# Connexions

You are here: Home » Content » Principles of Digital Communications » Diversity Techniques

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Tuan Do-Hong. E-mail the author

# Diversity Techniques

Module by: Sinh Nguyen-Le, Tuan Do-Hong. E-mail the authorsEdited By: Sinh Nguyen-Le, Tuan Do-HongTranslated By: Sinh Nguyen-Le, Tuan Do-Hong

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

The bit-error-probability, PB¯PB¯ 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 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 PB(x)PB(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=xSNR=x size 12{ ital "SNR"=x} {}, where x=α2Eb/N0x=α2Eb/N0 size 12{x=α rSup { size 8{2} } {E rSub { size 8{b} } } slash {N rSub { size 8{0} } } } {}, and p(x)p(x) size 12{p $$x$$ } {} is the pdfpdf size 12{ ital "pdf"} {} of xx size 12{x} {} due to the fading conditions. With EbEb size 12{E rSub { size 8{b} } } {} and N0N0 size 12{N rSub { size 8{0} } } {} constant, αα size 12{α} {} is used to represent the amplitude variations due to fading.

For Rayleigh fading, αα size 12{α} {} has a Rayleigh distribution so that α2α2 size 12{α rSup { size 8{2} } } {}, and consequently xx size 12{x} {}, have a chi-squared distribution:

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

where Γ=α2¯Eb/N0Γ=α2¯Eb/N0 size 12{Γ= {overline {α rSup { size 8{2} } }} {E rSub { size 8{b} } } slash {N rSub { size 8{0} } } } {} is the SNRSNR size 12{ ital "SNR"} {} averaged through the “ups and downs” of fading. If each diversity (signal) branch, i=1, 2, ..., Mi=1, 2, ..., M size 12{i=1," 2, " "." "." "." ", "M} {}, has an instantaneous SNR=γiSNR=γi size 12{ ital "SNR"=γ rSub { size 8{i} } } {}, and we assume that each branch has the same average SNRSNR size 12{ ital "SNR"} {} given by ΓΓ size 12{Γ} {}, then

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

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

P ( γ i γ ) = 0 γ p ( γ i ) i = 0 γ 1 Γ exp ( γ i Γ ) i 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 ) = 1 exp ( γ Γ ) size 12{ {}=1 - "exp" $$- { {γ} over {Γ} }$$ } {}

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

P ( γ 1 , . . . , γ M γ ) = 1 exp ( γ Γ ) M 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>γSNR>γ size 12{ ital "SNR">γ} {} is:

P ( γ i > γ ) = 1 1 exp ( γ Γ ) M 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 SNRSNR size 12{ ital "SNR"} {} is Γ=20 dBΓ=20 dB size 12{Γ="20"" dB"} {}, determine the probability that all four branches are received simultaneously with an SNRSNR size 12{ ital "SNR"} {} less than 10 dB10 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 dBγ=10 dB size 12{γ="10"" dB"} {}, and γ/Γ=10 dB20 dB=10 dB=0.1γ/Γ=10 dB20 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

SNRSNR size 12{ ital "SNR"} {} will drop below 10 dB10 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 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 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 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 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"} {}

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks