Numeri al results

2.4 IR of CPMs over Channel Ae ted by Phase Noise

2.4.4 Numeri al results

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)).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2 0 2 4 6 8 10

Information Rate [bit/ch.use]

E _b /N ₀

Coherent

Wiener (σ _∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64k Baud), D-DP

(a)

E b /N 0

^from^-2^to¹⁰^dB

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 0.5 1 1.5 2 2.5 3 3.5 4

Information Rate [bit/ch.use]

E _b /N ₀

Coherent

Wiener (σ _∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64 kBaud), D-DP

(b)

E b /N 0

^from⁰^to⁴^dB

Figure2.10: Informationratefor abinarymodulation with

2

^-RC

h = 1/3

^and

metri s omputation(see(2.38),(2.49),(2.53)).Insu haway,we onsiderthe

mismat hbyredu ingthe reliabilityofthedete torde isions and,byproperly

optimizingthe parameter

N _I

^,^we ân în
rease^theÎR.^Theôbtained ÎRîs ^still

a hievablebythedete torwhi h onsidersaredu edthermalnoisevarian e.In

thedete torasso iatedtotheWiener hannelinFig2.10,we hoose

N _I = 10 ⁻²

and

D = 101

quantizationlevels.Theinformation ratelosswithrespe ttothe oherent hannel (i.e., the IR loss due to the PN) an be of approximately

1

^dBât ^high ÎR ^vâlues. În ^Fig. ^2.10 ^we âlso ônsider ^SATMODE ^PN ^model

CT generated and ST generated, a symbol frequen y of

64

^kBaud ^and ^the

D-DP dete tor. In both ases we dis retize the two PN omponents in two

dierent ways. For the slow omponent we uniformly dis retized theinterval

[−π, π)

^with

D _a = 157

^levels, ^while ^for ^the ^fast ômponent ^we âdopt â ^non

uniform dis retization, but thi ker around the origin. For su h a omponent

just

D _b = 19

^levels âre ^su
ient. ^Hen
e, ^looking ât ^the ^two ûrves ôbtained

whenwe onsiderCT andSTdouble-AR1PNgeneration andD-DPdete tor,

we anverifythattheIRlossduetothePNispra ti allynegligible(

0.1

^dB^or

less). We understand thatthe ISIee t, presentintheCT generation ase, is

notrelevant,sin ethe urve orrespondingtosu ha aseisalmostoverlapped

to the urve asso iated to the ST ase. So in the following we will always

onsider justtheST double-AR1model. In on lusion,when we employa

D-DPdete tor,we seethatinprin iplewe ana hievealmostthesameIRofthe

AWGN hannel. However, the omputational omplexity (i.e., the number of

trellis state) of the D-DP is too high and hen e su h a dete tor annot be of

pra ti al interest.

Thus,inFig. 2.11we resort tosome simplied dete tors forthe same

s e-narioof Fig.2.10.Indetail, we generate thePN followingtheSTdouble-AR1

generation model but we employ a simple DP dete tor (with

D = 157

quan-tization levels,

N _I = 10 ⁻¹

^and

a = 1 − 18 · 10 ⁻⁶

^and

σ ² _a = 2.1625

^). ^W^e ^have

a redu tion in the omputational omplexity of the dete tor but we pay the

strongmismat h between the hannelandthedete torintermsofaboutthan

0.5

^dB ôf ÎR.^Moreover, ^from ^Fig ^2.12 ^we ^see ^that ÎR onstrained to theDP

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2 0 2 4 6 8 10

Information Rate [bit/ch.use]

E _b /N ₀

Coherent ST (64 kBaud), DP ST (64 kBaud), I-DP Slow-AR1 (64 kBaud), DP Fast-AR1 (64 kBaud), DP

(a)

E b /N 0

^from^-2^to¹⁰^dB

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

0.5 1 1.5 2 2.5 3 3.5

Information Rate [bit/ch.use]

E _b /N ₀

Coherent ST (64 kBaud), DP ST (64 kBaud), I-DP Slow-AR1 (64 kBaud), DP Fast-AR1 (64 kBaud), DP

(b)

E b /N 0

^from^0.5 ^to^3.5 ^dB

Figure2.11: Informationratefor abinarymodulation with

2

^-RC

h = 1/3

^and

is justfew tenthsof dB far from IR onstrained to the D-DP, sowe on lude

that asimplied dete torbased on a singlerst-order phasenoise model (see

forexampletheMM-DPalgorithminSe tion4.4.2) ana hieveapproximately

the same performan e of a more omplex algorithm base on D-DP.When we

employa I-DP,with

D = 157

quantization levels for theslow omponent, we deriveanIRslightlygreaterthantheIR onstrainedtotheDP.Inotherwords,

bysimplyin reasingthebran hmetri omputationoftheDP (asdone inthe

I-DP algorithm), we annot re over the few tenths of dB whi h separate the

IR urve onstrained to theDP, from that one onstrained to the D-DP.

Fi-nally,inFig.2.11wealso onsiderthe ase inwhi hboththe hannelandthe

dete torarebasedonasingle-AR1PNmodel,whi hisinturnmat hedtothe

slowor tothefastSATMODEPN omponents.We anseethattheIR urves

ofboththe slowandthe fast omponentsarepra ti ally overlapped totheIR

urve of the oherent ase. Hen e, for the lowest spe tral e ient CPM

for-matwe have onsidered, both omponents anbetra ked byadete torbased

on asingle-AR1PN modelmat hed to the hannel; thatresult wasexpe ted,

sin e alsothe performan eoftheD-DP dete torinFig.2.10isapproximately

overlapped to the oherent urve.

We now onsider a more spe trally e ient CPM s heme: a quaternary

2

^-RC ^with

h = 1/5

^and ^with ^symbol ^frequen
y ^of

64

^kBaud. ^In^Fig. ^2.13, ^we

present,for su h aquaternary s heme, the same urves reported inFig.2.10.

Alsointhis ase,thelargerIRloss(from

1

^to

2

^dB)^is^exhibited^by^the^Wiener

hannel, with

σ _∆ = 5

^both ⁱⁿ^the^hannel ^andⁱⁿ^the^DP ^dete
tor⁽

N _I = 10 ⁻²

and

D = 101

quantization levels). The IR onstrained to theD-DP dete tor (with

D _a = 101

^and

D _b = 19

⁾^is^less^than ^one^half^dB⁽

0.2

^dBapproximately) fromthe oherent IR,bothforthe hannelwithCT PNgenerationandforthe

hannelwithSTPNgeneration.Alsointhis asewe annegle tthedistortion

ee t due to the ontinuous-time PN generation (sin e the urve related to

theCT aseisoverlapped totheST urve),andwithaD-DPdete torwe an

obtain aperforman equite lose tothe oherent one.

In Fig. 2.14, we take into a ount redu ed- omplexity dete tors. First of

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2 0 2 4 6 8 10

Information Rate [bit/ch.use]

E _b /N ₀

Coherent ST (64 kBaud), DP ST (64 kBaud), D-DP

(a)

E b /N 0

^from^-2^to¹⁰^dB

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

0 0.5 1 1.5 2 2.5 3 3.5 4

Information Rate [bit/ch.use]

E _b /N ₀

Coherent ST (64 kBaud), DP ST (64 kBaud), D-DP

(b)

E b /N 0

^from^0.5 ^to^3.5 ^dB

Figure 2.12: Information rate for a binary modulation with

2

^-RC

h = 1/3

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

-2 0 2 4 6 8 10 12

Information Rate [bit/ch.use]

E _b /N ₀

Coherent

Wiener (σ _∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64 kBaud), D-DP

(a)

E b /N 0

^from^-2^to¹²^dB

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

Information Rate [bit/ch.use]

E _b /N ₀

Coherent

Wiener (σ _∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64 kBaud), D-DP

(b)

E b /N 0

^from²^to⁸^dB

Figure2.13: Informationratefora quaternarymodulationwith

2

^-RC

h = 1/5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-2 0 2 4 6 8 10 12

Information Rate [bit/ch.use]

E _b /N ₀

Coherent ST (64 kBaud), DP ST (64 kBaud), I-DP Slow-AR1 (64 kBaud), DP Fast-AR1 (64 kBaud), DP

(a)

E b /N 0

^from^-2^to¹²^dB

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Information Rate [bit/ch.use]

E _b /N ₀

Coherent ST (64 kBaud), DP ST (64 kBaud), I-DP Slow-AR1 (64 kBaud), DP Fast-AR1 (64 kBaud), DP

(b)

E b /N 0

^from²^to⁷^dB

Figure2.14: Informationratefora quaternarymodulationwith

2

^-RC

h = 1/5

all, by omparing the IR urves obtained by the DP dete tor operating over

a hannel ae ted by just one PN omponent (with

D _a = 157

^for ^the ^slow

omponent and

D _b = 19

^for ^the^fast^one), ^we ^derive^that ^while^the^slow

om-ponent isperfe tlytra kedbythe DPdete tor (andthus itsIRisthesame of

the oherent ase),the urverelatedtothefast omponentisslightlyseparated

from the oherent one. Hen e we derive that fast AR1 PN pro ess is harder

to be tra ked, and it is the only ause of the the

0.2

^dB ôf ÎR ^loss ôf ^D-DP

dete tor, remarked inFig.2.13.

If we employ a simple DP algorithm (with

D = 157

quantization levels,

N _I = 10 ⁻¹

^and

a = 1 − 18 · 10 ⁻⁶

^and

σ _a ² = 2.1625

⁾ ^on ^the ^double-AR1 ^PN

hannel, the IR losswith respe t to the oherent IR is approximately double

of the IR onstrained to the D-DP, as depi ted in Fig. 2.15. Moreover, IR

onstrainedtotheDPdegradesatlargeIRvalues,andthelossbe omesalmost

1

^dB.^Thus, ^passing^from âômplex ^D-DP^dete
tor ^toâ ^simpler ^DP^dete
tor,

we expe t a BER performan e degradation. We take into a ount the I-DP

algorithm (with

D _a = 157

⁾ ând âlso ⁱⁿ^that âse ît ^does ^not împrove ^the^DP

performan e.

Finally inFig. 2.16 and 2.17, we present the same urves of Fig. 2.14 for

two dierent symbol rate frequen ies,

256

^and

2048

^kBaud. ^What ^is ^relevant

is the ase at

256

^kBaud, ^where ^we ^an ^see ^that ^the ^IR degradation related to DP dete tion ofjust thefast omponent, isgreater than its ounterpart at

the

64

^kBaud^(of ^Fig.^2.14).^Thisîs ânûnexpe
ted^result, ^sin
eⁱⁿ^general ^the

phasenoise impairment is strongerat lowerbaudrates.

Nel documento Advanced Modulation/Demodulation Schemes For Wireless Communications (pagine 61-69)

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

2.4.4 Numeri al results

(σ n , θ (a) n )

θ n (b)

n

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2 0 2 4 6 8 10

Information Rate [bit/ch.use]

E b /N 0

Coherent

Wiener (σ ∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64k Baud), D-DP

E b /N 0

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 0.5 1 1.5 2 2.5 3 3.5 4

Information Rate [bit/ch.use]

E b /N 0

Coherent

Wiener (σ ∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64 kBaud), D-DP

E b /N 0

2

h = 1/3

N I

N I = 10 −2

D = 101

1

64

[−π, π)

D a = 157

D b = 19

0.1

D = 157

N I = 10 −1

a = 1 − 18 · 10 −6

σ 2 a = 2.1625

0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2 0 2 4 6 8 10

Information Rate [bit/ch.use]

E b /N 0

Coherent ST (64 kBaud), DP ST (64 kBaud), I-DP Slow-AR1 (64 kBaud), DP Fast-AR1 (64 kBaud), DP

E b /N 0

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

0.5 1 1.5 2 2.5 3 3.5

Information Rate [bit/ch.use]

E b /N 0

Coherent ST (64 kBaud), DP ST (64 kBaud), I-DP Slow-AR1 (64 kBaud), DP Fast-AR1 (64 kBaud), DP

E b /N 0

2

h = 1/3

D = 157

2

h = 1/5

64

1

2

σ ∆ = 5

N I = 10 −2

D = 101

D a = 101

D b = 19

0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2 0 2 4 6 8 10

Information Rate [bit/ch.use]

E b /N 0

Coherent ST (64 kBaud), DP ST (64 kBaud), D-DP

E b /N 0

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

0 0.5 1 1.5 2 2.5 3 3.5 4

Information Rate [bit/ch.use]

E b /N 0

Coherent ST (64 kBaud), DP ST (64 kBaud), D-DP

E b /N 0

2

h = 1/3

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

-2 0 2 4 6 8 10 12

Information Rate [bit/ch.use]

E b /N 0

2.4 IR of CPMs over Channel Ae ted by Phase Noise

(σ _n , θ ^(a) n )

θ n ^(b)

E _b /N ₀

Wiener (σ _∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64k Baud), D-DP

E _b /N ₀

Wiener (σ _∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64 kBaud), D-DP

N _I

N _I = 10 ⁻²

D _a = 157

D _b = 19

N _I = 10 ⁻¹

a = 1 − 18 · 10 ⁻⁶

σ ² _a = 2.1625

E _b /N ₀

E _b /N ₀

σ _∆ = 5

N _I = 10 ⁻²

D _a = 101

D _b = 19

E _b /N ₀

E _b /N ₀

E _b /N ₀

Wiener (σ _∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64 kBaud), D-DP

E _b /N ₀

Wiener (σ _∆ =5 deg.), DP CT (64 kBaud), D-DP ST (64 kBaud), D-DP

E _b /N ₀

E _b /N ₀

D _a = 157

D _b = 19

N _I = 10 ⁻¹

a = 1 − 18 · 10 ⁻⁶

σ _a ² = 2.1625

D _a = 157