Proposed Algorithm

4.3 Non-Bayesian Algorithms

4.3.2 Proposed Algorithm

Allproposednon-Bayesianalgorithmsprodu ethephaseestimates

{ˆ θ _n }

^based

onproperphase-lo kedloops(PLLs)[29℄.Wealso onsideredtheve tortra ker

strategy [69℄,thatistheestimationofthephasor

e ^jθ ⁿ

^instead^of^the^s
alar

θ _n

Sin e we didnot verifyanyperforman eimprovement, asexpe ted a ording

to [69℄,we fo usonPLL-based algorithmsdue to their lower omplexity.

First, let us onsider an ideal PLL, referred to as genie-aided (GA) PLL

in the following, whi h knows the a tual symbol

x _n

^and ^the ^a
tual ^state

σ _n

at ea h time epo h

n

^. ^The ^rst ^order² ^GA-PLL ^is ^governed ^by ^the ^following

updaterule[29℄

θ ˆ _n+1 = ˆ θ _n + γ Im n

y _n (x _n , σ _n )e ^{−j ˆ} ^θ ⁿ o

(4.7)

where

γ

îs ^the ^step ^size ôf ^the ^loop, ând

y n (x n , σ n )

^is ^dened ⁱⁿ ^(2.12). ^W^e

an employ a known preamble for ea h data symbol sequen e, in order to

initialize the re ursion (4.7) to the value orresponding to the

maximum-likelihood (ML) phase estimation over the preamble [29℄. This option, with

respe ttoarandomly-initializedPLL,ensuresamu hbetterperforman e[29℄.

Similarly to the forward update rule (4.7), we an dene a ba kward update

rule asfollows

θ ˆ _n−1 = ˆ θ _n + γ Im n

y _n (x _n , σ _n )e ^{−j ˆ} ^θ ⁿ o

.

^(4.8)

In this ase, sin e in many system standard (see for example the DVB-RCS)

no knownpostambleis present after thedata symbols,we an only randomly

initialize the re ursion (4.8). This motivates our hoi e of implementing the

GA-PLL using the forward update rule (4.7) only. As a ben hmark for all

non-Bayesian algorithms, we an onsider the BCJR algorithm des ribed in

Se tion 4.2,repla ingthea tualphaseosets

{θ n }

ⁱⁿ^the^bran
h ^metri
s^(4.5)

bytheestimates

{ˆ θ _n }

^produ
ed^by^the^GA-PLL.Îtîs^lear^thatâny^PLL-based

algorithm annot perform betterthan this ideal algorithm.

Let us now des ribe pra ti al algorithms, onsidering rst the

implemen-tation of theforward re ursion(2.17). Asstated before, we onsiderper-state

phasetra king, omputingone phaseestimate

θ ˆ _n (σ _n )

^for^ea
h^state

σ _n

^at ^ea
h

time epo h

n

^, ^and ^exploiting

θ ˆ _n (σ _n )

ⁱⁿ ^the omputation of the bran h met-ri s related to all trellis paths extending from

σ _n

^. ^For âll ^trial ^vâlues ôf ^the

initial state

σ ₀

^, ^the ^estimate

θ ˆ ₀ (σ ₀ )

^is initialized to the same value, possibly the ML phase estimation over the preamble. The point is how to derive an

updaterulesimilarto(4.7),takingintoa ountthatthea tualsymbol

x _n

^and

the a tual state

σ n

âreûnknown ^to ^the ^re
eiver. Ât âny^given ^timeêpo
h

n

Itis known[29℄ that higherorderPLLsdonotprovideany performan eimprovement

whenthefrequen yosetisnotpresent,asin(4.3).

let

θ ˇ n+1 (x n , σ n )

^be ^the^phaseêstimateât ^timeêpo
h

n + 1

^related^to ^the^trial

symbol

x _n

^and ^the^trial^state

σ _n

^,^dened ^as

θ ˇ _n+1 (x _n , σ _n ) = ˆ θ _n (σ _n ) + γ Im n

y _n (x _n , σ _n )e ^{−j ˆ} ^θ ⁿ ^(σ ⁿ ⁾ o

(4.9)

simplyextending (4.7) tothe aseofper-state tra king.

Itis worthto noti e

that, forea h valueofthefuturestate

σ n+1

^,^every ^ouple

(x n , σ n )

^ompatible

with

σ _n+1

ârriesâ^dierent êstimateôf^theâ
tual ^phaseôset

θ _n+1

^.^Now,^the

problemishowto ombinethevariousestimates

{ˇ θ _n+1 (x _n , σ _n )}

^to ^obtain^the

estimate

θ ˆ n+1 (σ n+1 )

^required ^to ^perform ^the ^forward ^re
ursion. ^In ^pra
ti
e,

we have to dene a riterion for properly weighting the dierent estimates.

A ouple of dierent solutions are dis ussed in the following. Both of them

require the denition, at ea h timeepo h

n

^and ^for ^ea
h ^trial^symbol

x _n

^and

trialstate

σ _n

^,^of^the^oe
ient

W _n+1 (x _n , σ _n , σ _n+1 ) = T (x _n , σ _n , σ _n+1 )F _n (x _n , σ _n )η _f,n (σ _n )

^(4.10)

whi h equals the ontribution given bythe onsidered trellis path to the

for-wardmetri

η _f,n+1 (σ n+1 )

^related^to^the orrespondingfuturestate

σ n+1

^(see^(2.17)).

We point out thatthe omputationof theterms(4.10) isdone while

perform-ingthe forward re ursion (2.17), and isnot to be onsidered an overheaddue

to phase tra king. The simplest solution, exploited in [66℄, onsists of

set-ting

θ ˆ _n+1 (σ _n+1 )

^equal ^to ^the^estimate

θ ˇ _n+1 (x _n , σ _n )

^related ^to ^the ^trellis^path

giving the largest ontribution to the forward metri

η _f,n+1 (σ n+1 )

^. ^F^ormally^,

for ea hgiven valueof

σ _n+1

^,^we^obtain

θ ˆ _n+1 (σ _n+1 ) = ˇ θ _n+1 (ˆ x _n , ˆ σ _n )

^(4.11)

with

(ˆ x _n , ˆ σ _n ) = arg max

(x n ,σ n ) {W n+1 (x _n , σ _n , σ _n+1 )} .

^(4.12)

Thephaseestimates

{ˇ θ n+1 (x n , σ n )}

ân^beâlsoûsedⁱⁿ^the omputationofthebran h metri s

{F n (x n , σ n )}

^,însteadôf^theêstimates

{ˆ θ n (σ n )}

^.^In^this^ase,^per-bran
h^phase

tra k-ingisperformed,sin edierentphaseestimatesforea htrellispathareexploited.

Alternatively, we an use the terms

{W ⁿ⁺¹ (x n , σ n , σ n+1 )}

^as ^weight ^fa
tors

for the omputation of the estimates

{ˆ θ _n+1 (σ _n+1 )}

âs âverages ôf ^the

esti-mates

{ˇ θ _n+1 (x _n , σ _n )}

^,^as^follows

θ ˆ _n+1 (σ _n+1 ) =

P

x n

P

σ n W _n+1 (x _n , σ _n , σ _n+1 ) ˇ θ _n+1 (x _n , σ _n ) P

x n

P

σ n W _n+1 (x _n , σ _n , σ _n+1 )

= P

x n

P

σ n W _n+1 (x _n , σ _n , σ _n+1 ) ˇ θ _n+1 (x _n , σ _n )

η _f,n+1 (σ _n+1 ) .

^(4.13)

A ording to the notation in [29℄ and widely adopted in the literature, both

theupdaterules(4.11) and(4.13)areinstan esoftrellis-basedphasetra king,

but the former is de ision-dire ted, while the latter is soft-de ision dire ted.

Interestingly,theupdaterule(4.11) anbeseenasanapproximationof(4.13),

obtainedbynegle tingallterms

{W ⁿ⁺¹ (x n , σ n , σ n+1 )}

^ex
ept^the^largest^one.⁴

Hen e,the latterstrategyisexpe tedtobemoreee tive.Ontheotherhand,

theaverageoperationin(4.13) learlyrequiresa omputationaloverheadwith

respe t to (4.11). The simulation results reported in Se tion 4.5 show that

su h an in rease in omplexity is often justied by a signi ant performan e

improvement. We will refer to the update rule (4.11) simply asPLL update,

whereas we will refer to the update rule (4.13) as weighted PLL (W-PLL)

update.

So far, we only onsidered the appli ation of per-state phase tra king to

the forward re ursion (2.17), but the same strategies an be adopted while

performingthe ba kward re ursion (2.18). It isindeed trivialto derive, based

on the GA ba kward update rule (4.8), the ba kward ounterparts of (4.9),

(4.11),and (4.13).Inpra ti e, aba kward re ursionwithjoint phasetra king

an beperformedsimilarly to theforward re ursion.It is worth noti ingthat

this approa h leadsto asetof phaseestimatesprodu ed during theba kward

re ursion whi h arenot onstrained to be equal to those produ ed duringthe

forward re ursion. In this ase, the ompletion stage (2.21) is not

unambigu-ously dened, sin e the bran h metri s

{F n (x _n , σ _n )}

^an ^be ^omputed ^based

Froma on eptual viewpoint,the abovementioneddieren esare similartothose

o - urringbetweentheViterbialgorithmandtheBCJRalgorithm[6,70℄.

onthe phaseestimatesprodu ed duringtheforward re ursionaswellasthose

produ edduringtheba kwardre ursion.Thisissuehasbeenaddressedin[66℄,

where the authors investigate several ways for ombining thephase estimates

produ edinthe re ursions.Weproposeanalternative solution,whi h onsists

of performing the ompletion stage basedon (4.6) insteadof (2.21).

The key

point is that the implementation of (4.6) allows to perform the ompletion

stage without omputing the bran h metri s, so that no phase estimate has

to beexploited. On e omputed theforward/ba kward state metri s, this

ap-proa h doesnot introdu e anyfurther approximationinthe ompletion stage,

asinstead done byall algorithms whi h perform the ompletion stage

expli -itly omputing the bran h metri s based on phase estimates. Hen e, this is

expe ted to be the most ee tive solution when the two re ursions perform

phasetra king independently ea h other.

Asanaldesignoption,we onsiderthepossibilityofexploitingthephase

estimates produ ed during the forward re ursion while performing the

ba k-ward re ursion and the ompletion stage, so that per-state phase tra king is

performed only on e. This approa h is suggested in [66℄, where it is shown

thatthe bidire tional phaseestimation suerstherotational invarian e ofthe

CPMs,sothatphaseslipso urmorefrequently.

Then,evenifwea eptthat

we have to perform monodire tionalphase tra king, itis natural to wonderif

there existsa physi al reason for doingthis duringthe forward re ursion and

notintheba kwardre ursion.Asigni antadvantageinfavourofperforming

forward phasetra king isdue to thefa tthat, asstated before, theestimates

anbereliablyinitializedtothevalueprodu edbyaMLphaseestimationover

the preamble, while the same annot be always done for the ba kward phase

tra king sin e frequently no postamblefollows the data stream. Hen e, when

Asexplained inSe tion 4.2, the ompletion (4.6) anbe adopted onlywhen the

on-sidered CPM formathas orrelation lengthlarger than one. This ondition holdsfor high

spe trale ien yCPMformatswewill onsider.

In[66℄, itisalsoshownthatthebidire tionalphaseestimationistobepreferredwhen

the phase osets are time-invariant. Anyway, this is not the ase of the satellite hannel

onsideredinthis hapter.

onsidering monodire tional phase tra king, we will always refer to forward

phasetra king.

Nel documento Advanced Modulation/Demodulation Schemes For Wireless Communications (pagine 156-161)

4.3 Non-Bayesian Algorithms

4.3.2 Proposed Algorithm

{ˆ θ n }

e jθ n

θ n

x n

σ n

n

θ ˆ n+1 = ˆ θ n + γ Im n

y n (x n , σ n )e −j ˆ θ n o

γ

y n (x n , σ n )

θ ˆ n−1 = ˆ θ n + γ Im n

y n (x n , σ n )e −j ˆ θ n o

.

{θ n }

{ˆ θ n }

θ ˆ n (σ n )

σ n

n

θ ˆ n (σ n )

σ n

σ 0

θ ˆ 0 (σ 0 )

x n

σ n

n

θ ˇ n+1 (x n , σ n )

n + 1

x n

σ n

θ ˇ n+1 (x n , σ n ) = ˆ θ n (σ n ) + γ Im n

y n (x n , σ n )e −j ˆ θ n (σ n ) o

σ n+1

(x n , σ n )

σ n+1

θ n+1

{ˇ θ n+1 (x n , σ n )}

θ ˆ n+1 (σ n+1 )

n

x n

σ n

W n+1 (x n , σ n , σ n+1 ) = T (x n , σ n , σ n+1 )F n (x n , σ n )η f,n (σ n )

η f,n+1 (σ n+1 )

σ n+1

θ ˆ n+1 (σ n+1 )

θ ˇ n+1 (x n , σ n )

η f,n+1 (σ n+1 )

σ n+1

θ ˆ n+1 (σ n+1 ) = ˇ θ n+1 (ˆ x n , ˆ σ n )

(ˆ x n , ˆ σ n ) = arg max

(x n ,σ n ) {W n+1 (x n , σ n , σ n+1 )} .

{ˇ θ n+1 (x n , σ n )}

{F n (x n , σ n )}

{ˆ θ n (σ n )}

{W n+1 (x n , σ n , σ n+1 )}

{ˆ θ n+1 (σ n+1 )}

{ˇ θ n+1 (x n , σ n )}

θ ˆ n+1 (σ n+1 ) =

P

x n

P

σ n W n+1 (x n , σ n , σ n+1 ) ˇ θ n+1 (x n , σ n ) P

x n

P

σ n W n+1 (x n , σ n , σ n+1 )

= P

x n

P

σ n W n+1 (x n , σ n , σ n+1 ) ˇ θ n+1 (x n , σ n )

η f,n+1 (σ n+1 ) .

{W n+1 (x n , σ n , σ n+1 )}

{F n (x n , σ n )}

{ˆ θ _n }

e ^jθ ⁿ

θ _n

x _n

σ _n

θ ˆ _n+1 = ˆ θ _n + γ Im n

y _n (x _n , σ _n )e ^{−j ˆ} ^θ ⁿ o

θ ˆ _n−1 = ˆ θ _n + γ Im n

y _n (x _n , σ _n )e ^{−j ˆ} ^θ ⁿ o

{ˆ θ _n }

θ ˆ _n (σ _n )

σ _n

θ ˆ _n (σ _n )

σ _n

σ ₀

θ ˆ ₀ (σ ₀ )

x _n

x _n

σ _n

θ ˇ _n+1 (x _n , σ _n ) = ˆ θ _n (σ _n ) + γ Im n

y _n (x _n , σ _n )e ^{−j ˆ} ^θ ⁿ ^(σ ⁿ ⁾ o

σ _n+1

θ _n+1

{ˇ θ _n+1 (x _n , σ _n )}

x _n

σ _n

W _n+1 (x _n , σ _n , σ _n+1 ) = T (x _n , σ _n , σ _n+1 )F _n (x _n , σ _n )η _f,n (σ _n )

η _f,n+1 (σ n+1 )

θ ˆ _n+1 (σ _n+1 )

θ ˇ _n+1 (x _n , σ _n )

η _f,n+1 (σ n+1 )

σ _n+1

θ ˆ _n+1 (σ _n+1 ) = ˇ θ _n+1 (ˆ x _n , ˆ σ _n )

(ˆ x _n , ˆ σ _n ) = arg max

(x n ,σ n ) {W n+1 (x _n , σ _n , σ _n+1 )} .

{W ⁿ⁺¹ (x n , σ n , σ n+1 )}

{ˆ θ _n+1 (σ _n+1 )}

{ˇ θ _n+1 (x _n , σ _n )}

θ ˆ _n+1 (σ _n+1 ) =

σ n W _n+1 (x _n , σ _n , σ _n+1 ) ˇ θ _n+1 (x _n , σ _n ) P

σ n W _n+1 (x _n , σ _n , σ _n+1 )

σ n W _n+1 (x _n , σ _n , σ _n+1 ) ˇ θ _n+1 (x _n , σ _n )

η _f,n+1 (σ _n+1 ) .

{W ⁿ⁺¹ (x n , σ n , σ n+1 )}

{F n (x _n , σ _n )}