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This section shows the error-performance improvements that can be obtained with the use of diversity techniques.

The bit-error-probability, P B ¯ size 12{ {overline {P rSub { size 8{B} } }} } {} ­­­­, averaged through all the “ups and downs” of the fading experience in a slow-fading channel is as follows:

P B ¯ = P B ( x ) p ( x ) dx size 12{ {overline {P rSub { size 8{B} } }} = Int {P rSub { size 8{B} } \( x \) p \( x \) ital "dx"} } {}

where P B ( x ) size 12{P rSub { size 8{B} } \( x \) } {} is the bit-error probability for a given modulation scheme at a specific value of SNR = x size 12{ ital "SNR"=x} {} , where x = α 2 E b / N 0 size 12{x=α rSup { size 8{2} } {E rSub { size 8{b} } } slash {N rSub { size 8{0} } } } {} , and p ( x ) size 12{p \( x \) } {} is the pdf size 12{ ital "pdf"} {} of x size 12{x} {} due to the fading conditions. With E b size 12{E rSub { size 8{b} } } {} and N 0 size 12{N rSub { size 8{0} } } {} constant, α size 12{α} {} is used to represent the amplitude variations due to fading.

For Rayleigh fading , α size 12{α} {} has a Rayleigh distribution so that α 2 size 12{α rSup { size 8{2} } } {} , and consequently x size 12{x} {} , have a chi-squared distribution :

p ( x ) = 1 Γ exp ( x Γ ) size 12{p \( x \) = { {1} over {Γ} } "exp" \( - { {x} over {Γ} } \) } {} x 0 size 12{x>= 0} {}

where Γ = α 2 ¯ E b / N 0 size 12{Γ= {overline {α rSup { size 8{2} } }} {E rSub { size 8{b} } } slash {N rSub { size 8{0} } } } {} is the SNR size 12{ ital "SNR"} {} averaged through the “ups and downs” of fading. If each diversity (signal) branch, i = 1, 2, . . . , M size 12{i=1," 2, " "." "." "." ", "M} {} , has an instantaneous SNR = γ i size 12{ ital "SNR"=γ rSub { size 8{i} } } {} , and we assume that each branch has the same average SNR size 12{ ital "SNR"} {} given by Γ size 12{Γ} {} , then

p ( γ i ) = 1 Γ exp ( γ i Γ ) size 12{p \( γ rSub { size 8{i} } \) = { {1} over {Γ} } "exp" \( - { {γ rSub { size 8{i} } } over {Γ} } \) } {} γ i 0 size 12{γ rSub { size 8{i} }>= 0} {}

The probability that a single branch has SNR size 12{ ital "SNR"} {} less than some threshold γ size 12{γ} {} is:

P ( γ i γ ) = 0 γ p ( γ i ) i = 0 γ 1 Γ exp ( γ i Γ ) i size 12{P \( γ rSub { size 8{i} }<= γ \) = Int rSub { size 8{0} } rSup { size 8{γ} } {p \( γ rSub { size 8{i} } \) dγ rSub { size 8{i} } = Int rSub { size 8{0} } rSup { size 8{γ} } { { {1} over {Γ} } "exp" \( - { {γ rSub { size 8{i} } } over {Γ} } \) dγ rSub { size 8{i} } } } } {}

= 1 exp ( γ Γ ) size 12{ {}=1 - "exp" \( - { {γ} over {Γ} } \) } {}

The probability that all M size 12{M} {} independent signal diversity branches are received simultaneously with an SNR size 12{ ital "SNR"} {} less than some threshold value γ size 12{γ} {} is:

P ( γ 1 , . . . , γ M γ ) = 1 exp ( γ Γ ) M size 12{P \( γ rSub { size 8{1} } , "." "." "." ,γ rSub { size 8{M} }<= γ \) = left [1 - "exp" \( - { {γ} over {Γ} } \) right ] rSup { size 8{M} } } {}

The probability that any single branch achieves SNR > γ size 12{ ital "SNR">γ} {} is:

P ( γ i > γ ) = 1 1 exp ( γ Γ ) M size 12{P \( γ rSub { size 8{i} }>γ \) =1 - left [1 - "exp" \( - { {γ} over {Γ} } \) right ] rSup { size 8{M} } } {}

This is the probability of exceeding a threshold when selection diversity is used.

Example: Benefits of Diversity

Assume that four-branch diversity is used, and that each branch receives an independently Rayleigh-fading signal. If the average SNR size 12{ ital "SNR"} {} is Γ = 20 dB size 12{Γ="20"" dB"} {} , determine the probability that all four branches are received simultaneously with an SNR size 12{ ital "SNR"} {} less than 10 dB size 12{"10"" dB"} {} (and also, the probability that this threshold will be exceeded).

Compare the results to the case when no diversity is used.

Solution

With γ = 10 dB size 12{γ="10"" dB"} {} , and γ / Γ = 10 dB 20 dB = 10 dB = 0 . 1 size 12{ {γ} slash {Γ} ="10"" dB" - "20"" dB"= - "10"" dB"=0 "." 1} {} , we solve for the probability that the

SNR size 12{ ital "SNR"} {} will drop below 10 dB size 12{"10"" dB"} {} , as follows:

P ( γ 1 , γ 2 , γ 3 , γ 4 10 dB ) = 1 exp ( 0 . 1 ) 4 = 8 . 2 × 10 5 size 12{P \( γ rSub { size 8{1} } ,γ rSub { size 8{2} } ,γ rSub { size 8{3} } ,γ rSub { size 8{4} }<= "10"" dB" \) = left [1 - "exp" \( - 0 "." 1 \) right ] rSup { size 8{4} } =8 "." 2 times "10" rSup { size 8{ - 5} } } {}

or, using selection diversity, we can say that

P ( γ i > 10 dB ) = 1 8 . 2 × 10 5 = 0 . 9999 size 12{P \( γ rSub { size 8{i} }>"10"" dB" \) =1 - 8 "." 2 times "10" rSup { size 8{ - 5} } =0 "." "9999"} {}

Without diversity,

P ( γ 1 10 dB ) = 1 exp ( 0 . 1 ) 1 = 0 . 095 size 12{P \( γ rSub { size 8{1} }<= "10"" dB" \) = left [1 - "exp" \( - 0 "." 1 \) right ] rSup { size 8{1} } =0 "." "095"} {}

P ( γ 1 > 10 dB ) = 1 0 . 095 = 0 . 905 size 12{P \( γ rSub { size 8{1} }>"10"" dB" \) =1 - 0 "." "095"=0 "." "905"} {}

Questions & Answers

how can chip be made from sand
Eke Reply
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Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
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Google
da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Source:  OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1
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