Formel KKE5201 5 Rev N A5 d

Planning digital radio-relay networks k&k engineering

Performance and unavailability

Principles & formulae

Version G.826

2 k&k engineering

2005-03-06 Rev. N-A5 TECHNICAL PAPER KKE 5201/5

1998-2005

Copying the contents of this booklet as well as translations to other languages, completely or partly, is not allowed without the permission of K&K Engineering HB. This includes any kind of copying by print, duplication, tape recording, electronic methods etc.

K&K Engineering HB, Box 2, S-610 54 NÄVEKVARN / Sweden Phone & Fax: +46-155-535 77 or: +46-8-532 51 888 E-mail: [email protected] Internet home page: http://www.KK-Engineering.a.se

050306

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TECHNICAL PAPER KKE 5201/5 Rev. N-A5 2005-03-06

Introduction K&K Engineering’s PC-based computer programs FORMULA and RLTool, are in-tended for the prediction of performance and availability of radio-relay paths and circuits. This paper, which is based on H. Karl’s booklets Planning and engineering of radio-relay networks and Performance and availability as applied to digital radio-relay systems {1,2}, describes the principles and formulae utilized in the program. In version 1 of this TECHNICAL PAPER KKE 5201/1, the formulae were mainly derived from CCIR Report 338. The ITU-R Recommendation P.530 has later on replaced this report. In September 1997, the ITU-R published version 7 of its Rec. P.530. This recommendation contains a complete new set of formulae for the prediction of both flat and selective multi-path fading, as well as for the improvement due to diversity. Also the formulae for the pre-diction of attenuation by atmospheric gases have been modified - Rec. P.676-3. This new formulae have been introduced in the above programs with effect from version 2.0 for FORMULA and version 2.20 for RLTool. During 2001, ITU-R introduced version 9 of Rec. P.530, which contains a complete new set of formulae for the prediction of multipath fading. This new formulae have been introduced in the above programs with effect from version 2.0 for FORMULA and version 3.0 for RLTool. This paper is based on the new versions of the above ITU-R recommendations.

Note: In some of the formulae, a distance parameter may be included. Dependent on the subject of the formula, this distance parameter may represent the geodetic distance, as read from a map, or the real distance of the radio beam between two antennas. To distinguish between these two distances, two different symbols are used: d... distance as read from a map in km, or: geodetic path length = plane projection of the

radio path

d*... real length of the radio beam between transmitter and receiver antenna in km = beam path length

d*can be calculated applying the following formula:

( ) 622* 10−⋅−+= BA hhdd

d* ... real length of the radio beam between transmitter and receiver antenna in km d ... geodetic path length in km = plane projection of the radio path hA ... height above sea level for station A in m hB ... height above sea level for station B in m

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Table of contents 1 Path geometry...................................................................................................... 7

1.1 Co-ordinates and bearing ............................................................................ 7 1.1.1 Calculation of great-circle distance and bearing ................................ 7 1.1.2 Determination of co-ordinates............................................................ 7

1.2 1st Fresnel zone radius................................................................................ 8 1.3 Calculation of antenna heights.................................................................... 8 1.4 Calculation of path clearance...................................................................... 9 1.5 Effective Earth radius factor k .................................................................. 10 1.6 Ground reflection and its calculation ........................................................ 10

1.6.1 Calculation of antenna heights ......................................................... 10 1.6.2 Location of reflection point.............................................................. 12 1.6.3 Difference in path length between direct and reflected ray .............. 13 1.6.4 The distance between receiver input level minima or maxima......... 13 1.6.5 Optimum antenna spacing with space diversity protection .............. 14 1.6.6 Efficiency of selected space diversity versus k-value variation ....... 14 1.6.7 Antenna discrimination .................................................................... 14

2 Path attenuation and receiver input level ........................................................... 16 2.1 Total path attenuation during fading-free time.......................................... 16 2.2 Free-space basic attenuation ..................................................................... 16 2.3 Additional attenuation(s) .......................................................................... 16 2.4 Gain or loss in a passive repeater, antenna back-to-back .......................... 16 2.5 Gain or loss in a passive repeater, plane reflector ..................................... 17

2.5.1 Check of far-/near-field operation:................................................... 17 2.5.2 Angle in space.................................................................................. 17 2.5.3 Repeater gain in far field.................................................................. 18 2.5.4 Half-power (3 dB) beam width ........................................................ 19

2.6 Losses due to atmospheric gases............................................................... 20 2.7 Receiver input level during fading-free time ............................................ 21

3 Overall performance of a digital radio-relay link during fading-free time and time of shallow fading ................................................................................ 22

4 Overall performance of a digital radio-relay link during fading - Performance calculation .................................................................................... 22

4.1 General...................................................................................................... 22 4.2 The multipath occurrence factor ............................................................... 22

4.2.1 Prediction formula............................................................................ 22 4.2.2 Path inclination ................................................................................ 23 4.2.3 Geoclimatic factor K........................................................................ 23

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4.3 Performance prediction considering multipath fading and related mechanisms.................................................................................................24

4.3.1 Prediction formulae ..........................................................................24 4.3.2 Fading margin...................................................................................26 4.3.3 Paths going via passive repeaters......................................................26

4.4 Performance prediction considering distortions due to propagation effects (selective fading) .....................................................................26

4.4.1 Prediction formulae ..........................................................................26 4.4.2 Prediction procedure for path going via a passive repeater...............27

4.5 Small-time-percentage for exceeding the planning objectives due to attenuation caused by precipitation.........................................................................27

4.5.1 Attenuation caused by rain ...............................................................27 4.5.2 Attenuation coefficient .....................................................................28 4.5.3 Rainfall intensity...............................................................................28 4.5.4 Effective path length.........................................................................29 4.5.5 Fading probability due to rain for one path.......................................29 4.5.6 Prediction procedure for path going via a passive repeater...............29 4.5.7 Worst-month concept and average annual probability......................30

4.6 Improvement of the performance by diversity reception..........................31 4.6.1 Improvement by frequency diversity ................................................31 4.6.2 Improvement by space diversity .......................................................32 4.6.3 Improvement by combined frequency and space diversity - 2 Rx ....34 4.6.4 Improvement by combined frequency and space diversity - 4Rx.............35

4.7 Total performance with respect to the G.826 objectives........................................36 4.7.1 Calculation of the block-based severely errored seconds ratio

(SESR)..............................................................................................37 4.7.2 Fading exceeding the background block error ratio

(BBER) objective .............................................................................39 4.7.3 Fading exceeding the errored second ratio (ESR) objective.............41 4.7.4 Total performance for the circuit ......................................................41

5 Unavailability calculations for radio-relay systems............................................43 5.1 Unavailability and reliability of hardware .................................................43

5.1.1 Single (unprotected) structures .........................................................43 5.1.2 Duplicated (protected) structures......................................................44

5.2 Unavailability due to propagation disturbances.........................................46 5.3 Total unavailability....................................................................................46

6 Frequency planning............................................................................................48 6.1 The number of disturbing signals reaching a receiver ...............................48 6.2 General formula for the calculation of interfering signal levels ................48 6.3 Formulae for triangular network configuration .........................................49

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6.3.1 Nodal station disturbs outstation (TxA1 _ RxC).............................. 50

6.3.2 Outstation disturbs nodal point (TxC _RxA1) ................................. 51 6.4 Interference via passive repeater............................................................... 52

6.4.1 Passive repeater as first-source transmitter ...................................... 52 6.4.2 Passive repeater as receiver of interfering signals............................ 54

6.5 Total interference...................................................................................... 54 7 Bibliography...................................................................................................... 56 Appendix I 59 Appendix II 61 Appendix III 62

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Principles and formulae

1 Path geometry 1.1 Co-ordinates and bearing 1.1.1 Calculation of great-circle distance and bearing

[1] [ ]( )122121 coscoscossinsincos12.111 xxyyyyad −⋅⋅+⋅⋅=

d... great-circle distance in km x1... longitude for site A in degrees negative values for x2... longitude for site B in degrees W of Greenwich y1... latitude for site A in degrees negative values for y2... latitude for site B in degrees S of the equator and the antenna bearing in A is:

[2] ( )( ) �

��

��

��

cos.sin.cossinsincos'

⋅⋅⋅⋅−=Θ∠

for sin (x2- x1) > 0: ∠ Θ1 =

∠ Θ'1 for sin (x2 – x1) < 0: ∠ Θ1 =

360o - ∠ Θ'1

1.1.2 Determination of co-ordinates If the co-ordinates for one site, eg A, and the bearing and great-circle distance to the other site are known, the co-ordinates of that site can be calculated accordingly:

[3] [ ] [ ]( )dyyday ⋅⋅+⋅⋅⋅Θ= 0089992.0cossincos0089992.0sincossin 1112

[4] [ ]21

2112 cos.cos

sinsin0089992.0coscos

yyyydaxx ⋅−⋅

±=

for ∠ Θ1 < 180o: +acos for ∠ Θ1 > 180o: -acos

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1.2 1st Fresnel zone radius

[5] dfdd.r⋅⋅

= 211 317

r1... radius of the 1st Fresnel zone at a certain point in m d... radio beam length in km d1... distance from one site to that point in km d2 = d – d1, in km f... radio frequency in GHz

1.3 Calculation of antenna heights The below formula presumes the knowledge or the assumption of one antenna height. If the antenna height at A is the known one, the antenna height at B can be calculated according to:

[6]

[ ] ( ) ( )B

GAAOBST

GB hd

hhdd.kdddhrd

h −+⋅−−

⋅−++∆⋅

=1

1211

7412

hGA... height above ground level for antenna at A in m hGB... height above ground level for antenna at B in m hA... height above sea level for station A in m hB... height above sea level for station B in m hOBST... height above sea level for highest obstacle (with respect to propagation) in m d... distance A to B in km d1... distance A to obstacle in km k... effective earth radius factor ∆r1... required clearance above obstacle in m where:

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[7] 100

11 rrrr ∆⋅=∆

r1... radius at the 1st Fresnel zone in m ∆rr... required clearance above obstacle in %

If there is more than one obstacle which may influence the determination of antenna heights, the calculation will have to be repeated and the highest value for hGB chosen.

For calculation of hGA, if hGB is known, replace the indices 1 by 2, and A by B, and B by A.

1.4 Calculation of path clearance Referring to the same parameters as in formula [6] and the associated figure, the clearance above an obstacle is:

[8] ( ) ( )( ) ( )7412

11111

.kdddh

dhhddhhd

r OBSTGAAGBB

⋅−

−−+−++

=∆

For ∆r1> r1 the 1st Fresnel zone is free from intrusions For r1 > ∆r1 > 0 the 1st Fresnel zone is intruded, but there is still line-of-sight For ∆r1 < 0 no line-of-sight

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1.5 Effective Earth radius factor k The antenna heights according to the above sections have to be calculated for both the stan-dard atmosphere - k = 1.33 - and for kmin.

If kmin is not known, the below diagram may be used. Path lengths <20 km should be set to 20 km.

1.6 Ground reflection and its calculation

1.6.1 Calculation of antenna heights The below formulae require the antenna height above the reflection area. Reference should hereby be made to the figure, which shows the basic path geometry for a reflective path. Pa-rameters not stated below are according to section 1.3.

[9] υ⋅⋅+−+= tan1031 ooAGA xyhhh

[10] ( ) υ⋅⋅−−−+= tan1032 ooBGB xdyhhh

h1... height of antenna above reflection area at site A in m h2... height of antenna above reflection area at site B in m tan υ... inclination angle for sloping terrain (υ = 0 for horizontal terrain) according

to formula [11]

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[11] ( ) 3

12

12

10tan

⋅−

−=υ

xxyy

xo, yo... midpoint of the reflection area according to formulae [12] and [13]

[12] 2

121

xxxxo−

+=

[13] 2

121

yyyyo−

+=

x1... the distance from site A to the beginning of the reflection area in km x2... the distance from site A to the end of the reflection area in km y1... the altitude in m above sea level for point x1 y2... the altitude in m above sea level for point x2

Basic geometry for a reflective path

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1.6.2 Location of reflection point

[14] ( )Zdd += 121

[15] ( ) 12 12

ddZdd −=−=

d1 and d2 are the distances in km to the reflection point from either side of the path accord-ing to Figure 2.

[16] 21

21hhhhq

+−

=

q... parameter to be used in formula [18] h1... height of antenna above reflection area at site A in m h2... height of antenna above reflection area at site B in m

[17] ( )2

21

251

dhhkQ

⋅

+⋅⋅=

Q... parameter to be used in formulae [18]-[20] The other parameters have their previous significance.

[18]

Q

qV11+

=

[19] [ ] [ ] [ ]

+

++

++

++

++= ....

11111 4

8

3

6

2

42

QV

QV

QV

QVVZ

Since the series in the above formula converges quite rapidly, it can, with good approxima-tion, be terminated after the fourth term, and the formula can consequently be written as follows:

[20] [ ] [ ]

++

++

++≈ 3

6

2

42

1111

QV

QV

QVVZ

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1.6.3 Difference in path length between direct and reflected ray

[21] 322

221

1 1074.1274.12

2 −⋅

⋅−

⋅−=δ

kdh

kdh

d

δ ... difference in path length between direct and reflected ray in m The other parameters have their previous significance. Expressed in terms of wavelengths, this difference will be:

[22] 3.0f⋅δ=τ

τ ... difference in path length between direct and reflected ray in number of wavelengths

Each time the number of wavelengths, τ, is a positive integer (1, 2, etc), the receiver input level passes through a minimum. The receiver input level will pass through more than one minimum when k is varying.

1.6.4 The distance between receiver input level minima or maxima

The pitch, ϑ1 (or ϑ2), i.e. the distance between adjacent minima or maxima in the input level, can be calculated using the formulae below:

[23] 322

2

1 10

74.12

115.0 ⋅

⋅−

⋅⋅=ϑ

kdh

fd

Receiver input level vs k value variation

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[24] 321

1

2 10

74.12

115.0 ⋅

⋅−

⋅⋅=ϑ

kdh

fd

1.6.5 Optimum antenna spacing with space diversity protection Optimum spacing between the antennas, for a certain k value, is obtained by dividing the pitch ϑ1 and ϑ2 respectively by a factor 2, i.e.:

[25] ( )( )2

2121

ϑ=∆h

∆h1(2)... antenna spacing between diversity antennas in m at station A or B respectively

ϑ1(2) ... as above

1.6.6 Efficiency of selected space diversity versus k-value variation

[26] 322

21

1 1074.123,0

2 −⋅

⋅−

⋅∆⋅⋅

=τ∆k

dhdhf

[27] 321

12

2 1074.123,0

2 −⋅

⋅−

⋅∆⋅⋅

=τ∆k

dhdhf

∆τ1(2)... space diversity efficiency at station A or B respectively: ∆τ = 0.5 corresponds to optimum efficiency

The other parameters have their previous significance.

1.6.7 Antenna discrimination On steep paths or paths with large clearance it is sometimes possible to take advantage of the radiation pattern of the antennas to discriminate the reflected signal. Then the angles α1

and α2 in the figure on page 11 must be determined. With these values we can enter the radiation pattern for the used antennas.

[28] 3221

1

11 10

74.12180 −⋅

⋅−

−−

π=α

kd

dhh

dh

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[29] 3112

2

22 10

74.12180 −⋅

⋅−

−−

π=α

kd

dhh

dh

α1(2) ... angles between direct and reflected ray in degrees according to the figure in section 1.6.1 All other parameters have their previous significance.

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2 Path attenuation and receiver input level 2.1 Total path attenuation during fading-free time

[30] RBWWgAoL GGGAAAAAAA −−−+++++= 2121

AL .... total (or net) path attenuation in dB Ao .... free-space basic attenuation in dB AA ... additional attenuation(s) in dB Ag ... attenuation due to atmospheric gases in dB AW1,2 ... antenna feeder attenuation at the transmitting (1) and receiving (2) end, in dB AB ... attenuation in the RF-branching assembly of the radio-relay equipment in dB G1,2 ... antenna gain at the transmitting (1) and receiving (2) end, in dB GR ... gain in a passive repeater in dB

2.2 Free-space basic attenuation

[31] fdAo lg20lg204.92 * ⋅+⋅+=

d*... length of the radio beam between transmitter and receiver antenna in km f ... radio frequency in GHz

2.3 Additional attenuation(s) The additional attenuation can be caused by: - RF attenuators - obstacles, - partial clearance, - periscopic antennas, - passive repeaters in the near-field of the closest antenna. The first four values have to be given as fixed input data, the computer program is not de-signed to determine one of these values. The program, however, deals with passive repeat-ers, - see next section.

2.4 Gain or loss in a passive repeater, antenna back-to-back In formula [30] the free-space basic attenuation Ao is replaced by:

oBoA AA +

where: AoA... free-space basic attenuation between station A and the repeater site

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AoB... free-space basic attenuation between station B and the repeater site and GR by:

BA GG +

where: GA... antenna gain in the passive repeater for the antenna directed towards site A GB... antenna gain in the passive repeater for the antenna directed towards site B

2.5 Gain or loss in a passive repeater, plane reflector 2.5.1 Check of far-/near-field operation:

[32]

2cos

75 *

ψ⋅⋅

⋅π⋅=

Yf

ds sZ

d*s ... the shorter one of the two partial paths (legs) in km f... radio frequency in GHz Y... reflector area in m2

ψ... angle in space at repeater in degrees For sZ > 2.5 ⇒ far-field condition For sZ < 2.5 ⇒ near-field condition

2.5.2 Angle in space

[33] ( ) ( )( )[ ]( ) [ ]( )2'622'62

'''1

62'

1010

cos10cos

BRBARA

ABARBAAR

hhdhhd

hhhhddhh

−+⋅−+⋅

−−−ψ⋅⋅+−=ψ

[34] ψ=ψ∠ cosa

h’A... height above sea level for the antenna at station A in m (= hA+hGA) h’B... height above sea level for the antenna at station B in m (= hB+hGB)

hR... height above sea level for passive repeater1 in m dA... distance station A to passive repeater in km (in plane projection)

dB... distance station B to passive repeater in km (in plane projection)

1This height is the sum of ground level and reflector height

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ψ1... angle at repeater point in plane projection in degrees ψ... angle in space at repeater point in degrees Y... reflector area (physical area) in m2

bmax... largest side dimension (width or height) of the reflector in m f... radio frequency in GHz

2.5.3 Repeater gain in far field In formula [30] the free-space basic attenuation Ao is replaced by:

oBoA AA +

where: AoA... free-space basic attenuation between station A and the repeater site AoB... free-space basic attenuation between station B and the repeater site

and GR is calculated according to:

[35]

ψ⋅⋅⋅⋅=2

cos5.139lg20 2 YfGR

2.5.3.1 Repeater loss in near field

In formula [30] the free-space basic attenuation Ao is replaced by:

olA

where: Aol... free-space basic attenuation for the longer of the two legs

The repeater loss is obtained from the below, computerized diagram. The help parameters are as per formulae [32] and [36]

[36]

2cos4 ψ⋅⋅

π=ηY

Das

Read AA from the above diagram and insert it in formula [30].

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2.5.4 Half-power (3 dB) beam width

[37]

2cos

3.152max

3 ψ⋅⋅≈Θ

bfdB

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2Θ3dB ... half-power or 3 dB beam width in degrees bmax... largest side of reflector in m


2.6 Losses due to atmospheric gases

[38] *dA gg ⋅γ=

Ag... attenuation in dB due to absorption by oxygen and water vapour

γg... specific attenuation in dB/km d*... length of the radio beam between transmitter and receiver antenna in km and:

[39] wog γ+γ=γ

γo... specific attenuation in dB/km for dry air γw... specific attenuation in dB/km for water vapour

∗ For f = radio frequency < 57 GHz:

[40] [ ]

322252222257 10

44.2575.7

351.0

27.7 −− ⋅⋅⋅⋅

⋅⋅+−+

⋅⋅+

⋅=γ tp

tptp

to rrf

rrfrrf

r

∗ For 57 < f < 63 GHz:

[41]

( ) ( ) ( ) ( )( )( )

63

5.825760

186360

635766.118

6360

−

−−

γ−−

+−⋅−⋅⋅⋅−γ−⋅−=γ

o

tpoo

ff

ffrrff

∗ For 63 < f < 350 GHz

[42] [ ]

[ ]

[ ]

3222

222

2

5225.155.14

63 10

84.275.118

28.0

5.1634102.11102

−

−−

− ⋅⋅⋅⋅

⋅⋅+−

⋅

+⋅⋅+−

+⋅⋅−⋅⋅

=γ tp

tp

t

tpt

o rrf

rrf

r

rrffr

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[43] t

rt +=

273288

[44] 1013

prp =

t... average lowest temperature in °C p... air pressure in hPa

[45] ρ... water vapour density in g/m3. (If no measured data are available for the water

vapour density, approximate values can be obtained from the charts in Appendix I.)

2.7 Receiver input level during fading-free time

[46] ATPCALL LTxRx −−=

LRx... receiver input level in dBm during fading-free time LTx... transmitter output level in dBm AL... total path attenuation in dB during fading-free time acc. to formula [30] ATPC … control range of the adaptive transmitter power control in dB

[ ]

[ ] [ ]

42

2222

225.04

7

10

44.10153.32501.4

85.1131.18373.11

81.9235.2279.3107.700167.00327.0

−

−

⋅ρ

+−+

+−+

+−+⋅+ρ+

=γ tp

tp

t

tp

t

tpp

tt

w rrf

rrfr

rrfr

rrff

rrr

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3 Overall performance of a digital radio-relay link during fading-free time and time of shallow fading

During fading-free time, the performance is determined by the background bit-error ratio, BBER. This is also valid for the time of shallow fading.

4 Overall performance of a digital radio-relay link during fading - Performance calculation

4.1 General

∗ The calculation with respect to the small-time-percentage objective is car-ried out individually for each path.

∗ The small-time-percentage objectives only take account of multipath fad-ing through the troposphere, of precipitation and of the influence of inter-fering signals. Other fading types, such as two-way propagation by ground-reflected waves, ducting etc are assumed to be compensated for by appropriate engineering, such as the selection of suitable antenna heights and/or sites, diversity reception, etc.

∗ For the SESR objective - rain attenuation is assumed to exceed the avail-able fading margin for at least 10 consecutive seconds. It is thus consid-ered as unavailability. For the ESR and BBER performance objective, however, all rain fading, irrespective its duration, has to be treated as a performance influencing parameter.

∗ Multipath propagation and precipitation appear uncorrelated. The total time percentage during which the planning objectives are not met is the sum of two independent contributions.

4.2 The multipath occurrence factor 4.2.1 Prediction formula For detailed planning:

[47] ( ) Lhfoi dKP ⋅−⋅+−− ⋅ε+⋅= 00085.0032.0297.02.3* 101

For approximate planning:

[48] ( ) Lhfoi dKP ⋅−⋅+−− ⋅ε+⋅= 001.0033.022.10.3* 101

Poi... multipath occurrence factor for the individual radio hop

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hL ... the lower of the two antenna altitudes in m above sea level, i.e. hA+hGA or hB+hGB

K... geoclimatic factor f... frequency in GHz d*... length of the radio beam between transmitter and receiver antenna in km ε... hop inclination in milliradians i... serial number of the individual hop (i = 1...n)

4.2.2 Path inclination

The path inclination, ε, is the angle between the line-of-sight and the horizontal. Its absolute value, calculated according to equation [49], is used in formulae [47] and [48].

[49] ( ) ( )d

hhhh GBBGAA −−+≈ε

ε... inclination in milliradians d... path length in km hA... elevation in m above sea level for the left-hand site hB... elevation in m above sea level for the right-hand site hGA... antenna height in m above the ground for the left-hand site hGB... antenna height in m above the ground for the right-hand site

4.2.3 Geoclimatic factor K If no fading data are available for the area concerned, the factor K can be estimated follow-ing the below procedure: For detailed planning (formula [47]):

[50] 1003.09.342.0 10 dNaK ⋅−−− ⋅σ=

For approximate planning (formula [48]):

[51] 10029.02.410 dNK ⋅−−=

The parameters have the following significance: dN1... The point refractivity gradient in the lowest 65 m of the atmosphere not

exceeded for 1% of an average year. The figure can be obtained on a 1.5o

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grid resolution in latitude and longitude from a database2 available from ITU-R (see also ITU-R Rec. 453-8).

σa... The area terrain roughness, defined as the standard deviation in m of the terrain heights (in m) within a 110 km x 110 km area with a 30” resolution. The area should be aligned with the longitude, such that two equal halves of the area are on each side of the longitude going through the hop’s midpoint. Terrain data are available from Internet, eg the Globe gtopo30 data.

The standard deviation can be calculated applying the following formula:

[52] ( )1

2

11

2

−κ⋅κ

−⋅κ

=σ

∑∑κ

=

κ

= jj

jj

a

hh

For σa < 1, set σa = 1. hj... altitude a.s.l. in m for the individual height sample

κ... total number of samples j... ordinal number of the individual sample (j = 1...κ) For the calculation of a hop’s midpoint, and for the bilinear interpolation in order to obtain the correct figure for dN1, reference should be made to the Annex of this booklet, page 57.

4.3 Performance prediction considering multipath fading and related mechanisms

4.3.1 Prediction formulae From the multipath occurrence factor, Po, calculated according to either formula [47] or [48], a fading depth M (dB) is calculated:

[53] oPM lg2.14.27 ⋅+=

If M is less or equal than the available fading margin, MF, i.e.

[54] FMM ≤

the probability, that the available fading margin is exceeded is calculated according to the below formula

2 The corresponding data files, DNDZ_01.txt, DNDZ_LAT.txt and DNDZ_LON.txt can be downloaded from ITU-R’s website. A table - dN_1.xls - showing dN1 versus longitude and latitude can be downloaded from K&K Engineering’s website (see page 2).

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[55] 1010 FMoFi PP −⋅=

PFi... probability rate for exceeding the planning objective, defined by the avail-able fading margin, MF, for one radio hop during the average worst month

Po... multipath occurrence factor for the respective radio hop as per formula [47] or [48]

MF... fading margin in dB i... serial number of the individual hop (i = 1...n) For fade depths, M, larger than the available fading margin, MF, the following method is recommended: (i) Use formula [55] above, calculate PFiM for the fade margin M as obtained

by formula [53]. (ii) Calculate parameter qa for the same fade margin, M, and the associated

value for PFiM from:

[56] ( )[ ]FiMa PM

q −−⋅−= 1lnlg20

(iii) If PFi is very small, your calculator may round:

1 - PFiM

to become ln 1 = 0. To avoid that, set the quotient to the highest value for 0.999..., which still is considered by your calculator as an

ln 0.999... ≠ 0

(iv) Calculate parameter qt for the same fade margin, M, from:

[57] ( )

+⋅−⋅⋅+

−= −

⋅−− 800103.4

10103.01

2 2016.020

Mqq M

MMa

t

(v) Finally, calculate the probability, PFi, that the planned fading margin, MF, is exceeded:

[58] ( )SFi eP −−= 1

where:

[59] 2010 FMqS ⋅−= and:

[60] [ ]

++⋅⋅++= −⋅−−800

103.410103.012 20016.020 FMt

MM Mqq FFF

26 k&k engineering


The parameters PFi and MF have their previous significance.

4.3.2 Fading margin

[61] DLATPCLLATPCLM TeRxTeIRxF −−+=−+=

MF... flat-fading margin in dB LRx... receiver input level in dBm during fading-free time

LTr... receiver threshold level in dBm for the planning criterion and for an undis-turbed receiver (CIR = ∞)

LTrI... receiver threshold level in dBm for the planning criterion and for a disturbed receiver (CIR ≠ ∞)

D... receiver threshold degradation in dB due to interfering signals ATPC … selected control range of the adaptive transmitter power control in dB

4.3.3 Paths going via passive repeaters The total probability rate for exceeding the fading margin, MF, is the sum of the percentage of time that the fading margin, MF, is exceeded for each leg:

[62] 21 legFilegFiFi PPP −− +=

4.4 Performance prediction considering distortions due to propagation effects (selective fading)

4.4.1 Prediction formulae

[63] 32

20 10103.4 −− ⋅ττ

⋅⋅⋅η⋅=ref

mBSi WP

with:

[64] 3.1

507.0

=τ

∗dm

and:

[65] 75.02.01 oPe ⋅−−=η

PSi... probability that one radio hop exceeds the planning criterion due to distor-tions during the average worst month

η... multipath activity factor

27k&k engineering


Po. multipath occurrence factor acc. to formula [47] or [48] W... mean value of the width of the signature in MHz B... mean value of the signature (or notch) depth in dB τref... reference delay in ns used to obtain the signature (W and B) d*... beam path length in km In case the manufacturer submits its equipment data separately for minimum phase (MPh) and non-minimum phase (NMPh) fading, the mean value can be calculated as

[66] 2

NMPhMPh WWW +=

for the signature width, and

[67]

+⋅=21010lg20

2020 NMPhMPh BBB

for the notch depth, and

[68] 2

,, NMPhrefMPhrefref

τ+τ=τ

for the reference delay.

4.4.2 Prediction procedure for path going via a passive repeater The statement given in section 4.3.3 is also valid here, i.e. the fading contribution due to selective multipath propagation will be calculated individually for each leg, applying formu-lae [63] to [65]. The fading margin, MF, is, again, that for the total path length, and will thus be the same for both legs. The total percentage of time for selective fading is thus:

[69] 21 legSilegSiSi PPP −− +=

4.5 Small-time-percentage for exceeding the planning ob-jectives due to attenuation caused by precipitation

The influence of rain is predicted by calculating the rain attenuation for 0.01% of the time. Relating the rain attenuation to the available flat-fading margin, the percentage of time dur-ing which the fading margin is exceeded is calculated.

4.5.1 Attenuation caused by rain The attenuation caused by rain is:

[70] effRR dA ⋅γ=01.0

28 k&k engineering


AR0.01... attenuation due to rainfall in dB during 0.01% of time

γR... rain attenuation coefficient in dB/km deff... the path length in km influenced by rain - the effective path length

4.5.2 Attenuation coefficient

The attenuation coefficient, γR, versus radio frequency, f, for various clock-minute rainfall rates during 0.01% of time, J0.01, is calculated from formula [71]:

[71] α⋅Τ=γ 01.0JR

J0.01... clock-minute average annual rainfall rate (or rainfall intensity) in mm/h exceeded for 0.01% of the time, see section 4.5.3

Τ and α are frequency- and polarization-dependent parameters, which are to be obtained from the table in Appendix III

4.5.3 Rainfall intensity If no measured data are available, the rainfall intensity can be estimated from 3 parameters, Pr6, Ms and Mc. Their data can be found in the corresponding data files esarainPR6.txt,

esarain_Mc.txt and esarain_Ms.txt3. The data are extracted the following way: For each of the 3 parameters, Pr6, Ms and Mc, the figures for the 4 grid points surrounding the hop’s midpoint are used in order to calculate the corresponding figures for the midpoint applying bilinear calculation - see formula [174] on page 58. The midpoint’s rainfall intensity, J0,01, is then calculated with the help of the midpoint figures for Pr6, Ms and Mc:

[72] ( )

+⋅−+−

+⋅⋅=

−

AMMC.

BBMM

A.J cs

cs.

42

4010

109361100331

[73] ( )60117.06 1 rs PM

r ePA ⋅−−=

3 The corresponding data files, ESARAINPR6.txt, ESARAIN_Mc.txt and ESARAIN_Ms.txt can be

downloaded from ITU-R’s website. Alternatively, tables - P-Pr6.xls, P-MC.xls and P-MS.xls - showing the corresponding parameters versus longitude and latitude can be downloaded from K&K Engineering’s website (see page 2).

29k&k engineering


[74] CA

MMB cs ⋅+

⋅⋅+= −3103736.111.1

[75]

=A

C 01.0ln

Appendix II at the end of this handbook shows rainfall intensity charts based on the above data.

4.5.4 Effective path length

[76] o

eff dddd

+=

1

[77] 01.0015.035 Jo ed ⋅−⋅=

d... geodetic path length in km deff... effective path length in km

Note: For J0.01 > 100 mm/h, use J0.01 = 100 mm/h in formula [77].

4.5.5 Fading probability due to rain for one path The percentage of time during which the rain attenuation exceeds the available flat-fading margin, MF, is estimated to be:

[78] [ ]( )

≥= ⋅⋅++− 154023.010 01.012.0lg172.029812.0546.0628.11 01.0

FRMA

Ri MAp FR

pRi... fading probability in % of time for a radio hop due to rain

MF... fading margin in dB

Equation [78] converges quickly to % as the factor decreases and approaches 0.154. For values <0.154024, a figure of 0.155 is used for AR0.01/MF in the above equation, giving a pRi of 8⋅10-7%.

4.5.6 Prediction procedure for path going via a passive repeater Plane reflector type The rain fading probability is calculated applying the same calculation method as described in sections 4.5.1-4.5.5, but using

[79] 21 legleg ddd += in formula [76]

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Antenna back-to-back type

∗ If both legs utilise the same polarisation, the calculations is as described for the plane reflector type

∗ If both legs operate at different polarisations, proceed as follows: −

apply the formulae in sections 4.5.1-4.5.5 plus formula [79] for verti-cal polarisation and obtain pR-V

−

perform the same for horizontal polarisation and obtain pR-H − obtain the final rain fading probability, pR:

[80] d

dpdpp HlegHRVlegVR

Ri−−−− ⋅+⋅

=

4.5.7 Worst-month concept and average annual probability 4.5.7.1 Average annual probability:

[81] RiRai pP ⋅= 01.0

pRi... fading probability according to formula [78] PRai... average annual probability, during which the rain attenuation exceeds the

available fading margin. When the fading margin, MF, in formula [78] refers to the BERSES threshold level of the re-ceiver, the above formula [81] changes to

[82] 100

01.0 UPpP SESRaiSESRai ⋅⋅= −−

pRai-SES... fading probability in % according to formula [78], with fading margin, MF, referred to the BERSES threshold level of the receiver

PRai-SES... resulting average annual probability rate, during which the rain attenuation exceeds the available fading margin

UP... portion in percentage of the average annual probability rate, which lasts longer than 10 consecutive seconds and, thus, has to be treated as unavail-ability

4.5.7.2 Worst month probability

[83] ( ) 87.033.301.0 RiRwmi pP ⋅⋅=

31k&k engineering


PRwmi... average worst month probability, during which the rain attenuation exceeds the available fading margin

Consequently, that part, UP, of the average annual probability rate, which lasts shorter than 10 consecutive seconds, has also to be converted to an average worst month probability rate, applying formula [83]:

[84] 87.0

10010033.301.0

−⋅⋅⋅= −−UPpP SESRaiUPRwmi

PRwmi-UP... average worst month probability, during which the portion (100-UP) of the rain attenuation exceeds the available fading margin to the BERSES thresh-old level of the receiver

pRai-SES and UP as above

4.6 Improvement of the performance by diversity reception 4.6.1 Improvement by frequency diversity 4.6.1.1 For flat fading

[85] 102 1080 FM

fidffI ⋅

⋅∆⋅= ∗

Ifi... improvement factor due to frequency diversity for the individual hop f... band centre frequency in GHz ∆f... frequency separation between the two diversity paths, r.f.1 – r.f.2 , in GHz d*... beam path length in km MF... flat fading margin according to section 4.3.2. In case the main and the diver-

sity path have different fading margins (due to different Tx output levels, etc.), the lower of the two fading margins has to be used.

The above formula is verified by measurements for the following data ranges: 2 < f < 13 GHz 30 < d* < 75 km ∆f / f < 0.05

If ∆f > 0.5 GHz, use ∆f = 0.5 The validity of formula [85] outside these ranges is not yet sufficiently proved. Calculate the improved fading probability by applying:

[86] fi

FidfFi I

PP =

32 k&k engineering


PdfFi... probability for the worst month for exceeding the planning criterion due to fading for a 1+1 frequency-diversity configuration for the individual hop

PF... probability in for the worst month for exceeding the planning criterion due to fading for an unprotected configuration according to equations [55] or [58] for the individual hop

4.6.1.2 For distortions

[87] ( )2

2

1 fSi

SidfSi

k

PP

−η=

PdfSi... probability for the worst month for exceeding the planning criterion due to distortions for a 1+1 frequency-diversity configuration for the individual hop

PSi... probability for the worst month for exceeding the planning criterion due to distortions fading for an unprotected configuration according to equation [63] for the individual hop

η... multipath activity factor, see equation [65] for the individual hop

[88] 8238.02 =fSik for rwi < 0.5

[89] ( ) ( )wirwifSi rk −−−−= 1lg13.0109.02 1195.01 for 0.5 < rwi < 0.9628

[90] ( ) 5136.02 13957.01 wifSi rk −−= for rwi > 0.9628

[91] ( ) 17.2219746.01 fFiwi kr −−= for k2fFi < 0.26

[92] ( ) 034.1216921.01 fFiwi kr −−= for k2fFi > 0.26

[93] η⋅

−= FififFi

PIk 12

4.6.2 Improvement by space diversity 4.6.2.1 For flat fading:

With f and d* having their previous significance, the equation for the space-diversity im-provement factor can be written as follows:

33k&k engineering


[94] 101034.3 10104.148.012.087.04

Fo MPdfhsi eI ⋅

−=−∗−− ⋅⋅⋅∆⋅⋅−

Isi... improvement factor due to space diversity for the individual hop ∆h... vertical spacing of receiving antennas, centre-to-centre, in m MF... flat fading margin in dB according to section 4.3.2. In case the main and the

diversity path have different fading margins (due to different antenna sizes, waveguide length, etc.), the fading margin has to be corrected accordingly:

[95] GMM mFF ∆−= −

[96] dWmWdm AAGGG −− +−−=∆

If ∆G < 0 → ∆G = 0 MF-m... flat fading margin in dB for the main-antenna path Po... multipath occurrence factor according to formula [47] or [48] Gm... gain in dB for the main antenna Gd... gain in dB for the diversity antenna AW-m... waveguide attenuation dB for the main-antenna path AW-d... waveguide attenuation dB for the diversity-antenna path

The above formula is verified by measurements for the following data ranges: 2 < f < 11 GHz 43 < d< 240 km 3 < ∆h < 23 m The validity of the formula outside these ranges is not yet sufficiently proved. Calculate the improved flat-fading probability by applying:

[97] si

FidsFi I

PP =

PdsFi... probability rate for the worst month for exceeding the planning criterion due to fading for a space-diversity configuration for the individual hop

PFi... probability rate for the worst month for exceeding the planning criterion due to fading for an unprotected configuration according to equations [55] or [58] for the individual hop

4.6.2.2 For distortions

[98] ( )2

2

1 sSi

SidsSi

kP

P−η

=

34 k&k engineering


PdsSi... probability for the worst month for exceeding the planning criterion due to distortions for a frequency-diversity configuration for the individual hop

PSi... probability for the worst month for exceeding the planning criterion due to

distortions for an unprotected configuration according to equation [63] for the individual hop

η... multipath activity factor, see equation [65] for the individual hop

[99] 8238.02 =sSik for rwi < 0.5

[100] ( ) ( )wirwisSi rk −−−−= 1lg13.0109.02 1195.01 for 0.5 < rwi < 0.9628

[101] ( ) 5136.02 13957.01 wisSi rk −−= for rwi > 0.9628

[102] ( ) 17.2219746.01 sFiwi kr −−= for k2sFi < 0.26

[103] ( ) 034.1216921.01 sFiwi kr −−= for k2sFi > 0.26

[104] η⋅

−= FisisFi

PIk 12

4.6.3 Improvement by combined frequency and space diversity - 2 Rx

4.6.3.1 For flat fading:

The flat fading improvement, Is,f-2, and the improved probability, pds,fF-2, is obtained by using the same formulae [94]-[97] as for space diversity. The limitations apply also here

4.6.3.2 For selective fading:

Also here, the same formulae as for space diversity are valid, i.e. formulae [98]-[104]. The flat-fading correlation coefficient, ksFi, in formulae [102] and [104], however, has to be replaced by:

[105] sFifFisFif kkk ⋅=,

with ksFi according to formula [104], and kfFi according to formula [93].

35k&k engineering


4.6.4 Improvement by combined frequency and space diversity - 4Rx 4.6.4.1 For flat fading:

[106] D

FiisFdf m

PP

44, =−

[107] ( )( )223 11 sFifFiD kkm −−⋅η=

Pdf,sFi... probability rate for the worst month for exceeding the planning criterion due to fading for a combined frequency/space-diversity configuration with 4 Rx

PFi... probability rate for the worst month for exceeding the planning criterion due to fading for an unprotected configuration according to equations [55] or [56]

η... multipath activity factor acc. to equation [65] ksFi... flat fading correlation coefficient for space diversity configuration according

to formula [104] kfFi... flat fading correlation coefficient for frequency diversity configuration ac-

cording to formula [93]

4.6.4.2 For distortions:

[108] ( )[ ] 22

4

4,1 sSi

SiisSdf

k

PP

−η=−

Pdf,sS-4i... probability rate for the worst month for exceeding the planning criterion due to distortions for a combined frequency/space-diversity configuration with 4 Rx

PSi... probability rate for the worst month for exceeding the planning criterion due to distortions for an unprotected configuration according to equation [63]

ksS... selective fading correlation coefficient for space diversity configuration according to formulae [99] to [104]

36 k&k engineering


4.7 Total performance with respect to the G.826 objectives. Table 1 BERSES and block sizes...

Path type Bit rate

(Mbit/s)

BERSES

(Note ii) Blocks/s (Note ii)

n

Bits/block (Note ii)

NB

...for various PDH systemsNote i

E1 2 4x10-4 2 000 1 120

2xE1 2x2 2x10-4 2 000 2 000

E2 8 1.1x10-4 2 000 4 224

8xE1 8x2 8.8x10-5 4 000 5 170

E3 34 6.5x10-5 8 000 6 120

...for various SDH paths and sections

VC-11 1.5 5.4x10-4 2 000 832

VC-12 2 4.0x10-4 2 000 1 120

VC-2 6 1.3x10-4 2 000 3 424

VC-3 34 6.5x10-5 8 000 6 120

VC-4 140 2.1x10-5 8 000 18 792

STM-1 155 2.3x10-5

2.33x10-4 8 000

192 000 19 940

801

i No figures are stated so far for PDH systems. P.530 advises to select the BERSES closest to the SDH transmission rate. This applies for 2 and 34 Mbit/s systems. For the other PDH capacities, the author proposes the above figures.

37k&k engineering


ii The BERSES is the bit-error ratio for which the number of errored blocks within 1 second exceeds 30%. The figures stated assume a Poisson distribution of errors.

The Block/s is defined in Rec. G.826 for SDH paths, and in G.829 for SDH sections. Some STM-1 equipment might be designed with 8000 blocks/s (19 940 bits/block), but Rec. G.829 defines the block rate and size to be 192 000 block/s and 801 bits/block respectively.

4.7.1 Calculation of the block-based severely errored seconds ratio (SESR)...

4.7.1.1 ... for an unprotected hop

The total fading probability rate for the individual, unprotected hop is:

[109] SESSiSESFiSESMi PPP −−− +=

PMi-SES... probability rate for the worst month for exceeding BERSES on an unpro-

tected hop due to multipath propagation PFi-SES... probability rate for the worst month for exceeding BERSES on an unpro-

tected hop due to multipath fading; calculated acc. to formulae [55] or [58], applying the relevant fading margin to BERSES

BERSES... the bit-error ratio for which the number of errored blocks within one second exceeds 30% (Table 1)

PSi-SES... probability rate for the worst month for exceeding BERSES on an unpro-tected hop due to distortions; calculated acc. to formula [63], applying the relevant signature data for BERSES

i... ordinal No for the individual hop

4.7.1.2 ...for a diversity-protected hop

The total fading probability for the individual, protected hop is:

[110] ( ) 3475.075.0SESsSiSESdFiSESMi PPP −−− +=

PMi-SES... probability rate for the worst month for exceeding BERSES on a diversity-

protected hop due to multipath propagation PdFi-SES... probability rate for the worst month for exceeding BERSES on a diversity-

protected hop due to multipath fading; calculated acc. to formulae [55] or [58], applying the relevant fading margin to BERSES

BERSES... the bit-error ratio for which the number of errored blocks within one second exceeds 30% (Table 1)

PSi-SES... probability rate for the worst month for exceeding BERSES on a diversity-protected hop due to distortions; calculated acc. to formula [63], applying the relevant signature data for BERSES

38 k&k engineering



4.7.1.3 Distribution between performance and unavailability

Performance part A part of the above excess probability - UM% - (formulae [109] and [110]) may last longer than 10 consecutive seconds and have to be treated as unavailability. The above figure for PMi-SES has, thus, to be reduced to:

[111] 100

100'

UMPP SESMiSESMi−⋅= −−

PMi-SES-1’... resulting probability rate for the worst month for exceeding BERSES on a

hop due to multipath propagation PMi-SES... probability rate for the worst month for exceeding BERSES

on a hop due to multipath propagation as per formulae [109] or [110]

UM... part in percentage of the probability rate, which has to be treated as unavail-ability

Unavailability part

[112] 1010100

GSESMiuMi

UMPP ∆−−− ⋅⋅=

[113] ( ) ( )ε+⋅+⋅−ξ±⋅−=∆ ∗ 1lg7.1lg7.22cos1.1lg6.55.10 7.0 dG

For ∆G > 10,8: use 10,8 PMi-u... average annual unavailability rate on a hop due to multipath propagation

PMi-SES and UM as above

ξ... latitude in degrees + 1 decimal ±... + for ξ≤45ο − for ξ >45o N or S of the Equator d*... hop length in km ε... path inclination in mrad (formula [49])

4.7.1.4 Resulting block-based severely errored seconds ratio (SESR)

[114] UPRwmiSESMiSESMi PPP −−− += '"

PMi-SES ”... final probability rate for the worst month for exceeding BERSES on a hop

due to multipath propagation

39k&k engineering


PMi-SES ’.. resulting probability rate for the worst month for exceeding BERSES on a

hop due to multipath propagation acc to formula [111] above PRwmi-UP... probability rate for the worst month for exceeding BERSES

on a hop due rain fading acc to formula [84]

4.7.2 Fading exceeding the background block error ratio (BBER) objective

This fading event is caused both by multipath propagation and rain

4.7.2.1 Prediction of BBER due to multipath propagation

The following prediction model is recommended:

[115] ( )18.2 12

1−⋅α⋅

α⋅= −− m

PP SESMiBBEMi

[116] SESMiRBERMi

SESPP

BERRBERm

−− −−

=lglg

lglg1

PMi-BBE... BBER probability rate for the worst month due to multipath propagation

PMi-SES... probability rate for the worst month for exceeding BERSES due to multipath propagation acc. to formula [109], [110] or [111]

RBER... residual bit-error ratio BERSES... the bit-error ratio for which the number of errored blocks within one second

exceeds 30% (Table 1) PMi-RBER... from the next formula:

[117] RBERSiRBERFiRBERMi PPP −−− +=

PFi-RBER... probability rate for the worst month for exceeding RBER on a hop due to fading; calculated acc. to formulae [55] or [58], applying the relevant fading margin to RBER

PSi-RBER... probability rate for the worst month for exceeding RBER on a hop due to distortions; calculated acc. to formula [63], applying the relevant signature data for RBER

α1... number of error/burst for the BER in the range between BER = 10-3 and BERSES; normally between 10 and 30

α2... number of error/burst for the BER in the range between BERSES; and RBER; normally between 1 and 10


40 k&k engineering


4.7.2.2 Prediction of BBER due to rain fading

Use again formula [115] to obtain PRwmi-BBER, but written as:

[118] ( )18.2 22

1−⋅α⋅

α⋅= −− m

PP SESRwmiBBERwmi

PRwmi-BBE...BBER probability rate for the worst month due to rain PRwmi-SES... probability rate for the worst month for exceeding BERSES due to rain, cal-

culated acc. to formula [78], applying the relevant fading margin to BERSES, and transferred to the worst-month value by formula [83]

RBER... residual bit-error ratio BERSES... the bit-error ratio for which the number of errored blocks within one second

exceeds 30% (Table 1) α1... number of error/burst for the BER in the range between BER = 10-3 and

BERSES; normally between 10and 30

α2... number of error/burst for the BER in the range between BERSES; and RBER; normally between 1and 20

whereby:

[119] 22

1 ≤αα

and:

[120] SESRwmiRBERRwmi

SESPP

BERRBERm

−− −−

=lglg

lglg2

PRwmi-RBER...probability rate for the worst month for exceeding RBER due to rain, calcu-lated acc. to formula [78], applying the relevant fading margin to BERRBER, and transferred to the worst-month value by formula [83]


4.7.2.3 Prediction of BBER due to equipment contribution

[121] RBERNP BBBEEi ⋅=−

PEi-BBE... BBER probability rate due to equipment contribution RBER... residual bit-error ratio NB... number of bits/block - see Table 1

41k&k engineering


4.7.3 Fading exceeding the errored second ratio (ESR) objective This fading event is caused both by multipath propagation and rain.

4.7.3.1 Prediction of ESR due to multipath propagation

[122] 1mSESMiESMi nPP ⋅= −−

PMi-ES... ESR probability rate for the worst month due to multipath fading n... number of block/s - see Table 1 m1... according to formula [116]

The other parameters have their previous significance

4.7.3.2 Prediction of ESR due to rain fading

[123] 2mSESRwmiESRwmi nPP ⋅= −−

PRwmi-ES... ESR probability for the worst month due to rain m2... according to formula [120]

The other parameters have their previous significance

4.7.3.3 Prediction of ESR due to equipment contribution

[124] RBERNnP BESEi ⋅⋅=−

PMi-ES... ESR probability rate due to equipment contribution NB... number of bits/block - see Table 1 n... number of block/s - see Table 1

4.7.4 Total performance for the circuit The total rate of time, Pc, during which the planning objectives are not met for the circuit is the sum of the cumulated rates, i.e.:

[125] ( )∑≤

=−−−− ++=

10

1

i

ixEixRwmixMixc PPPP

Pc-x... fading probability rate for exceeding SESR, ESR or BBER respectively for the radio circuit

PMi-x... fading probability rates for exceeding SESR, ESR or BBER respectively for the individual hop due to multipath propagation

42 k&k engineering


PRwmi-x... fading probability rates for exceeding ESR or BBER respectively for the individual hop due to rain fading

PEi-x... fading probability rates for exceeding ESR or BBER respectively for the individual hop due equipment contribution

Remember: Concerning the SESR, it should be observed, that:

PRwmi-SES = PEi-SES = 0

x... either SESR, ESR or BBER

The values for Pc-x should not exceed the planning objectives, i.e.:

[126] xplxc PP −− ≤

Pc-x... predicted total probability rate during which the planning objective is not met for a radio-relay circuit

Pp-xl... allowed probability rate for exceeding the planning objective for a radio-relay circuit, i.e. the planning objective

x... either SESR, ESR or BBER Formula [125] can be expressed in time:

[127] xcUNx Phs −⋅

⋅−⋅=12

360010628.2 6

sx... total time for exceeding the corresponding planning objective in sec / aver-age month

hUN... from formula [138]

43k&k engineering


5 Unavailability calculations for radio-relay systems 5.1 Unavailability and reliability of hardware 5.1.1 Single (unprotected) structures

[128] SrS

SrSuS M

MP

λ⋅+λ⋅

=1

PuS... unavailability rate for the single structure

λS... failure rate (failures per time unit) - the sum of the failure rates for the indi-vidual units, λ i, connected in tandem:

[129] ∑=

λ=λn

iiS

1

MrS... mean-time-to-repair (MTTR) for the single structure, in same time unit as the failure rate.

The mean-time-to-repair figures do no include any waiting time for spare parts. It is thus as-sumed that there is always access to spare parts when a fault occurs.

The failure rate, λ, can also be expressed in terms of mean-time-between-failure (MTBF):

[130] λ

= 1MTBF

MTBF... mean-time-between-failure for the single structure, in same time unit as the failure rate

44 k&k engineering


5.1.2 Duplicated (protected) structures 5.1.2.1 The duplicated structures are of the same type

The unavailability of a this type of duplicated structure, including the protection switching facilities, is calculated according to the following formula:

[131]

+

λ⋅+λ⋅λ⋅=

rND

rSNDrND

SrSSrSuD

MM

MMMP1

PuD... unavailability rate for the duplicated structure as per above figure

λS... failure rate (failures per time unit) for one of the duplicated equipment (= single structure as per above figure

λND... failure rate (failures per time unit) for the (non-duplicated) splitting and switching device proper - see also below

MrS... mean-time-to-repair for one of the duplicated equipment, in same time unit as the failure rate

MrND... mean-time-to-repair for the (non-duplicated) splitting and switching device proper, in same time unit as the failure rate

Formula [131] is only valid for systems using optional switching. This type of switching means that a failure in the switch element will not cause system failure unless switching is required. Consequently, the failure rate, λND, includes only the values for the splitting and switching elements themselves, together with those for the switch’s logic and control unit (L in the above figure), while the level and impedance interfacing elements of the splitting and switching units are a part of the failure rate of the duplicated equipment, λS, and of the sin-gle (i.e. non-protected) interface units (I), respectively, as they cause interruption of the traffic. The MTTR figures for these traffic-interrupting parts are the same as for the dupli-cated equipment, or = MrS.

For the complete path, including the non-protected interface units (I), equation [132] is ex-tended to:

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[132] 21 uIuIuDuP PPPP ++=

where PuI(1,2) are calculated according to equation [128], using the same value for MrS as in formula [131], and the failure rate, λI(1,2), instead of λS.

5.1.2.2 The duplicated structures are different equipment types

For this type of configuration, the following formula can apply for the calculation of the unavailability, PuD

[133] PPP NDuD +=

According to the figure below, these parameters have the following significance: P … unavailability rate for the two single structures in parallel according to for-

mula:

[134] 21 uSuS PPP ⋅=

PuS1,2… unavailability rate for single structure No 1 or 2 resp. acc. to formulas [128] and [129].

If one of the single structures itself consists of an 1+1-protected structure, its unavailability

PuD

M

M

L

T

T R

R

D

D

I1 I2

P PND PND

PuP

Structural type: single non-dupl. two parallel single non-duplicated single

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rate (PuS1 or PuS2 in formula [134]) has to be obtained from formula [131] or [132] resp.

PND is calculated applying formula [128], with the failures rates for the (non-protected) switching and splitting elements proper, and the corresponding mean-time-to-repair. (Con-cerning the distribution of the different hardware parts of the switching and splitting device to the non-duplicated and duplicated structures, reference should be made to the previous section 5.1.2.1.) For the complete hardware structure, including the non-protected interface units (I), equa-tion [133] is extended to:

[135] 21 uIuIuDuP PPPP ++=

where PuI1,2 is calculated according to equation [128], using the same value for MrS as in formula [133], and the failure rate, λI1,2, instead of λS.

5.2 Unavailability due to propagation disturbances The unavailability due to propagation disturbances, PRa, consists of contributions from rain and from multipath fading:

[136] uMiSESaiRai PRP −− +=

PRai... probability rate for a radio hop due to rain for the average annual year PRi-SES... average annual probability rate, during which the rain attenuation exceeds

the available fading margin acc to formula [82] PMi-u... average annual unavailability rate on a hop due to fading and distortions acc

to formula [112] i... ordinal No for the radio hop

5.3 Total unavailability The total unavailability of a radio circuit, URt, is the sum of the contributions from the hard-ware and the rain. It should be observed, however, that the unavailability of the hardware has to be considered for both the go and the return direction of transmission, i.e. twice, while that for rain is counted only once:

[137] ∑=

+⋅=n

iRaiut PPUR

12

URt... total unavailability rate of a radio circuit

Pu... unavailability rate of the total hardware, according to section 5.1 PRai... unavailability due to precipitations, see above n... number of radio hops included in the circuit

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Formula [137] expressed in time:

[138] 8760⋅= tUN URh

hUN... total unavailability in hours / average year

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6 Frequency planning 6.1 The number of disturbing signals reaching a receiver

[139] yxNn

−

= ∑

1

N... number of disturbing signals at each receiver n... number of hops in the area concerned x... number of parallel r.f. channels on the individual hop y... number of parallel r.f. channels on the own hop and the number of total interferences possible is

[140] nNK ⋅=

K... number of interference connections

6.2 General formula for the calculation of interfering sig-nal levels

[141] GARx,TxRx,BTxRx,WTxoTxIi GAAAAAATPCLI +−−−−−−=

LIi ... level of a single interfering signal in dBm LTx ... operating max output level of the disturbing transmitter in dBm ATPC … selected control range of the adaptive transmitter power control in dB Ao ... free-space attenuation in dB between disturbing transmitter and disturbed

receiver AWTx,Rx.. waveguide attenuation in dB in the transmitting alt. receiving station ABTx,Rx branching attenuation in dB in the transmitting alt. receiving station AWTx,Rx r.f. attenuators in dB in the transmitting alt. receiving station AA... additional attenuation in dB due to non-clearance of the interference path

and/or other attenuations in the interference path GG ... total antenna gain in dB(i) for angles ψ1 and ψ2, according to the following

formula [142]:

[142] ( ) ( )GRxGTxRxTxGRxGTxG AAGGGGG +−+=+=

Note: In case the transmitter and the receiver antenna operate at different polarization planes, the two possible H/V combinations have to be considered. In this case, the next equations apply in stead:

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[143] ( )

+++=

+= ⊥⊥⊥ −−⊥ 1010101010lg101010lg10

AARxTx

GGG AGGG

[144] ⊥⊥ += RxTx GGG

[145] RxTx GGG += ⊥⊥

[146] ⊥⊥ += RxTx AAA

[147] RxTx AAA += ⊥⊥

GTx... antenna gain for the main direction in dB(i) of the transmitting an-tenna, referred to an isotropic radiator

GRx... ditto for the receiving antenna GTx||... antenna gain in dB(i) of the transmitting antenna for angle ψ2 and parallel

polarisation, referred to an isotropic radiator GRx⊥ ... ditto for the receiving antenna for angle ψ1 and cross polarisation

GTx⊥ ... ditto for the transmitting antenna cross polarisation, GRx||... ditto for the receiving antenna and parallel polarisation

ATx|| ... antenna discrimination in dB of the transmitting antenna for angle ψ2 and parallel polarisation, referred to the antenna gain in the direction of trans-mission

ARx⊥ ... ditto for the receiving antenna for angle ψ1 and cross polarisation ATx⊥ ... ditto for the transmitting antenna cross polarisation

ARx||... ditto for the receiving antenna and parallel polarisation

6.3 Formulae for triangular network configuration

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6.3.1 Nodal station disturbs outstation (TxA1

_ RxC)

[148] WTxBTxTxTxAGCRxCIi AAAATPCLGALL ∆−∆−∆−∆−∆+∆+−= −−− 1

[149] 21 AA GGG −=∆

[150] 21 ATxATxTx LLL −− −=∆

[151] 21 ATxATxTx AAA −− −=∆

[152] 21 ABTxABTxBTx AAA −− −=∆

[153] 21 AWTxAWTxWTx AAA −− −=∆

[154] 21 AA ATPCATPCATPC −=∆

LIi ... level of a single interfering signal in dBm i ... ordinal number of the interfering signal LRx-C ... received level of the wanted carrier signal in dBm during fading-free time at

disturbed receiver C ∆G .. antenna discrimination or side- and back lobe attenuation for angle �in dB,

for antenna A1 in the nodal station, considering the polarisation for the dis-turbed and disturbing signal

GA1... antenna gain in dB for the disturbing transmitter A1 in the nodal station GA2 ... antenna gain in dB for the transmitter A2 in the nodal station LTx-A1... output level in dBm for the disturbing transmitter A1 LTx-A2... output level in dBm for the transmitter A2 ATx-A1... RF attenuator in dB in the disturbing transmitter A1 ATx-A2... RF attenuator in dB in the transmitter A2 ABTx-A1.. branching attenuator in dB in the disturbing transmitter A1 ABTx-A2.. branching attenuator in dB in the transmitter A2 AWTx-A1.. waveguide attenuation in dB in the disturbing transmitter A1

AWTx-A2.. waveguide attenuation in dB in the transmitter A2

ATPCA1… automatic transmitter power control range for transmitter A1 ATPCA2… automatic transmitter power control range for transmitter A2 If there is only one interference path to receiver C, equation [148] can be used to select the antenna A1 by writing:

[155] WTxBTxTxTxFiITrIG AAAATPCLGMLLA −−∆−∆−∆+∆++−=

Applying the definition of CIR, this equation can be expressed as:

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[156] WTxBTxTxTxFiG AAAATPCLGMCIRA −−∆−∆−∆+∆++=

This equation shows, that the higher the CIR, and the higher the flat fading margin, MFi, the higher the antenna discrimination necessary.

6.3.2 Outstation disturbs nodal point (TxC _RxA1)

[157] WRxBRxRxRxAGARxAIi AAALGALL ∆−∆−∆−∆+∆+−= −−− 111

[158] 12 ARxARxRx LLL −− −=∆

[159] 21 ARxARxRx AAA −− −=∆

[160] 21 ABRxABRxBRx AAA −− −=∆

[161] 21 AWRxAWRxWRx AAA −− −=∆

LRx-A1... receiver input level of the wanted signal in dBm during fading-free time at disturbed receiver A1

LRx-A2... receiver input level of the wanted signal in dBm during fading-free time at receiver A2

ARx-A1... RF attenuator in dB in the disturbed receiver A1 ARx-A2... RF attenuator in dB in the receiver A2 ABRx-A1.. branching attenuation in dB in the disturbed receiver A1 ABRx-A2.. branching attenuation in dB in the receiver A2 AWRx-A1.. waveguide attenuation in dB in the disturbed receiver A1 AWRx-A2.. waveguide attenuation in dB in the receiver A2


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Also equation [157] can be expressed with the antenna discrimination as the unknown pa-rameter:

[162] WRxBRxRxRxFiG AAALGMCIRA −−∆−∆+∆++=

6.4 Interference via passive repeater 6.4.1 Passive repeater as first-source transmitter

[163] SGAWRxWTxooTxIi GGAAAAAATPCLL ++−−−−−−= 221

For GG, equation [142] is valid.

Ao1... free-space attenuation in dB between PR and its associated transmitter, Tx Ao2... free-space attenuation in dB for the interference path between PR and the

disturbed receiver, Rx AA2... (eventual) additional attenuation in dB due to obstacle in the interference

path to Rx GS... passive repeater gain in dB for the interfering signal at angle ΘS

- for antenna back-to-back:

[164] 221 GSSS AGGG −+=

- for plane reflector:

[165] S

S bfYGΘ⋅

⋅⋅+=sin

lg205.22

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[166] If : RSRS GGYfGG =⇒

ψ⋅⋅⋅⋅=≥2

cos5.139lg20 2

GR.... gain in passive repeater, plane reflector type, in dB for the wanted signal, and angle ψ

GS1... passive repeater - antenna back-to-back type - gain in dB for the parabolic antenna towards Tx (main direction)

GS2... ditto for the parabolic antenna towards Rx'

AG2... antenna discrimination in dB for angle ΘS for the passive repeater antenna towards Rx'

f..... radio frequency in GHz Y..... physical area of the plane reflector in m2 b..... largest side dimension of the reflector in m ΘRx... angle in degrees between the wanted and the interfering signal paths for

receiver Rx ΘS.... angle in degrees between the reflected ray and the interfering signal path

ψ..... angle in space, in degrees, between the incident and reflected ray

The significance of the remaining parameters is according to formula [141]

Formula [165] is valid for:

2

90 ψ−<ΘS , if ΘS is outside the reflection angle ψ

2ψ≤ΘS , if ΘS is inside the reflection angle ψ.

If 2

90 ψ−≥ΘS , and outside ψ:

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[167]

ψ⋅⋅⋅+=2

coslg1025.21 2fYGS

6.4.2 Passive repeater as receiver of interfering signals

The level of the interfering signal, LIi is calculated according to the same formula [163] as above, but with changed Tx-Rx co-ordination. Formulae [164] to [167] are also valid here, including the conditions for their application, as well as their limitations.

6.5 Total interference

If more than one interfering signal has to be considered at a receiver’s input, the various contributions are added together on a power law basis:

[168] ( )∑=

−∆−⋅=n

i

ACIRLI

jiIilgL1

101010

LI. combined level in dBm of all interfering signals LIi... level in dBm of an individual interfering signal Aj... adjacent-channel attenuation in dB of the interfering signal in the receiver,

see above. For co-channel interference (∆f = 0): Aj = 0.

∆CIR… power density compensation in dB: The influence of interfering signals on the wanted signal depends on the power distribution within the spectrum of the interfering signal. When summing up the various interferer signal levels as per formula [168] we have to compensate for the different power densities by in-

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troducing the parameter ∆CIR. This parameter has to be calculated for each interfering signal; it is the difference between the highest CIR and the CIR of the individual interferer:

[169] irefi CIRCIRCIR −=∆

CIRref… the CIR figure in dB for an interfering signal from a transmitter of the same system type as the disturbed receiver

CIRi… the CIR figure of the individual interferer in dB

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7 Bibliography {1} Heinz Karl, Performance and availability as applied to digital radio-relay

systems, K&K Engineering {2} Heinz Karl, Planning and engineering of radio-relay networks, K&K Engi-

neering

@ Heinz Karl, 2005

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Annex 1. The co-ordinates for the hop’s midpoint are calculated as

follows: - for the longitude:

[170] 2

21 xxxo+

=

- for the latitude:

[171] 2

21 yyyo+

=

xo... longitude for the midpoint in degrees x1... longitude for site A in degrees negative figures for

x2... longitude for site B in degrees ⌡ W of Greenwich

yo... latitude for the midpoint in degrees y1... latitude for site A in degrees negative figures for y2... latitude for site B in degrees ⌡ S of the equator

2. The co-ordinates for the corners of the 110x110 km area around a hop’s midpoint:

[172] ( )oC yyay sin999925.0cos0122165.0cossin 11 ⋅+⋅⋅Θ=

[173] Co

CooC yy

yyaxx

coscossinsin999925.0

cos⋅

⋅−+=

xo and yo are the co-ordinates in degree of the hop’s midpoint as calculated above.

∗ NE corner: Θ1 = 45o cos Θ1 = 0.707107 acos = +

∗ SE corner: Θ1 = 135o cos Θ1 = - 0.707107 acos = +

∗ SW corner Θ1 = 225o cos Θ1 = - 0.707107 acos = -

∗ NW corner Θ1 = 315o cos Θ1 = 0.707107 acos = -

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3. Bilinear interpolation The co-ordinates of a hop’s midpoint, xo and yo, have been calculated according to section 1 above. This midpoint is located between 4 grid points of a digital map, points I, II, III and IV, eg those with a mutual distance of 1.5o - see the below figure.

x and y are the longitudes and latitudes in degree, z11-22 are the co-ordinated third parame-ters, eg the refractivity gradient, dN1 - section 4.2.3 - or the rainfall intensity, J0.01 - section 4.5.1. The unknown parameter, zo, is obtained by the following calculation:

[174]

( )( ) ( )( )

( )11211222

1112112111

zzzzyyyy

xxxx

yyzzyy

xxzzxx

zz

ab

ao

ab

ao

ab

ao

ab

aoo

+−−−−

−−

+

+−

−−+

−−−

+=

1.5 o

1.5 o

GP III xb yb z22

xo yo zo

GP II xa yb z12

GP I xa ya z11 GP IV xb yb z21

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Appendix I Water vapour density - a World atlas The data below are an extract from ITU-R Rec. P.836 and show the water vapour density in g/m3 for two months of the year for the various regions of the world. These charts should be used the following way:

∗ for the small time-percentage calculation, use the higher of the two values for the location concerned,

∗ for the fading-free time calculation, use the lower of the two values.

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Appendix II Rain intensity data - a World atlas The data below are an extract from ITU-R Rec.P.837-3 and refer to the annual average clock-minute rainfall rates in mm/h for 0.01% of the time. The figures in the below chart have been derived from the data and equations shown in chapter 4.5.3. For the charts of other regions of the World, reference should be made to bibliography {1} or to ITU-R Rec.P.837-3.

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Appendix III Rain attenuation coefficient, parameters ΤΤΤΤ and αααα

Regression coefficients for estimating specific attenuations in equation [71]

Frequency

(GHz) ΤH αH ΤV αV

7 8

10 12 15 20 25 30 35 40 35 50

0.00301 0.00454 0.0101 0.0188 0.0367 0.0751 0.124 0.187 0.263 0.350 0.442 0.536

1.332 1.327 1.276 1.217 1.154 1.099 1.061 1.021 0.979 0.939 0.903 0.873

0.00265 0.00395 0.00887 0.0168 0.0335 0.0691 0.113 0.167 0.233 0.310 0.393 0.479

0.00265 0.00395 0.00887 0.0168 0.0335 0.0691 0.113 0.167 0.233 0.310 0.393 0.479

Raindrop size distribution according to Laws and Parsons, [1943] Terminal velocity of raindrops according to Gunn and Kinzer, [1949] Index of refraction of water at 20°C, see Ray, [1972] Values of ΤH, ΤV, αH and αV for spheroidal drops [Fedi, 1979; Maggiori, 1981] based on regression for the range 1 to 150 mm/h.

Formel KKE5201 5 Rev N A5 d

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