AIAA JOURNAL Vol. 51, No. 4, April 2013
Technical Notes Compressive Sensing and Reconstruction in Measurements with an Aerospace Application
I.
C
Xun Huang∗ Peking University, Beijing 100871, People’s Republic of China
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DOI: 10.2514/1.J052227
Nomenclature a, b arg min B C0 CM E f g g^ min J k K M Mj m N n p r u, v, w V x θ θK ρ ω
= = = = = = = = = = = = = = = = = = = = = = = = = =
integers the argument of the minimum blade number speed of sound, m∕s a constant approximation error a test function approximation of f ^ 1 arg min kgk Bessel function of the first kind wave number compressive sample number number of nonzero entries Mach number circumferential mode dimension of a signal integer pressure, Pa radial axis velocity, m∕s vane number axial axis circumferential angle, rad circumferential position of Kth microphone, rad density, kg∕m3 angular frequency, rad∕s
= =
stationary value variables related to the mth mode
Subscripts 0 m
Superscripts ^ 0
= =
Introduction
OMPRESSIVE sensing is a newly emerging signal-processing method [1,2] in information technologies that could extensively reduce sampling efforts, which significantly impact scientific and engineering areas. In this note, we introduce the fundamentals of compressive sensing and demonstrate its usage by studying a linear-duct acoustic problem that is a classical topic in aerospace [3–6]. To deepen our understanding in subtle fluid mechanisms, an everincreasing number of sensors have been deployed in flow measurements to satisfy the Nyquist–Shannon sampling theorem, which states that a spatial wave with no wavenumber higher than BW (denoting bandwidth) can be perfectly reconstructed from the measurements of those sensors spaced 1∕2BW apart. As a result, in a spatial region, many sensors are required to achieve a fine-scale knowledge over a large flowfield. For example, hundreds of microphones have been used to measure flow-induced noise in aviation industry [7]. The Nyquist–Shannon sampling theorem sets the most pessimistic criterion. If some prior knowledge, such as an inherent model of the fluid process of interest, can be incorporated, it is possible to employ fewer sensors [8]. On the other hand, Candes et al. [1] proposed that it is possible to perfectly reconstruct a discrete-time signal f ∈ CN with M nonzero sparse items in a Hilbert basis from the randomly and partially accomplished samples. As a sparse signal, M should be much less than N. The so-called compressive sensing is also known as compressive sampling and compressed sensing [2,9,10]. These terms will be used interchangeably throughout this note. According to our best knowledge, compressive sensing has not been adopted thus far in the area of fluid mechanisms. Hence, this note attempts to introduce the fundamentals of this new strategy to colleagues in aerospace. The classical problem of spinning-mode flow sound radiation from an unflanged straight duct [3,4,11] is used as an example to illustrate the potential of compressive sensing in aerospace applications. This problem is an approximation of a current turbofan engine duct. The development of high-bypass-ratio turbofan engines has led to relatively more prominent tonal noise by interaction of fan and stator assembly [5,12]. This tonal frequency sound propagates through a slowly varying engine duct [4], radiates to the far field, and finally irritates communities local to airports. To resolve this stringent environmental issue, extensive efforts have been invested on silentengine design. Experimental investigation of the propagation of tonal spinning-mode noise is valuable to understand inherent fluid mechanics, to evaluate numerical models, and to verify and validate suitable low-noise design methods [8,13]. In this note, we illustrate that the advanced signal-processing technology holds the potential to extensively simplify experimental apparatus. This note is organized as follows. A short introduction to compressive sensing is given in Sec. II. The duct acoustic problem is summarized in Sec. III. After that, compressive sensing is conducted for this particular case in Sec. IV, where numerical simulations [14] are performed, and the performance of compressive sensing is discussed. A summary and concluding remarks can be found in Sec. V. Finally, a very succinct MATLAB program is provided in the Appendix to illustrate the operation and performance of compressive sensing. We hope our colleagues can use and extend the code in their more complicated aerospace problems.
Fourier transform disturbance
Received 4 August 2012; revision received 10 November 2012; accepted for publication 12 November 2012; published online 28 February 2013. Copyright © 2012 by Xun Huang. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-385X/13 and $10.00 in correspondence with the CCC. *Professor, State Key Laboratory of Turbulence and Complex Systems, College of Engineering, Department of Aeronautics and Astronautics;
[email protected]. Senior Member AIAA.
II.
Compressive Sensing
We first consider a spatial signal fθ cos18θ 2 cos25θ 0.5 cos35θ as a tutorial example, where θ ∈ 0; 2π. It is easy to see that f is sparse in wavenumber space. The maximal wavenumber 1011
1012
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TECHNICAL NOTES
5
5
Original Reconstructed
4
3
Pressure amplitude
Pressure amplitude
3 2 1 0
1 0 −1
−2
−2 −3 0
1
2
3
θ
4
5
6
7
Fig. 1 The spatial profiles of the test signal fθ−, and the reconstructed signal (∘) from the equally spaced samples (∘).
is 35. According to the Nyquist–Shannon sampling theorem, we have to place at least 70 sensors equidistantly throughout θ. Otherwise, we would have aliasing, which prevents complete reconstruction of the original signal. For example, Fig. 1 shows f and its incorrect reconstruction g from 24 equally spaced samples. ^ Figure 2 compares Fourier coefficients, where fn ∫ π−π fθ exp−inθ dθ. The distinctive peaks in f^ represent the three dominant and sparse modes. Small oscillations in other Fourier coefficients come from digital errors. Theoretically, they are zero. Figure 2 also shows the Fourier coefficients of g recovered from 24 equally spaced samples. It is easy to see that the outcome is overwhelmed by aliasing and is completely incorrect. In contrast, compressed sensing could yield a sub-Nyquist sampling with correct reconstruction. Suppose a discrete spatial signal fθ ∈ CN with M sparse Fourier coefficients. We can randomly sample fθ at K different spatial locations. If K ≥ CM · M · log
N M
(1)
where CM is a constant, compressive sensing theory ensures a perfect recovery of the original signal “with overwhelming probability” [15]. The reconstruction g is achieved by solving the following linear programming [1]: ^ 1; arg min kgk
subject to gθi fθi ;
i 1; : : : ; K (2)
0
1
2
3
θ
4
5
6
7
Fig. 3 The spatial profiles of the test signal fθ−, the random samples (×), and the reconstructed signal (∘) from the randomly compressive samples.
where g^ ∈ CN are Fourier coefficients of g, and arg min stands for the argument of the minimum (i.e., arg min kgk ^ 1 : g^ min , that is, for all ^ 1 ). The previous linear programming ^ kg^ min k1 ≤ kgk possible g, based on L1 minimization is a convex problem that can be solved ^ efficiently using available convex optimization tools [16]. Once g, ^ is achieved, the original signal f can be the approximation of f, reconstructed by an inverse Fourier transform. Figure 3 shows 24 samples at randomly chosen angles. Now, we would be able to understand that the term of compressive sampling was coined to denote the sensing way, taking account of the fact that the number of samples is much smaller than the desired Nyquist rate. Figure 3 shows that the reconstruction signal g from 24 compressive samples agrees well with the original signal f. Moreover, Fig. 4 compares those signals in the frequency domain. It can be seen that the recovered amplitudes and positions of sparse Fourier coefficients ^ suggesting the working of the of g^ perfectly match those of f, compressive sensing based on the L1 optimization. The digital error is increased for subsampling operations but remains insignificant. In this numerical demonstration, we found that a smaller CM might work for this test case as well, whereas a bigger CM always leads to an exact reconstruction with a higher probability. An extensively simplified code of the previous case is placed in the Appendix for readers who have the interest to apply the method in their own aerospace investigations. The calculation takes approximately 3 s using a medium-specification laptop computer (1.7 GHz Intel Core i5 with 4 GB DDR3 memory). We have to admit that only the preliminary introduction of the compressive sampling theory has been given previously. A more
Equi
10 5
10 5
Sinusoid−based coefficients
Original Reconstructed
Sinusoid−based coefficients
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2
−1
−3
Original Compressive samples Reconstructed
4
10 0
10 −5
10 −10
10 −15
0
10
20
30
40
50
60
70
Mode
Fig. 2 The Fourier coefficients of f × and the Fourier coefficients of its spatial samples by 24 equally spaced samples (∘).
Original 10 0
Reconstructed
10 −5
10 −10
10 −15
0
10
20
30
40
50
60
70
Mode
Fig. 4 The Fourier coefficients of the original test signal f× and the reconstructed signal g∘.
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generic expression can be found in the literature. For beginners, the references by Candes and Wakin [15] and Romberg [9] are recommended. For more competent theoretical framework, readers can refer to the literature by Candes et al. [1] and Candes [2].
In practical fluid tests, N dynamic pressure sensors are equidistantly deployed on the duct wall (r R) to determine spinning modes. The previous formulation becomes pm0 x0 ; R; t ≈
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III.
Fan-Noise Formulations
The most dominant noise of turbofan engine during takeoff and landing comes from the interaction between rotating fans and nearby stator vanes. The vortical flow in the rotor wakes impinges the stator surface, leading to the generation of noise, which mainly consists of tonal spinning modes. The sound pressure, acoustic density, and associated particle velocities, (p 0 , ρ 0 , u 0 , v 0 , w 0 ) are generally small compared with the background mean flow variables (p0 , ρ0 , u0 , v0 , w0 ). Sound-wave propagation can thus be approximately modeled by linearized Euler equations (LEEs) [6,12,17]. The acoustic disturbances can be represented by Fourier series in terms of circumferential (azimuthal) modes m [11]. For example, the sound pressure satisfies ∞ X
p 0 x; r; θ; t
pm0 x; r; te−imθ
(3)
m−∞
In addition, the mean flow in an unflanged straight cylindrical duct at the operating Reynolds numbers can be assumed stationary and almost one-dimensional. Then, the mean flow is presumably (u0 ; 0; 0). Hence, the three-dimensional LEE governing a single mth spinning-mode sound propagation is 0 ∂ρm0 ∂ρ 0 ∂um ∂vm0 v0 ∂w 0 u0 m ρ0 m m 0 ∂t ∂x ∂x ∂r r r∂θ
(4)
N 1 X 2π p 0 x0 ; R; θK ; teimθK 2π K1 N
N 1X p 0 x0 ; R; θK ; teimθK N K1
(5)
∂vm0 ∂v 0 ∂p 0 u0 m m 0 ∂t ∂x ρ0 ∂r
(6)
∂wm0 ∂w 0 ∂pm0 u0 m 0 ∂t ∂x ρ0 r∂θ
(7)
where all variables are nondimensionalized with respect to a reference length, a reference speed, and a reference density. The values of m depend on the numbers of fan blades B and stator vanes V. A beautiful manipulation based on Fourier series has given the following relationship [18]:
(10)
where θK 2πK∕N (for K 1; : : : ; N), and x0 and R are the axial and radial positions of sensors. According to the Nyquist and Shannon sampling theorem, the detectable azimuthal modes should fall within the range of b−N∕2c and bN∕2c. In other words, for the two dominant spinning modes (m −33 and 22) at x0 , we need at least 66 sensors based on traditional signal-processing technologies. Furthermore, hundreds of microphones forming a sensor cage have to be used to reconstruct spatial profiles of spinning modes at various axial positions.
IV.
Compressive Sensing-Based Measurements
Compressive sensing is proposed here for the duct acoustic case to demonstrate its potential in aerospace. The sketch of a possible test setup is shown in Fig. 5. The sensory array with randomly deployed sensors is installed on the inlet wall to recover the spinning-mode profiles. The demonstration depends on analytical solutions that convey clear physical insights. For a tonal fan noise of the frequency ω propagating with the idealized geometry (a straight, unflanged, and semi-infinite duct), Eqs. (4–7) generate the following analytical solution: pm0 x; r; θ; t cJm kr reiωt−kx x−mθ
∂um0 ∂u 0 ∂p 0 u0 m m 0 ∂t ∂x ρ0 ∂x
m aV bB
1013
TECHNICAL NOTES
(11)
where Mj u0 ∕C0 , C0 is the speed of sound; c is the amplitude of the acoustic perturbation (the nondimensional value is set to 10−3 herein); Jm is the mth-order Bessel function of the first kind; ω is the angular frequency; and kr is the nth radial wavenumber, where dJm Rkr ∕dr 0 for the hard-wall engine case, and R is the radius of the outer duct wall. The axial wavenumber kx of the mth spinning mode can be subsequently calculated using kx −Mj q 1 − k2r 1 − M2j ∕k2 k∕1 − M2j , where k ω∕C0 is the wavenumber; and the sign of depends on the upstream/downstream direction of the spinning wave. In this work, we assume that the fan noise source is at x 0, and the sensory array is installed at x 0.1, where almost all spinning
Engine duct
(8) r
where a is any integer, and b denotes the bth harmonic of the bladepassing frequency. A quiet engine design can be simply performed based on the observation that the decay rate of a spinning mode increases with increasing m. As a result, the preferred vane number should be more than double of the blade number (i.e., min jmj B). Consider a practical turbofan engine case as an example (B 22 and V 55); the possible spinning modes are m : : : ; −88 ; −33; 22; 77; : : : etc. Generally, m −33 and m 22 are dominant modes at the working conditions of airworthiness certifications [8]. Hence, p0 is sparse in m wavenumber space, where the mth Fourier coefficient pm 0 x; r; t satisfies pm0 x; r; t
1 2π
Z
2π 0
p 0 x; r; θ; teimθ dθ
(9)
u0
x
Fig. 5 The sketch of a duct acoustic test using compressive sensing, where circles represent sensors.
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E
^ 2 kjp^ 0 j − jgjk kjp^ 0 jk2
(12)
where j · j is the amplitude of the Fourier coefficients, and k · k2 is the L2 norm. Figure 7 shows E with respect to the number of compressive samples, which vary from 1 to the Shannon sampling rate (≥66). For each sampling value, we repeat the simulation 10 times with random sensor locations and various white noise waveforms. As a result, different reconstruction errors could appear. For the same sample
10 0
10 −5
Analytical Measurements Reconstructed
10 −10
10 −15
10 −20
0
10
20
30
40
50
60
Mode
Fig. 6 Fan noise in frequency domain: analytical solution (−), polluted measurements from a single sensor (×), and reconstructed result from 24 sensors using compressive sensing (∘).
TECHNICAL NOTES
10 0
Errors
modes, except m −33 and m 22, are cut off and have been largely attenuated [8]. Figure 6 shows the Fourier coefficients of analytical solutions at x 0.1. In practical experiments, test-facility noise, self-noise of sensors, and other flow-induced noise will pollute measurement results. To represent those background interferences, a white noise with autocorrelation of 10−9 is added to the analytical solution. Figure 6 shows the Fourier coefficients of the polluted measurements from a single sensor. The outcome is still dominated by the two spinning modes. However, the dynamic range is extensively reduced, which will lead to indistinctive spatial profiles. As an aside comment, a background noise with larger autocorrelation will overwhelm the two spinning modes, leading to the failure of the present recovery method. Compressive sensing was conducted in the following steps. 1) Check that Fourier coefficients are sparse and supported on a set of size M. 2) Randomly choose K sensor locations; K satisfies Eq. (1). 3) Take time domain samples yk t from kth sensor, k 1; : : : ; K. ^ 1 ∈ CN , subject to gk yk ; 4) Solve Eq. (2), that is, arg min kgk k 1; : : : ; K. 5) Maintain M dominant entries in g^ and set the rest to zero. 6) Calculate the inverse Fourier transform of g^ to recover the original signal. Figure 6 shows the Fourier coefficients of the recovered signal, which agree perfectly well with the analytical solution. It seems that the interference from background noise has been largely suppressed. In this simulation case, 24 sensors are randomly deployed throughout [0; 2π]. It can be noticed that the dynamic range of the recovered signal is better than the previous case in Fig. 4 because only the most dominant entries remain in this case. In addition, it is obviously impossible to complete a good signal reconstruction from so few sensors if the classical Nyquist and Shannon sampling methods are adopted. The present spatial sampling rate (24) is almost three times smaller than the Nyquist rate (≥66). We wonder whether it is still possible to have a more compressed sampling. The reconstruction error by the following nondimensional definition is examined:
Sinusoid−based coefficients
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1014
10 −1
10 −2
10 −3
0
1
10
10
2
10
Number of sensors
Fig. 7 The reconstruction error with respect to the number of spatial samples.
number, the sensory layouts with the smallest error could be optimal. This Monte Carlo simulation results in Fig. 7 show that E decreases almost linearly when the spatial samples increase from 1 to 10. Beyond that sampling rate, the distribution of E gradually condenses on the region between E 0.01 and E 0.1. Figure 7 confirms the working of compressive sensing and its potentials in fluid and acoustic measurements. It shows that 12 spatial samples might already be sufficient for the present case, five times less than the Shannon sampling value. It is worthwhile to mention that Figs. 6 and 7 are the results at one specific time step. The same findings were discovered for all time steps in the simulation.
V.
Conclusions
Compressive sensing is a newly emerging method in information technology that could significantly impact scientific research and engineering applications in aerospace as well. In this note, the fundamentals of compressive sensing are introduced and its usage by studying a linear-duct acoustic problem is demonstrated. Numerical simulations have been used to demonstrate the satisfactory performance of compressive sensing. The required number of sensors can be reduced by approximately five times. Moreover, some comments are made next. 1) An entirely random sensory position might be inconvenient in a practical setup. In this simulation, 128 spots are equidistantly spaced throughout 0; 2π. K spots are thereafter randomly chosen for compressive sensing. 2) The value of CM depends on the coherence of the sampling matrix and the sparsity basis. In this work, CM 3 works well. 3) A bigger CM may be chosen to achieve a good reconstruction with an overwhelmingly high probability. As a result, there is a tradeoff between probability and samples number. Monte Carlo simulations can be conducted to single out possibly optimal sensor layouts. On the other hand, a spiral shape of sensory position has been regarded as the optimal layout in aeroacoustic array beamforming [19]. 4) The two cases studied in this work are sparse in a Fouriertransformed basis. For more complicated aerospace cases, a different sparse representation (e.g., in a wavelet basis or a proper orthogonal decomposition basis) might be considered.
Appendix A: Demonstration Code The following MATLAB code (Fig. A1) is extensively simplified for brevity [20]. CVX toolbox should be installed before running the code.
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function CSV1
% Developed by Prof Xun Huang, Peking University, 2012
clc; clear all; close all
[3]
% Initialization n=128; m=24;
% m: Number of random measurements
theta = [0:2*pi/n:2*pi-2*pi/n]’;
[4]
% Circumferential angles
f = cos(18*theta) + 2*cos(25*theta) + 0.5*cos(35*theta); [A,r] = get random meas(n,m); y = A*f;
TECHNICAL NOTES
% Test signal
% Random measurement locations
[5]
% Take the random measurements
Psi = inv(fft(eye(n))); x0 = pinv(A*Psi)*y;
[6] % Optimization based on cvx toolbox cvx begin
[7]
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variable xp(n) complex;
[8]
minimize(norm(xp,1)); subject to
[9]
A*Psi*xp==y;
[10]
cvx end % Display
[11]
figure(1); F=fft(f); semilogy(1:n/2,abs(F(1:n/2)),’-kx’); hold on; yF= fft(Psi*xp);
[12]
semilogy(1:n/2,abs(yF(1:n/2)),’-ro’); % Generate measurements at random locations
[13]
function [A,r] = get random meas(n,m) r = randi([1 n],m,1);
[14] r = sort(r); A = zeros(m,n);
[15]
for i=1:m A(i,r(i))=1; end
[16] Fig. A1
MATLAB code. [17]
Acknowledgments This research was supported by the National Natural Science Foundation Grant of China (grant 11172007) and Science Foundation of Aeronautics of China (20101271004).
[18] [19]
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C. Bailly Associate Editor
