ASYMPTOTIC METHOD FOR PREDICTING ERROR PERFORMANCE PARAMETERS AND OBJECTIVES FOR CONSTANT BIT RATE DIGITAL PATHS CARRIED BY DIGITAL RADIO-RELAY SYSTEMS

"JOURNAL OF RADIO ELECTRONICS" N 2, 2000

ASYMPTOTIC METHOD FOR PREDICTING ERROR PERFORMANCE PARAMETERS AND OBJECTIVES FOR CONSTANT BIT RATE DIGITAL PATHS CARRIED BY DIGITAL RADIO-RELAY SYSTEMS

Vladimir Minkin, Alexander Voschinin

Radio Research & Telecommunication Institute (NIIR)

Received February 22, 2000

At present there are two concepts to evaluate the error performance parameters and objectives for digital radio-relay systems forming a part of SDH-network. The parameters and objectives for systems below the primary rate are defined in ITU-T Recommendation G.821 [1] while those for systems at or above the primary rate are specified in ITU-T Recommendation G.826 [2]. For the time being the design of radio-relay links at or above the primary rate was based on ITU-T Recommendation G.821 approach. For both design and maintenance of the digital links it is important to determine converting statistics and establish relations between error performance objectives according ITU-T Recommendations G.821 and G.826. In the present paper this problem is solved with asymptotic approach based on normal approximation to the binomial distribution. Under this, for time-invariant number of errors per burst and bit error ratio, BER(t), distribution, multipliers for converting parameters in accordance with Recommendation G.821 into parameters in accordance with Recommendation G.826 are found. For the complex case of time-variant characteristics the asymptotic method for predicting error performance parameters is developed. Proposed evaluating algorithm is numerically verified with estimation of block error performance parameters on a design of radio-relay hop. The BER(t) distribution mask that could allow meeting error performance objectives for a hop has been defined using different propagation and equipment parameters.

INTRODUCTION

Error performance objectives for a constant bit rate digital path at or above the primary rate carried by digital radio-relay systems which may form part of the international and national portions of a 27 500 km hypothetical reference path are defined by ITU-R Recommendations F.1092 [3] and F.1189 [4] based on ITU-T Recommendations G.826.

To simplify in-service measurements the Recommendation G.826 is based upon a block-based measurement concept using error detection codes inherent to the path under test and considers the following block-based events:

Errored Block (EB): a block in which one or more bits are in error;

Errored Second (ES): A one second period with one or more errored blocks or at least one defect;

Severely Errored Second (SES): a one-second period, which contains ³ than 30% errored blocks or at least one defect. SES is a subset of ES. (It should be mentioned that the same definition could be used not only for a path but also for a regeneration section, for example, for one hop, while for a multiplex section the value 30% should be change provisionally with 15%);

Background Block Error (BBE): an errored block not occurring as a part of SES.

Error performance parameters related with above events are defined as:

Errored Second Ratio (ESR): the ratio of ES to total seconds in available time during a fixed measurement interval;

Severely Errored Seconds Ratio (SESR): the ratio of SES to total seconds in available time during a fixed measurement interval;

Background Block Error Ratio (BBER) is the ratio of errored blocks to total blocks during a fixed measurement interval, excluding all blocks during SES and unavailable time

A path fails to satisfy this Recommendation if any parameter exceeds the allocated objective in either direction at the end of the given evaluation period. The suggested evaluation period is 1 month.

At the same time the radio-relay equipment specifications define receiver level threshold based on Bit Error Ratio (BER) criterion (BER=10^-3). This approach was widely used for a path design in ITU-R Recommendation P.530-7 [6] because old ITU-R Recommendation F.594 [7], F.634 [8], F.696 [9] and F.697 [10] were based on ITU-T Recommendation G.821 where BER criteria were used for error event definitions and SES event was declared at BER=10^-3. Also for the time being there have been published only a few experimental measurement results based upon errored blocks and there are a lot of publications containing experimental results related with BER. As it is mentioned in the ITU-R Recommendation P.530-7 if requirements in Recommendation G.826 have to be met there is a need for prediction methods that are based on estimating block errors rather than bit errors could allow to calculate the ESR, BBER and SESR objectives.

For this it is important to have a method and procedure to convert BER statistics into block-based event statistics. A number of such methods have been recently proposed [11], [12] and [13]. Basing on different sets of assumptions and input data above methods provide numerical evaluation of error performance parameters. Being tested with real data these methods proved to be quit convenient. However, to investigate the basic properties of different error performance parameters it is important to establish analytical relations between different criteria.

In this paper based on a normal approximation to the binomial distribution the asymptotic approach to this problem is developed. The received equations give simple relationships between block based error performance parameters and could be useful for the amendment of ITU-R Recommendation P.530. The BER (t) distribution mask has been developed to estimate possibility for a hop with known propagation and equipment parameters to meet the error performance objectives.

The paper is organized as follows. In section 2 asymptotic formulas for probability of block-based events as a function of BER are derived. Time dependence of BER and probability of basic events is investigated in section 3. Asymptotic formulas for ESR, SESR and BBER are obtained in section 4. Under assumption that mean error burst length and BER distribution are time invariant a simple converting multipliers are derived in section 5. For more complicated case of time variant characteristics the prediction method for error performance parameters is developed in section 6 where the BER(t) distribution mask has been defined. Concluding remarks are presented in section 7.

2. ASYMPTOTIC PROBABILITY OF BASIC EVENTS

Relationship between error performance parameters and BER can be evaluated employing either Poisson distribution or Neyman-A contagious distributions of bit errors within a burst. It is shown in [11] that either approach leads practically to identical results. Since Poisson distribution is more pragmatic and provides useful results with minimal complexity henceforth Poisson distribution is applied.

2.1. Probability of an errored block P_EB

Assuming a Poisson distribution of errored bursts probability of an errored block P_EB can be expressed in the form

P_EB=1-exp(-N_B BER/a )=1-exp(-BER/t ),

(1)

where BER is a bit error ratio, N_B is a number of bits per block, a is an average number of errors per burst (a ³ 1), constant t = a /N_B is used to evaluate a change rate of the exponent. Basing on the fact that exponent runs from 0 up to 0.95 when its argument changes from 0 up to 3t we can state that

under changing of BER from 0 up to 3a /N_B probability of an errored block P_EB runs from 0 up to 0.95.

This statement, which explicitly connects working range of P_EB with block size and number of errors per burst is useful when numerical integration should be performed.

Due to above given definition of SES this event occurs when number of errored blocks per second is greater than 30%, i.e. when P_EB³0.3. It allows to write an expression for equivalent BER threshold B_SES for SES to occur as

B_SES=a [-(ln0.7)/N_B] =t | ln0.7|

(2)

To compare different distribution models numerical values of BER threshold normalized to the mean burst a =1 are given in Table 1 for different block sizes under Poisson, binomial and uniformed distribution.

Table 1. Threshold BER for a SES to occur for different block size and distribution models

Distribution models	N_B=18 792 bit	N_B=6120 bit	N_B=2048 bit
Poisson	1.83 10^-5	5.50 10^-5	1.74 10^-4
Binomial	1.88 10^-5	5.64 10^-5	1.78 10^-4
Uniform	1.54 10^-5	4.63 10^-5	1.46 10^-4

From the Table 1 it can be seen that the difference between BER threshold values under Poisson and binomial distribution is insignificant. As to the uniform distribution it yields more stringent requirements for system. However, the data in the Table 1 shows relatively low sensitiveness of threshold to the type of distribution.

2.2. Probability of an errored second P_ES

With probability of errored blocks given by equation (1) it is possible to define a probability P_ES of an errored second as

P_ES=1-exp(-nP_EB)

(3)

where n is a number of blocks per second. Under this, based on the “3t rule” we can conclude that

P_ES=0.95 if P_EB=3/n ,

(4)

i.e. the probability of an errored second P_ES runs interval 0£ P_ES£0.95 where probability of errored blocks P_EB has order 1/n. Under this a linear approximation to equation (1) is valid and hence we can write expression

P_{ES @} 1-exp(-n N_B BER/a ) = 1-exp(-nBER/t ) ,

(5)

which is practically accurate (error of approximation is less than 1/n²). Comparing expression (1) and (5) we can conclude that for any fixed value of BER and other things equal the following relation is valid

P_EB(BER)=P_ES(BER/n)

(6)

With equation (6) it is easy to set the correspondence between different values of BER which yields the same probability of errored blocks and errored second P_EB=P_ES. For example, for path VC-4 with N_B=18720, n=8000 assigning P_EB=P_ES=0.3 from Table 1 for Poisson distribution we have P_EB(BER=1.83 10^-5)=0.3. Dividing argument by n=8000 in accordance with equation (6) we get

P_EB(1.83 10^-5)=P_ES(2.3 10^-9)=0.3

(7)

2.3 Probability of severely errored seconds P_SES

To evaluate probability of errored seconds it is necessary first to define the probability P_nk of k errored blocks in total of n blocks. It can be found by use of binomial distribution as

P_nk=[n!/(n-k)!][(1-P_EB^n-k)(P_EB)^k]

(8)

It is known [14] that normal distribution with mean m and variance s² defined as

m =n P_EB, s ²=n P_EB(1-P_EB)

(9)

provides a very accurate approximation to the binomial distribution (8) when n is large. It allows to write the limiting form of equation (8) when n tends to infinity as

(10)

Here random normal variable x in expression (11) has a sense of number of errored blocks per second. Considering all cases when ratio k/n is greater than or equal to 0.3 the probability P_SES can be calculated as a sum

P_SES=,

(11)

which with equation (10) can be replaced by integral, i.e. expression for P_SES can be written as

P_SES=

(12)

As before, f(x) is the probability density function of normal random variable x with mean m and variance s ².

The properties of integral (12) have been investigated in details in Appendix A. Main conclusions can be summarized in the set of following statements:

Probability P_SES can be expressed in terms of normal distribution as

P_SES=1-F(u₁), u₁= (0.3n-m )/s = ,

(13)

where F(u) is cumulative standard normal distribution of random variable u with zero mean and variance equal to 1, u₁ is normalized lower limit of integral (12).

For any path type is valid the relation P_EB(B_SES)=0.3, P_SES(B_SES)=0.5 , where B_SES denotes the BER threshold defined by equation (2).

Probability P_SES runs within interval [0.005, 0.995] if probability of errored blocks P_EB changes as 0.3± 3. Corresponding interval of BER is defined as

0.005£ P_SES£0.995 under BER= B_SES ± 3(a /N_B)

(14)

Expressions (13) and (14) reveal that P_SES curve that is exponential in shape changes from 0 to 1 in very narrow BER interval. For example, for VC-4 path with N_B=18720 bits, n=8000 blocks/s, P_SES changes from 0.005 to 0.995 when BER=1.83 10^-5±7.7 10^-7. For path with N_B=2048 bits and n=1000 blocks/s respective interval is defined as BER=1.74 10^-4±2.1 10^-5

Thus, expression (14) analytically verifies usually accepted assumption that function P_SES is nearly step function. The P_EB,P_ES and P_SES curves vs. BER are shown for VC-4 path in the Figure 1.

3. BER AND PROBABILITY OF BASIC EVENTS AS FUNCTIONS OF TIME

3.1. Cumulative BER distribution

In real radio-relay links BER depends on propagation conditions and can be represented by a stochastic process. Cumulative distribution t (BER) of this process evaluates worst month time (taken as relative time t, which is equal to 1 for the whole observation interval) when BER exceeds the given BER value.

A precise relationship between time t and BER can be obtained only from direct measurements on considered radio link. Another way is to derive the distribution t (BER) from distribution t (V) of fade margin V that could be known or proposed for given path.

It is shown in [15] that for low fraction of the distribution of fade margin V

(for ½V½> 25 dB) can be described as

(15)

where t is relative worst month time when input fade margin V exceeds the given value,
k_p- coefficient adjusted to considered path,
l - coefficient which depends on frequency range and diversity mode.

From real link measurements it could be consider that, for example,
l = 5 for multipath fading and space diversity,
l = 10 for multipath fading and single reception as well as for fading due to rain at some paths in frequency ranges about 11 GHz,
l = 20-25 for fading due to rain at some paths in frequency range 18 GHz and higher.

Let also assume that the relationship between BER and fade margin V_[dB] could be approximated by equation

Log(BER)=j - b |V|,

(16)

where j is a constant adjusted to the link, coefficient b reflects the rate of change of the function BER(V).

In Figure 2 some typical curves described by equitation (16) are presented. For a generality on an abscissa axis we shall consider relative meanings ½V- V₀½ , where V₀ - fade margin,dB, appropriate to a receiver input level at B₀=10^-3.

Figure 2.

For all curves j = -3 and the curve slope b is defined as

c) only fir systems with FEC with a large code gain.

For prediction of error performance parameters it is necessary to derive distribution t(BER) from expressions (15) and (16). To solve this problem let’s suppose that there is given a reference point (t₀, V₀, B₀) on the distribution curves (15) and (16). Under this assumption after several manipulations it is possible to avoid unknown coefficients k and j from expressions (15) and (16) and to combine them into the unique distribution t(BER).

Indeed, substituting the pair of coordinates (t₀, V₀) into equation (15) we can write for unknown coefficient k

With this expression we can represent distribution (15) in the form

(17)

which does not include coefficient k.

Using the pair (V₀, B₀) after similar manipulation equation (16) can be written as

log(BER) - log(B₀) = - b {| V_0|-| V| }.

(18)

Comparing equation (17) and (18) we can see that both of them include the term {| V_0|-| V| }. Its explicit expression can be easily found from equation (18). Substituting it into (17) and taking inverse function we can write resulting distribution BER(t) as

BER (t)=B₀(t₀/t)^m,

(19)

where the index m=b l is supposed to be a known positive number greater than one.

As it follows from the above given expression the BER value falls m decades when relative time changes for one decade, i.e. in loglog scale relation (19) is represented by a straight line with the slope m.

As a rule distribution (19) could not be valid for whole range of relative time [t₀, 1]. For this reason to describe actual relation BER(t) more precise, the whole range of relative time [t₀, 1] can be divided into several line segments with different corresponded slope m. Resulting relation BER(t) in this case is represented by piece- wise function (Application of 3-points distribution will be given in section 6).

3.2 Probability of basic events as a function of time

If relation BER (t) is known, the probability of block events can be represented as a function of time percentage t. For example, if BER distribution BER (t) is given by equation (19) then by substitution in it BER in expressions (1), (3) and (13) we get respective probability as a function of time.

P_EB(t)=1-exp[-N_B (B₀/a )(t₀/t)^m]	(20)
P_ES(t)=1-exp[-n N_B (B₀/a )(t₀/t)^m]	(21)
P_SES(t)=1-F[u₁(t)],	(22)

where u₁(t) =

In Fig. 3 the curves P_EB(t), P_ES(t) and P_SES(t) are shown for VC-4 path with B₀=10^-3, t₀=0.0001, a =1 and m=10.

Figure 3

Three specific points are shown in Fig.3, namely point t₀, corresponded to the maximum BER value equal to B₀=10^-3, point t_SES corresponded to the threshold B_SES where P_EB=0.3 and point t_ES where P_ES=0.3.

4. DETERMINATION OF ERROR PERFORMANCE PARAMETERS

Under explicitly given time dependent probabilities P_EB(t), P_ES(t) and P_SES(t) evaluation of error performance parameters can be done by using integration of respective functions over relative time interval [t₀, 1]. Indeed, for ESR and SESR parameters we can write

ESR =P_ES(t)dt,	(23)
SESR =P_SES(t)dt,	(24)

where the lower limits of integrals t₀ is the fraction of time when BER exceeds its highest value equal to B₀=10^-3, and the upper limit, which is equal to one, corresponds to the total relative observation time (any month).

Recommendation ITU-T G.826 define BBER as the ratio of errored blocks to total blocks during available seconds, excluding all blocks during SES. In Appendix B with asymptotic approach it is shown that BBER can be evaluated as integral

BBER= P_EB(t)dt,

(25)

where the lower integration limit is taken as time percentage t₁ corresponded to the SES threshold B_SES where P_EB=0.3. For t>t_SES there are no SES. The multiplier 1/(1-SESR) in expression (25) reflects the fact that all blocks during SES are excluded from consideration. Equation (25) is useful for two reasons. First, it yields the implicit, easily interpreted relationship between BBER and probability of errored blocks P_EB and the second, it essentially simplifies numerical integration.

Geometrically, integrals (23), (24) and (25) correspond to the area bounded by the respective curves P_ES(t), P_SES(t) and P_EB(t) shown in Fig. 3.

Taking into account that P_SES is nearly step function a simple relation can be applied to evaluate SESR, namely

SESR=t_{SES ,}

(26)

which is valid with high accuracy. SESR value due to equation (26) equal to area of rectangle with base t_SES and height equal to one. It can be seen from Fig. 2 that it is very close to the area bounded by curve P_SES(t), i.e. to the exact SESR value. The value of t_SES related with SES threshold can be found for given m with equation (19) by substitution BER=B_SES as

t_SES=t₀(B₀/B_SES)^1/m

(27)

Concerning BBER parameter it can be observed that for P_EB<0.3 linear approximation to probability P_EB given by expression (1) is valid , i.e. we can write

P_EB (t)=N_B BER(t)/a

(28)

Assuming distribution BER(t) in the form of (19) and using approximated relation (28) integral (25) can be evaluated analytically as

BBER=

[N_B (B_SES/a )(t_SES/t)^m]dt =

[N_B (B_SES/a)

(t_SES)^m][(t_SES)^-(m-1) -1] @N_B (B_SES/a )t_{SES @}

(29)

Final form of equation (29) is obtained with equation (2) taking into account that the value of integral under its upper limit is negligible small in comparison with its value under the lower limit. (That is correct if t_SES and the impact from BER “tail” (when, e.g., BER < 10^-12) are negligible small and m and a are constant over the interval [t_SES, 1]).

Figure 4

In Fig. 4 the exact curve P_EB(t) and its linear approximation (28) are shown for the same data as for Fig. 3.

The shadowed area in Fig. 4 corresponds to the exact value of BBER. Linear approximation of P_EB(t) represented by upper curve adds the narrow band to this area. Resulting overestimation of BBER for either path type does not exceed 7%.

It should be mentioned that using distribution BER(t) in the form of (19) we practically have considered only part of the BER(t) distribution without BER “tail”.

4. DETERMINATION OF ERROR PERFORMANCE PARAMETERS

ESR =P_ES(t)dt,	(23)
SESR =P_SES(t)dt,	(24)

BBER= P_EB(t)dt,

(25)

Geometrically, integrals (23), (24) and (25) correspond to the area bounded by the respective curves P_ES(t), P_SES(t) and P_EB(t) shown in Fig. 3.

Taking into account that P_SES is nearly step function a simple relation can be applied to evaluate SESR, namely

SESR=t_{SES ,}

(26)

t_SES=t₀(B₀/B_SES)^1/m

(27)

Concerning BBER parameter it can be observed that for P_EB<0.3 linear approximation to probability P_EB given by expression (1) is valid , i.e. we can write

P_EB (t)=N_B BER(t)/a

(28)

Assuming distribution BER(t) in the form of (19) and using approximated relation (28) integral (25) can be evaluated analytically as

BBER=

[N_B (B_SES/a )(t_SES/t)^m]dt =

[N_B (B_SES/a)

(t_SES)^m][(t_SES)^-(m-1) -1] @N_B (B_SES/a )t_{SES @}

(29)

Figure 4

In Fig. 4 the exact curve P_EB(t) and its linear approximation (28) are shown for the same data as for Fig. 3.

It should be mentioned that using distribution BER(t) in the form of (19) we practically have considered only part of the BER(t) distribution without BER “tail”.

5. SIMPLE CONVERTING METHOD FOR CONSTANT m and a

Above given asymptotic expressions (23), (26) and (29) provide theoretical base to develop asymptotic prediction method of error performance parameters due to ITU-T Recommendation G.826 [1]. To improve the accuracy of method any available information related with given link should be included into consideration. For example, as much as possible precise presentation of BER(t) in the form of piece wise function is of great importance. It is also necessary to take into account time dependence of average number of errors per burst a . Due to the simple structure of integrals (23), (24) and (26) this information can be involved in calculation without any difficulties.

The problem of converting of bit error-based measurements into ESR, SESR and BBER parameters can be reduced to the determination of converting multipliers k, k₁(k_ES) and k₂(k_BBE) such that

SESR@ kt₀=t_SES, ESR@ k₁t₀=k_ES t_SES, BBER@ k₂t₀=k_BBE t_SES

(30)

Let’s find k, k_1, k₂, k_ES and k_BBE under the following assumptions:

Time dependence of BER is described by the relation (19) with index m constant over time, maximum value of BER is taken as B₀=10^-3 and related time percentage t₀ is known,
The average number of errors per burst a is constant over period of observation,
The unavailability and SES threshold B_SES is defined with equation (2), corresponded time percentage t_SES is defined with equation (27).

Note: It should be mentioned again that for ESR and BBER relationships (30) are valid under an assumption that the impact of BER “tail” is negligible small.

Below in this section simple converting method based on multipliers k, k₁ and k₂ is considered.

5.1 SESR evaluation.

Using expression (26) for SESR and expression (2) and (27) for t₁ we can readily get the converting relation

SESR=t_SES= k t₀

(31)

_{with converting multiplier k defined as

k=[N_BB₀/(-a
ln0.7)]^1/m

(32)

The values of k calculated for selected block size N_B,
index m and mean burst a are presented in the
Table 2

(B₀= 10^-3).
Table 2. Converting
multiplier “k” for SESR

N_B=18 792 bit

N_B=6120 bit

N_B=2048 bit

m

5

10

20

5

10

20

5

10

20

a = 1

2.2

1.5

1.2

1.8

1.3

1.15

1.4

1.2

1.09

a = 2

1.9

1.4

1.18

1.5

1.24

1.1

1.1

1.1

1.05

a = 5

1.6

1.3

1.12

1.3

1.13

1.0

1.0

1.0

1.0

a = 10

1.4

1.2

1.09

1.1

1.05

1.0

0.9

0.9

0.97

The SESR estimation given by equation (31)
is nearly exact and for this it can be applied even for an accurate evaluation
of SESR parameter.
The equations (31) and (32)
allow to solve the reverse problem - to estimate what should be the t₀
value if a path meets the SESR₀ value that denotes the
required one for SESR objective
t_{0 ³}SESR₀
/k = SESR₀=
SESR₀ .
The t₀ value may be used for path design
purposes using the known received threshold at B₀= 10^-3for
V_mincalculations in ITU-R Recommendations P.530[2]
and F.1093[3].

5.2 ESR evaluation.

The exact value of ESR parameter can be found with equation (23)
which even under above assumptions is not linear with respect to time t₀.
For this reason to find multiplier k₁ the following rough
estimation of integral (23) is applied

ESR=t_{ES ,}

(33)

where t_ES is time percentage when
probability of error seconds P_ES(t) exceeds the value 0.3.
Replacement of integral (23) by the equation (33)
is equivalent to the approximation of exponent P_ES(t) in figure
3 by the step function with the jump at the point t_ES.
Under this, instead of area bounded by the curve P_ES(t) that is exact
value of ESR, we have an area of rectangle with the base t_ES. This
yields to slight overestimation of ESR value.
To calculate the value t_ES the relationship (6)
can be applied. From equation (6) it follows that P_ES=0.3
if BER value is equal to B₂=B_SES/n. Substituting it in
distribution (19) we have

t_ES= t₀(B₀/B₂)^1/m= t₀(nB₀/B_SES)^1/m= t_SESn^1/m

(34)

Simple relation between t_ES and t_SES
together with equation (31) yields the following converting
formula for ESR

ESR = t_SESn^1/m = k_ES
t_SES= k₁ t₀, k_ES= n^1/m, k₁ = k
k_ES

(35)

where k is given by expression (31).
The values of multiplier k_ES for selected combinations of n and
m are presented in the Table 3.
Table 3.
Converting multiplier k_ES for ESR

n (blocks/s)

1000

8000

m= 5

4

6

m=10

2

2.5

m=20

1.4

1.6

The k₁ values can be easily found by multiplying
columns of Table 2 by appropriate factor k_ES from Table 3 or
directly from expression (32) and (35).
Though ESR estimation due to (33) have been referred as a
rough one, practically in worst-case error does not exceed 20%.
If we have defined the m and a
values correctly the receiving result should be less than measured ESR value and
the difference between estimated and measured ESR values could be used for real
residual BER “tail” (RBER) impact estimation.

5.3 BBER evaluation.

For determination of multipliers k₂ and k_BBE
in expression (30) the approximation (29)
for BBER can be used. Under this, using expression (2) for B_SES
and (31) for t_SESwe can represent BBER as

BBER = N_B (B_SES/a
)t_SES= t_SES=k_BBE t_SES = k_BBE
SESR=k₂t₀,

(36)

where multiplier k_BBE is defined as

,
k₂= k k_BBE

(37)

The values of k_BBEare given in the Table
4.
Table 4.
Converting multiplier k_BBE for BBER

m

5

10

20

k_BBE

0.09

0.04

0.019

The k₂ values can be easily found by multiplying
columns of Table 2 by appropriate factor k_BBE from Table 4. As
it was mentioned before, the evaluation of BBER with use of converting formula
(36) for worst case does not exceed 7% (without taking into account the RBER
“tail” impact).

5.4 Relations between error performance parameters.

Simple manipulations with equations (31),
(35) and (37) reveal the following
relations:

(38)

,

(39)

,

(40)

which are invariant to the time percentage t₀,
block size N_B and average number of errors per burst a
.
For example, the ESR/SESR and BBER/SESR relations between
error performance parameters under index m=5 and m=10 are
calculated below:
Table 5.
Relationship between error performance parameters

m

5

10

n

1000

8000

1000

8000

ESR/ SESR

4

6

20

2.5

BBER/ SESR

0.09

0.09

0.04

0.04

ESR/ BBER

44

67

50

62

Let’s compare equations (38) and (39)
using the relationship between objectives SESR₀, ESR₀and
BBER₀taken from Table 1/ G.826 in the ITU-T Recommendation
G.826.
Table 6. Relations
between error performance parameters according ITU-T Recommendation G.826

Rate (kbit/s)

2048

48 960

(VC-3)

150 336

(VC-4)

Block size

2048 bits

6 120 bits

18 792 bits

Blocks/second

1 000

8 000

8 000

ESR₀/ SESR₀

20

37.5

80

BBER₀/ SESR₀

0.1

0.1

0.1

It could be seen from Table 6 that the most critical is the
relationship BBER/ SESR.
Suppose that BER(t) distribution is described by expression (19)
with a constant index m and a path meets SESR objective. Under this
assumption using the relation (39) and BBER₀/
SESR₀ ratio from Table 6, we can state that the path will meet
the BBER₀objective if the following inequalities are
satisfied.

Experimental date shows that the index could vary in the
range m = 5…20. It means that the relationship BBER/ SESR
could not be less than 0.019 for radio-relay systems without space diversity and
not less than 0.04 for radio-relay systems with space diversity (the BER
“tail” impact is here ignored).
6. PREDICTION METHOD FOR TIME-VARIANT m and a
The simple converting method developed in section 5 is based
on assumptions that index m and mean burst a are
constant. However, in real radio-relay links both index m of BER
distribution and average number of errors per burst a
are time variant. Under this it should be developed the prediction method with
greater level of complexity to take into consideration detail information
related with mean burst a and distribution BER(t). It
is also important to evaluate the contribution of the ‘tail’ of BER(t)
distribution in error performance parameters. Usually, a ‘tail’ is referred
to that part of BER(t) where BER(t) is constant. Usually mainly an equipment
residual BER and some type of interference but also a number of unpredictable
reasons define this part.
To take into consideration all these effects to estimate errors performance
parameters of given path with known block size N_Band number
of blocks per second n the following assumptions are made

BER (t) distribution is a 3-point piece-wise function with characteristic
points at t₀, t₁ and t_S,where

t₀   is
percentage-of time corresponded to the highest BER value B₀,

t_SEScorresponds to the SES threshold B_SES
and

t_Sis a
‘tail’ starting point related with B_S=10^-10…10^-13
.
Each part of distribution is described by equation (19)
with slope m_i and average number errors per burst a
_i, i.e.

under t_0£t£
t_SES

,

(41)

under t_SES£t£
t_S

,

under t_S£t

6.1 Evaluating algorithm.

Let’s evaluate error performance objectives for an one hop path under given
relation (41), known indexes m₁, m₂, m₃
and average number of error per burst a ₁,
a ₂, a _{3 .}It is also necessary to define the coordinates of characteristic point. It
is mentioned in the current ITU-R Recommendation P.530 that the prediction
methods are all based on a definition of outage as BER above a given value, i.e.,
10^-3, for meeting requirements set out in Recommendation G.821[4].
If requirements in Recommendation G.826 have to be met there is a need for
prediction methods that are based on estimating block errors rather than bit
errors. In order to meet the requirements in G.826 the link should be designed
to meet another BER than 10^-3.
It means that using the above mentioned prediction methods we could receive
for the path under consideration for B₀=10^-3 only
the coordinate of characteristic point t₀ and the coordinates
of another characteristic point [B_SES, t_SES] should
be calculated.
In this case the following step-by-step evaluating procedure is given.

Step 1. SESR evaluation. With equations (2),
(28) and given B₀, t₀
calculate SES threshold B_SES and corresponding
percentage-of-time t_SES. With relation (27)
determine SESR parameter

,
.

(42)

If it is possible using the prediction methods to calculate t_SES
based on the B_SES threshold known from DRRS equipment
specification we could take them as co-ordinates of the characteristic point [B_SES,
t_SES].

In both cases we could continue the further consideration from the point with
coordinates [B_SES; t_SES] where t_SES=
SESR.

Step 2. ESR evaluation. Using 2-point distribution and
numerical solution of integral (23) evaluate parameter
ESR parameter as

ESR =(1-exp[-n
N_B (B₀/a ₁)(t₀/t)^m1])dt
+(1-exp[-n
N_B (B₀/a ₂)(t₀/t)^m2])dt

(43)

Approximate value of ESR can be found in accordance with (35)
as

(44)

Step 3. BBER evaluation. Calculate BBER with equation (29)
as

BBER = (N_B t_SES B_SES)/[a
₂(m₂-1)] = SESR

(45)

Step 4. Contribution of the BER “tail” (BER< B_S).
Calculate relative time t_S when BER value is equal to B_S
as

t_S=t_SES(B_SES/B_S)^1/m2=
SESR.

(46)

The value B_S can be considered as 10^-10¸10^-13. Geometrically contribution of the tail is equal to the
area of rectangle with base (1-t_S) and the height equal to the
appropriated probability at the point t_S. Under this, the
contributions of the tail to the error performance parameters following
equations are applied

D SESR=0,
D BBER=N_B B_S (1-t_S),
D ESR=N_B n B_S (1-t_S)

(47)

The contribution to SESR is equal to zero because P_SES is nearly
step function.
Then using approximate relation (44) and relations (45),
(46) and (47) we could estimate ESR and
BBER taking into account the “tail” effect and considering ts<<1

,

(48)

(49)

Step 5. Compare resulting SESR, ESR and BBER values with objectives
required by ITU-R Recommendation F.1092 or F.1189 (Recommendation ITU-T
G.826). If all parameters do not exceed required values the given path meets
the objectives.

6.2 Design of radio-relay link

Above procedure can be basically applied also for the design of radio-relay
links. It is assumed that BER (t) distribution is a 3-point piece-wise function
with each part of the distribution described by equation (41).

Then suppose that a path has been designed to meet SESR (=t_SES)
objective according ITU-T Recommendation G.826 or respective ITU-R
Recommendations.

It is necessary to determine a bit-error probability mask BER (t), defined by
values of m_iand a _i in
the distribution (41), in a such way that digital
transmission would meet the above mention objectives taking into account the
relationship from Table 6.

Step-by-step procedure

Calculations are accomplished from small values of
percentage-of-time to the higher one when the BER “tail” effect is neglected
during the first steps.

Step 1. SESR estimation. The contribution of the mask to SESR
is defined by its first portion because P_SESis insignificant whent>t_SES. To defined the characteristic points [B₀,
t₀] and [B_SES, t_SES] we get the
requirements for parameters m₁ and a _{1
,}taking into account the equations (42) and consider
t_SES = SESR and B₀=10^-3 known, using
the following relations

,
.

(50)

Values for B_SES /a ₁and k could be taken from Tables 1 and 2
respectively.

Step 2. ESR estimation. Using (47) we
could write

(51)

For of m₂
we have <
6.
Under this the ratio ESR/SESR £ 80 for VC-4, £
37,5 for VC-3 and £ 20 for 2048 kbit/s are met
when

<
(0.5;0.6 or 8)10^-6
t_SES(respectively)

(52)

Step 3. BBER evaluation. The main contribution of mask to BBER
is related with its second portion which starts from point with coordinates [B_SES,
t_SES] defined in the Step 1. Using (49) we
could write

(53)

First of all let’s estimate under what value B_S / a
₃ we could consider the second part of the sum negligible for N_B
= 18792 , 6120 or 2048.

,
if <
(0.5;1.6 or 4.8)10^-6
t_SES(respectively)

(54)

Then taking it into account we could receive that
,
if ,
for a ₁= a ₂

     a ₁=2a ₂}

Comparing inequalities (50) and (52) one could see that for paths VC-4/VC-3 and 2048 kbit/s ITU-T Recommendation G.826 objectives could be met if

< (0.5; 0.6 or 4.8)10^-6 t_SES (respectively)

(55)

Step 4. BER (t) mask.

Let consider BER(t) mask for a VC-4 path under different a and m.

In the Table 7 relative values of t₀/t_SES and t_s/t_SESare given for

m=5, 10 and 20 under

1) a ₁=a ₂=a ₃=1 (B₀=10^-3; B_SES=2x10^-5;)
2) a ₁=a ₂=10, a ₃=1 (B₀=10^-3; B_SES=2x10^-4; B_s=10^-12).

The B_s=10^-12 value is taken as the threshold value that could allow to meet (53) even for a 50 km hop from 50-hops 2500 km international VC4 path, block size N_B=18792 bits, number of blocks n=8000 blocks/s.

Error performance objectives for given path is defined as SESR₀=2.4 10^-6, ESR₀=1.9 10^-4, BBER₀=2.4 10^-7.

For other paths this value could be more than 10^-12.

Relative values t₀/t_SES and t_s/t_SESfor a 50 km hop from 50-hops 2500 km international VC4 path for variants 1) and 2) could be found in Table 7.

Table 7. Relative values t₀/t_SES and t_s/t_SES

	t₀/t_SES			t_s/t_SES
m	5	10	20	5	10	20
for 1)	0.46	0.68	0.82	30	5.5	2.3
for 2)	0.72	0.85	0.92	47	6.8	2.6

The BER (t) mask based on Table 7 is given in Fig. 5.

Considering the Figure 5 it could be seen that the average number errors per burst a _i has less impact on the mask than the slope m_iespecially with m_i increasing.

As an example the dotted line in the Figure 5 shows the BER(t) mask for following parameter:

m₁ = 7.5; a ₁=10                                   t Ì [t₀,t_SES]
m₂ = 10;   a ₂= vary from 10 to 2         t Ì [t_SES,t_S]                              (56)
                a ₃=2                                     t ³ t_S

The above-considered results have shown that it will be useful to include in DRRS technical specification the following additional parameters:

the receiver threshold at SES level,
error burst length (a _i) at different BER levels.

7. CONCLUSIONS.

We have presented the asymptotic method for predicting error performance parameters of terrestrial digital radio-relay links based on normal approximation to the binomial distribution. With analytical expressions obtained for probability of basic events it has been proved that probability P_SES is equal to 0.5 under 30% P_EB threshold. The simple relation between 30% thresholds P_EB and P_ES has been established. It was shown that SES(BER) is practically a step function and strongly depends on average error burst value for BER=10^-3 …10^-6 and a block size.

The simple converting method has been developed for time variant case with constant index of distribution BER (t) and invariant mean burst length. Under this, analytical relations depending on number of blocks per second and index of BER distribution between ESR, SESR and BBER have been found.

The prediction method and evaluating step-by-step procedure have been developed for piece wise distribution BER (t) with time variant average number of errors per burst has been also elaborated.

The numerical relationship between propagation and equipment parameters that could allowed to meet error performance objectives for 2048 kbit/s, VC-3 and VC-4 paths according ITU Recommendations have been also presented as well as the relative BER(t) mask.

It was shown that if the residual BER < 10^-12:

the relationship between BBER and SESR is practically defined mainly by the BER(t) slope and also the ratio between average error burst values for BER~10^-4 and for BER~10^-6…10^-10 and does not depend on block size and blocks/s values.The BBER objective could be the most critical for DRRL to meet ITU Recommendations objective;

the relationship between ESR and SESR is defined practically by the BER(t) slope and blocks/s values and does not depend on block size values and average error burst values.The value of the residual BER could be critical for ESR objective.

References

1	ITU-T Recommendation G.826	Error performance parameters and objectives for international bit rate digital paths at or above the primary rate. ITU. Geneva. 1999.
2	ITU-T Recommendation G.821	Error performance of an international connection forming part of an integrated services digital network ITU. Geneva. 1996.
3	ITU-R Recommendation F.1092	Error performance objectives for constant bit rate digital paths at or above the primary rate carried by digital radio-relay systems, which may form part of the international portion of a 27500 km hypothetical reference path. ITU. Geneva. 1997.
4	ITU-R Recommendation F.1189	Error performance objectives for constant bit rate digital paths at or above the primary rate carried by digital radio-relay systems which may form part of the national portion of a 27500 km hypothetical reference path. ITU. Geneva. 1997.
5	ITU-R Recommendation F.1093	Effects of multipath propagation on the design and operation of line-of-sight digital radio-relays systems. ITU. Geneva. 1997.
6	ITU-R Recommendation P.530	Propagation data and prediction methods required for the design of terrestrial line-of-sight systems. ITU. Geneva. 1997.
7	ITU-R Recommendation F.594	Error performance objectives of the hypothetical reference digital path for radio-relay systems providing connections at a bit rate below the primary rate and forming part or all of the high grade portion of an integrated services digital network. ITU. Geneva. 1997.
8	ITU-R Recommendation F.634	Error performance objectives for real digital radio-relay links forming part of the high-grade portion of international digital connections at a bit rate below the primary rate within an integrated services digital network ITU. Geneva. 1997.
9	ITU-R Recommendation F.696	Error performance and availability objectives for hypothetical reference digital sections forming part or all of the medium-grade portion of an ISDN connection at a bit rate below the primary rate utilizing digital radio-relay systems. ITU. Geneva. 1997.
10	ITU-R Recommendation F.697	Error performance and availability objectives for the local-grade portion at each end of an ISDN connection at a bit rate below the primary rate utilizing digital radio-relay systems. ITU. Geneva. 1997.
11	ITU-R Recommendation S.1062	Allowable error performance for a hypothetical reference digital path operating at or above the primary rate. ITU. Geneva. 1995.
12	I. Ruis de Gauna, M. Spini	Methods for predicting error performance and availability on terrestrial SDH radio links. Proc. of 6^th ECRR’98, Bergen, Norway, 1998, pp.324-329.
13	U. Johansson, B. Danielsson and G. Salomonsen,	Estimated performance of terrestrial SDH radio links based on measured data from Norway, Spain and Sweden. Proc. of 6^th ECRR’98, Bergen, Norway, 1998, pp.310-315.
14	R.E. Walpole, R.H.Myers	Probability and statistics for engineers and scientists. Macmillan publishing company, N.Y., 1990
15	H.L Van Trees	Detection, estimation and modulation theory. Part I, Wiley, N.Y., 1968

Appendix A Asymptotic properties of probability P_SES as a function of BER

It was shown in section 2.3 that probability P_SES can be represented by the integral

(A-1)

where f(x) probability density function of normal random variable x with m and covariance s ² defined as

m =n P_EB, s ²=n P_EB(1-P_EB).

(A-2)

where probability of errored blocks P_EB described by expression (1).

Using standardization u=(x-m )/s integral (A-1) can be expressed through in terms of standard normal distribution of random variable u as

(A-3)

where F(u) is tabulated cumulative standard normal distribution, u₁=(0.3n-m )/s , u₂=(n-m )/s are standardized lower and upper limits of integral (A-1).

It is easy to show that the upper limit of integral

u₂=(n-n P_EB)/[n P_EB(1-P_EB)]^0.5=[n(1-P_EB)/P_EB]^0.5.

(A-4)

for any type path exceeds value 40 when probability of errored blocks P_EB is close to the value 0.3 that defines SES threshold. Under this, bearing in mind the property of standard normal distribution we can write the expression

F(u₂)=F(40)@ F(¥ )=1,

(A-5)

which is accurate with 9 digits. Thus, probability P_SES given by expression (À-3) can be presented in more compact form

P_SES=1-F(u₁)

(A-6)

Considering the lower limit

u₁=(0.3n-m )/s =(0.3n-nP_EB)/[nP_EB(1-P_EB)]^0.5=(0.3-P_EB)/[P_EB(1-P_EB)/n]^0.5

(A-7)

with P_EB=0.3 one can readily obtain the important relation between P_EB and P_SES

if P_EB=0.3 , then u₁=0, F(0)=0.5 and P_SES=0.5.

(A-8)

Bearing in mind that P_EB=0.3 under SES threshold of BER denoted earlier as B₁, relation (A-8) means that

P_EB(B₁)=0.3, P_SES(B₁)=0.5.

(A-9)

Applying 3s -rule for normal standard distribution we can conclude that

0.005£ P_SES£0.995 if -3£ u_1£3,

(A-10)

It is useful to express this range in terms of BER in the neighbourhood of SES threshold B₁. At this point P_EB=0.3 and due to (A-2) variance of variable u is equal to

s ₁=[P_EB(1-P_EB)/n]^0.5=

(A-11)

If so, we can state that

-3s _1£P_EB(B₁)-P_EB £ 3s ₁, if -3£ u_1£3.

(A-12)

Let’s use a linear approximation to P_EB described by equation (1)

P_EB (BER)=N_B BER₁/a =BER/t

(A-13)

where t =(a /N_B). Under this, the linear relation between P_EB increment and that of BER can be found

P_EB(B₁)± 3s ₁=(B_{1±
D}B)/t .

(A-14)

Using expression (A-11) after simple manipulations with (A-14) we obtain the following expression for increment D B

D B=± 3t s ₁=± 3(a /N_B)

(A-15)

Summarizing all above results related with expression (A-10) we could state that

0.005£ P_SES£0.995 under BER=B_1±3(a /N_B)

(A-16)

Appendix B Asymptotic presentation of BBER parameter

The Recommendation ITU-T G.826 defines BBER as the ratio between errored blocks to total blocks during available seconds, excluding all blocks during SES. Probability P_nk of k errored blocks in total of n blocks can be found by use of binomial distribution [1]

P_nk=[n!/(n-k)!][(1-P_EB^n-k)P_EB)^k]

(B-1)

Using expression (B-1) average number of blocks that not occurring as a part of a SES can be found as

(B-2)

It is known that normal approximation to the binomial is quite accurate for n³ 1000. Under this the sum in expression (B-2) can be replaced by integral

(B-3)

where f(x) is a probability density function (PDF) of normal random variable x with time variant mean and variance defined as

m (t)= n P_EB(t), s ²(t)=n P_EB(t)[1-P_EB(t)].

(B-4)

Standardized value of average number errored blocks, which are not a part of SES, can be expressed in terms of standard normal distribution as

(B-5)

where j (u) is a probability density function of standard normal variable and limits of integral is given as

u₁=(1-m )/s , u₂= [(0.3n-1)-m ]/s .

(B-6)

Using expression (B-5) average number of background errored blocks can be presented as

N_EB=m +s =n {P_EB+[P_EB (1-P_EB)/n]^0.5}.

(B-7)

From expression (B-3) it follows that for any digital pat with n³ 1000 under P_EB(t)£ 0.3 the inequality ½ ½ <0.5 is valid. Under this the contribution of second term in equation (B-7) is negligible small. Thus, instead of (B-7) we can write

N_EB(t)=nP_EB(t),

(B-8)

Thus, for any fixed percentage-of-time average number errored blocks, which are not a part of SES, is a simple product of number of blocks/s n and probability of errored block.

Using expression (B-8) BBER parameter can be easily found in accordance with its definition as

(B-9)

It is worth mentioning that expression (B-9) provides an explicit relation between BBER parameter and probability of errored blocks.

contents

discussion

m	5		10
n	*1000*	*8000*	*1000*	*8000*
ESR/ SESR	4	6	20	2.5
BBER/ SESR	0.09	0.09	0.04	0.04
ESR/ BBER	44	67	50	62

*Rate (kbit/s)*	*2048*	48 960 (VC-3)	150 336 (VC-4)
Block size	2048 bits	6 120 bits	18 792 bits
Blocks/second	1 000	8 000	8 000
ESR₀/ SESR₀	20	37.5	80
BBER₀/ SESR₀	0.1	0.1	0.1