Dete tors for IR Computation over Channels Ae ted

2.4 IR of CPMs over Channel Ae ted by Phase Noise

2.4.3 Dete tors for IR Computation over Channels Ae ted

in the MAP sense. Moreover, it is interesting also to evaluate the IR by a

DP-BCJR operatingovera CT orST hannel generation, sin e we derive the

maximumperforman ewe ana hieve,whenemploying adete tionalgorithm

arried out by starting from a single rst order PN model (Wiener or AR1),

overa hannel ae tedby SATMODEPN.Finally,thethird typeof dete tor

is parti ularly suitable to ompute the IR for double-AR1 PN generation at

low baudrate values. Looking to the double-AR1 parameters in Table 2.3 at

64

^kBaud, ^for êxample,^we ^see ^that ^the^fastômponents êxhibits â

b

^value^so

lose tozerothatinthedete torderivation, we an onsiderits orresponding

PN samples as almost independent. In su h a way, in the dete tion

deriva-tion, we take into a ount both the two AR1 omponent but we also a hieve

a redu tioninthedete tor omplexity.

trialphase value

θ n

^,^we ^dene ^the^sample

r n (x n , σ n , θ n )

^as

r _n (x _n , σ _n , θ _n ) ,

Z _(n+1)T

nT r(t)es ^∗ (t − nT ; x n , σ _n )e ^−jθ ⁿ dt

= y _n (x _n , σ _n )e ^−jθ ⁿ

^(2.36)

where the ve tor

y

^olle
ting

y _n (x _n , σ _n )

^(2.12) ^represents ^a ^set ^of ^su
ient

statisti of the re eived signal.

Finally we all

r

^the ^ve
tor ^olle
ting ^the

sample

r _n (x _n , σ _n , θ _n )

^for ^ea
h ^trial ^symbol

x _n

^,^ea
h ^trial ^CPM^state

σ _n

^, ^and

phasevalue

θ _n

^,^and ^ea
h dis rete-time instant.

We now derive the MAP symbol dete tion through FG and SPA. To this

purpose, we rstfa torize the joint distribution

p(x, σ, θ|r)

^as

p(x, σ, θ|r) ∝ p(r|x, σ, θ)P (σ|x)P (x)p(θ)

^(2.37)

= P (σ ₀ )P (θ ₀ )

N −1 Y

n=0

F _n (x _n , σ _n , θ _n )T (x _n , σ _n , σ _n+1 )P (x _n )p(θ _n+1 |θ n )

where

θ

^olle
ts

θ _n

^samples ⁽

n

^from

0

^to

N

^),

T (x _n , σ _n , σ _n+1 )

îs ân îndi
ator

fun tion, equal to one if

x n , σ n

^and

σ n+1

^satisfy ^the ^CPM ^trellis onstraints andtozerootherwise,and

F _n (x _n , σ _n , θ _n )

^is^the^bran
h^metri^fun
tion,^dened

F _n (x _n , σ _n , θ _n ) = exp

1 N ₀ Re n

y _n (x _n , σ _n )e ^−jθ ⁿ o

.

^(2.38)

Finally,asstated inSe tion 2.4.1,

p(θ n+1 |θ ⁿ )

ⁱⁿ^(2.37) ^is ^a^Gaussian ^pdf⁴

p(θ _n+1 |θ n ) = 1

q 2πσ ² _∆

exp

− (θ _n+1 − θ n ) ² 2σ _∆ ²

.

^(2.39)

In orderto have a nitestate representation for su h a hannel, we dis retize

thevalues thatsamples

θ _n

^mayâssume. Ôbviously^,^weâre approximatingthe 3

Notethatalsointhis ase, aswellas inthe oherent optimaldete tor,theve tor

y

^is

obtainedfromasetofjust

M ^L

^lters.

Notethat,sin ethe hannelphaseisdenedmodulo

2π

^,^theprobabilitydensityfun tion

p(θ n+1 |θ n )

^an^beapproximatedasGaussianonlyif

σ ∆ ≪ 2π

x _n η _f,n+1 η _b,n

P (x _n ) (σ _n , θ _n )

P (x n )

(σ _n+1 , θ _n+1 ) F _n T P _θ

P _e (x n )

Figure2.7: Fa tor graphofDP-BCJR inthe

n

^-thtime-interval.

real hannel and, by in reasing thenumber of quantization levels, better

ap-proximations anbea hievedatthepri eofagreater hannelstate ardinality.

In parti ular, we propose an uniform dis retization, bywhi h samples

θ _n

^are

onsidered belongingto thebelowalphabet

θ _n ∈

0, 2π

D , 2 2π

D , . . . , (D − 1) 2π D

(2.40)

where

D

^is^the^number^ofquantizationlevels.Hen e,thepdf

p(θ n+1 |θ ⁿ )

ⁱⁿ^(2.39)

mustrepla edbythepmf

P _θ (θ _n+1 |θ n )

^,^whi
h ân ^be ^hosen^for êxample âs:

P _θ (θ _n+1 |θ n ) =

Z _θ _n+1 ₊ ^π

D

θ n+1 − _D ^π

q 1 2πσ _∆ ²

exp

− (x − θ n ) ² 2σ _∆ ²

dx .

^(2.41)

The FG orrespondingto (2.37) has y les.However, by lustering [7℄the

variables

θ _n

^and

σ _n

^,^weôbtain ^the^FGⁱⁿ^Fig.^2.7.^Thus, ^byâpplying^to ît^the

SPAbyusinga non-iterative forward-ba kward s hedule, we obtaintheexa t

a posteriori probabilities

P (x n |r)

^ne
essary ^to ^implement ^the ^MAP ^symbol

dete tion strategy. With referen e to the messages in Fig. 2.7, the resulting

forwardba kward algorithm is hara terized by the following re ursions and

ompletion:

η _f,n+1 (σ _n+1 , θ _n+1 ) = X

σ n

X

θ n

X

x n

P (x _n )F _n (x _n , σ _n , θ _n )T (x _n , σ _n , σ _n+1 ) P _θ (θ _n+1 |θ n )η _f,n (σ _n , θ _n )

(2.42)

η _b,n+1 (σ _n , θ _n ) = X

σ n+1

X

θ n+1

X

x n

P (x _n )F _n (x _n , σ _n , θ _n )T (x _n , σ _n , σ _n+1 ) P _θ (θ _n+1 |θ n )η _b,n+1 (σ _n+1 , θ _n+1 )

(2.43)

P e (x n ) ∝ X

σ n

X

σ n+1

X

θ n

X

θ n+1

F n (x n , σ n , θ n )T (x n , σ n , σ n+1 )P _θ (θ n+1 |θ ⁿ ) η _f,n (σ _n , θ _n )η _b,n+1 (σ _n+1 , θ _n+1 )

(2.44)

with the following initial onditions for the two re ursions:

η _f,0 (σ ₀ , θ ₀ ) = P (σ ₀ )/(2π)

^and

η _b,N (σ _N , θ _N ) = P (σ _N )/(2π)

^. Îtîs ^lear ^thatâ ^better

approx-imation of the real hannel, whi h is a hieved by in reasing the number of

quantizationlevels

D

^,^determinesâlsoânîn
reased^dete
tionômplexity^,^sin
e

the ardinality ofthe trellisstate

(σ _n , θ _n )

^isproportionalto

D p M ^L−1

Finally, we an extend the DP-BCJR algorithm to all ases in whi h the

dis rete-time PNpro ess isaGaussian AR1model(2.25).Insu ha ase, itis

su ient to repla ethepmf

P _θ (θ _n+1 |θ n )

ⁱⁿ^(2.41) ^by

P _θ (θ _n+1 |θ n ) =

Z _θ _n+1 ₊ ^π

D

θ n+1 − _D ^π

p 1

2π(1 − a ² )σ _a ² exp

− (x − a θ n ) ² 2(1 − a ² )σ _a ²

dx .

^(2.45)

Double-DP-BCJR (D-DP)

Wenow onsiderthedouble-AR1phasenoisemodeldes ribedinSe tion2.4.2.

Thus,theaboveDP-BCJRalgorithmderivation anbegeneralizedto the ase

ofa hannelwhosePNmodelisgivenbythesumoftwodis rete-timepro esses

θ ^(a) n

^and

θ n ^(b)

^for ^whi
h ^the ^following ^re
ursive ^denitions ^exist

θ _n+1 ^(a) = a θ ^(a) _n + v _n ^(a)

^(2.46)

θ _n+1 ^(b) = b θ _n ^(b) + v _n ^(b)

^(2.47)

x _n η f,n+1

P (x n ) P (x n )

(σ _n , θ ^(a) _n , θ _n ^(b) ) η _b,n F _n T P _θ ^(a) P _θ ^(b) (σ _n+1 , θ ^(a) _n+1 , θ ^(b) _n+1 )

P _e (x _n )

Figure 2.8:FGofDouble-DP-BCJR dete torinthe

n

^-th time-interval.

where

v ^(a) n

^and

v ^(b) n

^areindependent andidenti allydistributedGaussian zero-meanrandomvariablesofvarian e

(1−a ² ) σ _a ²

^and

(1−b ² ) σ _b ²

^,respe tively.Also inthis ase, the two phase values an be uniformly dis retized inthedomain

and we denote by

D _a

^and

D _b

^the ^number ^of quantization levels for

θ n ^(a)

^and

θ ^(b) _n

^,respe tively.

In order to apply the MAP symbol dete tion strategy, we fa torize the

globalpmf

P (x, σ, θ ^(a) , θ ^(b) |r)

ⁱⁿ^a^way^similar^to ^(2.37)^and^we^derive^the^FG

inFig.2.8.TheSPAallows usto omputethetwo re ursionsand ompletion;

herewereport justtheexpressionfor theforward re ursion:

η _f,n+1 (σ _n+1 , θ ^(a) _n+1 , θ _n+1 ^(b) ) = X

σ n

X

θ ^(a) n

X

θ ^(b) n

X

x n

F _n (x _n , σ _n , θ _n ^(a) , θ ^(b) _n )T (x _n , σ _n , σ _n+1 )

P (x _n )P _θ ^(a) (θ _n+1 ^(a) |θ n ^(a) )P _θ ^(b) (θ _n+1 ^(b) |θ n ^(b) )η _f,n (σ _n , θ ^(a) _n , θ ^(b) _n )

(2.48)

where we have dened

F _n (x _n , σ _n , θ _n ^(a) , θ ^(b) _n ) = exp

1 N ₀ Re n

y _n (x _n , σ _n )e ^−j(θ ^(a) ⁿ ^+θ ^(b) ⁿ ⁾ o

(2.49)

and

P _θ ^(a) (θ ^(a) _n+1 |θ ^(a) ⁿ )

P _θ ^(b) (θ ^(b) _n+1 |θ ^(b) ⁿ )

^are^the ^pmfs^des
ribing ^the^hannel^phase

transition, dened asin(2.45).

In su h a ase, the trellis state is

(σ n+1 , θ _n+1 ^(a) , θ ^(b) _n+1 )

^and ^its ardinality results

D _a D _b pM ^L

Improved-DP-BCJR (I-DP)

Looking at the parameters des ribing the faster PN omponent inTable 2.3,

we see that its

b

^parameter ^is ^very ^lose ^to ^zero, ^espe
ially ^for ^low ^baudrate

values. In other words, PN samples

θ ^(b) _n

^an ^be ^onsidered independent from ea h otherand we assumethefollowing model:

θ ^(b) _n = v _n ^(b)

^(2.50)

obtained repla ing

b = 0

ⁱⁿ^(2.46); ^hen
e

P (θ ^(b) _n ) =

N −1 Y

n=0

P _θ ^(b) (θ ^(b) _n ) .

^(2.51)

where

P _θ ^(b) (θ ^(b) n )

êxpression îs êquivalent ^to ^(2.45) ^with

a = 0

^. ^By fa torizing the pmf

P (x, σ, θ ^(a) , θ ^(b) |r)

^, ^we ^derive â ^new ^dete
tion âlgorithm ^based ôn

assumption (2.50), whose fa tor graph is provided in Fig 2.9. The forward

re ursion expressionobtained thought theSPAis

η _f,n+1 (σ _n+1 , θ ^(a) _n+1 ) = X

σ n

X

θ ^(a) n

X

x n

P (x _n )F _n ^(a) (x _n , σ _n , θ ^(a) _n )T (x _n , σ _n , σ _n+1 )

P _θ ^(a) (θ _n+1 ^(a) |θ n ^(a) )η _f,n (σ n , θ _n ^(a) ).

(2.52)

where the bran h metri is

F _n ^(a) (x _n , σ _n , θ _n ^(a) ) , X

θ ^(b) n

F _n (x _n , σ _n , θ ^(a) _n , θ _n ^(b) )P _θ ^(b) (θ _n ^(b) )

^(2.53)

and

F _n (x _n , σ _n , θ _n ^(a) , θ _n ^(b) )

îs^providedⁱⁿ^(2.49).^Hen
e,^weân^see^thatêmploying

the assumption in (2.50), the double-DP-BCJR omplexity is redu ed sin e

η _f,n+1 η _b,n

P (x _n ) (σ n , θ _n ^(a) )

P (x _n )

(σ n+1 , θ ^(a) _n+1 ) F n T P _θ ^(a)

P (θ ^(b) _n )

x _n P (θ ^(b) _n )

θ ^(b) _n

P _e (x _n )

Figure2.9: FGof I-DP-BCJRdete torinthe

n

^-thtime-interval.

here the trellis state is the same of the DP-BCJR, i.e.

(σ _n , θ ^(a) n )

^. ^However,

with respe t to the DP-BCJR algorithm we take into a ount a se ond PN

omponent,

θ n ^(b)

^,independent from ea h interval

n

^, ^and ^so ^the^bran
h ^metri

omputation isin reased (see (2.53)).

Nel documento Advanced Modulation/Demodulation Schemes For Wireless Communications (pagine 54-61)

Dete tors for IR Computation over Channels Ae ted

2.4 IR of CPMs over Channel Ae ted by Phase Noise

2.4.3 Dete tors for IR Computation over Channels Ae ted

64

b

θ n

r n (x n , σ n , θ n )

r n (x n , σ n , θ n ) ,

Z (n+1)T

nT r(t)es ∗ (t − nT ; x n , σ n )e −jθ n dt

= y n (x n , σ n )e −jθ n

y

y n (x n , σ n )

r

r n (x n , σ n , θ n )

x n

σ n

θ n

p(x, σ, θ|r)

p(x, σ, θ|r) ∝ p(r|x, σ, θ)P (σ|x)P (x)p(θ)

= P (σ 0 )P (θ 0 )

N −1 Y

n=0

F n (x n , σ n , θ n )T (x n , σ n , σ n+1 )P (x n )p(θ n+1 |θ n )

θ

θ n

n

0

N

T (x n , σ n , σ n+1 )

x n , σ n

σ n+1

F n (x n , σ n , θ n )

F n (x n , σ n , θ n ) = exp

 1 N 0 Re n

y n (x n , σ n )e −jθ n o

.

p(θ n+1 |θ n )

p(θ n+1 |θ n ) = 1

q 2πσ 2 ∆

exp



− (θ n+1 − θ n ) 2 2σ ∆ 2



.

θ n

y

M L

2π

p(θ n+1 |θ n )

σ ∆ ≪ 2π

x n η f,n+1 η b,n

P (x n ) (σ n , θ n )

P (x n )

(σ n+1 , θ n+1 ) F n T P θ

P e (x n )

n

θ n

θ n ∈

 0, 2π

D , 2 2π

D , . . . , (D − 1) 2π D



D

p(θ n+1 |θ n )

P θ (θ n+1 |θ n )

P θ (θ n+1 |θ n ) =

Z θ n+1 + π

D

θ n+1 − D π

q 1 2πσ ∆ 2

exp



− (x − θ n ) 2 2σ ∆ 2



dx .

θ n

σ n

P (x n |r)

η f,n+1 (σ n+1 , θ n+1 ) = X

Dete tors for IR Computation over Channels Ae ted

2.4 IR of CPMs over Channel Ae ted by Phase Noise

2.4.3 Dete tors for IR Computation over Channels Ae ted

r _n (x _n , σ _n , θ _n ) ,

Z _(n+1)T

nT r(t)es ^∗ (t − nT ; x n , σ _n )e ^−jθ ⁿ dt

= y _n (x _n , σ _n )e ^−jθ ⁿ

y _n (x _n , σ _n )

r _n (x _n , σ _n , θ _n )

x _n

σ _n

θ _n

= P (σ ₀ )P (θ ₀ )

F _n (x _n , σ _n , θ _n )T (x _n , σ _n , σ _n+1 )P (x _n )p(θ _n+1 |θ n )

θ _n

T (x _n , σ _n , σ _n+1 )

F _n (x _n , σ _n , θ _n )

F _n (x _n , σ _n , θ _n ) = exp

1 N ₀ Re n

y _n (x _n , σ _n )e ^−jθ ⁿ o

p(θ n+1 |θ ⁿ )

p(θ _n+1 |θ n ) = 1

q 2πσ ² _∆

− (θ _n+1 − θ n ) ² 2σ _∆ ²

θ _n

M ^L

x _n η _f,n+1 η _b,n

P (x _n ) (σ _n , θ _n )

(σ _n+1 , θ _n+1 ) F _n T P _θ

P _e (x n )

θ _n

θ _n ∈

0, 2π

p(θ n+1 |θ ⁿ )

P _θ (θ _n+1 |θ n )

P _θ (θ _n+1 |θ n ) =

Z _θ _n+1 ₊ ^π

θ n+1 − _D ^π

q 1 2πσ _∆ ²

− (x − θ n ) ² 2σ _∆ ²

θ _n

σ _n

η _f,n+1 (σ _n+1 , θ _n+1 ) = X

P (x _n )F _n (x _n , σ _n , θ _n )T (x _n , σ _n , σ _n+1 ) P _θ (θ _n+1 |θ n )η _f,n (σ _n , θ _n )

η _b,n+1 (σ _n , θ _n ) = X

P (x _n )F _n (x _n , σ _n , θ _n )T (x _n , σ _n , σ _n+1 ) P _θ (θ _n+1 |θ n )η _b,n+1 (σ _n+1 , θ _n+1 )

F n (x n , σ n , θ n )T (x n , σ n , σ n+1 )P _θ (θ n+1 |θ ⁿ ) η _f,n (σ _n , θ _n )η _b,n+1 (σ _n+1 , θ _n+1 )

η _f,0 (σ ₀ , θ ₀ ) = P (σ ₀ )/(2π)

η _b,N (σ _N , θ _N ) = P (σ _N )/(2π)

(σ _n , θ _n )

D p M ^L−1

P _θ (θ _n+1 |θ n )

P _θ (θ _n+1 |θ n ) =

Z _θ _n+1 ₊ ^π

θ n+1 − _D ^π

2π(1 − a ² )σ _a ² exp

− (x − a θ n ) ² 2(1 − a ² )σ _a ²

θ ^(a) n

θ n ^(b)

θ _n+1 ^(a) = a θ ^(a) _n + v _n ^(a)

θ _n+1 ^(b) = b θ _n ^(b) + v _n ^(b)

x _n η f,n+1

(σ _n , θ ^(a) _n , θ _n ^(b) ) η _b,n F _n T P _θ ^(a) P _θ ^(b) (σ _n+1 , θ ^(a) _n+1 , θ ^(b) _n+1 )

P _e (x _n )

v ^(a) n

v ^(b) n

(1−a ² ) σ _a ²

(1−b ² ) σ _b ²

D _a

D _b

θ n ^(a)

θ ^(b) _n

P (x, σ, θ ^(a) , θ ^(b) |r)

η _f,n+1 (σ _n+1 , θ ^(a) _n+1 , θ _n+1 ^(b) ) = X

θ ^(a) n

θ ^(b) n

F _n (x _n , σ _n , θ _n ^(a) , θ ^(b) _n )T (x _n , σ _n , σ _n+1 )

P (x _n )P _θ ^(a) (θ _n+1 ^(a) |θ n ^(a) )P _θ ^(b) (θ _n+1 ^(b) |θ n ^(b) )η _f,n (σ _n , θ ^(a) _n , θ ^(b) _n )

F _n (x _n , σ _n , θ _n ^(a) , θ ^(b) _n ) = exp

1 N ₀ Re n