Bayesian Algorithms based on double-AR1 Phase Noise

4.4 Bayesian Algorithms

4.4.4 Bayesian Algorithms based on double-AR1 Phase Noise

Model

All Bayesian algorithms derived until now, are arried out by onsidering a

rstorderphasenoisemodel,i.e.,adis rete-time Wienermodel.Ontheother

hand, sometimes hannels of pra ti al interest are ae ted by a phase noise

whi h anbemodeledasthesumoftworstorderauto-regressivelters.This

is the ase of the SATMODE phase noise, whi h impairs satellite

ommuni- ations hara terized bylow- ost equipment, andwhose double-AR1modelis

des ribed inSe tion 2.4.2. In Chapter 2, numeri al results show that the

de-te tor deriveduntil now, basedonjustasingle-AR1PNgeneration,exhibitsa

signi ant performan e degradation when employed on double-AR1 PN

gen-eration (see Se tion 2.4.4). Thus, it an be useful to derive some Bayesian

algorithms basedon su hdouble-AR1 PN model, asinthe aseof the

D-DP-BCJRalgorithmdes ribedinSe tion2.4.3.Se tion2.4.4,demonstratethatthe

information rate lossa hieved bythese algorithms withrespe t to the

oher-ent ase, is lose to zero. However, the good result in performan e is rea hed

at the pri e of an in reased trellis state omplexity sin e the system state is

omposedbytheunionoftheCPMstate,withthestateofboththetwophase

noise omponents.Thusthesealgorithms annotbetakenintoa ountfor

sys-temsofpra ti alinterest.FinallyinSe tion2.4.3,wealsoproposeanimproved

version of the DP algorithm (the so alled I-DP), suitable when onsidering

double-AR1 PN with a very fast omponent, whi h an be approximated as

jitter,independent fromsampleto sample.ThetrellisoftheI-DPalgorithm is

basedon thefull- omplexityCPMtrellisbasedon theRimoldide omposition

and thus is omposed by

D pM ^L−1

^states. ^We ^an ^redu
e ^the ^omplexity ^by

applying the same te hnique des ribed in Se tion 2.4.3 to the MM-DP

algo-rithm of Se tion 4.4.2, whi h is based on the MM de omposition and ounts

just

D

^trellis ^states. ^The ^way ^to ^obtain ^the Improved-MM-DP (I-MM-DP) starting from MM-DP is very similar to thatused to derive the I-DPin

Se -tion 2.4.3.Inparti ular, rstlywe justneed to hange themean and varian e

of the Gaussian pdf in (4.22) representing phase noise probability transition

p(θ n+1 |θ ⁿ )

^, ^sin
e ^the ^PN

θ n

îs ^no ^more ^modeled âs â ^Wiener ^model ^but âs

an AR1 pro ess. Se ondly, in order to take into a ount also the se ond PN

omponent

θ _f,n

^(i.e., ^the ^faster omponent), we ompute the bran h metri

H _n (x _n , ψ _n−1 , θ _n )

ⁱⁿ^(3.11) ^byâveraging ôver ^su
h â ômponent. ^Sin
e ^su
h â

omponent is a zero mean Gaussian random variable, with varian e

σ ² _f

^, ^the

bran h metri be omes:

H n (x n , ψ _n−1 , θ n ) = Z

θ f,n

exp (

− θ _f,n ² 2σ _f ²

)

· exp ( 1

N 0

Re

" _{M −2} X

k=0

p _k,n a ^∗ _k,n (x _n , ψ _n−1 )e ^−j(θ ⁿ ^+θ ^f,n ⁾

#)

dθ _f,n .

^(4.45)

wheretheintegralin(4.45) anbeevaluatedasasimplesumbydis retizingthe

faster phase omponent

θ _f,n

^as ^done ^for ^the^slower ^omponent

θ _n

^.^Hen
e ^the

I-MM-DP algorithm isobtained byjust in reasing thebran h metri

omple-tionoftheMM-DPalgorithm.However,Se tion 2.4.4showthatimproved-DP

algorithm doesnot get signi ant information rateimprovement with respe t

to the simple DP algorithm. We will verify this statement in theSe tion 4.5,

bynumeri al results.

Dete tion Algorithm Based on Phase Noise Linear Predi tion

In the following, we propose a Bayesian algorithm basedon linearpredi tion

of the phase noise pro ess. In parti ular, we extend the approximate linear

predi tive approa hdes ribed in[71℄,fora generaltime-varyingphasepro ess

θ _n

^.^W^e^assume

θ _n

^stationary^and^des
ribed^by^a^givenauto orrelationsequen e of thephasorpro ess

h _n = e ^jθ ⁿ

^,^denoted ^by

R _h (k) = E n

e ^jθ ^n+k ^e ^−jθn o

.

^(4.46)

Firstofall,wefo usontheve tor

y

^whi
h^olle
ts^the^set

y _n (x _n , σ _n )

^of

suf- ient statisti softhere eivedsignal

r(t)

^,^for ^the^ideal^oherent ^dete
tor^(see

Se tion4.2).Thesestatisti saregeneratedasindi atedin(2.12),i.e.,byabank

of lters mat hed to all possible length-

T

^CPM ^waveforms

{¯s(t; x n , ω _n )}

^. ^In

thepresentSe tion, wepropose adierent setof su ient statisti s, olle ted

inthe ve tor

r

^,^obtained ^proje
ting^the^re
eived ^signal

r(t)

^over^analternative orthonormal base. In detail, the statisti s are obtained by oversampling the

re eived signal by an oversampling fa tor

N _s

^whi
h ^respe
ts ^theanti-aliasing

ondition;hen e,byoversamplingthesignal

r(t)

ⁱⁿ^(4.3)^at

t _n,k = n T +k T /N s

we get

r _n,k = es n,k (x _n , σ _n )e ^jθ ⁿ + w _n,k , n = 0, 1, . . . , N − 1 k = 0, 1, . . . , N _s − 1

(4.47)

where we have dened

r _n,k , r(t _n,k )

e s _n,k (x _n , σ _n ) , es(t n,k ; x _n , σ _n )

^and

w _n,k , w(t _n,k )

^. ^In ^other ^words,

n

^is ^the ^symbol ^index, ^while

k

^is ^the oversampling index.Itistrivialtodemonstratethat

{w n,k } n,k

^areindependentanduniformly distributed omplexGaussianrandomvariables,withzeromeansandvarian e

σ _w ² = 2 N ₀ N _s /T

^.^From^that onsideration, wendthefollowing expressionfor thepdf

p(r|x)

p(r|x) =

N −1 Y

n=0

exp (

− 1 σ _w ²

N X s −1 k=0

r n,k − es n,k (x _n , σ _n )e ^jθ ⁿ ²

)

.

^(4.48)

We dene

r _n,k ^′ (x n , σ n ) , r _n,k e

s _n,k (x n , σ n ) = r _n,k e s ^∗ _n,k (x n , σ n )

^(4.49)

sothat, repla ing(4.49) in(4.47), when onsidering thetransmitted sequen e

we get

r ^′ _n,k (x _n , σ _n ) = e ^jθ ⁿ + w ^′ _n,k

^(4.50)

where

w ^′ _n,k , w _n,k s e ^∗ _n,k (x n , σ n )

^isstatisti allyequivalentto

w _n,k

^.^Hen
e,^we^an

exploitthe re eived samples(4.50) toperform

e ^jθ ⁿ

estimation, basedonlinear predi tion ltering. We derive

e ^{j ˆ} ^θ ⁿ = P C

i=1 p _i r ^′ _n−i,0 (x _n−i , σ _n−i )

P _C

i=1 p _i r ^′ _n−i,0 (x _n−i , σ _n−i )

(4.51)

where

C

âssumes^the^meaningôf^predi
tionôrder ând

{p i } ^C i=1

^are^the^predi
tor

oe ients. The predi tor oe ients an be omputed by solving a

Wiener-Hopf linear system

R p = b

^, ^where

p , (p ₁ , p ₂ , . . . , p _C ) ^T

^is ^the ^unknown

ve tor 10

b = [R _h (1), R _h (2), . . . , R _h (C)] ^T

^and

R

^is ^a ^square

C × C

^matrix

Thenotation

(.) ^T

^indi
ates^thetranspositionoperator

whoseelementsaredened astheauto orrelationfun tionofthesamples

r _n,k ^′

and hasthe followingexpression

R(ℓ, m) =

( R _h (|ℓ − m|)

^if

ℓ 6= m R _h (0) + σ _w ²

^if

ℓ = m .

(4.52)

Repla ing(4.51) in(4.48), we derive

p(r|x) ≃

N −1 Y

n=0

exp (

− 1 σ _ǫ ²

N X s −1 k=0

|G n (x _n , . . . , x _n−C , σ _n , . . . , σ _n−C )| ² )

(4.53)

where

σ ² _ǫ

îs ^the ^mean ^square êrror ^predi
tion êrror, ^whi
h ân ^be êxpressed

as[71℄

σ ² _ǫ = R _h (0) + σ _w ² − X C

i=0

p _i R _h (i) .

^(4.54)

and

G _n (x _n , . . . , x _n−C , σ _n , . . . , σ _n−C ) , r _n,k −es n,k (x _n , σ _n ) P _C

i=1 p _i r _n−i,0 ^′ (x _n−i , σ _n−i )

P C

i=1 p _i r _n−i,0 ^′ (x _n−i , σ _n−i ) .

(4.55)

By employing (1.16), we an writethe

s e _n,k (x _n , σ _n )

^denition^as

e s _n,k (x _n , σ _n ) = ¯ s(t _n,k ; x _n , ω _n ) e ^jπ ⁿ

= ¯ s _n,k (x _n , ω _n ) e ^jπ ⁿ

^(4.56)

where

s(t ¯ _n,k ; x _n , ω _n )

^(1.17)îs^theômponentôf^the^CPM^waveform,^depending

on just the present symbol

x _k

^and ^on ^the orrelative state

ω n

^(1.4);

π n

^(1.5)

is the phasestate and we dened

s ¯ _n,k (x _n , ω _n ) , ¯ s(t _n,k ; x _n , ω _n )

^. ^Thus,

repla -ing(4.56)and (4.49)in(4.55),we ndthatthe

G _n (x _n , . . . , x _n−i , σ _n , . . . , σ _n−i )

fun tion be omes

G _n (x _n , . . . , x _n−i , σ _n , . . . , σ _n−i ) =

^(4.57)

= r _n,k − ¯s n,k (x _n , σ _n ) P _C

i=1 p _i r _n−i,0 ¯ s ^∗ _n−i,0 (x _n−i , ω _n−i ) e ^j(π ⁿ ^−π ⁿ⁻ⁱ ⁾

P _C

i=1 p _i r _n−i,0 s ¯ ^∗ _n−i,0 (x _n−i , ω _n−i )

= r _n,k − ¯s n,k (x _n , ω _n ) P _C

i=1 p _i r _n−i,0 s ¯ ^∗ _n−i,0 (x _n−i , ω _n−i ) e ^jπh ^P ⁱ⁻¹ ^m=0 ^x ^n−L−m

P C

i=1 p _i r _n−i,0 s ¯ ^∗ _n−i,0 (x _n−i , ω _n−i )

.

(4.58)

Sin ethe orrelative stateis omposedbythe

L − 1

^most ^re
ent ^past^symbol,

we understand that the fun tion

G _n

^just ^depends ^on ^the ^present ^symbol

x _n

andonasetof

L − 1 + C

^past^symbols^given^by

(x _n−1 , . . . , x _{n−(L−1+C)} )

^.^Thus,

bydeninga newsystemstate

µ _n , x _n−1 , . . . , x _{n−(L−1+C)}

(4.59)

we have

G n (x n , . . . , x _n−i , σ n , . . . , σ _n−i ) ≡ G ⁿ (x n , µ n ) ,

^(4.60)

and the pdf in(4.53) anbe written as

p(r|x) ≃

N −1 Y

n=0

V _n (x _n , µ _n )

^(4.61)

where

V _n (x _n , µ _n ) , exp (

− 1 σ ² _ǫ

N X s −1 k=0

|G n (x _n , µ _n )| ² )

(4.62)

Finally, we an derive the MAP symbol dete tion strategy by marginalizing

p(x, µ|y)

^(where^ve
tor

µ

^olle
ts

µ _n

^elements,^with

n

^from

0

^to

N − 1

⁾^by^FG

and SPA. Indetail, byBayesrule

p(x, µ|r) ∝ p(r|x, µ) P (µ|x) P (x)

= P (µ 0 )

N −1 Y

n=0

V n (x n , µ n )I(x n , µ n , µ n+1 )P (x n )

^(4.63)

where

I(x n , µ n , µ n+1 )

îs ^the^trellis îndi
ator ^fun
tion, êqual ^to ône îf

x n

µ n

and

µ _n+1

^satisfy^the ^trellis onstraints and to zero otherwise. It is lear that fa torization (4.63) is equivalent to the fa torization in (2.14) of the optimal

oherent dete tor, where here the state denition

µ _n

^repla
es ^the ^CPM^state

σ _n

^(and ^thus, âlso ^the ^trellis îndi
ator ^fun
tion îs ^dierent), ând ^where ^the

bran h metri

V n (x n , µ n )

^repla
es

F n (x n , σ n )

^.^Thus, ^forward ^re
ursion,

ba k-ward re ursion and ompletion stage are performed as des ribed in (2.17),

(2.18), (2.21), respe tively. From (4.59), we know that the BCJR algorithm

we have derived isbased on atrellis of

M ^L−1+C

^states. ^So,ît îs^lear ^that îts

omputational omplexity rapidly in rease with the parameters

L

^and

C

^. ^In

the present work,we do not address the problemof omplexity redu tionfor

the des ribed algorithm, but we think that a possible solution is represented

byredu ed-sear h te hniques(des ribed inSe tion3.6).

Predi tor Coe ients for Wiener and double-AR1 PN Models

In order to ompute the predi tor oe ients

p

^by ^the Wiener-Hopf linear system

R p = b

^, ^we ^need ^to ^assume ^a statisti al model for the phase noise

θ _n

^.^From^su
h ^model ^we ân ^derive ân êxpression^for

R _h (k)

^dened ⁱⁿ^(4.46),

ne essaryfor the omputation ofmatrix

R

^(4.52).

Ifwemodelthephasenoiseasadis rete-timeWienerpro essofin remental

varian e

σ _∆ ²

^,ît îsêasy ^to ^verify^that

R _h (k) = e ^−|k|σ ² ^∆ ^/2 .

^(4.64)

Ontheotherhand,whenwemodelthePNbythedouble-AR1modeldes ribed

inSe tion 2.4.2,we an derive

R _h (k)

âs^follows.^Firstôf âll,^we^re
all^that^the

hara teristi fun tion ofanyrandom variable

β

^is^given ^by

γ _β (t) = E n

e ^jtβ o

.

^(4.65)

Hen e,wenotefrom(4.46),that

R _h (k)

^is^equivalent^to^the hara teristi fun -tionoftherealrandomvariables

(θ _k+n −θ n )

^,^omputed^for

t = 1

^.^The

dis rete-timephasenoisepro ess

θ _n

^we^are onsidering,isthesumoftwoGaussianrst

order auto-regressive pro esses,

θ _a,n = a θ _a,n−1 + v _a,n θ _b,n = b θ _b,n−1 + v _b,n

where

v _a,n

^and

v _b,n

âre^two ^Gaussian ^zero-mean ^random ^variables ôf ^vârian
e

σ _v,a ²

^and

σ _v,b ²

^,respe tively.

θ a,n

^and

θ _b,n

^are^still ^Gaussian ^random^variables ^of

varian e

σ ² _a

^and

σ _b ²

^,respe tively.Thus,it is easy to prove that

(θ _k+n − θ n )

^is

stillaGaussianvariable,withmeanzeroandvarian e

2[R _θ (0) − R θ (k)]

^,^where

R _θ (k)

^is^the auto orrelation sequen e

R _θ (k) = E{θ n θ _n+k } .

^(4.66)

Thus, sin e the hara teristi fun tion of a zero-mean Gaussian random v

ari-able

β

^with^varian
e

σ ²

^is ^[72℄

γ _β (t) = e ^−σ ² ^/2

^(4.67)

we derive that

R _h (k) = e ^[R ^θ ^(k)−R ^θ ^(0)] .

^(4.68)

Finally,sin e the auto orrelationfun tion

R _θ (k)

^is^the ^sum ^of ^the

auto orre-lationfun tionsof its AR1 omponents

θ _a,n

^and

θ _b,n

^,^we ^get

R _h (k) = exp

σ ² _a a ^|k| + σ _b ² b ^|k|

exp

σ _a ² + σ _b ² .

^(4.69)

Nel documento Advanced Modulation/Demodulation Schemes For Wireless Communications (pagine 171-178)

Bayesian Algorithms based on double-AR1 Phase Noise

4.4 Bayesian Algorithms

4.4.4 Bayesian Algorithms based on double-AR1 Phase Noise

D pM L−1

D

p(θ n+1 |θ n )

θ n

θ f,n

H n (x n , ψ n−1 , θ n )

σ 2 f

H n (x n , ψ n−1 , θ n ) = Z

θ f,n

exp (

− θ f,n 2 2σ f 2

)

·

exp ( 1

N 0

Re

" M −2 X

k=0

p k,n a ∗ k,n (x n , ψ n−1 )e −j(θ n +θ f,n )

#)

dθ f,n .

θ f,n

θ n

θ n

θ n

h n = e jθ n

R h (k) = E n

e jθ n+k e −jθn o

.

y

y n (x n , σ n )

r(t)

T

{¯s(t; x n , ω n )}

r

r(t)

N s

r(t)

t n,k = n T +k T /N s

r n,k = es n,k (x n , σ n )e jθ n + w n,k , n = 0, 1, . . . , N − 1 k = 0, 1, . . . , N s − 1

r n,k , r(t n,k )

e s n,k (x n , σ n ) , es(t n,k ; x n , σ n )

w n,k , w(t n,k )

n

k

{w n,k } n,k

σ w 2 = 2 N 0 N s /T

p(r|x)

p(r|x) =

N −1 Y

n=0

exp (

− 1 σ w 2

N X s −1 k=0

r n,k − es n,k (x n , σ n )e jθ n 2

)

.

r n,k ′ (x n , σ n ) , r n,k e

s n,k (x n , σ n ) = r n,k e s ∗ n,k (x n , σ n )

r ′ n,k (x n , σ n ) = e jθ n + w ′ n,k

w ′ n,k , w n,k s e ∗ n,k (x n , σ n )

w n,k

e jθ n

e j ˆ θ n = P C

i=1 p i r ′ n−i,0 (x n−i , σ n−i )

P C

i=1 p i r ′ n−i,0 (x n−i , σ n−i )

C

{p i } C i=1

R p = b

p , (p 1 , p 2 , . . . , p C ) T

b = [R h (1), R h (2), . . . , R h (C)] T

R

C × C

(.) T

r n,k ′

R(ℓ, m) =

D pM ^L−1

p(θ n+1 |θ ⁿ )

θ _f,n

H _n (x _n , ψ _n−1 , θ _n )

σ ² _f

H n (x n , ψ _n−1 , θ n ) = Z

− θ _f,n ² 2σ _f ²

" _{M −2} X

p _k,n a ^∗ _k,n (x _n , ψ _n−1 )e ^−j(θ ⁿ ^+θ ^f,n ⁾

dθ _f,n .

θ _f,n

θ _n

θ _n

θ _n

h _n = e ^jθ ⁿ

R _h (k) = E n

e ^jθ ^n+k ^e ^−jθn o

y _n (x _n , σ _n )

{¯s(t; x n , ω _n )}

N _s

t _n,k = n T +k T /N s

r _n,k = es n,k (x _n , σ _n )e ^jθ ⁿ + w _n,k , n = 0, 1, . . . , N − 1 k = 0, 1, . . . , N _s − 1

r _n,k , r(t _n,k )

e s _n,k (x _n , σ _n ) , es(t n,k ; x _n , σ _n )

w _n,k , w(t _n,k )

σ _w ² = 2 N ₀ N _s /T

− 1 σ _w ²

r n,k − es n,k (x _n , σ _n )e ^jθ ⁿ ²

r _n,k ^′ (x n , σ n ) , r _n,k e

s _n,k (x n , σ n ) = r _n,k e s ^∗ _n,k (x n , σ n )

r ^′ _n,k (x _n , σ _n ) = e ^jθ ⁿ + w ^′ _n,k

w ^′ _n,k , w _n,k s e ^∗ _n,k (x n , σ n )

w _n,k

e ^jθ ⁿ

e ^{j ˆ} ^θ ⁿ = P C

i=1 p _i r ^′ _n−i,0 (x _n−i , σ _n−i )

P _C

i=1 p _i r ^′ _n−i,0 (x _n−i , σ _n−i )

{p i } ^C i=1

p , (p ₁ , p ₂ , . . . , p _C ) ^T

b = [R _h (1), R _h (2), . . . , R _h (C)] ^T

(.) ^T

r _n,k ^′

( R _h (|ℓ − m|)

ℓ 6= m R _h (0) + σ _w ²

− 1 σ _ǫ ²

|G n (x _n , . . . , x _n−C , σ _n , . . . , σ _n−C )| ² )

σ ² _ǫ

σ ² _ǫ = R _h (0) + σ _w ² − X C

p _i R _h (i) .

G _n (x _n , . . . , x _n−C , σ _n , . . . , σ _n−C ) , r _n,k −es n,k (x _n , σ _n ) P _C

i=1 p _i r _n−i,0 ^′ (x _n−i , σ _n−i )

i=1 p _i r _n−i,0 ^′ (x _n−i , σ _n−i ) .

s e _n,k (x _n , σ _n )

e s _n,k (x _n , σ _n ) = ¯ s(t _n,k ; x _n , ω _n ) e ^jπ ⁿ

= ¯ s _n,k (x _n , ω _n ) e ^jπ ⁿ

s(t ¯ _n,k ; x _n , ω _n )

x _k

s ¯ _n,k (x _n , ω _n ) , ¯ s(t _n,k ; x _n , ω _n )

G _n (x _n , . . . , x _n−i , σ _n , . . . , σ _n−i )

G _n (x _n , . . . , x _n−i , σ _n , . . . , σ _n−i ) =

= r _n,k − ¯s n,k (x _n , σ _n ) P _C

i=1 p _i r _n−i,0 ¯ s ^∗ _n−i,0 (x _n−i , ω _n−i ) e ^j(π ⁿ ^−π ⁿ⁻ⁱ ⁾

P _C

i=1 p _i r _n−i,0 s ¯ ^∗ _n−i,0 (x _n−i , ω _n−i )

= r _n,k − ¯s n,k (x _n , ω _n ) P _C

i=1 p _i r _n−i,0 s ¯ ^∗ _n−i,0 (x _n−i , ω _n−i ) e ^jπh ^P ⁱ⁻¹ ^m=0 ^x ^n−L−m

i=1 p _i r _n−i,0 s ¯ ^∗ _n−i,0 (x _n−i , ω _n−i )

G _n

x _n

(x _n−1 , . . . , x _{n−(L−1+C)} )

µ _n , x _n−1 , . . . , x _{n−(L−1+C)}

G n (x n , . . . , x _n−i , σ n , . . . , σ _n−i ) ≡ G ⁿ (x n , µ n ) ,

V _n (x _n , µ _n )

V _n (x _n , µ _n ) , exp (

− 1 σ ² _ǫ

|G n (x _n , µ _n )| ² )

µ _n

µ _n+1

µ _n