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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
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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
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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]
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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%.
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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
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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:
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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.
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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.
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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
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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
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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):
95% Confidence level biased-estimator (not exceeded 95% of
the time):
99% Confidence level biased-estimator (not exceeded 99% of
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.
Solution for Sample Problem 2:
8/21/2019 Jimenez, Gonzales, Serrano
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.
Solution for Sample Problem 3:
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.
Solution for Sample Problem 4:
In this case, the approximation form gives
Sample Problem 5:
Given the system below, determine the network failure rate.
Solution for Sample Problem 5:
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.