Abstract

Robust defect detection in the presence of grain noise originating from material microstructures is a challenging yet essential problem in ultrasonic non-destructive evaluation (NDE). In this paper, a novel method is proposed to suppress the gain noise and enhance the defect detection and imaging. The defect echo and grain noise are distinguished through analyzing the spatial location where the echo is originating from. This is achieved by estimation of the angle of arrival (AOA) of the returned echo and evaluation of the likelihood that the echo is reflected from the point where the array is focused or otherwise from the random reflectors like the grain boundaries. The method explicitly addresses the statistical models of the defect echoes and the spatial noise across the array aperture, as well as the correlation between the flaw signal and the interfering echoes; estimates the AOA and the likelihood in a dimension-reduced beam space via a linear transformation; and determines a weighting factor based on the mean likelihood. The factors are then normalized and utilized to correct and weigh the NDE images. Experiments on industrial samples of austenitic stainless steel and INCONEL Alloy 617 are conducted with a 5 MHz transducer array, and the results demonstrate that the grain noise is reduced by about 20 dB while the defect reflection is well retained, thus the great benefits of the method on enhanced defect detection and imaging in ultrasonic NDE are validated.

1 Introduction

Robust defect detection in the presence of grain noise is a challenging yet essential problem in ultrasonic non-destructive evaluation (NDE), which has attracted significant attention in recent decades [110]. A variety of coarse-grained materials such as alloy and austenitic stainless steel offer attractive properties like high-temperature strength or excellent resistance to a corrosive environment and thus are widely used to build components like ducting, combustion cans, and transition liners in a range of key industrial sectors such as energy, transportation, oil and gas, nuclear, and aerospace. However, when these materials and the structures made of these materials are inspected using ultrasound, the flaw echoes are usually contaminated by high-level grain noise originating from the material microstructures; furthermore, the grain noise is time-invariant, highly correlated with the flaw echoes and showing similar spectral characteristics; as a result, the grain noise cannot be removed by using the time-averaging or the spectral filtering techniques.

Several techniques have been investigated to reduce the grain noise and enhance defect detection through exploiting the key differences between the defect echoes and grain noise. The defect and grain boundaries usually have different sizes, thus their scattering properties and responses to the broadband excitation signal demonstrate different spectral characteristics [2]. These differences have motivated temporal-spectral processing, such as Weiner filtering [3,4], split-spectrum processing [5], sub-spectrum phase coherence factoring [6], spectral distribution similarity analysis [7], and matched filtering [8]. On the other hand, the defect echo typically has a coherent structure with energy mainly scattered from a single spatial point, while the grain noise is spatially distributed throughout the insonified resolution cell. This observation has motivated spatial diversity and processing, such as adaptive beamforming [1,9] and generalized coherence factoring [10].

In this paper, a novel approach is proposed to enhance the defect detection in ultrasonic imaging and non-destructive inspection based on the estimation of the arrival angle of the echo impinging on the transducer array. With the angle of arrival (AOA) estimation, this method distinguishes a flaw echo from the grain noise via evaluating the likelihood that the echo is reflected from the point of interest or from other spatial locations due to random reflectors like the grain boundaries. Different from the beamforming and coherence factoring techniques, this method explicitly addresses the statistical models of the flaw echoes and the noise across the array aperture as well as the correlation between the flaw signal and the interfering echoes from random reflectors, and estimates the likelihood that the echo originates from the point where the array is focused on. In particular, the AOA estimation is performed in a dimension-reduced beam space (BS), which is achieved by applying a well-designed linear transformation to the array data vectors. As the signal processing complexity is directly related to the dimension of the data; obviously, the beam-space processing may significantly reduce the computation cost. Furthermore, as demonstrated in the literature [11], a well-designed beam-space transformation may improve the AOA estimation performance in terms of a reduced estimation bias, a lower resolution threshold, and a lower sensitivity to the wave-front distortion, which are all important for ultrasonic NDE applications.

The paper is organized as follows: Section 2 presents the data model for array processing in ultrasonic NDE, the beam-space transformation design for data reduction while maintaining the AOA estimation accuracy and discusses the performance index evaluation based on the maximum likelihood estimator in order to distinguish a flaw echo from the grain noise. Section 3 presents two experiments on real industrial samples of austenitic stainless steel and INCONEL Alloy 617 to analyze the performance of the proposed method on ultrasonic NDE inspection and imaging against the standard total focusing method (TFM) [1] and to evaluate the impact of the BS transformation matrix on the image formation. Section 4 concludes the paper.

2 Materials and Methods

2.1 Transducer Array Data Model.

The sensor array has been used in a variety of systems and applications including ultrasonic NDE, medical imaging, underwater sonar, modern radar, and wireless communications. A standard data model for the problem of AOA estimation of L planar narrowband waves impinging on a passive array of M (M > L) narrowband sensors with arbitrary locations and directional characteristics is represented by the equation [12] 
x(t)=A(θ0)s(t)+n(t),t=1,,N
(1)
where x(t) is the array output data vector, n(t) is an immeasurable noise process, s(t) is a vector of signal waveforms, and N is the number of snapshots or data samples. The matrix A(θ0) is the array steering matrix with a structure 
A(θ0)=[a(θ1)a(θL)]
(2)
where a(θi) denotes a steering vector toward the direction θi, and θ0 = {θ1, θ2, …, θL} are the parameters of interest or AOAs. The exact form of a(θi) depends on the transducer spectral and spatial characteristics and the array geometry. In this work, the source signal vector s(t) and the additive noise n(t) are assumed to be independent, stationary, zero-mean, and Gaussian processes, and the covariance matrix of the array output vector x(t) is given by 
Rx=E[x(t)xH(t)]=A(θ0)PsAH(θ0)+Q
(3)
where E[•] stands for the expectation operator, (•)H denotes the complex conjugate transpose, Ps and Q represent the covariance matrix of s(t) and n(t), respectively.
If we introduce a linear M × J transformation T covering a spatial sector Θ (one spatial sector or a union of multiple separate spatial sectors of interest), and the mapping x(t)z(t)=THx(t) from element space (ES) to the beam space (BS), a set of BS observations is obtained 
z(t)=THx(t)=THA(θ0)s(t)+THn(t)
(4)
and the covariance matrix of z(t) is given by 
Rz=E[z(t)zH(t)]=THA(θ0)PsAH(θ0)T+THQT=Ab(θ0)PsAbH(θ0)+THQT
(5)
where Ab(θ0) = THA(θ0) is the associated array steering matrix in the beam space.

2.2 Beam-Space Transformation Design.

A key challenge in beam-space array processing is the design of the linear transformation T. Several approaches to transformation design have been proposed based on the respective criterion. The maximum estimation accuracy (MEA) technique [13] evaluates the distance between the full-dimension element-space Cramer-Rao bound (CRB) and the associated beam-space CRB and derives a transformation where the ES CRB is asymptotically attained in the beam space. The method of spheroidal sequences (MSS) [14] creates a transformation that provides excellent coverage of the spatial sector of interest. Both MEA and MSS produce excellent BS AOA estimation but are sensitive to the out-of-sector interference. To address the undesired interferers, Eriksson and Viberg [15] propose an adaptive transformation design approach, which effectively rejects interference based on the information in the data covariance matrix; however, the performance degrades significantly in hard conditions involving a small number of snapshots or highly correlated sources, which is unfortunately exactly the case in ultrasonic NDE, where the defect echo and the echoes reflected from the grain boundaries are highly correlated or even coherent, and only a single data vector is returned from the point of interest. To better address the challenge, Li and Lu [16] propose a subspace projection-based technique to reject interference by incorporating nulls toward the undesired emitters, which is robust and effective if the interferers are closely clustered.

To illustrate the effectiveness of BS processing for ultrasonic NDE and imaging, in this paper, we employ the MEA technique for BS transformation design, which is simple and straightforward to implement, although it is not optimal in the ultrasonic NDE scenarios due to the strong out-of-sector interference. Essentially, MEA evaluates the transformation’s in-sector AOA estimation accuracy and is defined by measuring the distance between the full dimension ES CRB and the associated BS CRB obtained with T over the spatial sector of interest Θ [13], 
η(T)=(ΘCRBES(θ)dθΘCRBBS(θ,T)dθ)1/2
(6)
The AOA estimation CRB is well studied and defined in the literature, and the element-space stochastic CRB is given by [17] 
CRBES(θ)=σ22N[Re{[DHPAD][PsAHRx1APs]T}]1
(7)
where XY denotes the Hadamard product, Re{•} denotes the real part of the complex variable, and 
D=[θa(θ)|θ=θ1,,θa(θ)|θ=θL]
(8)
 
PA=IA[AHA]1AH
(9)
 
Ps=E[s(t)sH(t)]
(10)
 
Rx=E[x(t)xH(t)]=APsAH+σ2I
(11)
and σ2 denotes the noise variance. For simplicity, the dependence on θ for A and D has been suppressed. The expression for CRBBS(θ, T) is achieved by substitution of A, D, and Rx by THA, THD, and THRxT, respectively. In general, 0 ≤ η(T) ≤ 1, and η(T) = 1 is desired and attainable when the ES CRB is obtained in the beam space.

2.3 Factor Evaluation and Imaging.

The maximum likelihood estimation algorithm is employed in this work, considering its excellent performance to address AOA estimation in a harsh environment involving highly correlated (or even coherent) sources, low signal-to-noise ratio (SNR), and a short number of snapshots (there is a single snapshot in the ultrasonic NDE scenario) [18]. An index is introduced to evaluate the likelihood that the echo is originating from the array focusing point. As an example, the spatial spectrum (likelihood) obtained from a data vector is normalized and illustrated in Fig. 1. The minimizing solution is achieved at 90 deg corresponding to the array focusing point. The BS spatial sector is defined and illustrated by the dashed line, and the AOA estimation is only evaluated within the spatial sector of interest, i.e., at [70, 110] deg (which thus requires less computation). Assume that the array focal region is defined and illustrated by the dotted line, for example, at [85, 95] deg, a factor is defined by 
F=MeanfitnesswithinthefocalregionMeanfitnessoutsidethefocalregion
(12)

The factor achieves a higher value if the data vector is returned from a point with a major reflector; on the contrary, if the echo comes from spatially distributed sources, the value of the factor will be lower.

3 Results and Discussion

The performance of the proposed method for enhanced defect detection in ultrasonic NDE inspection and imaging is demonstrated and analyzed in this section. The experimental apparatus consists of the test samples, the ultrasonic transducer array, the phased array control system, and a personal computer. A 64-element transducer array with a 0.7 mm element pitch and 5 MHz central frequency (Vermon, Tours, France) is utilized in contact with the test sample upper surface with gel coupling, as shown in Fig. 2(a). An OPEN ultrasonic phased array control system with 64 independent parallel channels and 16-bit resolution (LeCouer, Chuelles, France) is connected with the transducer array for excitation and data acquisition, as shown in Fig. 2(b).

A personal computer is connected to the OPEN system to control the excitation sequence and record the return signals for post-processing and imaging. A matlab routine is developed to implement the full matrix capture (FMC) data acquisition, where each transducer element is excited sequentially and the echoes received by all the array elements are recorded [1]. As a result, a complete FMC data set is composed of M2 A-scan waveforms, where M is the number of array elements. The A-scan waveform is recorded at a sampling rate of 100 MHz, and each waveform is band-pass filtered to remove the DC drift and high-frequency noise. The industrial test samples with different materials and characteristics are employed in the following two experiments.

The beam-space transformation is designed using the MEA technique [13]. The BS spatial focusing sector is specified to be an interval [70, 90] deg. As the echoes are supposed to be symmetrical around the array center, a scan in the angle range [0, 90] deg is utilized (rather than [0, 180] deg) to save the computation while maintaining the same performance. The proper dimension of the beam space is determined by evaluating the performance measure (6) over the spatial interval of interest. The beam-space transformations with different dimensions are constructed, and the performance index η(T)(6) over the interval [70, 90] deg is evaluated and illustrated in Fig. 3. As can be seen from Fig. 3, with the BS dimension k increasing, the BS performance gets closer to the element space performance in terms of the AOA estimation CRB. When k = 16, η(k = 16) = 1, thus the BS dimension is chosen to be 16.

The beam pattern of the generated BS transformation is demonstrated in Fig. 4. The echoes within the spatial interval [70, 90] deg will be passed through the transformation with a unit gain, and the interference impinging from other directions out of the spatial sector will be attenuated. The further away from the focusing sector, the more attenuation will be applied, for instance, the echoes from the interval of [110, 180] deg will be attenuated by 15–20 dB, which thus have less impact on the AOA estimation within the BS focusing sector.

3.1 Experiment I.

A test sample from a coal-fueled power plant generator end ring with a thickness of around 55 mm made of austenitic stainless steel is utilized in the experiment. Due to the size of microstructure grains, the stainless steel is demonstrated to be highly scattering to the 5 MHz ultrasound and results in dominant and significant grain noise and pretty low SNR.

Figure 5(a) shows the image obtained with the TFM [1] using the band-pass filtered FMC data set in a dynamic range of 40 dB. In order to apply the proposed method to the FMC data, each A-scan waveform is appropriately delayed to focus at the point of interest (an imaging point), the pre-focused data vector is processed with the BS transformation whose beam pattern is demonstrated in Fig. 4 to reduce the dimension from 64 to 16 (while maintaining the same AOA performance), and then the maximum likelihood AOA estimator is applied to the dimension-reduced BS data vector and the likelihood is measured for the spatial interval of [70, 90] deg. In the experiment, the focal region is set to be 5 deg, which means that any echoes within the angle range of [85, 90] deg are considered to be on-axis, and the others are considered to be off-axis. The factor is evaluated using the Eq. (12) for each point in the region and then normalized and illustrated in Fig. 5(b). In addition, the amplitudes of the TFM image in Fig. 5(a) are multiplied with the weighting factors in Fig. 5(b) point by point, and the corrected image is shown in Fig. 5(c).

As can be seen from Fig. 5, the TFM image of the stainless steel sample in Fig. 5(a) is quite noisy when the 5 MHz ultrasound is utilized. The back-wall reflection is visible, but the SNR is not high. There are a lot of points with high amplitude values that are similar to the back-wall reflection as the result of significant scattering noise from the microstructure grain boundaries. In Fig. 5(b), the factors corresponding to the back-wall region demonstrate higher values, and the other areas produce relatively lower coefficients, due to the fact that in the back-wall region, a larger portion of the received energy originates from the point where the array is focused, but in a region without dominant reflectors, the received energy is more evenly distributed spatially across the angle domain. It is interesting to note that in the back-wall area with the lateral distance around 0 and the depth around 55 mm (highlighted by the dashed line box in Figs. 5(a) and 5(b)), the reflection in the TFM image of Fig. 5(a) is not well visible, probably due to the partial cancellation by the multipath interference and fading; however, as shown in Fig. 5(b), in this region, there are more on-axis echoes impinging on the array than the off-axis echoes, although the overall summed amplitude is not high. It seems that the factors computed from the AOA of the echoes may represent a different set of information against the standard echo amplitude-based methods, like the TFM. When the factors are multiplied with the TFM amplitudes and in the corrected image of Fig. 5(c), the grain noise is greatly reduced in comparison to the raw TFM image in Fig. 5(a), and at the same time, the back wall is well retained with consistent strength.

To quantitatively characterize the benefits of the method on enhanced defect detection and imaging, the SNRs of the images are evaluated and compared. Assume that the signal level is defined by the back-wall reflection, and the noise level is evaluated by averaging the echo amplitude within an area without major reflectors between the points (−10 mm, 25 mm) and (0 mm, 35 mm), as highlighted by the solid line box in Figs. 5(a) and 5(c), the SNR of the raw TFM image in Fig. 5(a) is measured to be 26.7 dB and that of the corrected image in Fig. 5(c) is 41.9 dB. The method results in a further grain noise reduction of about 15 dB, through estimating the spatial location where the echo is originating from.

In order to evaluate the impact of the beam-space transformation on the noise reduction and imaging, another BS transformation of the same dimension designed using the MSS [14] is applied to the pre-focused data vectors to reduce the data dimension from 64 to 16, and then the maximum likelihood AOA estimator is employed with the same parameters and the factors are calculated from the mean likelihood using Eq. (12). The normalized factors and the weighted TFM image (the TFM amplitude is multiplied with the weighting factor point by point) are illustrated in Fig. 6. When comparing Fig. 6 with the counterparts Figs. 5(b) and 5(c), it is obvious that if the BS transformation is not properly designed, the out-of-sector interference is not well attenuated and the in-sector AOA estimation is inferior; as a result, the images of the weighting factors and the corrected TFM are very noisy, where the back wall is not well represented and the strong noise appears in other regions.

3.2 Experiment II.

A test sample made of INCONEL Alloy 617 with a thickness of around 148 mm is considered in this experiment. As described in the datasheet, Alloy 617 is a solid-solution, nickel–chromium–cobalt–molybdenum alloy with an exceptional combination of high-temperature strength and oxidation resistance, and it is readily formed and welded by conventional techniques [19]. Due to the attractive properties, Alloy 617 is widely used for components such as ducting, combustion cans, and transition liner in gas turbines as well as power-generating plants, both fossil-fueled and nuclear. The scenario seems more challenging than that in Experiment I, where the material is also highly scattering to 5 MHz ultrasound, and the thickness is much larger, as a result, the grain noise is dominant and the SNR is pretty low.

Figure 7(a) illustrates the image obtained with the TFM and displayed in the dynamic range of 40 dB. The grain noise is quite significant, and the back-wall reflection is not highly dominant, whose amplitude is at a similar level as that of the grain noise. The image is very noisy and the back wall is barely visible at the depth around 148 mm. To form the image in Fig. 7(b), the pre-focused array data vector is processed with the BS transformation, then the maximum likelihood AOA estimation is applied to the dimension-reduced data vectors, and the factor is calculated from the spatial likelihood spectrum. As shown in Fig. 7(b), the back-wall reflection is visible, but the distinction against the background is not so significant as demonstrated in Fig. 5(b) in Experiment I, which means that the factors for the back wall and the other regions have similar values. This phenomenon is probably due to the greater depth of the back wall observed in this scenario. Because of the numerous scattering from the microstructure grain boundaries, when the array is focused on the back wall, the interference is significant or even dominant in terms of the amplitude, and the summed interference tends to be more coherent rather than just being random noise. Figure 7(c) shows the corrected TFM image, where the TFM image is multiplied with the weighting factors point by point. In comparison with the original TFM image in Fig. 7(a), clearly the grain noise is significantly reduced and the back-wall reflection becomes more visible. Again, the SNR is compared quantitatively. Assume that the signal is defined by the back-wall reflection, and the noise level is evaluated by averaging the amplitude within an area without major reflectors between the points (−10 mm, 110 mm) and (10 mm, 130 mm) highlighted by the solid line box in Figs. 7(a) and 7(c), the SNRs of the raw TFM image in Fig. 7(a) and the weighted image in Fig. 7(c) are measured to be 21.9 dB and 43.8 dB, respectively, which means a grain noise reduction of about 22 dB.

To better understand and evaluate the impact of the beam-space transformation on the noise reduction, the BS transformation designed using the MSS in Experiment I is applied to this dataset. The resultant images of the weighting factors and the corrected TFM are illustrated in Fig. 8. As shown in Fig. 8(a), the factors for the back-wall region show similar values as the other regions because the AOA cannot be accurately estimated since the data vector is somewhat distorted by the inferior BS transformation. The back-wall reflection is barely invisible in the factor image. As compared with the images in Figs. 7(b) and 7(c), the corrected TFM image in Fig. 8(b) is much noisier than the image of Fig. 7(c), the back-wall is less dominant against the strong grain noise, which makes it more difficult to conduct a reliable inspection.

In addition to the performance advantage of non-destructive evaluation and imaging, the beam-space transformation and processing may help to reduce the computation, which is also a promising and demanding feature since the array processing is usually computationally intensive and time-consuming. As observed in Experiments I and II, the BS-based imaging is roughly six times faster than the element space-based method where the AOA estimation is applied directly to the pre-focused array data vectors.

4 Conclusion

This paper presents a novel approach to suppress the gain noise and improve the defect detection in ultrasonic NDE and imaging, based on the maximum likelihood AOA estimation in a dimension-reduced beam space. The method distinguishes the flaw echoes from grain noise in the spatial domain via analyzing the spatial location where the returned echo is originating from. The technique is validated with experiments on the industrial samples of austenitic stainless steel and INCONEL Alloy 617 with a 64-element 5 MHz transducer array. It has been observed that the echoes from the back wall of the specimen produce much higher factor values than that of grain noise due to the fact that the echo mainly originates from the point where the array is focused on, and the TFM images are improved by more than 20 dB when the amplitudes are weighted and corrected using the normalized factors. The computation cost is reduced by about six times as demonstrated in the experiments, as a result of transforming the data into a lower dimension space. In future work, the optimal beam-space transformation design and robust AOA estimation will be explored for the evaluation of the factors on various industrial samples.

References

References
1.
Li
,
M.
, and
Hayward
,
G.
,
2012
, “
Ultrasound Non-Destructive Evaluation (NDE) Imaging With Transducer Arrays and Adaptive Processing
,”
Sensors
,
12
(
1
), pp.
42
54
. 10.3390/s120100042
2.
Papadakis
,
E. P.
,
1965
, “
Ultrasonic Attenuation Caused by Scattering in Poly-Crystalline Metals
,”
J. Acoust. Soc. Am.
,
37
(
4
), pp.
703
710
. 10.1121/1.1909399
3.
Neal
,
S. P.
,
Speckman
,
P. L.
, and
Enright
,
M. A.
,
1993
, “
Flaw Signature Estimation in Ultrasonic Nondestructive Evaluation Using the Wiener Filter With Limited Prior Information
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
40
(
4
), pp.
347
353
. 10.1109/58.251283
4.
Izquierdo
,
M. A. G.
,
Hernández
,
M. G.
,
Graullera
,
O.
, and
Ullate
,
L. G.
,
2002
, “
Time-Frequency Wiener Filtering for Structural Noise Reduction
,”
Ultrasonics
,
40
(
1–8
), pp.
259
261
. 10.1016/S0041-624X(02)00148-8
5.
Amir
,
I.
,
Bilgutay
,
N. M.
, and
Newhouse
,
V. L.
,
1986
, “
Analysis and Comparison of Some Frequency Compounding Algorithms for the Reduction of Ultrasonic Clutter
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
33
(
4
), pp.
402
411
. 10.1109/T-UFFC.1986.26848
6.
Gongzhang
,
R.
,
Li
,
M.
,
Xiao
,
B.
,
Lardner
,
T.
, and
Gachagan
,
A.
,
2014
, “Robust Frequency Diversity Based Algorithm for Clutter Noise Reduction of Ultrasonic Signals Using Multiple sub-Spectrum Phase Coherence,”
Review of Progress in Quantitative Nondestructive Evaluation
, Vol.
33
,
D. E.
Chimenti
,
L. J.
Bond
and
D. O.
,
Thompson
, eds., (
American Institute of Physics 1581
,
Melville, NY
), pp.
1948
1955
.
7.
Xiao
,
B.
,
Li
,
M.
,
Gongzhang
,
R.
,
O’Leary
,
R.
, and
Gachagan
,
A.
,
2014
, “Image de-Noising via Spectral Distribution Similarity Analysis for Ultrasonic Non-Destructive Evaluation,”
Review of Progress in Quantitative Nondestructive Evaluation
, Vol.
33
,
D. E.
Chimenti
,
L, J.
Bond
and
D. O.
Thompson
, eds., (
American Institute of Physics 1581
,
Melville, NY
), pp.
1941
1947
.
8.
Li
,
M.
, and
Hayward
,
G.
,
2016
, “Optimal Matched Filter Design for Ultrasonic NDE of Coarse Grain Materials,”
Review of Progress in Quantitative Nondestructive Evaluation
, Vol.
35
,
D. E.
Chimenti
and
L. J.
Bond
, eds., (
American Institute of Physics 1430
,
Melville, NY
), p.
020011
.
9.
Synnevag
,
J. F.
,
Austeng
,
A.
, and
Holm
,
S.
,
2007
, “
Adaptive Beamforming Applied to Medical Ultrasound Imaging
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
54
(
8
), pp.
1606
1613
. 10.1109/TUFFC.2007.431
10.
Li
,
P.-C.
, and
Li
,
M.-L.
,
2003
, “
Adaptive Imaging Using the Generalized Coherence Factor
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
50
(
2
), pp.
128
141
. 10.1109/TUFFC.2003.1182117
11.
Li
,
F.
, and
Liu
,
H.
,
1994
, “
Statistical Analysis of Beam-Space Estimation for Direction-of-Arrivals
,”
IEEE Trans. Signal Process.
,
42
(
3
), pp.
604
610
. 10.1109/78.277852
12.
Li
,
M.
, and
Lu
,
Y.
,
2008
, “
Maximum Likelihood DOA Estimation in Unknown Colored Noise Fields
,”
IEEE Trans. Aerosp. Electron. Syst.
,
44
(
3
), pp.
1079
1090
. 10.1109/TAES.2008.4655365
13.
Anderson
,
S.
,
1993
, “
On Optimal Dimension Reduction for Sensor Array Signal Processing
,”
Signal Process.
,
30
(
2
), pp.
245
256
. 10.1016/0165-1684(93)90150-9
14.
Forster
,
P.
, and
Vezzosi
,
G.
,
1987
, “
Application of Spheroidal Sequences to Array Processing
,”
ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing
,
Dallas, TX
, pp.
2267
2271
,
15.
Eriksson
,
J.
, and
Viberg
,
M.
,
2000
, “
Adaptive Data Reduction for Signal Observed in Spatially Colored Noise
,”
Signal Process.
,
80
(
9
), pp.
1823
1831
. 10.1016/S0165-1684(00)00091-8
16.
Li
,
M.
, and
Lu
,
Y.
,
2006
, “
Dimension Reduction for Array Processing with Robust Interference Cancellation
,”
IEEE Trans. Aerosp. Electron. Syst.
,
42
(
1
), pp.
103
112
. 10.1109/TAES.2006.1603408
17.
Stoica
,
P.
, and
Nehorai
,
A.
,
1989
, “
MUSIC, Maximum Likelihood, and Cramer-Rao Bound
,”
IEEE Trans. Acoust. Speech Signal Process.
,
37
(
5
), pp.
720
741
. 10.1109/29.17564
18.
Li
,
M.
, and
Lu
,
Y.
,
2007
, “
A Refined Genetic Algorithm for Accurate and Reliable DOA Estimation With a Sensor Array
,”
Wireless Pers. Commun.
,
43
(
2
), pp.
533
547
. 10.1007/s11277-007-9248-5
19.
INCONEL 617 Technical Data
, http://www.hightempmetals.com/techdata/hitempInconel617data.php, Accessed in Oct. 2019.