 (︀ 1,
0 )︀ ︀( 1, 0 )︀ .
Thus, each element in the top-left -by- matrix of and ( )− 1 is identical. This indicates that the noise suppression performance in the decision interval can be evaluated by is a symmetric Toeplitz matrix, as follows: investigating the first to -th diagonal components of ( )− 1.
= ⎜⎜⎜ 3
. ⎛ 1 ⎜⎜ 2 ⎜ ⎜⎜⎝ ...
′ · · · · · · 2 1 2 . . .
3 · · · · · · 2 . . . . . . . .
. . . . . . .
.
3 . 2 . 2 . .
. 1 2 ′ ⎞ . .. ⎟⎟ ... ⎟⎟⎟
⎟ 3 ⎟⎟ 2 ⎠ ⎟ ⎟ 1 All the elements of this matrix are integers and are represented as follows: = {︃ − 0 − 1 ℎ (( − 1) mod ) = 0 ∧ > −1 .
Note that the inverse matrix of symmetric Toeplitz matrix can be obtained by the Trench algorithm. (14) (15) (16)
We then examine the diagonal components of inverse matrix ( )− 1. Let each of these elements be defined as follows: ⎜ 2,1 ( )− 1 = ⎝⎜⎜ ... This matrix satisfies ( )( )− 1 = .
We consider the first diagonal component 1,1 of ( )− 1. For the first column of ( )− 1, we assume that 1 + ( − 1) ( = 1, 2, . . . , ⌈ ′ ⌉)-th rows are real numbers, and all other rows are zero. ⌈·⌉ denotes the ceiling function. Here, in -th rows of , all columns except the (( − 1) mod ) + 1 + ( − 1))-th column are zero, and thus the inner product of the first column of ( )− 1 and all rows except the (1 + ( − 1))-th row of are zero. From the above discussion, the product of and the first column of ( )− 1 can be expressed as: ⎛ ⎜ ⎜⎜⎜ ⎜ ⎝
1,1 1+,1 1+2,1 . .
. 1+(− 1),1 ⎞ ⎛1⎞ ⎟ ⎜0⎟ ⎟ = ⎜⎜0⎟⎟ ⎟ ⎟⎟⎠ ⎜⎝⎜ ... ⎟⎠⎟
0 ⎛ ⎜ ⎜ ⎜ = ⎜⎜ ⎜ ⎜ ⎜ ⎝
1 (1+) (1+2) . . . . . .
(1+)
1 (1+)
. . . (1+(− 1)) · · ·
· · · (1+) 1 . . .
. . . · · · · · · · · · · · · . . . . . .
· · · where = ⌈ ′/⌉.
When = 1, 1,1 = 1/ according to Eq. (18). becomes one only when the transmission period is greater than ′, which corresponds to the averaging method.
When > 1, 1,1 is the (
⎜ ⎜ ⎜ = ⎜ ⎜ ⎜ ⎜ ⎝ · · · · · · · · · . . .
− 1 − + 1⎞ . . . . .
. − 1 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (17) (18) (19) (20) Theorem 1. When ≥ − 1 ≥
1, the inverse matrix of is as follows. · · · · · · · · · · · · 0 . . . · · · · · ·
· · · · · · · · ·
. . . · · ·
· · · · · · . . . . . . Proof. We show that Eq. (21) is ()− 1 = . For the first column of ()− 1, the inner product of the first row of is (2 − + 2) − + 1 4 − 2 + 2 − 0.5( − 1) + 4 − 2 + 2 = 1.
The inner product of with the ( = 2, 3, . . . , )-th row is: ( − + 1)(2 − + 2) 4 − 2 + 2 − 0.5( − + 2) +
− + 4 − 2 + 2 = 0.
For the ( = 2, 3, . . . , − 1)-th column of ()− 1, the inner product of the -th row of is
− 0.5( − 1) + − 0.5( − 1) = 1.
For the inner product of the ( = 2, 3, . . . , − 1)-th column of ()− 1 and ( ̸= )-th row of , if > ,
− 0.5( − 1 + − ) + ( − 2 + − ) − 0.5( − 3 + − ) = 0.
If < , then
− 0.5( − 1 + − ) + ( + − ) − 0.5( + 1 + − ) = 0.
We considered the -th column of ()− 1. Each of and ()− 1 is equal to the matrix that inverts the row and column orders. Therefore, by the same reasoning as in the first column, the inner product of the -th column of ()− 1 with the -th row of is 1, and the other rows of are 0, which concludes the proof.
This indicates that the noise suppression performance 1,1 can be expressed in the form 1 (Eq. (22)).
The above discussion is for the first column of ( )− 1, but holds for the second and subsequent columns, and the value of each element can be calculated using ()− 1. Therefore, we can represent the noise suppression performance in the decision interval as 1 (Eq. (22)).
Figure 1 shows the results of calculating the (
In the following, we show that the above properties are maintained under the condition ≥ − 1 ≥ 1. We will discuss the case < − 1 in future work.
First, we investigate the noise suppression performance when increases. If ≥ − 1 ≥ 1 is satisfied, the noise suppression performance can be expressed by Eq. (22), and thus the diference between the noise suppression performance when ′ and ′+1 ( ≥ ′+1 > ′ ≥ − 1 ≥ 1) is This expression can be simplified as 2 − (′ + 1) + 2 2 − ′ + 2 4 − 2(′ + 1) + 2 − 4 − 2′ + 2 The equation is positive because ≥ ′ ≥ 1. In other words, as increased, the noise suppression performance decreased.
The value of increased as the transmission period decreased. Therefore, the noise suppression performance decreased as the transmission period was shortened. 1) and ′ + 1 are
Next, we investigate the noise suppression performance as the number of transmissions increases. The diference between 1 when the number of transmissions ′ (′ ≥ ≥ − 1 ≥ 2(′ + 1) − + 2 4(′ + 1) − 2 + 2 − 2′ − + 2 4′ − 2 + 2 =
1. Therefore, the noise suppression performance improves when the number of transmissions is increased. In addition, from Eq. (22), it can be observed that 1 converges to 0.5 as is increased. To summarize the above, it can be said that under the condition that ≥ − 1 ≥
1, the noise suppression performance can be improved by increasing the number of transmissions , but there is an lower limit of 0.5.
The lower limit of 1 0.5 is the same as that of the noise suppression performance of the averaging method when the transmission period is set to more than and the number of transmissions is set to two, as described in Section 2.3.2. In other words, when the proposed method was used with a transmission period of or less, its noise suppression performance was less than or equal to that of the averaging method under the aforementioned conditions. In contrast, the proposed method has the advantage of reducing the time required for the signal transmission. Specific numerical examples and performance comparisons are presented in Section 4.3. 3.4. Relationship between the starting point of the cutout and the error caused by the projection In this section, we consider the case in which does not contain a part of . There are two cases in which the cutout is made either too early or too late, and we discuss them sequentially.
If the starting point of the cutout is too early, and does not contain the last components of , is represented as follows: = ︂( matrix on the right-hand side of Eq. (32) can be expressed as 1,− is the first to ( − )-th rows of . Let be the -th column of matrix , then, the ︀( 1+ 2+ · · · ′ ′+1 · · · ′+︀) rows of − . The product of this and is: Here, ( > ′) is a vector consisting of a -by-1 zero vector, and the first to ( ′ − )-th ⎛ 1 ⎞ ⎜ . ⎟ ⎝ . ⎠ ︂( columns of . Thus, ′ can be represented as: This indicates that the first to ( ′ − )-th columns are identical to the ( − 1)-th to ′-th ′ =( )− 1
+ = ︂( ′′ ′− 2 ︂( ′′ is the -by-( ′− ) zero matrix. 1 and 2 are -by- and ( ′− )-by- matrices, respectively. From Eq. (34), 1 and 2 are not zero matrices. Therefore, it can be observed that ′ contains an error that is the product of the last components of and 1, 2. Note that this error does not occur when 2 ≥ because the last 2 components of are zero (Eq. (11)).
If the starting point of the cutout is too late, and does not contain the first components of , then is represented as follows: = ︂( +1,︂) + .
′ ︀( 1− 2− · · · 1 · · · ′− ︀) .
The matrix on the right-hand side of this equation can be written as Here, ( < 1) is a vector comprising the -th to ′-th rows of + and -by-1 zero vectors. The product of this and is: ⎛ 1 ⎞ ⎜ . ⎟ ⎝ . ⎠ ︂( +1,︂) ′ = ⎜ ⎜ 2 ⎟⎟ (︀ 1− 2− · ′− ︀) .
. ′ =( )− 1
′ ︂( +1,︂) + = 4
′′ ︂( 3 ′− ︂) + .
It can be observed that ( + 1)-th to ′-th columns of this matrix are identical to first to ( ′ − )-th columns of . Thus, ′ can be represented as: 3 and 4 are ( ′ − )-by- and -by- matrices, respectively. From Eq. (38), 3 and 4 are not zero matrices. Therefore, this indicates that ′ contains errors owing to the product of the ifrst components of and 3, 4.
In this section, numerical experiments were conducted to verify the efectiveness of the proposed method. In Experiment 1, we compared the performances of the conventional and proposed methods when the number of delay waves was varied. Experiment 2 was a comparative evaluation of the required transmission time and noise suppression performance. (35) (36) (37) (38) (39)
This section describes the experimental setup common to Experiments 1 and 2. The list of parameters is presented in Table 1. If the parameters are not specified, the values listed in Table 1 are used.
The transmitted signal was a sine wave in Eq. (
For the simulated delayed waves, the amplitudes of each delayed wave were uniformly
random numbers, ranging from 0 to 0.8. The received time of each delayed wave was set to a
uniform random number in a range from the received time of the direct wave to 0.08 s.
(
The threshold (
In this section, we compare the performance of each method when the number of delay waves is varied. In the proposed method, the number of signal transmissions is set to four. The following three methods were evaluated using conventional methods.
The first is the maximum peak method described in Section 2. Note that for transmission, the signal length () is set to = 0.004 s, and the number of transmissions is set to one. The reason for this was to make the energy identical to that of the signal transmitted using the proposed method.
The second is the leading-edge method described in Section 2. Here, in Section 2.3.1 is calculated from the false-alarm probabilities (2) and noise variance.
The third is the leading-edge method using the threshold (2). Hereafter, this method will be referred to as the low-threshold leading-edge method. The diference between this method and the proposed method is that detection is performed using the threshold (2) without the noise (a) Number of delay waves: 0
(b) Number of delay waves: 25 (c) Number of delay waves: 50 suppression process described in Section 3. This method was evaluated to show that the number of false alarms increased when the threshold was lowered, without the noise suppression process of the proposed method.
The evaluation results of each method are shown in Figures 2a, 2b, and 2c. In these figures, the horizontal axis shows the error of the estimated index and the vertical axis shows the cumulative distribution function (CDF). The number of false alarms for each condition and method is shown in Figure 3. False alarms were defined as errors greater than 15.
Figure 2a shows that, in the absence of delay waves, the maximum peak method yields the best performance, as described in Section 2. However, in the maximum peak method, the false alarm rate increases as the number of delay waves increases (Figures 2b, 2c, and 3).
For the methods based on leading-edge detection (leading-edge, low-threshold leading-edge, and proposed methods), the performance was not significantly degraded even when the number of delay waves increased, as shown in Figures 2a, 2b, 2c, and 3. Moreover, Figures 2a, 2b, and 2c show that the proposed method reduces the error around the true value compared to the leading-edge method. Comparing the low-threshold leading-edge method and the proposed method, although the errors around the true value are not significantly diferent, the number of false alarms increases in the low-threshold leading-edge method compared to the proposed method, as shown in Figure3.
This section presents a comparative study of the relationship between the required transmission time and noise suppression performance. In this experiment, the number of delay waves is set to 25.
For the proposed method, the number of transmissions was set to one, two, three, and four. Their required transmission times are 0.13, 0.17, 0.21, and 0.25 s, respectively. For the averaging method, because is unknown, the transmission period is set to ′ × = 0.13 s, and the number of transmissions is set to two. In this case, the required transmission time was 0.26 s. As described in Section 2.3.2, under these conditions, the noise suppression performance of the averaging method was 0.5, which is identical to the performance limit of the proposed method shown in the 3.3.2 section.
The results of the experiment are shown in Figure. 4. As aforementioned, for the proposed method, the required transmission time increases as the number of transmissions increases. However, the performance of the estimation of the received time improves as the number of transmissions increases. The averaging method had the best performance in estimating the received time, but the required transmission time was the largest. These results confirm that the proposed method can provide noise suppression with reduced transmission time if the required noise suppression performance is greater than 0.5.
In this study, we proposed a noise suppression method that can be applied to multiple signal transmissions without waiting for the disappearance of reverberation. Numerical experiments were conducted to compare the efectiveness of the proposed method with that of conventional methods in estimating the received time under multiple reverberation conditions.
The relationship between noise suppression performance and and is discussed under the conditions = 1 or ≥ − 1 ≥ 1. The case in which < − 1 will be discussed in our future work.
This work was supported by JSPS KAKENHI Grant Numbers 20K23313 and 21K17797. ultrasonic transducer positioning, IEEE Sensors Journal 17 (2017) 3832–3843.
C. Schindelhauer, Acoustic self-calibrating system for indoor smartphone tracking (ASmethod using short-time and narrow-band dual-carrier acoustic signals, IEEE Sensors panding the positioning area for acoustic localization using cots mobile devices, in:
We show that is full rank. Because is a Gram matrix, it is suficient to show that all the column vectors of are linearly independent. First, for the (1 ≤ ≤ vector, the -th row is one, but all -th rows of column vectors except -th column vector are )-th column zero, and thus they are linearly independent.
For the ( > )-th column vector, let Λ be the index set of the column vector whose -th row is one, except for the -th column as follows: ︂⌈ ⌉︂ In the following, let 1, 2, 3 ∈ Λ: For 1-th rows, the value is zero in -th columns but one only in 2 ( 1 ≥ 2)-th columns. Therefore, if the -th column vector is linearly dependent, then the linear sum of ⌈/⌉ −
1 column vectors must satisfy the ⌈/⌉ equality constraint, -th column vectors are linearly independent. Thus, ( ) is shown to have full rank. where 3-th rows are zero and -th row is one. Because this is impossible, it can be said that all