#
Mitigating Wind Induced Noise in Outdoor Microphone Signals Using a Singular Spectral Subspace Method^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Rationale

## 3. The Method

#### 3.1. Overview

#### 3.2. The SSA Theory

#### 3.3. Mathematical Formulation of the SSA Method

#### 3.3.1. Embedding Process

- Step One: Vector RepresentationIn practice, the SSA is nonparametric spectral method based on embedding a given time series {X(t): t = 1, …, N
_{t}} in a vector space. The vector X, whose entries ${N}_{t}$ are the data points of a time series, can clearly define and describe this time series at regular intervals [27,30,40]. If we consider a real-valued time series X(t) = (x_{1}, x_{2}, …, x_{Nt}) of length ${N}_{t}$ and x_{1}, x_{2}, …, x_{Nt}data points, therefore, the given time series can simply be represented as a column vector as shown in Equation (1):$${X}^{T}=({x}_{1},{x}_{2}\dots \dots .{x}_{{N}_{t}}),$$This column vector shows the original time series at zero lag (i.e., when there is no delay, $k=0$). At a given window length m, the lag k can therefore be expressed in the range from 0 to m − 1 in a 1 lag shifted version as in Equation (2):$$k=0,1,\dots \dots ,m-1,$$Hence, the window length m should be suitably identified in order to obtain the lag k which needed to construct a new matrix according to delay coordinates.In SSA jargon, this matrix is called “embedded time series” or trajectory matrix and denoted by Y. The window length is also called embedding dimension and it represents the number of time-series elements in each snapshot [27,30,33,40].The whole procedure of the SSA method depends upon the best selection of this parameter as well as the grouping criteria. These two key aspects are very important to develop the concept of reconstructing noise free series from a noisy series. Different rank-one matrices obtained from the SVD can be selected and grouped in order to be processed separately. If the groups are properly partitioned, they will reflect different components of the original time record [27,40]. - Step Two: Trajectory Matrix ConstructionThe trajectory matrix contains the original time series in the first column and a lag 1 shifted version of that time series for each of the next columns. We can obviously understand from Equation (2) that the column vector shown in Equation (1) is $X(t)$ when $k=0$. As explained in [31], according to delay coordinates, we will obtain a total number of column vectors equals m. Importantly, these vectors are similar in size to the first column vector but with a 1 lag shift at $k=1$, $k=2$, up to $k=m-1$. This assumption is given when the last rows are supplemented by 0s based on the delay as a first method.In the second method, arranging the snapshots of any given time series as row vectors can lead to construct the trajectory matrix when the last rows are not supplemented by 0 s [27,40]. To simplify, we assume for example, an embedding dimension $m=4$, therefore according to Equation (2), only lags of $k=0$, 1, 2 and 3 will be considered. However, in this case, a trajectory matrix Y of size N × m will be constructed as in Equation (4). For more clarification, we consider a representation of any given time series $X(t)$ according to our assumption $m=4$ as depicted in Figure 4.The coordinates of the phase space can be defined by using lagged copies of a single time series [33]. The trajectory matrix corresponds to a sliding window of size m that moves along the time series $X(t)$ [12,40]. Since the sliding window has an overlap equals $m-1$ as shown in Figure 1 and values of k according to Equation (2), therefore the number of rows of Y which can be filled with the values of $X(t)$ is denoted by N and can be calculated by Equation (3):$$N={N}_{t}-\left(m-1\right)={N}_{t}-m+1,$$As explained in [33], the snapshots of a given record when considering only number of rows of Y which can be filled with the values of $X(t)$ according to Equation (3) are vectors and can be seen as ${V}_{1}^{T}=\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right)$, ${V}_{2}^{T}=\left({x}_{2},{x}_{3},\dots ,{x}_{m+1}\right)$ up to ${V}_{N}^{T}=\left({x}_{N},{x}_{N+1},\dots ,{x}_{{N}_{t}}\right)$. Hence, the trajectory matrix can be constructed by arranging the snapshots as row vectors and only N
_{t}− m + 1 rows can be filled with values of $X(t)$ as in Equation (4):$$Y=1/\sqrt{N}\left[\begin{array}{c}{V}_{1}^{T}\\ {V}_{2}^{T}\\ {V}_{3}^{T}\\ \vdots \\ {V}_{N}^{T}\end{array}\right]=1/\sqrt{N}\left[\begin{array}{ccccc}{x}_{1}& {x}_{2}& {x}_{3}& \dots & {x}_{m}\\ {x}_{2}& {x}_{3}& {x}_{4}& \dots & {x}_{m+1}\\ {x}_{3}& {x}_{4}& {x}_{5}& \cdots & {x}_{m+2}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {x}_{N}& {x}_{N+1}& {x}_{N+2}& \cdots & {x}_{{N}_{t}}\end{array}\right],$$The constructed trajectory matrix includes the complete record of patterns that have occurred within a window of length m. To generalise, we assume that $X(t)$ is a given time series and t = 1, 2, 3, …, N_{t}, the augmented or trajectory matrix is constructed as in Equation (5):$$Y=\left[\begin{array}{ccc}{X}_{1}:& \dots & :{X}_{m}\end{array}\right]={({x}_{ij})}_{i,j=1}^{N,m},$$The arrangement of entries ${x}_{ij}$ of the trajectory matrix depends on the lag, considering that the trajectory matrix has dimensions N by m. The trajectory matrix and its transpose ${Y}^{T}$ are linear maps between the spaces ${R}^{m}$ and ${R}^{N}$ [40]. Two important properties of the trajectory matrix are stated in [49], the first is that both the rows and columns of Y are subseries of the original series. The second is that Y has equal elements on anti-diagonals; makes it a Hankel matrix (i.e., all the elements along the diagonal i + j = const are equal).

#### 3.3.2. Covariance Matrix Construction

_{ij}of this matrix depend only on the lag $\left|i-j\right|$ as in Equation (6) [46].

_{ij}when $\left|i-j\right|=0$ for $i=j$, are the entries across the main diagonal (i.e., their values are typically closed to 1).

#### 3.3.3. Computing the Eigenvalues and Eigenvectors

- Finding the EigenmodesFinding the eigenvalues and eigenvectors or the so-called eigenmodes is based on the fundamental question of eigenvector decomposition. In general, this question is for what values is the matrix $A-\lambda I$ singular? Such question of singularity regarding matrices can be answered with determinants [33,40]. Using determinants, however, the fundamental question which just has been asked above can be reduced to; for what values of $\lambda $ is the determinant of the matrix $A-\lambda I$ equals to zero? or as in Equation (8):$$\mathrm{det}\left(A-\lambda I\right)=0,$$This is called the characteristic equation for the matrix A where ${\lambda}^{\prime}s$ that make the matrix $A-\lambda I$ singular are called eigenvalues [33]. The eigenvalues of this matrix can therefore be found by solving the characteristic equation.For each of these special values there is a corresponding set of vectors called the eigenvectors of A and should satisfy Equation (9):$$\left(A-\lambda I\right)X=0,$$Equation (9) can be written as $AX=\lambda X$. Vector $X$ represents a set of eigenvectors ${X}_{n}$ correspond to ${\lambda}^{\prime}s$.When representing the eigenvectors geometrically, they can be considered as the axes of a new coordinate system. Hence, any scalar multiple of these eigenvectors is also an eigenvector of matrix A [27]. To simplify, all eigenvectors of the form $C{X}_{n}$ ($C$ is scalar) will form an eigen-subspace spanned by ${X}_{n}$, which means that the eigen-subspace is one-dimensional and is spanned by ${X}_{n}$. In this case, only the scale of the eigenvectors is changing while their direction remains unchanged. The process of decomposition can be simplified as matrix A is usually symmetric with real coefficients. The eigenvalues and eigenvectors can be seen as a way to express the variability of a set of data [50].
- Diagonal Form of the Covariance MatrixThe covariance matrix C is an n by n symmetric matrix with n linearly independent eigenvectors $({e}_{i})$, $i=1,\dots ,n$; hence a matrix E, whose columns are the eigenvectors of C, can be constructed and satisfies Equation (10):$${E}^{-1}CE=\mathsf{\Lambda},$$The product on the left-hand side of Equation (10) is called the diagonal form of C and therefore Λ is a diagonal matrix whose nonnegative entries are the eigenvalues of C.The eigenvectors of C should be linearly independent in order to make C diagonalisable in this way. Also matrix E is not unique because the eigenvectors can always be multiplied by a constant scalar preserving their nature as eigenvectors [33].
- Spectral DecompositionWe assume C is a real symmetric matrix where $C={C}^{T}$. Now, in this case, every eigenvalue of C is also real and if all eigenvalues are distinct, then their corresponding eigenvectors are orthogonal. Our real, symmetric matrix C can therefore be diagonalised by an orthogonal matrix E whose columns are the orthonormal eigenvectors of C [44]. It is important now to state a principal theorem when there is a diagonalizable matrix E whose columns are orthonormal of a real and symmetric matrix C.Since ${E}^{T}E=I$ and ${E}^{T}={E}^{-1}$, then Equation (10) can be rewritten as:$$C=E\mathsf{\Lambda}{E}^{T},$$Matrices Λ and E can be seen as:$$\mathsf{\Lambda}=\left[\begin{array}{cccc}{\lambda}_{1}& 0& \dots & 0\\ 0& {\lambda}_{2}& \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0& 0& \cdots & {\lambda}_{m}\end{array}\right],E=\left[\begin{array}{cccc}{e}_{1}^{1}& {e}_{1}^{2}& \dots & {e}_{1}^{m}\\ {e}_{2}^{1}& {e}_{2}^{2}& \cdots & {e}_{2}^{m}\\ \vdots & \vdots & \ddots & \vdots \\ {e}_{m}^{1}& {e}_{m}^{2}& \cdots & {e}_{m}^{m}\end{array}\right],$$Matrix Λ is symmetric with entries ${\lambda}_{i}$ along the leading diagonal for $i=1,\dots \dots ,m$, however ${E}^{k}$ is the corresponding normalised column eigenvector for $k=1,\dots \dots ,m$ as well. The eigenvectors matrix consists of a set of column vectors with entries ${e}_{j}^{k}$ that represent the j
^{th}component of the k^{th}eigenvector. Once we conserve our matrices square, then we have $j=k=1,\dots \dots ,m$. The diagonal matrix Λ consists of ordered values $0\le {\lambda}_{1}\le {\lambda}_{2}\le \dots \le {\lambda}_{m}$ [21,22,27]. - The Singular SpectrumThe square roots of the eigenvalues of matrix C are called the singular values of the trajectory matrix [27,28,33]. These ordered singular values are referred to collectively as the singular spectrum. From the (SVD), the trajectory matrix can be written as:$$Y=US{E}^{T}$$The singular spectrum of Y consists of the square roots of the eigenvalues of C which called the singular values of Y with the singular vectors being identical to the eigenvectors that given in matrix E [33]. The decomposition of matrix C can also be performed by substituting Equation (13) in the form $C={Y}^{T}Y$ to yield $C={(US{E}^{T})}^{T}(US{E}^{T})$$=ES{U}^{T}US{E}^{T}$. Since ${U}^{T}U=I$, we find:$$C=E{S}^{2}{E}^{T}$$For the decomposition being unique, it follows that ${S}^{2}=\mathsf{\Lambda}$. The right singular vectors of Y are the eigenvectors of C and the left singular vectors of Y are the eigenvectors of the matrix $Y{Y}^{T}$ [27,40]. Importantly, the number of eigenvalues is equal to the window length and in turn the number of the associated eigenvectors that matrix E contains [33].

#### 3.3.4. Principle Components

^{th}component of the k

^{th}eigenvector in the matrix E [21,22].

#### 3.3.5. Reconstruction of the Time Series

## 4. The SSA Algorithm

#### 4.1. Description of the SSA Algorithm

#### 4.1.1. Signal Decomposition

#### 4.1.2. Computing the Diagonal Matrix C

#### 4.1.3. Performing SVD of Matrix C

#### 4.1.4. Computing the Principle Components and Performing Grouping

#### 4.1.5. Reconstruction the One-Dimensional Series

#### 4.2. Parameters of the SSA Algorithm

#### 4.3. Grouping and Separability

## 5. Experimental Procedure

#### 5.1. Description of Experiments

#### 5.2. Dataset

#### 5.3. Window Length Optimisation

## 6. Results

## 7. Discussion and Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 4.**Representation of a given time series when considering for example a window length m = 4 to construct the trajectory matrix.

**Figure 8.**The leading four principle components used for grouping and reconstruction and the record of birds’ chirps with additive wind noise: (

**a**) Reconstruction with the first two leading pairs of the principle components; (

**b**) Reconstruction with the third and fourth principle components pairs.

**Figure 10.**The de-noised record was separated by retaining only the leading two pairs of the eigenvalues.

**Figure 11.**A combination of our signals used in the experiments: (

**a**) Noisy record (birds’ chirps and wind noise); (

**b**) Clean record of birds’ chirps.

**Figure 13.**Matrix of w-correlations of the selected 50 eigenvectors of the SVD of the trajectory matrix.

**Table 1.**SNR measure applied for evaluating the SSA method for wind noise reduction Measurement cases and difference.

The Objective Measure for Evaluating the Method | Before | After | Difference |
---|---|---|---|

SNR in dB | 0 dB | 9.47 dB | 9.47 dB |

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**MDPI and ACS Style**

Eldwaik, O.; F. Li, F. Mitigating Wind Induced Noise in Outdoor Microphone Signals Using a Singular Spectral Subspace Method. *Technologies* **2018**, *6*, 19.
https://doi.org/10.3390/technologies6010019

**AMA Style**

Eldwaik O, F. Li F. Mitigating Wind Induced Noise in Outdoor Microphone Signals Using a Singular Spectral Subspace Method. *Technologies*. 2018; 6(1):19.
https://doi.org/10.3390/technologies6010019

**Chicago/Turabian Style**

Eldwaik, Omar, and Francis F. Li. 2018. "Mitigating Wind Induced Noise in Outdoor Microphone Signals Using a Singular Spectral Subspace Method" *Technologies* 6, no. 1: 19.
https://doi.org/10.3390/technologies6010019