jimenez, gonzales, serrano

8/21/2019 Jimenez, Gonzales, Serrano

1/12

Microwave

Genina Teriz M. Gonzales, Maria Christina D. Jimenez and Mari Fatima P. Serrano

Department of Electronics Engineering

Faculty of Engineering, University of Santo Tomas

Sampaloc, Manila, Philippines

[email protected], [email protected], [email protected]

Keywords — diversity; fading; reliabi li ty

1.

FADING

There are two types of fading in a radio propagation

channel: large scale fading and small scale fading. Large scale

fading deals with attenuation due to path loss over large

distances and shadowing effects while small scale fading deals

with distances in the range of the signal wavelength, this is

mainly caused by multipath interferences.[1]

1.1

Large Scale Fading

This type of fading is caused by path loss over large

distances and shadowing by obstructions such as buildings and

mountains. This is useful in estimating the coverage area of a

transmitter and is used in cell-site planning and is typically

frequency independent. [1]

1.2

Small Scale Fading

This type of fading is caused by the rapid fluctuations of

the amplitude of a signal over distances in the order of the

signal wavelength or short period of time and is frequencydependent. It is also caused by multipath interferences

between two or more versions of the transmitted signal

combining at the receiver giving a resultant signal varying in

amplitude and phase. [1]

Effects

a)

Rapid changes in signal strength over a small distance or

period of time

b)

Random frequency modulation due to varying Doppler

Shifts on different multipath signals

c)

Time dispersion (echoes) caused by multipath

propagation delays

Influencing Factors

a)

Multipath propagation – The multiple versions of the

transmitted signal combining at the receiver and are

displaced with respect to time and spatial orientation are

caused by the presence of reflecting objects in the channel

changing the signal in amplitude, phase and time. These

random changes cause fluctuations in signal strength

(small-scale fading).

b)

Speed of the mobile - The random frequency modulation

due to varying Doppler Shifts on each of the multipath

signals are caused by the relative motion between the base

station and mobile, the Doppler shift can be positive or

negative whether the receiver is moving towards or away.

c) The transmission bandwidth of the signal – If the

transmitted bandwidth of the signal is greater than the

bandwidth of the multipath channel, the received signal

will be distorted, but the signal strength will not fade

much over a local area.

Time Dispersion Parameters

a)

Based on power delay profile (PDP)

b)

Mean excess delay

c)

RMS delay spread,

Fig. 1.

Typical measured values of RMS delay spread

d) Coherence bandwidth, BC = 1/50

Frequency Dispersion Parameters

a)

Doppler spread, B D – equivalent to maximum

Doppler shift, f m=

The operation depends whether the Doppler shift is

positive or negative respectively.

b)

Coherence time, T C = 1/B D


2/12

Practical examples include temporary failure ofcommunication due to a severe drop in the channel signal tonoise ratio and FM radio transmission experiencingintermittent loss of broadcast when away from station.

1.3

Types of Small Scale Fading

The transmitted signals will undergo different types offading depending on the nature of the transmitted signal

(bandwidth, period) and characteristics of the channel (RMSdelay spread and Doppler spread). Fig.2 shows a tree of thefour different types of fading. The multipath delay spread leadsto flat fading and frequency fading while Doppler spread leadsto slow and fast fading. These two mechanisms areindependent. [1]

Fig. 2.

Different types of fading

Fading due to Multipath delay spread

A. Flat Fading

The most common type of fading, this occurs

when the channel has a constant gain and linear

phase response over a bandwidth greater than the

bandwidth of the transmitted signal. In flat fading,the strength of the received signal changes with time

caused by the fluctuations in the gain of the channel. [1]

Fig. 3 shows the characteristics of a flat fadingchannel which is also known as amplitude varyingchannels or narrowband channels. It can be see that achange in amplitude occurs in the received signalwhen the channel gain changes over time. Thespectral characteristics are preserved in the receiver.In this type of fading, the transmitted signal’sreciprocal bandwidth is much greater than the

Fig. 3.

Characteristics of flat fading channel

multipath delay spread of the channel, and can beapproximated as having no excess delay. These channels causedeep fades and usually requires 20 or 30 dB more transmitter power to achieve low bit error rates compared to non-fadingchannels. [1]

Thus, flat fading occurs when

where T S is the reciprocal bandwidth, BS is the bandwidth, BCis the coherence bandwidth and is the rms delay spread.

Types of flat fading include rain fading and diffraction fading . Above 10 GHz temporal variation in path loss is due torain attenuation – the process depending on instantaneousrainfall rate. When the atmosphere is sufficiently sub-refractive(large positive values of the gradient of refractive index, low k-factor values), the ray paths will be bent in such a way that theearth appears to obstruct the direct path between transmitterand receiver, giving rise to the kind of fading called diffraction

fading. [2]B.

Frequency Selective Fading

This occurs when the channel has a constant gain

and linear phase response over a bandwidth smaller

than the bandwidth of the transmitted signal and is

caused by multipath delays exceeding the time of the

transmitted signal. In this type of fading, the

multipath delay spread of the channel is greater than

the transmitted signal’s reciprocal bandwidth. [1]The multiple versions of the transmitted signal

combined at the receiver and are displaced withrespect to time and attenuated (faded) and with thistime dispersion, intersymbol interference occurs and

as time varies, the channel also varies in gain and phase, distorting the received signal. [1]

Fig. 4.

Characteristics of frequency selective channel

Fig. 4 shows the characteristics of a frequency selective

fading channel or also known as wideband channels. S(f)

represents the spectrum of the transmitted signal, and in this

type of fading, it’s bandwidth is greater than the coherence

bandwidth BC of the channel. The channel becomes selective

where the gain is different for different frequency


3/12

components. This channel is harder to model than flat fading

since each channel must be considered a linear filter and each

multipath signal must be modeled. [1]

Thus, frequency selective fading occurs when

Rule of thumb is used in this type of fading, it is said to be frequency selective if .

Fading due to Doppler spread

Fading can be fast or slow depending on how

rapidly the transmitted signals changes compared to

the channel. When a channel is said to be fast or slow

fading, it does not specify if it is flat or frequency

selective. It only deals with the rate of change of the

channel due to motion. [1]

A. Fast Fading

Fast fading is also known as time selective

fading because it causes frequency dispersion. In this

type of fading, the channel impulse response changes

rapidly and occurs for very low data rates as shown

in Fig. 5. The coherence time is small relative to the

period of the transmitted signal. In this case, the

amplitude and phase change imposed by the channel

varies considerably over the period of use. In the

frequency domain, the distortion caused by fast

fading increases as Doppler spread increases. [1]

Fig. 5.

Characteristics of frequency selective fading channel

Thus, fast fading occurs when

The rate of change of the channel is higher than

the signal period and the channel changes over one

period. T C is related to the Doppler spread, fm, as

A higher Doppler spread results in a smaller

coherence time. In dealing with flat fading, we

approximate the impulse response to be a delta

function (no delay). If a channel is said to be flat and

fast fading, it implies that the amplitude of the delta

function varies faster than the transmitted signal

while it is the amplitude, phase and time delay of any

of the multipath components that vary faster in a

frequency selective and fast fading channel. [1]

B.

Slow Fading

In this type of fading, the channel impulse

response changes at a rate much lower than the

transmitted signal. In the frequency domain, slow

fading is expected with low Doppler spread. [1]

Thus, slow fading occurs when

The channel coherence time is larger than the

symbol period and the channel remains static. Fig. 6summarizes the relationship between the types of

fading due to multipath delay spread and due to

Doppler spread. [1]

Fig. 6.

Characteristics of flat fading channel

1.4

Fade Margin

It is the difference between the nominal receive level and

the receive threshold level. It should match the availability and

performance objectives set. It also considers the non-ideal and

less predictable characteristics of radio wave propagation such

as multi-path loss and terrain sensitivity. [2]


4/12

Barnett-Vignant Equation

where:

30 log D = multi-path effect

10 log (6ABf) = terrain sensitivity10 log (1 – R) = reliability objectiveness

FM - Fade Margin

D - Distance (km)

f - Frequency (GHz)

R - Reliability

(1 – R) – Reliability objective

A – roughness factor

B – factor to convert a worst month

probability to an annual

probability

Table 1. A and B table

a 4 for very smooth terrain including overwater

1 for average terrain with some roughness

0.25 for mountainous, very rough or very dry

b 0.5 Gulf coast or similar hot, humid areas

0.25 normal interior temperate or northern

0.125 mountainous or very dry

1.5

Problems

I. Consider a transmitter which radiates a sinusoidalcarrier frequency of 1850 MHz. For a vehicle moving60 mph, compute the received carrier frequency if the

mobile is moving (a) directly towards the transmitter,(b) directly away from the transmitter, (c) in adirection which is perpendicular to the direction ofarrival of the transmitted signal.

Given:

Carrier frequency, f c : 1850 MHz

Vehicle speed v = 60 mph = 26.82 m/s

(a)

The vehicle is moving directly towards thetransmitter. The Doppler shift is positive and the

received frequency is:

(b)

The vehicle is moving directly away from thetransmitter. The Doppler shift is negative and thereceived frequency is:

(c) The vehicle is moving perpendicular to the angleof arrival of the transmitted signal.

In this case, , and there is no

Doppler shift. The received signal frequency is

the same as the transmitted frequency of 1850

MHz.

II.

A wireless system operates at frequency fc = 1GHz,for each case below, determine what type of small-scale fading occurs (fast or slow; flat or frequency-selective).

a. User browses at data rate R = 1Mbps, in a car

moving at 60 mphGiven:f c = 1 GHzVehicle speed v = 60 mph = 26.82 m/s

BD is much smaller than the signal

bandwidth 1 Mbps. Therefore it is a slow fading

channel.

b.

User is on a voice call at data rate R = 5kbps,driving on a highway at 60 mph

Given:f c = 1 GHzFor suburban environment, the value ofRMS delay spread is 200 ns.

BC is greater than the bandwidth 5kHz.Therefore, it is a flat fading channel.

III.

Determine the fade margin for the followingconditions; distance between sites, D = 60 km;frequency, f = 2.5 GHz; smooth terrain; humidclimate; and a reliability objective of 99.999%.


5/12

2. DIVERSITY

Different forms of diversity are available to the path

designer. The ITU gives some useful data on different diversity

schemes. Diversity schemes that can be used on point-to-point

microwave links are the following:

Angle diversity;

Space diversity with RF or IF combiners, which

can be minimum dispersion or maximum power;

Space diversity with baseband switching;

Frequency diversity (in-band or cross-band; 1 + 1,

or n + 1);

Hybrid diversity (space diversity and frequency

diversity with two or four receivers)

2.1.

Angle Diversity

Angle diversity has been quoted in some literature as

performing well against selective fading. However, it is not

implemented yet because it has not been conclusively proved to

be efficient in practice.

Fig. 2.1 Angle Diversity Configuration

Angle diversity can be done using two different antenna

configurations: Two separate antennas mounted side by side or a

single antenna with dual-beam feedhorn. These can be operatedin numerous ways. For the configuration with two separate

antennas, the most general alignment is for one antenna to be

bore-sighted on the primary (normal) transmitted signal and the

second antenna to have an elevation angle slightly greater than

the primary antenna. The second antenna is aligned such that the

primary transmitted signal arrives at the first low side null of the

main beam. The second antenna theoretically will only see

signals that have a higher angle of arrival than the primary

signal. This second antenna will have no received signal during

regular propagation conditions but should have a significant

signal when the secondary signals are causing cancellation of the

primary signal at the first antenna. While this is optimum

alignment, because it forces the second antenna receiver to be inconstant alarm, this alignment is not well accepted with

maintenance staff. If the transmission path has excessive terrain

clearance, the receive antenna may experience significant

secondary signals from below the normal receive path angle. In

this case, the second antenna is either pointed above or below

the main antenna, depending on whether the secondary signal is

expected to be normal atmospheric multipath or ground

reflections. Sometimes, this decision must be made based on

experience.

During normal propagation conditions, with the second

antenna aligned on the first low side null of the upper beam as

specified above, the secondary antenna may not be receiving a

significant signal. For this reason, some operators change thealignment of the second antenna back toward the main signal

path to keep the diversity

2.2. Frequency Diversity

Frequency Diversity is more intricate and more costly than

space diversity. It has advantages as well as disadvantages

Frequency diversity needs two transmitters at the near end of the

link. The transmitter is modulated simultaneously by the same

signal but transmit on different frequency. Frequency separation

must be at least two percent (2%) but five percent (5%) is

preferable. Figure 2.2 shows an example of a frequency –

diversity configuration. The two diversity paths are derived inthe frequency domain. When a fade take place on one frequency,

it will probably not occur in the other frequency. The more

frequency is separated from the other the less chance there is

that fade will occur simultaneously on each path.

Fig. 2.2 Frequency Diversity Configuration

Frequency diversity is more expensive but there is greater

guarantee of path reliability. It offer full and simple equipment

redundancy and has the great operational advantage of two

complete end to end electrical paths. In this case crash of one

transmitter or one receiver will not interrupt service and the

transmitter and/or a receiver can be taken out of service for

maintenance. The primary disadvantage of frequency diversity

is that it doubles the amount of frequency spectrum required in

this day and age when spectrum is at premium. In many cases, it

is forbidden by national licensing authorities. For example, the

FCC does not allow frequency diversity for industrial users. It

also should be appreciated that it will not be difficult to get the

required frequency spacing.

Cross Band Diversity


6/12

Cross-band frequency diversity is a very effective

method from a propagation point of view, but is not

very spectrum efficient because it requires the

availability of two frequency bands. One could use a

high-frequency band such as eighteen (18) GHz as the

protection channel assuming that when rain affects this

band, there is no multipath fading on the lower

frequency band, for example, 6 GHz (the turbulent

conditions associated with rain do not favourmultipath). This would permit one to use the high-

frequency band over much larger distances than

normal. Although this may sound interesting in theory,

it is improbable to be used in practice because it is so

wasteful of valuable frequency spectrum.

In Band Diversity

In-band frequency diversity is the most familiar

form of diversity because when an n + 1 system is

configured, one of the channels can be used for

protection. A dedicated protection channel such as a 1

+ 1 system is not as frequency efficient but affords ahigh level of protection. One can also put lesser priority

traffic on the protection channel that can be dropped

when switching takes place, thus improving the spectral

efficiency. Frequency diversity is not endorsed in many

countries due to the extra spectrum usage.

Frequency Diversity Outage

For frequency diversity the improvement factor is

directly proportional to the frequency separation. The

improvement factor is given by:

IFD= (80/fd) (∆f/f) (10F/10)

Where ∆ f is the frequency separation in gigahertz, f

is the carrier frequency, d denotes the hop distance in

kilometers, and F is the fade margin in decibels. The

outage P with frequency diversity is given by:

PFD= P/ IFD

2.3. Space Diversity

Space diversity is very spectrum efficient and provides

excellent performance against multipath fading. The concept is

to separate the two antennas in the vertical plane such that when

there is phase cancellation on the main path due to multipathfading, the diversity path is not influenced due to the extra path

length. Normally, provided there are at least 200 wavelengths of

separation between the antennas, the two paths will not be

linked. Due to the improvement factor of space diversity,

smaller antennas can be used, which counterbalance the

additional cost of extra antennas. The degree of improvement

when using any of the diversity options depends on the amount

of uncorrelation between the main channel and the diversity

channel.

Space diversity generally provides superior improvement, in

practice. If one equates the correlation factors for comparison

purposes, one can determine that at, for example, 2 GHz, 10m of

space diversity spacing is equivalent to 14 MHz of frequencyseparation. At 7 GHz, the same spacing is equivalent to 610

MHz of spacing. One can see therefore that in-band frequency

diversity is more effective at lower frequencies. Tower height

can be a limiting factor for space diversity, and in the end a

solution needs to be found depending on the particular situation

rather than by rules of thumb. Despite this, it can generally be

stated that at higher frequencies space diversity is more effective

than in-band frequency diversity given the spacing is not limited

as shown previously. As a rule of thumb, the spacing of the

antennas should be separated by 200 wavelengths to ensure the

two signals are not linked. Although for digital radio systems

that are affected by selective fading, there is a advantage in

having the antennas spaced closer together, because most of theoutage is still due to flat fading, due to the success of adaptive

equalizers in handling selective fading effects, it is still

recommended to increase spacing for improved overall

performance.

Fig. 2.3 Space Diversity System

Space Diversity Outage

Historically, for space diversity using baseband

switching, the improvement factor has been based on

the Vigants formula:

Where s denotes the antenna separation in meters, γdB is

the difference between the main and diversity receive levels

in decibels (20 log (γ)), f is the frequency in gigahertz, and d

is the path length in kilometres.

The improvement factor recommended by the ITU is

Where:


7/12

A = fade depth in decibels

p0 = multipath occurrence factor (%)

S = vertical separation of receiver antennas in m

f = frequency in gigahertz

d = path length in kilometres

V = |G1 − G2 |

G1, G2 = gain of the two antennas in dBi

The outage P with space diversity is given by:

2.4.

Hybrid Diversity

A cost-effective and very efficient method for 1 + 1 systems

is hybrid diversity, where the frequency diversity switch is used

to switch two channels separated spatially over the link. To

accomplish this, at one end two antennas are employed, each

connected to the respective main and standby transmitters and

receivers. At the far end one antenna is used but the receivers are

switched by the frequency diversity switch. Space and frequency

diversity are thus attained in both directions of propagation.

Fig. 2.4 Hybrid System

SAMPLE COMPUTATION

Sample Problem 1

Consider a frequency diversity microwave radio systemoperating at an RF frequency of 8 GHz. The fade margin of the

system is 37.88 dB. The frequency separation is 0.20 GHz and

the distance between stations is 40 km. Find the improvement

factor.

Solution:

IFD= (80/ (8) (40)) (0.2/8) (1037.88/10)

IFD= 38.36≈39

Sample Problem 2

Consider a space diversity microwave radio system operating at

an RF frequency of 8 GHz. The fade margin of the system is37.88 dB. The difference between the main and receive levels is

13 Mhz. The antenna is separated to 10 m spacing. The fade

depth is 10 dB. The multipath occurrence factor is 0.1% and fade

depth is equal to 20 dB. The difference in gain between two

antennas is 10 dB. Find the improvement factor.

Solution:

γdB= 20 log(13MHz)= 142.28 dB

ISD= (1-exp (-0.04x(10)0.87(8)0.12(40)0.48(0.1)-1.04)(10(20-10)/10)

ISD= 10

Sample Problem 3

If the improvement factor was adjusted to 50, what is the

frequency separation?

20= (80/ (8) (40)) (∆f//8) (1037.88/10)∆f= 0.261 GHz

Sample Problem 4

Consider a frequency diversity microwave radio system

operating at an RF frequency of 1.8 GHz. The fade margin of the

system is 31.4 dB. The frequency separation is 0.1 GHz and the

distance between stations is 40 km. Find the improvement factor.

Solution:

IFD= (80/ (1.8) (40)) (0.1/1.8) (1031.4/10)

IFD= 85.21

Sample Problem 5

Consider a space diversity microwave radio system operating at

an RF frequency of 8 GHz. The fade margin of the system is

37.88 dB. The difference between the main and receive levels is

13 Mhz. The antenna is separated to 10 m spacing. The fade

depth is 10 dB. The multipath occurrence factor is 0.1% and fade

depth is equal to 20 dB. The difference in gain between two

antennas is 10 dB. Find the improvement factor.

Solution:

γdB= 20 log(13MHz)= 142.28 dB

ISD= (1-exp (-0.04x(10)0.87(8)0.12(40)0.48(0.1)-1.04)(10(20-10)/10)

ISD= 10

3. RELIABILITY

System reliability is defined as the ability of an item to

perform a required function, under given environmental and

operational conditions and for a stated period of time. It is also

the probability that an item will operate when needed and theaverage fraction of time that a system is expected to be in an

operating condition.


8/12

3.1 HARDWARE R ELIABILITY

Electronic system reliability analysis is based on component

failures without considering the physical failure mechanisms:

(1) A piece of equipment can be in only one of two states:

failed or not failed

(2) The failure of any component is independent of the

failure of other components

(3) The probability of equipment failure is quite small(allowing second-order probability factors to be ignored)

(4) The equipment is assumed to have aged beyond the

infant mortality period (it is mature) but is not worn out

(5) The equipment failure statistics are constant.

(6) The r epaired equipment is “good as new”

U = equipment unavailability = outage time/total time;

MTBF(hours) = mean time between (device two way) failure;

MTTR(hours) = mean time to restore (mean downtime).

RT(hours) = mean time to detect, diagnose, and report an alarm

to the appropriate repair person;

TT(hours) = mean travel time;

PS = probability of having a working spare module;

TR(hours) = mean time to obtain a spare module from an outside

source if no spare is available locally;

MTR(hours) = mean time to replace (or repair) failed module

and restore equipment.

Rare equipment failures meeting the previous assumptions

are modeled as a homogeneous Poisson process with

exponential failure distribution. Subsystem (module) failure rate

is assumed to be constant and defined by λ.

Device reliability R(t) is the probability that the device will

perform without failure over the time period 0 to t (h) when the

device is operated within its intended environment. R(t), the

time integral of the failure probability density function λe−λt , isalso called the survivor probability.

e = Napier’s constant (Euler’s number) ∼= 2.7182818.

An interesting aspect of the Poisson distribution is, as it is

exponential, most failures occur earlier in time than the MTBF

On average, half of the units will fail by 0.61 MTBF, the median

failure time. By the MTBF time, it is expected that 63%

(essentially 2/3) of the modules will have failed.

MTBF represents the statistics of rare random failures of the

entire population of similar devices. Mean time to failure

(MTTF) or mean life (ML) are terms used to describe the

average period until the device is worn out. They should not be

confused with MTBF.


9/12

3. 2. SYSTEM R ELIABILITY

On the basis of predicted or measured circuit packs or

modules, network element reliability is determined by

estimating the availability. Many methods are used: Markov

modeling, Monte Carlo simulation, or graph theory (flow

networks). A popular approach (suitable for static networks with

rare and unrelated failures) is by analyzing the network

element’s reliability block diagram. The reliability diagram is a

block diagram showing the failure dependency of the various packs or modules. On the basis of the reliability diagram,

analysis is made to determine the availability or unavailability of

the entire network element. The following network element

subsystems are often encountered.

Figure 1. (a,b) Various interconnected systems

Equipment in Series (System A, Fig. 1)

Multiple Equipment in Parallel (Systems B and C,

Fig. 1)

Consider n separate identical working devices (channels)

operated with m separate identical standby (backup) devices

(channels), where all n lines (signal paths through a working or

standby device) must operate.

Atotal = the availability of the total system consisting of all

the working devices;

Aw = the availability of one of the working devices;

Ap = the availability of one of the standby (protection)

devices;

Utotal = the unavailability of the total system consisting of

all the working devices;

Uw = the unavailability of one of the working devices;

Up = the unavailability of one of the standby (protection)devices.

Consider m = 1 and n = 1 (typical hot standby or frequency

diversity)

Consider n > 1, m = 1 (typical multiline) case (System C, Fig.

15.1):

Nested Equipment (System D, Fig. 1)

Atotal = the availability of all devices as a group;

An = availability of the device n.

Meshed Duplex Configuration (Systems E and F,

Fig. 1)

This configuration is common in high reliability computer

and digital cross-connect systems (System E, Fig.1). It is

functionally equivalent to the reliability block diagram (System

F, Fig. 1). It is evaluated by reducing the parallel components to

equivalent series elements and then evaluating the reliability of

the series units.

3.3. COMMUNICATION SYSTEMS

The evaluation of reliability of communication systems is

similar to the evaluation of network elements. The system

reliability block diagram of the system is drawn using the


10/12

availability values of the network elements. Those values are

determined from two-way MTBF and MTR values.

The system is then evaluated using the same techniques

described earlier.

3.4. APPLICATION TO R ADIO CONFIGURATIONS

Radio system hardware reliability is usually specified as

MTBO (system failure). The system planner wants to know

what the equipment availability (or unavailability) is. If the

MTBO is one or two way, the availability is one or two way,

respectively. Notice that MTTR (a function of failure detection,

reporting, spare availability and maintenance technician training,

and reliability) has significant impact on achieved availability or

required MTBF.

The previous formulas may be applied to determine the

effect of different r adio configurations on radio system hardware

availability.

Multiline radio systems, as they use an unprotected radio

channel to protect multiple working radio channels, areinherently less reliable than hot standby or frequency diversity

configures. Cross-polarization multiline systems typically use

two protection channels, one for vertical channels and the other

for horizontal channels. These systems would be analyzed as

one for N, where N is the number of vertical or horizontal

working channels being protected.

3.5. SPARE U NIT R EQUIREMENTS

Maintainability is often defined as “The ability of an item,

understated conditions of use, to be restored to a state in which it

can perform its required functions when maintenance is

performed understated conditions and using prescribed

procedures and resources.” Usually, maintenance is performed by storing a number of repairable spare units (e.g., modules,

cards, plug-ins, or blades) in reserve. These units are used to

repair the system. The problem to be solved is to determine the

number of spare units N required to support a system of Q

operational units with a probability of success P.

For complicated systems with interacting failure

mechanisms, a Markov model analysis is necessary. For systems

with rare failures occurring with a constant rate λ, the m ean

number of equipment failures is given by λt (with λ and t as

previously defined). In this situation, a homogeneous Poisson

process model is generally applied.

Pn= for n = 0, 1, 2, 3,...

with λand t as previously defined

Pn= the probability that a failure occurs exactly n times in

the time interval 0 to t ;

S = the number of spares normally on hand;

PS= the probability of a successful repair when a failure

occurs;

= the probability that not more than n failures have

occurred in the time interval 0 to t ;

= the probability of having a spare available if S spares

are normally available.

3.6. BER ESTIMATION

The fundamental quality of digital payloads, which may

have a predefined number of states at any instant of time, is

characterized by the probability of message error. BER

estimation is a statistical measurement that attempts to measurecurrent average error performance within a defined confidence

range by measuring errors within a defined time interval. The

classical approach is to use binomial distribution sampling

theory applied to a predefined sample size [although there are

some time advantages on using the slightly more complicated

negative binomial sampling]. Direct binomial sampling theory

yields the following results.

Assume N digits (N > 104) have been transmitted and

E digits (E


11/12

The above is the intuitive result. Less intuitive is the

expected BER if N digits (N > 10 ) have been transmitted and

zero digits were found to be in error. Not finding an error in the

sampled data is not to say, an error would not be found if the

testing lasted long enough.

Median-unbiased maximum likelihood estimator (not

exceeded 50% of the time):

90% Confidence level biased-estimator (not exceeded 90% of

the time):


the time):


the time):

Sample Problem 1.

Consider a system consisting of two identical

communication set where at least one must be operational at the

conclusion of a mission for success. Both units are operational

throughout the mission except when a transition is made to the

failed state. A reliability block diagram is shown below:

The formula and estimations for specific situations are

listed in the table below.

Solution for Sample Problem 1:

Assume that the mission period for the system is 20 hours. The

reliability model is

R(t) = 2 e –λt – e –2λt

using the approximation shown in the table below:

R(t) = 1 – (λt)2

and substituting in values

R(20) = 1 – (0.0005 X 20)2

R(20) = 1 – (0.01)2

R(20) = 0.9999

Sample Problem 2.

If the probabilities for A, B, and C for success are the 0.9, 0.8

and 0.7, respectively, determine the system reliability of the

system functions, given the success diagrams for Function 1,

Function 2, and the system.



12/12

Function 1 = 0.9 + 0.8 – (0.9)(0.8) = 0.98

Function 2 = 0.8 + 0.7 – (0.8)(0.7) = 0.94

* System reliability cannot be derived by multiplying function

reliabilities because of the common element B

System Reliability ≠ (0.98)(0.94) = 0.9212

System Reliability = PB + PA PC – PA PB PCSystem Reliability = 0.8 + (0.9)(0.7) – (0.9)(0.8)(0.7)

System Reliability = 0.926

Sample Problem 3:

Consider the availability model:

Availability=

where MTBF is mean time between failures and MTTR is mean

time to repair. Assume that the MTBF of the system is 100 hours

and the MTTR is 0.5 hours. Calculate the availability using the

approximation form.


A=

A= 1-

A= 1-

A= 0.995

A = 0.995

Sample Problem 4.

Consider the availability model for two-unit redundant system

with the standby unit operational.

Where is the failure rate of each unit and is the repair of

each unit. This equation is of the same form as

A=

Let equal 2X10-2 failures per hour and let equal one repair

per hour. Determine the availability.


In this case, the approximation form gives

Sample Problem 5:

Given the system below, determine the network failure rate.


REFERENCE[1] G. Kizer, Digital Microwave Communication:

Engineering Point-to-Point Microwave Systems. New Jersey,

John Wiley & Sons, Inc., 2013.

R EFERENCES

[1] R. L. Feeman, Fundamentals of Telecommunications, Wiley

1999.

[2] D. Blake, Communications.

[3] G. Kizer, Digital Microwave System, Wiley, 2013.

[4] T. Manning, Microwave Radio Transmission Design Guide

Artech House, 2009.

[5] Theodore S. Rappaport, Wireless Communications:

rinciples and Practice, 2nd ed.: Prentice Hall, 2002.

[6] G. Kizer, Digital Microwave Communication

Engineering Point-to-Point Microwave Systems. New

Jersey, John Wiley & Sons, Inc., 2013.

jimenez, gonzales, serrano

Documents