# TSaT-MUSIC: a novel algorithm for rapid and accurate ultrasonic 3D localization

- Kyohei Mizutani
^{1}Email author, - Toshio Ito
^{1}, - Masanori Sugimoto
^{1}and - Hiromichi Hashizume
^{2}

**2011**:101

https://doi.org/10.1186/1687-6180-2011-101

© Mizutani et al; licensee Springer. 2011

**Received: **9 November 2010

**Accepted: **10 November 2011

**Published: **10 November 2011

## Abstract

We describe a fast and accurate indoor localization technique using the multiple signal classification (MUSIC) algorithm. The MUSIC algorithm is known as a high-resolution method for estimating directions of arrival (DOAs) or propagation delays. A critical problem in using the MUSIC algorithm for localization is its computational complexity. Therefore, we devised a novel algorithm called Time Space additional Temporal-MUSIC, which can rapidly and simultaneously identify DOAs and delays of mul-ticarrier ultrasonic waves from transmitters. Computer simulations have proved that the computation time of the proposed algorithm is almost constant in spite of increasing numbers of incoming waves and is faster than that of existing methods based on the MUSIC algorithm. The robustness of the proposed algorithm is discussed through simulations. Experiments in real environments showed that the standard deviation of position estimations in 3D space is less than 10 mm, which is satisfactory for indoor localization.

## Keywords

## 1 Introduction

In recent years, localization techniques have attracted considerable attention in ubiquitous computing communities.There have been many studies on localizing objects by using ultrasonic signals; for example, indoor positioning [1–3] or robotics [4, 5]. There are several requirements for localization techniques, including accuracy, robustness, and ease of deployment.

We propose a new localization technique using the Time Space additional Temporal MUSIC (TSaT-MUSIC) algorithm, a variant of the MUltiple SIgnal Classification (MUSIC) algorithm [6]. The principle advantage of the TSaT-MUSIC algorithm is its low computational complexity compared with other variants of the MUSIC algorithm.

The MUSIC algorithm is a well-known method for direction of arrival (DOA) or propagation delay estimations. As the algorithm conducts null steering of incoming waves, it shows higher resolution than the main beam steering methods such as the delay and sum beamforming methods. There have been many studies on 3D localization using MUSIC algorithm; for example, [7]. The Spatial-MUSIC (S-MUSIC) algorithm, which is just called the MUSIC algorithm, is used for DOA estimations. Another variant of the MUSIC algorithm called Temporal-MUSIC (T-MUSIC) offers propagation delay estimates. By using these algorithms, we can identify either the DOA or the delay but not both simultaneously.

To estimate the DOA and delay simultaneously using these algorithms, two main famous approaches have been proposed. The first applies the MUSIC algorithm to spatial- and frequency-domain data at the same time; for example, 2D-MUSIC [8], 2D-TDM MUSIC [9], and JADE-MUSIC [10]. This approach conducts a 2D search of angle and time. Thus, its computational complexity becomes very large. The second approach integrates other DOA or delay estimation methods with the MUSIC algorithm, such as [11]. For instance, TST-MUSIC [12] uses beamforming and temporal filtering methods. Compared with the first approach, the second approach has less computational complexity. However, it still requires high computation times, because the computational complexity increases in proportion to the number of incoming waves.

Our proposed algorithm can estimate DOA and delay values simultaneously in an entirely different manner from the existing approaches. The procedure of the TSaT-MUSIC algorithm is as follows. First, sets of DOA and delay values are estimated using the S-MUSIC and T-MUSIC algorithms, respectively. Next, the true pairs of DOA and delay values are decided by applying the T-MUSIC algorithm at a sensor different from the sensor used in the first step. As a result, we can estimate DOAs and delays with only three MUSIC algorithm executions. The original TSaT-MUSIC algorithm is for DOA-delay estimation in a 2D space and can easily be extended to a localization method in a 3D space.

One advantageous point of MUSIC algorithms used for 3D localization is that we can design a compact receiver array. In this study, a small L-shaped receiver array (about 36 mm × 36 mm) is implemented to evaluate the performance of the TSaT-algorithm for 3D localization.

The objective of this paper is to prove that the proposed method reduces the computational complexity of sound source localization and still retains the satisfactory level of accuracy. Thus, we conduct comparative evaluations using the TST-MUSIC algorithm, which is one of the fastest and most accurate localization methods using MUSIC algorithms.

In this paper, a brief introduction of the MUSIC algorithm is first presented. then, the proposed algorithm and its 3D localization method are explained. Subsequently, the results of computer simulations, and experiments in real environments using the TSaT-MUSIC algorithm are reported.

## 2 The MUSIC algorithm

### 2.1 Data model

*K*and

*M*, respectively. By using the Fourier transform for received signals at each sensor, we obtain a received data matrix X. The dimension of X is

*K*×

*M*. We define the received data for the mth frequency at the

*k*th sensor as

*x*

_{k,m}, so X can be written as:

_{ m }(

*m*= 1, 2,...,

*M*) is the received data vector at the mth frequency, and a vector T

_{ k }(

*k*= 1, 2,...,

*K*) is the data received at the

*k*th sensor. Thus, S

_{ m }and T

_{ k }can be expressed as:

^{ T }denotes the matrix transpose operation. Next, we introduce the mode vector a

_{ m }(

*θ*

_{ l }) (

*m*= 1, 2,...,

*M*), where

*θ*

_{l}is the angle of the

*l*th wave.

*a*

_{ m }(

*θ*

_{ l }) is defined as:

**Ψ**

_{m,k}(

*θ*

_{ l }) (

*k*= 1,2,...,

*K*) is the phase of the

*l*th wave of the

*m*th frequency at the

*k*th sensor and is expressed as:

*c*is the velocity of sound,

*f*

_{ m }is the

*m*th frequency, and

*d*

_{ k }is the distance between the

*k*th sensor and the 1st sensor (as shown in Figure 1). Assuming that the receiving waves are plane waves, S

_{ m }can be written as:

where *F*_{
l
}is the complex amplitude of the *l* th wave, and N is the Gaussian noise vector with zero means and equal variances *σ*^{2}.

_{ k }(

*τ*

_{ l }) (

*k*= 1, 2,...,

*K*), where

*τ*

_{ l }is the propagation delay time of the lth wave. g

_{ k }is defined as:

_{ k }, T

_{ k }can be written as:

_{ k }is:

### 2.2 The S-MUSIC algorithm

^{ H }denotes the Hermitian operation. The eigenvectors of Rss are the orthogonal direct sum of the signal subspace and the noise subspace. Assuming that the incoming waves are incoherent and that the value of

*L*is smaller than that of

*K*, we can derive the MUSIC spectrum by using the eigenvectors u

_{ i }(

*i*=

*L*+ 1,...,

*K*) that span the noise subspace of Rss. The MUSIC spectrum can be written as:

The DOAs can be obtained as the peak values of *Ps*(*θ*) by changing the angle *θ*. As can be seen in Equation (3), the number of incoming waves *L* is given. If *L* is unknown, Akaike Information Criteria (AIC) or Minimum Description Length (MDL) [13] can be used to estimate *L*. When the incoming waves are coherent, the S-MUSIC Algorithm does not work properly. Therefore, spatial smoothing preprocessing (SSP) [14] is used to suppress the coherence.

### 2.3 The T-MUSIC algorithm

The T-MUSIC algorithm is for propagation delay estimations. It differs from the S-MUSIC algorithm in that the T-MUSIC algorithm uses only one sensor and multiple-frequency waves. We therefore use a received data vector T_{
k
}.

*Pt*(

*τ*) in the same way as in the S-MUSIC algorithm. When the eigenvectors of the correlation matrix calculated using T

_{ k }are defined as v

_{ i }(

*i*= 1, 2,...,

*M*),

*Pt*(

*τ*) can be described as:

The propagation delays are obtained by finding the peak values of *Pt*(*τ*) in the same way as in the S-MUSIC algorithm.

When the bandwidth of the multicarrier waves is *f*_{
d
}, the T-MUSIC algorithm can estimate a delay time to an accuracy of 1/*f*_{
d
}.

## 3 The TSaT-MUSIC algorithm

### 3.1 Principle

*L*incoming waves at sensor A in the sensor array their DOA and propagation delay values are described as (

*θ*

_{1},

*θ*

_{2},...,

*θ*

_{L}) and (

*τ*

_{1},

*τ*

_{2},...,

*τ*

_{ L }) as shown in Figure 2. By applying T-MUSIC again at sensor B in the same sensor array, propagation delays can be estimated as (

*D*

_{1},

*D*

_{2},...,

*D*

_{ L }). The path length of the

*l*th incoming wave arriving at sensor A is

*d*sin

*θ*

_{ l }/

*c*longer than that of the wave arriving at sensor B, where

*d*is the distance between the two sensors. Therefore, we can plot

*L*

^{2}points as possible DOA-delay pairs in the (

*d*sin

*θ*,

*cτ*) space. These points are called "candidate points". Here, the following equation must be fulfilled:

*d*sin

*θ*,

*cτ*) space as "path difference lines". Theoretically, there can be only one point that represents a correct DOA-delay pair on each line. In the situation shown in Figure 3, for example, the pair of DOA and delay values are estimated as (

*θ*

_{1},

*τ*

_{2}), (

*θ*

_{2},

*τ*

_{3}), (

*θ*

_{3},

*τ*

_{1}).

*i*,

*j*,

*l*) between each candidate point (

*d*sin

*θ*

_{ i },

*cτ*

_{ j }) and the path different line

*cτ*-

*d*sin

*θ*=

*cD*

_{ l }as:

Then, the points with the minimum distance are selected as true points.

### 3.2 3D localization using the TSaT-MUSIC algorithm

*θ*

_{ a }and

*θ*

_{ b }, and one time delay

*τ*

_{ c }by using two sensor arrays Aa and Ab, and the sensor Sc, respectively. The pairs of

*θ*

_{ a }and

*τ*

_{ c }, and

*θ*

_{ b }and

*τ*

_{ c }can be decided by TSaT-MUSIC. As shown in Figure 5, the position of the transmitter from Sc is described as:

Hence, we can estimate the transmitter's position by using the TSaT-MUSIC algorithm.

### 3.3 Computational complexity

The procedure of the TSaT-MUSIC algorithm includes one S-MUSIC calculation and two T-MUSIC calculations. The computational complexity of the S-MUSIC algorithm can be written as max (O (K^{3}), O (hK^{2})), where *h* is the number of searches conducted along the DOA axis, O (K^{3}) is the computational complexity of the eigenvalue decomposition of Rss, and O (hK^{2}) is that of the 1D spatial search. In the same way, the computational complexity of T-MUSIC can be expressed as max (O (M^{3}), O (h_{t}M^{2})), where *h*_{
t
}is the number of searches conducted along the time delay axis. Because *K* is generally smaller than *M*, the computational complexity of the TSaT-MUSIC algorithm is given bymax(O(M^{3}),O(h_{t}M^{2})).

On the other hand, the computational complexity of the 2D-MUSIC algorithm is max (O ((KM)^{3}), O (hh_{t}(KM)^{2})) and that of the TST-MUSIC algorithm is max (O (LM^{3}), O (Lh_{t}M^{2})). This means that the TST-MUSIC algorithm must perform the T-MUSIC calculation at least *L* times. Consequently our proposed algorithm is theoretically faster than the existing method using the MUSIC algorithm.

## 4 Simulations

### 4.1 Simulation setting

*x*(

*t*) can be expressed as:

where *N* is the number of subcarriers (here, *N* = 6), *f*_{0} is the center frequency, Δ*f* is the interval of the subcarriers, *a*_{
i
}is the amplitude of the *i* th carrier, and *ϕ*_{
i
}is its initial phase. In this simulation, all the subcarriers have the same amplitude, and their initial phases are given so that the transmitted wave has a peak amplitude at the half time of the transmission period (0.5 ms). The frequencies of the subcarriers are 34-39 kHz (their interval was 1 kHz). Because the bandwidth of the wave was 5 kHz, the theoretical time resolution was 0.2 ms. In the simulations, DOA-delay estimations in the 2D space were conducted using a linear array sensor consisting of 16 receiver sensors at 5-mm intervals. The SNR (Signal Noise Ratio) of the multicarrier wave is set to 20 dB by adding Gaussian noise.

As we plan to use the TSaT-MUSIC algorithm for rapid ultrasonic imaging by detecting reflectors, all sound sources transmit the same signal in this simulation. Therefore, receiving waves at a receiver array become coherent. To control the effect of the coherence, we use the spatial smoothing method [14] in S-MUSIC and T-MUSIC algorithm, respectively. In 2D localization simulations, the number of sensors in a subarray (*K*), the number of frequencies (*M*), and the number of searches conducted along the time delay axis (*h*_{
t
}) were set to 11, 15, and 1,000, respectively.

From the principle of the TSaT-MUSIC algorithm as described in the previous section, its angular and range resolutions are same as those of the 1D S-MUSIC and T-MUSIC algorithms, respectively. Therefore, comparative evaluations between the proposed method and the TST-MUSIC algorithm are conducted through computer simulations shown in this section and experiments in real environments shown in the next section.

### 4.2 Computational complexity and accuracy

*L*) of incoming waves. The simulations of the TSaT-MUSIC and TST-MUSIC algorithms for each

*L*were conducted 1,000 times, and their average computation times and values of the root mean square error (RMSE) were measured. Figure 6a shows the simulation results of the computation times. From the figure, we find that the TSaT-MUSIC algorithm can estimate DOAs and delays in almost a constant time. On the other hand, the computation time of the TST-MUSIC algorithm increases linearly as

*L*increases. This is consistent with the theory discussed in Section 3.3. Therefore, the computational complexity of the TSaT-MUSIC algorithm is proved to be less than that of the TST-MUSIC algorithm. Figure 6b shows the simulation results of the RMSE. These values were calculated by estimated positions of a sound source at (-60°, 0.6 ms). From the figure, we find that the values of RMSE using the TSaT-MUSIC algorithm is higher than that using TST-MUSIC algorithm regardless of the number of waves. However, all the RMSE values of the TSaT-MUSIC algorithm are less than 6 mm, which indicates the satisfactory level of accuracy.

The DOA and delay time values of incoming waves

DOA [degrees] | -60 | -45 | -30 | -15 | 0 | 15 | 30 | 45 | 60 |
---|---|---|---|---|---|---|---|---|---|

Delay time [ms] | 0.6 | 1.2 | 1.8 | 2.4 | 3.0 | 3.6 | 4.2 | 4.8 | 5.4 |

### 4.3 Robustness

### 4.4 3D localization simulations

In this simulation, the L-shaped sensor array consisting of 15 receiver sensors at 5-mm intervals was used. The same signals were transmitted from transmitters, and the spatial smoothing method was applied in the same way as in the 2D localization simulations. The SNR was set to 20 dB. Because the L-shaped sensor array had 8 sensors on each side, the number of waves which could be classified was up to 4. Hence, the number of transmitters was set to 3.

The transmitters were placed at points Ta (800, -300, 400), Tb (0, 200, 2,000), and Tc (-900, -850, 700) (unit: mm). The L-shaped sensor was placed in the *x*-*y* plane in the same way as in Figure 5. The average and standard deviations of the transmitters' positions were calculated for 25 simulations.

Results of the computer simulation (unit: mm)

x | y | z | ||
---|---|---|---|---|

True position | 800.00 | -300.00 | 400.00 | |

Ta | Average position | 799.16 | -310.17 | 393.58 |

Standard deviation | 1.12 | 3.84 | 4.34 | |

True position | 0.00 | 200.00 | 2000.00 | |

Tb | Average position | 0.25 | 200.92 | 1999.70 |

Standard deviation | 3.66 | 5.51 | 0.66 | |

True position | -900.00 | -850.00 | 700.00 | |

Tc | Average position | -905.48 | -837.14 | 708.40 |

Standard deviation | 2.50 | 6.56 | 8.37 |

## 5 Experiments

### 5.1 Configuration of experimental system

### 5.2 3D localization using TSaT-MUSIC

3D localization experiments in real environments were conducted by using the TSaT-MUSIC algorithm to estimate the positions of three ultrasonic transmitters. These transmitters were set at the same positions as in the simulations in Section 4.

Results of the experiment in real environments (unit: mm)

x | y | z | ||
---|---|---|---|---|

True position | 800.00 | -300.00 | 400.00 | |

Ta | Average position | 836.00 | -320.06 | 401.20 |

Standard deviation | 0.415 | 1.71 | 2.2 | |

True position | 0.00 | 200.00 | 2000.00 | |

Tb | Average position | -23.73 | 251.65 | 2036.57 |

Standard deviation | 2.91 | 1.19 | 8.40 | |

True position | -900.00 | -850.00 | 700.00 | |

Tc | Average position | -918.74 | -867.48 | 702.68 |

Standard deviation | 0.47 | 2.36 | 3.02 |

### 5.3 Comparisons with TST-MUSIC

Through experiments in real environment, we compared TSaT-MUSIC with TST-MUSIC in terms of RMSE values and computation times. Because TST-MUSIC algorithm is 2D localization algorithm, we conducted 2D localization experiment.

Obtained values of RMSE (unit: mm)

TSaT-MUSCI | TST-MUSIC | |
---|---|---|

Ta | 17.1 | 14.3 |

Tb | 21.1 | 28.3 |

Tc | 81.1 | 75.0 |

## 6 Conclusions and future work

We have proposed a new DOA-delay estimation algorithm called the TSaT-MUSIC algorithm. The remarkable feature of the algorithm is its small computational complexity compared with existing algorithms based on the MUSIC algorithm. This feature was proved through computer simulations. Moreover, the 3D localization technique using the TSaT-MUSIC algorithm was verified to show its satisfactory accuracy using computer simulations and experiments in real environments. There are several remaining issues to be investigated. Improving the performance of 3D localization using the proposed algorithm is one of the most important future tasks. Another important task is to improve the robustness of the TSaT-MUSIC algorithm when a true point is not clearly identified due to placements of transmitters and low SNRs. We also plan to develop applications using our 3D localization technique, such as indoor positioning and robot navigation systems.

## Declarations

## Authors’ Affiliations

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