Improvement of orthogonal matching pursuit deconvolution beamforming method for acoustic source identification

Abstract

The deconvolution approach for the mapping of acoustic sources (DAMAS) based on orthogonal matching pursuit (OMP-DAMAS) has attracted much attention due to its advantages of high spatial resolution and excellent capability to suppress spurious sources. In this paper, we propose an improved version of OMP-DAMAS based on fast Fourier transformation (FFT), abbreviated as FFT-OMP-DAMAS. This method assumes that the array point spread functions (PSFs) are spatially shift-invariant. The sum of the product of the acoustic source distribution and PSFs is converted into a convolution form, which is further converted into a product in the wave number domain. With these conversions, FFT can be used to improve the solving speed. Both simulations and experiments show that the proposed method not only inherits the advantages of OMP-DAMAS in terms of spatial resolution and spurious source suppression but also significantly improves the computational efficiency compared with OMP-DAMAS.

Keywords

Acoustic source identification microphone array deconvolution orthogonal matching pursuit fast Fourier transformation

Introduction

Noise control is an important research work in the fields of automobiles,^1–2 aero-engines,^3–5 high-speed rail,^6–7 propeller plane,^8–9 and other fields. How to accurately identify noise sources is the premise and foundation of noise control. Beamforming noise source identification technology based on microphone array measurement has attracted much attention due to its fast measurement speed and convenient layout.^10–11 Delay and sum (DAS)¹² is one of the classic beamforming algorithms. However, it cannot clearly locate the sound source due to the poor spatial resolution at low frequencies and the heavy pollution of spurious sources at high frequencies.

To overcome the above issues of DAS, the deconvolution approach for the mapping of acoustic sources (DAMAS) was first proposed.^13–14 The method removes beamforming features from the output representation and employs a unique system of linear equations to account for interactions at different locations within the measurement region of the array. It is solved by introducing positive constraints in the iterative process to extract the real acoustic source information. The spatial resolution is significantly improved, and the spurious sources caused by the high-level side lobes are effectively suppressed. Once DAMAS was proposed, it attracted extensive attention and continuous research from scholars, and a series of deconvolution methods have been proposed one after another, such as non-negative least-squares (NNLS),¹⁵ Richardson-Lucy (RL),^15–17 sparse constrained-deconvolution acoustic source imaging (sparsity constrained-DAMAS, SC-DAMAS),^18–19 covariance matrix fitting (CMF),^18–19 fast iterative shrinkage-thresholding algorithm (FISTA),^20–21 orthogonal matching pursuit deconvolution acoustic source imaging (OMP-DAMAS),²² and so on. Among the above methods, OMP-DAMAS is based on the theory of compressive sensing and utilizes the assumption of sparsity of acoustic source distribution to reconstruct the original signal. It not only improves the spatial resolution and suppresses spurious sources but also has significant advantages in computational efficiency compared with other deconvolution methods mentioned above.

This paper is devoted to further improving the performance of OMP-DAMAS and proposes FFT-OMP-DAMAS based on fast Fourier transformation (FFT). This method assumes that the array point spread functions (PSFs) are spatially shift-invariant and converts the sum of the product of the acoustic source distribution and the PSFs into a convolution form. Then, the Fourier transform is used to further convert the convolution into a product in the wave number domain. Consequently, the size of the matrix involved in the operation is dramatically reduced, greatly improving the solving speed. FFT-OMP-DAMAS not only inherits the advantages of high spatial resolution and strong spurious source suppression capability of OMP-DAMAS but also significantly improves the computational efficiency compared with OMP-DAMAS, which brings great convenience to practical engineering applications.

The paper is organized as follows: The Theory section presents the theory of the proposed FFT-OMP-DAMAS; the Numerical Simulation section and the Experiment section perform the numerical simulations and experiments, respectively; the final section summarizes the full text.

Theory

DAS

Figure 1 shows the model of beamforming acoustic source identification. The symbol “•” represents the microphone. $r_{m}$ represents the position vector of the mth microphone. When beamforming is utilized to postprocess the acoustic pressure signals received by microphones, the output around the location of the acoustic source is enhanced, forming a “main lobe” that is larger than the “side lobe” output formed elsewhere, helping to indicate the acoustic source. The output of DAS beamforming can be expressed as

b (r) = \frac{1}{M} \frac{v^{H} (r) C v (r)}{{| | v (r) | |}_{2}^{2}} .

(1)

where

r

represents the position vector of the focus point,

M

is the total number of microphones,

C \in ℂ^{M \times M}

is the cross-spectral matrix of the acoustic pressure signals received by the microphones,

ℂ

represents the set of complex numbers,

| | \cdot | |_{2}

represents the 2-norm of a vector, and the superscript “^H” represents conjugate transpose operator.

v (r) = {[v_{1} (r), v_{2} (r), \cdot \cdot \cdot, v_{M} (r)]}^{T}

is a column vector composed of the transfer functions from the acoustic source at the position

r

to each microphone, and the superscript “^T” means transpose operator. The expression for the element

v_{m} (r)

v (r)

v_{m} (r) = \frac{e^{- j k | r - r_{m} |}}{| r - r_{m} |},

(2)

where

k = 2 π f / c

is wave number,

f

is frequency,

c

is acoustic velocity, and

j = \sqrt{- 1}

is imaginary unit.

Figure 1.

Beamforming acoustic source identification model.

Assuming that the acoustic sources are independent of each other, the cross-spectrum of the acoustic pressure signals received by the microphones is equal to the linear superposition of the cross-spectra generated by each acoustic source at the microphones, so there is

C = \sum_{r_{s}} s (r_{s}) v (r_{s}) v^{H} (r_{s}),

(3)

where

r_{s}

is the position of the acoustic source and

s (r_{s})

is the source strength.

DAMAS

DAMAS establishes the following system of linear equations

\begin{matrix} b=Aq & subject to \end{matrix} q \geq 0,

(4)

where

b \in ℂ^{G \times 1}

is the beamforming output column vector,

q \in ℂ^{G \times 1}

is the column vector formed by the average acoustic pressure contribution of the acoustic source at each focus point, and the term “average acoustic pressure contribution (AAPC)” refers to the average of the acoustic pressure generated by the acoustic source at each microphone.

A \in ℂ^{G \times G}

is a matrix consisting of the PSFs of the acoustic sources at all focus points, and

G

is the total number of focus points. The elements of

A

can be expressed as

p s f (r | r_{s}) = \frac{v^{H} (r) v (r_{s}) v^{H} (r_{s}) v (r)}{{| | v (r_{s}) | |}_{2}^{2} {| | v (r) | |}_{2}^{2}} .

(5)

DAMAS imposes positive constraints on the components of $q$ and continuously extract the real acoustic source information $q$ from $b$ using the Gauss–Seidel iteration method.^13–14 The operations are as follows.

Initialize $q^{(0)} = 0$ . The specific steps in the $γ th$ iteration are:

1. Calculate the residual $μ^{(γ - 1)} (r)$

μ^{(γ - 1)} (r) = \sum_{r_{s} \in D} q^{(γ)} (r_{s}) p s f (r| r_{s}) + \sum_{r_{s} \in D_{N}} q^{(γ - 1)} (r_{s}) p s f (r| r_{s}) - b (r),

(6)

where

D

is a set of focus position vectors that have completed

γ

iterations,

D_{N}

is a set of focus position vectors that have completed

γ - 1

iterations,

D \cup D_{N} = F

F

is the set composed of all focus point position vectors, and

\cup

represents to union.

2. Calculate $q^{(γ)} (r_{s})$

q^{(γ)} (r_{s}) = \max (q^{(γ - 1)} (r_{s}) - \frac{μ^{(γ - 1)} (r)}{p s f (r| r_{s})}, 0) .

(7)

Then, $q^{(γ)}$ is obtained by calculating the values of all focus points, and a clear acoustic source imaging result can be obtained after an appropriate number of iterations.

OMP-DAMAS

For the system of linear equations $b=Aq$ , when the unknown source vector $q$ contains only a few non-zero elements, the source distribution is sparse and the orthogonal matching pursuit (OMP) method can be used. According to the idea of OMP,²² when the column of $A$ best matches the beamforming output $b$ (i.e., the column in $A$ that makes the maximum value of $A^{T} b$ appear), the source corresponding to that column contributes the most to $b$ , so the position of the source can be identified by searching for the maximum value of $A^{T} b$ . Next, the residual vector $η$ is obtained by subtracting the identified source contributions from $b$ and is used to participate in the iterative operation instead of $b$ . Finally, the unknown source vector can be reconstructed after iterating the above procedure K times (K is the number of acoustic sources to be identified). The operations are as follows.

Initialize $q^{(0)} = 0$ , index set $I^{(0)} = \emptyset$ , and residual $η^{(0)} = b$ . The specific steps in the $γ th$ iteration are:

1. Determine the index of the $γ th$ iteration

i^{(γ)} = \arg \max_{γ} | A^{T} η^{(γ - 1)} | .

(8)

2. Update the index set

I^{(γ)} = I^{(γ - 1)} \cup i^{(γ)} .

(9)

3. Calculate the projection matrix

D^{(γ)} = A_{New}^{(γ)} A_{New}^{(γ) +},

(10)

where

A_{New}^{(γ) +} = {(A_{New}^{(γ) H} A_{New}^{(γ)})}^{- 1} A_{New}^{(γ) H}

A_{New}^{(γ)} = A (:, I^{(γ)})

is the PSF matrix restricted to the index set

I^{(γ)}

, and the superscript “⁺” indicates the Moore–Penrose pseudo-inverse.

4. Update the residual vector

η^{(γ)} = b - D^{(γ)} b .

(11)

Return to step 1) and stop iterating until $γ = K$ . The final source vector is

q_{I} = A_{New}^{(K) +} b .

(12)

FFT-OMP-DAMAS

Define the function $p s f_{s} (Δ r)$ as

p s f_{s} (Δ r) = p s f_{s} (r- r_{c}) = p s f (r | r_{c}),

(13)

where

r_{c}

is the position vector of the central focus point. If the PSFs of all acoustic sources satisfy

p s f (r | r_{s}) = p s f_{s} (r - r_{s}),

(14)

then the PSF is considered to be spatially shift-invariant. That is, the PSF depends on the relative position between the observation point and the acoustic source, rather than their specific positions.¹⁵

The array PSF of the acoustic source located in the center of the calculation plane, which is denoted by $A_{c o n s t} = [p s f (r | r_{c}) | r \in F] \in R^{N_{r} \times N_{c}}$ , is used to be the spatially shift-invariant PSF, that is, replace the PSFs at other positions to calculate their outputs. Here, $N_{r}$ represents the row number of the focus point set, $N_{c}$ represents the column number of the focus point set, and $N_{r} \times N_{c} = G$ , and $R$ is the real number set.

Rotate $A_{c o n s t}$ by 180° to get $A_{R}$ , and construct the matrix $Q$ , $Q= mat (q) \in R^{N_{R} \times N_{C}}$ , $mat (\cdot)$ means to reshape the vector in the parentheses into a matrix. Then, as shown in Figure 2, the element at the center of $A_{R}$ is aligned with the element at position $r$ of $Q$ , and then the elements in $A_{R}$ are multiplied with corresponding elements in $Q$ and added together, resulting in $\hat{b} (r)$

\hat{b} (r) = \sum_{r_{s} \in G} q (r_{s}) p s f (r - r_{s} + r_{c} | r_{c}),

(15)

where

G

represents the set of some or all focus point position vectors. According to equation (13), equation (15) can be further written as

\hat{b} (r) = \sum_{r_{s} \in G} q (r_{s}) p s f_{s} (r - r_{s}) .

(16)

Figure 2.

Schematic diagram of spatial convolution of $A_{R}$ and $Q$ .

Obviously, $\hat{b} (r)$ is the result of the 2D convolution of $Q$ and $A_{c o n s t}$ . Since the PSF is spatially invariant, $\hat{b} (r)$ is the output at position $r$ of all acoustic sources in the blue area in Figure 2.

Further converting the convolution of the spatial domain to the product of the wave number domain, equation (16) can be written as

\hat{b} = vec (Q * A_{c o n s t}) = vec (F^{-1} (F (Q) \circ F (A_{c o n s t}))),

(17)

where

\hat{b} = [\hat{b} (r) | r \in F] \in R^{G}

, “

vec (\cdot)

” represents the vectorization of the matrix in the parentheses. “

F (\cdot)

” and “

F^{- 1} (\cdot)

,” respectively, represent forward Fourier transform and inverse Fourier transform on the matrix in parentheses, the symbol “

*

” denotes convolution operator, and the symbol “

°

” denotes Hadamard product operator.

Similarly, for arbitrary variable $x (r)$ about $r$ and $y (r_{s})$ about $r_{s}$ , if

y (r_{s}) = \sum_{r \in G} x (r) p s f_{s} (r - r_{s}),

(18)

then there is

Y = vec (X * A_{R}) = vec (F^{- 1} (F (X) ° F (A_{R}))),

(19)

where

Y = [y (r_{s}) | r_{s} \in F] \in R^{G}

X = [x (r) | r \in F] \in R^{N_{r} \times N_{c}}

Let

J = A^{T} η .

(20)

According to shift-invariance of PSF, the element in $J$ can be approximately replaced by the following formula

J (r_{s}) \approx \sum_{r \in G} η (r) p s f_{s} (r - r_{s}) .

(21)

Combine equations (18), (19), and (21), equation (20) can be written in the following form

J = vec (F^{- 1} (F (N) ° F (A_{R}))),

(22)

where

N = mat (η) \in C^{N_{r} \times N_{c}}

Consequently, equation (8) can be converted into

i^{(γ)} = \arg \max_{γ} | vec (F^{- 1} (F (N^{(γ - 1)}) ° F (A_{R}))) | .

(23)

In summary, converting the product operation between the large-dimensional matrix ( $G \times G$ dimension) and vector ( $G \times 1$ dimension) in equation (8) into the small-dimensional matrix ( $N_{r} \times N_{c}$ dimension) operation in equation (23) is the key to improve the computational efficiency.

Numerical simulation

In order to verify the advantages of the FFT-OMP-DAMAS method proposed in this paper, simulations are first performed. A 36-channel sector wheel array with a diameter of 0.65 m is used. The measurement distance is 1 m. The imaging plane of acoustic sources is 1.6 m $\times$ 1.6 m, with 65 $\times$ 65 irregular focusing grid points distributed.^24–25 The number of iterations of DAMAS is set to 200.

Figure 3 shows the results of acoustic source imaging based on the existing DAS, DAMAS, OMP-DAMAS methods, and the proposed FFT-OMP-DAMAS method, respectively. The coordinates of four acoustic sources are $(- 0.2, 0) m$ , $(0.2, 0) m$ , $(- 0.2, 0.6) m$ , and $(0.2, 0.6) m$ , respectively. The imaging frequency bands are the 1/3-octave bands with center frequencies of 2000 Hz, 3150 Hz, and 5000 Hz, respectively, and these 1/3-octave bands are synthesized from narrow bands of 4 Hz. The dynamic range is set to 20 dB. As can be seen from Figure 3(a)–(c), at low frequencies, the main lobes of DAS are very wide, which is not conducive to locating the exact position of the acoustic source; at high frequencies, severe side lobe contaminations lead to unclear imaging results. Figure 3(d)–(f) shows the imaging results of DAMAS, which can accurately identify all four acoustic sources, overcoming the wide low-frequency main lobes and severe high-frequency side lobes of DAS. The imaging results of OMP-DAMAS are shown in Figure 3(g)–(i). In each frequency band, the main lobes of OMP-DAMAS are further narrowed, and the spatial resolution is further improved. Figure 3(j)–(l) show the imaging results of the proposed FFT-OMP-DAMAS, which is close to OMP-DAMAS in terms of spatial resolution and spurious source suppression. The real AAPCs of acoustic sources for each frequency band are 114.24 dB, 116.22 dB, and 118.22 dB, respectively. The AAPC errors of DAMAS are 6.98 dB, 3.91 dB, and 0.82 dB, respectively, while those of OMP-DAMAS are 1.32 dB, 0.00 dB, and 0.00 dB, respectively, and those of FFT-OMP-DAMAS are 0.71 dB, 0.00 dB, and 0.00 dB, respectively. Obviously, both OMP-DAMAS and FFT-OMP-DAMAS significantly improve the quantification accuracy, while FFT-OMP-DAMAS seems to perform better at low frequencies.

Figure 3.

Acoustic source imaging results of simulations. The 1/3 octave band center frequencies from left to right are 2000 Hz, 3150 Hz, and 5000 Hz, respectively. (a)–(c) DAS, (d)–(f) DAMAS, (g)–(i) OMP-DAMAS, and (j)–(l) FFT-OMP-DAMAS are used.

Table 1 compares the computational time consumed by DAMAS, OMP-DAMAS, and FFT-OMP-DAMAS. DAMAS consumes the longest time. Compared to DAMAS, OMP-DAMAS consumes much less time. The proposed FFT-OMP-DAMAS further reduces the consumed time by less than 50% of OMP-DAMAS.

Table 1.

Consuming time of the deconvolution methods in the simulations.

1/3 octave band center frequency	2000 Hz, s	3150 Hz, s	5000 Hz, s
DAMAS	6097.60	9941.81	15,336.67
OMP-DAMAS	78.82	123.06	202.15
FFT-OMP-DAMAS	35.57	57.13	89.13

DAMAS: deconvolution approach for the mapping of acoustic sources; OMP-DAMAS: DAMAS by using orthogonal matching pursuit; FFT-OMP-DAMAS: OMP-DAMAS by using fast Fourier transformation.

In summary, the proposed FFT-OMP-DAMAS not only retains the advantages of high spatial resolution, strong spurious source suppression capability, and high quantification accuracy of OMP-DAMAS but also enjoys significantly higher computational efficiency than OMP-DAMAS.

Experiment

Verification experiments based on loudspeakers

In order to verify the effectiveness of the proposed FFT-OMP-DAMAS and the conclusions of the above numerical simulations, a loudspeaker-based validation experiment was carried out in a semi-anechoic chamber. The experimental layout is shown in Figure 4. The four loudspeakers are located approximately at $(- 0.2, 0) m$ , $(0.2, 0) m$ , $(- 0.2, 0.6) m$ , and $(0.2, 0.6) m$ , and they are excited simultaneously by steady white noise signals. A 0.65 m diameter 36-channel sector wheel array with Type 4958 microphones from Brüel and Kjær is used to capture the acoustic pressure. The microphone array is about 1 m away from the acoustic source plane. The imaging plane of the acoustic sources and the number of iterations of DAMAS are the same as in the previous simulations.

Figure 4.

Experimental layout.

The acoustic source identification results using DAS, DAMAS, OMP-DAMAS, and FFT-OMP-DAMAS at 1/3 octave bands with the center frequencies of 2000 Hz, 3150 Hz, and 5000 Hz are shown in Figure 5, respectively.

Figure 5.

Acoustic source imaging results of loudspeaker-based experiments. The 1/3 octave band center frequencies from left to right are 2000 Hz, 3150 Hz, and 5000 Hz, respectively. (a)–(c) DAS, (d)–(f) DAMAS, (g)–(i) OMP-DAMAS, and (j)–(l) FFT-OMP-DAMAS are used.

In Figure 5(a), the main lobes of DAS are so wide that the spatial resolution is low. In Figure 5(b) and (c), although the width of main lobes of DAS is decreased with increasing frequency, the side lobes increase significantly in the rest of the imaging region. Figure 5(d)–(f) show the imaging results of DAMAS. It is clear that although DAMAS accurately identifies all sound sources, there are many spurious acoustic sources at low frequencies. In Figure 5(g)–(l), the main lobe peaks of OMP-DAMAS and the proposed FFT-OMP-DAMAS in each frequency band appear at the real positions of loudspeakers. Moreover, the imaging results of both methods are free of spurious sources. The imaging results of OMP-DAMAS and FFT-OMP-DAMAS are very close because they have almost the same shape of the main lobe and only the peak of the main lobe is slightly different. This indicates that the proposed FFT-OMP-DAMAS is not inferior to OMP-DAMAS in terms of sound source identification performance. In addition, Table 2 illustrates that time consumed by FFT-OMP-DAMAS is less than 50% of that by OMP-DAMAS, which is consistent with the previous simulation results.

Table 2.

Consuming time of the deconvolution methods in the loudspeaker-based experiments.

1/3 octave band center frequency	2000 Hz, s	3150 Hz, s	5000 Hz, s
DAMAS	6389.41	9251.61	14,939.51
OMP-DAMAS	77.34	122.06	192.65
FFT-OMP-DAMAS	34.93	56.37	90.90

DAMAS: deconvolution approach for the mapping of acoustic sources; OMP-DAMAS: DAMAS by using orthogonal matching pursuit; FFT-OMP-DAMAS: OMP-DAMAS by using fast Fourier transformation.

In each frequency band, the maximum main lobe peaks of DAS are 74.70 dB, 74.17 dB, and 74.93 dB, respectively. When the sound sources are separated, the main lobe peak of the DAS can be used to measure the quantification accuracy of the deconvolution method.²³ As can be seen from Figure 5, the AAPC errors of DAMAS are 8.10 dB, 6.58 dB, and 2.67 dB; the AAPC errors of OMP-DAMAS are 4.73 dB, 2.02 dB, and 1.76 dB; and the AAPC errors of FFT-OMP-DAMAS are 4.65 dB, 1.46 dB, and 0.62 dB. Obviously, the proposed FFT-OMP-DAMAS has the smallest AAPC errors and more accurate quantification.

In summary, the verification experiments show that the proposed FFT-OMP-DAMAS not only has high acoustic source identification performance due to the narrow main lobes and strong side lobes suppression but also significantly improves the computational efficiency.

Application experiment for a car dash panel

The proposed method is further used to identify weak acoustic insulation areas of a car dash panel. The experiment is conducted in a reverberation chamber-anechoic chamber kit, which is specifically designed to measure the acoustic insulation performance of a panel. Figure 6 shows the experimental layout with the dash panel mounted on the window between the reverberation and anechoic chambers.²⁶ As shown in Figure 6(a), an omnidirectional sound source is used to generate random sound in the reverberation chamber. As shown in Figure 6(b), a 36-channel sector wheel microphone array with a diameter of 0.65 m is placed in an anechoic chamber to capture the sound passing through the panel. The distance between the array and the window surface is approximately 1.2 m. The imaging plane of acoustic sources is 1.6 m $\times$ 0.6 m, with 65 $\times$ 25 irregular focusing grid points distributed.

Figure 6.

Application experiment layout.

Figure 7 shows the acoustic imaging results of DAS, DAMAS, OMP-DAMAS, and FFT-OMP-DAMAS at 2000–3000 Hz, with a display range of 10 dB. The results of all four methods show the presence of acoustic hot spots to the left of the air conditioning intake in the dash panel, indicating that this area is a weak point for sound insulation. We deduce that the reason for the poor sound insulation in this area is the poor sealing of the internal and external circulation switching valves and intake valve ports of the air conditioning system. The imaging result of FFT-OMP-DAMAS is similar to that of OMP-DAMAS, and both are cleaner than the imaging results of DAS and DAMAS. This case demonstrates the effectiveness and superiority of the proposed FFT-OMP-DAMAS in identifying acoustic sources. In addition, in terms of computational consumption time to obtain the above imaging results, DAMAS takes 219.31 s, OMP-DAMAS takes 5.15 s, and FFT-OMP-DAMAS takes only 2.45 s. Among these three deconvolution methods, FFT-OMP-DAMAS enjoys the highest computational efficiency.

Figure 7.

Results of identifying weak sound insulation areas for a car dash panel. (a) DAS, (b) DAMAS, (c) OMP-DAMAS, and (d) FFT-OMP-DAMAS are used.

Conclusion

Compared with the traditional DAS beamforming method, the deconvolution methods can greatly improve the spatial resolution and reduce the side lobe contamination, thus making the acoustic source identification more accurate. In this paper, we propose the FFT-OMP-DAMAS deconvolution method, which is obtained by improving the existing OMP-DAMAS deconvolution method using spatial shift-invariant point spread function and fast Fourier transform. Compared with OMP-DAMAS, the proposed FFT-OMP-DAMAS has comparable acoustic source imaging performance and enjoys significantly higher computational efficiency. The conclusion is verified by numerical simulations and experiments.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The study was supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202103206) and the Graduate Research and Innovation Foundation of Chongqing, China (Grant No. CYB22012).

ORCID iDs

Xiaoguang Yang

Zhigang Chu

Appendix 1

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