Beam-space spatial spectrum reconstruction under unknown stationary near-field interference: Algorithm design and experimental verification

Chu, Jichen; Cheng, Lei; Xu, Wen

doi:10.1121/10.0030334

In acoustic array signal processing, spatial spectrum estimation and the corresponding direction-of-arrival estimation are sometimes affected by stationary near-field interferences, presenting a considerable challenge for the target detection. To address the challenge, this paper proposes a beam-space spatial spectrum reconstruction algorithm. The proposed algorithm overcomes the limitations of common spatial spectrum estimation algorithms designed for near-field interference scenarios, which require knowledge of the near-field interference array manifold. The robustness and efficacy of the proposed algorithm under strong stationary near-field interference are confirmed through the analysis of simulated and real-life experimental data.

1. Introduction

Stationary near-field interferences are frequently encountered in many underwater systems. For instance, the array towed behind the ship inevitably receives the strong near-field interferences caused by the loud vibration noise of the towing vessel, which can be regarded as stationary within a certain observation period. Unlike spatial white noise, near-field interference tends to have directionality, high energy (often significantly greater than that of the targets due to its proximity to the array), and various other spatial and statistical properties. These characteristics pose great challenges for spatial spectrum estimation algorithms. Specifically, strong near-field interference can create masking regions¹ in the spatial spectrum for conventional beamforming (CBF), resulting in substantial estimation errors, and even failed detection of targets located within the masking region. Some high-resolution spectrum searching algorithms, including minimum-variance distortionless response (MVDR)² and multiple signal classification (MUSIC),³ can effectively alleviate these issues. However, they are sensitive to signal mismatch.^4,5 Furthermore, they experience compromised estimation accuracy when the near-field interference closely resembles far-field targets, such as when both are situated near the endfire of an array.

Recently, algorithms targeting near-field interference environments have been proposed, such as matrix filter techniques,⁶ which eliminate interference prior to conventional spectrum estimation algorithms, and multi-snapshot sparse Bayesian (MSBL) algorithms that incorporate the array manifold of near-field interference into the dictionary matrix.⁷ However, these algorithms face the same challenge of requiring knowledge of the array manifold of near-field interference, which is equivalent to knowing the position of near-field interference if the near-field interference follows spherical wave propagation. Although stationary near-field interferences allow for pre-estimating the position of near-field interference, the algorithms' performance still experiences a significant decline. This is due to the estimation results of the near-field interference often being accompanied by significant errors, especially in scenarios involving multiple near-field interferers.

To tackle the stationary near-field interferences more effectively, we propose a systematic approach, called beam-space spatial spectrum reconstruction algorithm. Initially, we designed a sample matrix inversion (SMI) adaptive beamformer^4,5 using pre-collected snapshots of interference plus noise. This beamformer serves as a spatial filter to eliminate strong near-field interferences, deriving the beam-space signal. Subsequently, inspired by the remarkable performance of sparse Bayesian learning (SBL) framework in sparse reconstruction⁸ and the intrinsic nonnegative property of the spatial spectrum, this paper exploits the non-negative fast sparse Bayesian learning (NN-FSBL) algorithm to reconstruct the spatial spectrum from the beam-space signal. The salient features (e.g., super-resolution and robustness) of the proposed algorithm are demonstrated by simulation and real-life experimental results.

2. Problem formulation

2.1 Array signal model

Consider a single frequency case with K far-field targets and J discrete near-field interferers. They impinge on a uniform linear array (ULA) with M sensors. The received array signal model for the tth frequency-domain snapshot is

s (t) = \sum_{k = 1}^{K} a_{k} (t) v_{F} (θ_{k}) + \sum_{j = 1}^{J} b_{j} (t) v_{N} (r_{j}, θ_{j}) + n (t),

(1)

where t is the index of the frequency-domain snapshot,

a_{k} (t)

and

b_{j} (t)

are the amplitudes of the kth target source and the jth interferer, respectively at the tth frequency-domain snapshot.

v_{F} \in C^{M}

and

v_{N} \in C^{M}

are array manifold vectors of the far-field target source and near-field interferer, respectively.

n (t) \in C^{M}

is the additive noise, identically following complex Gaussian

C N (0, σ_{n}^{2} I_{M})

across snapshots.

r_{j}

represents the radial distance from the jth near-field interferer to the center of the array, and

θ_{j}

denotes the angle between the line connecting the jth near-field interferer and the array center and the horizontal direction of the array.

2.2 Adaptive beamformer design

Analogous to frequency domain filters, we consider designing a spatial filter to eliminate near-field interference from the received array signals. The optimum beamformer designed based on the maximum output signal-to-noise ratio (SINR) criterion is a typical type of spatial filter that has an approximate zero response to interference, known as a null.^4,5 The steering vector of the optimum beamformer steered at

ϕ

is defined as^4,5

c (ϕ) = α R_{i + n}^{- 1} v_{F} (ϕ),

(2)

where

α

is an constant that can be specified according to the different requirements of the beamformer,

R_{i + n}

is the interference-plus-noise covariance.

In practice, the actual

R_{i + n}

is unknown. However, in scenarios where training data containing only near-field interference and noise is available, we turn to the maximum-likelihood (ML) estimate of the

R_{i + n}

in Eq. , resulting in the sample matrix inversion (SMI) adaptive beamformer.⁹

R_{i + n}

is given by the average of outer products of T snapshots,⁴

{\hat{R}}_{i + n} = \frac{1}{T} \sum_{t = 1}^{T} s_{i + n} (t) s_{i + n}^{H} (t),

(3)

where

s_{i + n} (t)

denotes the interference-plus-noise component of the array signal.

Consider uncorrelated zero-mean random sources, which are commonly encountered in passive localization scenarios when receiving the self-noise from non-cooperative targets.⁵ The ideal output beam power of the adaptive beamformer steered at angle

ϕ

for the received array signal can be expressed as

\begin{matrix} y (ϕ) = E {{| c_{SMI}^{H} (ϕ) s (t) |}^{2}} = c_{SMI}^{H} (ϕ) R_{s} c_{SMI} (ϕ) \\ = \sum_{k = 1}^{K} σ_{k}^{2} {| c_{SMI}^{H} (ϕ) v_{F} (θ_{k}) |}^{2} + σ_{n}^{2} c_{SMI}^{H} (ϕ) c_{SMI} (ϕ) \end{matrix},

(4)

where

R_{s}

is the covariance matrix of the received array signal. The equation in the second line of Eq. is derived based on the assumption that the SMI adaptive beamformer can completely suppress the near-field interference using sufficient interference-plus-noise snapshots, i.e.,

{| c_{SMI}^{H} (ϕ) v_{N} (r_{j}, θ_{j}) |}^{2} = 0

for

j = 1, \dots, J

⁠. This assumption holds under the condition that the near-field interferers are stationary since the null of the optimum beamformer is sensitive to the direction-of-arrival (DOA) changes.^4,5

Imposing the constraint of unit white noise gain, i.e.,

c_{SMI} {(ϕ)}^{H} c_{SMI} (ϕ) = 1

⁠, Eq. is simplified as

y (ϕ) = \sum_{k = 1}^{K} σ_{k}^{2} \underset{≜ B p (ϕ | θ_{k})}{\underset{︸}{{| c_{SMI}^{H} (ϕ) v_{F} (θ_{k}) |}^{2}}} + σ_{n}^{2},

(5)

where

c_{SMI} (ϕ) = \frac{{\hat{R}}_{i + n}^{- 1} v_{F} (ϕ)}{\sqrt{v_{F}^{H} (ϕ) {\hat{R}}_{i + n}^{- 1} {\hat{R}}_{i + n}^{- 1} v_{F} (ϕ)}},

(6)

and

B p (ϕ | θ_{k})

denotes the beam power as a function of beam direction

ϕ

for the far-field source at

θ_{k}

with unit power.

Consider a measurement vector composed of N sequentially arranged beam outputs at different $ϕ_{n}$ s, that is $y = {[y (ϕ_{1}), \dots, y (ϕ_{n})]}^{T} \in R_{+}^{N}$ (⁠ $R_{+}$ is the field of nonnegative real numbers). Based on Eq. (5), $y$ can be expressed as the linear combination of the beam power vectors for varying arriving angles, i.e., ${h_{k} = {[B p (ϕ_{1}, θ_{k}), \dots, B p (ϕ_{N}, θ_{k})]}^{T}}_{k = 1, \dots, K}$ ⁠, with the signal power $σ_{k}^{2}$ being the associated coefficient. This interpretation motivates a sparsity-aware problem formulation, which has witnessed great success in modeling element-space measurements.^10,11

Consequently, we discretize the entire angular space into L (⁠

L ≫ K

⁠) grids, thus obtaining a dictionary matrix

H \in R_{+}^{N \times L}

⁠, of which the lth column is

h_{l}

⁠. Considering the non-negativity of the spatial spectrum, let vector

x = {[x_{1}, \dots, x_{L}]}^{T} \in R_{+}^{L}

comprise the source powers at all the grid points, of which non-zero entries correspond to grid points where actual sources lie. We refer to the space spanned by the output of the beamformer as the beam-space, then the beam-space data model can be expressed as

\begin{matrix} y = Hx + σ_{n}^{2} 1 = \bar{H} \bar{x} \end{matrix},

(7)

where

\bar{H} = [H, 1] \in R_{+}^{N \times (L + 1)}

and

\bar{x} = [x; σ_{n}^{2}] \in R_{+}^{L + 1}

⁠. Equation implies that both spatial spectrum and noise power can be simultaneously estimated, thereby enhancing estimation accuracy, particularly in target strength.

The equality in Eq. holds only if

y

is computed using the actual covariance matrix

R_{s}

in Eq. , which, however, is challenging to meet. In practice, the sample covariance matrix

{\hat{R}}_{s}

is instead used to calculate the beam output in Eq. , and is computed by replaced

s_{i + n}

with

s

in Eq. . Thus we have

\hat{y} = \bar{H} \bar{x} + e, \bar{x} \in R_{+}^{L + 1},

(8)

where

e

denotes the model errors induced by using

{\hat{R}}_{s}

⁠.

3. Non-negative fast sparse Bayesian learning

To exploit the nonnegative characteristics of the spatial spectrum, we incorporate a non-negativity constraint in the off-the-shelf fast sparse Bayesian learning (FSBL)¹² algorithm, leading to the development of the NN-FSBL algorithm.

First, consider Eq. without the nonnegative constraint on

\bar{x}

⁠. under SBL framework, the likelihood function of the observed data is modeled as

p (\hat{y} | \bar{x}, σ^{2}) = \frac{1}{{(2 π σ^{2})}^{N / 2}} exp {- \frac{{‖ \hat{y} - \bar{H} \bar{x} ‖}^{2}}{2 σ^{2}}},

(9)

and

\bar{x}

follows a Gaussian prior with hyper-parameters

{α_{l}}_{l}

⁠,

p (\bar{x} | α) = {(2 π)}^{- (L + 1) / 2} \prod_{l = 1}^{L + 1} α_{l}^{1 / 2} exp (- \frac{α_{l} {\bar{x}}_{l}^{2}}{2}) .

(10)

Then using Bayes' theorem, the minimum mean square error (MMSE) estimate of

\bar{x}

is given by the posterior mean of

\bar{x}

⁠,

\hat{\bar{x}} = σ^{- 2} Σ {\bar{H}}^{T} \hat{y},

(11)

where

Σ = {(A + σ^{- 2} {\bar{H}}^{T} \bar{H})}^{- 1}

denotes the posterior covariance, and

A = diag (α_{1}, \dots, α_{L + 1})

⁠.

The hyper-parameters in Eq. can be determined via maximizing the marginal likelihood,

L (α, σ^{2}) ≜ log p (\hat{y} | α, σ^{2}) = - [N log 2 π + log | C | + {\hat{y}}^{T} C^{- 1} \hat{y}],

(12)

where

C

can be decomposed as

C = \underset{≜ C_{- i}}{\underset{︸}{σ^{2} I + \sum_{l \neq i}^{} α_{l}^{- 1} {\bar{h}}_{l} {\bar{h}}_{l}^{T}}} + α_{i}^{- 1} {\bar{h}}_{i} {\bar{h}}_{i}^{T},

(13)

where the term

C_{- i}

is irrelevant to

α_{i}

⁠,

{\bar{h}}_{l}

is the lth column of

\bar{H}

⁠.

Substituting Eq. into Eq. yields

L (α, σ^{2}) = L_{- i} (α, σ^{2}) + \frac{1}{2} [log α_{i} - log (α_{i} + s_{i}) + \frac{q_{i}^{2}}{α_{i} + s_{i}}],

(14)

where

L_{- i} (α, σ^{2})

contains the terms not relevant to

α_{i}

⁠, and

s_{i} = \frac{α_{i} S_{i}}{α_{i} - S_{i}}, q_{i} = \frac{α_{i} Q_{i}}{α_{i} - S_{i}},

(15)

with

S_{i} = {\bar{h}}_{i}^{T} C^{- 1} {\bar{h}}_{i}, Q_{i} = {\bar{h}}_{i}^{T} C^{- 1} \hat{y} .

(16)

To maximize the marginal likelihood in Eq. , the optimal

α_{i}

and

σ^{2}

satisfies¹²

\begin{matrix} α_{i} = {\begin{array}{l} \frac{s_{i}^{2}}{q_{i}^{2} - s_{i}}, & q_{i}^{2} > s_{i}, \\ \infty, & q_{i}^{2} \leq s_{i}, \end{array} σ^{2} = \frac{{‖ \hat{y} - \bar{H} \bar{x} ‖}_{2}^{2}}{N - \sum_{l = 1}^{L + 1} (1 - α_{l} Σ_{l l})}, \end{matrix}

(17)

where

Σ_{l l}

is the lth diagonal element of the posterior covariance

Σ

⁠.

Given the values of these hyper-parameters, Eq. (17) leads to a birth-and-death procedure in FSBL for determining the values of hyper-parameters ${α_{i}}_{i}$ along with adding/deleting one basis vector in each iteration (for more insight, see Ref. 12). Fortunately, this scheme of FSBL provides a viable way to enforce the non-negativeness of $\bar{x}$ after certain modifications, while still guaranteeing the increment of the objective function $L (α, σ^{2})$ ⁠.

Specifically, we calculate the potential $\bar{x}$ and the increment of the marginal likelihood (i.e.,the objective function) of FSBL for each basis candidate in the dictionary matrix. Then, in each iteration, we select the candidate that ensures both the non-negativity of $\bar{x}$ and the largest increment in marginal likelihood for the update. This procedure provides a viable way to enforce the non-negativeness of $\bar{x}$ while preserving the algorithm's convergence. For further details, please refer to Ref. 13 where the NN-FSBL algorithm was first proposed.

4. Spatial spectrum reconstruction using beam-space signal

After introducing the SMI adaptive beamformer design and the NN-FSBL algorithm, we are now ready to present the developed beam-space spatial spectrum reconstruction method. Following the consecutive blocks in Fig. 1, the proposed method can be summarized in the following steps:

Fig. 1.

View large Download slide

Block diagram of beam-space spatial spectrum reconstruction.

Step 1: Design an SMI adaptive beamformer using interference-plus-noise snapshots.
Step 2: Let the received array signal mixed with target sources and near-field interferences pass the adaptive beamformer to obtain the beam-space signal.
Step 3: Use the NN-FSBL algorithm to reconstruct the spatial spectrum from beam-space signal.

It is worth noting that the adaptive beamformer design and sparse learning are not two independent modules that are simply combined together. The adaptive beamformer not only enables the suppression of near-field interference but also leads to spectrum estimation in the beam space. Compared with previous approaches that directly perform sparse reconstruction algorithms on the array signals, the proposed algorithm offers a more concise solution by requiring much fewer unknowns to be determined. Specifically, in element space, the variables to be solved are complex signal amplitudes across multiple snapshots, with the total number of variables being the product of the number of grid points and the number of snapshots, i.e., $L \times T$ ⁠. However, since our proposed algorithm operates in beam space, the variables to be solved are L nonnegative real numbers representing the signal power at grid points and the number of these variables is independent of the number of received snapshots, T.

Moreover, the dimension of the measurement vector $y \in R_{+}^{N}$ can be flexibly adjusted by artificially tuning the number of steering directions, $ϕ$ ⁠. This strikes an excellent balance between reconstruction probability and computational burden, a feat difficult to achieve with classical sparse learning methods performed in the element space, where the dimension of the measurement vector is fixed to the number of array sensors, M. In scenarios where the number of sensors is large, which is common with towed arrays, this property is highly valuable, as the computation cost can be excessively high for certain sparse reconstruction algorithms such as the basis pursuit (BP) algorithm.

5. Simulation

In this section, we validate the effectiveness of the proposed algorithm using simulated data and compare it with classical and recent state-of-the-art algorithms, including CBF, MVDR, and a variant of MSBL designed for near-field interference scenarios⁷ (mentioned in Sec. 1, represented as MSBL-1), to demonstrate its robustness against signal mismatch.

The simulation considers two cases: with no signal mismatch and with signal mismatch for three far-field targets at

40 °

⁠,

60 °

⁠, and

80 °

⁠, respectively, measured from the endfire of the array, and two interferers at (r,

θ

⁠) = (10 m,

45 °

⁠) and (r,

θ

⁠) = (8 m,

60 °

⁠) with INR = 20 dB. We assume the signal mismatch results from changes in the gain and phase of the array sensors or from imperfect gain and phase in the processor filters. Denoting the mth component of

v_{F}

as

v_{m}

⁠, we can write the nominal sensor response as

v_{m} = g_{m} * e^{j Ψ_{m}},

(18)

where

g_{m}

and

Ψ_{m}

represent the amplitude and phase of the mth array sensor, respectively. The actual sensor response can be written as

v_{m} = g_{m} * (1 + Δ g_{m}) * e^{j (Ψ_{m} + Δ Ψ_{m})},

(19)

where

{Δ g_{m}}_{m}

and

{Δ Ψ_{m}}_{m}

are the changes in the gain and phase of the array sensors. Assuming they are independent and identically distributed (i.i.d.) Gaussian variables, following

N (0, λ_{g})

and

N (0, λ_{Ψ})

⁠, respectively. We set

λ_{g} = 0.1

and

λ_{Ψ} = 0.05

for the situation with signal mismatch.

In simulation, we assume that the SNR of the target at $80 °$ is 15 dB, and the other targets are 10 dB, and 16-element ULA with inter-element spacing of 0.75 m is applied to receive signals at a frequency of 1000 Hz. One-thousand snapshots of interference plus noise are used to construct the SMI adaptive beamformer, while 200 snapshots containing the signals of interest are utilized for subsequent spatial spectrum estimation processing. To fairly compare the performances, CBF deals with the case assuming no near-field interference, since it does not have mechanisms for interference suppression.

Comparing Figs. 2(a) and 2(b), the MVDR algorithm is sensitive to signal mismatches, leading to a significant performance degradation by underestimating the target strength by 15 dB. Meanwhile, MSBL-1 encounters severe basis mismatch problems¹⁴ under near-field interference, resulting in substantial estimation errors and numerous false targets.

Fig. 2.

View large Download slide

Spatial spectrum estimation results of CBF, MVDR, MSBL-1, and the proposed method for three targets at $40 °$ ⁠, $60 °$ ⁠, and $80 °$ with (a) no signal mismatch, and (b) with signal mismatch. The red plus signs denote the actual targets. The two near-field interferers locate at (r, $θ$ ⁠) = (10 m, $45 °$ ⁠) and (r, $θ$ ⁠) = (8 m, $60 °$ ⁠). Histograms of (c)–(f) power estimates for CBF, MVDR, MSBL-1, and the proposed method, obtained from 1000 Monte Carlo runs. The red dashed lines in (c)–(f) show the true powers.

In contrast, the proposed algorithm achieves satisfactory spatial spectrum reconstruction results both without and with signal mismatch, respectively. It can be seen from Fig. 2(b) that the proposed algorithm can achieve much more accurate strength estimation in the presence of signal mismatch, depending on the available prior information about the near-field interference. It is noteworthy that the snapshots of interference plus noise required by the proposed algorithm are often milder conditions and more easily achievable compared to the specific prior knowledge of the location of near-field interference required by MSBL-1. Figures 2(c)–2(f) further prove that our proposed algorithm achieves better robustness in power estimation compared to the MVDR and MSBL-1 algorithms. Note that, in addition to benefiting from the robustness of the SBL framework,¹¹ the superior robustness against signal mismatch, comparable to that of the CBF algorithm, primarily originates from the beamforming operation within the algorithm. This highlights a significant advantage of processing signals in the beam space over the element space.

6. Experimental data analysis

In this section, the effectiveness and superior performance of the proposed method are validated using experimental data.

6.1 Laker experimental data

The experimental setting is illustrated in Fig. 3(a). In the experiment, the same ULA used in the previously noted simulation is deployed horizontally, positioned 10 m underwater. Two far-field target sources are positioned approximately 40 m below the surface and at a horizontal distance of 100 m from the array, at angles of $θ = 90 °$ and $θ = 120 °$ relative to the array end-fire direction, respectively. With the array center as the origin of the coordinates, the near-field interference is suspended from the experiment platform at (r, $θ$ ⁠) = (5 m, $90 °$ ⁠).

Fig. 3.

View large Download slide

Laker experiment. (a) The diagram of the experiment setup. The 16-element ULA with inter-element spacing of 0.75 m is deployed horizontally, positioned 10 m underwater. Two far-field target sources are located at angles of $θ = 90 °$ and $θ = 120 °$ relative to the array end-fire direction, respectively. With the array center as the origin of the coordinates, the near-field interference is suspended from the experiment platform at (r, $θ$ ⁠) = (5 m, $90 °$ ⁠). (b) The normalized spatial spectrum estimation results of CBF, MVDR, MSBL-1, and the proposed method. The red plus signs in (b) denote the actual targets.

Before receiving signals from two far-field targets, we recorded a 30-s signal containing only near-field interference and noise. Subsequently, two target sources started transmitting a single-frequency signal at 1000 Hz and the signal mixed with both far-field target sources and near-field interference was recorded for 25 s. A 50% overlapping FFT is performed on the received two segments of signals with a sliding time window of 0.2 s, resulting in two datasets containing 299 and 249 snapshots respectively. The 299 snapshots of interference plus noise are used to construct the SMI adaptive beamformer, while the 249 snapshots, containing the signals of interest, are utilized for subsequent spatial spectrum estimation processing. The angular space from $0 °$ to $180 °$ is uniformly discretized at $0.1 °$ in the data processing.

6.2 Results

Figure 3(b) shows the normalized spatial spectrum estimation results of CBF, MVDR, MSBL-1, and the proposed method. It should be noted that spatial smoothing and forward-backward averaging techniques⁵ are employed in the CBF, MVDR, and the proposed algorithm to mitigate the correlation of the sources, making them approximately satisfy the assumption of uncorrelated sources.

As observed in Fig. 3(b), near-field interference creates a masking region from approximately $30 °$ to $90 °$ for CBF, making it challenging to observe the target at $90 °$ ⁠. Since the other far-field target is located outside the masking region at $120 °$ ⁠, a peak can be observed in this direction. However, next to this peak, there is an obvious false peak with comparable energy (reduced by less than 2 dB) due to the energy leakage caused by the high sidelobes of CBF. While MVDR can mitigate this deficiency, Fig. 3(b) shows that many false targets are still noticeable. MSBL-1 reduces the number of false targets, however, its estimation errors for the far-field target at $90 °$ ⁠, near the direction of the near-field interferer, are significantly increased, especially in terms of target strength. This can be attributed to mismatches between the dictionary atoms and the actual array manifold, particularly with the manifold of the near-field interference.

Compared with the algorithms mentioned previously, the proposed algorithm demonstrates superior estimation accuracy, both in terms of the DOA and strength of targets, see Fig. 3(b). Additionally, the proposed algorithm exhibits the sharpest peaks and cleanest backgrounds, showcasing the super-resolution capability and excellent denoising performance, owing to the sparsity and robustness provided by the NN-FSBL algorithm.

7. Conclusions

In this paper, to address the problem of far-field spatial spectrum reconstruction under strong stationary near-field interferers, we design a systematic solution for scenarios where training data containing only near-field interference and noise is available in advance. The proposed algorithm demonstrates superior robustness against signal mismatch resulting from changes in the gain and phase of the array sensors, particularly in terms of power estimation. Additionally, it also inherits the advantages of SBL, including super-resolution and denoising capability, thereby producing sharp peaks and a clean background. In future research, we will explore extending the proposed algorithm to scenarios where near-field interference moves rapidly relative to the receiving array.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 61831020 and the National Key Research and Development Program of China Grant No. 2016YFC1400100.

Author Declarations

Conflict of Interest

The authors confirm that there are no conflicts of interest associated with the publication of this research article. Any potential conflicts arising from funding sources or affiliations have been disclosed accordingly.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Beam-space spatial spectrum reconstruction under unknown stationary near-field interference: Algorithm design and experimental verification

1. Introduction

2. Problem formulation

2.1 Array signal model

2.2 Adaptive beamformer design

3. Non-negative fast sparse Bayesian learning

4. Spatial spectrum reconstruction using beam-space signal

5. Simulation

6. Experimental data analysis

6.1 Laker experimental data

6.2 Results

7. Conclusions

Acknowledgements

Author Declarations

Conflict of Interest

Data Availability

References

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Beam-space spatial spectrum reconstruction under unknown stationary near-field interference: Algorithm design and experimental verification

1. Introduction

2. Problem formulation

2.1 Array signal model

2.2 Adaptive beamformer design

3. Non-negative fast sparse Bayesian learning

4. Spatial spectrum reconstruction using beam-space signal

5. Simulation

6. Experimental data analysis

6.1 Laker experimental data

6.2 Results

7. Conclusions

Acknowledgements

Author Declarations

Conflict of Interest

Data Availability

References

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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