Compressive sensing method with enhanced sparsity for aeroengine duct mode detection

Bu, Huanxian; Huang, Xun; Zhang, Xin

doi:10.1121/1.5116893

This Letter proposes an improved compressive-sensing-based methodology for aeroengine fan noise detection to guarantee the performance under low signal-to-noise ratio conditions. The principle behind the proposed method is to modify the mode power spectrum in a way with increased sparsity, which is demonstrated in both numerical studies and experimental validations. Overall, the proposed method could be applicable to aeroengine testing applications.

1. Introduction

Compressive sensing,^1,2 a technique that has been evolved from the field of information technology, has demonstrated great potential in acoustics. There exists a number of recent studies^3–9 to extend and apply this generic signal processing method to acoustic applications in general and aeroengine fan noise mode detection in particular, with the aim of significantly reduced sampling efforts of conventional methods and improving mode detection capability.

The premise of such a compressive sensing method is sparsity of the mode spectrum to be detected. In practical tests, the mode sparsity would be easily destroyed either due to strong noise interference or attenuation of spinning modes. The former is reasonable since realistic testing environments contain various noise interference, such as flow-induced noise, testing facility noise, and reflection and scattering waves from the test system. The latter is exactly the case in far-field mode measurements or liner assessment tests. To avoid performance deterioration or failure of the compressive sensing approach in those specific cases, we present an improved algorithm to improve the required sparsity. The performance of the new method is validated by both simulations and measured aeroacoustic test data.

2. Statement of the problem

2.1 Signal-to-noise ratio (SNR) and sparsity (s)

First, the conventional mode detection method¹⁰ is introduced briefly for completeness. Given a classical sensor array with a total of N sensors deployed equidistantly along the azimuthal direction, the sound pressure signals at a specified frequency constitute the matrix $P = {[P_{1} (ω), P_{2} (ω), \dots, P_{N} (ω)]}^{T}$ ⁠, where ${(\cdot)}^{T}$ denotes transpose and $P_{j} (ω) (j = 1, 2, \dots, N)$ stands for the complex sound pressure in the angular frequency domain from the jth sensor. Then, the complex mode amplitude a_m can be calculated by the spatial Fourier decomposition¹⁰

a_{m} = \frac{1}{N} \sum_{j = 1}^{N} P_{j} (ω) e^{im φ_{j}},

(1)

where $φ_{j}$ is the azimuthal location of the jth sensor.

To further restrain potential background interference, four improved methods based on the calculation of the mode power A_m can be implemented.¹⁰ For simplicity, the conventional root-mean-square (rms)-averaging method is adopted in this work,

A_{m} = \frac{1}{2} ⟨ {| a_{m} |}^{2} ⟩ = \frac{1}{2} ⟨ a_{m} a_{m}^{*} ⟩ = \frac{1}{2 N^{2}} ⟨ \sum_{j = 1}^{N} \sum_{k = 1}^{N} e^{im φ_{j}} P {(ω)}_{j} P {(ω)}_{k}^{*} e^{- im φ_{k}} ⟩,

(2)

where (⋅)* stands for the complex conjugation, $⟨ \cdot ⟩$ represents the ensemble average, and $P {(ω)}_{j} P {(ω)}_{k}^{*}$ is the cross-power matrix. Physically, A_m can be interpreted as the sound power of mode m.

Represented mode spectra of $a_{m}$ and $A_{m}$ with a dominant mode m = 29 are presented in Fig. 1, where the y axis variables of the spectra are normalized by the maximum value. Those results are obtained from synthesized time-domain data based on the analytical solution of the spinning wave propagation model. Background noise is also numerically included; hence both spectra contain noise modes in Fig. 1. In our previous works,^5,6 the narrowband SNR is defined to measure the overall interference of the noise modes as

SNR = 10 {log}_{10} (\frac{\sum {| a_{m} |}^{2}}{\sum {| a_{Noise}^{2} |}^{2}}),

(3)

where a_m and a_Noise are the amplitudes of the dominant mode and the noise modes. Figure 1(a) shows the mode spectrum $a_{m}$ of $SNR = + \infty, - 10$ and −20 dB. Overall, the noise mode amplitudes are enhanced with the decreasing of SNR, but significantly suppressed in the corresponding mode power spectrum $A_{m}$ in Fig. 1(b).

Fig. 1.

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(Color online) Comparison of (a) the complex mode spectrum $a_{m}$ and (b) the mode power spectrum $A_{m}$ at different SNR, and (c) comparison of the mode sparsity s.

To apply the compressive-sensing-based mode detection method, it is necessary to estimate the sparsity s of the spectra of $a_{m}$ and $A_{m}$ ⁠. Strictly, the definition of sparsity is based on $ℓ_{0}$ norm, which is, however, not feasible for data with noise. More relaxed definitions of sparsity are proposed.^11,12 In the current work, we adopt the so-called effective sparsity,¹² which describes an alternative function counting the effective number of coordinates

s (a_{m}) : = \frac{{‖ a_{m} ‖}_{1}^{2}}{{‖ a_{m} ‖}_{2}^{2}} and s (A_{m}) : = \frac{{‖ A_{m} ‖}_{1}^{2}}{{‖ A_{m} ‖}_{2}^{2}},

(4)

where $| | \cdot | |_{1}$ and $| | \cdot | |_{2}$ are $ℓ_{1}$ norm and $ℓ_{2}$ norm. The sparsity s of $a_{m}$ and $A_{m}$ is compared in Fig. 1(c). Both $s (a_{m})$ and $s (A_{m})$ grows with the decreasing of SNR, indicating that the mode sparsity deteriorates with the enhancement of noise interference. This explains the reason why SNR influences the performance of the compressive sensing method in Refs. 5 and 6. At a fixed SNR, $s (A_{m})$ is much smaller than $s (a_{m})$ ⁠, which is due to two reasons. First, the noise is efficiently suppressed.¹⁰ Second, the dominant mode of $A_{m}$ becomes comparatively outstanding than that of $a_{m}$ since $a_{m} / a_{Noise} < {(a_{m} / a_{Noise})}^{2}$ ⁠. Therefore, the mode power spectrum $A_{m}$ is sparser and more appropriate for the compressive sensing reconstruction than the complex mode spectrum $a_{m}$ ⁠.

Previous works have proposed the compressive sensing method that can detect $a_{m}$ with a maintained sparsity. However, the mode sparsity $s (a_{m})$ is easily destroyed in practical tests, either due to enhanced noise interference or weakened spinning modes. Figure 2 from Ref. 10 shows a comparison of the in-duct and far-field mode detection results (m = 10) at Mach number of 0.6 and frequency of 5600 Hz. Due to a long propagation distance, the detected spinning mode in the far-field becomes feeble and less outstanding over the whole spectrum, resulting in a loss of sparsity. Similar situations would appear in liner-evaluation tests where the dominant mode could be significantly attenuated or unwanted scattered modes are generated by liner splices.¹³ Therefore, to guarantee a satisfied performance in those challenging occasions, this work develops an improved compressive sensing method based on the reconstruction of $A_{m}$ ⁠.

Fig. 2.

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(Color online) A comparison of in- and out-duct mode detection from Ref. 10.

2.2 Compressive sensing method with enhanced sparsity

Our previous compressive sensing method (CSM-I) detecting $a_{m}$ (Refs. 4–6) is briefly introduced. The procedures of implementation are divided into four steps.

First, we determine the number of sensors and select their positions randomly, and prepare the 1–0 measurement matrix B according to selected sensor locations.^4–6 To guarantee a satisfactory sampling, B should satisfy the restricted isometry property (RIP). It is difficult to prove the RIP for a given measurement matrix construction method. However, a random measurement matrix B will usually satisfy the RIP with a high statistical probability.¹

The second step is to build the compressive sensing measurement y in the frequency domain. The matrices y, B, and P satisfy

y = BP .

(5)

As explained by Ref. 5, P itself is usually not sparse, whereas the corresponding azimuthal modes are possibly sparse, especially at each tonal frequency. Utilizing an orthogonal Fourier basis $Ψ_{F}$ (obtained by discrete Fourier transform of an N-dimensional identity matrix), P can be projected into the sparse modal space by $P = \sum_{i = 1}^{N} φ_{i} x = Ψ_{F} x$ ⁠, here $x = {[x_{1}, x_{2}, \dots, x_{N}]}^{T}$ is an intermediate variable in the compressive sensing calculation. x and $a_{m}$ are the projection of P in different Fourier bases, hence different in values. Then we have

y = B Ψ_{F} x = G_{I} x,

(6)

where $G_{I} = B Ψ_{F} \in ℂ^{M \times N}$ ⁠.

Next in the third step, we recover x by $ℓ_{1}$ -optimization,

minimize | | \hat{x} | |_{1}, subject to | | G_{I} \hat{x} - y | |_{2} = 0 .

(7)

Finally, the frequency signal of the whole array $P (= Ψ_{F} x)$ and the cross-power matrix of $P P^{*}$ are obtained, based on which, the mode amplitude $a_{m}$ and mode power $A_{m}$ can be calculated according to Eqs. (1) and (2).

The new compressive sensing mode detection method (CSM-II) is to directly recover the covariance $x x^{*}$ to obtain the $P P^{*}$ thanks to the improved sparsity. A similar strategy has been used in Ref. 14 for a totally different application, where the compressive sensing was applied to acoustic beamforming. From Eq. (5), we have

y y^{*} = B Ψ_{F} x x^{*} Ψ_{F}^{*} B^{*} = G_{I} x x^{*} G_{I}^{*} .

(8)

$y y^{*}$ is expressed as

y y^{*} = [\begin{matrix} y_{1} y_{1}^{*} & y_{1} y_{2}^{*} & \dots & y_{1} y_{M}^{*} \\ y_{2} y_{1}^{*} & y_{2} y_{2}^{*} & \dots & y_{2} y_{M}^{*} \\ ⋮ \\ y_{M} y_{1}^{*} & y_{M} y_{2}^{*} & \dots & y_{M} y_{M}^{*} \end{matrix}] .

(9)

With a reshape of $y y^{*}$ where each row is stacked behind the previous ones, we can obtain

Y = {[y_{1} y_{1}^{*}, y_{1} y_{2}^{*}, \dots, y_{1} y_{M}^{*}, \dots, y_{M} y_{1}^{*}, \dots, y_{M} y_{M}^{*}]}^{T} .

(10)

It should be noticed that $Y \in ℂ^{M^{2} \times 1}$ is a very long vector.

Assume the modal components of x are uncorrelated, $E {x_{i} x_{j}} = 0, \forall i \neq j$ ⁠. Then we have

x x^{*} G_{I}^{*} = [\begin{matrix} x_{1} x_{1}^{*} & 0 & \dots & 0 \\ 0 & x_{2} x_{2}^{*} & \dots & 0 \\ 0 & 0 & ⋮ & 0 \\ 0 & 0 & \dots & x_{N} x_{N}^{*} \end{matrix}] [\begin{matrix} G_{11}^{*} & \dots & G_{M 1}^{*} \\ ⋮ & ⋮ \\ G_{1 N}^{*} & \dots & G_{MN}^{*} \end{matrix}] = [\begin{matrix} x_{1} x_{1}^{*} G_{11}^{*} & \dots & x_{1} x_{1}^{*} G_{M 1}^{*} \\ ⋮ & ⋮ \\ x_{N} x_{N}^{*} G_{1 N}^{*} & \dots & x_{N} x_{N}^{*} G_{MN}^{*} \end{matrix}],

(11)

\begin{matrix} G_{I} x x^{*} G_{I}^{*} = [\begin{matrix} G_{11} & \dots & G_{1 N} \\ ⋮ & ⋮ \\ G_{M 1} & \dots & G_{MN} \end{matrix}] [\begin{matrix} x_{1} x_{1}^{*} G_{11}^{*} & \dots & x_{1} x_{1}^{*} G_{M 1}^{*} \\ ⋮ & ⋮ \\ x_{N} x_{N}^{*} G_{1 N}^{*} & \dots & x_{N} x_{N}^{*} G_{MN}^{*} \end{matrix}] \\ = [\begin{matrix} x_{1} x_{1}^{*} G_{11} G_{11}^{*} + \dots + x_{N} x_{N}^{*} G_{1 N} G_{1 N}^{*}, & \dots, & x_{1} x_{1}^{*} G_{11} G_{M 1}^{*} + \dots + x_{N} x_{N}^{*} G_{1 N} G_{MN}^{*} \\ ⋮ & ⋮ \\ x_{1} x_{1}^{*} G_{M 1} G_{11}^{*} + \dots + x_{N} x_{N}^{*} G_{MN} G_{1 N}^{*}, & \dots, & x_{1} x_{1}^{*} G_{M 1} G_{M 1}^{*} + \dots + x_{N} x_{N}^{*} G_{MN} G_{MN}^{*} \end{matrix}] . \end{matrix}

(12)

Once again, reshaping each row of the above matrix, we have,

Y = [\begin{matrix} G_{11} G_{11}^{*} & G_{12} G_{12}^{*} & \dots & G_{1 N} G_{1 N}^{*} \\ G_{11} G_{21}^{*} & G_{12} G_{22}^{*} & \dots & G_{1 N} G_{2 N}^{*} \\ ⋮ \\ G_{M 1} G_{M 1}^{*} & G_{M 2} G_{M 2}^{*} & \dots & G_{MN} G_{MN}^{*} \end{matrix}] [\begin{matrix} x_{1} x_{1}^{*} \\ x_{2} x_{2}^{*} \\ ⋮ \\ x_{N} x_{N}^{*} \end{matrix}] = G_{II} X,

(13)

where $G_{II} \in ℂ^{M^{2} \times N}, X = {[x_{1} x_{1}^{*}, x_{2} x_{2}^{*}, \dots, x_{N} x_{N}^{*}]}^{T} \in ℝ^{N \times 1}$ ⁠. Similarly, we solve

minimize {‖ \hat{X} ‖}_{1}, subject to {‖ G_{II} \hat{X} - Y ‖}_{2} = 0.

(14)

After $\hat{X}$ is obtained, then we can calculate $P P^{*} (= Ψ_{F} \hat{X} Ψ_{F}^{*})$ and the mode power spectrum $A_{m}$ by Eq. (2). Note that the dimension of $G_{II}$ and Y in Eq. (13) is much larger than $G_{I}$ and y in Eq. (6), therefore the calculation speed of the CSM-II method is lower than that of the CSM-I method.

Finally, a simulation case with three dominant modes of m = −8, 4, and 28 at SNR = -20 dB obtained by different methods is presented in Fig. 3. The compressive sensing methods adopt only half sensors of the full array (64 sensors). Both the conventional rms-averaging method and the CSM-I method detect the mode power spectra with rich noise modes. In comparison, the CSM-II method reconstructs the mode distribution with suppressed noise interference.

Fig. 3.

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(Color online) Detection of the mode power $A_{m}$ by (a) the conventional rms-averaging method, (b) the CSM-I method, and (c) the CSM-II method.

3. Experimental validation

To validate the newly proposed method, we perform the mode detection experiments on a duct acoustic test rig in an anechoic wind tunnel. The test setup has been previously used to demonstrate the CSM-I method for in-duct azimuthal mode detection,⁶ which is brought here for out-duct mode measurements. Figure 4 presents the design of the whole experimental system. The testing rig consists of an aeroacoustic wind tunnel, a duct system, a spinning mode synthesizer system, and a data-acquisition system. The design parameters and the performance of each sub-system are included in Fig. 4. More details can be found in Ref. 6. The major difference between the previous test rig and current version is the addition of an out-duct array with 32 sensors.

Fig. 4.

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(Color online) Sketch of the experimental setup.

The out-duct detection of mode power spectra $A_{m}$ with dominant mode m = 6 at the flow speed of 40 m/s by different methods are compared in Fig. 5. The compressive sensing methods adopt half sensors of the full array. Moreover, the effect of the jet shear layer is not considered. Consistent with the simulation results in Fig. 3, only the CSM-II method obtains a very “clean” mode spectrum with little noise interference, thus demonstrating the good performance.

Fig. 5.

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(Color online) Out-duct detection of mode power spectra $A_{m}$ with dominant mode m = 6 at the flow speed of 40 m/s by (a) the conventional method, (b) the CSM-I method, and (c) the CSM-II method.

4. Summary

This paper presents an improved compressive-sensing-based method for aeroengine fan noise detection. We first show that the former methodology is primarily aimed at recovering the mode spectrum, which is difficult to retain sparse under low SNR conditions. In contrast, the current work is based on the reconstruction of the mode power spectrum whose sparsity is significantly enhanced due to two reasons. First, the influence of background noise on the power spectrum is diminished. Next, the predominant mode becomes more distinctive as the ratio of the dominant mode amplitude and the noise mode amplitude is squared.

Both simulation studies and experimental validations of out-duct mode measurements show a very clean mode spectrum with little noise interference, confirming the improved performance. This technique is considered as a supplement and extension of the previous method (CSM-I), both of which demonstrate the potential of the compressive sensing for aeroengine testing applications.

Acknowledgments

This work is supported by the Research Grants Council of the Hong Kong Special Administrative Region (Grant No. 16205317), Ministry of Industry and Information Technology of China (Grant No. MJ-2015-F-012-03), and National Science Foundation of China (Grant No. 11772005).

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Acoustical Society of America

Compressive sensing method with enhanced sparsity for aeroengine duct mode detection

1. Introduction

2. Statement of the problem

2.1 Signal-to-noise ratio (SNR) and sparsity (s)

2.2 Compressive sensing method with enhanced sparsity

3. Experimental validation

4. Summary

Acknowledgments

References and links

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Compressive sensing method with enhanced sparsity for aeroengine duct mode detection

1. Introduction

2. Statement of the problem

2.1 Signal-to-noise ratio (SNR) and sparsity (s)

2.2 Compressive sensing method with enhanced sparsity

3. Experimental validation

4. Summary

Acknowledgments

References and links

Citing articles via

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