SATMODE Phase Noise Model

We now onsidera more realisti PN model, i.e., theSATMODE phasenoise

whi hae tsthesatellite ommuni ationsystemswithlow- ostequipment[22℄.

Itspowerspe traldensityisknownanditis representedinTable.2.1. Evenif

inprin iple thatmask refersto the phasor

e ^jθ(t)

^{,it hasbeen assumedthat it}

approximately orresponds to the PSD of thephase pro ess

θ(t)

^{.In order to}

Frequen y [Hz℄ PSD [dB/Hz℄

10 ¹

^-23

10 ²

^-48

10 ³

^-68

10 ⁴

^-68

10 ⁵

^-80

10 ⁶

^-100

Table2.1: SATMODEPSD of the ontinuous-time phasepro ess

θ(t)

^.

ompute the IR for CPM modulation s hemes over SATMODE hannel and

to derive some mat hed dete tionalgorithms, we needto derive anite-order

statisti al des ription of the PN. Sin e we want to deal with a dis rete-time

pro ess, we sample the phase pro ess with a sampling period

T _S = 10 ⁻⁷

ⁱⁿ

orderto avoid aliasing;hen eweobtain adis rete-timepro ess withthesame

-100 -90 -80 -70 -60 -50 -40 -30 -20

10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴ 10 ⁵ 10 ⁶ 10 ⁷

PSD [dB/Hz]

f [Hz]

SATMODE PSD mask

Figure 2.2:SATMODEPSD of the ontinuous-time phase pro ess

θ(t)

^.

PSD maskof the ontinuous-time pro ess.

ByanalyzingtheSATMODEPSD,representedinFig2.2,wenotethatthe

mask anbeseenasthesumoftwodierent slopesand hen ethePN pro ess

an be well approximated as the sum of two 1st order auto-regressive (AR1)

Gaussian pro esses, instead of just one AR1. In detail, one AR1 des ribes

the PN behaviour at low frequen ies (slow pro ess hara terized by higher

power)whiletheotheronerepresentsthePNathigherfrequen ies(fastpro ess

hara terized bylowerpower).

In general aGaussian AR1 pro ess isdened as

u _k = a u _k−1 + v _a,k

^(2.25)

where

v _a,k

^{are zero-mean} independent and identi ally distributed Gaussian random variables, with varian e

σ ² _v

^{and where}

a

^{is a real value su h that}

|a| < 1

^(toensure stability). The varian e of

u _k

^isthenrepresentedby

σ ² _a , E 

|u k | ²

= σ ² _v

1 − a ²

^(2.26)

and itsPSD is

S u (f ) = σ ² _a (i − a ² )

1 + a ² − 2a cos(2πf T S ) , f ∈



− 1 2T _S , 1

2T _S



.

^(2.27)

Thus, inour asewe have

θ _k , u _a,k + u _b,k

^(2.28)

where

u _a,k = a u _a,k−1 + v _a,k u _b,k = b u _b,k−1 + v _b,k .

(2.29)

Hen e, our aim isto nd the four parameters

a, b, σ _a ² , σ ² _b

^{su h that thetarget}

PSD of the dis rete-timephase pro ess(Fig2.2) iswell approximated by

S _θ (f ) = σ ² _a (i − a ² )

1 + a ² − 2a cos(2πf T S ) + σ _b ² (i − b ² )

1 + b ² − 2b cos(2πf T S ) ,

^(2.30)

f ∈



− 1 2T _S , 1

2T _S

 .

InFig2.3wereporttheSATMODEPSD maskalongwiththePSDsresulting

from the double-AR1 approximations with three dierent set of parameters,

reported in Table 2.2. The approximations denoted as UB (upper bound)

and LB (lower bound) represent respe tively an overestimated and an

un-derestimated approximation of the target PSD, while the IA (intermediate

approximation) isaPSD representation fallingbetween thetwoextreme

rep-resentations. Fromnow on,thePN random pro ess will be always generated

a ording to the IA double-AR1 approximations.Moreover, sin e thePSD of

the oversampled dis rete-time phase pro ess we generate by su h parameters

is equal to that of the ontinuous-time pro ess up to

1/T S

^{, we will denote}

su h PN generation method as the ontinuous-time (CT) generation, in

on-trast withthe symbol-time generation (ST) we will des ribe in thefollowing.

a σ ² _a b σ ² _b

LB

1 − 10 ⁻⁷ 1.40 0.980 8 · 10 ⁻³

IA

1 − 10 ⁻⁷ 2.45 0.985 16.2 · 10 ⁻³

UB

1 − 10 ⁻⁶ 1.00 0.987 2 · 10 ⁻²

Table2.2:Parameters oftheoversampled ontinuous-time phasenoise

double-AR1 approximation.

-100 -90 -80 -70 -60 -50 -40 -30 -20

10 ¹ 10 ² 10 ³ 10 ⁴ 10 ⁵ 10 ⁶

PSD [dB/Hz]

f [Hz]

SATMODE PSD mask Intermediate Approximation Lower Bound

Upper Bound

Figure 2.3: SATMODE PSD mask and PSDs obtained from various

double-AR1 approximations.

-140 -120 -100 -80 -60 -40 -20 0 20

10 ^-1 10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴ 10 ⁵ 10 ⁶ 10 ⁷

PSD [dB/Hz]

f [Hz]

PSD of slow AR1 component PSD of fast AR1 component double-AR1 PSD

Figure 2.4:PSDs ofslowand fastAR1 obtained withIAparameters.

In order to better understand the ee ts of the two PN omponents, we

re-porttwo gures. Inthe rst one,Fig 2.4, we seethe PSD of ea h singleAR1

omponents evaluated by repla ing in (2.27) the IA parameters of Table 2.2:

ifone AR1 (denoted by

a

^{) determinestheresulting phase noise PSD at lower}

frequen ies, the other one (denoted by

b

^{) determines the phase noise PSD at}

higher frequen ies. Fig 2.5 shows the time-domain snapshot of the two

om-ponentsofthe ontinuous-time phasenoise

θ(t)

^{,i.e.,theslowoneandthefast}

one, generated following the double-AR1approximations withIA parameters

of Table2.2. The behaviourof thetwo PN omponentsis lear:thefast

om-ponent is ae ted by very qui k and low power u tuations, while the slow

omponent is hara terized byhighvarian e and slowu tuations. Notethat

in pra ti e it is very di ult to tra k the faster omponent, therefore it is

-3 -2 -1 0 1 2 3

0 1 2 3 4 5 6 7

θ (t)

t [s]

Slow AR1 component Fast AR1 component

Figure2.5:Time snapshotofthetwo phasenoiseAR1 omponents, generated

a ordingto theIA parameters.

themajor reasonofperforman edegradation.Wedis ussinthefollowing,the

ee t of phasenoise on digital ommuni ations. Given the CPMsignal (1.1),

theuseful re eived signalis

r(t) = s(t; x)e ^jθ(t) .

We donot areaboutthermalnoise sin eitsstatisti sarenotae tedbyPN.

Atthe

k

^{-thsymbolintervaltheoutput ofthebankof mat hed ltersis}

e ^{−jπ n}

Z (n+1)T nT

s(t; x)¯ s ^∗ (t − nT ; x ⁿ , ω n )e ^−jθ(t) dt

^(2.31)

wheretheCPMwaveform

¯ s ^∗ (t −nT ; x n , ω _n )

^{andthephasestate}

π _n

^aredened

in(1.17) and(1.5)respe tively.Duetothepresen eofthePN

θ(t)

^,theoutput

of the mat hed lter

s ¯ ^∗ (−t; x ⁿ , ω n )

^{is no more a su ient statisti but it is}

a orrupted version of the su ient statisti

y _n (x _n , σ _n )

^{dened in (2.12). In}

other words, the phase noise auses a sort of inter-symbol interferen e (ISI)

whi h,inprin iple, anhave anextremely harmfulee t whena re eiverthat

negle ts itisemployed.

However,we veriedthatthisisnotthe aseinall s enariosofinterest(as

shown later). Hen e, we negle t its ee t and we also provide a dierent PN

generation,denotedassymbol-time(ST)generation,bywhi hjustonesample

isgenerated atea hsymbolinterval,withouttakingintoa ountthemat hed

ltering distortionpresentsinthe ontinuous-time (CT)generation. Thenthe

mat hed ltersoutput(2.31) an beapproximated as

e ^{−jπ n}

Z (n+1)T nT

s(t; x)¯ s ^∗ (t − nT ; x n , ω _n )e ^jθ(t) dt ≃ y n (x _n , σ _n )e ^{jψ n}

^(2.32)

where

ψ n = 1 T

Z (n+1)T nT

θ(t)dt.

^(2.33)

Thus the PN pro essat symboltimeisapproximately awindowed and

down-sampled version of the original PN pro ess. In order to obtain a statisti al

hara terization of

ψ _n

^{asdone for the} ontinuous-time pro ess

θ(t)

^{,we derive}

theauto orrelation fun tion(ACF)of

ψ _n

R _ψ (m) = E {ψ n ψ _n−m } .

^(2.34)

In Fig 2.6 the auto orrelation fun tion (2.34) for a signal rate of

256

^kBaud

is arried out by numeri al simulation: we have generated various

θ(t)

se-quen es employing IA parameters, derived the orresponding

ψ _n

^{sequen es}

following (2.33) when the symbol rate is

R = 1/T = 256

^{kBaud and then}

omputed

R _ψ (m)

^{byaveraging overall simulated}

ψ _n

^{sequen es.Wenotefrom}

Fig 2.6 that the symbol-time phase noise pro ess

ψ _n

^{an again be}

approxi-mated as the sum of two independent AR1 pro esses, a slowly varying (i.e.,

highly orrelated) omponent anda rapidly varyingand low varian e

ompo-nent.Thesetofparameters

a, b, σ ² _a , σ _b ²

^{bywhi hwederivedtheACFinFig2.6}

2.134 2.136 2.138 2.14 2.142 2.144 2.146 2.148 2.15 2.152

0 10 20 30 40 50 60

R φ (m)

m

ACF at 256 kBaud

ACF of the double AR1 approximation

Figure 2.6: Auto orrelation fun tion of the ontinuous-time PN pro ess

θ(t)

and ofthe symbol-timePN pro ess

ψ _n

^at

256

^kBaud.

are listed in Table 2.3, together with the same parameters obtained for a set

of sele ted baudrates. As mentioned, themajor reason of performan e

degra-Baudrate [KBaud℄

a σ ² _a b σ ² _b

64 1 − 1.76 · 10 ⁻⁵ 2.163 0.263 8.51 · 10 ⁻³ 128 1 − 8.9 · 10 ⁻⁶ 2.143 0.447 1.150 · 10 ⁻² 256 1 − 4.45 · 10 ⁻⁶ 2.136 0.604 1.399 · 10 ⁻² 512 1 − 2.15 · 10 ⁻⁶ 2.140 0.758 1.558 · 10 ⁻² 1024 1 − 1.05 · 10 ⁻⁶ 2.126 0.871 1.611 · 10 ⁻² 2048 1 − 4.7 · 10 ⁻⁷ 2.144 0.942 1.609 · 10 ⁻²

Table2.3: Parameters ofthesymbol-time phasenoise double-AR1

approxima-tion, provided for dierent baudrate values.

dation is the faster omponent, whi h is very di ult to tra k. For example,

from Fig.2.6 itis lear that su ha omponent at

256

^{KBaud has astandard}

deviation of about

4

^{degrees and the} orresponding samples at symbol-time arealmost independent.

Taking advantage of the double-AR1 PN des ription, it is interesting to

investigate the IR of a hannel ae ted by SATMODE phase noise, in order

to quantify the mutual information rate loss due to su h a PN. In parti ular

we evaluate IR a ording to the Arnold and Loeliger method, for dierent

hannel representations and dierent dete tion algorithms (ne essary for the

forward metri omputation). In parti ular, we onsider two possible hannel

generation models:

ˆ the real hannel, whi h works in a pra ti ally ontinuous-time setting

(i.e.,samplingtime

10 ⁻⁷

^{s),byaddingan}oversampledversionofthePN pro ess

θ(t)

^{generated a ording to theCT double-AR1}approximation, with IA set of parameters; insu h a ase an ISI ee t on the su ient

statisti smayo urs;

ˆ the auxiliary hannel, whi h introdu es a dis rete-time PN pro ess

ψ _n

onstant over ea hsymbolperiod and following the STdouble-AR1

ap-proximation.

The dete tion algorithms we propose are all based on MAP symbol

dete -tion, but ingeneral they annotbe mat hed to the hannelgenerationmodel.

Moreover they areall derived following a Bayesian approa h, i.e., we assume

a proper probabilisti model for the phase noise

{θ n }

^{, and we exploit it for}

deriving algorithmsfor MAP symboldete tion. Indetailwe onsider:

ˆ DP-BCJR,alreadyanti ipatedinSe tion2.4.1anddetailedinSe tion2.4.3;

inthe algorithm derivation we onsider a rst order auto-regressive PN

modelor a WienerPN model.

ˆ Double-DP-BCJR, whi h is a generalization (provided inSe tion 2.4.3)

ofthe DP-BCJR;inthealgorithmderivationwe onsidertworst order

auto-regressive omponentsfor the PN modeland not justone;

ˆ Improved-DP-BCJR (Se tion 2.4.3), whi h assumes both the two AR1

omponents, but onsiders all the sample of one omponent (the faster

one)independent to ea h others.

We notethatwhenthereal hannelgenerated bytheCTdouble-AR1appro

x-imationis onsidered,thereisnoonedete torthat anbe onsideredmat hed

to the hannel sin e all algorithms are based on a phase dis retization

te h-nique (even if the number of levels is very large) and all algorithms onsider

justone PN samplefor ea h symbolinterval (and not a ontinuous-time

gen-eration). Howeverit isinteresting toevaluate theIRfor thereal hannelwith

theDouble-DP-BCJR inorder to quantify theee t of theISIdistortion. On

theotherhand,when we take into a ount theST hannelgeneration, ex ept

for the phase dis retization (whi h auses justa little mismat h between the

dete tor and the hannel model), the double-DP-BCJR is the optimum one

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 example,we see that thefast omponents exhibits a}

b

^valueso

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.

Nel documento Advanced Modulation/Demodulation Schemes For Wireless Communications (pagine 44-54)