handbook for radio planners
TRANSCRIPT
Handbook for Radio PlannersVersion 1.0
Date: 12 March’2003
First Edition
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Table of Contents
1. WHAT IS RF..............................................................................................................................4
RF BAND DEFINITION.......................................................................................................................5PROPAGATION AND ATTENUATION OF ELECTROMAGNETIC WAVES.................................................6
ANTENNA PARAMETERS.............................................................................................................7
IMPEDANCE OF AN ANTENNA............................................................................................................7ANTENNA CONSTRUCTION:...............................................................................................................7STOCKING (PILING) OF ANTENNAS....................................................................................................7ANTENNA RELATED DEFINITIONS.....................................................................................................8
Front to back ratio:......................................................................................................................8Half Power Beamwidth:...............................................................................................................8Directivity of antenna D:..............................................................................................................8Gain of the antenna:.....................................................................................................................8EIRP & ERP:................................................................................................................................9Polarization:.................................................................................................................................9Impedance:...................................................................................................................................9Bandwidth:...................................................................................................................................9
POLARIZATION:.................................................................................................................................9When to use circular polarization?............................................................................................10Polarization diversity Vs Space diversity...................................................................................10
PRINCIPLE OF SWITCHED-BEAM SMART ANTENNAS: SPOTLIGHT® GSM...............11
FIGURE 2 SPOTLIGHT® GSM BEAM-SWITCHING SCHEMEGSM FREQUENCY BANDS..............................................................................................................................................12
GSM FREQUENCY BANDS..........................................................................................................13
RADIO PROPAGATION................................................................................................................13
FADING AND FADING MARGIN.......................................................................................................13CALCULATION OF FADING MARGIN VALUE....................................................................................15ABOUT DIFFRACTION......................................................................................................................16
LINK BUDGET................................................................................................................................16
CALCULATION OF MAXIMUM ACCEPTABLE PATH LOSS IN UP-LINK AND DOWN-LINK ...................16CELL RADIUS..................................................................................................................................17COVERAGE AREA OF A 3 SECTORED SITE.......................................................................................18
CELLULAR TRAFFIC...................................................................................................................18
CELLULAR TRAFFIC...................................................................................................................18
FREQUENCY PLANNING............................................................................................................20
AUTOMATIC FREQUENCY PLANNING..............................................................................................21THE 4 S PRINCIPLES OF MANUAL FREQUENCY PLANNING..............................................................23FREQUENCY HOPPING.....................................................................................................................26
Baseband Frequency Hopping:..................................................................................................27Synthesizer Frequency Hopping.................................................................................................28Frequency Planning Ideology for Synthesizer hopping:............................................................29
APPENDIX: 1...................................................................................................................................33
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OKUMURA HATA PROPAGATION MODEL TO TORNADO(PLANET) CONVERSION..........................33
APPENDIX 2....................................................................................................................................34
JAKES CURVES:...............................................................................................................................34
APPENDIX 3....................................................................................................................................35
ERLANG – TABLES.......................................................................................................................35
APPENDIX 4....................................................................................................................................39
Normal Distribution chart..............................................................................................................39
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1. What is RFLet us start the introduction to RF by first reviewing the Electric field and the magnetic field, many points cannot be discussed in detail and are assumed that as studied in university (school).
Electric Field:When two metallic plates are isolated to each other and are charged with positive and negative charge, a homogeneous electric field exists between them. The isolation between plates can be done by dry air or a non-metallic material, known as dielectric ε.
An electric field has an electric voltage. To size and compare the field’s strength, One unit of voltage and distance between charged poles is taken.
Mathematically it can be represented asE= V / d
Where E= electric field strength.d= distance between charged platesV= Voltage between charged plates.
A very famous electronic device that can store electric energy is the capacitor.
Capacitor: Capacity C = Charge / Voltage = (Dielectrium * Area of plates)/distance of plates
Formula: C = Q/ U = ε*F/d
Where C= CapacityQ= ChargeU=electric field. ε = DielectriumF= Area.d=distance
Magnetic Field:A magnetic field is found between two polarized magnets, attracting each other by magnetic force and therefore a magnetic field between positive and negative poles are found. Magnetic field is represented by field lines. These field lines describe a magnetic voltage to size and compare magnetic field strength.
Mathematically it can be represented asH= Vmag / d
Where H= Magnetic field strength.d= distance Vmag = Magnetic Voltage.A famous device for magnetism is the magnet. Also when an electric current flows through a wire a magnetic field is created around the wire.
Electromagnetic field.
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When electric and magnetic fields are linked together and stand orthogonal (perpendicular) to each other, we refer to as an electromagnetic field. Once the coupling between the two fields is established it is fixed and cannot be destroyed, but can only be distorted.
The Table below shows the band plan of the electromagnetic waves. RF is also a type of EMW
RF Band Definition3 - 30Hz30 - 300Hz Extremely Low Frequency ELF300 - 3000Hz Voice Frequency VF3 - 30kHz Very Low Frequency VLF30 - 300kHz Low Frequency LF300 - 3000kHz Medium Frequency MF3 - 30MHz High Frequency HF30 - 300MHz Very High Frequency VHF300 - 3000MHz Ultra High Frequency UHF3 - 30GHz Super High Frequency SHF30 - 300GHz Extremely High Frequency EHF
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These electric and magnetic field vary in intensity in a periodic manner, the number of these periodic changes in a second is referred to as the frequency of the electromagnetic wave.
These electromagnetic waves travel with the speed of light and the wavelength and frequency are related to each other by a factor. The speed of light c =300,000 km/se
And c= λ.ν.c= speed of lightλ= Wavelength ν = frequency.
Propagation and Attenuation of electromagnetic waves.
Propagation is the force to move a wave forward. The propagation force is a result of vector E and H, known as representing vectors of electric and magnetic fields linked orthogonal together as electromagnetic wave.
The radiated wave has the following properties. It travels with the speed of light. Radiated wave propagates spherical into the room Field strength of wave reduces cubic with distance.
Attenuation is known as the reduction of HF –energy. Attenuation can be caused by: Free space propagation attenuation. Atmospheric attenuation Material in which the wave propagates.
The free space propagation loss is represented by:
F(dB) = 121.98 – 20 log (λ in cm) + 20*log(R in km) = 32.45+20*log(ν in MHz) + 20*log (R in Km)
Where λ = Wavelengthν = frequencyR= Distance at which the field strenth is measured.
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Antenna parameters
Impedance of an antenna.We know that the electromagnetic waves travel with the speed of light in every direction (spherical) from the source of radiation. But, the atmosphere also poses some impedance to the propagation which can be deduced as follows.
The impedance of an electromagnetic wave is similar to that of a resistor: Z= V/I (V=voltage and I= current)Z= V/I =E/H Which means that the electric field component E divided by the magnetic field component H of an electromagnetic wave gives us the impedance Z.
The free space (atmosphere) impedance will then be: Z˚= E/H = √(μ/ε) = 120*π = 377Ω.
Therefore to achieve full efficiency of the power fed into the antenna, it must have an impedance equal to free space, 377 Ω at their dipoles. But our transmitters are connected to the feeding point and not to the emitting dipoles itself, so the feeding point must match our cable impedance in order to be efficient, usually 50 Ω or 75 Ω
So an antenna should at least fulfill the following two tasks Provide low loss impedance matching from cable feeding point impedence to free space ( 50
Ω/75 Ω to 377 Ω). Change type of electromagnetic wave from transversal (TEM-wave type) to free space wave
type.
Antenna Construction: Broadly speaking there are only two types of antenna, Isotropic antenna and Dipole antenna. An isotropic antenna is an infinitely small radiation source; you cannot build an isotropic antenna in reality, so it is used for calculation only. The radiation pattern of an isotropic antenna is a complete sphere. A Dipole is the simplest antenna that you can build, the radiation pattern looks like a “8”, in the vertical plane with a 3dB beamwidth of 90˚ and as a circle in the horizontal plane.
The gain relationship between dBi and dBd can be defined as : dBi = dBd + 2.15
Stocking (piling) of antennas.Stocking means to take 2,3,4….n antennas of the same type, placing them in horizontal and or vertical lines with a spacing of λ/2 and feeding them commonly. Sometimes this measure is called piling. When antennas are placed vertically. The whole piled antenna system becomes a
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Isotropic antenna Dipole antenna
“new”antenna, with its own characterstics, such as radiation pattern, gain, impedance, F/B ratio e.t.c.
By doing this, the radiation pattern becomes more narrow, which means that the main lobe becomes longer and smaller. The increase in directivity of our main lobe increases gain. Basically speaking, double the number of antennas will increase the gain of antenna system by 3dB (double gain) and decrease half-power beamwidth by half. Or
1 antenna = nominal gain nominal beamwidth2 antennas = nominal gain + 3dB nominal beamwidth/24 antennas = nominal gain + 3dB +3dB nominal beamwidth/48 antennas = nominal gain + 3dB +3 dB + 3 dB nominal beamwidth/8…..2ⁿ antennas = nominal gain + n*3dB nominal beamwidth/2ⁿ
The spacing of dipoles in case of piling and the feeding phase can change the radiation pattern of the antenna dramatically. Usually, in some antennas with adjustable downtilt the feeding phase of the dipoles is altered using a adjustable length dipole feeding cable.
Antenna related definitions
Front to back ratio: Describes the difference in radiation intensity between main lobe and back lobe
F/B ratio = 10 * log (Pmain /Pback) dB
Half Power Beamwidth:Describes the angle at which the radiation intensity is reduced by 50%, half of maximum value, or expressed in Decibel means –3 dB of maximum value. Therefore, expression “3dB-Beamwidth” is commonly used.
Directivity of antenna D:Describes how well directed the radiation of an antenna is. A high Directivity D means that the lobe (usually the main lobe) is very narrow; the beam is directed and has high radiation intensity. So, the narrower the lobe (beam) the smaller is the Beamwidth. When we have a clearly formed main lode and no side or back lobes, we can calculate the Directivity D approximately by the 3 dB-Beamwidth angles.
Gain of the antenna:Describes the efficiency of antenna related to a reference antenna. There are different ways to calculate and measure gain od antennas. Calculation can be done by Directivity D of antennas, Measurements by a reference antenna. To Indicate from which method gain comes from, an index is added to unit, e.g. dBd means measured with Dipole reference, dBi means calculated with isotropic reference. Accuracy of measurements is very low, so ∓ 0.5 dB is considered good.
Gain = 10* log D ( dBi)
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EIRP & ERP:
Effective Radiated Power (ERP) is a product of directivity and power, which is fed into antenna. This number describes the power which is emitted into free space. That leads to field strength calculation. EIRP is the Effective Isotropic Radiated Power, where the directivity is related to Isotropic.
Polarization:
Describes the relationship between the electric field component of the emitted wave in comparison to the Horizon. When the E- Field id parallel to the horizon, we refer to it as Horizontal polarization, when the E-field is vertical to the horizon, we call it vertical polarization. When the E-field rotates left or right, we refer to it as circular polarization.
Impedance:
Describes the feeding point impedance of an antenna. Usually the commercial antenna has been matched to 50 Ω or 75 Ω already.
Bandwidth:
Describes in which frequency range we find good matching to our nominal impedance (50 Ω/75 Ω). We must define where we set the maximum mismatch, Usually, a mismatch value of 14dB (SWR= 1:1.5) or 9.5 dB(SWR=1:2.0) is used for defining bandwidth of and antenna.
Polarization:The polarization of an antenna is defined as the electric field vector E compared to the horizon. But, the antennas are Vertically, Horizontally or circularly polarized. The choice of polarization will depend on the type of application.
In cellular networks we generally prefer vertical polarization due to the following reasons We want to have a omni directional reception by the user and very low effor in antenna
system, therefore only ground plane antenna suite the need perfectly. Ground planes are vertically polarized. Car antennas are mounted vertically on the roof. Mobile phones with antenna integrated in devices are held vertically on the head.
One may also choose to horizontal polarization, usually when we want to oprate another RF service in the same frequency band as others. The cross-polarization attenuation (horizontal to vertical) has a practical value of 20dB, under laboratory conditions about 60dB and theoretically infinite (+) attenuation. So 20 dB means that the signal is reduced 100 times from the other service operator. The same is true for left and right polarization in case of circularly polarized antennas. When we want to minimize trouble with other users of the same frequency, we better choose the opposite polarization.
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When to use circular polarization?. Circular polaraization ( left or right polarization) can be used when we must receive both horizontal and vertical polarized signals and do not want to install 2 different antennas. The off-polarization attenuation between exact linear to circular polarization is only 3dB. But our antenna gain is quite high so the 3dB reduction from the polarization difference does not have a major effect.
Another big advantage with circular polarization is that it seems that wave becomes less attenuated in forest, mountains and metropolitan areas compared to linear polarized signals. These are empirical data and cannot be explained exactly by theory.One of the theories states however that when a wave is reflected, the wave’s polarity distorts, which means we do not have exact linear polarization anymore, maybe -23° offset. The more the reflection, the higher the possibility of coming out of exact linear polarization (distortion) & the bigger the offset angle. So when we use a linear polarized antenna (horizontal/vertical) we receive a weak signal because of the cross-polarization attenuation. The more offset- angle we have, the higher is the cross-polarization, at +- 90° (270°) it reaches maximum.
When we use a circular polarized antenna, we receive all possible angles of wave (360°). So the wave is not attenuated which we measure as higher signal strength.
The signal reduction of 3 dB only appears when we have exact linear polarized waves compared with a circular polarized antenna. Hence it is sometimes advantageous to use circularly polarized antennas.
Another method of increasing the reception is known as “space diversity”. In space diversity the antennas are placed with some gap (>1*λ) and “polarization diversity” where antennas with different polarity are used.
Polarization diversity Vs Space diversity.
Since the mobile handhelds have a limited battery capacity available therefore we need to reduce their output power to a minimum, we need a good antenna installation on Base station side to receive those weak uplink signals.
High gain antennas are usually large and have an undesired radiation pattern. To counter this problem small antennas (less gain and wider radiation pattern) are used, usually 2 or more of the same type antennas are mounted with generally 10 λ gap between them. This measure is called “Space diversity”.
But since you need 2 or more antennas of the same type for the receiving path, the installation becomes a heavy construction to fight against windload and other environmental problems.
Hence sometime it may be preferable to use Polarization diversity.
The probability that a wave comes to the antenna in exactly the same polarization as the antenna’s radiated polarization is very small. One can increase this rate by “Space Diversity” or by using dual Polarized antennas and then we call it “Polarization diversity”.
Advantages and disadvantages of polarization diversity.
Some of the advantages of using polarization diversity over space diversity are.
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Advantages Since the dualpolarized antenna is a single physical antenna, the wind load of the antenna is
quite small. Because dual polarized antennas can receive waves with different polarization, they are as
good as two single polarized antennas used in a space diversity configuration. This makes the construction smaller because less antennas are needed and wind load reduces. This saves cost in material of the mast structure, foundation and erection. On top of that this makes the site aesthetically more pleasant.
Cost of a dual polarized antenna are quite the same as single polarized antennas. By reducing installation cost and time one can save some money. Some analyst has concluded that you can save as much as 40% costs for antenna installation.
Disadvatages One major disadvantage of polarization diversity is that transmission gain is not as high as
in space diversity, when dual polarized antennas are used for transmission from base station to mobiles.
For indoor installations such as the shopping malls, schools, universities and offices there is absolutely no difference between space and polarization diversity. In cities and suburban areas with mainly concrete buildings polarization diversity is just as good as space diversity, the measured difference is approximately –2dB in field strength compared to space diversity.
Smart antennas:
Principle of GSM switched-beam antennas Smart antennas, as one of the most promising technologies in the cellular area, are rapidly becoming an integral part of both analog and digital cellular networks. The intelligence of the antenna system resides in a high-speed DSP algorithm that is constantly monitoring the RF environment and controlling the adjustment of the antenna beams on a time-slot basis. By spatially isolating the serviced mobile in a narrower beamwidth the carrier-to-interference ratio is increased, network interference level is reduced, and dropped call rates due to call quality are reduced.
Figure 1 Interference Potential Comparison for 3-Sector GSM Antenna System
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Sector antennaSees interferers in 120°area
4 - beam antennaSees interferers in 30°area
The switched-beam smart antenna replaces traditional sectors with a high gain, narrow-beam phase array antenna. A multi-beam antenna panel consisting of three or four 30-degree beams improves reception of the mobile’s signal while receiving significantly less interference than the standard sector antenna.
According to theoretical calculation, a multi-beam panel consisting of four, switched, narrow-beam antennas can increase the average carrier to interference ratio (C/I) by 6dB over conventional 3-sector systems. The following formula shows the C/I gain of a 30-degree narrow beam antenna versus 120-degree sector antenna.
Where: G: C/I gain for narrow beam antenna system
C: carrier signal level
I30: interference level of 30-degree antenna
I120: interference level of a sector antenna
The smart antenna features an advanced beam-switching algorithm that selects the beams that are best for uplink and downlink. The system continuously updates beam selection on a slot-by-slot basis (Error: Reference source not found), ensuring that subscribers experience greatly improved call quality for the duration of their call.
Figure:2; Principle of beam switching in GSM antennas.
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GSM Frequency Bands
Standard or primary GSM 900 Band, P-GSM: ARFCN : 1 to 124890 - 915 MHz: mobile transmit, base receive 935 - 960 MHz: base transmit, mobile receive
Extended GSM 900 Band, E-GSM (includes Standard GSM 900 band): ARFCN for E- GSM Frequencies : 975 - 1023880 - 915 MHz: mobile transmit, base receive 925 - 960 MHz: base transmit, mobile receive
Railways GSM 900 Band, R-GSM (includes Standard and Extended GSM 900 Band); ARFCN for GSM-R Frequencies : 955 - 974876 - 915 MHz: mobile transmit, base receive 921 - 960 MHz: base transmit, mobile receive
DCS 1800 Band: ARFCN for DCS1800 Frequencies : 512 - 8851710 - 1785 MHz: mobile transmit, base receive 1805 - 1880 MHz: base transmit, mobile receive
PCS 1900 Band : ARFCN for PCS1900 Frequencies : 512 - 8101850 - 1910 MHz: mobile transmit, base receive 1930 - 1990 MHz: base transmit, mobile receive
Radio Propagation
According to Yoshihisa Okumura the land-mobile service is burdened with peculiar complications such as,
1. The antenna height of a mobile body with which communication is held is very low, usually not more than 1-3 Mts above the ground.
2. Between the base station and the mobile or between such mobiles themselves are ever changing, infinitely large number of propagation paths formed due to movement from place to place.
3. This causes the clearance of the propagation paths to be lost, while the field strength, hindered by terrain irregularities and other obstacles, suffers great attenuation and location variability all the time.
Fading and Fading MarginWe know that the formula for free space propagation is given as
Path Loss = 32.45 + 20* Log (frequency) + 20 * Log (distance)
Where, “frequency “ is given in = MHzAnd “distance” is given in = Km.
This formula is valid for Line of sight Communication between the transmitter and the receiver, and there is no consideration of the multipath propagation.
In practice however as suggested by Okumura the signal from the transmitter to the receiver does not always follow a single line of sight communication. The signal also undergoes reflection from
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the various sources. These Incident and reflected signals may add up “ In Phase” or “Out of phase” when they reach the receiver thus contributing to what is know as “Fading”.
Reflections and multi-path propagation can cause positive and negative effects. Coverage Extension
Multipath propagation allows the radio signals to reach behind buildings and into tunnels.The latter is known as ducting.
Ducting may occur in tunnels, valleys, buildings canyons and in the atmosphere if the boundaries (walls, steep hillsides, atmospheric layers) are good reflectors for the radio waves.
VHF signals do not propagate in long tunnels, but higher frequencies (>800Mhz) follow the tunnel like waveguide.If the coverage in a tunnel needs an enhancement. A repeater station at the tunnel entrance radiating into the tunnel may help.
Constructive and destructive interference
The interference due to multipath propagation manifests in the following three most important ways.
- Random Phase shifts create rapid fluctuations in the signal strength known as Rayleigh fading.- A delay spread in the received signal causes each symbol to overlap with adjacent symbol:
intersymbol interference.- Random frequency modulation due to different doppler shifts on different paths.
Practically there are two types of fading,
The Log Normal Fading or better know as the Long term fading, caused primarily due to the terrain contour variation between the MS and the BS. The fading effect is caused by the shadowing and diffraction.Log- normal fading follows the log-normal distribution curve N (, ²) with the mean µ and the standard deviation of about 8 dB.
The Short Term Fading, or the Multi Path Fading or also known as the Rayleigh Fading. It is variation is due to the near surroundings of the receiver.Multipath fading follows the rayleigh distribution curve with a standard distribution of about 6-7 dB.
To remove the uncertainty caused by the fading to the signals, a concept of fading margin is formulated; the idea is to reserve enough power to overcome the potential fading. Hence the
Preceived = Ptransmitted – Fading margin – Path Loss
The fade margin is normally equal to the maximum expected fade or to a smaller value. The value is chosen in such a way that the threshold value is undershot in only a short interval of time. For this reason it is necessary to know the “probabilty distribution function” of fading.
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-
Calculation of Fading Margin value.Area probability can be converted to edge probability using equations in Jake’s book. Standard deviation and Slope of Propagation model is needed.•Required edge probability gives a correction factor from the normal table.•Fading Margin = Standard Deviation * Correction Factor.Typical penetration loss has to be added to fading margin for In Car and In Building Coverage.•Penetration Loss for a In car is typically 6 dB.•Penetration Loss for a In Building is typically 6 dB.•Penetration Loss up to 40 dB for rooms/areas deep inside buildings, More than One concrete wall between room/area and outside. Small windows.
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6,18,30,4210,22,34,462,14,26,385,17,29,41
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Figure shows the multipath propagation conditions in an urban environment
.μ-Kσ .μ .μ+Kσ
Fading Margin
.μ-Kσ .μ .μ+Kσ
Fading Margin
Fading Margin Example
Steps in brief 1) Find total by the formula given above. {(sigma)²+ (Penetration Loss)²}2) Find Edge Probability by the Jakes Curves using the Value of / n and area probability.3) Using the Value of the edge Probability, find out the correction factor on the log normal
distribution table.4) Fading Margin = total * correction Factor + Penetration Loss.
About diffraction.
The Longer the wavelength the higher the diffraction, hence the higher the efficiency to reach the valleys. Or putting it mathematically
Diffraction (wavelength)
Link budget
A link budget is used to determine if the acceptable RF signal can be made available at the receiver, in the case of cellular communication the receiver can be the base station in case of uplink and a mobile station in the case of downlink. The link budget thus, helps determine if the selected type of RF power amplifier, antenna type, cable lengths, environment losses and penetration losses are sufficient for link stability. Usually, the weaker link of the uplink or the downlink is considered as the minimum acceptable signal level.
Calculation of maximum acceptable path loss in up-link and down-link .
•Input required:–MS Power class ( and Sensitivity)–MS Antenna Gain and body Loss–BS maximum output power–BS combiner, duplexer and feeder losses–BS Antenna and diversity gain–Any other gains and losses.
BS Output Power is used to balance up and downlink.Output from the Link Budget is the minimum signal strength at the receiver input
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Combiner
TX RX
Path Loss Fading Margin Penetration Loss
Feeder Loss
Base Station
P.A. Power
Antenna Gain Antenna Gain
Mobile
Sensitivity
Combiner
TX RX
Path Loss Fading Margin Penetration Loss
Feeder Loss
Base Station
P.A. Power
Antenna Gain Antenna Gain
Mobile
Sensitivity
•For each coverage environment the fading Margin (including possible penetration loss) is added to the minimum level required at the receiver input.•Coverage levels are inputted under Settings => Mobile Types in TORNADO.
Cell Radius.The Maximum cell radii for a coverage environment can be calculated using the propagation model, the BS EIRP and the minimum required coverage level ( PTx = PRx + Pathloss). The area covered by an Omni site is then calculated as the area covered by a circle with the above radius. Finally, using trigonometry it is also easy to calculate the area covered by one cell in a three sector site.•Corrections has to be made for the antenna pattern
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Base
A
B CD
Base
A
B CD
How to calculateArea of the D ABC = 2 * Area of D ABD Area of D ABD = ½ *AD * DCCos 30 = Base / Hypoteneous = AD / ACAD = 3 /2 * AC = 3/2 * RSin 30 = Perpendicular / Hypoteneus = DC / ACDC = ½ * AD = ½ * RTherefore total area of D ABC= 2 *(1/2)* ( 3/2 * R) * (1/2 *R)= ( 3/2 * 1/ 2 *R²)Total area of Hexagon is 6 * area of D ABC= 6 * (1/ 2 * 3/2 * R²)= 3* 3/2 * R²= 2.598 * R²
Coverage Area of a 3 sectored siteSector range is improved with directional antennas. Assuming that the omni antenna gain is G0, the directional antenna gain GD and the slope of the signal attenuation is “n”, If omni range is R0 then the sector range will be R0 * 10^ {(GD - G0)/n.}. For 65 deg. Antenna gain is reduced by 2.7 dB in the direction 30 deg. from antenna pointing direction giving an range in the direction of R0 *10 ^ {(GD - G0-2.7)/n}.If hexagon grid is to be maintained without coverage holes in the corners the longer dimension of the hexagon is limited to R0 *10^ (GD - G0-2.7)/n / Cos( 30 deg).
Example•G0 = 11 dBi•GD = 18 dBi•Slope of the signal attenuation 3.5•Longer dimension Rs =[R0 * 10^ (4.3)/3.5] / COS ( 30 Deg) = 1.532 * R0•Area covered by 3 sector site ( replace R0 with 0.5 * 1.532 R0)= 3 * (0.5*Rs)^2 * 2.598 = 3 * (0.5*1.532 *Rs)^2 * 2.598= 1.76 * 2.598 R0 ^2•Area covered by 3 sector site (65 Deg) = 1.76 * area covered by an omni cell.
Cellular Traffic
Cellular Traffic
Usually the cellular networks will be coverage limited or capacity limited, in other words the access to the cellular system depends on the RF signal availability and its capacity. We have already seen the system design criteria to design an RF network with limited uncertainties due to system coverage. Now, we look at the network dimensioning using subscriber density data. Let us start with the basic characteristics of cellular traffic.
Telephone networks are planned so that even during the periods of heaviest traffic, the "busy hours", the calls made by the subscribers have a good chance of success. The amount of switching equipment and the number of resources to be provided for handling the telephone traffic are therefore normally calculated so that during the busy hours only a small but usually predetermined proportion of the desired connection cannot be established at all, i.e. the calls are lost, in case of loss systems this is called the grade of service (GOS), thus, the grade of service is the probability that an offered call will be rejected, (lost). This parameter is also called the loss probability. it is usually expressed in proportion to the total call offered.
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Rs = 1.532 R0Antenna
30 Deg
Rs = 1.532 R0Antenna
30 Deg
Clover Leaf Layout
The theoretical solution of system design problems of this kind belongs to the field of telephone traffic theory of which the Danish mathematician A. K. Erlang considered to be the founder. A. K. Erlang has given the following expression:
E A
A
n
AA A
n
n
n
n1 21
2
,!
! !
where A is the flow of traffic offered expressed in erlang. “n” is the number of devices and quantity is E1, n (A). In unitary terms the 1 unit Erlang (E) is defined as 1 device (n) occupied for one hour time period.
The figure above shows the relation between A, E and blocking B , it can be inferred from the above graph that in order to offer higher Erlang traffic for the same grade of service the number of n (or N) should be increased or else the system blocking will increase.
In TDMA&FDMA systems such as GSM system radio interface (which is of primary concern for this handbook), the number of device “n” (or N) is actually physical timeslots, which can be further logically subdivided into timeslots for signalling and timeslots for traffic. It must be noted that the GOS for the signalling and the traffic channels would be different, usually the signalling (SDCCH) load will also depend on the network design e.g the number of location updates, amount of call-setups, SMS’s, IMSI attach/detach e.t.c. While the traffic (TCH) load is only a function of subscriber behaviour.
In Siemens European traffic model the Erlang per subscriber for traffic is assumed to be 25 milli Erlang with GOS of 2% or 5%, and the erlang traffic for signalling is considered as 4 milli-Erlang with GOS of 2 % blocking. These assumptions are used to dimension the TRX capacity of each base station transmitted.
e.g: If we have to dimension a system for 2000 subscribers with 2% GOS for traffic and signalling and 25 Milli Erlang for traffic and 4 Milli-erlang for signalling, then two options of achieving this configuration. Could be having 2 base station site with 2/2/2 capacity. Calculated as follows
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=> total Erlang offered = sites * sectors * Erlang per sector.In this case we know that the total Erlang is = Erlang per subscriber * number of subscribers = 2000 * 0.025 = total 50 erlangs, if our frequency spectrum only allows a maximum of 2 TRX per cell then we can conclude from looking at the Erlang table we have
50 = sites * 3 * 9.0096 …… (9.0096 based on 15 timeslots per cell, since BCCH timeslot is not used for signalling)
therefore the number of sites is = 1.85 which is rounded off to 2 sites.
Alternatively, If the frequency spectrum allows and the coverage requirement is not high then we could have used an omni site with 8 TRXs (total 50.589 erlang from 61 channels for TCH and 3 for SDCCH) saving the cost of infrastructure and equipment, this additional gain in capacity of an omni site is attributed to the “trunking efficiency” of an omni site (see above graph, and trunking efficiency formula: trunking efficiency= Total erlangs/total number of Channels) . However, omni sites result in poor frequency reuse, reduced flexibility for coverage adjustments (downtilting) and lesser coverage.
Frequency Planning.
Frequency spectrum is a very precious resource that needs to be utilised very prudently while ensuring minimum degradation of connection quality, all this requires efficient frequency planning. In GSM system the process of frequency planning is simplified by assigning a unique number to each of the Uplink and downlink frequency pair, this number is called the ARFCN (Or Absolute Radio Frequency Carrier Number).
Usually, each mobile network operator would be allocated only a part of the total radio spectrum specified for a particular frequency band. E.g. a GSM 900 operator may be typically allocated 12 Mhz pair for operation. This means that the remaining band may be either already occupied or it can be auctioned to another operator for a network within the same geographical area. However, the network capacity requirement may far exceed the available spectrum, this necessitates the need for frequency reuse. Frequency reuse is achieved by breaking down the entire coverage area in the form of small clusters that are spread all over the network. A cluster is a set of cells in which the entire frequency spectrum may be spread, however each frequency in one cluster is unique and not repeated within the same cluster.
There are some standard frequency clusters that have been proven to minimize co-channel and adjacent channel interference, the precondition is however that the network should have a homogeneous structure with sites in a grid pattern and BTS at more or less the same heights with minimum coverage overlaps.
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The figures above show the two most frequently used clusters the 4/12 reuse pattern (Left) which known to provide a C/I ratio > 12 dB and the 3/ 9 reuse pattern (Right) which is know to provide a C/I ratio of > 9 dB. A 4/12-reuse pattern means that we have 4 three sectors site supporting 12 cells.
Usually, the tight re-use pattern such as the 3/9 patterns are used in conjunction with interference reduction features such as power control, frequency hopping and or DTX. Some planners may also use 4/12 patterns for BCCH TRX planning and 3/9-reuse pattern for the TCH TRXs, this helps in protecting the signaling information which is transmitted on the BCCH.
Other re-use patterns such as 7/21 can be used for systems that more vulnerable to interference and have spare bandwidth available for a loose frequency reuse.
The table above shows the case of 3/ 9 reuse pattern for 36 frequency network. One point worth noting is that the re-use pattern frequency planning also provides protection against the constraints of combiner properties. A filter combiner requires 600 KHz separations between the combining frequencies and a hybrid combiner 400 KHz separation.
Automatic Frequency Planning.There are various tools to aid a radio planner to plan all frequencies in a GSM network, some of the most popular tools used at Siemens are AFP of TORNADO and FAT (Frequency Assignment Tool).
Though the process of making an automatic frequency plan through these tools is explained in detail in their individual manuals, but to expect good results it is important to understand the algorithm behind the frequency allocation. Here I will mention only a brief about the AFP algorithm of TORNADO.
Best signal strength Sc is created of pixel-by-pixel basis for a particular cell from the coverage array. This server signal strength is also compared with the potential interfere signal strength Si
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from overlapping predictions. The difference between the Sc and Si is the C/I on each pixel. Or mathematically C/I = Sc-Si. These values are summed over squares to calculate 3 items these are a) the total coverage area of each cell, b) cell pairs affected by C/Ic (Co-channel interference) and c) cell pairs affected by C/Ia (adjacent channel interference).
To calculate the percentage of are on per pixel basis the following table may be used by TORNADO.
Similar to the coverage calculation above on pixel basis, another calculation is made on traffic density which may be derived from I) Traffic package or II) Clutter type. The traffic density is also calculated in the Pixel by pixel basis and then summed over squares to achieve d) Total traffic for each cell e) cell pairs affected by C/Ic (Cochannel Interference) due to trafiic f) cell pairs affected by C/Ia (adjacent channel interference) due to traffic. The above a), b), c), d). e) and f) constitute the contents of Interference table as prepared in Tornado,
The contents of the interference table are then expressed in terms of percentage and compared with the “soft constraints” as configured by the user, the “soft constraints” can be % area affected, % of traffic affected, absolute affected area and absolute affected traffic. In addition to the soft constraints, during planning procedure a number of “hard constraints” can also be defined such as channel seperation between Neighbours of the cells (Neighbour list from the handover package is used for this purpose), Channel separation in co-cell (useful for specific combiner type), channel seperation between cells of the same site. It must be noted that although the soft constraints can be compromised during the planning iterations, the hard constraints on the other hand are never over-ridden. All these constraints are used to create what is called a separation martrix. The statistics of a separation matrix usually consists of the interferer, victim counts, priority lists and distributions.
If the separation matrix is symmetrical then the frequency assignment will start by first comparing the frequency assignments with forbidden frequencies, and then checking with existing carriers in the cells. At this time one can check the statistics to see the difficulties and assignment priority list and distribution.
Although there are many different algorithms within this basic algorithm and these would help in making an efficient plan, however a lot would depend on the user inputs and frequency spectrum
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User defined C/I table loaded at startup
C/I% calls affected bu C/Ic
% calls affected by C/Ia
0dB 100% 40%1dB 100% 30%2dB 95% 20%3dB 90% 10%4dB 75% 5%5dB 60% 0%6dB 50% 0%7dB 40% 0%8dB 25% 0%9dB 10% 0%10dB 0% 0%--- --- ---25dB 0% 0%
limitations. In such a case instead of planning all the frequencies TORNADO will plan only a part of the required frequency plans and would leave some frequencies blank. One can iteratively try to change the user inputs and rerun the AFP to replan the frequencies until maximum number of frequencies is planned without compromising too much on the network quality. Once the best suited incomplete plan is accepted by the RF planner the remaining frequencies can be planned manually. There are some short-cut methods that can help a radio planner in planning the deficient frequencies manually, these are introduced briefly here.
The 4 S principles of manual frequency Planning.
Swap: Swapping is most helpful in cases where it is difficult to assign a new frequency to the interfering frequency pair, hence we can only make a frequency reallocation by redistributing the allocated frequencies.
As an example consider the case below, The checkered arrow points to the adjacent channel interference in the neighboring cells. These problems are very apparent in the Multiple Reuse Pattern frequency planning, however sometimes these can be very obvious in Automatic Frequency planning as well. The Solid red arrow shows the possible scenario of swapping the two frequency groups to resolve this problem.
It may be worth noting that the swaps can be made not only between two adjacent sites but also between two cells of the same site.
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6,18,30,4210,22,34,462,14,26,385,17,29,41
6,18,30,4210,22,34,462,14,26,385,17,29,41
6,18,30,4210,22,34,462,14,26,385,17,29,41
6,18,30,4210,22,34,462,14,26,385,17,29,41
8,20,32,4412,24,36,48
4,16,28,40
9,21,33,45
1,13,25,37
35,23,29,47 7,19,31,43
3,15,27,39
Split: Frequency splitting is a very efficient method in case of tight reuse, sometimes it can also be used to allocate new frequencies in cells where the number of frequencies allocated are less than the number of frequencies required. It is usually useful if the results of the AFP are not very optimum.
It has been observed by many planners that AFP algorithm in tornado usually uses either the odd frequencies more than the even frequencies or the other way around, this allocation is because of the “hard constraints” setting during the AFP and the random seed selected for frequency planning. Therefore, sometimes the allocation through AFP is not paramount.
The figure above shows the example case of a frequency allocation, where the frequencies in the adjacent cells have co-channel interference. Such problem can easily be avoided using frequency splitting. The odd ARFCN “13” can be split into even frequency 12 and 14. Hence we can allocate ARFCN 12 in site-1 sector 2 and ARFCN 14 in Site-2 Sector 1 thereby easily avoiding co-channel and adjacent channel interference.
Search: This is most widely used practice in frequency planning, most of us use it without even thinking of a name of this practice. In networks with a wider frequency band available we can think about replacing the interfering frequency rather than spending time and effort in Swapping, Splitting or shifting it.
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5,13,29,41
2,13,26,38
10,22,34,46 6,18,30,428,20,32,44
12,24,36,48
4,16,28,40
9,21,33,45
1,17,25,37
35,23,29,47 7,19,31,43
3,15,27,39
Site-1
Site- 2
51, 55, 58,61
32,53,57,63
3, 25, 28,45 39,22,30,475,47,20,59
33,2,14,27
7,12,22,38
40,18,16,44
9,24,29,48
35,17,29,41 37,24,9,11
21,3,13,19
Figure above shows and example where frequency 57 and 58 are adjacent channels to each other. It seems from the frequency allocation that there is ample frequency band available for 4/4/4 configuration sites and hence we can think of replacing one of the two interfering frequencies by searching for a new frequency.
TIP: It sometimes helps to check each frequency in the chronological order to find the frequency that best fits to replace the victim or interfering frequency.
Shift. This case is most useful in case of the border sites of downtown, since the frequency configuration is very tight in the downtown areas hence it is difficult to make an ARFCN allocation in these sites, the idea is to borrow clean frequencies from the low configuration sites and shift the interfering frequency out of the downtown area since it is not difficult to assign a new frequency to the out-skirts sites with low configuration.
In the example below, if the cell with the Yellow dot (Checks) needs a new frequency assignment then it needs to take care of the adjacent channel and co-channel interference in the red dot cells (vertical lines), however it possible for the yellow dot cell to borrow the frequency from any of the red dot cells and shift the constraining frequency to the Blue dot cell (Horizontal Lines).
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Interference Reduction
Frequency Hopping
Frequency hopping is a technique in which the information carrier changes the modulation frequency within a specified band, this technique was use by the military to maintain confidentiality over their transmission and prevent their signals from being intercepted by the enemy. There are broadly two types of frequency hopping, namely slow frequency hopping and fast frequency hopping. If the frequency changes faster than the modulation rate then it is termed as fast frequency hoppping and otherwise it is called the slow frequency hopping. GSM applies only to the slow frequency hopping technique and this is further classified as Baseband Frequency Hopping and Synthesizer frequency hopping. The difference in the two techniques is as follows.
Advantages of Frequency Hopping: There are some basic advantages of frequency hopping.1) Frequency Hopping maintains confidentiality over the transmission, since the number of
frequencies in the hopping sequence are high therefore it is not possible to latch on the frequencies and therefore maintain confidentiality.
2) Rayleigh fading is frequency dependent and causes the fading dips for different frequencies to occur at different places, in case of Frequency Hopping network a slow moving mobile will not be stuck up in a deep fade for a long duration and hence will benefit more than a fast moving subscriber.
3) Co-channel or adjacent channel interference in reduced in case of hopping system due to the fact that a subscriber will not be latched on an persistently on an interfered frequency.
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D o w n T o w n
4) Due to advantage No. 3 frequency hopping allows tighter frequency re-use and helps realize efficient spectrum utilization.
Baseband Frequency Hopping:
The figure above shows the hopping sequence as seen in the base-band hopping mode, the t1….t6 are the sequences of the timeslots in different bursts, TPU is the transceiver processing unit, PA is the power amplifier, C1…..C6 are the resonant cavities in the combiner. F0, F3, F6, F9, F12, F15 are the ARFCN the circuit is tuned to.
The figure clearly shows that the TPU, the resonant cavity (C1..C6) and all the transceiver circuits are always tuned to only one frequency only, so in order for a timeslot to hop the timeslot is relayed to from one TPU to another on every burst basis.As per the example in the figure, the timeslot of a particular subscriber is at TPU 0 at the t1 instant of time. However in the next burst the same subscriber timeslot is at TPU 1 at t2 instant of time, and so on, so for this subscriber the timeslot is at different frequency in each burst, hence for this subscriber the frequency is hopping.
Advantages:
1) BCCH can also participate in the hopping sequence.2) Narrow band combiners such as Filter combiner (less combiner loss) can be used hence saving
the EIRP of transmission.3) (Siemens Advantage), no change in hardware required.
Disadvantages:
1) Hopping gain is negligible for less than 3 frequencies in the hopping sequence and therefore is not suitable for low TCH configurations.
2) The numbers of total TRX in the cell limits the maximum frequencies in the hopping sequence.3) Siemens disadvantage (till BR 3.7) If a TRX fails, frequency hopping is disabled.
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Synthesizer Frequency Hopping
Figure above shows a descriptive diagram of Synthesizer frequency hopping, the different colors of the subscriber timeslot signify different bursts, it should be noticed here that the TPU in this case will change the frequency at each burst. Therefore after each burst is transmitted the TPU should change the frequency to the new frequency in the hopping sequence. This is achieved by having two frequency tuned circuits in the same TPU , One RF circuit will prepare to change frequency as long as the other is already transmitting.
Advantages:
1) More frequencies than the total TRX in a cell are possible in the hopping sequence. Therefore allowing more hopping gain in the system.
2) Lower C/Ic ratio are possible in the system without compromising speech quality, this results in a tighter frequency re-use and hence a higher capacity gain.
3) TCH expansions are very easy.
Disadvantages:
1) Since the cavity in the combiner will be required to change frequency very fast, therefore the combiner such as FICOM cannot be used for Synthesizer hopping since these combiners need 2-3 seconds to tune to each frequency. This is a disadvantage because in higher configuration the FICOMS have less combiner compared to DUCOM or HYCOMs.
2) BCCH cannot participate in the hopping sequence. Since the total number of frequencies in the hopping sequence is more than the number of TRX required therefore no fixed frequency allocated for each TRX. However BCCH must always be transmitted, therefore BCCH allocation is done separately as a separate frequency.
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Hardware Requirements:
1) For SIEMENS Base Station Only: TPU 2 is mandatory to be used for synthesizer hopping.
2) All the combiners in the base station with synthesizer hopping should be wide band combiners, therefore FICOMS cannot be be used in base station with synthesizer hopping.
Frequency Planning Ideology for Synthesizer hopping:
Many frequency planing ideologies are propounded for the frequency planning of the synthesizer hopping, however nearly all of them state that thought the frequency plan is easy to generate but MAIO (Mobile Allocation Index Offset) planning is of crucial importance. Also the synthesizer hopping requires in some configurations that the Base station should be synchronized between them to avoid any frequency collisions.
MRP:- 1 X 3 Reuse Pattern.
A 1 X 3 reuse pattern is the frequency assignment in which the all the three sectors of a site have different frequency group, these frequency group may contain adjacent channel frequencies but no co-channel frequency, the adjacent channel interference can be avoided by intelligent allocation of MAIO.
Before we continue our discussion further, let us define a few new terms
MRP: (Multiple Reuse Pattern):a frequency re-use scheme in which the BCCH and TCH allocation is done by reserving separate band of frequencies for BCCH and separate band of frequencies for TCH.
MAIO: (Mobile Allocation Index offset): The MAIO defines the start frequency of the hopping sequence. The maximum value of MAIO is detemined by the total number of frequencies in the hopping sequence and not by the total number of TRX in a cell.
HSN: Hopping sequence number: It defines the the sequence of the frequencies while hopping. In Siemens base station it is possible to define 64 hopping sequences (0 - 63), where “0” defines cyclic hopping and 1 - 63 define un-correlated pseudo randon hopping sequences.
Channel Occupancy Rate: A term specifically used for synthesizer hopping which defines a ratio of total number of TCH frequencies in a cell to total number of frequencies in the hopping sequence, for a good network this ratio should not exceed 40%.
Mathematically: No. of TCH in a cell
No. Of Frequencies in the hopping sequence.
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In a 1 X3 Re-Use pattern the frequency re- use pattern will look like the following,
Where A, B, C are mutually exclusive groups however these groups have adjacent channel frequencies. A typical allocation of the frequencies may look like the following:
Group ARFCN Numbers allocated to the group.A 1 4 7 10 13B 2 5 8 11 14C 3 6 9 12 15
Therefore for a site the allocation may look like
In the above case we assume that the total site configuration is 3/3/3 and that BCCH frequency planning is done separately using a dedicated band therefore the remaining configuration is 2/2/2 for the TCH frequency allocation. Therefore if we have 5 frequencies in the hopping sequence then the Channel occupancy rate would be 2/5 = 40%, which is acceptable value.
Notice that in the above example, each sector has adjacent channel frequency within the same site, this could pose a serious problem. This is problem can be solved by carefully planning MAIO for
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C C C
AB
C AB
C
AB
C AB
C
AB
C AB
C
AB
C AB
CAB
C AB
AB
C AB
AB
C AB
AB
C AB
C
(1,4,7,10,13)
(2,5,8,11,14)
(3,6,9,12,15)
the above site we propose the following MAIO allocations. (notice that since there are 5 frequencies in the hopping sequence hence the Maximum number of MAIO can be 5 { MAIO= 0- 4})
(For reasons of simplicity we assume cyclic hopping in the above case.)
So consider the first stage of Hopping, in this case the first sector will transmit frequency 1 and 7 , sector 2 will transmit frequency 5 and 11 and sector 3 will transmit frequency 3 and 9, hence even-though we have adjacent channel frequencies in the same site still we can avoid adjacent channel interference by planning MAIO. For the case of neighboring sites with the same frequency allocation, to avoid the co-channel collision we must plan different Hopping Sequence Number (HSN).
MRP:- 1 X 1 Reuse Pattern.
For low configuration networks with limited frequency band, another idea can be proposed is that of a 1 X 1 Re-use pattern. In such a case a separate band can be reserved for BCCH alone and the other band can be used for a TCH in the hopping sequence, usually the allocation is such that the number of frequencies in the BCCH band is sufficient to avoid any possibility of collision.
It should be pointed out that the 1X 1 configuration requires that the BTS be synchronize between all the cells in one site, that mean with the existing BS 60 the maximum configuration possible with 1 X 1 hopping is 2/2/2, because the BS 60 is unable to synchronize beyond one rack. As a suggestion for configuration like 3/3/3 what can be done is to allocate a second band for this third cell which is in the extension BS60 rack, this will result in a 1 X 2 re-use pattern. However the BS 240 is capable of synchronizing between racks and therefore 1 X 1 can be implemented more easily in BS 240.
One can configure the network in such a fashion that all the cells in one site have the same hopping sequence number, however differ only in the MAIO allocation. This argument is valid both for 1 X 3 re-use pattern and also for 1X 1 re-use pattern also.
Impact on the Network:
Since the frequency re-use is very tight the effective BER ( Bit error rate, a measure of RX- Qual in GSM ) will be very high, however the quality as perceived by the subscriber is not a measure of BER but a measure of F.E.R (Frame Erasure Rate), in a good network the FER should not exceed 2 %.
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One important point to notice is that since the emergency (quality Based) hand-over parameters is the based on the quality as perceived by BER, therefore the value of the parameters related to quality handovers should be adjusted to avoid unnecessary handovers in the network..
Power control
..
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Appendix: 1
Okumura hata propagation model to TORNADO(PlaNet) conversion.The standard Okumura hata equation is as follows:
Path Loss = 69.55 + 26.16 log10 fc – 13.82 log 10 hb –a(hm) + (44.9 – 6.55 log10hb)log10 d
Where
fc = Frequency In MHz.
hb = Effective Height of the base station.(Mts.)
hm = Effective Height of the Mobile (MTs.)
d = Distance from the base station (Kms.)
a() = Mobile height correction factor.
However in the Planet format the frequencies should be entered in the Hertz and the Distance in Meters however the Units in the Okumura hata equation should not be changed. (i.e. It should be in MHz and Kms.)
Therefore the equation gets Modified as follows.
=69.55 + 26.16 *(log fc + log 10 -6)Mhz – 13.82 log hb + ( 44.9 –6.55 log hb) * (log d + log 10-3) Kms.
=69.55 + 26.16 * (log fc - 6)Mhz – 13.82 log hb + ( 44.9 –6.55 log hb) * (log d - 3) Kms
=69.55 + 26.16 * (log fc) Mhz – 156.96 – 13.82 log hb + ( 44.9 –6.55 log hb) * (log d) Kms – 44.9 *3 + 6.55 *3 * log hb
=-222.11 + 26.16 * (log fc) Mhz + 5.83 log hb + ( 44.9 log d –6.55 log hb* log d) Kms
=-222.11 + 26.16 * (log 925000000) Mhz + 5.83 log hb + ( 44.9 log d –6.55 log hb* log d) Kms
=12.44 + 5.83 log hb + 44.9 log d –6.55 log hb* log d Kms
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Appendix 2
Jakes curves:
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.μ-Kσ .μ .μ+Kσ
Fading Margin
.μ-Kσ .μ .μ+Kσ
Fading Margin
Appendix 3
ERLANG – Tables.
n Loss probability (E) n
0.00001 0.00005 0.0001 0.0005 0.001 0.002 0.003 0.004 0.005 0.0061 .00001 .00005 .00010 .00050 .00100 .00200 .00301 .00402 .00503 .00604 12 .00448 .01005 .01425 .03213 .04576 .06534 .08064 .09373 .10540 .11608 23 .03980 .06849 .08683 .15170 .19384 .24872 .28851 .32099 .34900 .37395 34 .12855 .19554 .23471 .36236 .43927 .53503 .60209 .65568 .70120 .74124 45 .27584 .38851 .45195 .64857 .76212 .89986 .99446 1.0692 1.1320 1.1870 56 .47596 .63923 .72826 .99567 1.1459 1.3252 1.4468 1.5421 1.6218 1.6912 67 .72378 .93919 1.0541 1.3922 1.5786 1.7984 1.9463 2.0614 2.1575 2.2408 78 1.0133 1.2816 1.4219 1.8298 2.0513 2.3106 2.4837 2.6181 2.7299 2.8266 89 1.3391 1.6595 1.8256 2.3016 2.5575 2.8549 3.0526 3.2057 3.3326 3.4422 910 1.6970 2.0689 2.2601 2.8028 3.0920 3.4265 3.6480 3.8190 3.9607 4.0829 1011 2.0849 2.5059 2.7216 3.3294 3.6511 4.0215 4.2661 4.4545 4.6104 4.7447 1112 2.4958 2.9671 3.2072 3.8781 4.2314 4.6368 4.9038 5.1092 5.2789 5.4250 1213 2.9294 3.4500 3.7136 4.4465 4.8306 5.2700 5.5588 5.7807 5.9638 6.1214 1314 3.3834 3.9523 4.2388 5.0324 5.4464 5.9190 6.2291 6.4670 6.6632 6.8320 1415 3.8559 4.4721 4.7812 5.6339 6.0772 6.5822 6.9130 7.1665 7.3755 7.5552 1516 4.3453 5.0079 5.3390 6.2496 6.7215 7.2582 7.6091 7.8780 8.0995 8.2898 1617 4.8502 5.5583 5.9110 6.8782 7.3781 7.9457 8.3164 8.6003 8.8340 9.0347 1718 5.3693 6.1220 6.4959 7.5186 8.0459 8.6437 9.0339 9.3324 9.5780 9.7889 1819 5.9016 6.6980 7.0927 8.1698 8.7239 9.3515 9.7606 10.073 10.331 10.552 1920 6.4460 7.2854 7.7005 8.8310 9.4115 10.068 10.496 10.823 11.092 11.322 2021 7.0017 7.8834 8.3186 9.5014 10.108 10.793 11.239 11.580 11.860 12.100 2122 7.5680 8.4926 8.9462 10.180 10.812 11.525 11.989 12.344 12.635 12.885 2223 8.1443 9.1095 9.5826 10.868 11.524 12.265 12.746 13.114 13.416 13.676 2324 8.7298 9.7351 10.227 11.562 12.243 13.011 13.510 13.891 14.204 14.472 2425 9.3240 10.369 10.880 12.264 12.969 13.763 14.279 14.673 14.997 15.274 2526 9.9265 11.010 11.540 12.972 13.701 14.522 15.054 15.461 15.795 16.081 2627 10.537 11.659 12.207 13.686 14.439 15.285 15.835 16.254 16.598 16.893 2728 11.154 12.314 12.880 14.406 15.182 16.054 16.620 17.051 17.406 17.709 2829 11.779 12.976 13.560 15.132 15.930 16.828 17.410 17.853 18.218 18.530 2930 12.417 13.644 14.246 15.863 16.684 17.606 18.204 18.660 19.034 19.355 3031 13.054 14.318 14.937 16.599 17.442 18.389 19.002 19.470 19.854 20.183 3132 13.697 14.998 15.633 17.340 18.205 19.176 19.805 20.284 20.678 21.015 3233 14.346 15.682 16.335 18.085 18.972 19.966 20.611 21.102 21.505 21.850 3334 15.001 16.372 17.041 18.835 19.743 20.761 21.421 21.923 22.336 22.689 3435 15.660 17.067 17.752 19.589 20.517 21.559 22.234 22.748 23.169 23.531 3536 16.325 17.766 18.468 20.347 21.296 22.361 23.050 23.575 24.006 24.376 3637 16.995 18.470 19.188 21.108 22.078 23.166 23.870 24.406 24.846 25.223 3738 17.669 19.178 19.911 21.873 22.864 23.974 24.692 25.240 25.689 26.074 3839 18.348 19.890 20.640 22.642 23.652 24.785 25.518 26.076 26.534 26.926 3940 19.031 20.606 21.372 23.414 24.444 25.599 26.346 26.915 27.382 27.782 4041 19.718 21.326 22.107 24.189 25.239 26.416 27.177 27.756 28.232 28.640 4142 20.409 22.049 22.846 24.967 26.037 27.235 28.010 28.600 29.085 29.500 4243 21.104 22.776 23.587 25.748 26.837 28.057 28.846 29.447 29.940 30.362 4344 21.803 23.507 24.333 26.532 27.641 28.882 29.684 30.295 30.797 31.227 4445 22.505 24.240 25.081 27.319 28.447 29.708 30.525 31.146 31.656 32.093 4546 23.211 24.977 25.833 28.109 29.255 30.538 31.367 31.999 32.517 32.962 4647 23.921 25.717 26.587 28.901 30.066 31.369 32.212 32.854 33.381 33.832 4748 24.633 26.460 27.344 29.696 30.879 32.203 33.059 33.711 34.246 34.704 4849 25.349 27.206 28.104 30.493 31.694 33.039 33.908 34.570 35.113 35.578 4950 26.067 27.954 28.867 31.292 32.512 33.876 34.759 35.431 35.982 36.454 5051 26.789 28.706 29.632 32.094 33.332 34.716 35.611 36.293 36.852 37.331 51
0.00001 0.00005 0.0001 0.0005 0.001 0.002 0.003 0.004 0.005 0.006
n Loss probability (E) n
Page: 35 of 39
n Loss probability (E) n
0.007 0.008 0.009 0.01 0.02 0.03 0.05 0.1 0.2 0.41 .00705 .00806 .00908 .01010 .02041 .03093 .05263 .11111 .25000 .66667 12 .12600 .13532 .14416 .15259 .22347 .28155 .38132 .59543 1.0000 2.0000 23 .39664 .41757 .43711 .45549 .60221 .71513 .89940 1.2708 1.9299 3.4798 34 .77729 .81029 .84085 .86942 1.0923 1.2589 1.5246 2.0454 2.9452 5.0210 45 1.2362 1.2810 1.3223 1.3608 1.6571 1.8752 2.2185 2.8811 4.0104 6.5955 56 1.7531 1.8093 1.8610 1.9090 2.2759 2.5431 2.9603 3.7584 5.1086 8.1907 67 2.3149 2.3820 2.4437 2.5009 2.9354 3.2497 3.7378 4.6662 6.2302 9.7998 78 2.9125 2.9902 3.0615 3.1276 3.6271 3.9865 4.5430 5.5971 7.3692 11.419 89 3.5395 3.6274 3.7080 3.7825 4.3447 4.7479 5.3702 6.5464 8.5217 13.045 910 4.1911 4.2889 4.3784 4.4612 5.0840 5.5294 6.2157 7.5106 9.6850 14.677 1011 4.8637 4.9709 5.0691 5.1599 5.8415 6.3280 7.0764 8.4871 10.857 16.314 1112 5.5543 5.6708 5.7774 5.8760 6.6147 7.1410 7.9501 9.4740 12.036 17.954 1213 6.2607 6.3863 6.5011 6.6072 7.4015 7.9667 8.8349 10.470 13.222 19.598 1314 6.9811 7.1155 7.2382 7.3517 8.2003 8.8035 9.7295 11.473 14.413 21.243 1415 7.7139 7.8568 7.9874 8.1080 9.0096 9.6500 10.633 12.484 15.608 22.891 1516 8.4579 8.6092 8.7474 8.8750 9.8284 10.505 11.544 13.500 16.807 24.541 1617 9.2119 9.3714 9.5171 9.6516 10.656 11.368 12.461 14.522 18.010 26.192 1718 9.9751 10.143 10.296 10.437 11.491 12.238 13.385 15.548 19.216 27.844 1819 10.747 10.922 11.082 11.230 12.333 13.115 14.315 16.579 20.424 29.498 1920 11.526 11.709 11.876 12.031 13.182 13.997 15.249 17.613 21.635 31.152 2021 12.312 12.503 12.677 12.838 14.036 14.885 16.189 18.651 22.848 32.808 2122 13.105 13.303 13.484 13.651 14.896 15.778 17.132 19.692 24.064 34.464 2223 13.904 14.110 14.297 14.470 15.761 16.675 18.080 20.737 25.281 36.121 2324 14.709 14.922 15.116 15.295 16.631 17.577 19.031 21.784 26.499 37.779 2425 15.519 15.739 15.939 16.125 17.505 18.483 19.985 22.833 27.720 39.437 2526 16.334 16.561 16.768 16.959 18.383 19.392 20.943 23.885 28.941 41.096 2627 17.153 17.387 17.601 17.797 19.265 20.305 21.904 24.939 30.164 42.755 2728 17.977 18.218 18.438 18.640 20.150 21.221 22.867 25.995 31.388 44.414 2829 18.805 19.053 19.279 19.487 21.039 22.140 23.833 27.053 32.614 46.074 2930 19.637 19.891 20.123 20.337 21.932 23.062 24.802 28.113 33.840 47.735 3031 20.473 20.734 20.972 21.191 22.827 23.987 25.773 29.174 35.067 49.395 3132 21.312 21.580 21.823 22.048 23.725 24.914 26.746 30.237 36.295 51.056 3233 22.155 22.429 22.678 22.909 24.626 25.844 27.721 31.301 37.524 52.718 3334 23.001 23.281 23.536 23.772 25.529 26.776 28.698 32.367 38.754 54.379 3435 23.849 24.136 24.397 24.638 26.435 27.711 29.677 33.434 39.985 56.041 3536 24.701 24.994 25.261 25.507 27.343 28.647 30.657 34.503 41.216 57.703 3637 25.556 25.854 26.127 26.378 28.254 29.585 31.640 35.572 42.448 59.365 3738 26.413 26.718 26.996 27.252 29.166 30.526 32.624 36.643 43.680 61.028 3839 27.272 27.583 27.867 28.129 30.081 31.468 33.609 37.715 44.913 62.690 3940 28.134 28.451 28.741 29.007 30.997 32.412 34.596 38.787 46.147 64.353 4041 28.999 29.322 29.616 29.888 31.916 33.357 35.584 39.861 47.381 66.016 4142 29.866 30.194 30.494 30.771 32.836 34.305 36.574 40.936 48.616 67.679 4243 30.734 31.069 31.374 31.656 33.758 35.253 37.565 42.011 49.851 69.342 4344 31.605 31.946 32.256 32.543 34.682 36.203 38.557 43.088 51.086 71.006 4445 32.478 32.824 33.140 33.432 35.607 37.155 39.550 44.165 52.322 72.669 4546 33.353 33.705 34.026 34.322 36.534 38.108 40.545 45.243 53.559 74.333 4647 34.230 34.587 34.913 35.215 37.462 39.062 41.540 46.322 54.796 75.997 4748 35.108 35.471 35.803 36.109 38.392 40.018 42.537 47.401 56.033 77.660 4849 35.988 36.357 36.694 37.004 39.323 40.975 43.534 48.481 57.270 79.324 4950 36.870 37.245 37.586 37.901 40.255 41.933 44.533 49.562 58.508 80.988 5051 37.754 38.134 38.480 38.800 41.189 42.892 45.533 50.644 59.746 82.652 51
0.007 0.008 0.009 0.01 0.02 0.03 0.05 0.1 0.2 0.4
n Loss probability (E) n
Page: 36 of 39
.s ICMn Loss probability (E) n
0.00001 0.00005 0.0001 0.0005 0.001 0.002 0.003 0.004 0.005 0.00651 26.789 28.706 29.632 32.094 33.332 34.716 35.611 36.293 36.852 37.331 5152 27.513 29.459 30.400 32.898 34.153 35.558 36.466 37.157 37.724 38.211 5253 28.241 30.216 31.170 33.704 34.977 36.401 37.322 38.023 38.598 39.091 5354 28.971 30.975 31.942 34.512 35.803 37.247 38.180 38.891 39.474 39.973 5455 29.703 31.736 32.717 35.322 36.631 38.094 39.040 39.760 40.351 40.857 5556 30.438 32.500 33.494 36.134 37.460 38.942 39.901 40.630 41.229 41.742 5657 31.176 33.266 34.273 36.948 38.291 39.793 40.763 41.502 42.109 42.629 5758 31.916 34.034 35.055 37.764 39.124 40.645 41.628 42.376 42.990 43.516 5859 32.659 34.804 35.838 38.581 39.959 41.498 42.493 43.251 43.873 44.406 5960 33.404 35.577 36.623 39.401 40.795 42.353 43.360 44.127 44.757 45.296 6061 34.151 36.351 37.411 40.222 41.633 43.210 44.229 45.005 45.642 46.188 6162 34.900 37.127 38.200 41.045 42.472 44.068 45.099 45.884 46.528 47.081 6263 35.651 37.906 38.991 41.869 43.313 44.927 45.970 46.764 47.416 47.975 6364 36.405 38.686 39.784 42.695 44.156 45.788 46.843 47.646 48.305 48.870 6465 37.160 39.468 40.579 43.523 45.000 46.650 47.716 48.528 49.195 49.766 6566 37.918 40.252 41.375 44.352 45.845 47.513 48.591 49.412 50.086 50.664 6667 38.677 41.038 42.173 45.183 46.692 48.378 49.467 50.297 50.978 51.562 6768 39.439 41.825 42.973 46.015 47.540 49.243 50.345 51.183 51.872 52.462 6869 40.202 42.615 43.775 46.848 48.389 50.110 51.223 52.071 52.766 53.362 6970 40.967 43.405 44.578 47.683 49.239 50.979 52.103 52.959 53.662 54.264 7071 41.734 44.198 45.382 48.519 50.091 51.848 52.984 53.848 54.558 55.166 7172 42.502 44.992 46.188 49.357 50.944 52.718 53.865 54.739 55.455 56.070 7273 43.273 45.787 46.996 50.195 51.799 53.590 54.748 55.630 56.354 56.974 7374 44.045 46.585 47.805 51.035 52.654 54.463 55.632 56.522 57.253 57.880 7475 44.818 47.383 48.615 51.877 53.511 55.337 56.517 57.415 58.153 58.786 7576 45.593 48.183 49.427 52.719 54.369 56.211 57.402 58.310 59.054 59.693 7677 46.370 48.985 50.240 53.563 55.227 57.087 58.289 59.205 59.956 60.601 7778 47.149 49.787 51.054 54.408 56.087 57.964 59.177 60.101 60.859 61.510 7879 47.928 50.592 51.870 55.254 56.948 58.842 60.065 60.998 61.763 62.419 7980 48.710 51.397 52.687 56.101 57.810 59.720 60.955 61.895 62.668 63.330 8081 49.492 52.204 53.506 56.949 58.673 60.600 61.845 62.794 63.573 64.241 8182 50.277 53.012 54.325 57.798 59.537 61.480 62.737 63.693 64.479 65.153 8283 51.062 53.822 55.146 58.649 60.403 62.362 63.629 64.594 65.386 66.065 8384 51.849 54.633 55.968 59.500 61.269 63.244 64.522 65.495 66.294 66.979 8485 52.637 55.445 56.791 60.352 62.135 64.127 65.415 66.396 67.202 67.893 8586 53.427 56.258 57.615 61.206 63.003 65.011 66.310 67.299 68.111 68.808 8687 54.218 57.072 58.441 62.060 63.872 65.897 67.205 68.202 69.021 69.724 8788 55.010 57.887 59.267 62.915 64.742 66.782 68.101 69.106 69.932 70.640 8889 55.804 58.704 60.095 63.772 65.612 67.669 68.998 70.011 70.843 71.557 8990 56.598 59.526 60.923 64.629 66.484 68.556 69.896 70.917 71.755 72.474 9091 57.394 60.344 61.753 65.487 67.356 69.444 70.794 71.823 72.668 73.393 9192 58.192 61.164 62.584 66.346 68.229 70.333 71.693 72.730 73.581 74.311 9293 58.990 61.985 63.416 67.206 69.103 71.222 72.593 73.637 74.495 75.231 9394 59.789 62.807 64.248 68.067 69.978 72.113 73.493 74.545 75.410 76.151 9495 60.590 63.630 65.082 68.928 70.853 73.004 74.394 75.454 76.325 77.072 9596 61.392 64.454 65.917 69.791 71.729 73.896 75.296 76.364 77.241 77.993 9697 62.194 65.279 66.752 70.654 72.606 74.788 76.199 77.274 78.157 78.915 9798 62.998 66.105 67.589 71.518 73.484 75.681 77.102 78.185 79.074 79.837 9899 63.803 66.932 68.426 72.383 74.363 76.575 78.006 79.096 79.992 80.760 99100 64.609 67.760 69.265 73.248 75.242 77.469 78.910 80.008 80.910 81.684 100101 65.416 68.589 70.104 74.115 76.122 78.364 79.815 80.920 81.829 82.608 101
0.00001 0.00005 0.0001 0.0005 0.001 0.002 0.003 0.004 0.005 0.006
n Loss probability (E) n
Created by Rupinder S. Kathuria Page 37 of 39
.s ICM
n Loss probability (E) n
0.007 0.008 0.009 0.01 0.02 0.03 0.05 0.1 0.2 0.451 37.754 38.134 38.480 38.800 41.189 42.892 45.533 50.644 59.746 82.652 5152 38.639 39.024 39.376 39.700 42.124 43.852 46.533 51.726 60.985 84.317 5253 39.526 39.916 40.273 40.602 43.060 44.813 47.534 52.808 62.224 85.981 5354 40.414 40.810 41.171 41.505 43.997 45.776 48.536 53.891 63.463 87.645 5455 41.303 41.705 42.071 42.409 44.936 46.739 49.539 54.975 64.702 89.310 5556 42.194 42.601 42.972 43.315 45.875 47.703 50.543 56.059 65.942 90.974 5657 43.087 43.499 43.875 44.222 46.816 48.669 51.548 57.144 67.181 92.639 5758 43.980 44.398 44.778 45.130 47.758 49.635 52.553 58.229 68.421 94.303 5859 44.875 45.298 45.683 46.039 48.700 50.602 53.559 59.315 69.662 95.968 5960 45.771 46.199 46.589 46.950 49.644 51.570 54.566 60.401 70.902 97.633 6061 46.669 47.102 47.497 47.861 50.589 52.539 55.573 61.488 72.143 99.297 6162 47.567 48.005 48.405 48.774 51.534 53.508 56.581 62.575 73.384 100.96 6263 48.467 48.910 49.314 49.688 52.481 54.478 57.590 63.663 74.625 102.63 6364 49.368 49.816 50.225 50.603 53.428 55.450 58.599 64.750 75.866 104.29 6465 50.270 50.723 51.137 51.518 54.376 56.421 59.609 65.839 77.108 105.96 6566 51.173 51.631 52.049 52.435 55.325 57.394 60.619 66.927 78.350 107.62 6667 52.077 52.540 52.963 53.353 56.275 58.367 61.630 68.016 79.592 109.29 6768 52.982 53.450 53.877 54.272 57.226 59.341 62.642 69.106 80.834 110.95 6869 53.888 54.361 54.793 55.191 58.177 60.316 63.654 70.196 82.076 112.62 6970 54.795 55.273 55.709 56.112 59.129 61.291 64.667 71.286 83.318 114.28 7071 55.703 56.186 56.626 57.033 60.082 62.267 65.680 72.376 84.561 115.95 7172 56.612 57.099 57.545 57.956 61.036 63.244 66.694 73.467 85.803 117.61 7273 57.522 58.014 58.464 58.879 61.990 64.221 67.708 74.558 87.046 119.28 7374 58.432 58.930 59.384 59.803 62.945 65.199 68.723 75.649 88.289 120.94 7475 59.344 59.846 60.304 60.728 63.900 66.177 69.738 76.741 89.532 122.61 7576 60.256 60.763 61.226 61.653 64.857 67.156 70.753 77.833 90.776 124.27 7677 61.169 61.681 62.148 62.579 65.814 68.136 71.769 78.925 92.019 125.94 7778 62.083 62.600 63.071 63.506 66.771 69.116 72.786 80.018 93.262 127.61 7879 62.998 63.519 63.995 64.434 67.729 70.096 73.803 81.110 94.506 129.27 7980 63.914 64.439 64.919 65.363 68.688 71.077 74.820 82.203 95.750 130.94 8081 64.830 65.360 65.845 66.292 69.647 72.059 75.838 83.297 96.993 132.60 8182 65.747 66.282 66.771 67.222 70.607 73.041 76.856 84.390 98.237 134.27 8283 66.665 67.204 67.697 68.152 71.568 74.024 77.874 85.484 99.481 135.93 8384 67.583 68.128 68.625 69.084 72.529 75.007 78.893 86.578 100.73 137.60 8485 68.503 69.051 69.553 70.016 73.490 75.990 79.912 87.672 101.97 139.26 8586 69.423 69.976 70.481 70.948 74.452 76.974 80.932 88.767 103.21 140.93 8687 70.343 70.901 71.410 71.881 75.415 77.959 81.952 89.861 104.46 142.60 8788 71.264 71.827 72.340 72.815 76.378 78.944 82.972 90.956 105.70 144.26 8889 72.186 72.753 73.271 73.749 77.342 79.929 83.993 92.051 106.95 145.93 8990 73.109 73.680 74.202 74.684 78.306 80.915 85.014 93.146 108.19 147.59 9091 74.032 74.608 75.134 75.620 79.271 81.901 86.035 94.242 109.44 149.26 9192 74.956 75.536 76.066 76.556 80.236 82.888 87.057 95.338 110.68 150.92 9293 75.880 76.465 76.999 77.493 81.201 83.875 88.079 96.434 111.93 152.59 9394 76.805 77.394 77.932 78.430 82.167 84.862 89.101 97.530 113.17 154.26 9495 77.731 78.324 78.866 79.368 83.134 85.850 90.123 98.626 114.42 155.92 9596 78.657 79.255 79.801 80.306 84.100 86.838 91.146 99.722 115.66 157.59 9697 79.584 80.186 80.736 81.245 85.068 87.826 92.169 100.82 116.91 159.25 9798 80.511 81.117 81.672 82.184 86.035 88.815 93.193 101.92 118.15 160.92 9899 81.439 82.050 82.608 83.124 87.003 89.804 94.216 103.01 119.40 162.59 99100 82.367 82.982 83.545 84.064 87.972 90.794 95.240 104.11 120.64 164.25 100101 83.296 83.916 84.482 85.005 88.941 91.784 96.265 105.21 121.89 165.92 101
0.007 0.008 0.009 0.01 0.02 0.03 0.05 0.1 0.2 0.4
Created by Rupinder S. Kathuria Page 38 of 39
.s ICMn Loss probability (E) n
Appendix 4
Normal Distribution chart.
Normal Distribution Chart for continuous random variable ZZ 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
Created by Rupinder S. Kathuria Page 39 of 39