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Part-1 Wave Phenomenon, Antennas & Radiation - Introduction | BASIL Networks Blog BN'B

25 Feb, 2020

Part-1 Wave Phenomenon, Antennas & Radiation - Introduction

Part 1: Wave Phenomenon, Antennas & Radiation - Theory and Practice
Introduction - Setting The Atmosphere For The Series

Excellence is not a gift but a skill that takes practice.  We do not act rightly because we are excellent, in fact we achieve excellent by acting rightly. - Plato (428/427 - 348/347 BCE)

Our human species resides on a cloud of knowledge between total knowledge at the top of the scale and the absolute basics at the bottom of the scale always exploring the constraints that hold us from expanding those boundaries.  This series explores the many ways to represent physical phenomenon, a direct simple represenation mathematically or a relational similar representation that easily becomes complex.  Many of the solutions in this series are based on convenience with the end result being the expansion of the current knowledge base.

Wave_Index Quick Links

What we want to cover in Part 1:
Starting a new blog series for us always means setting a solid foundation of reference for the series to transfer knowledge to our readers.  The intent and expectations for this series is the same for all our series - it is a living publication and will be updated and revised as we receive requests and contributions to the series.
Enjoy the series - Sal Tuzzo

Introduction:
The first blog category presented on BASIL Networks blog was the IoT Core Platform Development Project.  During the IoT Project blog many readers asked if I would do a blog on 5G and Radiation?  My immediate answer is No!  However, when I started to look into the massive amounts of information on the Internet it did not take long to realize that this is a series in itself just on any one of the following: WiFi, Bluetooth, MESH networks, 4G and 5G so here we are creating a separate series.  Our presentation methodology has been very effective so we will continue with our methodology, Overview %u2192 Basic %u2192 Detailed (OBD).  We will be addressing the many sections of wave phenomenon separately to keep the presentations at a reasonable length, design and application details will be presented during the actual hardware design and analysis sections.

Setting the Atmosphere for the Series
Setting the atmosphere for this series is a challenge since it encompasses several fields of interest in engineering technologies and encourages understanding multiple disciplines. However do not get discouraged, this series will as all of our series incorporates the OBD methodology Overview, Basics, Detailed to walk you through a new experience of ever changing field of wireless communications.

Overview
Technology Connects Everything; The market introduced listening and camera devices that are able to listen and monitor our movements and issue commands to devices connected to the Internet, not to ignore the smart phones already tracking data.   The things to keep in mind here is for these listening and camera devices the data is going to some third party and being sold to other third party marketers to send advertising on the go for "profit".  That means that your gestures and voices are stored and shared with multiple unknown third parties.  Now think of ordering products through this audio/visual device and giving personal information to these unknown third party data gatherers.  Oh by the way make sure the TV in the bedroom is really off, oops!

Now enter the Internet of Things, everything interconnected and beyond with IPv6 huge addressing capability, security and privacy is gaining high visibility as the predominate issue.  The privacy and security issue is not just for the home environment is extends to the industry, from transportation, retail, marketing, engineering and more.  The world of access through IoT sensors of all types, video, audio, control networks like traffic lights, electric grids and municipal utilities that monitor and control the safety of our everyday lives.  Ok, do not look at this as a Gloom and Doom issue, but as a challenge to insure privacy, security and safety of your (PII) personal identifiable information.

A brief introduction to this new series "Wavelength, Radiation and Antennas" is all about wireless communications and the radiated energy it transmits.  Radiation theory is basically broken down to two broad frequency bands or wavelength bands 'types', Non-Ionizing and Ionizing radiation.  All radiation is harmful to the human body if concentrated at high levels.

The Ionizing radiation bands effect the human DNA directly, Ultraviolet, X-Rays Gamma Rays which is why humans wear radiation badges when working in areas where radiation equipment is used.  The frequency range is 1014 Hz to 1021 Hz and beyond.   There is a "visible" waveband that is a small waveband from 380 to 740 nanometers or 430-770 Terahertz 1THz =  1012 Hz = 1000Giga-Hertz).  The non-ionizing radiation effects mostly living organs and have caused various types of mutations within these organs.  we will cover that later in the series.

Lets categorize these radiation fields, there are three Ionizing field groups, Ultraviolet: Energetic Light, X-Rays: Penetrating Radiation and Gamma Rays: Nuclear Energy." There are four Non-Ionizing radiation field groups, Radio Waves: Communications, Microwaves: Data and Heat, Infrared Waves: Invisible Heat, Visible Light Waves. The frequency of the Non-Ionizing band is everything below the Ionizing radiation band 0 HZ to just short of 1015 Hz.  The references for Radiation Waves are Wikipedia Electromagnetic Spectrum and NASA Website on Electromagnetic Spectrum.

Wireless Communication Networks
Now the unspoken good bad and ugly of any wireless network period.  As I stated before, all companies have a right to monitor their product usage, especially if there product entails Power, Time, and Bandwidth as product entities.  Encouraging the monitoring of competitive products brings the best results in a competitive free market environment.  What we also encourage is all products in use by the populous is the accountability for those products safety, privacy and security!

I have been following Bluetooth, WiFi, 3G, 4G and 5G network development for a some time and would not comment on it until there were substantial facts to present, vetted by recognized accountable scientists and medical professionals with undeniable proof of the results.  Well here is the 3rd annual update to our blog about the IoT and wireless communications in a new series called "Wavelength, Radiation & Antennas".

OK, the last thing we want here is another global controversy, lets stick to the scientific facts backed by absolute data only, to date (December 1, 2019) over 300 Scientists and doctors in 41 nations, along with a few hundred dead birds and cases of humans that suffered from disorientation and other medical ailments including cancer of the brain and other organs included in their published research findings on radiation from wireless technologies, vetted at several information websites, Wi-CancerCell Phones 5G Network Health RisksEMF WISE and the list goes on.

For companies that incorporate CDMA and other leased bands for their monitoring a list of cell towers in the area by physical address location Antenna Search by Physical Address Location, or by Provider Technology 3G,4G, 5G etc.  Antenna Search Globally by Provider CellMapper.  There are many more websites and some sites have been blocked from the connections that we use to vet information, which generally means that even registered businesses that purchase bandwidth with a fixed open IP address are blocked at the will of the Internet Provider.

OK, we went through this in previous presentations with 3G, 4G networks as stated in the publications, since 3G and 4G antennas are greater distances from residential areas.  There are several in the area where our servers are here, the closest one in the area of our lab is about  0.27 (1415 ft - 431 meters) away and is 200 feet high directed to cover a 1/2 mile area or more and far away from a direct concentration, however, there are a few buildings that are within 100 feet of the base of the antenna.  Finally there is conclusive diagnostic clinical evidence of health related cases with the wearable devices incorporating 3G and 4G, now a fact, from the released reports as well as scientific studies and is no longer an isolated issue just to a specific location.

What does this have to do with the IoT development project?
Yeh, Lots of questions still - OK, we have been living with wireless radio communications an cell phones for well over 60 years now and there have been some health issues as with every new technology that comes on board.  A problem with relating illness with a conclusive source is the variances in lifestyles as well as conditional parameters of use, diet etc. of all the individuals that use them.  So let me try and break this down to understand this a bit better.

To start I have many years in Pulsed NMR (Nuclear Resonance Spectroscopy, Dual Beam IR (Infrared Spectrophotometry) sonar, ultrasonic's and frequency modulation techniques used in the analog and digital world, (Whoopee, so what does that have to do with the facts?).  So with that whoopee button and the fact that many IoT applications will incorporate some from of wireless communications which has to "radiate" the signal this raises the concern that our device must be in a safe operating area as not to cause harm.   So my research went in a different direction for a while in order to gather the current scientific facts and data to insure our device will operate safely in these environments.

The following are vetted data from scientific studies on radiation effects and a safe amounts, yeh, yeh I hear it now,  "THERE ARE NO SAFE AMOUNTS OF RADIATION", this is true however, we are subjected to radiation just walking down the street, hopefully, only small enough amounts that the human body over powers and neutralizes for a given time frame.  Radiation effects on the human body are accumulative, the issue is how fast will the human body replenish the damage cells and stop the replications/mutation process.

Resonance and Its Effects on Matter:
All matter has what is called a resonance (parallel or series) frequency that will (absorb, pass through or reflect) energy waves, this includes all life as we know life that is a breathing biological "earth" entity.  Solids also have a resonance frequency as shown many times by the old breaking a glass with a sound-wave experiment.  So, if the energy is reflective or passive when it comes in contact with an objects surface there will be very little effect on the internal structure past the objects surface, hence: it either bounces off or just passes through the object without any absorption of the transmitted energy.   Just like the glass will only break if the resonant frequency is that of the glasses material structure to absorb energy.

At resonance the energy is absorbed the structure starts to vibrate, when enough energy is absorbed within the glass structure the structure becomes compromised and literally shakes itself apart and shatters.  Now keeping this concept in mind and applying it to biological life form structures is where we get similar disruptions with how the biological structures react to this intrusion of energy.

Biological's and Energy Absorption
Low power radiation produced by low power RF fields seems to be passive or is handled by the biological's anti-intrusion mechanism, that is until we start to focus the energy, like 24/7 connected wireless cell phone headsets while we holster the cell phone on our side day in and day out.  Because of the low power this would take years on some people to create a health issue if any depending on the individual diets, exercise etc., yes, diet is an effective means for neutralizing small doses of radiation poisoning, do an Internet search on Iodine and other minerals that the body absorbs.

Now the 4G and 5G broad band network, base frequencies for 3G range 2GHz to 8GHz, while 5G range 3GHz to 300GHz.  As we see there is a wide overlap of frequencies that are used to define the range characteristics of the technology as defined in http://www.rfwireless-world.com/Terminology/4G-vs-5G-difference-between-4G-and-5G.html.   Hence: 4G is incorporated as part of the 5G network to address various bandwidths within the communications range.   OK, now, when you start to get beyond reasonable RF frequencies, radiation tends to have a greater effect on the biological's of a living organism.   One of the major differences is 5G has a much shorter transmission range than 4G and require more antennas to cover the same range.

Electromagnetic Absorption by Water  and "wait for it" --  the body is made up of about 60% to 80% water variant of species, human to animals - Plasma is that pale yellow mixture of 90% water and the rest proteins and salt that make up 50% of the blood.  Plasma is the traffic highway that forms a conductive conduit for product delivery to all the organs, yes, including the brain - oops!.  The real issue arises with the 5G channel frequencies that are used within the Gigahertz frequency ranges of the technology,  the UHF, SHF and EHF Bands 2GHz to 100GHz that makes up the 2MHz to 4GHz digital bandwidth modulation schemes that fall in the [RF, EF, EMF] band.  These Basebands are modulated with a variety of schemes [EGDE, EGPRS, WCDMA, HSPA, HSPA , CDMA2000, WiMAX, LTE, LTE-Advanced, to name a few, which are controlled digitally being carried on the higher RF base frequency which generate many harmonic frequencies as part of the waveforms being transmitted.  These modulation frequencies add to the transmission energy fields being absorbed or reflected that pass through the air in all directions.

The intent of 5G technology because of its limited range of under 1000 Feet is to put a focused RF base station every 10 or 12 blocks on these transmission power line poles.  This is where most of the 5G experimentation has been performed prior to its official release and where much of the health issues have been researched and published.  For those of you that are interested in radiation and the body and how frequency and power play a roll in biological cells a great read is the research done by Dr. Royal Raymond Rife  original frequency to organ list and the Consolidated Annotated Frequency List  that shows the effects frequency modulation and radiation go hand in hand to disrupting biological cell structures and organs.

OK, nice introduction but what does this have to do with our IoT Platform Project?
The IoT core platform will incorporate wireless technology and will be reference to the new blog series Wavelength, Radiation & Antennas.  Designing wireless products entails a bit more research than that of direct connect (wired) devices.

Lets Get Started - Basics:
OK, lets set a Basic foundation of Facts/Givens that this series will reference, also for those that are very new to wave phenomenon as to set a solid foundation to relate as we move on through the series.  "If you wish to understand the universe, think of energy, frequency and vibration." - Nikola Tesla.   There two types of laws in the universe, OK, no I have not been to the universe but I am here on earth and that is in the universe, so lets move on.   These two Universal Laws are: Immutable and Mutable.  That easy enough right?  Immutable Laws are laws that can not be changed over time, unable to be changed which are absolute.  You remember growing up and your mom says "NO."Cool   Mutable Laws are laws that are liable to change either completely or slightly and effect our very existence, you know when you talk back to your parents and you get a response " keep it up and you will experience pain".Laughing

Frequency, Time, Wavelength:
To start lets look at some simple concepts we use regularly.  Frequency, which is the rate of which an event happens or for a waveform to travel one period of time where the vibration or amplitude of the waveform is at the same point from where it started as shown in Figure 1.0 below.

alt
Figure 1.0  Sinewave  Parameters

Frequency = 1/Time and is expressed in Hertz (Hz) or Cycles per Second (cps).   Wavelength is the distance one period travels and is generally expressed in, m (meters) which is all referenced to the speed of light relative to inside the atmosphere of the planet earth which is 2.998x108 meters/second and is given the label C, Therefore the wavelength of any frequency is defined by the equation below.

λ = c / f
{ (Lambda in meters)   λ =  c ( Speed of Light in Meters/Seconds) divided by f ( frequency in Hz ) }

Wave Phenomenon, Antennas, Radiation theory, analysis and real world applications are an on-going field with lots of bits and pieces of information on the interest.   Several Universities like MIT in Massachusetts offer all their courses on line free and if you wan credit it is a fraction of the tuition for the in-class course.  To maintain the scope of this blog we will only cover a few types of antennas that are used for our IoT project. Bluetooth, WiFi, Cell Phones and Cell Towers 4G and the latest 5G base antennas.

USA Frequency Allocation Chart:
The range of frequencies we are exposed to on a daily basis in the communications bands range from a couple of hertz to 300 Gigahertz (300,000,000,000 Hz).  The object of this series is to understand the different bands, where they are used then show how to analyze the radiation from the sources like WiFi routers, laptops, base antennas and smart phones and other devices.  Figure 1.1 below shows the United States Radio Frequency Allocation bands as of January 2016.  The chart is updated as technology changes and frequency bands are reassigned.

OK the funny looking chart below categorizes the primary and secondary bands,  all Capital Letters are Primary bands and 1st Capital with lower case letters are Secondary band allocations for the United States and is available to anyone for downloading.  The code of federal regulations Telecommunications Cfr Title 47 part 70-79 2019 is also available and is suggested for developers to insure that the product is in compliance.  The last one BASIL Networks purchased was parts Cfr 47 parts 20-69 Oct 1981 since the entire Cfr may be accessed on the web.

This chart and other documentation are available at the U.S. Department of Commerce January 2016, Frequency allocation chart  which shows a point in time of allotted frequencies that are in our air space at any one time depending on location on the planet.   It is recommended to download the table since it gives the exact frequencies allocated as they are reassigned and new ones allocated at  [PDF]Current allocated Frequency Table for each band.

USA_Communications_Frequency_Spectrup
Figure 1.1  United States Radio Frequency Allocation Spectrum January 2016
Click on the chart to download the PDF file enlarged view of the chat (42.00" x 26.88")

As we see from the communication frequency allocation chart that from 0-300 GHz (300,000,000,000) there are many bands that are allocated for many purposes.  The Allocation bands change as technology changes and are updated and published in table form above link.   The position of the radio frequency spectrum is shown in Figure 1.2 below referenced to the above chart and associate table.

[Radio_Spectrun_Position]
Figure 1.2  United States Radio Frequency Allocation Spectrum Position

Frequency Band Standards Groups
OK - one last bit of basic information before we get started and that is the names and acronyms assigned to the different bands, as shown in Table 1.0 below.  Some of the confusion is that there are several frequency bands that are associate with different countries and they are assigned different letter acronyms to represent specific bands.  The confussion is that the bands overlap each other depending on the accepted terminologies for the selected country or region.  The Frequency Allocation groups that label the frequency bands in their own sand box are listed below.

Standards Group Name Band Designations Frequency Range
EU - European Union A - M 1Hz    to 100GHz
NATO - North American Treaty Organization A - M 1Hz    to 100GHz
US ECM - United States of America A - M 1Hz    to 100GHz
IEEE -Institute of Electrical and Electronic Engineers HF,VHF,UHF,L,S,C,X,Ku,K,Ka,
V,W,nm
3MHz to 300GHz
ITU - International Telecommunications Union ELF,SLF,ULF,VLF,LF,MF,
HF,VHF,UHF,SHF,EHF,THF
3Hz    to 3THz

Table 1.0  Frequency Band Naming for Different Communications Organizations

ITU International Telecommunications Union Frequency Bands:
As the frequency allocations advanced it becomes evident that a standard is required to encourage some type of categorization of the frequency bands and the allocations of them.  For this series we will use the ITU (International Telecommunications Union)  based in Geneva, Switzerland since it includes the cooperation of many countries and it will give us a reference point when analyzing radiation and wave transmission. More information on the ITU is here https://en.wikipedia.org/wiki/International_Telecommuni cation_Union

Table 1.1 below list the frequency ranges assigned to the ITU bands.  As reassignment of the actual frequency for specific applications as technology advances changes will be reflected in the [PDF]Current allocated Frequency Table.

ITU  Radio Communications Frequency Bands
BAND Designation Frequency Range Hertz Wavelength Meters Application
ELF (Extremely Low Frequency) 3 Hz to 30 Hz 100'000 km to 10'000 km Submarine Comm
SLF (Super Low Frequency) 30 Hz to 300 Hz 10'000km to 1'000km Submarine Comm
ULF (Ultra Low Frequency) 300 Hz to 3000 Hz 1'000km to 100km Submarine Comm, within Mines
VLF (Very Low Frequency) 3 kHz to 30 kHz 100km to 10km Maritime Radio, Navigation
LF  (Low Frequency) 30 kHz to 300 kHz 10km to 1km Maritime Radio, Navigation
MF (Medium Frequency) 300 kHz to 3000 kHz 1km to 100m AM Radio, Shortwave Radio
HF (High Frequency) 3 MHz to 30 MHz 100m to 10m Shortwave Radio
VHF (Very High Frequency 30 MHz to 300 MHz 10m to 1m VHF TV, FM Radio
UHF (Ultra High Frequency) 300 MHz to 3000 MHz 1m to 10cm PMUHF TV,Mobil Phones,GPS,WiFi,4G
SHF (Super High Frequency) 3 GHz to 30 GHz 10cm to 1cm Satellite Comm, WiFi, 5G
EHF (Extra High Frequency) 30 GHz to 300 GHz 1cm to 1mm Radio Astronomy, Satellite, 5G
THF (Tremendously High Frequency) 300 GHz to 3000 GHz 1mm to 0.1mm  
Not Considered Part of the Radio Communications Bands {yet}
BAND Designation Frequency Range Hertz Wavelength Meters Application
INFRARED 300 GHz to 430 THz 1mm to 700nm  
VISIBLE 430 THz to 790 THz 700nm to 400nm  
ULTRAVIOLET 790 THz to 30 PHz 400nm to 10nm  
XRAY 30 PHz to 30 EHz 10nm to 0.01nm  
GAMMA-RAY > 30 EHz < 0.01nm  
COSMIC-RAY   Interstella Space Medium (ISM)  
MICROWAVE 300 MHz to 300 GHz    
       
LW (Long Wave) 153 kHz to 279 kHz    
MW (Medium Wave) 531 kHz to 1620 kHz  
SW (Short Wave) 2310 kHz to 25820 kHz    
       

Table 1.1  International Telecommunications Union (ITU) Frequency Bands

Energy Distribution Concepts Overview
Before discussing details about energy and radiation theory it is a good practice to look at the conceptual overviews that are used throughout the industry.   Figure 1.3 below shows the fundamental concept of radiant energy distribution in free space that follows the Inverse-Square Law that the further away from the source of the energy GM the lower the energy level.  Figure 1.3 is for ideal free space and the source radiates energy geometrically as a sphere and is a good place to start.  So, for the sphere as the radiant energy travels the energy decreases by the inverse of the square of the distance, why? because if the distance is two time the radius of the sphere the energy has to cover 4 times the area.  This is the defiinition of the Inverse-Square Law   So by the definition of the Inverse-Square Law the Intensity for a given area at a distance from the source is defined by I = GM / R².

IMAGE_Energy_Distribution
Figure 1.3  Inverse Square Law - Visible Spectrum

So if we analyze the entire sphere with a radius R where a spheres area is defined by [Area]with a geometric source S = alt radiating around the sphere, then by the Inverse-Square Law the Intensity of the radiant energy at the surface of the sphere is defined by Equation 1.0 below.

alt   alt      Equation 1.0  Intensity at surface of sphere at Radius R

So if R = 1 meter the Intensity at the 2R surface of the sphere from the source GM as a percentage by definition is I = 1 / Distant² of the source GM which yields 1/4 of the source energy or 25% of the source GM.  A quick check of the equation labels for the energy being watts of power yields [watt/ meter*meter = w/m².  If the source GM = 1 Watt, then the radiant energy at 2R would be 1/2² = 1/4 = 0.25 Watts/meter-squared = 25 mW/m².  As we see the radiant energy is reduced by the inverse square of the distance.  OK, that was simple at least for the ideal free space environment.

SUMMARY
The introduction to this series presents an overall view of just how wireless transmissionof energy interacts with all matter.  The Overview presents some basic of energy transmission without getting into advanced matematics in order to understand the basic concepts to use as a foundation as the series progresses.  At this point we are able to understand that energy transmission in free space does not maintain its energy level but decreases as it travels farther from its original source by the Inverse-Square Law.  From all the different organizations that have selected names for different frequency bands we see that many overlap so in order to maintain consistence the ITU - International Telecommunications Union frequency band identifications are used.  The foundation of names and frequency bands and the Inverse-Square Law of radiant energy in free space.

There are many other parameters that have to be added to this simple Inverse-Square Law where the disruptions in energy transfer by real world obsticals like buildings, reflective materials, wavelength, other energy phenomenon colliding in the same path, the height of the energy source from the ground, the type of energy source being a single rod type antenna or directional antenna being fed with other types of energy sources.

As the series progresses the author, Sal Tuzzo will be available for discussion through the BASIL Networks Contact Form for those that want to apply this series to conduct their own experiments.  I will always be appreciative for the private comments sent through the contact form for suggestions and advice during the development of this series.  This is a growing opportunity for everyone entering into product development as well as a great review for us "well seasoned" in the field to just refresh our human DRAM (Dynamic Random Access Memory).

It is recommended for those that have specific questions on the series to use the BASIL Networks Contact Form to separate them from getting lost in the general comments for each blog presentation.  For all specific design request or contracts please feel free to contact us.

 


Part 2...n  Preliminary Outline for the series "Basic Wave Phenomenon Antennas & Radiation" -Continued
There are many more Laws of energy transmission in the wireless arena that will be addressed as the series progresses showing the relationships between them and how they relate to the wireless communications as they are applied today.  Energy distribution laws we will address are:  Planks Law,   Stefan-Boltzmann Law,  Maxwell-Boltzmann Distribution Law,  Wien Displacement law,  Emissivity,  Kirchoff's Law,   Lambert's Law also know as the Beer-Lambert Law,  Ampere's Law, Coulomb's Law, Maxwell's Law, Orsted's Law and Biot-Savart's Law.  So as we see there are many theories that have been experimented with however, with all these laws there are still anomalous deviations that seem to fall outside the norm which we will discus later in the series.


Wave_Index Quick Links


Reference:
The books in these references are my Northeastern University college of engineering text books except for the Ultra-High-Frequency Techniques, that was a gift from a colleague.

Ultra - High - Frequency Techniques (1942  D. Van Nostrand Company)
    J.G. Brainerd,  Professor EE Univ. of Pennsylvania
    Glenn Koehler, Assistant Prof. EE Univ Wisconsin 
    Herbert J. Reigh,  Prof. EE Univ. Illinois
    L.F. Woodruff, Assoc Prof. EE  MIT Massachusetts

Wave Propagation and Antennas (1958 Library of Congress: 58-9431) by George B. Welch Professor of Physics Northeastern University and Professor Hollis S. Baird whom I have had the honor of being one of his students.
Basic Microwaves (1966 Library of Congress: 65-16814) Bernard Berkowitz
Physics (1966 -ISBN: 0 471 71715 0) Robert Resnic and David Halliday Part I
Physics (1960 -Library of Congress: 62-15336) ) Robert Resnic and David Halliday Part II
Fundamentals of Physics  Revised Printing (1974 ISBN 0471-34431-1) Robert Resnic and David Halliday
Signals in Linear Circuits (1974 ISBN 0-395-16971-2) Jose B. Cruz and M.E. Van Vlakenburg

Wikipedia - On-Line Knowledge Center


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[Sal_TUzzo]

Sal (JT) Tuzzo - Founder CEO/CTO BASIL Networks, PLLC.
Sal may be contacted directly through this sites Contact Form or
through LinkedIn

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