^{1}

^{*}

^{1}

Detection and localization of acoustic events in an environment are important to protect the military and civilian installations. While there are finite paths of wave propagation in simple or low reverberant environments, in complex environments (e.g. a complex urban environment) obstacles such as terrain or buildings introduce multipath propagations, reflections and diffractions which make source localization challenging. Therefore, numeric results of simulated models (simplified and Fort Benning urban models) of 3D complex environments can highly help in real applications. Some of the conventional beamformer algorithms have been used in order to localize point sound source. Analyzing results shows that MRCB beamformer has better performance than others in this issue and its accuracy superiority is more than 3 m in simplified urban model and 5 m in Fort Benning urban model with respect to the SOC. Moreover, due to possible uncertainties between the numerical model and the actual environment such as squall effect, temperature gradient etc., sensitivity of the beamformers to temperature gradient is investigated which shows higher robustness of SOC beamformer than the MRCB beamformer. According to the results, due to gradient temperature uncertainty the accuracy degradation of the SOC is about 1m while in MRCB it alters from 0.5 m to 20 m approximately at all SNRs. COMSOL Multiphysics has been used to numerically simulate the environment of wave propagation.

Source localization is one of the fundamental problems in sonar [

Sound source localization has several methods including direction of arrival (DOA) [

In RSS method, the received energy of the signal determines the source location while TDOA method uses the time delay of received signals by two sensors to estimate the source location. In TDOA, increasing the number of microphones leads to more computational complexity which can be considered as disadvantage of this method. The other method which is based on orientation of the ear system is HRTF. It is used in robots which have two sensors.

DOA method uses sensors to estimate the direction of the source. One of the techniques used in DOA method is beamforming [

Assume an array of M sensors. Beamformer output at the kth time instant is

b = w H x ( k ) (1)

where x ( k ) ∈ C M × 1 and w ∈ C M × 1 are the array snapshot and beamformer weight vectors, respectively, M is the number of sensors, C denotes the set of complex number and ( . ) H represents the Hermition transpose. The snapshot vectors are as follows:

x ( k ) = ∑ l = 1 N a l ( k ) s l ( k ) + n ( k ) (2)

where N is the number of sources, a l ∈ C M × 1 is the steering vector of the lth source, s l ( k ) is the baseband waveform of the lth source at the kth time instant, n ( k ) ∈ C M × 1 is the noise vector and ( . ) T represents the transpose. Assuming a main source and the other sources as interferers, the steering vector of the main source is a s and hence, the received snapshot vector can be formulated as

x ( k ) = x s ( k ) + x i ( k ) + n ( k ) (3)

where x s ( k ) is the desired signal and x i ( k ) is the interferers.

Let R denote the theoretical covariance matrix of the array output vector. Then the array covariance matrix can be expressed as

R = E [ x ( k ) x H ( k ) ] = P S a S a S H + R i + n (4)

where P s is the power of the main signal, E [ . ] denotes the statistical expectation and R i + n is the interference-plus-noise covariance matrix. The beamformer performance is commonly measured in terms of the output SINR, defined as [

S I N R = P S | w H a S | 2 w H R i + n w (5)

We can maximize the performance of the beamformer by minimizing the denominator of the equation subject to a distortionless constraint for the main signal. This can be formulated as

min w w H R i + n w s . t . w H a s = 1 (6)

The weight which is obtained from (6) is

w = R i − 1 a S a s H R i + n − 1 a S (7)

Because of (4) and the distortionless constraint in (6), replacing R i + n by R in the objective function of (6) yields an extra term of constant value. Thus, the weight vector of (6) does not get altered if R i + n is replaced by R. The array covariance matrix can be estimated as [

R ^ = 1 K ∑ k = 1 K x ( k ) x H ( k ) (8)

where K is the number of vectors in training snapshot. Replacing R i + n and a s in (7) by R ^ and the estimated signal steering vector a s , respectively, leads to the SCB [

w = R ^ − 1 a ^ s a ^ s J R ^ − 1 a ^ s (9)

It is reputed that estimation errors in R ^ and a s gives severe performance degradation of the SCB.

This type of beamforming is based on sum of the weighted microphone array signal, and hence, it is often referred to as a “delay-and-sum (DS) beamformer”. The weight vector of this beamformer is equivalent to the presumed signal steering vector [

Modeling of the actual desired signal steering vector is used to design the SOC beamformer. It is modeled as a sum of the estimated steering vector and a deterministic norm bounded mismatch vector δ :

a S = a ^ S + δ , ‖ δ ‖ ≤ ε (10)

where ε is a priori known bound and ‖ . ‖ represents the norm. Thus the SOC beamformer of [

min w , a ¯ s w H R ^ − 1 w s . t . | w H a ˜ s | ≥ 1 ∀ a ˜ s ∈ G a ˜ s (11)

The worst-case steering vector, which minimizes the objective function of (11), satisfies the constraints. It is assumed that ‖ a ^ s ‖ > ε [

| w H a ˜ s | = | w H a ^ s | − ε ‖ w ‖ 2 (12)

Thus (11) can be written as

min w , a ˜ s w H R ^ − 1 w s . t . | w H a ^ s | − ε ‖ w ‖ 2 ≥ 1 ∀ a ˜ s ∈ G a s (13)

(13) is a semi-infinite nonconvex quadratic program. It is reputed that the general nonconvex quadratically constrained quadratic programming (QCQP) problem is intractable. However, in [

The beamformer output power comprises noise, interferences and desired signal component. Minimizing output power of the beamformer in (11) diminishes the presence of the desired signal component and therefore it may lead to suppression of the desired signal component. Rubsamen proposes the Capon beamformer with minimizing the beamformer sensitivity [

min w , a ˜ s w H w | w H a ^ S | 2 s . t . w = R ^ − 1 a ˜ S a ˜ s H R ^ − 1 a ˜ s ∀ a ˜ S ∈ G a s (14)

By using the same uncertainty set, the robustness of the MRCB beamformer is larger than or equal to that of the SOC beamformer against model errors [

min a ˜ s a ˜ s H R ^ − 2 a ˜ s | a ˜ s H R ^ − 1 a ˜ s | 2 s . t . a ˜ S ∈ G a s (15)

The constraint of (15) is replaced by [

‖ a ˜ s β − a ^ s ‖ 2 ≤ ε (16)

β = a ˜ s H a ^ s / ‖ a ˜ s ‖ 2 2 minimizes (16). The constraint and the objective function in the optimization problem are invariant with respect to the scaling of a ˜ s . Then a ˜ s H R ^ − 1 a ^ S is scaled to one. The optimization problem of (15) leads to [

min a ˜ s a ˜ s H R ^ − 2 a ˜ s s . t . a ˜ s H R ^ − 1 a ˜ s = 1 α ‖ a ˜ S ‖ 2 ≤ a ˜ s H a ˜ S (17)

where

α = ‖ a ^ S ‖ 2 2 − ε 2 (18)

and the optimization problem of (17) can be solved using Lagrange duality [

In order to analyze localization error, the simulated models are considered as point grid which are spaced one meter from neighboring points and beamformers output are attained for these points. The beamformer output cut-off threshold b cut-off is used to determine the source location. b cut-off for the grid points with a beamforming output lower than the cut-off threshold are ignored. The source coordinate is estimated as

l j = ∑ j = 1 L b j l j ∑ j = 1 L b j (19)

where L is the number of grid points and l j = { x j , y j , z j } is the coordinate vector for the j th grid point, and b j is the corresponding beamforming output and b j > b cut-off . In the real scenario, it is barely possible that a grid point would be placed exactly in which the supposed source is situated. Because the simulations are restricted to the case where the source is placed on a grid point, selecting a weighted average of the coordinates of the grid points rather than pinpointing the single location with the largest beamforming output is reasonable. The source localization error can then be computed as the Euclidean distance between the true and the estimated source location. These simulations have been carried out in the frequency domain. The used noise is the uncorrelated noise with the same variances. For getting closer to the realistic conditions, two interferers are applied in θ i 1 = 45 ˚ and θ i 2 = 45 ˚ and also array includes six microphones which form diamond shape.

At the first case of theoretical investigation, we study the localization error of the beamformers versus the input SNR (averaged over the sensors) at two models of the city. It has been studied for the simplified urban model to test the performance of the beamformers in the complex environment. Then Fort Benning urban model is evaluated as a more complex environment.

In previous simulations, complete knowledge of the acoustic environment was assumed to attain the steering vector of the beamformers. It means that the localization performance of the beamforming methods is depended on the prior information of the environment to compute the steering vectors. However, there are always some uncertainties between the simulated model and the actual environment which causes error in localization of the source. In the second case, we study the beamformer performance in the presence of gradient temperature uncertainty in the simplified urban model. According to the experiments on the lowest 100 m of the atmosphere, the air can be separated into two parts: the part over the ground which the temperature gradient rate is log-linear and the second part which has a constant temperature gradient with height [

T 2 = T 1 + α ln ( z 2 z 1 ) (20)

where T 1 and T 2 are the absolute temperature in Kelvin at two different height z 1 and z 2 respectively and α is the profile constant.

To be consistent with the previous results, the array and source are located at the same positions. The baseline case without any uncertainty (i.e. 40˚C uniform temperature) is also presented in

SNRs, the degradation in accuracy due to uncertainty is between 0.5 m to 4 m. Additional error of the SOC beamformer is about 1 m in all SNRs because of the temperature gradient uncertainty.

In this literature, we see that the MRCB beamformer has better accuracy in complex environments than SOC and DS in two simulated models. Due to more complexity of the Fort Benning urban model, the degradation in accuracy of the beamformers can also be seen even with closer distance (about 25 m) between array and source in it than in the simplified urban model. Therefore, complexity of the models plays an important role in localization error of the beamformers.

Beamformer | SNR | Error (m) |
---|---|---|

SOC (Without gradient temperature uncertainty) | −5 | 8.22 |

0 | 7.74 | |

5 | 7.4 | |

10 | 7.27 | |

15 | 7.22 | |

20 | 7.22 | |

SOC (With gradient temperature uncertainty) | −5 | 8.68 |

0 | 8.56 | |

5 | 8.32 | |

10 | 8.25 | |

15 | 8.23 | |

20 | 8.26 | |

MRCB (Without gradient temperature uncertainty) | −5 | 6.53 |

0 | 6.13 | |

5 | 5.97 | |

10 | 5.72 | |

15 | 5.44 | |

20 | 5 | |

MRCB (With gradient temperature uncertainty) | −5 | 7.9 |

0 | 8.56 | |

5 | 6.92 | |

10 | 6.235 | |

15 | 25.72 | |

20 | 8.8 |

Secondly, the SOC and MRCB beamformers were tested with uncertainty of gradient temperature which is caused because of the difference between the numerical models and actual environments. The results show that temperature gradient uncertainty exerts more influence on MRCB than SOC. In future works, it is important to study the robustness of the beamformers to errors in the simulated models due to difference between actual environments and the models and also it is necessary to evaluate the topology effects of the source and array(s) on beamformers localization error in complex environments.

Nassaji, N. and Shafieian, M. (2018) Detection of Point Sound Source Using Beamforming Technique in Complex Environments. Open Journal of Acoustics, 8, 23-35. https://doi.org/10.4236/oja.2018.82003