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## **Meet the editor**

N. A. Ahmed has worked and published extensively on environmental issues of high practical significance. He has developed advanced flow control and flow diagnostic techniques that have opened up new horizons of research of diverse applications, ranging from safe operation of wind turbines to noise mitigation and control, from ventilation within enclosed spaces of building or

aircraft cabin to performance enhancement of aircraft flight, and so forth. He obtained his BSc (Hons) in Mechanical Engineering from Strathclyde University, UK, and his PhD in Compressor Aerodynamics from Cranfield University, UK. He has worked in the industry as a Design Engineer for Kent Industrial Measurements, UK, and Manager of Low Speed Compressor Test Facility of Cranfield. He was an Associate Professor and Head of Aerospace Engineering at the University of New South Wales, Australia. He is currently a Professor at the Faculty of Engineering and Built Environment of the University of Johannesburg, South Africa.

## Contents

Pérez, Pilar Candelas, Francisco Belmar and Antonio Uris

## Chapter 10 **Noise Reduction and Control in Hospital Environment: Design of the NeoNoise Project 229**

Carlos Carvalhais, Manuela V. Silva, Ana Xavier and Joana Santos

## Preface

Chapter 7 **Effects of Platform Screen Doors on Sound Fields in**

Chapter 8 **Noise Control by Suppression of Gas Pulsation in Screw**

Chapter 9 **Analysis of Acoustic Noise and its Suppression in Speech**

Jiří Přibil, Anna Přibilová and Ivan Frollo

**of the NeoNoise Project 229**

**Recorded During Scanning in the Open-Air MRI 205**

Chapter 10 **Noise Reduction and Control in Hospital Environment: Design**

Elvedin Mujić, Ahmed Kovačević, Nikola Stošić and Ian K. Smith

Carlos Carvalhais, Manuela V. Silva, Ana Xavier and Joana Santos

**Underground Stations 161** Yoshiharu Soeta and Yong Hee Kim

**Compressors 179**

**VI** Contents

While rapid industrialisation and urbanisation have underpinned the growth and character of modern day civilisation, they have also given rise to unsustainable energy usage and lifestyles that have the potential to destroy the very environment that supports life (Ahmed & Cameron, Renew‐ able & Sustainable Energy Reviews, vol. 38, pp 439–460, 2014; Ahmed, Procedia Engineering, Elsevier Publications, vol. 49, 2012, pp 338–347). Thus there are many issues that need urgent atten‐ tion and addressing the significant issue 'noise' from the operation of machines, transport and oth‐ er man-made systems is one of them.

Today, the term 'noise' has evolved to be more complex than 'unwanted sound', with its definition encompassing random fluctuations of signals originating from a variety of sources, such as human vocal cord, electronic devices, cars, aircrafts, trains, musical instruments and so forth. However, the detrimental effects of some forms of noise and the pace with which they have accelerated in the last few decades are alarming and have the potential to adversely impact our daily activities, health and well-being, prompting intense research to mitigate and control their detrimental effects. This is the background against which work on the present book was undertaken.

This book is a collection of works of eminent researchers from various countries of the world on some selected but important aspects of noise with a heavier emphasis on what can be described as noise pollution. The book is arranged in two sections, each comprising five chapters. For easy iden‐ tification, the chapters are numbered from one to ten. While there is some overlap of ideas and concepts, the first section, Section 1, broadly comprises chapters that deal with technologies or methodologies to advance the cause of noise analyses, while the second section, Section 2, contains chapters that are more geared to approaches towards noise mitigation and its control, specifically in transport, industry and health sectors.

#### Section 1: Noise Analysis

It is important to realise that in any research, the capacity to investigate and analyse has been greatly aided by modern computers and their ability to store increasingly greater amount of infor‐ mation, which itself relies heavily on advancements in electronics and their methods of produc‐ tion.

One way to achieve advancement in analysis is by reducing the 'noise' of electronics used in infor‐ mation gathering. Chapter 1 of Section 1, by Gaubert and Teramoto, deals with this aspect. It con‐ siders the reduction of noise levels of MOSFET or metal-oxide-semiconductor field-effect transistor, a type of transistor, which is widely used in amplifying or switching electronic signals. The authors show that low resistivity source and drain electrodes can greatly lower the low-fre‐ quency noise level of MOSFET by suppressing their contribution to the total noise. They show that new plasma processes having the advantages to work at low electron temperature can also achieve a further reduction through the fabrication of a better gate oxide as well as reduction of damages generally induced by conventional plasma processes. The newly developed MOSFET devices with a lower noise level then conventional structures are, therefore, a serious platform for the next gen‐ eration of on-board, battery-powered semiconductor chip or 'complementary metal-oxide semi‐ conductor (CMOS)' technology for storing information inside computers.

The following chapter, Chapter 2, considers the critically important aspect of data analysis. Data obtained in experimental situations, especially in real environments, are generally very complicat‐ ed, chaotic and heavily corrupted by noise. The authors Min Lei and Guang Meng propose a novel approach which they term 'symplectic principal component analysis (SPCA)' to reduce noise of continuous chaotic systems. The authors show that the SPCA method can yield more reliable re‐ sults for chaotic time series under the different data lengths and sampling times, especially with short data length and under-sampled sampling time, than the classic principal component analysis (PCA).

Chapter 3 by Thomas Hasse, entitled 'Optimal placement of sensors and actuators for feedforward noise and vibration control', contains a detailed analyses of parameters such as the causality, filter weights and the damping properties that have significant influence on the performance of a feed‐ forward control system. Studies of the past three decades suggest that to devise any successful practical system to reduce noise and vibration, it is imperative to look into the locations of detec‐ tors and actuators and how they are arranged. The author in this chapter demonstrates that a prior knowledge of the influences of these parameters should be established that will result in the opti‐ misation of the location of detectors and actuators for better performance.

Chapter 4 discusses variable drive of AC systems which are used in electromechanical systems that can range from small appliances to large mine mill drives and compressors and are integral parts of modern day life. The authors of this chapter, Brudny, Morganti and Lecointe, describe control strategies to reduce the noise of magnetic origin through the rejection at high frequencies of the voltage switching 'three-phase harmonic systems' (STPHSs) using a strategy called 'carrier-phase jump'. Their proposed calculation method is simple to implement and characterises STPHS phase sequences that have a direct impact on the definition of the force components at the origin of the magnetic noise generated by AC machines and subsequent control strategies for reducing this noise.

In Chapter 5, the final chapter of Section 1, Freitas, Lamas, Silva, Soares, Mouta and Santos detail a new methodology to assess the 'annoyance' aspect of road-traffic noise that predominantly occurs at speeds above 40 km/h. The negative impact of 'annoyance' on health is generally mitigated by optimising road pavement surface characteristics. However, experimental data linking surface characteristics and annoyance rate remains scarce. Moreover, because of the complexities of assess‐ ing annoyance by experimental means using real sounds, the authors have resorted to developing virtual sounds that have the real possibility to accurately establish requirements to control traffic noise based on indicators that better describe the annoyance.

#### Section 2: Noise Mitigation and Control

Section 2 contains advances in methods to control and mitigate the effects of noise. To maintain continuity with Section 1, the five chapters of Section 2 are numbered from Chapters 6 to 10. The first chapter of this section—Chapter 6—describes an approach that is based on the postulation that if noise cannot be reduced at source, then attempts to mitigate it during its transmission phase become appropriate. Consequently, for the transmission phase, the authors, Rubio, Ibariez, Perez, Candelas, Belmar and Uris, develop the concept of 'open acoustic screens'. These screens, or the 'subwavelength slit acoustic screens' as the authors call them, employ a new type of attenuation mechanism that are not based on crystalline geometry but consist of arrangements of isolated scat‐ terers. Such acoustic screens have interesting properties and can be considered as real alternatives to the classical screens formed by continuum rigid materials.

Noise reduction in underground stations using platform screen doors (PCD) forms the basis of Chapter 7. Underground stations are now an integral part of urban city transport systems of most parts of the world. The authors, Soeta and Kim, use both computer simulation and scale model testing of several PCDs of different configurations and sizes. They found that speech intelligibility and the sound pressure level were increased by most types of PSDs. These findings are significant and pave the way for the development of better public announcement systems in trains which will give safety and comfort in hearing of passengers.

The following chapter, Chapter 2, considers the critically important aspect of data analysis. Data obtained in experimental situations, especially in real environments, are generally very complicat‐ ed, chaotic and heavily corrupted by noise. The authors Min Lei and Guang Meng propose a novel approach which they term 'symplectic principal component analysis (SPCA)' to reduce noise of continuous chaotic systems. The authors show that the SPCA method can yield more reliable re‐ sults for chaotic time series under the different data lengths and sampling times, especially with short data length and under-sampled sampling time, than the classic principal component analysis

Chapter 3 by Thomas Hasse, entitled 'Optimal placement of sensors and actuators for feedforward noise and vibration control', contains a detailed analyses of parameters such as the causality, filter weights and the damping properties that have significant influence on the performance of a feed‐ forward control system. Studies of the past three decades suggest that to devise any successful practical system to reduce noise and vibration, it is imperative to look into the locations of detec‐ tors and actuators and how they are arranged. The author in this chapter demonstrates that a prior knowledge of the influences of these parameters should be established that will result in the opti‐

Chapter 4 discusses variable drive of AC systems which are used in electromechanical systems that can range from small appliances to large mine mill drives and compressors and are integral parts of modern day life. The authors of this chapter, Brudny, Morganti and Lecointe, describe control strategies to reduce the noise of magnetic origin through the rejection at high frequencies of the voltage switching 'three-phase harmonic systems' (STPHSs) using a strategy called 'carrier-phase jump'. Their proposed calculation method is simple to implement and characterises STPHS phase sequences that have a direct impact on the definition of the force components at the origin of the magnetic noise generated by AC machines and subsequent control strategies for reducing this

In Chapter 5, the final chapter of Section 1, Freitas, Lamas, Silva, Soares, Mouta and Santos detail a new methodology to assess the 'annoyance' aspect of road-traffic noise that predominantly occurs at speeds above 40 km/h. The negative impact of 'annoyance' on health is generally mitigated by optimising road pavement surface characteristics. However, experimental data linking surface characteristics and annoyance rate remains scarce. Moreover, because of the complexities of assess‐ ing annoyance by experimental means using real sounds, the authors have resorted to developing virtual sounds that have the real possibility to accurately establish requirements to control traffic

Section 2 contains advances in methods to control and mitigate the effects of noise. To maintain continuity with Section 1, the five chapters of Section 2 are numbered from Chapters 6 to 10. The first chapter of this section—Chapter 6—describes an approach that is based on the postulation that if noise cannot be reduced at source, then attempts to mitigate it during its transmission phase become appropriate. Consequently, for the transmission phase, the authors, Rubio, Ibariez, Perez, Candelas, Belmar and Uris, develop the concept of 'open acoustic screens'. These screens, or the 'subwavelength slit acoustic screens' as the authors call them, employ a new type of attenuation mechanism that are not based on crystalline geometry but consist of arrangements of isolated scat‐ terers. Such acoustic screens have interesting properties and can be considered as real alternatives

misation of the location of detectors and actuators for better performance.

noise based on indicators that better describe the annoyance.

to the classical screens formed by continuum rigid materials.

Section 2: Noise Mitigation and Control

(PCA).

VIII Preface

noise.

Chapter 8, by Mujic, Kovacevic, Stosic and Smith, identifies gas pulsations as the most important source of noise in screw compressors that can use a variety of fluids and are widely used in a num‐ ber of industrial and domestic applications. Using both numerical modelling and physical tests on industrial screw compressors, the authors demonstrate that the gas pulsations in the compressors and the consequent noise that they generate can be substantially reduced through appropriate changes to the shape of the compressor discharge port.

The last two chapters of section 2 tackle problems of noise that bear direct relations to the medical world. The authors, Pribil, Pribilova and Frollo, of Chapter 9 focus their attention on improving the performance and application of magnetic resonance imaging (MRI). The discovery and develop‐ ment of magnetic resonance imaging (MRI) scanners have revolutionised medical procedures and research around the world. The authors describe methods to reduce noise of the speech recorded in an open air MRI working in a weak magnetic field during human phonation for the vocal tract modelling. The achieved results will prove useful as a database in the design of filters for noise suppression in the speech signal recorded simultaneously with three-dimensional human vocal tract scanning.

The final chapter, or Chapter 10, is a 'neo-project' on noise reduction and control in a hospital envi‐ ronment. The authors, Carvalhais, Silva, Xavier and Santos, provide the groundwork for an inte‐ grated approach to minimise sound pressure levels in neonatal intensive care units where exposure to sound pressure levels can significantly influence the quality and well-being of the workers and others, as well as the recovery of premature infants who are hospitalised. The out‐ comes of this project will contribute towards educating health-care staff and providing good prac‐ tice guide in a daily basis and reduce and control noise production in hospital environments.

This book is intended to be a valuable addition to students, engineers, scientists, industrialists, con‐ sultants and others by providing greater insights into noise-related problems, particularly in noise pollution, and methodologies to study, mitigate or control them.

All chapters have been prepared by professionals who are experts in their respective research fields and the contents reflect their views. All chapters included in this book are a culmination of the interactions of the editor, publisher and authors.

The editor takes the opportunity to thank all the authors for their expert contributions and the pub‐ lisher for their patience and the hard work that has gone in producing this book of high practical significance.

#### **Noor Ahmed, BSc (Hons), PhD, CEng, Fellow IEF, Fellow IMechE**

Department of Mechanical Engineering Technology within the Faculty of Engineering and the Built Environment Auckland Park Kingsway Campus University of Johannesburg, South Africa

**Section 1**

## **Noise Analysis**

## **New Processes and Technologies to Reduce the Low‐ Frequency Noise of Digital and Analog Circuits New Processes and Technologies to Reduce the Low**‐ **Frequency Noise of Digital and Analog Circuits**

Philippe Gaubert and Akinobu Teramoto Philippe Gaubert and Akinobu Teramoto

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64630

## **Abstract**

The chapter is intended to provide the reader with means to reduce low‐frequency noise in Metal‐Oxide‐Semiconductor Field‐Effect‐Transistor (MOSFET). It is demonstrated that low‐resistivity source and drain electrodes can greatly lower the low‐frequency noise level by suppressing their contribution to the total noise. Furthermore, new plasma processes having the advantages to work at low electron temperature can achieve a further reduction, thanks to the fabrication of a better gate oxide and to a reduction of damages generally induced by conventional plasma processes. Reducing the impact of the traps on the carrier flowing inside the channel by burying the channel can also achieve a reduction of the noise level, but unfortunately at the cost of a degradation of the electrical performances. Finally, the noise analysis of the low‐ frequency noise in accumulation‐mode MOSFETs showed that these newly developed devices have a lower noise level than conventional structures, which, in addition to their superiority in term of electrical performances, establishes them as a serious platform for the next Complementary Metal‐Oxide‐Semiconductor Field‐Effect‐Transistor (CMOS) technology.

**Keywords:** low‐frequency noise, radical oxidation, silicide, series resistances, accumu‐ lation‐mode, MOSFET, buried‐channel, fabrication process

## **1. Introduction**

Since the dawn of electronics more than 50 years ago, manufacturers have been providing customers with faster and smaller chips by fabricating increasingly better devices and improving processes. The main strategy adopted has been to shrink the gate size of the MOSFETs to improve chip performances, especially speed. Since the signal‐to‐noise ratio was large enough, the noise

© 2016 The Author(s). Licensee InTech. This chapter is 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. © 2016 The Author(s). Licensee InTech. This chapter is 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.

was not an issue and its reduction dragged very little effort among the scientific community. After a steady working frequency doubling every year, the recent downscaling of the dimension has led to high stress and increased variability and, in turn, stagnation of performances of chips. There is no doubt that the increased noise is to blame for that standstill, even if other problems such as the doping concentration could be implicated as well. Nevertheless, the distinction between noise and signal has become critical and the noise issue must be tackled. A suppressed noise level should pave the way to once again lower biases leading to less heat, better reliability, and better performances.

Noise is a fluctuation of a quantity that shifts back and forth with uncertainty. In electronics, it is generally noted as a fluctuation of the voltage or current around its mean value and is ascribed to stochastic events which find their origin at a microscopic level trough the discrete nature of the transport or the Brownian nature of the carrier. There are several types of noises such as thermal noise, shot noise, generation‐recombination noise, inter‐band noise, and low‐ frequency noise. They are generally classified upon their origin. Among these, the thermal noise and low‐frequency noise are of paramount importance in MOSFETs, with the latter one being of most concern since its origin is still in debate and that its evaluation for a given technology is made extremely difficult. Even if the low‐frequency noise has been a limiting factor of performances for analog circuits for several years, it has recently become as well an issue for digital ones. Indeed, even though its limitation applies in the low range of frequency, it is up‐converted into phase noise leading to time domain instabilities and therefore problems in the high‐frequency range. Its reduction is therefore mandatory not only for analog but also for digital circuits.

In Section 1, the theory of the low‐frequency noise in MOSFETs is briefly reviewed. While Sections 2 and 3 present new technologies to suppress it by the means of, respectively, silicide and damage free processes, Sections 4 and 5 introduce improved MOSFETs. Thus, the results regarding buried‐channel and accumulation‐mode MOSFETs are reported, respectively, in Sections 4 and 5.

## **2. Low‐frequency noise in MOSFETs**

The MOSFET is a complex device composed of purely resistive parts surrounding the channel whose resistance is controlled by the bias applied at the gate electrode. It is therefore natural that the noise generated inside each region is propagating up to the source and the drain electrodes. However, the noise stemming from the channel is generally the most dominant one, even though the one coming from the surrounding areas can play an important role and can even take over as the main noise source [1]. **Figure 1(a)** shows a schematic of a MOSFET structure while **Figure 1(b)** represents its equivalent noise circuit. The source access resistance, the drain access resistance, and the channel are the three main regions the noise is coming from and the total measured noise *S*Id at a given frequency can be written [2].

New Processes and Technologies to Reduce the Low‐Frequency Noise of Digital and Analog Circuits http://dx.doi.org/10.5772/64630 5

**Figure 1.** Representation of (a) MOSFET and (b) its equivalent circuit noise.

was not an issue and its reduction dragged very little effort among the scientific community. After a steady working frequency doubling every year, the recent downscaling of the dimension has led to high stress and increased variability and, in turn, stagnation of performances of chips. There is no doubt that the increased noise is to blame for that standstill, even if other problems such as the doping concentration could be implicated as well. Nevertheless, the distinction between noise and signal has become critical and the noise issue must be tackled. A suppressed noise level should pave the way to once again lower biases leading to less heat, better reliability,

Noise is a fluctuation of a quantity that shifts back and forth with uncertainty. In electronics, it is generally noted as a fluctuation of the voltage or current around its mean value and is ascribed to stochastic events which find their origin at a microscopic level trough the discrete nature of the transport or the Brownian nature of the carrier. There are several types of noises such as thermal noise, shot noise, generation‐recombination noise, inter‐band noise, and low‐ frequency noise. They are generally classified upon their origin. Among these, the thermal noise and low‐frequency noise are of paramount importance in MOSFETs, with the latter one being of most concern since its origin is still in debate and that its evaluation for a given technology is made extremely difficult. Even if the low‐frequency noise has been a limiting factor of performances for analog circuits for several years, it has recently become as well an issue for digital ones. Indeed, even though its limitation applies in the low range of frequency, it is up‐converted into phase noise leading to time domain instabilities and therefore problems in the high‐frequency range. Its reduction is therefore mandatory not only for analog but also

In Section 1, the theory of the low‐frequency noise in MOSFETs is briefly reviewed. While Sections 2 and 3 present new technologies to suppress it by the means of, respectively, silicide and damage free processes, Sections 4 and 5 introduce improved MOSFETs. Thus, the results regarding buried‐channel and accumulation‐mode MOSFETs are reported, respectively, in

The MOSFET is a complex device composed of purely resistive parts surrounding the channel whose resistance is controlled by the bias applied at the gate electrode. It is therefore natural that the noise generated inside each region is propagating up to the source and the drain electrodes. However, the noise stemming from the channel is generally the most dominant one, even though the one coming from the surrounding areas can play an important role and can even take over as the main noise source [1]. **Figure 1(a)** shows a schematic of a MOSFET structure while **Figure 1(b)** represents its equivalent noise circuit. The source access resistance, the drain access resistance, and the channel are the three main regions the noise is coming from

and the total measured noise *S*Id at a given frequency can be written [2].

and better performances.

4 Advances in Noise Analysis, Mitigation and Control

for digital circuits.

Sections 4 and 5.

**2. Low‐frequency noise in MOSFETs**

$$\begin{split} S\_{\mathbf{I}\_{\downarrow}}\left(f\right) &= \frac{S\_{\mathbf{I}\_{\downarrow\_{\text{ch}}}}\left(f\right)}{\left[1 + g\_{m}R\_{s} + g\_{\text{ch}}\left(R\_{s} + R\_{d}\right)\right]^{2}}\\ &+ \frac{g\_{m}^{2}R\_{s}^{2}S\_{\mathbf{I}\_{\downarrow\_{\text{h}\_{\star}}}}\left(f\right) + g\_{\text{ch}}^{2}\left[R\_{s}^{2}S\_{\mathbf{I}\_{\downarrow\_{\text{h}\_{\star}}}}\left(f\right) + R\_{d}^{2}S\_{\mathbf{I}\_{\downarrow\_{\text{h}\_{\downarrow}}}}\left(f\right)\right]}{\left[1 + g\_{m}R\_{s} + g\_{\text{ch}}\left(R\_{s} + R\_{d}\right)\right]^{2}}\end{split} \tag{1}$$

where *S*Id,ch, *S*IRs, and *S*IRd are the noises generated, respectively, inside the channel, the source *Rs* and drain *Rd* access resistances. *Gm* and *g*ch are, respectively, the transconductance and conductance of the channel. The left‐hand side term is the contribution coming from the channel while the right‐hand side one is the contribution coming from the access resistances.

It is worth noticing that *S*Id can become equal to *S*Id,ch meaning that the measured noise will be the noise of the channel, whereas even if the series resistances are the main noise source, it is their contribution that will be measured and not their pure noise. The noise in a resistive material such as the access resistances of a MOSFET is either thermal noise or 1/*f* noise at low frequencies [3]. This 1/*f* noise is known to originate from the fluctuations of the fundamental mobility and it follows the Hooge's empirical formulae [3]:

$$\frac{S\_f\left(f\right)}{I^2} = \frac{\alpha\_H}{fN},\tag{2}$$

which represents the normalized noise of the current *I* while *N* is the number of carriers. αH is the Hooge parameter suggested to be constant and to reflect the quality of the crystal. After several decades of controversies between the Hooge theory and the Mc. Whorter one [4], the latter explaining the noise in terms of fluctuation of the carrier number, Mikoshiba [5] devel‐ oped a new theory able to gather all data into a single model to explain the noise stemming from the channel.

His theory has been confirmed afterward by several researchers and is now well accepted among the scientific community. Within this theory, the noise is given by [6]

$$\frac{S\_{\rm I\_d}\left(f\right)}{I\_d^2} = \frac{g\_m^2}{I\_d^2} \left(1 \pm \alpha \mu\_{\rm eff} C\_{\rm ox} \frac{I\_d}{g\_m}\right)^2 S\_{\rm V\_{\mu}}\left(f\right),\tag{3}$$

where *C*ox is the gate oxide capacitance, is the transconductance, and *μ*eff is the effective mobility of the carriers flowing inside the channel. This theory is known as the insulator and induced mobility fluctuations, and is ascribing the origin of the noise to the traps located inside the gate insulator near the interface. The constant dynamic capture and release of carriers from and to traps generate interfacial insulator charge variation and, in turn, fluctuation of the insulator charge. These fluctuations are equivalently generating flat band variations. These fluctuations are summarized on the left‐hand side term in Eq. (3) and are proportional to the flat band voltage fluctuations *S*Vfb, expressed as [6]

$$S\_{V\_{\rho}}\left(f\right) = \frac{\lambda k\_{\text{B}} T q^2 N\_{t}}{W L C\_{\text{ox}}^2 f},\tag{4}$$

with *k*B being the Boltzmann constant, *q* the electron charge, *T* the temperature, and *λ* the tunnel attenuation length of the traps in the insulator equal to 0.1 nm for SiO2. *Nt* is the interface trap density. In addition to the variation of the insulator charge, the capture and release mechanism is locally affecting the surface potential at the interface, resulting in a Coulomb interaction between the locally deformed potential surface and the carriers flowing inside the channel. The localized scattering rate will vary and will induce fluctuations of the mobility. These fluctuations are ascribed to the right‐hand side term in Eq. (3) and are related to the fluctuations of the insulator charge through the Coulomb parameter α, which measures the strength between both quantities. With nowadays miniaturization of the gate of the MOSFETs, two kinds of noises are at stake, although they are both explained in the frame of the previous theory. Indeed, it is obvious that in the case of very small gate size involving a very limited number of carriers, the removal or introduction of a single free carrier scared within the total number of free carriers involved in the conduction will have a tremendous impact on the current, making it jump between two or several levels like randomly disposed crenels. This type of noise is called random telegraph noise and is commonly an issue for sensors, especially optical ones [7]. However, when the number of traps is more significant and for a specific distribution of their energy within the bandgap, the resulting noise is commonly called 1/*f* or even Flicker noise and it follows a distribution, proportional to the inverse of the frequency when plotted as a function of the frequency [1]. This noise is an issue for analog circuits, and even for some digital ones, and will even impact at high frequency due to its conversion into phase noise. Finally, the noise in MOSFETs can be summarized as depicted in **Figure 2**. The noise measured at the electrode of a MOSFET is the sum of three terms: the fluctuation of the insulator charge, induced fluctuation of the mobility, and the contribution coming from the access resistances. It is worth mentioning that there is a fourth term, the cross‐correlation term between the fluctuation of the insulator charge and the induced mobility one even though it does not have a physical origin.

**Figure 2.** Normalized noise in a MOSFET. The several contributions of each noise sources have been reported with the non‐full lines.

## **3. Source and drain contacts**

oped a new theory able to gather all data into a single model to explain the noise stemming

His theory has been confirmed afterward by several researchers and is now well accepted

( ) ( ) <sup>d</sup> <sup>2</sup> <sup>2</sup>

> ( ) <sup>2</sup> B 2 ox ,

with *k*B being the Boltzmann constant, *q* the electron charge, *T* the temperature, and *λ* the tunnel

density. In addition to the variation of the insulator charge, the capture and release mechanism is locally affecting the surface potential at the interface, resulting in a Coulomb interaction between the locally deformed potential surface and the carriers flowing inside the channel. The localized scattering rate will vary and will induce fluctuations of the mobility. These fluctuations are ascribed to the right‐hand side term in Eq. (3) and are related to the fluctuations of the insulator charge through the Coulomb parameter α, which measures the strength between both quantities. With nowadays miniaturization of the gate of the MOSFETs, two kinds of noises are at stake, although they are both explained in the frame of the previous theory. Indeed, it is obvious that in the case of very small gate size involving a very limited number of carriers, the removal or introduction of a single free carrier scared within the total number of free carriers involved in the conduction will have a tremendous impact on the current, making it jump between two or several levels like randomly disposed crenels. This type of noise is called random telegraph noise and is commonly an issue for sensors, especially optical ones [7]. However, when the number of traps is more significant and for a specific distribution of their energy within the bandgap, the resulting noise is commonly called 1/*f* or even Flicker noise and it follows a distribution, proportional to the inverse of the frequency when plotted as a function of the frequency [1]. This noise is an issue for analog circuits, and even for some digital ones, and will even impact at high frequency due to its conversion into

*k Tq N S f WLC f* l

*t*

= (4)

is the interface trap

*fb*

attenuation length of the traps in the insulator equal to 0.1 nm for SiO2. *Nt*

*V*

2 2 eff ox 1 , *fb m d*

where *C*ox is the gate oxide capacitance, is the transconductance, and *μ*eff is the effective mobility of the carriers flowing inside the channel. This theory is known as the insulator and induced mobility fluctuations, and is ascribing the origin of the noise to the traps located inside the gate insulator near the interface. The constant dynamic capture and release of carriers from and to traps generate interfacial insulator charge variation and, in turn, fluctuation of the insulator charge. These fluctuations are equivalently generating flat band variations. These fluctuations are summarized on the left‐hand side term in Eq. (3) and are proportional to the

æ ö = ± ç ÷ è ø

*V*

(3)

among the scientific community. Within this theory, the noise is given by [6]

*d d m S f g I C Sf I I <sup>g</sup>* am

I

flat band voltage fluctuations *S*Vfb, expressed as [6]

from the channel.

6 Advances in Noise Analysis, Mitigation and Control

When it comes to low‐frequency noise, the contribution stemming from the source and drain access series resistances is generally overlooked. This negligence can have tremendous impact on the noise analysis especially at high gate voltage where their contribution will mostly take over as the dominant noise.

As a matter of fact, this is not the noise generated inside the source and drain access series resistances, which is at stake but their contribution. Rather than reducing the noise sources, reducing their contribution is more efficient and easier. Indeed, the reduction of the resistance of the source and drain access contacts does not only mean a better drivability and a better transconductance but can guarantee a reduction of the propagation of the noise generated by the noise sources inside the contacts and, in turn, a reduction of the contribution of the source and drain contact noise to the total measured noise [8]. The performance improvement of CMOS has become of paramount importance with scaled dimension. Much effort is being made to increase the carrier mobility by several means such as strained technology [9], different silicon orientation [10], or even different semiconductor [11]. The reduction of the source and drain electrode series resistances is another means to improve the drivability and silicide has already been used for such purposes. Nitride alloy silicide is widely used to lower the Schottky barrier height to either n+ or p+ silicon down with contact resistance as low as 2 × 10–9 Ω cm2 in the best case [12]. A new structure [13] and new processes [14] have been developed in order to further lower down the series resistances. Instead of using the same silicide for both p‐ and n‐MOSFETs, erbium has been selected to perfectly fit the requirement of n‐MOSFETs and palladium for the p‐MOSFETs. Additionally, tungsten metal stack above the thin silicide layer was introduced to reduce the sheet resistance and protect erbium from being oxidized. In order to confirm the above statement, two kinds of MOSFETs have been fabricated following the very same process flow, except during the source and drain contact fabrication stage. The source and drain contacts of the reference transistors have been fabricated with aluminum (Al), while erbium silicide associated with tungsten (ErSi2/W) has been used for the second set of transistors. The respective structures have been represented in **Figure 3(a)** and **(b)**. While both wafers followed the same process flow until the contact lithography step, the ErSi*x*/W wafer followed an advanced process entirely developed at New Industry Creation Hatchery Center (NICHe). In this advanced process, the wafer has been loaded in an N2 sealed cleaning chamber after a total room temperature five‐step cleaning, which has been followed by the dipping of the cleaned wafer in O3 dissolved ultra pure water in order to form a chemical oxide at the silicon surface. The removal of the chemical oxide has been carried out by diluted HF solution and the wafer has been then transferred in clustered sputter equipment, still in N2 ambient, where the formation of a thin film of erbium followed by the deposition of a tungsten capping layer has been done by Radio Frequency (RF) sputtering.

**Figure 3.** Schematic of the structure of (a) reference contacts fabricated with aluminum (Al) and (b) salicide contacts fabricated with erbium silicide (ErSi*x*) and tungsten (W) layer. Copyright 2011 The Japan Society of Applied Physics [8].

The wafer has been then loaded in lamp annealing chamber to finally form ErSi*x*. The wafer has been then brought back to the conventional process flow to finally form aluminum contacts. Electric characterization has been carried out and the main results are summarized in **Figure 4(a)** and **(b)**. As expected, the drivability has been improved by a factor of 2 on account of silicide contacts [8]. Furthermore, the maximum of the transconductance also increased and confirms the interest of low‐resistivity source and drain contacts to enhance performances of electronic circuits. Noise measurements have been performed in the linear regime and for *f* = 10 Hz. The result is presented in **Figure 5**. When compared with the noise level of the reference transistor, the noise level of the transistor featuring ErSi*x*/W contacts is greatly reduced for positive gate overdrive voltages while it remains equivalent for lower values. The same noise level below 0 V is explained by the fact that within this range of measurement the channel is exclusively contributing to the total noise and by the fact that both devices have indeed almost the same channel, since they followed the same process flow with regard to the fabrication of the gate stack. The noise modeling has been carried out and is reported with the lines in **Figure 5**. The modeling reveals that from 0 V the noise from the reference device moves away from the noise stemming from the channel. The contribution of the series resistances to the total noise is increasing and is ultimately taking over as the main contribution. However, the noise of the device featuring ErSi2/W contacts is following the curve depicting the noise stemming from the channel and starts to slightly move away at high voltages. The impact of the series resistances on the noise is barely visible. The use of low‐resistivity contacts allows a drastic reduction in the contribution of the series resistances to the total noise and let the channel be the sole source of noise over the entire measurement range [8]. About 10 times reduction of the access resistance has led to 100 times reduction of the noise level.

the best case [12]. A new structure [13] and new processes [14] have been developed in order to further lower down the series resistances. Instead of using the same silicide for both p‐

n‐MOSFETs, erbium has been selected to perfectly fit the requirement of n‐MOSFETs and palladium for the p‐MOSFETs. Additionally, tungsten metal stack above the thin silicide layer was introduced to reduce the sheet resistance and protect erbium from being oxidized. In order to confirm the above statement, two kinds of MOSFETs have been fabricated following the very same process flow, except during the source and drain contact fabrication stage. The source and drain contacts of the reference transistors have been fabricated with aluminum (Al), while erbium silicide associated with tungsten (ErSi2/W) has been used for the second set of transistors. The respective structures have been represented in **Figure 3(a)** and **(b)**. While both wafers followed the same process flow until the contact lithography step, the ErSi*x*/W wafer followed an advanced process entirely developed at New Industry Creation Hatchery Center (NICHe). In this advanced process, the wafer has been loaded in an N2 sealed cleaning chamber after a total room temperature five‐step cleaning, which has been followed by the dipping of the cleaned wafer in O3 dissolved ultra pure water in order to form a chemical oxide at the silicon surface. The removal of the chemical oxide has been carried out by diluted HF solution and the wafer has been then transferred in clustered sputter equipment, still in N2 ambient, where the formation of a thin film of erbium followed by the deposition of a tungsten capping

**Figure 3.** Schematic of the structure of (a) reference contacts fabricated with aluminum (Al) and (b) salicide contacts fabricated with erbium silicide (ErSi*x*) and tungsten (W) layer. Copyright 2011 The Japan Society of Applied Physics

The wafer has been then loaded in lamp annealing chamber to finally form ErSi*x*. The wafer has been then brought back to the conventional process flow to finally form aluminum contacts. Electric characterization has been carried out and the main results are summarized in **Figure 4(a)** and **(b)**. As expected, the drivability has been improved by a factor of 2 on account of silicide contacts [8]. Furthermore, the maximum of the transconductance also increased and confirms the interest of low‐resistivity source and drain contacts to enhance performances of electronic circuits. Noise measurements have been performed in the linear regime and for *f* = 10 Hz. The result is presented in **Figure 5**. When compared with the noise level of the reference transistor, the noise level of the transistor featuring ErSi*x*/W contacts is greatly reduced for positive gate overdrive voltages while it remains equivalent for lower values. The same noise

layer has been done by Radio Frequency (RF) sputtering.

8 Advances in Noise Analysis, Mitigation and Control

[8].

and

**Figure 4.** (a) Drain current and (b) transconductance versus gate overdrive voltage for n‐MOSFETs featuring Al and ErSi2/W contacts. Copyright 2011 The Japan Society of Applied Physics [8].

**Figure 5.** Normalized noise of n‐MOSFETs with contacts fabricated with either aluminum(Al) or erbium silicide/tung‐ sten(ErSi2/W). (W) layer. Copyright 2011 The Japan Society of Applied Physics [8].

## **4. Radical oxidation**

The gate stack, especially, the gate insulator, is the most critical part of the MOSFET, mainly because of the defects that can appear during the fabrication and its tremendous impact on the device performance [15]. It is absolutely true these days that the need of always‐faster devices and smaller chips also promote the appearance of undesirable effects such as increase of variability and random telegraph noise. Thermal oxidation has been the way, since the establishment of the MOSFET, to fabricate the gate insulator and while the generated SiO2 was at the beginning of poor quality, leading to high *S* parameters, the process has greatly evolved since and the growth of quality oxide can be achieved now [15]. Unfortunately, the dimension of nowadays MOSFETs has reached a threshold where *S* parameter and *V*th variability are a major issue, as well as, noise level prevision [16].

**Figure 6.** Diagrams of (a) single shower plate and (b) double shower plate equipment based radicals formed by the low electron temperature microwave high‐density plasma.

Thermal oxidation, from its intrinsic chemical reaction, cannot be optimized anymore and will always promote the formation of damage (either inside the gate stack or cap layers) and will partly invalidate the flattening process and, in turn, deteriorate the surface roughness of the wafer. Thus, new oxidation processes have been developed to avoid these issues. They are all based on radical oxidation rather than chemical reaction to form SiO2 [17]. The specificity of the damage‐free very low electron temperature microwave exited high‐density plasma is, as represented in **Figure 6**, that it can be employed for oxidation at low temperature, chemical vapor deposition, or even reactive ion etching. Very high quality gate insulator and reduced damages generally occurring during the etching and the fabrication of interconnect can be achieved thanks to this advanced process as shown in **Figure 7** [15, 18, 19]. Contrary to the thermal oxidation, the radical oxidation has an oxidation rate that is almost regardless of the orientation of the silicon crystal on which the oxide is grown [20]. Additionally, the radical oxidation does not only help reduce the interface trap density but also help preserve and even improve the flatness of the Si/SiO2 interface. Two sets of p‐MOSFETs have been fabricated in our clean room. They followed almost the same process flow; however, they differed in such a way that the first set featuring a radically grown oxide has been processed exclusively with advanced plasma equipment, while the second set has been processed using conventional processes, among which the thermal oxidation process.

**4. Radical oxidation**

10 Advances in Noise Analysis, Mitigation and Control

major issue, as well as, noise level prevision [16].

electron temperature microwave high‐density plasma.

The gate stack, especially, the gate insulator, is the most critical part of the MOSFET, mainly because of the defects that can appear during the fabrication and its tremendous impact on the device performance [15]. It is absolutely true these days that the need of always‐faster devices and smaller chips also promote the appearance of undesirable effects such as increase of variability and random telegraph noise. Thermal oxidation has been the way, since the establishment of the MOSFET, to fabricate the gate insulator and while the generated SiO2 was at the beginning of poor quality, leading to high *S* parameters, the process has greatly evolved since and the growth of quality oxide can be achieved now [15]. Unfortunately, the dimension of nowadays MOSFETs has reached a threshold where *S* parameter and *V*th variability are a

**Figure 6.** Diagrams of (a) single shower plate and (b) double shower plate equipment based radicals formed by the low

Thermal oxidation, from its intrinsic chemical reaction, cannot be optimized anymore and will always promote the formation of damage (either inside the gate stack or cap layers) and will partly invalidate the flattening process and, in turn, deteriorate the surface roughness of the wafer. Thus, new oxidation processes have been developed to avoid these issues. They are all based on radical oxidation rather than chemical reaction to form SiO2 [17]. The specificity of the damage‐free very low electron temperature microwave exited high‐density plasma is, as represented in **Figure 6**, that it can be employed for oxidation at low temperature, chemical vapor deposition, or even reactive ion etching. Very high quality gate insulator and reduced damages generally occurring during the etching and the fabrication of interconnect can be achieved thanks to this advanced process as shown in **Figure 7** [15, 18, 19]. Contrary to the thermal oxidation, the radical oxidation has an oxidation rate that is almost regardless of the orientation of the silicon crystal on which the oxide is grown [20]. Additionally, the radical oxidation does not only help reduce the interface trap density but also help preserve and even improve the flatness of the Si/SiO2 interface. Two sets of p‐MOSFETs have been fabricated in our clean room. They followed almost the same process flow; however, they differed in such a way that the first set featuring a radically grown oxide has been processed exclusively with

**Figure 7.** *V*th distributions of p‐MOSFETs fabricated by applying (a) conventional plasma processes and (b) radial line slot antenna plasma processes plotted for antenna ratio of 23, 100, 1000, and 10,000.

**Figure 8.** Normalized noise of p‐MOSFETs with a gate oxide fabricated by either thermal or plasma oxidation as a function of the gate overdrive voltage. The modeling, reported with the lines, refers exclusively to the transistor with a gate oxide thermally grown. Copyright 2011 The Japan Society of Applied Physics [8].

Noise measurements have been performed in the linear and saturated region and for different gate sizes. The noise analysis has been carried out at 10 Hz. Results are presented in **Figure 8**, and they clearly indicate that the p‐MOSFET with a gate oxide fabricated by radical oxidation has a lower noise level than when the thermal oxidation process is used used, with a maximal reduction of over a decade. As expected, the noise stemming from series resistances and the noise from the channel are both contributing to the total noise, with the latter one being ascribed to the insulator charge and induced mobility fluctuations. In order to understand the origin of the noise reduction, the modeling of the p‐MOSFETs, featuring a gate oxide, fabricated by radical oxidation has been carried out. The result is reported in **Figure 9** and it revealed an unexpected behavior, i.e., no induced mobility fluctuations. The contribution of the series resistances added to the sole insulator charge fluctuations has been enough to model the total noise. Even though the trapping/release mechanism at the origin of the 1/*f* noise induces fluctuation of the mobility, these fluctuations are, in the present case, too small to be visible when compared to the other fluctuations. The interface trap density has been extracted for both sets of devices and revealed a three‐time reduction in favor of the transistors fabricated using plasma processes with *N*d = 2 × 1016 cm–3 eV–1, testifying the high integrity of the oxide when fabricated by radical oxidation.

**Figure 9.** Normalized noise of p‐MOSFETs with a gate oxide fabricated by plasma oxidation as a function of the gate overdrive voltage. Copyright 2011 The Japan Society of Applied Physics [8].

## **5. Buried‐channel MOSFETs**

#### **5.1. Structure of buried‐channel MOSFET**

The low‐frequency noise, such as 1/*f* noise and random telegraph noise, is basically caused by the defects at the gate insulator and silicon interface in MOSFETs. To reduce the low‐frequency noise, the number of defects or the influence of the defects has to be reduced. Here, the channel of the buried‐channel MOSFETs is separated from the gate insulator/Si interface, and then the carriers in the channel are hard to be influenced by the defects at the interface. Usually, the buried‐channel MOSFETs are characterized as high mobility but weak short channel Field‐ Effect‐Transistor (FET) [21–23] because the carriers flow through the bulk. In this case, the separated channel location of buried‐channel MOSFETs is very important for reducing the low‐ frequency noise. **Figure 10** shows the schematic illustration of the buried‐channel n‐MOSFET. The buried layer was formed by the ion implantation at the *V*th adjustment process [24, 25]. **Figure 11** shows the band diagram of the conventional surface channel (a) and the buried‐ channel MOSFETs (b), respectively [25]. The buried layer depth was set as 170 nm from the SiO2/Si interface, and the channel was appeared at around 30 nm from the SiO2/Si interface at the bias conditions of back bias (*V*BS) =1.5 V and drain current (*I*DS) =100 nA for a gate length (*L*) of 0.22 μm and a gate width (W) of 0.28 μm MOSFETs.

**Figure 10.** Schematic illustration of the buried‐channel MOSFET.

has a lower noise level than when the thermal oxidation process is used used, with a maximal reduction of over a decade. As expected, the noise stemming from series resistances and the noise from the channel are both contributing to the total noise, with the latter one being ascribed to the insulator charge and induced mobility fluctuations. In order to understand the origin of the noise reduction, the modeling of the p‐MOSFETs, featuring a gate oxide, fabricated by radical oxidation has been carried out. The result is reported in **Figure 9** and it revealed an unexpected behavior, i.e., no induced mobility fluctuations. The contribution of the series resistances added to the sole insulator charge fluctuations has been enough to model the total noise. Even though the trapping/release mechanism at the origin of the 1/*f* noise induces fluctuation of the mobility, these fluctuations are, in the present case, too small to be visible when compared to the other fluctuations. The interface trap density has been extracted for both sets of devices and revealed a three‐time reduction in favor of the transistors fabricated using plasma processes with *N*d = 2 × 1016 cm–3 eV–1, testifying the high integrity of the oxide

**Figure 9.** Normalized noise of p‐MOSFETs with a gate oxide fabricated by plasma oxidation as a function of the gate

The low‐frequency noise, such as 1/*f* noise and random telegraph noise, is basically caused by the defects at the gate insulator and silicon interface in MOSFETs. To reduce the low‐frequency

overdrive voltage. Copyright 2011 The Japan Society of Applied Physics [8].

**5. Buried‐channel MOSFETs**

**5.1. Structure of buried‐channel MOSFET**

when fabricated by radical oxidation.

**Figure 11.** Diagrams of surface channel (a) and buried‐channel MOSFETs. The bias condition was set at *V*BS = 1.5 V and *I*DS = 100 nA.

#### **5.2. Low‐frequency noise characteristics**

**Figure 12** shows the 1/*f* noise characteristics for the surface channel and buried‐channel MOSFETs. Five samples were measured for each MOSFET. The size (*W*/*L*) of MOSFETs was 10 μm/5 μm and bias conditions were set as *V*DS = 1.5 V, *V*BS = 0 V, *I*DS = 1, 10, and 100 μA. *S*Id increases as the drain current increasing for both MOSFETs. The noise power *S*Id of the surface channel MOSFETs is proportional to 1/*f*. In contrast, *S*Id of the buried‐channel MOSFETs for the low *I*d cases is not proportional to 1/*f*, because the noise level was smaller than the floor noise of the measurement system. For the same drain current *I*d, the noise power of the buried‐ channel MOSFETs are less than that of the surface channel MOSFETs, especially their differ‐ ences are observed for the low drain current cases. In this experiment, the gate voltage controlled the drain current. When the gate voltage increases, the distance between the channel and the interface decreases, and then it influences the defects. This indicates that the noise reduction of buried channel is very effective for the low gate voltage conditions.

**Figure 12.** 1/*f* noise characteristics of (a) surface channel and (b) buried‐channel MOSFETs.

## **5.3.** *V***th variability**

In the previous section, it is described that the noise can be reduced by introducing the buried‐ channel MOSFETs because the channel is separated from the SiO2/Si interface. However, this means the gate capacitance becomes lower compared to the surface channel MOSFETs. Then, the *V*th control becomes difficult compared with the surface channel. **Figure 13** shows distributions of (a) *V*th and (b) the channel charge (*Q*ch) with 65536 MOSFETs for the buried‐ and surface channel MOSFETs [24]. These distributions are measured by the array test circuit, which can measure the *V*th variation and the random telegraph signal for many MOSFETs (>1 million MOSFETs) during very short time (<1 s) [26–29]. The horizontal axis of both the graphs shows the difference values between the average values of all MOSFETs (65536 MOSFETs). The *V*th variability of the buried‐channel MOSFET is larger than that the surface channel MOSFETs as shown in **Figure 13(a)**. It is considered that the gate capacitance of buried channel is smaller than that of the surface channel. Then, the horizontal axis is converted from *V*th to *Q*ch by using the gate‐channel capacitance. Almost the same distributions are observed, and then the variability is caused by the small capacitance between the gate and the channel. Moreover, it is noticed that the noise increases by the excess capacitance decrease with the other transistor characteristics degradation, such as short channel effect and subthreshold swing degradation [22, 30].

**Figure 13.** Distributions of (a) *V*th and (b) *Q*ch with 65536 MOSFETs for the buried‐ and surface channel MOSFETs.

## **6. Accumulation‐mode MOSFETs**

**5.2. Low‐frequency noise characteristics**

14 Advances in Noise Analysis, Mitigation and Control

**Figure 12** shows the 1/*f* noise characteristics for the surface channel and buried‐channel MOSFETs. Five samples were measured for each MOSFET. The size (*W*/*L*) of MOSFETs was 10 μm/5 μm and bias conditions were set as *V*DS = 1.5 V, *V*BS = 0 V, *I*DS = 1, 10, and 100 μA. *S*Id increases as the drain current increasing for both MOSFETs. The noise power *S*Id of the surface channel MOSFETs is proportional to 1/*f*. In contrast, *S*Id of the buried‐channel MOSFETs for the low *I*d cases is not proportional to 1/*f*, because the noise level was smaller than the floor noise of the measurement system. For the same drain current *I*d, the noise power of the buried‐ channel MOSFETs are less than that of the surface channel MOSFETs, especially their differ‐ ences are observed for the low drain current cases. In this experiment, the gate voltage controlled the drain current. When the gate voltage increases, the distance between the channel and the interface decreases, and then it influences the defects. This indicates that the noise

reduction of buried channel is very effective for the low gate voltage conditions.

**Figure 12.** 1/*f* noise characteristics of (a) surface channel and (b) buried‐channel MOSFETs.

In the previous section, it is described that the noise can be reduced by introducing the buried‐ channel MOSFETs because the channel is separated from the SiO2/Si interface. However, this means the gate capacitance becomes lower compared to the surface channel MOSFETs. Then, the *V*th control becomes difficult compared with the surface channel. **Figure 13** shows distributions of (a) *V*th and (b) the channel charge (*Q*ch) with 65536 MOSFETs for the buried‐ and surface channel MOSFETs [24]. These distributions are measured by the array test circuit, which can measure the *V*th variation and the random telegraph signal for many MOSFETs (>1 million MOSFETs) during very short time (<1 s) [26–29]. The horizontal axis of both the graphs shows the difference values between the average values of all MOSFETs (65536 MOSFETs). The *V*th variability of the buried‐channel MOSFET is larger than that the surface channel MOSFETs as shown in **Figure 13(a)**. It is considered that the gate capacitance of buried channel is smaller than that of the surface channel. Then, the horizontal axis is converted from *V*th to

**5.3.** *V***th variability**

The separation between the interface and the channel is effective in reducing the noise in buried‐channel MOSFETs; however, the controllability of *V*th is worse than conventional MOSFETs because the gate capacitance is reduced. Then the introduction of Silicon‐on‐ Insulator (SOI) wafer and a new structure can solve this issue [31, 32]. The so‐called accumu‐ lation‐mode MOSFET has been developed keeping this in mind. As depicted in **Figure 14**, the accumulation‐mode MOSFETs differ from MOSFETs in such a way that the type of the SOI layer is the same as that of the contact. Additionally, the type of polysilicon must also be adjusted as required. Although the working of the conventional inversion‐mode MOSFETs is based on the generation of an inversion layer made of the minority carrier, the accumulation‐ mode one is making use of an accumulation layer composed of majority carrier. Actually, the accumulation‐mode MOSFET without bias is at first on the off‐state since the SOI layer is completely depleted. When a bias is applied at the gate, a thin conductive layer of majority carrier is first generated at the back interface between the Buried Oxide (BOX) and the SOI. Then accumulation‐mode MOSFETs become on the on‐state. A current at the back interface flows from the source to the drain. A further increase in the bias makes this layer disappear and makes a short portion of the originally depleted SOI layer become neutral. A bulk current is generated on the BOX side. This current continues to increase with the expansion of the neutral region (due to the shrinking of the depleted region) inside the SOI until the bias applied at the gate reaches the flat‐band voltage.

**Figure 14.** Schematic representation of (a) inversion‐mode and (b) accumulation‐mode p‐MOSFETs.

**Figure 15.** Drivability of accumulation‐ and inversion‐mode p‐MOSFETs.

The current generated inside the accumulation layer then adds to the bulk current. Therefore, in addition to, among other things, having an improved reliability [33] and being immune to radiation effects [34], they also have a better drivability than inversion‐mode MOSFETs since the total current is the sum of those generated inside the SOI and the accumulation layer [35]. When the bulk current has reached its maximum, corresponding to the SOI completely neutral, the majority carrier accumulates at the front interface between the gate insulator and the SOI. Accumulation‐mode fully depleted SOI MOSFETs have been fabricated on Si(100) surface to investigate the noise characteristics. The SOI layer impurity has been adjusted by ion implan‐ tation to 2 × 1017 cm–3. The thickness of the SOI layer has been reduced down to 50 nm. In order to avoid the increase of noise due to the defects at the front interface and the impact of the surface roughness of the interface, the radical oxidation [17] added to the five‐step‐cleaning process [20] has been repeated four times until reaching a flattened interface with a roughness *Ra* of 0.08 nm. A 7‐nm gate oxide has been formed by radical oxidation using a low electron temperature microwave high‐density plasma process at 400°C. As expected and as shown in **Figure 15**, the accumulation‐mode MOSFETs feature a better drivability than those of the inversion‐mode MOSFETs. The better drivability does not only owe to two distinctive currents but also from the higher carrier mobility thanks to a lower transversal electric field at the front interface [35].

**Figure 14.** Schematic representation of (a) inversion‐mode and (b) accumulation‐mode p‐MOSFETs.

16 Advances in Noise Analysis, Mitigation and Control

**Figure 15.** Drivability of accumulation‐ and inversion‐mode p‐MOSFETs.

The current generated inside the accumulation layer then adds to the bulk current. Therefore, in addition to, among other things, having an improved reliability [33] and being immune to radiation effects [34], they also have a better drivability than inversion‐mode MOSFETs since the total current is the sum of those generated inside the SOI and the accumulation layer [35]. When the bulk current has reached its maximum, corresponding to the SOI completely neutral, the majority carrier accumulates at the front interface between the gate insulator and the SOI. Accumulation‐mode fully depleted SOI MOSFETs have been fabricated on Si(100) surface to investigate the noise characteristics. The SOI layer impurity has been adjusted by ion implan‐ tation to 2 × 1017 cm–3. The thickness of the SOI layer has been reduced down to 50 nm. In order to avoid the increase of noise due to the defects at the front interface and the impact of the surface roughness of the interface, the radical oxidation [17] added to the five‐step‐cleaning

**Figure 16.** Normalized noise of accumulation‐ and inversion‐mode p‐MOSFETs. Lines refer to the modeling of the noise of the sole accumulation‐mode device and represent the noise generated by each region.

The noise of the inversion‐ and accumulation‐mode MOSFETs has been reported in **Fig‐ ure 16**. Even though their noise level is similar at low gate overdrive voltage, the superiority of the accumulation‐mode MOSFET over the inversion mode one becomes clear for voltage over 0 V. A gap of more than 1 decade is achieved at the best. While the noise of the inversion‐ mode MOSFET can be ascribed to the interface traps [6] and series resistances at high bias [3], the origin of the noise for the accumulation‐mode one must be investigated with care. Indeed, as discussed earlier, three conduction mechanisms generate the current and are therefore generating their own noise. The modeling of each noise and finally the total modeled noise have been reported with lines in **Figure 16**. The noise stemming from the front and back interfaces is originating from the interface traps [5], like for the inversion‐mode MOSFET while the noise from the SOI layer and access resistances is explained in terms of fundamental fluctuations of the mobility of the Hooge model [3]. Contrary to the inversion‐mode MOSFET, the front interface does not contribute to the total noise. The lower noise can be attributed to a change in the origin of the noise stemming from the channel, with the SOI region becoming the main contributor to the noise with regard to the channel and a significant shift toward the high gate voltage to turn the accumulation layer on. It is worth mentioning that the advantage of the accumulation‐mode MOSFET is effective for high doping concentration; otherwise, the accumulation layer will act like an inversion one [36] since it implies no contribution from either the back interface or the SOI layer to the total current.

## **7. Conclusion**

In this chapter, we reviewed several ways to suppress the low‐frequency noise of MOSFETs and, in turn, the noise of analog and digital circuits. One of the most underrated approaches is to optimize the contacts and interconnects by the means of low‐resistive materials, so that their contribution to the total noise can be drastically reduced. It has also been shown that a great deal must be paid to the fabrication processes. Indeed, the use of processes demonstrating very low damage generation at all the stages of fabrication can lead to MOSFETs with better performances and especially reduced noise level due to a reduction of induced defects located at the gate stack and its surroundings. Additionally, these low‐defect processes based on the damage‐free, low‐temperature, high‐density plasma technology achieved a further reduction by means of the disappearance of one component of the low‐frequency noise, the induced mobility fluctuations one, bringing the noise of MOSFETs to the sole fluctuations of the insulator charges. Focusing on a different electronic structure can also achieve low‐noise MOSFETs. For example, minimizing the interaction carrier‐traps by moving away the channel – so that fewer traps are activated and less variations of the insulator charge are generated – can achieve a reduced noise. Unfortunately, this reduction is obtained at the cost of a degra‐ dation of the electrical performances. So, the most promising structure to suppress noise is in the form of the accumulation‐mode MOSFETs. Indeed, in addition to offer reduced low‐ frequency noise when compared to conventional MOSFETs, their electrical performances are greatly improved. These devices obviously feature various assets, which should consequently pave the road for a new era of very low noise and high‐performance MOSFETs and bring microelectronic manufacturers back to the realization of highly performances and high‐speed analog and digital circuits.

## **Author details**

Philippe Gaubert1\* and Akinobu Teramoto2

## **References**

high gate voltage to turn the accumulation layer on. It is worth mentioning that the advantage of the accumulation‐mode MOSFET is effective for high doping concentration; otherwise, the accumulation layer will act like an inversion one [36] since it implies no contribution from

In this chapter, we reviewed several ways to suppress the low‐frequency noise of MOSFETs and, in turn, the noise of analog and digital circuits. One of the most underrated approaches is to optimize the contacts and interconnects by the means of low‐resistive materials, so that their contribution to the total noise can be drastically reduced. It has also been shown that a great deal must be paid to the fabrication processes. Indeed, the use of processes demonstrating very low damage generation at all the stages of fabrication can lead to MOSFETs with better performances and especially reduced noise level due to a reduction of induced defects located at the gate stack and its surroundings. Additionally, these low‐defect processes based on the damage‐free, low‐temperature, high‐density plasma technology achieved a further reduction by means of the disappearance of one component of the low‐frequency noise, the induced mobility fluctuations one, bringing the noise of MOSFETs to the sole fluctuations of the insulator charges. Focusing on a different electronic structure can also achieve low‐noise MOSFETs. For example, minimizing the interaction carrier‐traps by moving away the channel – so that fewer traps are activated and less variations of the insulator charge are generated – can achieve a reduced noise. Unfortunately, this reduction is obtained at the cost of a degra‐ dation of the electrical performances. So, the most promising structure to suppress noise is in the form of the accumulation‐mode MOSFETs. Indeed, in addition to offer reduced low‐ frequency noise when compared to conventional MOSFETs, their electrical performances are greatly improved. These devices obviously feature various assets, which should consequently pave the road for a new era of very low noise and high‐performance MOSFETs and bring microelectronic manufacturers back to the realization of highly performances and high‐speed

either the back interface or the SOI layer to the total current.

**7. Conclusion**

18 Advances in Noise Analysis, Mitigation and Control

analog and digital circuits.

Philippe Gaubert1\* and Akinobu Teramoto2

\*Address all correspondence to<EMAIL_ADDRESS>
1 Graduate School of Engineering, Tohoku University, Sendai, Japan

2 New Industry Creation Hatchery Center, Tohoku University, Sendai, Japan

**Author details**

[26] Watabe S, Sugawa S, Teramoto A, Ohmi T. New Statistical Evaluation Method for the Variation of Metal‐Oxide‐Semiconductor Field‐Effect Transistors. Jpn J Appl Phys. 2007;46(4B):2054–7.

[14] Isogai T, Tanaka H, Goto T, Teramoto A, Sugawa S, Ohmi T. Formation and Property of Yttrium and Yttrium Silicide Films as Low Schottky Barrier material for n‐Type

[15] Gaubert P, Teramoto A, Hamada T, Yamamoto M, Kotani K, Ohmi T. 1/f Noise Sup‐ pression of pMOSFETs Fabricated on Si(100) and Si(110) Using an Alkali‐Free Cleaning

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20 Advances in Noise Analysis, Mitigation and Control

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## **Symplectic Principal Component Analysis: A Noise Reduction Method for Continuous Chaotic Systems Symplectic Principal Component Analysis: A Noise Reduction Method for Continuous Chaotic Systems**

Min Lei and Guang Meng Min Lei and Guang Meng

22 Advances in Noise Analysis, Mitigation and Control

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64410

## **Abstract**

In real environments, data obtained from a dynamic system were more or less contami‐ nated by noise. In addition, the dynamic system was often very complicated, non‐linear, even chaotic, and sometimes unknown. These might generally lead the measured data to be very complex and seemingly stochastic. In order to obtain the more intrinsic charac‐ teristics of the dynamical system, especially chaotic systems, from the measured data, how to effectively reduce noise was still a crucial issue. The aim of this chapter is to introduce a method of noise reduction based on symplectic geometry for the continuous chaotic systems with noise called symplectic principal component analysis (SPCA). The symplectic geometry was a kind of phase space geometry that could preserve the dynamical structure of the system, especially non‐linear structure. In symplectic space, the SPCA method could give the dominant principal component values of the data and the component values of the noise floor by the measure‐preserving symplectic transform. In the end, this chapter investigated the performance of the SPCA method and applied it to reduce noise in the chaotic time series and experimental data.

**Keywords:** symplectic geometry, noise reduction, chaotic continuous systems, sym‐ plectic principal component analysis (SPCA), principal component analysis (PCA), lo‐ cally projective non‐linear noise reduction (NNR), time series, sunspot, signal processing

## **1. Introduction**

Data measured from a system were often very complicated since the underlying dynamical system was usually complex, non‐linear, stochastic, or even unknown. And it was generally heavily corrupted by noise in experimental situations or real environments so that the data

© 2016 The Author(s). Licensee InTech. This chapter is 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. © 2016 The Author(s). Licensee InTech. This chapter is 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.

might be treated as noise and disregarded. Since chaotic phenomena had been discovered, interpretation of irregular dynamics of various systems as a deterministic chaotic process had been popular and widely studied in almost all fields of science and engineering, such as health sciences, nanosciences, physical sciences, economics, ecology, biomedicine, fault diagnosis, and so on. The seemingly random data had been reanalysed. A number of important algorithms based on chaos theory had been employed to distinguish between the chaotic data and noise, or reduce noise from the data, or infer the system dynamics from the data [1–6]. However, it was still challenging how to get the most information of the underlying dynamical system from a measured data. Meanwhile, how to appropriately reduce noise from a measured data was a first crucial issue.

At present, a number of noise reduction methods had been used for noisy contaminated chaotic times series [7–10]. These methods had mainly employed the delay‐embedded time series to reduce the noise of a measured data. In other words, the analysed data were firstly recon‐ structed into a phase space according to Takens' embedding theorem [11–12]. Then, various approaches were applied to deal with the reconstructed phase space, such as the singular value decomposition (SVD)‐based denoising method. In the study field of noise reduction, the SVD method was one of the most commonly used methods to reduce noise from a time series. However, the heart of the SVD was linear in nature so that it might become misleading technique when it dealt with a non‐linear time series [13]. For this, we proposed a novel method based on symplectic geometry, which was non‐linear in nature.

The symplectic geometry was a kind of phase space geometry that could preserve the system structure, especially non‐linear structure. Since Kang [14] had proposed a symplectic algorithm for solving symplectic differential, the symplectic geometry method had been widely used to investigate the equation solving problems of various complex dynamical systems in physics, mathematics, classical mechanics, quantum mechanics, elasticity and Hamiltonian mechanics, and so on [15–25]. For some bottleneck basic problems in elasticity, the novel symplectic approaches were explored and developed by Zhong and his group [17, 18] and references therein and Lim et al. [19–25] to study the numerical solutions of the plates and beams. Some researchers had also studied eigenvalue problems of the Hamilton matrix in symplectic geometry [26–34]. For main eigenvalues of a large Hamiltonian matrix, an inverse substation method and an adjoint symplectic inverse substation method had been proposed [26–27]. The symplectic elementary transformation had been used to solve the eigenvalues of the Hamilton matrices, because it could preserve the structures of the Hamiltonian matrices [28–33]. A new algorithm (SROSH) for computing the SR factorization was proposed by optimal symplectic Householder transformations [28]. A detailed error analysis of the (SROSH) method was described by Salam and Al‐Aidarous [30]. For sparse and large‐structured matrices, some modified versions were usually involved in structure‐preserving Krylov subspace‐type methods [33]. The SR decomposition could be obtained using a symplectic QR‐like decompo‐ sition [29–32] or symplectic Gram‐Schmidt algorithm [34]. More results on numerical aspects of symplectic Gram‐Schmidt algorithms could be found in the study of Salam [34]. These methods based on symplectic geometry had mainly been used to solve the eigenvalue problems of the 2*n* × 2*n* real matrices or Hamiltonian matrices in the systems of dynamical mechanics and control theory. To our knowledge, a few literatures had employed symplectic geometry theory to analyse the data generated from the non‐linear systems [35–39]. The purpose of this chapter was to use symplectic geometry method to reduce noise from the chaotic time series and experimental data.

The remainder of this chapter was organized as follows. Section 2 introduced the symplectic principal component analysis (SPCA) method based on Householder transform. Section 3 provides the reduced noise method based on symplectic geometry and its algorithm. In Section 4, numerical and experimental data were described. Section 5 was devoted to analysing the noise reduction of these data. In Section 6, a general conclusion was given.

## **2. Symplectic principal component analysis (SPCA)**

might be treated as noise and disregarded. Since chaotic phenomena had been discovered, interpretation of irregular dynamics of various systems as a deterministic chaotic process had been popular and widely studied in almost all fields of science and engineering, such as health sciences, nanosciences, physical sciences, economics, ecology, biomedicine, fault diagnosis, and so on. The seemingly random data had been reanalysed. A number of important algorithms based on chaos theory had been employed to distinguish between the chaotic data and noise, or reduce noise from the data, or infer the system dynamics from the data [1–6]. However, it was still challenging how to get the most information of the underlying dynamical system from a measured data. Meanwhile, how to appropriately reduce noise from a measured data was a first

At present, a number of noise reduction methods had been used for noisy contaminated chaotic times series [7–10]. These methods had mainly employed the delay‐embedded time series to reduce the noise of a measured data. In other words, the analysed data were firstly recon‐ structed into a phase space according to Takens' embedding theorem [11–12]. Then, various approaches were applied to deal with the reconstructed phase space, such as the singular value decomposition (SVD)‐based denoising method. In the study field of noise reduction, the SVD method was one of the most commonly used methods to reduce noise from a time series. However, the heart of the SVD was linear in nature so that it might become misleading technique when it dealt with a non‐linear time series [13]. For this, we proposed a novel method

The symplectic geometry was a kind of phase space geometry that could preserve the system structure, especially non‐linear structure. Since Kang [14] had proposed a symplectic algorithm for solving symplectic differential, the symplectic geometry method had been widely used to investigate the equation solving problems of various complex dynamical systems in physics, mathematics, classical mechanics, quantum mechanics, elasticity and Hamiltonian mechanics, and so on [15–25]. For some bottleneck basic problems in elasticity, the novel symplectic approaches were explored and developed by Zhong and his group [17, 18] and references therein and Lim et al. [19–25] to study the numerical solutions of the plates and beams. Some researchers had also studied eigenvalue problems of the Hamilton matrix in symplectic geometry [26–34]. For main eigenvalues of a large Hamiltonian matrix, an inverse substation method and an adjoint symplectic inverse substation method had been proposed [26–27]. The symplectic elementary transformation had been used to solve the eigenvalues of the Hamilton matrices, because it could preserve the structures of the Hamiltonian matrices [28–33]. A new algorithm (SROSH) for computing the SR factorization was proposed by optimal symplectic Householder transformations [28]. A detailed error analysis of the (SROSH) method was described by Salam and Al‐Aidarous [30]. For sparse and large‐structured matrices, some modified versions were usually involved in structure‐preserving Krylov subspace‐type methods [33]. The SR decomposition could be obtained using a symplectic QR‐like decompo‐ sition [29–32] or symplectic Gram‐Schmidt algorithm [34]. More results on numerical aspects of symplectic Gram‐Schmidt algorithms could be found in the study of Salam [34]. These methods based on symplectic geometry had mainly been used to solve the eigenvalue problems of the 2*n* × 2*n* real matrices or Hamiltonian matrices in the systems of dynamical

based on symplectic geometry, which was non‐linear in nature.

crucial issue.

24 Advances in Noise Analysis, Mitigation and Control

In the real life, the studied system could usually give one observable that was a noisy one‐ dimensional (1D) signal sampled with a finite precision. Little was known about a system equation or its phase portrait. One only could reconstruct the original system from the sampled noisy 1D signal to study the dynamical characteristics of the system. First, a time series was constructed into a trajectory matrix X (i.e., an attractor in the phase space) in terms of Takens' embedding theorem. Second, the reconstructed attractor was transformed into a Hamiltonian matrix in symplectic space. Then, the symplectic transformations were used to deal with the Hamiltonian matrix [40–41].

#### **2.1. Phase space reconstruction (Attractor reconstruction)**

A dynamical system was defined in phase space *Rd* . A discretized trajectory *X* at times *t* = *nt s*, *n* = 1, 2, … might be described by maps of the form

$$\mathbf{x}\_{n+1} = f(\mathbf{x}\_n) \tag{2.1}$$

where *x*1, *x*2, ⋯, *xn* is the measured data, that is, the observable of the system under study, *ts* is sampling interval, *n* is the number of samples, and *d* is the dimension of phase space. According to Takens' embedding theorem, if *d* was large enough, then under certain genericity assump‐ tions, a one‐to‐one image of the original set {*x*} was given from the time series. The time‐delay embedding approach could map the time series {*x*} into a *d*‐dimensional space *Rd* :

$$X = \begin{bmatrix} X\_1^T \\ X\_2^T \\ \vdots \\ X\_m^T \end{bmatrix} = \begin{bmatrix} \mathbf{x}\_1 & \mathbf{x}\_2 & \cdots & \mathbf{x}\_d \\ \mathbf{x}\_2 & \mathbf{x}\_3 & \cdots & \mathbf{x}\_{d+1} \\ \vdots & \vdots & \cdots & \vdots \\ \mathbf{x}\_m & \mathbf{x}\_{m+1} & \cdots & \mathbf{x}\_n \end{bmatrix} \tag{2.2}$$

where *m* = *n* ‐ *d* + 1 is the number of dots in *d*‐dimensional reconstruction attractor, *Xm*×*<sup>d</sup>* denoted the trajectory matrix of the dynamical system in phase space, that is, the attractor in phase space. The matrix *X* contained all the dynamical information of the system that generated the data *x*.

#### **2.2. Basic theory and concept of symplectic geometry**

In common, the referred space was Euclidean space *Rn*. The architecture of Euclidean space was dependent on the bilinear symmetrical non‐singular cross product:

If *x*≠ 0,

$$\begin{cases} \{\mathbf{x}, \mathbf{x}\} > 0\\ \mathbf{x}, \quad \|\mathbf{x}\| = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle} > 0. \end{cases} \tag{2.3}$$

This showed that the measurement of Euclidean space was length scale. Unlike Euclidean space, the concept of orthogonal cross course existed in symplectic space. Symplectic space was the space with a special symplectic structure and dependent on a bilinear antisymmetric non‐singular cross product—symplectic cross product:

$$\{\mathbf{x}, \mathbf{y}\} = \{\mathbf{x}, \mathbf{b}\},\tag{2.4}$$

where

$$J = J\_{2n} = \begin{bmatrix} \theta & +I\_n \\ -I\_n & \theta \end{bmatrix}.$$

When *n* = 1, there was

$$
\begin{bmatrix} \mathbf{x}, \mathbf{y} \end{bmatrix} = \begin{vmatrix} \mathbf{x}\_1 & \mathbf{y}\_1 \\ \mathbf{x}\_2 & \mathbf{y}\_2 \end{vmatrix}.
$$

In traditional meaning, odd dimension concept did not exist in the non‐singular antisymmetric matrix, and hence, symplectic space was even dimension.

Therefore, the measurement of symplectic space was area scale. The length of arbitrary vector in symplectic space always equalled to zero and without any signification. This was essential difference between symplectic and Euclidean spaces [14–16, 42]. In symplectic geometry, the symplectic similar transform was the regular transform, which could preserve measure and keep the essential character of the primary time series unchanged, so it was fit to analyse non‐ linear system process.

**Definition 2.1.** *Let S be a matrix, if JSJ*‐1 = *S-\*, then S was a symplectic matrix.*

**Definition 2.2.** *Let H be a matrix, if JHJ*‐1 = ‐*H\* , then H was a Hamilton matrix.*

**Theorem 2.1.** *Any n × n matrix could be made into a Hamilton matrix. Let a matrix as An×n, if the matrix M could be constructed as follows:*

$$M = \begin{pmatrix} A & 0 \\ 0 & -A^\* \end{pmatrix} \tag{2.5}$$

*M is a Hamilton matrix*.

Proof:

(2.3)

(2.4)

where *m* = *n* ‐ *d* + 1 is the number of dots in *d*‐dimensional reconstruction attractor, *Xm*×*<sup>d</sup>* denoted the trajectory matrix of the dynamical system in phase space, that is, the attractor in phase space. The matrix *X* contained all the dynamical information of the system that generated the

In common, the referred space was Euclidean space *Rn*. The architecture of Euclidean space

, 0

\= > *x x x xx*

> [ ] *x, y = x, Jy ,*

*J = J =*

*2n*

>

This showed that the measurement of Euclidean space was length scale. Unlike Euclidean space, the concept of orthogonal cross course existed in symplectic space. Symplectic space was the space with a special symplectic structure and dependent on a bilinear antisymmetric

, 0.

é ù ê ú ë û *n*

2 2 , . *x y x y*

 *0*  *.*

*n*

*0 +I*

*-I*

[ ] 1 1

In traditional meaning, odd dimension concept did not exist in the non‐singular antisymmetric

Therefore, the measurement of symplectic space was area scale. The length of arbitrary vector in symplectic space always equalled to zero and without any signification. This was essential difference between symplectic and Euclidean spaces [14–16, 42]. In symplectic geometry, the symplectic similar transform was the regular transform, which could preserve measure and keep the essential character of the primary time series unchanged, so it was fit to analyse non‐

*x y* =

**2.2. Basic theory and concept of symplectic geometry**

26 Advances in Noise Analysis, Mitigation and Control

non‐singular cross product—symplectic cross product:

matrix, and hence, symplectic space was even dimension.

was dependent on the bilinear symmetrical non‐singular cross product:

data *x*.

If *x*≠ 0,

where

When *n* = 1, there was

linear system process.

$$\begin{aligned} &\stackrel{\cdot}{\cdot} \qquad J\begin{pmatrix} A & 0 \\ 0 & -A^\* \end{pmatrix} J^{-1} \\ &= \begin{pmatrix} 0 & I\_n \\ -I\_n & 0 \end{pmatrix} \begin{pmatrix} A & 0 \\ 0 & -A^\* \end{pmatrix} \begin{pmatrix} 0 & I\_n \\ -I\_n & 0 \end{pmatrix}^{-1} \\ &= \begin{pmatrix} -A^\* & 0 \\ 0 & A \end{pmatrix} \\ &= -\begin{pmatrix} A & 0 \\ 0 & -A^\* \end{pmatrix} \end{aligned}$$

∴ According to Definition 2.2, *M* was a Hamilton matrix.□

**Theorem 2.2.** *Hamilton matrix kept unchanged at symplectic similar transform.*

Proof:

Let *S* as a symplectic transform matrix, *H* as a Hamilton matrix. Then, *S*‐1 is also symplectic matrix. According to Definitions 2.1 and 2.2, there was

$$J\left(SHS^{-1}\right)J^{-1}$$

$$=J\Sigma J^{-1}JHf^{-1}JS^{-1}J^{-1}$$

$$=S^{-\*}\left(-H^\*\right)S^\*$$

$$=-\left(SHS^{-1}\right)^\*$$

∴ *SHS*‐1 is also a Hamilton matrix.

## ∴ *SHS*‐1 ~ *H*

So, Hamilton matrix *H* kept unchanged at symplectic similar transform.

**Theorem 2.3.** *Let M* ∈ *C*<sup>2</sup>*d* × 2*<sup>d</sup> be Hamilton matrix*, *so eM was symplectic matrix.*

**Theorem 2.4.** *The product of sympletcic matrices was also a symplectic matrix.*

Proof:

Let *S*1, *S*2, …, *S*n as symplectic matrix, respectively. According to Definition 2.1, there were

$$\begin{aligned} J S\_1 J^{-1} &= S\_1^{-\*} \\ J S\_2 J^{-1} &= S\_2^{-\*} \\ \cdots \\ J S\_n J^{-1} &= S\_n^{-\*} \\ \\ J \left( S\_1 S\_2 \cdots S\_n \right) J^{-1} \\ &= J S\_1 J^{-1} J S\_2 J^{-1} J \cdots J^{-1} J S\_n J^{-1} \\ &= S\_1^{-\*} S\_2^{-\*} \cdots S\_n^{-\*} \\ &= \left( S\_1 S\_2 \cdots S\_n \right)^{-\*} \end{aligned}$$

So, the product of sympletcic matrixes was also a symplectic matrix.

**Theorem 2.5** *Suppose a Householder matrix H was*

$$H = H(k, o) = \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix},\tag{2.6}$$

$$\begin{aligned} P &= I\_{\boldsymbol{\pi}} - \frac{2\boldsymbol{\sigma}\boldsymbol{\sigma}\boldsymbol{\sigma}}{\boldsymbol{\sigma}^\*\boldsymbol{\sigma}}, \\ \boldsymbol{\varpi} &= \left(0, \cdots, 0; \boldsymbol{\alpha}\_{\boldsymbol{\ell}}, \cdots, \boldsymbol{\alpha}\_{\boldsymbol{\iota}}\right)^{\boldsymbol{\tau}} \neq 0. \end{aligned}$$

So, *H* is a symplectic unitary matrix. \* is conjugate transposition.

Proof:

In order to prove that the matrix *H* was symplectic matrix, we only needed to prove *H*\* *JH* = *J*.

$$\begin{aligned} \stackrel{\circ}{H}^\* \stackrel{\circ}{J} &= \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix}^\* J \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} \\ &= \begin{pmatrix} 0 & P^\*P \\ -P^\*P & 0 \end{pmatrix} \end{aligned} \tag{2.7}$$

Symplectic Principal Component Analysis: A Noise Reduction Method for Continuous Chaotic Systems http://dx.doi.org/10.5772/64410 29

$$
\cdots \colon P = I\_n - \frac{2\sigma\sigma\sigma^\*}{\sigma^\*\sigma}
$$

$$
\therefore \quad P^\* = P
$$

$$\begin{split}P\,^\*P &= P^2\\ &= \left(I\_n - \frac{2\sigma\sigma\sigma^\*}{\varpi^\*\varpi}\right)\left(I\_n - \frac{2\sigma\sigma\sigma^\*}{\varpi^\*\varpi}\right)\\ &= I\_n - \frac{4\sigma\sigma\sigma^\*}{\varpi^\*\varpi} + \frac{4\sigma(\varpi^\*\varpi)\varpi^\*}{(\varpi^\*\varpi)(\varpi^\*\varpi)}\\ &= I\_n\end{split}\tag{2.8}$$

where = (0, ⋯, 0; *ωk* ,... , *ωn*) *<sup>T</sup>* ≠ 0

**Theorem 2.3.** *Let M*

Proof:

∈

28 Advances in Noise Analysis, Mitigation and Control

 *C*<sup>2</sup>*d* × 2*<sup>d</sup> be Hamilton matrix*, *so eM was symplectic matrix.*

Let *S*1, *S*2, …, *S*n as symplectic matrix, respectively. According to Definition 2.1, there were

( )

*J SS S J*

L

*SS S SS S*

L L

*n*

1 2

= = =

So, the product of sympletcic matrixes was also a symplectic matrix.

*H Hk*

*n*

= -

*P I*

v

\*

**Theorem 2.5** *Suppose a Householder matrix H was*

So, *H* is a symplectic unitary matrix. \*

Proof:

L

1

*n n*

*JS J JS J J J JS J*

11 11

0

*P*

*T*

is conjugate transposition.

è ø (2.6)

*JH* = *J*.

(2.7)

(, ) , <sup>0</sup> *P*

( )

= L L ¹

 ww

0, ,0; , , 0.

*k n*

\* \* <sup>2</sup> ,

vv

v v

In order to prove that the matrix *H* was symplectic matrix, we only needed to prove *H*\*

\*

æ ö <sup>=</sup> ç ÷ è ø -

*P P*

0

*H JH J*

\*

æ öæ ö <sup>=</sup> ç ÷ç ÷ è øè ø

0 0

*P P*

\*

0

0 0

*P P P P*

w

æ ö = = ç ÷

*n*

L

1 1 1 1 2 2

*JS J S JS J S*

= =

1

*JS J S*

*n n*

=

( )

**Theorem 2.4.** *The product of sympletcic matrices was also a symplectic matrix.*

Plugging Eq. (2.8) into Eq. (2.7), we had:

$$\text{H}^\* \text{J} \text{H} = \text{J} \tag{2.9}$$

∴ *H* is a symplectic matrix.

$$\begin{aligned} \boldsymbol{H}^\* \boldsymbol{H} &= \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} \\ &= \begin{pmatrix} P^\*P & 0 \\ 0 & P^\*P \end{pmatrix} \\ &= \boldsymbol{I}\_{2n} \end{aligned} \tag{2.10}$$

∴ *H* is also a unitary matrix.

∴ The Householder matrix *H* is a symplectic unitary matrix.

In the real calculation, the above householder matrix *H* could be constructed from a time series according to Theorem 2.6.

**Theorem 2.6.** *Suppose x and y were two unequal n dimension vectors, and* <sup>2</sup> <sup>=</sup> <sup>2</sup>, *so there was elementary reflective array* = 1 − 2 , *which made Hx = y, where*  = − − <sup>2</sup> *.*

For a non‐zero *n* dimension vector *x* = (*x*1, *x*2, ⋯, *xn*) *<sup>T</sup>* we could note = <sup>2</sup>. Then, there was

$$H\mathbf{x} = \alpha \,\, \mathbf{e}\_1,\tag{2.11}$$

$$\begin{cases} H = 1 - 2\sigma \sigma \sigma^T \\ \mathbf{e}\_1 = (1, 0, \dots, 0)^T \\ \sigma = \frac{1}{\rho} (\mathbf{x} - \alpha \mathbf{e}\_1) \\ \rho = \left\| \mathbf{x} - \alpha \mathbf{e}\_1 \right\|\_2 \end{cases} \tag{2.12}$$

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

It was easy to testify that the elementary reflective array *H* was symmetry matrix (*HT* = *H*), orthogonal matrix (*HTH* = 1) and involution matrix (*H*<sup>2</sup> = 1).

#### **2.3. Building Hamiltonian matrix**

For a time series *x*, the covariance matrix *A* of the matrix *X* (see Eq. (2.2)) could first be given as follows:

$$A = \overline{X} \cdot \overline{X}^r \tag{2.13}$$

Where was a mean‐centred matrix by removing the mean values of the columns of the data matrix *A* is a *d × d* symmetrical matrix. Then, the Hamiltonian matrix *M* could be constructed by Eq. (2.5).

#### **2.4. Symplectic principal component analysis (SPCA)**

For Hamilton matrix *M* (2*n* × 2*n*), its eigenvalues could be obtained by symplectic similar transform, such as symplectic *QR* decomposition. At present, there had been some algorithms for symplectic *QR* decomposition [28–34, 43]. Here, we could use symplectic Householder transform instead of the matrix *Q* to decompose the matrix *M*:

$$M = \underline{Q}R\tag{2.14}$$

where *Q* is a Householder matrix, that is, a symplectic unitary matrix (the eigenvector matrix). *R* is an upper triangular matrix. It was easy to prove that the Householder matrix *Q* was a symplectic unitary matrix according to Theorem 2.5.

Symplectic Principal Component Analysis: A Noise Reduction Method for Continuous Chaotic Systems http://dx.doi.org/10.5772/64410 31

In theory, for a Hamiltonian matrix = 0 <sup>0</sup> − (see Eq. (2.5)), there was a Householder matrix *H*, which made *HMHT* as an upper Hessenberg matrix, namely

$$
\hat{H}HM^{\top} = \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} \begin{pmatrix} A & 0 \\ 0 & -A^{\top} \end{pmatrix} \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix}^{\top} \tag{2.15}
$$

$$
\begin{aligned}
&= \begin{pmatrix} PAP^{\top} & 0 \\ 0 & -PA^{\top}P^{\top} \end{pmatrix} \\ &= \begin{pmatrix} B & 0 \\ 0 & -B^{\top} \end{pmatrix} \\ &\ddots \quad \mathcal{A}(A) = \mathcal{A}(B) \tag{2.16}
\end{aligned}
$$

Thus, the primary 2*n*‐dimension space could be transformed into *n*‐dimension space to resolve. The symplectic eigenvalues (**μ** = {*μ*1, *μ*2, ⋯, *μ*2*<sup>d</sup>*}) of the matrix *M* could be composed of those (*λ*(*A*) = {*μ*1, *μ*2, ⋯, *μd*}) of the matrix *A*. The symplectic eigenvectors of the Householder matrix *H* could consist of those of the matrix *P* corresponding to the symplectic eigenvalues of the matrix *A*. These eigenvalues were given by descending order as following:

$$
\mu\_1 > \mu\_2 > \cdots > \mu\_l > \mu\_{l+1} \ge \cdots \ge \mu\_d \tag{2.17}
$$

The above symplectic eigenvalue method could be used to analyse the principal components of a system, called symplectic principal component analysis (SPCA) [40].

## **3. Noise reduction based on SPCA**

<sup>1</sup> *Hx e* = a

> 1 2 (1,0, ,0)

1

<sup>í</sup> = - <sup>ï</sup>

r

1

v

ï <sup>ï</sup> = - <sup>î</sup>

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

orthogonal matrix (*HTH* = 1) and involution matrix (*H*<sup>2</sup>

**2.4. Symplectic principal component analysis (SPCA)**

symplectic unitary matrix according to Theorem 2.5.

transform instead of the matrix *Q* to decompose the matrix *M*:

**2.3. Building Hamiltonian matrix**

30 Advances in Noise Analysis, Mitigation and Control

as follows:

by Eq. (2.5).

r

*H e*

<sup>ì</sup> = - <sup>ï</sup> <sup>ï</sup> <sup>=</sup> <sup>ï</sup>

( )

 a

*x e*

vv

L

*x e*

 a

1 2

It was easy to testify that the elementary reflective array *H* was symmetry matrix (*HT* = *H*),

For a time series *x*, the covariance matrix *A* of the matrix *X* (see Eq. (2.2)) could first be given

Where was a mean‐centred matrix by removing the mean values of the columns of the data matrix *A* is a *d × d* symmetrical matrix. Then, the Hamiltonian matrix *M* could be constructed

For Hamilton matrix *M* (2*n* × 2*n*), its eigenvalues could be obtained by symplectic similar transform, such as symplectic *QR* decomposition. At present, there had been some algorithms for symplectic *QR* decomposition [28–34, 43]. Here, we could use symplectic Householder

where *Q* is a Householder matrix, that is, a symplectic unitary matrix (the eigenvector matrix). *R* is an upper triangular matrix. It was easy to prove that the Householder matrix *Q* was a

= 1).

*<sup>T</sup> A XX* = × (2.13)

*M QR* = (2.14)

1

*T T*

, (2.11)

(2.12)

For the above Hamiltonian matrix *M*, its symplectic eigenvalues **μ** = {*μ*1, *μ*2, ⋯, *μ*2*<sup>d</sup>*} could be got by using symplectic *QR* decomposition in Section 2.4. According to Eq. 2.17, the dominant symplectic eigenvalues *μi* (*i* = 1, …, *l*) of *A* could be obtained, that is

$$
\mu\_1 > \mu\_2 > \dots > \mu\_l \tag{3.1}
$$

when *μi* >> *μi+*1. The lower symplectic eigenvalues *μi* , (*i* =*l*+1, …, *d*) got into a noise floor. Consequently, noise could be reduced from the data *x* by eliminating the lower eigenvalue components *μi* , (*i* =*l*+1, …, *d*). Corresponding to the dominant components *μi* , (*i* = 1, …, *l*), the truncation matrix *W* could be got from the Householder matrix *H*. The new data with reduced noise could be generated using *W* to transform the original data *x*. The procedure of the SPCA method was given as follows:

$$A = \begin{pmatrix} a\_{11} & a\_{12} & \cdots & a\_{1n} \\ a\_{21} & a\_{22} & \cdots & a\_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ a\_{n1} & a\_{n2} & \cdots & a\_{nn} \end{pmatrix} = \begin{pmatrix} a\_{11} & A\_{12}^{(1)} \\ a\_{21}^{(1)} & A\_{22}^{(1)} \end{pmatrix} \tag{3.2}$$

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

$$S^{(1)} = \left(a\_{11}^{(1)}, a\_{21}^{(1)}, \dots, a\_{n1}^{(1)}\right)^{\Gamma} = \left(a\_{11}, a\_{21}, \dots, a\_{n1}\right) \tag{3.3}$$

Then, the elementary reflective matrix *P*(1) is computed by:

$$P^{(l)} = I - \Im \sigma^{(l)} (\varpi^{(l)})^T \tag{3.4}$$

where

$$\begin{cases} \begin{aligned} \alpha\_1 &= \left\| S^{(1)} \right\|\_2 \\ E^{(1)} &= \left( 1, \, 0, \, \cdots, 0 \right)^T \qquad n \times 1 \\ \rho\_1 &= \left\| S^{(1)} - \alpha\_1 E^{(1)} \right\| \\ \sigma^{(1)} &= \frac{1}{\rho\_1} \left( S^{(1)} - \alpha\_1 E^{(1)} \right) \end{aligned} \tag{3.5}$$

So, *A* is changed to a matrix *A*(2):

Symplectic Principal Component Analysis: A Noise Reduction Method for Continuous Chaotic Systems http://dx.doi.org/10.5772/64410 33

$$P^{(1)}A = \begin{pmatrix} \sigma\_1 & a\_{12}^{(2)} & \cdots & a\_{1s}^{(2)} \\ 0 & a\_{22}^{(2)} & \cdots & a\_{2s}^{(2)} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & a\_{n2}^{(2)} & \cdots & a\_{nn}^{(2)} \end{pmatrix} = A^{(2)} \tag{3.6}$$

where the first element of the first column is *σ*1. Other elements were zeros. Then, the second column vector of *A*(2) is also given like the above way. Let

$$S^{(2)} = \left(0, a\_{22}^{(2)}, \cdots, a\_{n2}^{(2)}\right)^{\mathrm{T}} \tag{3.7}$$

construct *P*(2) matrix:

$$P^{(2)} = I - \Im \alpha^{(2)} (\varpi^{(2)})^T \tag{3.8}$$

where

noise could be generated using *W* to transform the original data *x*. The procedure of the SPCA

**3.** Calculate the matrix *P* of the Householder matrix *H* from the matrix *A* [37]. Let *A* as

21 22 2 11 12

ç ÷ æ ö = = ç ÷

th column of 2

( ) ( ) (1) (1) (1) (1) 11 21 1 11 21 1 , ,, , ,, *<sup>T</sup>*

> (1) (1) (1) 2()*<sup>T</sup> P I* = v v

> > ( )

*E n S E*

<sup>ï</sup> = ´ <sup>ï</sup>

L

(1) (1) 1 1 (1) (1) (1)

 a

(1) <sup>1</sup> <sup>2</sup>

*S*

1

r

ï = -

1

<sup>í</sup> = - <sup>ï</sup>

( )

 a

*S E*

1

1, 0, , 0 1

*T*

*aa a a A <sup>A</sup>*

L L M MLM L

*n n*

(1)

(3.2)

(3.5)

(1) (1) 21 22

è ø

a

*A*

<sup>1</sup> ≠ 0, otherwise this column would be skipped and the next column

*n n S a a a aa a* = = L L (3.3)

<sup>1</sup> ≠ 0. Set first column vector of *A*:

(3.4)

**1.** Reconstruct the trajectory matrix *X* from the raw data *x* in terms of Eq. (2.2);

11 12 1

æ ö

*aa a*

*n n nn*

è ø

*aa a*

1 2

Then, the elementary reflective matrix *P*(1) is computed by:

(1)

r

v

ï

î

So, *A* is changed to a matrix *A*(2):

a

<sup>ì</sup> <sup>=</sup> <sup>ï</sup>

**2.** Build the real *d* × *d* symmetry matrix *A* (see Eq. (2.13));

method was given as follows:

32 Advances in Noise Analysis, Mitigation and Control

First, suppose 21

where

would be considered until the *i*

follows:

$$\begin{cases} \begin{aligned} \alpha\_2 &= \left\| S^{(2)} \right\|\_2 \\ E^{(2)} &= \left( 0, 1, 0, \cdots, 0 \right)^T \end{aligned} & n \times 1 \\ \rho\_2 &= \left\| S^{(2)} - \alpha\_2 E^{(2)} \right\| \\ \left\| \boldsymbol{\sigma}^{(2)} = \frac{1}{\rho\_2} \left( S^{(2)} - \alpha\_2 E^{(2)} \right) \right\| \end{aligned} \tag{3.9}$$

Thus, the second column of *A*(2) is also changed to all zero vector except the first and second elements. *A*(3) is obtained:

$$P^{(2)}A^{(2)} = A^{(3)} \tag{3.10}$$

By repeating above‐mentioned method, the matrix *P* could be obtained until *A*(*n*) became an upper triangle matrix:

$$P = P^{(n)} P^{(n-1)} \cdots P^{(1)} \tag{3.11}$$

It was easy to show that *P* was a symmetrical, orthogonal and involution matrix.

**4.** Use the matrix *P* to transform the matrix *A* into the upper triangular matrix *T*. The absolute values of the diagonal components *Tii* by descending order were called as the symplectic eigenvalues of the matrix *A* (see Eq. (2.17));

$$Q = \begin{pmatrix} W & 0 \\ 0 & W \end{pmatrix} \tag{3.12}$$

$$S = \begin{pmatrix} X & 0 \\ 0 & -X \end{pmatrix} \tag{3.13}$$

where *X* is given by Eq. (2.2).

**7.** Use the *Q* project *S* into *Y*,

$$Y = \underline{Q}^{\overline{r}} \, S \tag{3.14}$$

**8.** Reestimate the with reduced‐noise,

$$
\hat{X} = QY \tag{3.15}
$$

The reduced‐noise data could be given by the first row of the matrix .

For the heavily noisy time series, the first estimation of data was usually not good. Here, one could repeat the above steps 7 and 8 several times. Generally, the second or third reconstructed data would be better than the first reconstructed data.

## **4. Numerical and experimental data**

This chapter employed the three typical chaotic equations [41].

**1.** Lorenz equation

$$\begin{cases} \dot{\mathbf{x}} = \sigma \left( \mathbf{y} - \mathbf{x} \right) \\ \dot{\mathbf{y}} = \mathbf{y}\mathbf{x} - \mathbf{y} - \mathbf{x}\mathbf{z} \\ \dot{\mathbf{z}} = -bz + \mathbf{x}\mathbf{y} \end{cases} \tag{4.1}$$

where *σ* = 10, *b* = 8/3, and *γ* = 28.

Symplectic Principal Component Analysis: A Noise Reduction Method for Continuous Chaotic Systems http://dx.doi.org/10.5772/64410 35

**2.** Duffing equation

**5.** Corresponding to the dominant symplectic principal component values *μi*

**6.** Construct the symplectic transform matrix *Q* and the Hamiltonian matrix *S*, i.e.

0 *W <sup>Q</sup> <sup>W</sup>* æ ö <sup>=</sup> ç ÷

> 0 *X*

*S*

The reduced‐noise data could be given by the first row of the matrix .

data would be better than the first reconstructed data.

This chapter employed the three typical chaotic equations [41].

**4. Numerical and experimental data**

where *σ* = 10, *b* = 8/3, and *γ* = 28.

**1.** Lorenz equation

(*i*=*l*+1,…,*d*), the other vectors of *W* were zeros;

where *X* is given by Eq. (2.2).

34 Advances in Noise Analysis, Mitigation and Control

**8.** Reestimate the with reduced‐noise,

**7.** Use the *Q* project *S* into *Y*,

the first *l* column vectors of *W* as those of *P*. Corresponding to the lower eigenvalues *μi*

0

0

*X* æ ö <sup>=</sup> ç ÷

For the heavily noisy time series, the first estimation of data was usually not good. Here, one could repeat the above steps 7 and 8 several times. Generally, the second or third reconstructed

> *x yx* ( ) *y x y xz z bz xy*

s

ì= -

ï

& & &

g

í = - <sup>ï</sup> =- + <sup>î</sup>

è ø (3.12)

è ø - (3.13)

*<sup>T</sup> Y QS* = (3.14)

*X QY* ˆ = (3.15)

, (*i*= 1,…, *l*), let

,

(4.1)

$$
\ddot{\mathbf{x}} + c\dot{\mathbf{x}} - \varepsilon\mathbf{x}(\mathbf{l} - \mathbf{x}^2) = A\cos(\alpha t) \tag{4.2}
$$

where *ε*=1, *c*=0.4, *A*=0.4, and *ω*=1.

**3.** Chua's equation

$$\begin{aligned} \dot{x} &= a(y - x - f(x)) \\ \dot{y} &= x - y + z \\ \dot{z} &= -\beta y \\ f(\mathbf{x}) &= b\mathbf{x} + \frac{a - b}{2}(|\mathbf{x} + 1| - |\mathbf{x} - 1|) \end{aligned} \tag{4.3}$$

where *α* = 15.6, *β* = 28.58, *a* = ‐8/7, and *b* = ‐5/7.

A measurement noise *e* was used because all the real measurements were polluted by noise. Here, the Gaussian white noise *e* as a measure noise was added to the clean signal *x* generated from the above chaotic equations. The contaminated data *xn* is obtained as follows:

$$\mathbf{x}\_u(t) = \mathbf{x}(t) + \mathbf{e}(t) \tag{4.4}$$

The signal‐to‐noise ratio was defined by the following:

$$\text{SNR} = 10 \log(\sigma\_x^2 / \sigma\_s^2) \tag{4.5}$$

where *σx* and *σn* are the standard deviation of the clean signal *x* and the noise *e*, respectively. The more details of noise notions were referred to the literature [44–46]. Here, *SNR* is 10 dB.

As for a practical example of noise reduction, we chose the sunspot number data obtained from monthly observations. Sunspot number series were short, highly non‐linear and noisy [49]. It was hard to predict accurately the sunspot period, although Wolf had reported the well‐known 11‐year cycle.

## **5. Noise reduction analysis based on SPCA**

#### **5.1. Performance evaluation of SPCA**

SPCA, like PCA, could not only represent the original data by capturing the relationship between the variables but also reduce the contribution of errors in the original data. Here, the performance evaluation of SPCA was first given based on the analysis of Lorenz chaotic time series [39].

#### *5.1.1. Performance evaluation on the representation analysis*

Since the real systems were usually unknown, it was necessary to study the influence of sampling time, data length, and noise on the representation analysis based on the SPCA approach. For a clean chaotic time series, the root mean square error (RMSE) as a measure was employed in order to estimate the difference between the clean original data and the SPCAfiltered data:

$$RMSE = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} \left[ \chi(i) - \hat{\chi}(i) \right]^2} \tag{5.1}$$

where *x*(*i*) and are the clean original data and the estimated data, respectively.

In order to analyse noisy data, the percentage of principal components (PCs) was defined to study the occupancy rate of each PC in order to reduce noise as follows:

$$P\_i = \frac{\mu\_i}{\sum\_{i=1}^d \mu\_i} \times 100\% \tag{5.2}$$

where *d* is the embedding dimension and *μi* is the *i*th principal component value.

**Figure 1.** RMSE versus sampling time curves for the SPCA and PCA.

Here, we took the Lorenz system as an example. The dimension of the Lorenz system was 3, then, the embedding dimension *d* of the matrix *A* was chosen as 8. For the clean Lorenz time series generated by Eq. (4.1) (i.e., no noise *e* = 0 in Eq. (4.4)), when *k* = *d*, the estimated data were obtained by SPCA and PCA with *k*=*d*, respectively. For the different sampling time *T*s, the RMSE values are calculated in **Figure 1** by Eq. (5.1). The RMSE values of SPCA were lower than 10−14 for the different sampling time (see **Figure 1**). The results showed that the SPCA method was better than the PCA. **Figure 2** shows the RMSE values of the different data lengths for SPCA and PCA, respectively. For the different data lengths, the RMSE values of SPCA were also lower than 10−14 (see **Figure 2**). From the **Figures 1** and **2**, we could see that the sampling time and the data length had less effect on SPCA method in the case of free noise.

**Figure 2.** RMSE versus data length curves for the SPCA and PCA.

performance evaluation of SPCA was first given based on the analysis of Lorenz chaotic time

Since the real systems were usually unknown, it was necessary to study the influence of sampling time, data length, and noise on the representation analysis based on the SPCA approach. For a clean chaotic time series, the root mean square error (RMSE) as a measure was employed in order to estimate the difference between the clean original data and the SPCA-

> 1 <sup>1</sup> [ ( ) ( )] <sup>ˆ</sup> *<sup>N</sup>*

In order to analyse noisy data, the percentage of principal components (PCs) was defined to

*<sup>i</sup>* 100%

*<sup>i</sup> RMSE x i x i*

where *x*(*i*) and are the clean original data and the estimated data, respectively.

1 *i d*

m

*i i*

m= = ´

study the occupancy rate of each PC in order to reduce noise as follows:

*P*

where *d* is the embedding dimension and *μi*

**Figure 1.** RMSE versus sampling time curves for the SPCA and PCA.

2

*<sup>N</sup>* <sup>=</sup> <sup>=</sup> å - (5.1)

å (5.2)

is the *i*th principal component value.

*5.1.1. Performance evaluation on the representation analysis*

36 Advances in Noise Analysis, Mitigation and Control

series [39].

filtered data:

**Figure 3.** The percentage of PCs for the SPCA and PCA.

From **Figure 3**, all of the principal components were given by Equation. (5.2) for the clean Lorenz time series according to the SPCA and PCA methods, respectively. We could see that the first largest symplectic principal component (SPC) of the SPCA was a little larger than that of the PCA. For the SPCA method, the first largest SPC was almost possessed of all the proportion of the SPCs. Next, the reduced space spanned by a few largest SPCs was explored to estimate the chaotic Lorenz time series without noise. For the different data length, we gave the RMSE values between the original data and the data estimated from the first seven largest SPCs and PCs, respectively, that is, in the case of *k* = 7 (see **Figure 4**). The sampling time is 0.1. The result showed that the data length had less effect on the SPCA than the PCA. **Figure 5** shows the effect of sampling time on different number of PCs for the SPCA and the PCA methods, respectively. When the PCs number *k* =1 and *k* =7, respectively, the change of RMSE values with the sampling time is given in **Figure 5**. We could see that the RMSE values of the SPCA were smaller than those of the PCA. The sampling time had also less impact on the SPCA than the PCA.

**Figure 4.** The RMSE versus the data length for the SPCA and PCA, where *k* =7. The sampling time is 0.1.

**Figure 5.** The RMSE values versus the sampling time for the SPCA and PCA, where (a) the PCs number *k* =7; (b) *k* =1.

Furthermore, the estimated data based on the first three largest SPCs are calculated in **Figure 6**, where the original data *x* are given with a sampling time of 0.01 from chaotic Lorenz system. The average error between the original data and the estimated data was −6.55e‐16. The corresponding standard deviation was 1.03e‐2. The estimated data were very close to the original data not only in time domain (see **Figure 6a**) but also in phase space (see **Figure 6b**).

**Figure 6.** Chaotic signal reconstructed by the proposed SPCA algorithm with *k*=3, where (a) the time series of the origi‐ nal Lorenz data *x* without noise and the estimated data; (b) phase diagrams with *L* = 11 for the original Lorenz data *x* without noise and the estimated data. The sampling time *Ts* = 0.01.

To sum up, we could see that the SPCA method preserved the essential dynamical character of the primary time series generated from chaotic continuous systems. These indicated that the SPCA could reflect intrinsic non‐linear characteristics of the original time series. The SPCA could elucidate the dominant features in the observed data. It was feasible for the SPCA to study the principal component analysis (PCA) of time series. Moreover, the SPCA would help to retrieve dominant patterns from the noisy data.

#### *5.1.2. Performance evaluation on noise reduction*

From **Figure 3**, all of the principal components were given by Equation. (5.2) for the clean Lorenz time series according to the SPCA and PCA methods, respectively. We could see that the first largest symplectic principal component (SPC) of the SPCA was a little larger than that of the PCA. For the SPCA method, the first largest SPC was almost possessed of all the proportion of the SPCs. Next, the reduced space spanned by a few largest SPCs was explored to estimate the chaotic Lorenz time series without noise. For the different data length, we gave the RMSE values between the original data and the data estimated from the first seven largest SPCs and PCs, respectively, that is, in the case of *k* = 7 (see **Figure 4**). The sampling time is 0.1. The result showed that the data length had less effect on the SPCA than the PCA. **Figure 5** shows the effect of sampling time on different number of PCs for the SPCA and the PCA methods, respectively. When the PCs number *k* =1 and *k* =7, respectively, the change of RMSE values with the sampling time is given in **Figure 5**. We could see that the RMSE values of the SPCA were smaller than those of the PCA. The sampling time had also less impact on the SPCA

**Figure 4.** The RMSE versus the data length for the SPCA and PCA, where *k* =7. The sampling time is 0.1.

**Figure 5.** The RMSE values versus the sampling time for the SPCA and PCA, where (a) the PCs number *k* =7; (b) *k* =1.

than the PCA.

38 Advances in Noise Analysis, Mitigation and Control

For the performance evaluation on noise reduction, the SPCA method was further studied by dealing with the noisy Lorenz data *x* with Gaussian white noise of zero mean and one variance. The phase diagrams of the noisy and clean data are given in **Figure 7a** and **b**. The clean data *x* were obtained by the sampling time 0.01 from the chaotic Lorenz system with noise‐free. The noisy data were the chaotic Lorenz data *x* with Gaussian white noise of zero mean and one variance (Eqs. (4.1) and (4.4)). The time delay *L* is 11 in **Figure 7**. It was obvious that noise is very strong (see **Figure 7a**). Here, we first built an attractor *X* with the embedding dimension of 8. Then, the transform matrix *W* was constructed when *k*=1. The first denoised data are generated according to Section 3 (see **Figure 7c** and **d**). In **Figure 7c**, the first denoised data are compared with the noisy Lorenz data *x* from the view of time field. **Figure 7d** shows the corresponding phase diagram of the first denoised data. Compared with **Figure 7a**, the first denoised data could basically give the structure of the original system. In order to obtain better results, these denoised data were reduced noise again by the step (8). For the second noise reduction, the results were greatly improved in **Figure 7e** and **f**. Comparing by **Figure 7c**, **d**, **e** and **f**, the curves of the second denoised data were better than those of the first denoised data whether in time domain or in phase space. **Figure 7g** shows that the PCA technique gave the first denoised result. Like **Figure 7e**, the second denoised data are obtained by the PCA (see **Figure 7h**). Although some of noise had been further reduced in **Figure 7h**, the curve of PCA was not better than that of SPCA in **Figure 7e**. The reason was that the PCA was a linear method indeed. When non‐linear structures had to be considered, it could be misleading, especially in the case of a large sampling time (see **Figure 8**). The used program code of the PCA came from the TISEAN tools (http://www.mpipks– dresden.mpg.de/~tisean).

**Figure 7.** The noise reduction analysis of the proposed SPCA algorithm and PCA for the noisy Lorenz time series, where *L*=11. (a) Phase diagram for noisy data, (b) phase diagram for the clean data, (c) Lorenz time series, (d) phase diagram for the SPCA, (e) phase diagram for the SPCA, (f) the second denoised data, (g) phase diagram of the first denoised data for PCA and (h) phase diagram of the second denoised data for PCA.

In order to evaluate the effectiveness of noise reduction, the correlation dimension *D*<sup>2</sup> was estimated by the Grassberger‐Procaccia's algorithm [47–48] because it manifested non‐linear properties of chaotic systems. The variations of correlation dimension *D*2 with embedding dimension *d* were given for the clean, noisy, and denoised Lorenz data (see **Figure 8**). The sampling time was 0.1. The results showed that, for the clean and SPCA denoised data, the trend of the curves had a platform and tended to smooth in the vicinity of 2. For the noisy data, the trend of the curve was constantly increasing and had no platform. For the PCA denoised data, the trend of the curve was also increasing and trended to a platform with 2. However, this platform was smaller than that of SPCA. The PCA algorithm was less effective than the SPCA algorithm. The result indicated that it was difficult for the PCA to describe the non‐linear structure of a system.

**Figure 8.** *D*2 versus embedding dimension *d*.

reduction, the results were greatly improved in **Figure 7e** and **f**. Comparing by **Figure 7c**, **d**, **e** and **f**, the curves of the second denoised data were better than those of the first denoised data whether in time domain or in phase space. **Figure 7g** shows that the PCA technique gave the first denoised result. Like **Figure 7e**, the second denoised data are obtained by the PCA (see **Figure 7h**). Although some of noise had been further reduced in **Figure 7h**, the curve of PCA was not better than that of SPCA in **Figure 7e**. The reason was that the PCA was a linear method indeed. When non‐linear structures had to be considered, it could be misleading, especially in the case of a large sampling time (see **Figure 8**). The used program code of the PCA came from

**Figure 7.** The noise reduction analysis of the proposed SPCA algorithm and PCA for the noisy Lorenz time series, where *L*=11. (a) Phase diagram for noisy data, (b) phase diagram for the clean data, (c) Lorenz time series, (d) phase diagram for the SPCA, (e) phase diagram for the SPCA, (f) the second denoised data, (g) phase diagram of the first

denoised data for PCA and (h) phase diagram of the second denoised data for PCA.

the TISEAN tools (http://www.mpipks– dresden.mpg.de/~tisean).

40 Advances in Noise Analysis, Mitigation and Control

Besides, it was necessary to show that the method of the locally projective non‐linear noise reduction (NNR) in the TISEAN package (http://www.mpipks‐dresden.mpg.de/~tisean) could not give the better result than SPCA and PCA [41].

## **5.2. Noise reduction based on SPCA**

In the noise level *SNR* = 10 dB, the noisy Duffing chaotic data (see **Figure 9a**) and the noisy Chua's chaotic data (see **Figure 9c**) show reduced noise by applying the SPCA. Embedding dimension *d*=8 had been used with the time delay *k*=1. The reduced noise results of SPCA are shown in **Figure 9**. The third denoised data of the noisy Duffing chaotic data are shown in **Figure 9b**. For the noisy Chua's chaotic data, the third denoised data are given in **Figure 9d**. Obviously, a lot of noise had been removed from the noisy time series. Here, the number of dominant components was chosen as one according to the curves of SPCs in **Figures 10** and **11**.

**Figure 9.** The noise reduction for the noisy Duffing chaotic data and the noisy Chua's chaotic data based on the SPCA. (a) Duffing chaotic data with SNR=10 dB, (b)The third denoised data for Duffing data, (c)Chua's chaotic data with SNR=10 dB and (d) The third denoised data for Chua's data.

**Figure 10.** The symplectic principal component analysis of the noisy Duffing chaotic data *x*.

Symplectic Principal Component Analysis: A Noise Reduction Method for Continuous Chaotic Systems http://dx.doi.org/10.5772/64410 43

**Figure 11.** The symplectic principal component analysis of the noisy Chua's chaotic data *x*.

**Figure 9.** The noise reduction for the noisy Duffing chaotic data and the noisy Chua's chaotic data based on the SPCA. (a) Duffing chaotic data with SNR=10 dB, (b)The third denoised data for Duffing data, (c)Chua's chaotic data with

**Figure 10.** The symplectic principal component analysis of the noisy Duffing chaotic data *x*.

SNR=10 dB and (d) The third denoised data for Chua's data.

42 Advances in Noise Analysis, Mitigation and Control

**Figure 12.** The noise reduction analysis of the monthly sunspot number based on the SPCA. (a) The monthly sunspot number, (b) the second denoised data, (c) phase diagram for SPCA, (d) the yearly sunspot data and the denoised monthly sunspot data.