Deconvolved frequency-difference beamforming for a linear array Open Access

Improvement of spatial resolution and reduction of sidelobe levels in underwater acoustic broadband beamforming while maintaining robustness is of much interest. Conventional broadband beamforming¹ (Conv-BF), which has been widely used in underwater acoustic signal processing, confronts several serious challenges: (1) the beam is wide and has a low spatial resolution at relatively low frequency, (2) the array is sparse for high frequency components, and (3) the wavefront of a high frequency signal is easily mismatched and signal spatial coherence decreases rapidly.² Many high-resolution algorithms, such as the minimum-variance distortionless response³ and multiple signal classification,⁴ have been proposed in previous studies for superior resolution. Unfortunately, these high-resolution algorithms are sensitive to the wavefront mismatch and fail to work at low signal-to-noise ratios (SNRs), which are often encountered in sonar signal processing. The frequency-difference beamforming (FDB) proposed by Abadi and coworkers^5,6 conducts the beamforming by using the autoproduct calculated through the quadratic product of frequency domain signal amplitudes, $X(f1)*X(f2)$⁠, at difference frequency $f2−f1$⁠. FDB is more robust by shifting the beamforming to a lower frequency (in-band or out-of-band of the signal) where the array is not sparse and wavefront mismatch has a less adverse effect on beam output.⁷ However, this approach is detrimental to the beamformer as manifested by the increased beamwidth, i.e., the increased robustness of FDB is achieved at the expense of reduced spatial resolution.⁸ This outcome may ultimately lead to an unsuccessful detection of two targets within the same beam.

Recently, deconvolved conventional beamforming (Dv-CBF) was proposed by Yang based on “image deblurring” techniques, which provides a narrow beam and low sidelobe levels.⁹ In his method, the “original source distribution” is obtained by deconvolving the beam output of narrowband conventional beamforming (CBF) iteratively, with the shift-invariant beam pattern of CBF as the point spread function (PSF). Motivated by this work, a deconvolution technique is applied to FDB in this letter to achieve a high spatial resolution and low sidelobe levels while retaining robustness. The resulting method is named deconvolved frequency-difference beamforming (Dv-FDB).

The rest of the letter is organized as follows. The mathematical descriptions of the FDB for a linear array received signal from a distant point source are stated in Sec. 2. The shift invariance of the FDB beam pattern is proved and an analytic expression similar to that of narrowband CBF is provided, and then Dv-FDB is proposed in Sec. 3. Section 4 compares the performances of Conv-BF, FDB, and Dv-FDB based on numerical simulations and experimental results. Finally, we provide a short summary and draw conclusions in Sec. 5.

To simplify discussions, we consider a uniform linear array (ULA) with $M$ equally spaced elements as the receiving array. The signal from a distant point source is assumed to arrive as a plane wave with wide frequency band $[fL, fH]$⁠. The frequency-domain received signal on the mth element from the ith source can be written as

where f is the signal frequency, $Ai(f)$ is the frequency spectrum of the signal from the ith source, $j=−1$⁠, $d$ is the spacing between adjacent elements, $θi$ is the direction of arrival (DOA) measured from the broadside of the array, and $c$ is the sound speed.

The frequency-difference autoproduct¹⁰ $AXmi$ between $Xmi(f¯)$ and $Xmi(f¯+Δf)$ is given by

where the superscript asterisk denotes a complex conjugate, $Δf$ is the difference frequency satisfying $1/T<Δf<fH−fL$ for signal recordings with a duration $T$⁠, $Δf≤c/(2d)$ is required to avoid grating lobes, and $f¯$ is the frequency under examination with $fL≤f¯≤fH−Δf$⁠.

FDB conducts the delay-and-sum beamforming on $AXmi$'s at different elements with the weighting at the mth element as

and the complex beam output at frequency $f¯$ with the difference frequency $Δf$ is given by

From Eqs. (3) and (4), it is observed that $wm(Δf, sin θ)$ is similar to the weighting of narrowband CBF, and FDB is actually a CBF (in frequency domain) on $AXmi(f¯,Δf)$⁠.

Substituting Eqs. (2) and (3) into Eq. (4), the complex beam output of FDB is given by

For a given $Δf$⁠, the wideband beam output of FDB can be obtained by an incoherent average of either Eq. (4) or Eq. (5) over frequency $f¯$⁠, as

where $⟨·⟩f¯$ indicates the average over the whole range of $f¯$⁠. $PFDB(sin θ)$ is known as the spatial spectrum of $AXmi(f¯,Δf)$ and bearings corresponding to the peaks in the spectrum are taken as the DOA estimates of incoming signals. FDB provides a robust estimation by conducting the beamforming at a user-selectable $Δf$ which can be lower than the in-band signal frequencies where the effects of array sparseness and wavefront mismatch on the beam output are more prominent. However, this causes the beamwidth and the sidelobes to increase, thus decreasing the ability for source discrimination and interference rejection. Our main idea here is how to reduce the FDB's beamwidth and sidelobe levels while maintaining its robustness.

In this section, we consider using the deconvolution technique as postprocessing for the FDB beam output. We prove that the beam pattern of FDB for a distant point source is shift invariant, so that it can be used as a PSF in the deconvolution technique. Then, the deconvolution technique is applied to the FDB beam output, and the procedure for implementing Dv-FDB is presented.

For a given $Δf$⁠, Eq. (6) can be rewritten as

Denoting $APi≜⟨|Ai(f¯)Ai(f¯+Δf)|2⟩f¯$ as the signal level of the ith source, and after some manipulations, the beam output of FDB in Eq. (7) can be simplified as

where

is the FDB beam pattern steering to direction $θi$ with the difference frequency $Δf$⁠. We note that $BpFDB$ is shift invariant when expressed in $sin θ$ and can be regarded as PSF that denotes the spatial response of FDB to the incoming signal.

Assuming that the signals from $K$ sources with different DOAs are uncorrelated, the total output of FDB can be obtained by summing the FDB's outputs from individual sources

Following the work of Yang,⁹ Eq. (10) can be written in an integral form as

where $S(sin ϑ)≜Σi=1KAPiδ(sin θ−sin ϑ)$ is the source spatial distribution with the Dirac delta function $δ(sin θ−sin ϑ)$ defined as

Examining Eq. (11) from the perspective of image processing, $PFDB(sin θ)$ can be regarded as a “fuzzy image” produced by the convolution of $S(sin ϑ)$ (the original image) with the beam pattern $BpFDB$ (PSF). Since $BpFDB$⁠, used here as the PSF, is known for a given $Δf$⁠, the deconvolution technique can be applied to better recover the “original image” which is the source spatial distribution in the problem under consideration. This process is known as “deblurring” in image restoration. And the resulting method is addressed as the deconvolved frequency-difference beamforming (Dv-FDB) in this letter.

In this letter, we choose the Richardson-Lucy (R-L) algorithm^11,12 for the deconvolution in Eq. (11). Other iterative techniques¹³ are also applicable.

Taking $BpFDB$ as the PSF, the iterative procedure of the R-L algorithm is given by

where the superscript $(i)$ is the iteration number and symbol “$⊗$” denotes the convolution operator. The beam output of FDB is used as the initial estimate for source distribution

We set the maximum number of iterations for deblurring as $(I)$⁠. The deconvolution is terminated when $(I)$ iterations are finished, and the beam output of Dv-FDB is given by

According to the above analysis, the implementation of Dv-FDB can be divided into the following four steps:

1. apply Fourier transform to the received data of the array;

2. choose the difference frequency $Δf$⁠, and obtain the beam pattern of FDB according to Eq. (9) (obtain PSF);

3. calculate the beam output $PFDB(sin θ)$ given by Eq. (6) (the “fuzzy image”);

4. set the maximum number of iterations $(I)$⁠, and the beam output of Dv-FDB yields after $(I)$ iterations are executed, according to Eq. (13).

Compared to FDB, Dv-FDB demonstrates a “clearer” spatial spectrum by a narrower beamwidth and lower sidelobe levels. We will verify the superior performance of Dv-FDB using numerical simulations and experimental data.

In the simulations, we consider a ULA with 31 elements receiving signals from distant point sources with the same frequency band of 50–1000 Hz. The interelement spacing is 6 m. To avoid the grating lobes, the maximum difference frequency $Δf$ is calculated as 125 Hz with a sound speed of 1500 m/s. We assume that the signals arrive as plane waves and the ambient noise is white Gaussian. The sampling frequency is 4096 Hz.

In the first simulation, we consider the noise-free situation and investigate the $−3 dB$ beamwidths (the half-power beamwidth)¹⁴ of FDB and Dv-FDB. The signals are incident onto the array within the range of $[−60°,60°]$ (i.e., $−0.87≤sin θ≤0.87$⁠) with $1°$ interval. We set $Δf$ to cover the 30–125 Hz range and $(I)=10$⁠. The beamwidths are calculated as functions of $Δf$ and $sin θ$⁠, as shown in Fig. 1. As seen from Fig. 1, the beamwidths of both FDB (the top panel) and Dv-FDB (the bottom panel) become narrower with increasing $Δf$⁠, i.e., a high resolution can be achieved with a large $Δf$⁠. For a given $Δf$⁠, the beamwidths are the narrowest when their beams steer toward to the broadside, and then the beamwidths are widened as the steering angles approach the end-fire direction. It can be seen that the beamwidth of Dv-FDB is noticeably narrower than that of FDB with the same $sin θ$ and $Δf$⁠. Taking the maximum $Δf=125 Hz$ as an example, the beamwidth of FDB varies from $3.26°$ to $6.57°$⁠, while the corresponding beamwidth of Dv-FDB varies from $1.10°$ to $2.21°$⁠. It can be inferred that Dv-FDB yields a narrower beamwidth leading to a higher spatial resolution, as will be verified below.

In the second simulation, we investigate the performances of FDB and Dv-FDB in the scenario where there are four broadband signals arriving at the array in the directions of $−30°$⁠, $2°$⁠, $5°$⁠, and $30°$⁠, respectively. The SNR of the received data on the element is set as 10 dB. $Δf=125 Hz$ is used to achieve a high resolution in comparing the performances. $(I)=50$ iterations are executed in Dv-FDB. The spatial spectra estimated by FDB and Dv-FDB are shown in Fig. 2. For comparison, the result of Conv-BF is also shown in the figure. It is seen from Fig. 2 that Dv-FDB provides an accurate estimate of the DOAs of all four sources, while FDB only gives the correct DOA estimate of two sources from directions $−30°$ [$sin (−30°)=−0.5$] and $30°$ [$sin (30°)=0.5$]. Due to the wide beamwidth, FDB fails to resolve other two sources from the directions of $2°$ [$sin (2°)=0.04$] and $5°$ [$sin (5°)=0.09$] which are discriminated by Conv-BF, though with biased DOAs estimates. However, Conv-BF fails to indicate the presence of the other two sources, from directions of $−30°$ and $30°$⁠, in this example. The reason is that the spectrum of Conv-BF in the figure is the incoherent average over the whole frequency band, with a wide beamwidth in low frequency sub-band and grating lobes in high frequency sub-band, resulting in a bad estimation of DOAs for directions far from the broadside. Actually, the performance of FDB is equivalent to that of Conv-BF when the difference frequency $Δf$ used for FDB matches the absolute frequency used for Conv-BF.⁷ In addition, as shown in Fig. 2, the sidelobe levels of Dv-FDB are 3–7 dB lower than those of FDB.

The simulations demonstrate that as an estimator of DOA, Dv-FDB yields better performance than FDB and Conv-BF. This improvement occurs because the beam output of FDB is the convolution of the beam pattern $BpFDB$ with the “true” signal spatial distribution $S(sin θ)$⁠, and Dv-FDB recovers the original signal distribution by applying the deconvolution technique.

An experiment was carried out in South China Sea in July 2014. A horizontal linear array (HLA) with 43 elements and 6 m interelement spacing was towed with a speed of 4.4 kn at a depth of 120 m. The sampling frequency was 12 kHz. The time window length was 80 s, during which time signals radiated from two sources near the broadside of the array were recorded. The beam outputs of FDB and Dv-FDB were incoherently averaged over the frequency band of 20–600 Hz where the input SNRs were high. $Δf=125 Hz$ was used to achieve a high resolution, the same as in the simulations, and so was $(I)=50$ in Dv-FDB. For comparison, Conv-BF was also conducted using exactly the same frequency. Figure 3 shows the bearing time recordings (BTRs) obtained using Conv-BF (the top slice), FDB (the middle slice), and Dv-FDB (the bottom slice), respectively. It is seen that Conv-BF and FDB fail to estimate the DOAs of sources A and B due to their low spatial resolution. However, from the BTR of Dv-FDB, the bearing recordings of sources A and B are clearly distinguished, which are maintained around $−13.4°$ [$sin (−13.4°)=−0.23$⁠, source A] and $−10.6°$ [$sin (−10.6°)=−0.18$⁠, source B] during the analyzed time interval, respectively. In addition, compared the BTRs in Fig. 3, the BTR of Conv-BF presents the lowest sidelobe levels. A high-level sidelobe is found at $−7°$ [$sin (−7°)=−0.12$] near source B in the BTR of FDB. Nevertheless, the sidelobe is much suppressed in the BTR of Dv-FDB as a result of the lower sidelobe levels provided by the deconvolved beam output.

From both numerical simulations and the experimental results, it can be seen that the true source distribution is better recovered by Dv-FDB, providing not only lower sidelobe levelsbut also higher spatial resolution and robustness.

In this letter, the Dv-FDB algorithm is proposed to improve the beamwidth and suppress the sidelobe levels associated with the FDB algorithm. We prove that the beam pattern of FDB is shift invariant and hence can be regarded as a PSF corresponding to FDB beam output. On this basis, a “clearer” spatial spectrum of Dv-FDB is achieved by deconvolution using the R-L algorithm. Simulations and experimental results both demonstrate that Dv-FDB outperforms FDB in terms of higher resolution and lower sidelobe levels while maintaining robustness.

For simplicity, we have considered a uniform linear array in this letter. The Dv-FDB can be extended to a non-uniform array, in which case “deblurring” algorithms for a shift-variant PSF should be used in recovering original signal distribution.

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11534009, 11904342.