Determining the geographic location of signal sources is necessary for many applications such as robot localization and navigation [1,2] and user localization in wireless networks [3]. In the case of source localization at a base station with multiple antennas, main techniques are used, such as time of arrival (ToA) [4], angle of arrival (AoA) [5,6], and received signal strength indicator (RSSI) [7]. These techniques are used in various approaches in order to achieve the goal of localizing a wave source.

Fingerprinting approaches that use unique features and parameters are suggested for source localization in indoor and outdoor environments [8,9]. In Ref. [10], Savic and Larsson used the received signal level vector for constructing fingerprints which are used with the Gaussian process regression method for determining the position of users.

The lens is also proposed to improve the positioning accuracy due to their ability to focus the energy of all of the wavefronts, which is incident with a single AoA to the lens surface, into one focal point [11]. In Ref. [12], AoA is estimated by combining lens antennas with a massive multiple-input multiple-output (MIMO) antenna array under multipath conditions. Shaikh and Tonello proposed simplified antenna selection methods for selecting and processing signals from focused subsets of antennas, and a low-complexity ToA estimation scheme for determining the first arrival path that contains information about the actual AoA. Xu et al. in Ref. [13] developed new algorithms for AoA estimation with planar electromagnetic lens arrays only using signal energy with low computational complexity.

Direct approaches [14] in which the user location is determined by searching for the position that best fits the measurement data are also used. In Ref. [15], a direct localization approach based on a compressed sensing framework is proposed for localizing users by jointly processing the observations obtained at distributed massive MIMO base stations. Direct approaches are also used for indoor localization. Akbari and Valaee proposed a method to co-process the received signal of all the access points using the compressed sensing structure which is transformed to an Ising energy minimization problem [16].

Distributed sensors are also used to improve the accuracy of localization in wireless networks [17]. Hong and Ohtsuki used an array of sensors to improve the accuracy of detection and localization of passive devices [18]. Distributed sensors receive signals at different points in space and time. These temporal and spatial features are used with the support vector machine classification to enhance positioning accuracy.

In addition to its contribution to improving communications, intelligent reflecting surfaces (IRSs) technology has found its wide applications for source localization [19]. Nasri et al. used the existing IRSs in the propagation environment to provide at least three different paths between the base station and the user location [20]. Then, by estimating the distances from the locations of the base station and IRSs to the user location, the user location is determined.

Some of the above-mentioned methods demand more than one base station to achieve user localization [14,15]. However, this requires synchronization between base stations and users. In addition, multiple base stations may not be available, as the case in urban areas. The other approaches use one base station but deploy additional elements, especially to improve localization accuracy, such as lenses [11–13], sensors [17,18], and distributed antennas [10]. Although a single base station is used in Ref. [20], in addition to exploiting the presence of IRS to achieve user localization, time synchronization and knowledge of the locations of the base station and IRS elements are required.

User localization using one base station and without time synchronization is the research problem defined in our previous work [21]. We exploited the existing passive repeaters to ensure obtaining unique field signatures at the base station for different user locations. The generated field signatures are normalized to the first antenna at the base station and then used as fingerprints for user localization. The proposed approach provides the ability to localize users only using received signals without the need for synchronization or calculating additional parameters such as ToA or AoA.

In this article, the method presented in Ref. [21] is generalized by using the existing scatterers in the target area to provide a unique field signature at the base station for each source location. Then the proposed approach can be applied in any environment containing a sufficient amount of scatterers. Since the Helmholtz-Kirchhoff (H-K) integral is widely used to study the scattering of acoustic waves from rough surfaces [22], the H-K integral is used for modeling the scattered field created by scatterers. Field signatures are normalized in phase to the first antenna and in amplitude to the maximum amplitude value among all antennas to avoid the problem when the amplitude at the first antenna is very small. In addition, the principle of using field signatures for source localization has been experimentally demonstrated with ultrasonic waves in air in laboratory conditions, and the results show that it is possible to localize an ultrasonic source by the field signature using one base station and without time synchronization. The proposed method can be used for indoor robot localization and navigation. The idea is to equip the robot with an ultrasonic source. The area over which the robot is planned to move is divided into a grid with a step of less than half the wavelength. Initially, the robot performs a full scan of the target area, and the field signature corresponding to each grid node is normalized and stored as a fingerprint in the base station. Using the proposed method, the base station can determine which node the robot is now around and then instruct the robot to move to the appropriate next node along the predefined trajectory.

The problem of source localization with one base station and without time synchronization is considered. The fingerprinting approach based on the generated field signatures at the base station is used to determine the geographic location of a source. A grid configuration with $ L $ nodes and arbitrary scatterers is proposed, as shown in Fig. 1.

Signals from a source located at a location with coordinates $ \mathbf{u}=(x{\mathrm{,}}\; y) $ impinge on M objects distributed around the positioning area. Each object m ($ 1 \leq m \leq M $) is characterized by the point scatterers located on its surface $ S_{m} $. The scattered signals from the objects are received by a uniform linear array with N antennas located at coordinates $ \mathbf{B}_{n}=\left(x_{n}{\mathrm{,}}\; y_{n}\right){\mathrm{,}}\; 1 \leq n \leq N $. The field signature generated at the nth antenna in the base station can be expressed using H-K integral [22] as

$ A_{n}(\mathbf{u}{\mathrm{,}}\; f)=-2 \sum_{m=1 }^{M} {\iint_{S_{n}}}G\left(\mathbf{r}_{m}{\mathrm{,}}\; \mathbf{B}_{n}\right) \frac{\partial}{\partial n} \frac{\exp \left[\mathrm{i} k\left(\left|\mathbf{u}-\mathbf{r}_{m}\right|\right)\right]}{\left|\mathbf{u}-\mathbf{r}_{m}\right|} \mathrm{d} S_{m} $ (1)

where $ \mathbf{r}_{m} $ is a point scatterer on the surface $ S_{m} $ of the mth object; $ \left|\mathbf{u}-\mathbf{r}_{m}\right|=\sqrt{\left(x-x_{m}\right)^{2}+\left(y-y_{m}\right)^{2}} $ is the distance between the source coordinates and the point source $ \mathbf{r}_{m} $; $ G\left(\mathbf{r}_{m}{\mathrm{,}}\; \mathbf{B}_{n}\right) $ is Green’s function for a point source $ \mathbf{r}_{m} $ on the surface $ S_{m} $ at the antenna $ \mathbf{B}_{n} $ in the base station; $ \partial / \partial n $ is the derivative with respect to the surface normal; $ k=2 \pi f / c=2 \pi / \lambda $ is the wave number. $ A_{n}(\mathbf{u}{\mathrm{,}}\; f) $ is different for different frequencies and different source coordinates. For a large enough number of frequencies and scattering objects, $ A_{n}(\mathbf{u}{\mathrm{,}}\; f) $ is unique for each source location.

The field signatures corresponding to the grid nodes $ \left\{A_{1 \leq n \leq N}\left({\mathbf{u}}_{l}{\mathrm{,}}\; f\right)\right\}{\mathrm{,}}\; 1 \leq l \leq L $ are stored as fingerprints in the base station. When a source with coordinates $ \mathbf{u}=(x{\mathrm{,}}\; y) $ transmits signals to the base station, the generated field signature is correlated with all stored fingerprints. The correlation process is given by

$ p\left(\mathbf{u}_{l}\right)=\sum_{n} A_{n}(\mathbf{u}{\mathrm{,}}\; f) A_{n}^{*}\left(\mathbf{u}_{l}{\mathrm{,}}\; f\right) $ (2)

where * is the conjugate function.

The maximum value of $ p({{\mathbf{u}}_{{l}}}){\mathrm{,}}{\text{ }}1 \leq l \leq L $ is given by the node to which the source is the closest. By normalizing the phase of the stored fingerprints to the first antenna in the base station, and the amplitude to the maximum amplitude among all antennas, the need for time synchronization is avoided and only spatial coherence is used for localization. The field signature generated by the target source is also normalized in phase to the first antenna, and in amplitude to the maximum amplitude among all antennas and then correlated with all stored fingerprints to determine the source location using (2). Multiple frequencies are used to construct the field signatures to increase the differentiation between these fields based on (1). Using N_f frequencies for constructing the field signatures and without time synchronization, (2) becomes

$ p\left(\mathbf{u}_{l}\right)=\sum_{i=1}^{N_{f}}\left| \sum_{n} A_{n}\left(\mathbf{u}{\mathrm{,}}\; f_{i}\right) A_{n}^{*}\left(\mathbf{u}_{l}{\mathrm{,}}\; f_{i}\right)\right|^{2} $ (3)

where $ A_{n}\left(\mathbf{u}{\mathrm{,}}\; f_{i}\right) $ is the field signature calculated using (1) for the frequency $ f_{i} $ and coordinates u. Without time synchronization, signals in (3) are summed in the intensity as incoherent signals.

In the proposed method, the accurate localization depends on the uniqueness of the field signatures $ \left\{A_{1 \leq n \leq N }\left(\mathbf{u}{\mathrm{,}}\; f_{i}\right)\right\}{\mathrm{,}}\; 1 \leq i \leq N_{f} $ for different coordinates u as shown in Ref. [21]. By increasing the heterogeneity around the positioning area and simultaneously increasing the number of used frequencies, the discrimination of the correlation coefficient corresponding to the node (which is the closest to the source location) relative to the other correlation coefficients increases.

To demonstrate the concept of the proposed method, numerical simulations were carried out at ultrasonic frequencies at the speed of sound (340 m/s), considering the scenario presented in Fig. 1. The simulated positioning area was 40 cm×40 cm. A broadband source emitting ultrasonic waves with frequencies in the range of 38 kHz–43 kHz was considered as the target of localization. The simulated base station consisted of N = 8 antennas with an interelement spacing of 5 cm. The scatterers around the target area were chosen arbitrarily with shape modifications to increase the heterogeneity. Point scatterers at the surfaces of the scatterers were used to simulate the scattered fields based on the H-K integral.

The simulated area was divided into L = 512×512 nodes. Then, the distance between successive nodes was $ \lambda / 10 $ (the wavelength $ \lambda $ is about 8 mm). N_f = 30 frequencies were used to construct field signatures. It is the maximum number of frequencies that allows to obtain nonrepeating signals in the considered range of frequencies. The placement of the scatterers and the scattered fields they created are shown in Fig. 2.

The ultrasonic source was localized at different locations. At each location, the source transmitted a broadband signal with 30 frequencies. The scattered waves from the scatterers were received by the antenna array of the base station. The field signature corresponding to each source location was normalized and correlated with the stored fingerprints. Fig. 3 shows the correlation coefficients obtained by localizing the ultrasonic source existed at the node (0, 0). Numerical simulation results show that it is possible to localize the source using broadband field signatures without the need for time synchronization between the base station and the target source.

In the simulation, the scatterers were modeled as two-dimensional objects. In real experiments, since the scattered fields are three-dimensional, the field interference at each source location is more distinct and the correlation coefficient corresponding to the closet node is also more distinct from the correlation coefficients of other nodes.

The principle of using field signatures for source localization was investigated experimentally. The experimental setup is shown in Fig. 4. As scatterers, objects that can be found in any laboratory (keyboard, boxes, cans, etc.) were used. A uniform linear antenna array of 8 antennas with a step of 5 cm was used as the receiving antenna array of the base station. During the experiment, the ultrasonic source was moved by a two-dimensional scanner over an area of 40 cm×40 cm with a step of 3.125 mm (the step is chosen to be slightly less than half the wavelength to minimize the measurement time).

The measurement area was divided into 128×128 nodes. Directional sensors were used in the work, so the source was directed towards the scatterers. At each node, the source transmitted an ultrasonic pulse with a frequency band from 38 kHz to 43 kHz. The scattered signals were received by the antenna array. N_f = 30 frequencies were used to construct the fingerprints.

The designed base station is based on MA40S4/R ultrasonic sensors, LM837 amplifiers, and STM32F407 microcontroller as shown in Fig. 5. Signals from the 8 receiving channels are digitized using a multi-channel analog-to-digital converter (ADC) of the STM32F407 microcontroller. The sampling frequency of the received signals is 164 kHz. The digitized data is transferred from the microcontroller to the computer for processing via the Ethernet interface based on the user datagram protocol (UDP). The system provides continuous digitization and data transmission.

At the first stage, a complete scan of the measurement area was performed and the corresponding field signatures of the grid nodes were normalized and then stored in the base station as fingerprints. The uniqueness of the stored fingerprints was verified by performing the correlation process between the stored fingerprints. Fig. 6 shows an example of the correlation coefficients based on the fingerprint related to the grid node (0, 0) and all stored fingerprint signals. The results of the correlation process show that the fingerprint signals are unique.

After the fingerprints were stored, one more measurement was carried out to investigate the possibility of source localization anywhere in the positioning area. The field signatures from the second measurement were used to localize the ultrasonic source using the stored fingerprints from the first measurement. Fig. 7 shows the results of the correlation between the field signature generated by the source at a location around the (0, 0) node and the stored fingerprints. The results show that the source is accurately localized. Experimental results confirm the feasibility of using field signatures to localize signal sources.

In the ideal case, the distribution of the $ p\left(\mathbf{u}_{l}\right) $ values is the Dirac delta function, where there is a nonzero correlation coefficient at the corresponding node and zeros for the remaining nodes. In the practical experiment, the distribution of the correlation coefficients is shown in Fig. 8.

The correlation coefficient corresponding to the target node is considered as the desired signal, whereas the remaining correlation coefficients are considered as noise. Then, the signal-to-noise ratio (SNR) can be defined as

$ \mathrm{SNR}=\frac{S^{2}}{\sigma^{2}} $ (4)

where $ S $ is the desired signal magnitude and $ \sigma^{2} $ is the noise variance.

SNR is used as a metric to certify the significance of using more frequencies for constructing field signatures. Fig. 9 shows the change of SNR with the number of used frequencies. It can be seen that the value of SNR increases as the number of used frequencies (N_f) increases.

The ability of tracking the movement of the ultrasonic source by determining its location with field signatures is experimentally investigated. This is very important for localizing and navigating indoor robots using the proposed approach. The ultrasonic source was moved manually according to an arbitrary trajectory as shown in Fig. 10. The ultrasonic source sent signals from different locations labeled by the sign (×) in Fig. 10.

The correlation coefficients obtained by localizing the ultrasonic source at each location labeled by the sign (×) are shown in Fig. 11.

The base station is able to localize the ultrasonic source in every location from which the source transmits signals. Depending on the determined location, the base station instructs the robot, on which the ultrasonic source is installed, to move to the next location along the given path.

An approach to source localization based on field signatures generated at the base station was proposed. The presence of scatterers in the positioning area was used to obtain a unique field signature for each source location in a grid with a step of less than half the wavelength. The field signatures corresponding to the grid nodes were stored in the base station as fingerprints. The need for time synchronization between the base station and the target source was avoided by normalizing the field signatures in phase and amplitude. The numerical simulations and the results of experiments at ultrasonic frequencies showed that the ultrasonic source was successfully localized with accuracy of the grid step (half wavelength in the experiment) when broadband field signatures and one base station with a linear receiving array of 8 sensors were applied. The proposed method can be used for indoor localization and navigation of mobile robots. It is obvious that the proposed approach works accurately when the environment is stable and there is sufficient scattering in the positioning area. However, once a significant and permanent change occurs, a new scan is needed to modify the stored fingerprints. To overcome this limitation, in future work, it is possible to apply dynamic scanning, in which fingerprints are updated at any time once the robot moves (not just in the first scan). In this way, it is viable to avoid the need for a new scan to modify the stored fingerprints.

The authors declare no conflicts of interest.