To improve the estimation accuracy, a novel time delay estimation (TDE) method based on the closed-form offset compensation is proposed. Firstly, we use the generalized cross-correlation with phase transform (GCC-PHAT) method to obtain the initial TDE. Secondly, a signal model using normalized cross spectrum is established, and the noise subspace is extracted by eigenvalue decomposition (EVD) of covariance matrix. Using the orthogonal relation between the steering vector and the noise subspace, the first-order Taylor expansion is carried out on the steering vector reconstructed by the initial TDE. Finally, the offsets are compensated via simple least squares (LS). Compared to other state-of-the-art methods, the proposed method significantly reduces the computational complexity and achieves better estimation performance. Experiments on both simulation and real-world data verify the efficiency of the proposed approach.

The time delay estimation (TDE) is a key issue of the time difference of arrival (TDOA) method [

Generalized cross correlation (GCC) [

In this paper, we propose an efficient TDE method for TDOA-based passive localization, which overcomes the problem of limited sampling intervals. The contributions of our work are summarized as follows:

Based on the first-order Taylor expansion on the vector containing initial TDEs, high accurate time delays are estimated by adding closed-form offset compensation obtained via least squares (LS).

Theoretical analysis and simulation experiments are conducted, which demonstrate that the proposed method has lower computational complexity and more accurate estimation compared with SDC [

Experiments on real-world scenarios demonstrate that the proposed method significantly improves the performance of source localization.

Notation: Vectors are represented by lowercase bold characters and matrices by uppercase bold characters.

To better explain the TDOA localization principle,

Suppose that

Just like the system model introduced above, a passive positioning system is composed of

Generally, the received signal at first node

Assuming that

According to Wiener-Khinchin theorem, the discrete Fourier transform (DFT) of the correlation function is the corresponding power spectrum. In the GCC method, the received signal is weighted in frequency domain to enhance the frequency component with high SNR and suppress the noise power. Here the PHAT is chosen due to its robustness, whose weighting function is

Therefore, the initial TDEs are obtained by

Using the signal model shown in

The cross-power spectrum of the two observed signals has the following relationship with the power spectrum of

Using

It can be seen that the normalized cross-power spectrum model is equivalent to the parametric model of line spectra. Therefore, the method of super-resolution spectrum estimation can be applied to estimate the time delay, such as MUSIC algorithm. The covariance matrix of

It can be divided into signal subspace

According to the orthogonal relation, the spectral function of time delay estimation is obtained

Then

As a matter of fact, the subspace-based TDE methods need a wide time frame search, while direction of arrival (DOA) estimation only requires an angle search of

Since the GCC method cannot overcome the limitation that the resolution is the sampling period, the time delay

Next, we apply the first order Taylor expansion to

The offset

By compensating offset

In most cases, iterations are necessary during Taylor expansion method implementation.

Then we list the corresponding hyperbolic equations after acquiring more accurate TDEs based on

Generally, the method in this paper mainly includes the following steps:

Select the received signal

The TDE model is derived from the normalized cross-power spectrum of observed signals as

According to the orthogonal relation of

Calculate the fine time delay estimations

This section is presented to discuss the computational complexity of different methods. The computation cost of GCC-PHAT is about

This part mainly presents simulation experiments to test the methods. Assuming that the nodes number is

Meanwhile, the time delay estimation comparison of different methods between signal source arrival at

In this paper, the measured data collected in real-world experiments is used to further prove the validity of the proposed method. As shown in

To process the collected measured data, we choose anchor 3 as the reference anchor, and then estimate the time differences among the signal radiation reaching the anchor 3 and others. The radiation source is a wideband signal modulated by QPSK, the center frequency is 700 MHz with 30 MHz bandwidth, and the sampling frequency is

Methods | Time delay |
Time delay |
---|---|---|

proposed | 2.07592955890347 | −1.08428903505303 |

MUSIC | 8.66166432759801 | −1.79276782595522 |

SDC | 12.6616643275980 | 2.20723217404471 |

GCC | 16.6616643275980 | 6.20723217404472 |

This paper presents a high precision TDE method based on closed-form offset compensation. The initial TDEs are estimated by the GCC method. Then we make use of the orthogonality of the noise subspace and the delay vector to obtain equation. To compensate the error caused by limited resolution in GCC, the first-order Taylor expansion is considered. The improved estimations are achieved by adding closed-form offsets, which can be computed by simple LS. Simulation experiments show that our method offers more accurate TDE results as well as lower computational complexity. Finally, we make experiments under the real field condition, which demonstrates that our method provides high-precision TDE.

The authors would like to thank the editor of this article and the anonymous reviewer for their valuable suggestions and comments which greatly improved the presentation of this paper.