**Apendix A**

**Theorem 1 proof:** 

$$\begin{aligned} &\cdots \quad J\begin{pmatrix} A & 0 \\ 0 & -A^\* \end{pmatrix} \mathbf{J}^{-1} \\ &= \begin{pmatrix} 0 & I\_n \\ -I\_n & 0 \end{pmatrix} \begin{pmatrix} A & 0 \\ 0 & -A^\* \end{pmatrix} \begin{pmatrix} 0 & I\_n \\ -I\_n & 0 \end{pmatrix}^{-1} \\ &= \begin{pmatrix} -A^\* & 0 \\ 0 & A \end{pmatrix} \\ &= -\begin{pmatrix} A & 0 \\ 0 & -A^\* \end{pmatrix}^\* \end{aligned} \tag{81}$$

$$\text{1.: According Definition 2.} \begin{pmatrix} A & 0 \\ 0 & -A^\* \end{pmatrix} \text{ is a Hamiltonian matrix.} \begin{pmatrix} \Delta \end{pmatrix}$$

#### **Theorem 2 proof:**

Computational Intelligence in Electromyography Analysis – 166 A Perspective on Current Applications and Future Challenges

experiment data.

bionics of robot limb motion.

**Apendix A** 

**Theorem 1 proof:** 

The surrogate data method and VWK model mehtod are used to detect the surface EMG signal for arm movement and muscle fatigue. The results show that the surface EMG signal has deterministic nonlinear components. Moreover, our algorithm of surrogate data based on the null hypothesis 3 is proved that can completely satisfy the requirment of the null hypothesis 3. The VWK method with surrogate data can illuminate that not only the action

Chaotic analysis techniques are reviewed and applied to investigate the surface EMG signals. The results show that the surface EMG signals have high-dimension chaotic dynamics by using correlation dimension and largest Lyapunov expoent techniques. For the estimation of embedding dimension, symplectic principal component analysis method is introduced and discussed. In comparison with correlation dimension algorithm and SVD analysis, symplectic geometry analysis is both very simple and reliable. The results show that symplectic geometry method is useful for determining of the system attractor from the

The fractal theory is applied to study the fractal feature of the action surface EMG signal collected from forearm of normal person. The results show that he action surface EMG signal possesses the self-affine fractal characteristic. So, it is difficult to describe the surface EMG signals by using a single fractal dimension. The multifractal dimensions are used to analyze the action surface EMG signals during the arm movements. The results indicate that the surface EMG signals are non-uniform fractal structure signals. The multifractal dimension values can be used to identify the surface EMG signals for different movements. For the nonlinear characteristics of the surface EMG signals, chaos and fractal theories will play the leading role in the nonlinear study of the surface EMG signals. The related methods need to be further researched and developed although these techniques have been applied to analyze the surface EMG signals. It provides a new way for the study of the quantitative analysis of physiology and pathology, sports medicine, clinical medical diagnostics and

> \* \*

  0

*A*

 − 

0

−

 

*J A*

0

*A*

0

*A*

 

0

*n*

*I*

\*

*A*

 − 

0

*I*

*n*

1 \*

−

0

1

(81)

−

0

*n*

*I*

 

\*

*A*

0

*A J*

 

0

*A*

 <sup>−</sup> <sup>=</sup> <sup>−</sup>

 <sup>−</sup> <sup>=</sup>

 <sup>−</sup> <sup>=</sup>

0

0

*I*

*n*

but also fatigue surface EMG signals contain nonlinear dynamic properties.

Let *S* as a symplectic transform matrix, *M* as a Hamilton matrix. Then −<sup>1</sup> *S* is also symplectic matrix. According Definition 1 and 2, there is

$$\begin{aligned} &f\left(\text{SMS}^{-1}\right)f^{-1} \\ = &f\text{SJ}^{-1}f\text{MJ}^{-1}f\text{S}^{-1}f^{-1} \\ = &S^{-\circ}\left(-\text{M}^{\circ}\right)\text{S}^{\circ} \\ = &-\left(\text{S}\text{MS}^{\circ -1}\right)^{\circ} \end{aligned} \tag{82}$$

<sup>−</sup><sup>1</sup> ∴ *SMS* is also a Hamilton matrix.

$$\dots \text{SMS}^{-1} \sim M \tag{83}$$

So Hamilton matrix *M* keeps unchanged at symplectic similar transform. 

#### **Theorem 5 proof:**

Let *S*1, *S*2,…, *S*n as symplectic matrix, respectively. According Definition 1, there are

$$\begin{aligned} J S\_1 J^{-1} &= S\_1^{\rightarrow} \\ J S\_2 J^{-1} &= S\_2^{\rightarrow} \\ \cdots \\ J S\_n J^{-1} &= S\_n^{\rightarrow} \end{aligned} \tag{84}$$

$$\begin{aligned} J \{ S\_1 S\_2 \cdots S\_n \} J^{-1} \\ &= J S\_1 J^{-1} J S\_2 J^{-1} J \cdots J^{-1} J S\_n J^{-1} \\ &= S\_1^{\rightarrow} \, ^0 S\_2 \stackrel{\rightarrow}{\cdots} \cdots \, S\_n \stackrel{\rightarrow}{\to} \\ &= \{ S\_1 S\_2 \cdots S\_n \stackrel{\rightarrow}{\to} \cdots \stackrel{\rightarrow}{\to} S\_n \stackrel{\rightarrow}{\to} \cdots \end{aligned} \tag{85}$$

So the product of sympletcic matrixes is also a symplectic matrix.

#### **Theorem 6 proof:**

In order to prove that *H* matrix is symplectic matrix, we only need to prove *H JH* = *J* \* .

$$\begin{aligned} \prescript{}{}{H}^\* J H &= \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix}^\* J \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} \\ &= \begin{pmatrix} 0 & P^\* P \\ -P^\* P & 0 \end{pmatrix} \end{aligned} \tag{86}$$

Computational Intelligence in Electromyography Analysis – 168 A Perspective on Current Applications and Future Challenges

$$\begin{aligned} \text{\textbullet: } P &= I\_n - \frac{2\sigma\varpi^\*}{\varpi^\*\varpi} \\ &\quad \therefore \quad P^\* = P \\\ P^\* P &= P^2 \\ &= \left( I\_n - \frac{2\sigma\varpi^\*}{\varpi^\*\varpi} \right) \left( I\_n - \frac{2\sigma\varpi^\*}{\varpi^\*\varpi} \right) \\ &= I\_n - \frac{4\sigma\varpi^\*}{\varpi^\*\varpi} + \frac{4\sigma\left( (\varpi^\*\varpi)\varpi^\* \right)}{(\varpi^\*\varpi)(\varpi^\*\varpi)} \\ &= I\_n \end{aligned} \tag{87}$$

where (0, ,0; , , ) 0 *<sup>T</sup>* ϖ ωω*k n* = ≠ .

Plugging Eq.(87) into Eq.(86), we have:

$$H^\star J H = J \tag{88}$$

Nonlinear Analysis of Surface EMG Signals 169

(95)

(97)

(99)

(101)

(98)

(2)

(2) = 0, ,, (100)

*e* (94)

 

22

*A a A*

*<sup>n</sup> S* = *a a a* = *a a a* (96)

(1) 11 12

2 1

 = 

≠ , otherwise this column will be skipped and the next column will be

*<sup>i</sup>* . Set first column vector of *A*:

ϖ

(1) 1

1, 0, , 0 1

*T*

(1) 1

> (2) 2

*n n*

2

=

 

(2) 1

α

(1) 21

*n n*

( ) ( ) <sup>11</sup> <sup>21</sup> <sup>1</sup> (1) 1

(1) , , , , , , *<sup>n</sup> T*

> *<sup>T</sup> H I* 2 ( ) (1) (1) (1) = − ϖ

> > 2 (1)

*S E E n*

= ×

*S E*

α

(2) (2)

*n nn*

 

( )*<sup>T</sup> <sup>n</sup> <sup>S</sup> <sup>a</sup> <sup>a</sup>*(2)

ϖ

*a a*

*a a a a*

<sup>0</sup> *<sup>A</sup>*

2

(2) 22

*<sup>T</sup> H I* 2 ( ) (2) (2) (2) = − ω

0

 

=

σ

Second, the same method is adopted to the second column vector of *A*(2), let

(2) 22

(2) 1 12

( )

(1)

so, after *H*(1) transform, *A* is changed to a matrix with the first column is all zero except the

α

= −

= −

*S*

1

(1)

1

ρ

α

It's easy to testify, elementary reflective array *H* is symmetry matrix (*H H*) *<sup>T</sup>* = , orthogonal

For real symmetrical matrix *A*, Householder matrix *H* can be constructed as follows[73].

= (1)

ρ= *x* −

= , and *H* is elementary reflective array.

 

*A*

1 2

<sup>2</sup> ≠ α

(1) 21 (1) 11

=

21 22 2 11 12 1

*a a a a a a*

*a a a*

*n n nn*

 

matrix (*H H* =1) *<sup>T</sup>* and involution matrix ( 1) <sup>2</sup> *H* = .

Then <sup>1</sup> <sup>2</sup> ω

Notes *A*:

First, suppose 0 (1)

first element is

construct *H*(2) matrix:

where,

α

21 α

select elementary reflective array *H*(1):

where ( )

<sup>1</sup> , namely:

 

ϖ

(1)

1

ρ

(1)

α

1

 

(1)

*H A*

considered until the *i*th column of 0 (1)

∴ *H* is symplectic matrix.

$$\begin{aligned} \boldsymbol{H}^\* \boldsymbol{H} &= \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} \\ &= \begin{pmatrix} P^\*P & 0 \\ 0 & P^\*P \end{pmatrix} \\ &= \boldsymbol{I}\_{2n} \end{aligned} \tag{89}$$

∴ *H* is also unitary matrix.

∴ *H* is symplectic unitary matrix. 

#### **Apendix B**

**Theorem 7** suppose *x* and *y* are two unequal *n* dimension vectors, and <sup>2</sup> <sup>2</sup> *<sup>x</sup>* <sup>=</sup> *<sup>y</sup>* , so there is elementary reflective array *<sup>T</sup> H* = 1− 2ωω , which make *Hx* = *y* , where <sup>2</sup> *x y x y* − − ω= .

It can be easily deduced from theorem 5, for non zero *n* dimension vector *T <sup>n</sup> <sup>x</sup>* (*<sup>x</sup>* , *<sup>x</sup>* , , *<sup>x</sup>* ) <sup>=</sup> <sup>1</sup> <sup>2</sup> , notes <sup>2</sup> α= *x* , there is

$$H\mathbf{x} = \alpha \mathbf{e}\_1 \tag{90}$$

$$H = 1 - 2\varpi \sigma \sigma^T \tag{91}$$

$$e\_1 = (1, 0, \cdots, 0)^T \tag{92}$$

$$
\sigma \varpi = \frac{1}{\rho} \left( x - \alpha e\_1 \right) \tag{93}
$$

Nonlinear Analysis of Surface EMG Signals 169

$$\rho = \left\| x - \alpha e\_1 \right\|\_2 \tag{94}$$

Then <sup>1</sup> <sup>2</sup> ω= , and *H* is elementary reflective array.

It's easy to testify, elementary reflective array *H* is symmetry matrix (*H H*) *<sup>T</sup>* = , orthogonal matrix (*H H* =1) *<sup>T</sup>* and involution matrix ( 1) <sup>2</sup> *H* = .

For real symmetrical matrix *A*, Householder matrix *H* can be constructed as follows[73]. Notes *A*:

$$A = \begin{pmatrix} a\_{11} & a\_{12} & \cdots & a\_{1n} \\ a\_{21} & a\_{22} & \cdots & a\_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ a\_{n1} & a\_{n2} & \cdots & a\_{nn} \end{pmatrix} = \begin{pmatrix} a\_{11} & A\_{12}^{(1)} \\ a\_{21}^{(1)} & A\_{22}^{(1)} \end{pmatrix} \tag{95}$$

First, suppose 0 (1) 21 α ≠ , otherwise this column will be skipped and the next column will be considered until the *i*th column of 0 (1) <sup>2</sup> ≠ α*<sup>i</sup>* . Set first column vector of *A*:

$$S^{(\rm 1)} = \left(a\_{11}^{(\rm 1)}, a\_{21}^{(\rm 1)}, \dots, a\_{n1}^{(\rm 1)}\right)^{\rm T} = \left(a\_{11}, a\_{21}, \dots, a\_{n1}\right) \tag{96}$$

select elementary reflective array *H*(1):

$$H^{(\text{l})} = I - 2\varpi^{(\text{l})} (\varpi^{(\text{l})})^T \tag{97}$$

$$\begin{array}{ll} \text{where} & \begin{cases} \boldsymbol{\alpha}\_{1} = \left\| \boldsymbol{S}^{(1)} \right\|\_{2} \\ \boldsymbol{E}^{(1)} = \left( \mathbf{l}, \mathbf{0}, \cdots, \mathbf{0} \right)^{T} & n \times 1 \\ \boldsymbol{\mathcal{P}}\_{1} = \left\| \boldsymbol{S}^{(1)} - \boldsymbol{\alpha}\_{1} \boldsymbol{E}^{(1)} \right\| \\ \boldsymbol{\mathcal{O}}^{(1)} = \frac{1}{\boldsymbol{\mathcal{P}}\_{1}} \left( \boldsymbol{S}^{(1)} - \boldsymbol{\alpha}\_{1} \boldsymbol{E}^{(1)} \right) \end{cases} \end{array} \tag{98}$$

Computational Intelligence in Electromyography Analysis – 168 A Perspective on Current Applications and Future Challenges

where (0, ,0; , , ) 0 *<sup>T</sup>*

∴ *H* is symplectic matrix.

∴ *H* is also unitary matrix.

**Apendix B** 

∴ *H* is symplectic unitary matrix.

*T <sup>n</sup> <sup>x</sup>* (*<sup>x</sup>* , *<sup>x</sup>* , , *<sup>x</sup>* ) <sup>=</sup> <sup>1</sup> <sup>2</sup> , notes <sup>2</sup>

is elementary reflective array *<sup>T</sup> H* = 1− 2

α

 ωω *k n* = ≠ . Plugging Eq.(87) into Eq.(86), we have:

ϖ

ϖ ϖ

 <sup>−</sup> 

2 2

4 4 ( )

ϖ ϖ ϖ ϖ

*P P*

\*

 

0 0

 

*P*

\*

ϖ ϖ ϖ ϖ

( )( )

*P*

, which make *Hx* = *y* , where

 

0

\* \* \* \*

*P* = *In* −

*n*

*I*

*H H*

\*

=

*n*

= − +

ϖ ϖ

*n I*

 =

 =

=

ωω

= *x* , there is

2

*P P*

0

**Theorem 7** suppose *x* and *y* are two unequal *n* dimension vectors, and <sup>2</sup> <sup>2</sup> *<sup>x</sup>* <sup>=</sup> *<sup>y</sup>* , so there

It can be easily deduced from theorem 5, for non zero *n* dimension vector

<sup>1</sup> *Hx* =α

*<sup>T</sup> H* = 1− 2ϖϖ

ρ

ϖ

( ) <sup>1</sup> 1 *x*

α*e*

\*

*P*

0

*P*

0

*I*

 <sup>=</sup> <sup>−</sup>

*P P P*

=

\* 2

∴ *P* = *P* \*

*n n*

\* \*

ϖϖ

*I I*

\* \*

ϖϖ

ϖ ϖ ϖϖ

\* \* 2

> 

*H JH* = *J* \* (88)

(87)

(89)

<sup>2</sup> *x y x y* − −

= .

ω

*e* (90)

*<sup>T</sup> e* (1,0, ,0) <sup>1</sup> = (92)

= − (93)

(91)

\* \*

ϖϖ

ϖ ϖ

> so, after *H*(1) transform, *A* is changed to a matrix with the first column is all zero except the first element is α<sup>1</sup> , namely:

$$H^{(1)}A = \begin{pmatrix} \sigma\_1 & a\_{12}^{(2)} & \cdots & a\_{1n}^{(2)} \\ 0 & a\_{22}^{(2)} & \cdots & a\_{2n}^{(2)} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & a\_{n2}^{(2)} & \cdots & a\_{nn}^{(2)} \end{pmatrix} = A^{(2)} \tag{99}$$

Second, the same method is adopted to the second column vector of *A*(2), let

$$S^{(2)} = \left(0, a\_{22}^{(2)}, \cdots, a\_{n2}^{(2)}\right)^{\mathrm{T}} \tag{100}$$

construct *H*(2) matrix:

$$H^{(2)} = I - 2\alpha^{(2)} (\mathfrak{w}^{(2)})^T \tag{101}$$

where,

Computational Intelligence in Electromyography Analysis – 170 A Perspective on Current Applications and Future Challenges

$$\begin{cases} \begin{aligned} \alpha\_2 &= \left\| S^{(2)} \right\|\_2 \\ E^{(2)} &= \left( 0, 1, 0, \cdots, 0 \right)^T \\ \rho\_2 &= \left\| S^{(2)} - \alpha\_2 E^{(2)} \right\| \\ \sigma^{(2)} &= \frac{1}{\rho\_2} \left( S^{(2)} - \alpha\_2 E^{(2)} \right) \end{aligned} \tag{102}$$

Nonlinear Analysis of Surface EMG Signals 171

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Using *H*(2), the second column of *A*(2) can be changed to all zero vector except the first and second elements, namely:

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Householder matrix *H* can be obtained by repeating above mentioned method until *A*(*n*) becomes an upper triangle matrix:

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### **Author details**

Min Lei and Guang Meng

*Institute of Vibration, Shock and Noise, State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, P R China* 

#### **Acknowledgement**

This work was supported by the National Natural Science Foundation of China (No. 10872125 and No.69675002), Science Fund for Creative Research Groups of the National Natural Science Foundation of China(no. 50821003), State Key Lab of Mechanical System and Vibration,Project supported by the Research Fund of State Key Lab of MSV, China (Grant no. MSV-MS-2010-08) , Key Laboratory of Hand Reconstruction, Ministry of Health, Shanghai, People's Republic of China, Shanghai Key Laboratory of Peripheral Nerve and Microsurgery, Shanghai, People's Republic of China, Science and Technology Commission of Shanghai Municipality (no.06ZR14042). We also thank Chinese Academy of Engineering GU Yudong and Professor Zhang Kaili very much for providing related data and valuable discussions.

#### **6. References**


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Computational Intelligence in Electromyography Analysis – 170 A Perspective on Current Applications and Future Challenges

second elements, namely:

**Author details** 

Min Lei and Guang Meng

**Acknowledgement** 

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**6. References** 

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ϖ

(2)

2

ρ

(2)

α

2

 

( )

*S E E n*

2 (2)

= ×

*S E*

α

(2) 2

*T*

0,1, 0, , 0 1

(2) 2

(102)

(2) (2) (3) *H A* = *A* (103)

( ) ( 1) (1) *H H H H* = *<sup>n</sup> <sup>n</sup>*<sup>−</sup> (104)

( )

(2)

Using *H*(2), the second column of *A*(2) can be changed to all zero vector except the first and

Householder matrix *H* can be obtained by repeating above mentioned method until *A*(*n*)

*Institute of Vibration, Shock and Noise, State Key Laboratory of Mechanical System and Vibration,* 

This work was supported by the National Natural Science Foundation of China (No. 10872125 and No.69675002), Science Fund for Creative Research Groups of the National Natural Science Foundation of China(no. 50821003), State Key Lab of Mechanical System and Vibration,Project supported by the Research Fund of State Key Lab of MSV, China (Grant no. MSV-MS-2010-08) , Key Laboratory of Hand Reconstruction, Ministry of Health, Shanghai, People's Republic of China, Shanghai Key Laboratory of Peripheral Nerve and Microsurgery, Shanghai, People's Republic of China, Science and Technology Commission of Shanghai Municipality (no.06ZR14042). We also thank Chinese Academy of Engineering GU Yudong and Professor Zhang Kaili very much for providing related data and valuable

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α

= −

= −

*S*

1

ρ

(2)

2

=


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**Chapter 7** 

© 2012 Halaki and Ginn, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Halaki and Ginn, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Normalization of EMG Signals: To Normalize or** 

Electromyography (EMG) has been around since the 1600s [1]. It is a tool used to measure the action potentials of motor units in muscles [2]. The EMG electrodes are like little microphones which "listen" for muscle action potentials so having these microphones in different locations relative to the muscle or motor units affects the nature of the recording [3]. The amplitude and frequency characteristics of the raw electromyogram signal have been shown to be highly variable and sensitive to many factors. De Luca [4] provided a detailed account of these characteristics which have a "basic" or "elemental" effect on the signal dividing them into extrinsic and intrinsic sub-factors. Extrinsic factors are those which can be influenced by the experimenter, and include: electrode configuration (distance between electrodes as well as area and shape of the electrodes); electrode placement with respect to the motor points in the muscle and lateral edge of the muscle as well as the orientation to the muscle fibres; skin preparation and impedance [5, 6]; and perspiration and temperature [7]. Intrinsic factors include: physiological, anatomical and biochemical characteristics of the muscles such as the number of active motor units; fiber type composition of the muscles; blood flow in the muscle; muscle fiber diameter; the distance between the active fibers within the muscle with respect to the electrode; and the amount of tissue between the surface of the muscle and the electrode. These factors vary between individuals, between days within an individual and within a day in an individual if the electrode set up has been altered. Given that there are many factors that influence the EMG signal, voltage recorded from a muscle is difficult to describe in terms of level if there is no reference value to which it can be compared. Therefore, interpretation of the amplitude of the raw EMG signal is problematic unless some kind of normalization procedure is performed. Normalization refers to the conversion of the signal to a scale relative to a known and repeatable value. It has been reported [8] that normalized EMG signals were first presented by Eberhart, Inman & Bresler in 1954 [9]. Since then, there have been a number of methods used to normalize EMG signals with no consensus as to which method is most

**Not to Normalize and What to Normalize to?** 

Mark Halaki and Karen Ginn

http://dx.doi.org/10.5772/49957

**1. Introduction** 

Additional information is available at the end of the chapter

