MAP Symbol Dete tion Based on the MM De omposition100

3.3 Algorithms Based on MM De omposition

3.3.2 MAP Symbol Dete tion Based on the MM De omposition100

De omposi-tion

We represent the re eived signal (3.1) onto an orthonormal base and denote

r

^its ^ve
tor representation. Exploiting the onstant envelope of the CPM signal, we anwrite [48℄

p(r|x) = p(r|x, ψ) ∝ ^∼ Y

n

G _n (x _n , ψ _n−1 )

^(3.10)

where

ψ = {ψ n }

^and

G _n (x _n , ψ _n−1 ) = exp ( 1

N ₀ Re

" _{M −2} X

k=0

p _k,n a ^∗ _k,n

#)

(3.11)

p _k,n = r(t)⊗p k (−t)| t=nT

^being^theôutputôfâ^lter^mat
hed^to^the^pulse

p _k (t)

sampled at time

nT

^. ^The ^symbol

∝ ^∼

^has ^been ^used ⁱⁿ ^(3.10) ^to ^denote ^an

approximate proportionality relationship. The approximation here is related

to the fa t thatwe are onsidering the prin ipal omponents only. We noti e

that the fun tions

G _n (x _n , ψ _n−1 )

^are ^not probability density fun tions (pdfs) nor they are proportional to pdfs. In the arguments of

G _n

^, ^we ^omitted ^the

dependen eonthere eivedsignalandexploited theabovementionedproperty

thatallsymbols

{a k,n } ^{M −2} k=0

ⁱⁿ^(3.11) ^depend^on

x _n

^and

ψ _n−1

ônly^.Înôrder^to

implement theMAPsequen edete tionstrategy,themaximizationof thepdf

p(r|x)

ân ^be ^performed ^by ûsing ^the ^Viterbi âlgorithm ^with ^bran
h ^metri
s

In [48℄ the reader will nd the so- alled likelihood fun tion, whi h is proportional to

ln p(r|x)

given by

ln G n (x n , ψ _n−1 ) ∝ Re[ P _{M −2}

k=0 p _k,n a ^∗ _k,n ]

^. ^Hen
e, ^the ^resulting ^re
eiver

works on a trellis whose state is dened by

ψ _n−1

^, ^and ^thus ^the ^number ^of

trellisstatesis

p

^[48℄.^So,^it^is ^veryinteresting to notethatthedete torbased onMMde ompositionrstofallredu estheFE omplexitybyapproximating

the signal with only its prin ipal omponents and, asse ondary ee t of FE

redu tion, an automati trellis state redu tion also arises. Among all CPM

signalde ompositions we willpresent,theMMistheonlyone whi h attainsa

DA omplexityredu tion.

Inthefollowing,weextendthete hniquefor omplexityredu tionofMAP

sequen etoMAPsymboldete tion(seealso[53℄).Thejoint aposteriori

prob-ability massfun tion(pmf)

P (x, ψ|r)

ân^be êxpressedâs

P (x, ψ|r) ∝ p(r|x, ψ)P (ψ|x)P (x) .

^(3.12)

By using(3.10) and observing thatwe an further fa tor theterms

P (x)

^and

P (ψ|x)

ⁱⁿ^(3.12) ^as

P (x) =

N −1 Y

n=0

P (x _n )

^(3.13)

P (ψ|x) = P (ψ −1 )

N −1 Y

n=0

I _n (ψ _n , ψ _n−1 , x _n )

^(3.14)

where

I _n (ψ _n , ψ _n−1 , x _n )

îs ân îndi
ator ^fun
tion ^to ône îf

x _n

ψ _n−1

^, ^and

ψ _n

respe tthe onstraint (3.9) and equal to zerootherwise, we obtain

P (x, ψ|r) ∝ P (ψ ^∼ −1 )

N −1 Y

n=0

I _n (ψ _n , ψ _n−1 , x _n )P (x _n )G _n (x _n , ψ _n−1 ) .

^(3.15)

The orrespondingFG,depi tedinFig.3.2,is y le-free.Hen e,theappli ation

oftheSPAwithanon-iterative forward-ba kward s hedule,produ estheexa t

marginal a posteriori probabilities (APPs) of symbols

{x n }

^(ex
ept ^for ^the

approximation related to the use of the prin ipal omponents only). In the

gure, we dened

P _e (x _n )

^as ^the ^extrinsi information on

x _n

^, ^i.e.,

P _e (x _n ) =

P (x _n |r)/P (x n )

^. ^With ^referen
e ^to ^the ^messages ⁱⁿ ^Fig. ^3.2, ^by ^applying ^the

x _n µ _b,n−1

P (x _n ) P (x n ) P _e (x n )

G _n I _n ψ _n

ψ _n−1

µ _f,n

Figure3.2:Fa torgraph orrespondingtoeqn.(3.15),inthe

n

^-th^time^interval.

updating rules of the SPA, messages

µ _f,n (ψ _n )

^and

µ _b,n (ψ _n )

^an ^be ^omputed

bymeansof thefollowing forward and ba kward re ursions

µ _f,n (ψ n ) = X

ψ n−1

X

x n

µ _f,n−1 (ψ _n−1 )G n (x n , ψ _n−1 )I n (ψ n , ψ _n−1 , x n )P (x n )

= X

x n

µ _f,n−1 ( ˇ ψ _n−1 )G _n (x _n , ˇ ψ _n−1 )P (x _n )

^(3.16)

µ _b,n−1 (ψ _n−1 ) = X

ψ n

X

x n

µ _b,n (φ _n )G _n (x _n , φ _n−1 )I _n (ψ _n , ψ _n−1 , x _n )P (x _n )

= X

x n

µ _b,n ( ˇ ψ ˇ _n )G _n (x _n , ψ _n−1 )P (x _n )

^(3.17)

where in(3.16)

ψ ˇ _n−1 = [ψ _n − ¯x n ] _p

(i.e., given

ψ _n

^and

x _n

ψ ˇ _n−1

^is ^su
h ^that

I _n (ψ _n , ˇ ψ _n−1 , x _n ) = 1

^), ^whereas ⁱⁿ

(3.17)

ˇ ˇ

ψ _n = [ψ _n−1 + ¯ x _n ] _p

(i.e., given

ψ _n−1

^and

x _n

ψ ˇ ˇ _n

^is ^su
h ^that

I _n ( ˇ ψ ˇ _n , ψ _n−1 , x _n ) = 1

^), ^and ^with^the

following initial onditions

µ _f,−1 (ψ ₋₁ ) = P (ψ ₋₁ )

^(3.18)

µ _{b,N −1} (ψ _{N −1} ) = 1/p .

^(3.19)

Finally,the extrinsi APPsofsymbols

{x n }

ân ^beômputed ^by^meansôf^the

following ompletion

P _e (x _n ) = X

ψ n−1

X

ψ n

µ _f,n−1 (ψ _n−1 )µ _b,n (φ _n )I _n (ψ _n , ψ _n−1 , x _n )G _n (x _n , ψ _n−1 )

= X

ψ n

µ _f,n−1 ( ˇ ψ _n−1 )µ _b,n (ψ _n )G _n (x _n , ˇ ψ _n−1 ) .

^(3.20)

Hen e, as already stated, the MM dete tion algorithm not only redu es the

FE omplexity by employing just the lters mat hed to the prin ipal pulses,

butalsotheDA omplexityisredu edbyafa torequalto

pM ^L−1 /p = M ^L−1

thatisthe ratiobetweenthenumberofstatesoftheoptimalre eiverandthat

of the proposed re eiver. Among all CPM signal de ompositions, the MM is

theonly one bringingto aDA omplexity redu tion.

Extension of the Algorithm

When the onsidered CPM format is su h that

L > 2

^, ^several ^simulation

resultsshowthatthe redu ed- omplexitydete torbasedontheprin ipal

om-ponents exhibits a signi ant performan e loss with respe t to the optimal

dete tor. Thisisdueto thepresen eof somenon-prin ipal omponents ofthe

MM representation, denoted as se ondary omponents, with a non negligible

power. In other words, the performan e degradation is not due to the

dete -tion algorithm but to the ina urate signal approximation. In parti ular, for

all onsideredCPMformatswith

L = 3

^and

M = 4

^,^we ^found^eight ^se
ondary

pulses

{p k (t)}

^with^anon-negligiblepower,aswe anobservefromtheexample reportedinFig.3.3.Whenthesese ondary omponentsaretakenintoa ount,

the fun tions

G _n

ⁱⁿ ^(3.11) ^depend ^also ^on

x _n−1

^,^sin
e ^symbols

{a k,n }

^related

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 0.5 1 1.5 2 2.5 3 3.5 4

t/T

components Principal

components Secondary p k (t )

Figure 3.3:Signal pulsesfor a 3RC modulation with

h = 2/7

^and

M = 4

to the onsidered sele tedse ondary omponents requiretheknowledgeofthe

pastinformation symbol

x _n−1

^[16℄.^Hen
e,

G _n

ⁱⁿ^(3.11) ^is^repla
ed^by

G ^′ _n (x _n , x _n−1 , ψ _n−1 ) = exp

( 1 N ₀ Re

"

X

k∈K

p _k,n a ^∗ _k,n

#)

(3.21)

where

K

îs ^the ^set ôf âll ônsidered ^pulses, î.e., ^the

M − 2

^prin
ipal ^pulses

and the

8

^sele
ted ^se
ondary^pulses. ^Thus, ^extending^the^derivation^des
ribed

before, we an derive a forward-ba kward dete tion algorithm in whi h the

trellisstate isdened asthe ouple

(ψ _n−1 , x _n−1 )

^and ^whose^FG^is ^provided ⁱⁿ

Fig 3.4. In this way, we obtain a soft-output dete tor working on

pM

^states,

with a redu tion fa tor with respe t to the optimal s heme equal to

M ^L−2

Hen e, even ifthe FE and DA omplexityredu tion isnot lower than thatin

the ase of MMalgorithm withjust theprin ipal pulses, MM algorithm with

some se ondary pulsesallows us to obtain more or lessthesame performan e

of the FC dete tor when

L = 3

^with ^a ^omplexity ^redu
tion ⁱⁿ ^both ^the ^FE

and DA.

x _n

P (x _n ) G ^′ _n I _n

P _e (x n ) (ψ _n−1 , x _n−1 ) µ _f,n

µ _b,n−1

(ψ _n , x _n )

P (x n )

Figure 3.4: Fa tor graph orresponding to MM based algorithm with some

sele ted se ondarypulses, inthe

n

^-th^time ^interval.

Afurtherredu tion anbeobtainedbyresortingtoaproperredu ed-sear h

te hnique,similartotheFTalgorithmwewilldes ribeinSe tion3.6,explained

inthefollowing.Atea htimeepo h

n

^,^we^sele
t

p

^forward^metri
s^and^explore

onlythepathsextendingfromtherelatedstates.Then,onlythepathssele ted

in this way are onsidered while performing the ba kward re ursion and the

ompletion stage.Inthis ase,theredu tionfa torwithrespe ttotheoptimal

s hemeisequalto

M ^L−1

^,êxa
tlyâsⁱⁿ^the^basi^versionôf^theâlgorithm,^that

is when only the prin ipal omponents are onsidered. As a riterion for the

redu ed sear h, we found onvenient not to sele t the

p

^states

(ψ _n−1 , x _n−1 )

with the best forward metri s, but to sele t, for ea h of the possible

p

^values

ψ _n−1

^,^the^value^of

x _n−1

^providing^the^best^metri
.

3.4 Complexity-Redu tion Based on the G

De om-position

As already anti ipated inSe tion 3.3.2, in ontrast withthe MM

de omposi-tion, the CPMde ompositions in [42,54,56,57℄ allow to redu e thefront end

omplexity but not the state omplexity of the DA. They have in ommon

that the redu tion in the front end omplexity is rea hed by omputing the

su ient statisti s in(2.12) using alternative representations of the omplex

envelopeofaCPMsignal.TheCPMdete tor,i.e.,theDA,remainsun hanged

withrespe tto the FCdete tor.

Nel documento Advanced Modulation/Demodulation Schemes For Wireless Communications (pagine 116-122)

MAP Symbol Dete tion Based on the MM De omposition100

3.3 Algorithms Based on MM De omposition

3.3.2 MAP Symbol Dete tion Based on the MM De omposition100

r

p(r|x) = p(r|x, ψ) ∝ ∼ Y

n

G n (x n , ψ n−1 )

ψ = {ψ n }

G n (x n , ψ n−1 ) = exp ( 1

N 0 Re

" M −2 X

k=0

p k,n a ∗ k,n

#)

p k,n = r(t)⊗p k (−t)| t=nT

p k (t)

nT

∝ ∼

G n (x n , ψ n−1 )

G n

{a k,n } M −2 k=0

x n

ψ n−1

p(r|x)

ln p(r|x)

ln G n (x n , ψ n−1 ) ∝ Re[ P M −2

k=0 p k,n a ∗ k,n ]

ψ n−1

p

P (x, ψ|r)

P (x, ψ|r) ∝ p(r|x, ψ)P (ψ|x)P (x) .

P (x)

P (ψ|x)

P (x) =

N −1 Y

n=0

P (x n )

P (ψ|x) = P (ψ −1 )

N −1 Y

n=0

I n (ψ n , ψ n−1 , x n )

I n (ψ n , ψ n−1 , x n )

x n

ψ n−1

ψ n

P (x, ψ|r) ∝ P (ψ ∼ −1 )

N −1 Y

n=0

I n (ψ n , ψ n−1 , x n )P (x n )G n (x n , ψ n−1 ) .

{x n }

P e (x n )

x n

P e (x n ) =

P (x n |r)/P (x n )

x n µ b,n−1

P (x n ) P (x n ) P e (x n )

G n I n ψ n

ψ n−1

µ f,n

n

µ f,n (ψ n )

µ b,n (ψ n )

µ f,n (ψ n ) = X

ψ n−1

X

x n

µ f,n−1 (ψ n−1 )G n (x n , ψ n−1 )I n (ψ n , ψ n−1 , x n )P (x n )

= X

x n

µ f,n−1 ( ˇ ψ n−1 )G n (x n , ˇ ψ n−1 )P (x n )

µ b,n−1 (ψ n−1 ) = X

ψ n

X

x n

µ b,n (φ n )G n (x n , φ n−1 )I n (ψ n , ψ n−1 , x n )P (x n )

= X

x n

µ b,n ( ˇ ψ ˇ n )G n (x n , ψ n−1 )P (x n )

ψ ˇ n−1 = [ψ n − ¯x n ] p

ψ n

p(r|x) = p(r|x, ψ) ∝ ^∼ Y

G _n (x _n , ψ _n−1 )

G _n (x _n , ψ _n−1 ) = exp ( 1

N ₀ Re

" _{M −2} X

p _k,n a ^∗ _k,n

p _k,n = r(t)⊗p k (−t)| t=nT

p _k (t)

∝ ^∼

G _n (x _n , ψ _n−1 )

G _n

{a k,n } ^{M −2} k=0

x _n

ψ _n−1

ln G n (x n , ψ _n−1 ) ∝ Re[ P _{M −2}

k=0 p _k,n a ^∗ _k,n ]

ψ _n−1

P (x _n )

I _n (ψ _n , ψ _n−1 , x _n )

I _n (ψ _n , ψ _n−1 , x _n )

x _n

ψ _n−1

ψ _n

P (x, ψ|r) ∝ P (ψ ^∼ −1 )

I _n (ψ _n , ψ _n−1 , x _n )P (x _n )G _n (x _n , ψ _n−1 ) .

P _e (x _n )

x _n

P _e (x _n ) =

P (x _n |r)/P (x n )

x _n µ _b,n−1

P (x _n ) P (x n ) P _e (x n )

G _n I _n ψ _n

ψ _n−1

µ _f,n

µ _f,n (ψ _n )

µ _b,n (ψ _n )

µ _f,n (ψ n ) = X

µ _f,n−1 (ψ _n−1 )G n (x n , ψ _n−1 )I n (ψ n , ψ _n−1 , x n )P (x n )

µ _f,n−1 ( ˇ ψ _n−1 )G _n (x _n , ˇ ψ _n−1 )P (x _n )

µ _b,n−1 (ψ _n−1 ) = X

µ _b,n (φ _n )G _n (x _n , φ _n−1 )I _n (ψ _n , ψ _n−1 , x _n )P (x _n )

µ _b,n ( ˇ ψ ˇ _n )G _n (x _n , ψ _n−1 )P (x _n )

ψ ˇ _n−1 = [ψ _n − ¯x n ] _p

ψ _n

x _n

ψ ˇ _n−1

I _n (ψ _n , ˇ ψ _n−1 , x _n ) = 1

ψ _n = [ψ _n−1 + ¯ x _n ] _p

ψ _n−1

x _n

ψ ˇ ˇ _n

I _n ( ˇ ψ ˇ _n , ψ _n−1 , x _n ) = 1

µ _f,−1 (ψ ₋₁ ) = P (ψ ₋₁ )

µ _{b,N −1} (ψ _{N −1} ) = 1/p .

P _e (x _n ) = X

µ _f,n−1 (ψ _n−1 )µ _b,n (φ _n )I _n (ψ _n , ψ _n−1 , x _n )G _n (x _n , ψ _n−1 )

µ _f,n−1 ( ˇ ψ _n−1 )µ _b,n (ψ _n )G _n (x _n , ˇ ψ _n−1 ) .

pM ^L−1 /p = M ^L−1

G _n

x _n−1

x _n−1

G _n

G ^′ _n (x _n , x _n−1 , ψ _n−1 ) = exp

( 1 N ₀ Re

p _k,n a ^∗ _k,n

(ψ _n−1 , x _n−1 )

M ^L−2

x _n

P (x _n ) G ^′ _n I _n

P _e (x n ) (ψ _n−1 , x _n−1 ) µ _f,n

µ _b,n−1

(ψ _n , x _n )

M ^L−1

(ψ _n−1 , x _n−1 )

ψ _n−1

x _n−1