An adaptive multi-taper spectral estimation for stationary processes

doi:10.1016/j.ymssp.2022.109629

机械系统与信号处理

第 183 卷, 2023 年 1 月 15 日, 109629

Mechanical Systems and Signal Processing

工程技术TOPEI检索SCI升级版工程技术1区IF 7.9

https://doi.org/10.1016/j.ymssp.2022.109629 获取权利和内容

高亮

•

自适应多窗方法在频谱估计中采用每个频率点变化的窗数量。
•

所提出的方法在两个具有不同光谱特性的例子中被证明是有效的。
•

所提出的方法可以平衡光谱的局部分辨率和方差。

摘要

多窗谱估计方法在减少平稳随机过程谱的偏差和方差方面是有效的。该方法中的窗数量（NoT）控制着谱的偏差和方差之间的权衡。通常，NoT 是经验确定的，并且在所有频率下保持恒定，限制了局部偏差和方差的调整。本文提出了一种自适应多窗方法，该方法在每个频率点使用不同的 NoT 来估计多元平稳过程的功率谱密度（PSD）和相干函数。提出了一种带有停止准则的迭代过程，以在每个频率下优化 NoT，而无需手动调整。采用具有简单解析表达式和满意的泄漏保护的正弦窗。所提出的自适应多窗方法应用于三个数值示例：建筑物模型的结构响应、风速过程和自回归过程，这些示例表现出不同的谱特征。所有示例的结果表明，所提出的方法在估计功率谱密度（PSD）和相干性时优于 Welch 方法和传统多谱图方法，具有更小的偏差和方差。

关键词

多谱图方法

功率谱密度

相干性

平稳过程

1. 引言

各种具有随机特征的物理和工程事件，如风速和在随机激励下的结构响应[1]，[2]，[3]，[4]，[5]，可以视为平稳随机过程的一种实现。与时间域方法相比，通过频谱分析获得的频域表示更具信息量，允许分析能量分布 [6]，[7]，[8]。

由于频域表示通常是从有限数据估计得出的，所获得的频谱通常会受到偏差和误差方差的影响[9]。几种非参数频谱估计器，包括 Welch 方法[10]和多谱估计方法[11]，已被广泛用于减少估计频谱的偏差和方差[12]，[13]，[14]，[15]。不幸的是，这些方法涉及偏差-方差权衡的困境，即低偏差估计器往往具有高方差，而低方差估计器通常具有高偏差[16]。多谱估计方法采用多个正交序列（加窗函数）进行频域平均以估计频谱 [17]。它相对于 Welch 方法表现出更好的性能，因为每个加窗的周期图都是从所有数据中获得的[18]，[19]，[20]。在实践中，Welch 方法可以被视为多窗谱估计技术的一种特殊情况，其中窗函数是相同的但时间上是移位的[20]。尽管这些方法可以缓解偏差和方差的问题，但它们通常包含难以调整的特定参数，例如 Welch 方法中的合适分段数和多窗方法中的窗函数数。

由自谱和互谱导出的相干性是量化两个时间序列在频域中线性耦合强度的重要工具 [21]。频谱中的小低方差可能会导致较大的相干性方差。在工程应用中，准确估计相干性（如风相干性）具有挑战性，因为 Welch 方法和多谱估计方法是频率无关的，从而无法适应不同的频率分辨率 [22]。

在最近几十年里，人们提出了几种方法来解决多谱估计法的偏差-方差权衡问题[19]，[23]，[24]，[25]，[26]，[27]，[28]。Thomson 提出了离散球面椭圆序列（也称为 Slepian 序列）作为一组具有较小泄漏的正交谱估计器，用于估计平稳过程的频谱[11]，[29]。在设定时频积之后，可以计算出实现最佳方差的谱估计器数量。然而，控制频率分辨率和方差减少的带宽通常是由用户定义的，而没有系统的选取方法。为解决这一问题，Haley 和 Anitescu [26] 提出了一种通过最小化对数谱的均方误差（MSE）来确定最优带宽的优化方法。尽管 Thomson 的多谱估计方法应用广泛，但 Slepian 窗的计算需要进行数值特征值分解，对于长时间序列来说计算成本较高 [30]。这种窗还需要在不同的带宽下重新计算。Riedel 和 Sidorenko [23] 提出了多谱估计中的正弦窗，它具有简单的解析表达式，无需设置和求解特征值方程。特别地，正弦窗的带宽仅取决于所用窗的数量。这意味着谱的分辨率和方差减少之间的内在权衡可以通过仅改变正弦窗的数量来控制。具体来说，较大的窗数量可以减少方差，但会伴随较高的偏差。 Riedel 和 Sidorenko [23] 还给出了通过最小化均方误差（MSE）来确定每个频率下使用的谱线数的公式。利用这个公式，Barbour 和 Parker [30] 开发了一种迭代程序来优化每个频率下采用的谱线数。尽管他们的方法生成了令人满意的频谱，但迭代次数是由用户根据经验确定的。最近，一种多谱线 S 变换（MTST）方法被提出用于平稳和非平稳过程的谱估计，该方法结合了正交时频 Hermite 窗口和 S 变换方法 [27], [28]。该方法通过改变 Hermite 窗口的形状来降低不同频率下的谱和相干性的偏差和方差。然而，MTST 方法在各种频谱形状上的适用性是有限的，因为其性能主要取决于定义的窗口形状函数，并且 NoT 在所有频率上仍然是恒定的。

在这项研究中，我们提出了一种自适应多谱估计方法，用于多变量平稳过程的谱估计和相干性估计，无需人工干预。与传统多谱估计方法固定 NoT 不同，我们提出了一种带有停止准则的优化策略来选择频率依赖的 NoT。因此，它使谱估计器能够在局部分辨率和谱方差之间取得平衡。通过三个数值例子验证了自适应多谱方法在处理不同谱特征方面的有效性，即建筑物模型的结构响应、风速过程和自回归(AR)过程。本方法减少了 NoT 选择的主观性，并平衡了局部谱的偏差-方差权衡。它还为具有各种谱形状的工程应用中的谱估计和相干性估计提供了一种有前景的方法。

2. 静态过程的多窗谱估计

2.1. 多变量平稳过程

设

X (t) = {[X_{1} (t), X_{2} (t), \dots, X_{N} (t)]}^{T}

为一个零均值多变量平稳过程，具有

E [{|X (t)|}^{2}] < \infty

，

t / Δ t \in Z

。存在一个定义在区间

[- f_{N}, f_{N}]

上的正交过程

Z (f)

，使得[16] (1)

X (t) = \int_{- f_{N}}^{f_{N}} e^{i 2 π f t} d Z (f)

其中 T 表示转置算子，

Δ t

为采样间隔，

f_{s} = 1 / Δ t

为采样频率，

f_{N} = f_{s} / 2

是奈奎斯特频率。

Z (f) = {[Z_{1} (f), Z_{2} (f), \dots, Z_{N} (f)]}^{T}

是一个复值正交增量过程，具有以下性质[16] (2)

E [d Z (f)] = 0

(3)

E [d Z^{*} (f) d Z^{T} (f^{'})] = δ_{f f^{'}} d S^{(I)} (f)

(4)

d Z^{*} (f) = d Z (- f)

其中*表示复共轭操作，

δ_{f f^{'}}

是克罗内克δ函数（当

f = f^{'}

时其值为 1，否则为 0），并且

S^{(I)} (f)

被假定为纯粹连续的。

The power spectral density (PSD) matrix of

X (t)

is defined as [16](5)

S (f) = d S^{(I)} (f) / d f

and the coherence function of

X (t)

is given by(6)

C_{ij} (f) = \frac{S_{ij} (f)}{\sqrt{S_{ii} (f) S_{jj} (f)}}

where

S_{ij} (f)

is the ijth element of

S (f)

2.2. Multi-taper spectral estimation

Tapering is an effective way to balance the broad- and narrow-band bias of the spectral estimation. Let

X (t)

be a discrete parameter stationary process with L observations. Assuming

ψ_{m} (t)

is the m-order taper, the multi-taper spectral estimator has the following form [27](7)

{\hat{S}}_{X} (f) = \frac{1}{M} \sum_{m = 0}^{M - 1} s_{X, m}^{*} (f) s_{X, m}^{T} (f)

and(8)

s_{X, m} (f) = \sum_{r = 1}^{L} ψ_{m} (r Δ t) X (r Δ t) e^{- i 2 π f r Δ t} Δ t

where M is the maximum order of tapers.

ψ_{m} (t)

satisfies the orthogonal condition(9)

\int_{- \infty}^{+ \infty} ψ_{i} (t) ψ_{j} (t) d t = δ_{ij}

The selection of tapers significantly affects the resultant spectral estimation. The sine taper with a simple analytic expression is selected in this study. It is expressed as [23](10)

ψ_{m} (t) = \sqrt{\frac{2}{L + 1}} \sin \frac{π m t}{L + 1}

3. Adaptive multi-taper approach

3.1. Formula of the adaptive multi-taper method

As mentioned previously, the NoT of the multi-taper-based approaches controls the bias-variance trade-off in the spectral estimation. Usually, the NoT is fixed to be a constant integer empirically for all frequency points. The multi-taper spectral estimator with the frequency-dependent NoT is calculated as(11)

\hat{S} (f) = \sum_{m = 0}^{M (f) - 1} \frac{s_{X, m}^{*} (f) s_{X, m}^{T} (f)}{M (f)}

The estimator of the coherence is obtained as(12)

{\hat{C}}_{ij} (f) = \frac{{\hat{S}}_{ij} (f)}{\sqrt{{\hat{S}}_{ii} (f) {\hat{S}}_{jj} (f)}}

3.2. Bias and variance of the adaptive multi-taper method

The overall spectral window

V (f)

of the adaptive multi-taper estimator can be written as [18](13)

V (f) = \frac{1}{M (f) Δ t} \sum_{m = 0}^{M (f) - 1} {|V_{m} (f)|}^{2}

with(14)

V_{m} (f) = Δ t \sum_{r = 1}^{L} ψ_{m} (r Δ t) e^{- i 2 π f r Δ t}

where the spectral window has the requirement that

\int_{- f_{N}}^{f_{N}} V (f) d f = 1

The univariate stationary process is employed here for the convenience of demonstration. The bias of the multi-taper method is calculated by [16](15)

\begin{matrix} Bias [\hat{S} (f)] = E \{\hat{S} (f)\} - S (f) = \int_{- f_{N}}^{f_{N}} V (f - f^{'}) S (f^{'}) d f^{'} - S (f) \end{matrix}

Letting

ϕ = f^{'} - f

, with the requirement of

\int_{- f_{N}}^{f_{N}} V (ϕ) d ϕ = 1

, Eq. (15) becomes(16)

Bias [\hat{S} (f)] = \int_{- f_{N}}^{f_{N}} V (ϕ) [S (f + ϕ) - S (f)] d ϕ

Assuming

S (\cdot)

is double continuously differentiable and can be expanded in a second-order Taylor series with respect to f:(17)

S (f + ϕ) \approx S (f) + ϕ S^{'} (f) + \frac{ϕ^{2}}{2} S^{″} (f)

V (\cdot)

of the sine taper is approximately rectangular over the resolution band

|f| ⩽ W

, then [18], [26](18)

V (ϕ) \approx \{\begin{matrix} 1 / (2 W) & if |ϕ| ⩽ W \\ 0 & otherwise \end{matrix}

where W is the bandwidth. Substituting Eqs. (17), (18) into Eq. (16), we can obtain(19)

Bias [\hat{S} (f)] ≈ \int_{- W}^{W} \frac{1}{2 W} [ϕ S^{'} (f) + \frac{ϕ^{2}}{2} S^{''} (f)] d ϕ = \frac{W^{2}}{6} S^{''} (f)

If the spectrum is infinitely differentiable, the variance of the adaptive multi-taper method can be written as [18](20)

Var [\hat{S} (f)] = \sum_{j = 0}^{M (f) - 1} \sum_{k = 0}^{M (f) - 1} \frac{Cov \{{\hat{S}}^{(k)} (f), {\hat{S}}^{(j)} (f)\}}{M {(f)}^{2}} \approx \frac{S^{2} (f)}{M (f)}

3.3. The optimal NoT

The MSE of the log spectrum estimated by the adaptive multi-taper method is given as [16], [26](21)

\begin{array}{l} MSE & = {Bias}^{2} [\log \hat{S} (f)] + Var [\log \hat{S} (f)] \\ \approx {\{\frac{Bias [\hat{S} (f)]}{E [\hat{S} (f)]}\}}^{2} + \frac{Var [\hat{S} (f)]}{E {[\hat{S} (f)]}^{2}} \\ \approx \frac{{[M (f)]}^{4}}{576 {(L Δ t)}^{4}} {[\frac{S^{''} (f)}{S (f)}]}^{2} + \frac{1}{M (f)} \end{array}

where

W \approx \frac{M (f)}{2 L Δ t}

is used [26].

The optimal NoT at each frequency is determined by setting the differentiation of Eq. (21) with respect to M as zero:(22)

M (f) = {[12 {(L Δ t)}^{2} \frac{S (f)}{S^{''} (f)}]}^{\frac{2}{5}}

To obtain the stable

M (f)

, the ratio

S (f) / S^{″} (f)

is transformed to the logarithm by setting(23)

y = \log S (f)

Consequently, the first-order and second-order derivatives of y with respect to f are(24)

y^{'} = \frac{S^{'} (f)}{S (f)}

(25)

y^{″} = \frac{S^{″} (f) S (f) - {[S^{'} (f)]}^{2}}{S^{2} (f)}

Then, we can obtain the ratio

S (f) / S^{″} (f)

as(26)

\frac{S (f)}{S^{″} (f)} = \frac{1}{{(y^{'})}^{2} + y^{″}}

As a result, Eq. (22) becomes(27)

M (f) = {[\frac{12 {(L Δ t)}^{2}}{{(y^{'})}^{2} + y^{″}}]}^{2 / 5}

According to Eq. (27), to find the optimal

M (f)

, one needs to know the true log spectrum and its first- and second-order derivatives. An iterative procedure is proposed to obtain the optimal NoT as follows.

Step I: One taper for all frequencies is employed to generate the initial spectral estimate with a high resolution.

Step II: Fit a quadratic function to the estimated spectrum of the previous iteration within an interval of

[f - W, f + W]

for each frequency f. The spectrum is extended by mirror symmetry to fit the spectrum near both ends of the frequency domain. The first- and second-order spectral derivatives of the log spectrum are then estimated and applied to Eq. (27) to calculate NoT.

Step III: From Eq. (22), it is noted that the NoT will increase to a large and even infinite value when the second-order derivative is small or vanish. In this regard, a forward and backward filtering technique is used to smooth the estimated NoT. A simple first-order low-pass filter is used as(28)

M_{ft} (f) = α M_{or} (f) + (1 - α) M_{ft} (f - 1)

where

α

is the filter coefficient,

M_{or}

is the original NoT before filtering at frequency f, and

M_{ft} (f)

is the filtered value at f. The smaller the

α

, the smoother the filtered spectrum is. In this study,

α

is set as 0.1 to efficiently mitigate the distortion of the estimated NoT induced by the small second-order derivative of the spectrum. After that, the following constraint is placed on the filtered NoT to ensure the difference of the NoT at consecutive frequencies within one:(29)

|M_{con} (f + 1) - M_{con} (f)| ⩽ 1

Step IV: A two-step updating strategy is used to obtain the satisfactory NoT. First, repeat Steps II and III until the mean relative difference (MRD) of the estimated NoT between consecutive iterations is less than 10 %, or the number of iterations reaches 20. The MRD is defined as follows:(30)

MRD = \frac{1}{J} \sum_{j = 1}^{J} |\frac{M_{con}^{s + 1} (f_{j}) - M_{con}^{s} (f_{j})}{M_{con}^{s} (f_{j})}| \times 100 %

where J is the number of frequencies, and

M_{con}^{s} (f_{j})

denotes the NoT at frequency

f_{j}

of the sth iteration.

Step V: As fewer tapers are required for peaks in the spectrum to ensure a sufficient frequency resolution, the second updating step is adopted to reduce the NoT. In practice, the second-order derivatives of the spectrum peak are negatively large. Thus, the NoT is reduced by half at each iteration if the second-order derivative is less than

μ_{driv} - σ_{driv}

, where

μ_{driv}

and

σ_{driv}

are the mean and standard deviation of the second-order derivative of the log spectrum, respectively. This correction step will be terminated until the MRD between consecutive iterations decreases to less than 1 %, or 30 iterations reach.

The flowchart of the method is displayed in Fig. 1.

The proposed method can calculate the optimal NoT at each frequency, which allows flexibility in adjusting the local bias and variance of various spectra. Three typical examples with different spectral characteristics are adopted to demonstrate the effectiveness of the method. The performance of the present adaptive multi-taper approach will be compared with Welch’s and traditional multi-taper methods.

4. Case study 1: A 4-DOF model

A four-story shear building model is used to evaluate the performance of the adaptive multi-taper method. The four degrees-of-freedom (DOF) structure is shown in Fig. 2, where the mass (m_s) of each story is 1.0 ton, and the stiffness (k_s) of each story is 180 kN/m. The natural frequencies of the building are 0.74, 2.13, 3.27, and 4.01 Hz.

The damping ratio of each mode is chosen to be 2 %. The building model is assumed to be subjected to a white noise excitation at each DOF. The spectral density matrix for the excitation is

S_{input} (f) = diag (100, 100, 100, 100)

, where

diag (\cdot)

denotes a diagonal matrix. The responses of the model with a 400-s duration at a sampling frequency of 10 Hz are calculated by the 4-order Runge-Kutta method.

4.1. The spectrum estimate

The PSD of the structural responses is estimated using Welch’s, the traditional multi-taper, and the present adaptive multi-taper methods for comparison. The window length of Welch’s method and NoT of the traditional multi-taper method are determined such that both can provide their best results. Specifically, the window length of Welch’s method is 40 s with 50 % overlapping. The first ten sine tapers are used in the traditional multi-taper method. The NoT of the adaptive muti-taper approach is determined by the proposed optimization procedure. The optimization is terminated after 12 iterations, and the results are shown in Fig. 3. The NoT varies between 5 and 150. Fewer tapers are required at the PSD peaks, indicating a higher frequency resolution there.

The estimated PSDs of the structural responses at the first two floors are compared in Fig. 4. The Welch’s and multi-taper methods provide sufficient accuracy around the PSD peaks at the natural frequencies but have a large variance over the whole frequency range. The variances can be reduced by increasing the number of segments in Welch’s method and tapers in the traditional multi-taper method, which will increase the bias and decrease the frequency resolution. The proposed method leads to a similar resolution as the traditional multi-taper method at natural frequencies because it adopts a small NoT. It uses a larger NoT at other frequencies to reduce the variance.

The corresponding decibel-scale errors (DSEs) are presented in Fig. 5. The DSE is defined as [27](31)

DSE = 10 \log_{10} [\hat{S} (f)] - 10 \log_{10} [S (f)]

The PSD estimates using the adaptive multi-taper method have smaller DSEs than those using Welch’s and traditional multi-taper methods over the whole frequency range.

The decibel-scale normalized relative error (DNRE) is then used to compare the errors quantitatively. It is expressed as [31](32)

DNRE = \frac{{∥10 \log_{10} [\hat{S} (f_{j})] - 10 \log_{10} [S (f_{j})]∥}_{2}}{{∥10 \log_{10} [S (f_{j})]∥}_{2}} \times 100 %

where

\hat{S} (f_{j})

and

S (f_{j})

are the estimated and theoretical PSDs at f_j, respectively, and

{∥\cdot∥}_{2}

denotes the Frobenius norm. In engineering applications, spectral moments are important to determine the survival probability and assess the reliability of structural systems [32]. The spectral moment is defined as [33](33)

λ_{p} = \int_{0}^{\infty} {(2 π f)}^{p} S (f) d f

where p is the order of the spectral moment. The relative percentage error (RPE) between the estimated and theoretical spectral moments is given by(34)

{RPE}_{p} = |\frac{{\hat{λ}}_{p} - λ_{p}}{λ_{p}}| \times 100 %

where

{\hat{λ}}_{p}

is the estimated p-order spectral moment.

The DNRE,

{RPE}_{0}

, and

{RPE}_{2}

are listed in Table 1. The adaptive multi-taper method exhibits the smallest DNRE and RPE values in comparison with the other two methods. Hence, using the adaptive NoT results in more accurate PSD estimates than the traditional multi-taper method using a fixed NoT (ten tapers in this case) at all frequencies. The result also verifies the effectiveness of the proposed optimization strategy.

Table 1. DNRE and RPE for different methods.

Empty Cell	Methods	DNRE (%)	${RPE}_{0}$ (%)	${RPE}_{2}$ (%)
First floor	Welch	4.95	3.02	13.79
	Traditional multi-taper	6.29	2.37	12.68
	Adaptive multi-taper	3.81	0.98	12.09
Second floor	Welch	4.64	4.77	4.88
	Traditional multi-taper	6.06	3.13	3.97
	Adaptive multi-taper	3.70	2.47	3.38

4.2. The coherence estimate

The accuracy of the coherence estimate depends on the estimated PSD. Even a small fluctuation in the PSD could lead to a significant variance in the coherence estimate. The estimated coherence of the first two floors’ responses is presented in Fig. 6. The adaptive multi-taper method results in a more accurate coherence with a smaller fluctuation over the whole frequency domain than the other two methods. This is because increasing the NoT can also reduce the coherence variance, according to the statistical properties of the coherence of the multi-taper method [27]. In summary, the adaptive multi-taper method exhibits superior performance in balancing the bias and variance of the spectrum and coherence of structural responses.

4.3. Multiple realizations

Averaging multiple independent realizations can reduce the random fluctuation in the estimated spectrum and coherence. To demonstrate the stability of the optimal NoT and the effectiveness of the optimization procedure, the proposed method applies to 100 random realizations independently. According to the proposed adaptive approach, the optimal NoT for each realization is obtained. All 100 sets of the NoT are averaged and then compared with the theoretical one in Fig. 7. The theoretical NoT is calculated based on the theoretical PSD and Eq. (27). The figure shows that the averaged optimal NoT is close to the theoretical counterpart, except near the natural frequencies where the theoretical one dramatically increases to a high value. This is because the first-order and second-order derivatives of the PSD at these frequencies are approximately zero, as shown in Fig. 8. Accordingly, the optimal NoT becomes very large, according to Eq. (27). To avoid this happening, the difference of NoT at consecutive frequencies can be limited to less than one, as defined in Eq. (29). The constrained theoretical NoT is presented in Fig. 9.

The constrained theoretical and averaged optimal NoT are then used to estimate the PSD and coherence of the 100 samples. The averaged PSD and coherence over 100 realizations are calculated and presented in Fig. 10, Fig. 11. The 95 % confidence intervals of the results estimated by the adaptive multi-taper with the optimal NoT are also shown in these figures. Slight differences are observed in the estimated PSD using the theoretical and optimal NoT. The biases can be found between the averaged spectra and theoretical ones at the natural frequencies. This is because the adaptive multi-taper method minimizes the variance on the premise of sufficient resolution, as shown in Eq. (21). The confidence interval of the PSD and coherence estimates is narrow since the variance decreases with the increase of the NoT, according to Eq. (20).

The decibel-scale averaged absolute bias (AAB) and averaged coefficient of variation (ACoV) of the PSD and coherence, and RPE of spectral moments over 100 realizations are calculated and listed in Table 2. The AAB and ACoV are defined as(35)

AAB = \frac{1}{J} \sum_{j = 1}^{J} |10 \log_{10} [\bar{Y} (f_{j})] - 10 \log_{10} [Y (f_{j})]|

(36)

ACoV = \frac{1}{J} \sum_{j = 1}^{J} \frac{\sqrt{\frac{1}{K} \sum_{k = 1}^{K} {\{10 \log_{10} [{\hat{Y}}_{k} (f_{j})] - 10 \log_{10} [\bar{Y} (f_{j})]\}}^{2}}}{10 \log_{10} [\bar{Y} (f_{j})]}

Table 2. AAB, ACoV, and RPE of the estimated results.

Estimate		Adaptive multi-taper
Estimate		Theoretical NoT	Optimal NoT
PSD for first floor	AAB	0.358	0.359
	ACoV	−0.078	−0.088
	${RPE}_{0}$ (%)	4.425	4.816
	${RPE}_{2}$ (%)	0.915	1.954
PSD for second floor	AAB	0.353	0.359
	ACoV	0.027	−0.004
	${RPE}_{0}$ (%)	5.776	5.828
	${RPE}_{2}$ (%)	2.939	3.736
Coherence	AAB	0.075	0.106
Coherence	ACoV	−0.088	−0.095

with(37)

\bar{Y} (f_{j}) = \frac{1}{K} \sum_{k = 1}^{K} {\hat{Y}}_{k} (f_{j})

where

{\hat{Y}}_{k} (f_{j})

denotes the kth realization at frequency

f_{j}

, and K is the number of realizations (here is 100). AAB denotes the discrepancy between the mean of the estimate and the true value, while ACoV quantifies the averaged coefficient of variation. As displayed in Table 2, the AAB, ACoV, and RPE of the results estimated by the adaptive multi-taper method using the constrained theoretical NoT are close to those using the averaged optimal NoT. Given the above results, it can be concluded that the adaptive multi-taper approach can reduce the bias and variance of the PSD and coherence estimates.

5. Case study 2: A wind speed process

In this case, a bivariate stationary wind speed process is simulated and used to validate the adaptive multi-taper method. Its PSD is expressed as [27](38)

S_{w} (f) = \frac{2193.28}{{[1 + 70.8 {(fl / \bar{U})}^{2}]}^{5 / 6}}

where

\bar{U}

= 14.59 m/s and l = 500 m.

The Krenk coherence of the wind speed process is given by [27], [34](39)

C_{w} (f) = [1 - 0.5 c |r_{v}| w (f)] \exp [i r_{u} \frac{2 π f}{\bar{U}} - c |r_{v}| w (f)]

with(40)

w (f) = \sqrt{\frac{4 π^{2} f^{2}}{{\bar{U}}^{2}} + l^{- 2}}

where

r_{u}

= 16.17 m,

r_{v}

= 7.91 m, and c = 0.45.

A single realization of the wind speed process (Fig. 12) is produced from the PSD in Eq. (38) with a duration of 30-min and a sampling frequency of 2 Hz.

The PSD and coherence of the simulated wind speed process are estimated using Welch’s, traditional multi-taper, and adaptive multi-taper methods. Also, the window length and NoT of the Welch’s and traditional multi-taper methods are determined to generate the best results. The window width of the Fourier transform for Welch’s method is 100 s with 50 % overlapping. The first twenty sine tapers are used in the traditional multi-taper method [27]. For the adaptive multi-taper approach, the optimization procedure for determining the NoT stops after 16 iterations. The results are shown in Fig. 13, and it shows that the optimal NoT increases from 9 to 500 approximately.

The PSD and the associated DSEs are compared in Fig. 14, Fig. 15. The DSEs of the adaptive multi-taper method are smaller than those of Welch’s method and the traditional multi-taper method in the entire frequency range. Moreover, as the frequency exceeds 0.01 Hz, the DSEs of the adaptive multi-taper approach are small and stable, indicating that the PSD estimate of this method has a smaller bias and variation.

The coherences estimated from the three methods are shown in Fig. 16. The coherence estimated by the traditional multi-taper method exhibits a more significant variation than the adaptive multi-taper approach. Although the traditional multi-taper method can lead to a smaller coherence variation by including more tapers, the frequency resolution of the PSD estimate will be reduced and insufficient at low frequencies. Instead of using the fixed NoT (twenty tapers in this case), the adaptive method adopts frequency-dependent NoT and provides sufficient resolution at low frequencies and reduces the variance of the coherence estimate at high frequencies. Thus, the proposed approach outperforms other methods in estimating the PSD and coherence of a single realization of the wind speed process.

Similar to Section 4.3, 100 realizations are conducted. The optimal NoT averaged over 100 realizations is shown in Fig. 17, in comparison with the theoretical one. The latter is calculated from Eqs. (27), (38). The averaged optimal NoT is close to the theoretical results at frequencies less than 0.8 Hz. When the frequency exceeds 0.8 Hz, the optimal NoT decreases. This is because we fit a quadratic function to the PSD in an interval of

[f - W, f + W]

for each frequency f to obtain the derivatives. When the frequency is larger than 0.8 Hz, the fitted interval exceeds the frequency range. This limitation is overcome by the symmetric extension of the spectrum, which leads to the inaccurate estimation of the PSD derivatives and then the NoT.

The theoretical and averaged optimal NoT are then used to calculate the PSD and coherence of the wind speed. The averaged PSD and coherence over 100 realizations using the adaptive multi-taper approach are calculated and shown in Fig. 18, Fig. 19, respectively. Using the adaptive multi-taper method, the theoretical and averaged optimal NoT result in similar PSD and coherence estimates. For the PSD estimates at low frequencies, even though the bias can be further reduced using fewer NoTs, the variance may become higher and even unacceptable for an individual sample. The corresponding AAB and ACoV of PSD and coherence, and RPE of spectral moments are listed in Table 3. In general, the averaged optimal and theoretical NoT cause very similar AAB, ACoV and RPE values.

Table 3. AAB, ACoV, and RPE of the estimated results over 100 realizations.

Estimate		Adaptive MT
Estimate		Theoretical NoT	Optimal NoT
PSD for Sample 1	AAB	0.129	0.169
	ACoV	8.610 × 10^-5	1.464 × 10^-4
	${RPE}_{0}$ (%)	1.816	0.002
	${RPE}_{2}$ (%)	3.841	2.721
PSD for Sample 2	AAB	0.121	0.161
	ACoV	0.002	0.015
	${RPE}_{0}$ (%)	1.870	0.094
	${RPE}_{2}$ (%)	3.656	2.553
Coherence	AAB	0.193	0.255
Coherence	ACoV	−0.131	−0.151

6. Case study 3: AR (4) process

The following AR (4) process with two closely spaced peaks is used to demonstrate the generality of the adaptive multi-taper method:(41)

R_{t} = 2.7607 R_{t - 1} - 3.8106 R_{t - 2} + 2.6535 R_{t - 3} - 0.9238 R_{t - 4} + ε_{t}

where

ε_{t}

is the zero-mean Gaussian white process with a unit variance. A single realization of the AR (4) process is generated with a duration of 2000 s and a sampling frequency of 1 Hz. Similar to previous sections, the PSDs are estimated by Welch’s, the traditional multi-taper, and the adaptive multi-taper methods for comparison. The window length of Welch’s method is 400 s with 50 % overlapping. The first ten sine tapers are used in the traditional multi-taper method. The optimal NoT of the adaptive multi-taper approach is shown in Fig. 20.

The results of different methods are compared in Fig. 21. All methods can provide sufficient resolution around the PSD peaks, whereas the estimated PSDs of Welch’s method exhibit an obvious bias at the frequencies between 0.2 Hz and 0.5 Hz. Besides, the PSDs estimated by Welch’s method and the traditional multi-taper method have a larger variance than those by the adaptive multi-taper method.

500 random samples of the AR (4) process are generated, and their corresponding NoTs are calculated and averaged. The averaged NoTs are then used to estimate the PSD of each sample. The estimated PSDs are averaged over 500 realizations and are shown in Fig. 22. Again, the difference between the estimated PSD using the averaged optimal and theoretical PSD is small.

7. Conclusions

An adaptive multi-taper approach is proposed for estimating the PSD and coherence stationary processes. The NoT of the multi-taper method controls the inherent trade-off between the resolution and variance of the PSD. Unlike the traditional multi-taper method, the adaptive multi-taper approach determines the frequency-dependent NoT without manual adjustment. The sine taper is employed in the proposed approach for its simple expression and computational efficiency. An optimization procedure is presented to determine the NoT by minimizing the MSE of the log spectrum. The NoT at each frequency depends on the ratio of the spectrum and its second-order derivative. The low-pass filtering method and constraint strategy are then combined to refine the estimated number affected by the extremely small second-order derivatives. The stopping criterion of the iterative procedure is defined according to the MRD of the NoT between consecutive steps.

The proposed adaptive multi-taper approach is then applied to three numerical examples, the structural response of a four-story building, a wind speed process, and an AR (4) process, to calculate their PSD and coherence. As compared with Welch’s and the traditional multi-taper method, the developed method results in a smaller bias and variance. The robustness of the optimal NoT is also demonstrated through multiple realizations of the stationary processes.

CRediT authorship contribution statement

Yi-Ming Zhang: Conceptualization, Methodology, Formal analysis, Software, Writing – original draft. Zifeng Huang: Writing – review & editing, Supervision. Yong Xia: Writing – review & editing, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge the support from the Ministry of Science and Technology of China through the National Key R&D Program (Project No. 2019YFB1600700) and the PolyU Postdoc Matching Fund Scheme (Project No. 1-W20H).

References

[1]
L. Comerford, I.A. Kougioumtzoglou, M. Beer
Compressive sensing based stochastic process power spectrum estimation subject to missing data
Probab. Eng. Mech., 44 (2016), pp. 66-76
View PDF View article View in Scopus Google Scholar
[2]
Y. Xia, Y. Fujino
Auto-parametric vibration of a cable-stayed-beam structure under random excitation
J. Eng. Mech., 132 (2006), pp. 279-286
View in Scopus Google Scholar
[3]
Y. Song, J. Chen, M. Beer, L. Comerford
Wind speed field simulation via stochastic harmonic function representation based on wavenumber–frequency spectrum
J. Eng. Mech., 145 (2019), p. 04019086
View in Scopus Google Scholar
[4]
M. Behrendt, M. Bittner, L. Comerford, M. Beer, J. Chen
Relaxed power spectrum estimation from multiple data records utilising subjective probabilities
Mech. Syst. Sig. Process., 165 (2022), Article 108346
View PDF View article View in Scopus Google Scholar
[5]
H. Wang, Y.M. Zhang, J.X. Mao
Sparse Gaussian process regression for multi-step ahead forecasting of wind gusts combining numerical weather predictions and on-site measurements
J. Wind Eng. Ind. Aerodyn., 220 (2022), Article 104873
View PDF View article View in Scopus Google Scholar
[6]
K. Gurley, A. Kareem
Applications of wavelet transforms in earthquake, wind and ocean engineering
Eng. Struct., 21 (1999), pp. 149-167
View in Scopus Google Scholar
[7]
P.D. Spanos, M. Beer, J. Red-Horse
Karhunen-Loève expansion of stochastic processes with a modified exponential covariance kernel
J. Eng. Mech., 133 (2007), pp. 773-779
View in Scopus Google Scholar
[8]
Y. Zhang, L. Comerford, I.A. Kougioumtzoglou, M. Beer
Lp-norm minimization for stochastic process power spectrum estimation subject to incomplete data
Mech. Syst. Sig. Process., 101 (2018), pp. 361-376
View PDF View article View in Scopus Google Scholar
[9]
B. Babadi, E.N. Brown
A review of multitaper spectral analysis
IEEE Trans. Biomed. Eng., 61 (2014), pp. 1555-1564
View in Scopus Google Scholar
[10]
P. Welch
The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms
IEEE Trans. Audio Electroacoust., 15 (1967), pp. 70-73
Google Scholar
[11]
D.J. Thomson
Spectrum estimation and harmonic analysis
Proc. IEEE, 70 (1982), pp. 1055-1096
View in Scopus Google Scholar
[12]
P.D. Spanos, G. Malara
Nonlinear random vibrations of beams with fractional derivative elements
J. Eng. Mech., 140 (2014), p. 04014069
View at publisher
Crossref View in Scopus Google Scholar
[13]
A. Jeyaseelan, R. Balaji
Spectral analysis of wave elevation time histories using multi-taper method
Ocean Eng., 105 (2015), pp. 242-246
View PDF View article View in Scopus Google Scholar
[14]
P.D. Spanos, Y. Sun, N. Su
Advantages of filter approaches for the determination of wind-induced response of large-span roof structures
J. Eng. Mech., 143 (2017), p. 04017066
View at publisher
Crossref View in Scopus Google Scholar
[15]
Y.M. Zhang, H. Wang, J.X. Mao, Z.D. Xu, Y.F. Zhang
Probabilistic framework with Bayesian optimization for predicting typhoon-induced dynamic responses of a long-span bridge
J. Struct. Eng., 147 (2021), p. 04020297
Google Scholar
[16]
D.B. Percival, A.T. Walden
Spectral analysis for physical applications
Cambridge University Press (1993)
Google Scholar
[17]
T.P. Bronez
On the performance advantage of multitaper spectral analysis
IEEE Trans. Signal Process., 40 (1992), pp. 2941-2946
View in Scopus Google Scholar
[18]
A.T. Walden
A unified view of multitaper multivariate spectral estimation
Biometrika, 87 (2000), pp. 767-788
View at publisher
Crossref View in Scopus Google Scholar
[19]
G. Prieto, R. Parker, D. Thomson, F. Vernon, R. Graham
Reducing the bias of multitaper spectrum estimates
Geophys. J. Int., 171 (2007), pp. 1269-1281
View in Scopus Google Scholar
[20]
T. Kinnunen, R. Saeidi, F. Sedlák, K.A. Lee, J. Sandberg, M. Hansson-Sandsten, H. Li
Low-variance multitaper MFCC features: a case study in robust speaker verification
IEEE Trans. Audio Speech Lang. Process., 20 (2012), pp. 1990-2001
View in Scopus Google Scholar
[21]
L. Faes, G.D. Pinna, A. Porta, R. Maestri, G. Nollo
Surrogate data analysis for assessing the significance of the coherence function
IEEE Trans. Biomed. Eng., 51 (2004), pp. 1156-1166
View in Scopus Google Scholar
[22]
Z. Midjiyawa, E. Cheynet, J. Reuder, H. Ágústsson, T. Kvamsdal
Potential and challenges of wind measurements using met-masts in complex topography for bridge design: Part II–Spectral flow characteristics
J. Wind Eng. Ind. Aerodyn., 211 (2021), p. 104585
View PDF View article View in Scopus Google Scholar
[23]
K.S. Riedel, A. Sidorenko
Minimum bias multiple taper spectral estimation
IEEE Trans. Signal Process., 43 (1995), pp. 188-195
View in Scopus Google Scholar
[24]
K.S. Riedel
Adaptive smoothing of the log-spectrum with multiple tapering
IEEE Trans. Signal Process., 44 (1996), pp. 1794-1800
View in Scopus Google Scholar
[25]
K.Q. Lepage, D.J. Thomson
Reduced mean-square error quadratic inverse spectrum estimator
IEEE Trans. Signal Process., 62 (2014), pp. 2958-2972
View in Scopus Google Scholar
[26]
C.L. Haley, M. Anitescu
Optimal bandwidth for multitaper spectrum estimation
IEEE Signal Process Lett., 24 (2017), pp. 1696-1700
View in Scopus Google Scholar
[27]
Z.F. Huang, Y.L. Xu
A multi-taper S-transform method for spectral estimation of stationary processes
IEEE Trans. Signal Process., 69 (2021), pp. 1452-1467
View at publisher
Crossref View in Scopus Google Scholar
[28]
Z.F. Huang, Y.L. Xu, T.Y. Tao
Multi-taper S-transform method for evolutionary spectrum estimation
Mech. Syst. Sig. Process., 168 (2022), Article 108667
View PDF View article View in Scopus Google Scholar
[29]
L.D. Abreu, J.L. Romero
MSE estimates for multitaper spectral estimation and off-grid compressive sensing
IEEE Trans. Inf. Theory, 63 (2017), pp. 7770-7776
View in Scopus Google Scholar
[30]
A.J. Barbour, R.L. Parker
Psd: adaptive, sine multitaper power spectral density estimation for R
Comput. Geosci., 63 (2014), pp. 1-8
View PDF View article View in Scopus Google Scholar
[31]
S. Tang, P.F. Craigmile, Y. Zhu
Spectral estimation using multitaper whittle methods with a Lasso penalty
IEEE Trans. Signal Process., 67 (2019), pp. 4992-5003
View at publisher
Crossref Google Scholar
[32]
Y. Zhang, L. Comerford, I.A. Kougioumtzoglou, E. Patelli, M. Beer
Uncertainty quantification of power spectrum and spectral moments estimates subject to missing data
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, 3 (2017), p. 04017020
View at publisher
Crossref View in Scopus Google Scholar
[33]
E.H. Vanmarcke
Properties of spectral moments with applications to random vibration
Journal of the Engineering Mechanics Division, 98 (1972), pp. 425-446
View at publisherCrossref View in Scopus Google Scholar
[34]
S.O. Hansen, S. Krenk
Dynamic along-wind response of simple structures
J. Wind Eng. Ind. Aerodyn., 82 (1999), pp. 147-171
View PDF View article View in Scopus Google Scholar

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View Abstract

[1] [1]
L. Comerford, I.A. Kougioumtzoglou, M. Beer
Compressive sensing based stochastic process power spectrum estimation subject to missing data
Probab. Eng. Mech., 44 (2016), pp. 66-76
View PDF View article View in Scopus Google Scholar

[2] [2]
Y. Xia, Y. Fujino
Auto-parametric vibration of a cable-stayed-beam structure under random excitation
J. Eng. Mech., 132 (2006), pp. 279-286
View in Scopus Google Scholar

[3] [3]
Y. Song, J. Chen, M. Beer, L. Comerford
Wind speed field simulation via stochastic harmonic function representation based on wavenumber–frequency spectrum
J. Eng. Mech., 145 (2019), p. 04019086
View in Scopus Google Scholar

[4] [4]
M. Behrendt, M. Bittner, L. Comerford, M. Beer, J. Chen
Relaxed power spectrum estimation from multiple data records utilising subjective probabilities
Mech. Syst. Sig. Process., 165 (2022), Article 108346
View PDF View article View in Scopus Google Scholar

[5] [5]
H. Wang, Y.M. Zhang, J.X. Mao
Sparse Gaussian process regression for multi-step ahead forecasting of wind gusts combining numerical weather predictions and on-site measurements
J. Wind Eng. Ind. Aerodyn., 220 (2022), Article 104873
View PDF View article View in Scopus Google Scholar

[6] [6]
K. Gurley, A. Kareem
Applications of wavelet transforms in earthquake, wind and ocean engineering
Eng. Struct., 21 (1999), pp. 149-167
View in Scopus Google Scholar

[7] [7]
P.D. Spanos, M. Beer, J. Red-Horse
Karhunen-Loève expansion of stochastic processes with a modified exponential covariance kernel
J. Eng. Mech., 133 (2007), pp. 773-779
View in Scopus Google Scholar

[8] [8]
Y. Zhang, L. Comerford, I.A. Kougioumtzoglou, M. Beer
Lp-norm minimization for stochastic process power spectrum estimation subject to incomplete data
Mech. Syst. Sig. Process., 101 (2018), pp. 361-376
View PDF View article View in Scopus Google Scholar

[9] [9]
B. Babadi, E.N. Brown
A review of multitaper spectral analysis
IEEE Trans. Biomed. Eng., 61 (2014), pp. 1555-1564
View in Scopus Google Scholar

[10] [10]
P. Welch
The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms
IEEE Trans. Audio Electroacoust., 15 (1967), pp. 70-73
Google Scholar

[11] [11]
D.J. Thomson
Spectrum estimation and harmonic analysis
Proc. IEEE, 70 (1982), pp. 1055-1096
View in Scopus Google Scholar

[12] [12]
P.D. Spanos, G. Malara
Nonlinear random vibrations of beams with fractional derivative elements
J. Eng. Mech., 140 (2014), p. 04014069
View at publisher
Crossref View in Scopus Google Scholar

[13] [13]
A. Jeyaseelan, R. Balaji
Spectral analysis of wave elevation time histories using multi-taper method
Ocean Eng., 105 (2015), pp. 242-246
View PDF View article View in Scopus Google Scholar

[14] [14]
P.D. Spanos, Y. Sun, N. Su
Advantages of filter approaches for the determination of wind-induced response of large-span roof structures
J. Eng. Mech., 143 (2017), p. 04017066
View at publisher
Crossref View in Scopus Google Scholar

[15] [15]
Y.M. Zhang, H. Wang, J.X. Mao, Z.D. Xu, Y.F. Zhang
Probabilistic framework with Bayesian optimization for predicting typhoon-induced dynamic responses of a long-span bridge
J. Struct. Eng., 147 (2021), p. 04020297
Google Scholar

[16] [16]
D.B. Percival, A.T. Walden
Spectral analysis for physical applications
Cambridge University Press (1993)
Google Scholar

[17] [17]
T.P. Bronez
On the performance advantage of multitaper spectral analysis
IEEE Trans. Signal Process., 40 (1992), pp. 2941-2946
View in Scopus Google Scholar

[18] [18]
A.T. Walden
A unified view of multitaper multivariate spectral estimation
Biometrika, 87 (2000), pp. 767-788
View at publisher
Crossref View in Scopus Google Scholar

[19] [19]
G. Prieto, R. Parker, D. Thomson, F. Vernon, R. Graham
Reducing the bias of multitaper spectrum estimates
Geophys. J. Int., 171 (2007), pp. 1269-1281
View in Scopus Google Scholar

[20] [20]
T. Kinnunen, R. Saeidi, F. Sedlák, K.A. Lee, J. Sandberg, M. Hansson-Sandsten, H. Li
Low-variance multitaper MFCC features: a case study in robust speaker verification
IEEE Trans. Audio Speech Lang. Process., 20 (2012), pp. 1990-2001
View in Scopus Google Scholar

[21] [21]
L. Faes, G.D. Pinna, A. Porta, R. Maestri, G. Nollo
Surrogate data analysis for assessing the significance of the coherence function
IEEE Trans. Biomed. Eng., 51 (2004), pp. 1156-1166
View in Scopus Google Scholar

[22] [22]
Z. Midjiyawa, E. Cheynet, J. Reuder, H. Ágústsson, T. Kvamsdal
Potential and challenges of wind measurements using met-masts in complex topography for bridge design: Part II–Spectral flow characteristics
J. Wind Eng. Ind. Aerodyn., 211 (2021), p. 104585
View PDF View article View in Scopus Google Scholar

[23] [23]
K.S. Riedel, A. Sidorenko
Minimum bias multiple taper spectral estimation
IEEE Trans. Signal Process., 43 (1995), pp. 188-195
View in Scopus Google Scholar

[24] [24]
K.S. Riedel
Adaptive smoothing of the log-spectrum with multiple tapering
IEEE Trans. Signal Process., 44 (1996), pp. 1794-1800
View in Scopus Google Scholar

[25] [25]
K.Q. Lepage, D.J. Thomson
Reduced mean-square error quadratic inverse spectrum estimator
IEEE Trans. Signal Process., 62 (2014), pp. 2958-2972
View in Scopus Google Scholar

[26] [26]
C.L. Haley, M. Anitescu
Optimal bandwidth for multitaper spectrum estimation
IEEE Signal Process Lett., 24 (2017), pp. 1696-1700
View in Scopus Google Scholar

[27] [27]
Z.F. Huang, Y.L. Xu
A multi-taper S-transform method for spectral estimation of stationary processes
IEEE Trans. Signal Process., 69 (2021), pp. 1452-1467
View at publisher
Crossref View in Scopus Google Scholar

[28] [28]
Z.F. Huang, Y.L. Xu, T.Y. Tao
Multi-taper S-transform method for evolutionary spectrum estimation
Mech. Syst. Sig. Process., 168 (2022), Article 108667
View PDF View article View in Scopus Google Scholar

[29] [29]
L.D. Abreu, J.L. Romero
MSE estimates for multitaper spectral estimation and off-grid compressive sensing
IEEE Trans. Inf. Theory, 63 (2017), pp. 7770-7776
View in Scopus Google Scholar

[30] [30]
A.J. Barbour, R.L. Parker
Psd: adaptive, sine multitaper power spectral density estimation for R
Comput. Geosci., 63 (2014), pp. 1-8
View PDF View article View in Scopus Google Scholar

[31] [31]
S. Tang, P.F. Craigmile, Y. Zhu
Spectral estimation using multitaper whittle methods with a Lasso penalty
IEEE Trans. Signal Process., 67 (2019), pp. 4992-5003
View at publisher
Crossref Google Scholar

[32] [32]
Y. Zhang, L. Comerford, I.A. Kougioumtzoglou, E. Patelli, M. Beer
Uncertainty quantification of power spectrum and spectral moments estimates subject to missing data
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, 3 (2017), p. 04017020
View at publisher
Crossref View in Scopus Google Scholar

[33] [33]
E.H. Vanmarcke
Properties of spectral moments with applications to random vibration
Journal of the Engineering Mechanics Division, 98 (1972), pp. 425-446
View at publisherCrossref View in Scopus Google Scholar

[34] [34]
S.O. Hansen, S. Krenk
Dynamic along-wind response of simple structures
J. Wind Eng. Ind. Aerodyn., 82 (1999), pp. 147-171
View PDF View article View in Scopus Google Scholar

大纲

被引用次数 (14)

图表 (22)

Tables (3)

机械系统与信号处理

高亮

摘要

关键词

1. 引言

2. 静态过程的多窗谱估计

2.1. 多变量平稳过程

2.2. Multi-taper spectral estimation

3. Adaptive multi-taper approach

3.1. Formula of the adaptive multi-taper method

3.2. Bias and variance of the adaptive multi-taper method

3.3. The optimal NoT

4. Case study 1: A 4-DOF model

4.1. The spectrum estimate

4.2. The coherence estimate

4.3. Multiple realizations

5. Case study 2: A wind speed process

6. Case study 3: AR (4) process

7. Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

References

Cited by (14)

Dynamic performance of ultra-long stay cable in small-scale extreme winds

An improved multi-taper S-transform method to estimate evolutionary spectrum and time-varying coherence of nonstationary processes

Fusing damage-sensitive features and domain adaptation towards robust damage classification in real buildings

Source-Space Brain Functional Connectivity Features in Electroencephalogram-Based Driver Fatigue Classification

Removal of Artifacts in EEG Signals Using Sign Based LMS Adaptive Filtering Techniques

Application of Spectral Entropy in Haul Truck Joint Damage Detection

用于平稳过程的自适应多窗谱估计

高亮

摘要

关键词

1. 引言

2. 静态过程的多窗谱估计

2.1. 多变量平稳过程

2.2. Multi-taper spectral estimation

3. Adaptive multi-taper approach

3.1. Formula of the adaptive multi-taper method

3.2. Bias and variance of the adaptive multi-taper method

3.3. The optimal NoT

4. Case study 1: A 4-DOF model

4.1. The spectrum estimate

4.2. The coherence estimate

4.3. Multiple realizations

5. Case study 2: A wind speed process

6. Case study 3: AR (4) process

7. Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

References