Analyse théorique du compromis entre le rendement et la...

Analyse theorique du compromis entre lerendement et la linearite de l’amplificateur de

puissance dans un contexte OFDM

O. Abel GOUBA

Signal Communications & Embedded ElectronicsIETR/SUPELEC, Rennes campus,

France

Seminaire SCEE03 Mai 2012

Background and motivation Linearity performance PA efficiency study Joint combination Conclusion References

Outline

1 Background and motivation

2 Linearity considering PAPR reduction and PredistortionPredistortion error ε definitionExpressions of 1st and 2nd order moment of εLinearity performance measured by EVM

3 PA efficiency considering PAPR reduction and PredistortionDefinition of PA efficiencyExpression of the power efficiency

4 Joint combination of PAPR reduction and PredistortionDiscussion and analysisSimulations and results

2/24


Background and motivation


2 Linearity considering PAPR reduction and Predistortion

3 PA efficiency considering PAPR reduction and Predistortion

4 Joint combination of PAPR reduction and Predistortion

3/24


Background

High power fluctuations cause OFDM problems

PA P

R

O

B

L

E

M

S OFDM signal with high peaks Non-linear device

0inV

outV

High power fluctuations of multi-carrier signals like OFDM,represented by Peak-to-Average Power Ratio (PAPR).

In-bound and out-of-bound distortions caused by transmitter’sPower Amplifier (PA) for a signal with high peaks.

4/24


Background

PA’s behavior to high peaks signals

IBO*

Signal to be amplified

A perfect linearity isobserved when thepower efficiency islow and vice versa.

Two solutions proposed separately:1 Linearization that compensates the PA non-linearities.

(eg: Predistortion, Feedback, etc.)2 PAPR reduction that reduces signal fluctuations and thus

increases PA efficiency. (eg: Clipping, Tone Reservation, etc.)

5/24


Background

Combination of PAPR reduction and Linearization

S

O

L

U

T

I

O

N

S OFDM signal with

high peaks

Power amplifier

0inV

outV

PAPR Reduction Linearization

Since PAPR reduction and Linearization are complementarysolutions, two approaches of combination:

1 Simple combination of a PAPR reduction technique followedby Linearization.

2 Joint combination that takes into consideration the mutualeffects of PAPR reduction and Linearization.

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Background


S

O

L

U

T

I

O

N

S OFDM signal with

high peaks

Power amplifier

0inV

outV


PAPR reduction and Linearization are designed and optimizedseparately:

A joint analysis of linearity and PA efficiency.A trade-off is needed for optimal combination.

Our approach: Theoretical analysis.

6/24


Background


S

O

L

U

T

I

O

N

S OFDM signal with

high peaks

Power amplifier

0inV

outV


PA model: Memory-less Solid State Power Amplifier (SSPA).

Linearization method selected: Predistortion.

PAPR reduction methods selected: Probabilistic methods andClipping technique.

6/24


Motivation

Study objectives

Theoretical expressions of the linearity performance measuredby EVM metric and the PA efficiency.

Analytical trade-off that ensures a good linearity withreasonable efficiency by combining PAPR reduction andPredistortion.

7/24


Linearity considering PAPR reduction and Predistortion


2 Linearity considering PAPR reduction and PredistortionPredistortion error ε definitionExpressions of 1st and 2nd order moment of εLinearity performance measured by EVM



8/24


Error ε definition

Simplified transmission scheme

)(0)()( tjetrtx θ= )(11

1)(~)(~ tjetrtx θ=

]~[ 1rp ]~[ 2rh

)(22

1)(~)(~ tjetrtx θ=PAPR

reduction Predistortion POWER AMPLIFIER

)(3

1)(~)( tjetrty θ=

h(r) is PA’s characteristic and p(r) the predistorter’s function:

h(r) =r(

1 +(rA

)2b) 12b

, p(r) =r(

1−(rA

)2a) 12a

, r ∈ [0, A[,

a and b are “knee factors”; A the maximum output amplitude.

9/24


Error ε definition


)(0)()(tj

etrtx

)(

111)(~)(~ tj

etrtx

]~[ 1rp ]~[ 2rh

)(

221)(~)(~ tj

etrtx

PAPR reduction

Predistortion POWER AMPLIFIER

)(

31)(~)(

tjetrty

measure

Predistortion error ε:

ε (t) = |x1 (t)− y (t)| = |r1 (t)− r3 (t)|

Upper bound considered (closed form)[1]:

ε (t) ≤ r1 (t)∣∣∣1− 2

b−a2ab

∣∣∣[1] O. A. Gouba and Y. Louet “Predistortion Performance considering Peak to Average Power Ratio

Reduction in OFDM context”, in IEEE WCNC 2012, Paris, France, Apr. 2012.

9/24


Error ε definition


)(0)()(tj

etrtx

)(

111)(~)(~ tj

etrtx

]~[ 1rp ]~[ 2rh

)(

221)(~)(~ tj

etrtx

PAPR reduction

Predistortion POWER AMPLIFIER

)(

31)(~)(

tjetrty

measure

OFDM signal x(t) is a stationary complex Gaussian process soits amplitude r(t) converges to Rayleigh Distribution:

pr (r) =2r

Pre

−r2Pr

The distribution of the signal after PAPR reduction is neededfor the calculation of 1st or 2nd order moment of ε.

9/24


1st and 2nd order moment of the Predistortion error

1st order moment of ε

m1 , E [ε (r1)]

2nd order moment of ε

m2 ,E[|ε (r1)|2

]The distribution of the signal after PAPR reduction is neededfor the calculation of 1st or 2nd order moment of ε.

The signal distribution depends on the PAPR reduction method.

10/24



3 top categories of PAPR reduction methods[2]:

PAPR reduction methods

Adding signal methods Probabilistic methods Coding methods

[2] C. Langlais, S. Haddad, Y. Louet and N. Mazouz “Clipping noise mitigation with the capacityapproaching FEC codes for PAPR reduction of OFDM signals”, in MC-SS 2011, Herrshing, Germany, May 2011.

10/24






Data

bits

Coding

I

F

F

T

Mapping

Serial /

Parallel

S/P

Example of

(1-3)Reed Muller

kX nx

Coding methods use structuredsequences of frequency symbols thatare no longer independent identicallydistributed, therefore the resultedsignal is not Gaussian.

10/24






S

E

L

E

C

T

I

O

N

IFFT

IFFT

IFFT

1

kV

2

kV

P

kV

kX nx

Side

information

Selected Mapping principle

Probabilistic methods proceed basicallyto a linear transformation by multiplyingthe data symbols by a deterministicvector send as side information, thereforethe resulted signal remains Gaussian.

10/24






N

IFFT

N

IFFT

0X

1X

1NX

0C

1C

1NC

)(tc

)(tx )()( tctx

Adding signal technique

Adding signal methods depend on theadditive signal, therefore, the resultedsignal is not Gaussian anymore.

For our study, we consider Amplitudeclipping with the clipping ratio givenby γ =

Aclip√Pr

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Distribution of the signal after clipping[3]:

υ (r) = pr (r) 1r≤Aclip + Pr{r > A}δ (r −Aclip)

Probability that r larger than clipping threshold Aclip:

Pr{r > Aclip} =

∫ +∞

Aclip

pr (r) dr = e−A2

clipPr .

[3] P. Banelli, G. Leus, and G. B. Giannakis, “Bayesian Estimation of Clipped Gaussian Processes withApplication to OFDM”, in Proc. EUSIPCO, vol.1, pp.181-184, Sep. 2002.

10/24



Probabilistic case[1]

m(prob)1 , E [ε (r1)] =

∫ rmax

rmin

ε (r) pr (r) dr

• Considering ε upper bound and ρ =r21

Pr1:

m(prob)1 max =

√Pr1

∣∣∣1− 2b−a2ab

∣∣∣ [Γinc(3

2, ρ

)]ρ=ρmaxρ=ρmin

,∀ρmin

• For ρmin = 0 :

m(prob)1 max =

√Pr1

∣∣∣1− 2b−a2ab

∣∣∣Γinc(3

2, PAPRr1

)• with :

Γinc (z, a) =

∫ a

0xz−1e−xdx

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m(prob)2 , E [|ε (r1)|2] =

∫ rmax

rmin

|ε (r) |2pr (r) dr.

• Considering ε upper bound and ρ =r21

Pr1:

m(prob)2 max = Pr1

(1− 2

b−a2ab

)2 [(ρ+ 1) e−ρ

]ρ=ρminρ=ρmax

, ∀ρmin

• For ρmin = 0 :

m(prob)2 max = Pr1

(1− 2

b−a2ab

)2 (1− (PAPRr1 + 1) e−PAPRr1

)

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0.8 0.9 1 1.1 1.2 1.3

−70

−60

−50

−40

−30

−20

−10

Error after Selected Mapping with {Vk}={±1,± j}, P=10 and IBO=7dB

a/b

Err

or [d

B]

m1 simulated(10)

m2 simulated(14)

m1_max(11)

m1_max(13),ρmin=0

m2_max(15)

m2_max(16),ρmin=0

1st and 2nd order moments ofε with Selected Mapping(SLM) as PAPR reductionmethod.

m(prob)1 max =

√Pr1

∣∣∣∣1 − 2b−a2ab

∣∣∣∣Γinc ( 3

2, PAPRr1

)

m(prob)2 max = Pr1

(1 − 2

b−a2ab

)2 (1 −

(PAPRr1 + 1

)e−PAPRr1

)10/24



Amplitude clipping case[1]

m(clip)1 , E [ε (r1)] =

∫ rmax

rmin

ε (r) υ (r) dr.

• Considering ε upper bound and ρclip =A2clip

Pr1:

m(clip)1 max =

√Pr

∣∣∣1− 2b−a2ab

∣∣∣ [Γinc(3

2, γρ

)]ρ=ρclipρ=ρmin

+ ε (Aclip) e−γρclip .

• For ρmin = 0 :

m(clip)1 max =

√Pr

∣∣∣1− 2b−a2ab

∣∣∣Γinc(3

2, γPAPRr1

)+ ε (Aclip) e

−γPAPRr1 .

10/24



0.8 0.9 1 1.1 1.2 1.3

−70

−60

−50

−40

−30

−20

−10

Error after clipping, CR= 3dB, IBO=6dB

a/b

Err

or [d

B]

m1 simulated(17)

m2 simulated(20)

m1_max(18)

m1_max(19),ρmin=0

m2_max(21)

m2_max(22),ρmin=0

1st and 2nd order moments ofε with amplitude clipping asPAPR reduction method.

mclip1 max =

√Pr

∣∣∣∣1 − 2b−a2ab

∣∣∣∣Γinc ( 3

2, γPAPRr1

)+ ε

(Aclip

)e−γPAPRr1

m(clip)2 max =

Pr1

γ

(1 − 2

b−a2ab

)2 (1 − e

−γPAPRr1)

10/24


Expressions of EVM of the amplified signal

Linearity metric EVM represents Error Vector Magnitude.

EVM expressions deducted from m2:

EVM =

√√√√√ E[|ε(t)|2

]E[|x1(t)|2

]=

√m2

Pr1.

11/24




EVM (prob)max =

∣∣∣1− 2b−a2ab

∣∣∣√1− (PAPRr1 + 1) e−PAPRr1 .


EVM (clip)max =

∣∣∣1− 2b−a2ab

∣∣∣√(1

γ

)(1− e−γPAPRr1

).

[4] O. A. Gouba and Y. Louet “Theoretical analysis of the trade-off between efficiency and linearity of theHigh Power Amplifier in OFDM context”, in European Wireless 2012, Poznan, Poland, Apr. 2012.

11/24



EVM metric depending of the“knee factors” ratio a/b whenamplitude clipping is used asPAPR reduction technique

12/24



EVM metric depending of the“knee factors” ratio a/b whenSLM is used as PAPRreduction technique

12/24



EVM metric depending onPAPRr1 for a/b = 0.65,0.875 and 0.975 whenAmplitude Clipping is used

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Recap

Theoretical expressions are relatively closed to the simulationaccordingly to the upper bound of the Predistortion error ε.

The linearity measured by EVM depends mainly onPredistortion performance but also on PAPR reduction.

Linearity (EVM equals zero) is achieved for an effective PAPRreduction and a perfect Predistortion (a = b).

12/24


PA efficiency considering PAPR reduction and Predistortion



3 PA efficiency considering PAPR reduction and PredistortionDefinition of PA efficiencyExpression of the power efficiency


13/24


PA efficiency

Power budget of a power amplifier

POWER AMPLIFIER

Pin

Input power Pout

Output power

PDC

DC power

Plost

Dissipated power

PA efficiency definition:

ηDC =PoutPDC

Expression of PA efficiency[5]:

ηDC = G · exp (−g · PAPRr1)

Examples of power efficiencyparameters[5]

Class G [%] gA 58.7 0.1247B 90.7 0.1202

[5] D. Wulich, “Definition of efficient PAPR in OFDM”, IEEE Com. Letters, vol. 9, n. 9, p. 832 - 834, sept.2005.

14/24


PA efficiency

Power efficiency depending ofPAPRr1 for classes A and BPA

15/24


PA efficiency

PA’s behavior to high peaks signals

IBO*

Signal to be amplified

PA efficiency mainly depends on PAPR reduction performance.

The maximum possible efficiency is achieved when the peakpower of amplified signal coincides with the saturation power.

16/24


Joint combination of PAPR reduction and Predistortion




4 Joint combination of PAPR reduction and PredistortionDiscussion and analysisSimulations and results

17/24


Discussion and analysis

EVM expression depends on signal PAPR and the “kneefactors” a and b.

Efficiency expression depends mainly on signal PAPR.

Relationship between EVM and PA efficiency is carried out bysubstituting these two expressions[4]:

EVM (prob)max =

∣∣∣1− 2b−a2ab

∣∣∣√

1−(ηDCG

) 1g

(1− 1

gln(ηDCG

))

18/24



EVM metric depends on thepower efficiency of class A PAfor different values of the“knee factors” ratio a/b whenSLM is considered as PAPRreduction technique

19/24



Recommendations for PAPR reduction and Predistortioncombination:

High PAPR reduction gain and minimum drawbacks

Effective Predistortion designed taking into account PAPRreduction (maximum linearity)

Input Back-Off (IBO) of the PA equal to the signal’s PAPR(maximum efficiency)

Adaptive Predistortion can also be considered.

19/24


Simulations and results

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

IBO[dB]

EV

M [%

]

EVM metric measured between the amplified signal and the PAPR reduced signal

Clipping+Predistortion+PA with PAPRtarget=5dB

SLM+Predistortion+PA with PAPRtarget=5dB

class A power efficiency

Good trade−off betweenefficiency and linearity

Trade-off between linearity andpower efficiency whenamplitude clipping and SLMare used with Predistortion andSSPA Class A power amplifier.

20/24


Simulations and results

Recap

The main goal of our trade-off analysis is to ensure goodlinearity with reasonable efficiency.

Relationship between EVM and PA efficiency.

Optimal combination of PAPR reduction and Predistortion.

20/24


Conclusion

EVM = f (PAPR, a, b)

The quality in term of linearity and efficiency of thetransmitters depends on PAPR reduction and Linearization.

The trade-off in OFDM context can be estimated without theneed of extensive simulations

21/24


References I

[1] O. A. Gouba and Y. Louet “Predistortion Performanceconsidering Peak to Average Power Ratio Reduction in OFDMcontext”, in IEEE WCNC 2012, Paris, France, Apr. 2012.

[2] C. Langlais, S. Haddad, Y. Louet and N. Mazouz “Clippingnoise mitigation with the capacity approaching FEC codes forPAPR reduction of OFDM signals”, in MC-SS 2011,Herrshing, Germany, May 2011.

[3] P. Banelli, G. Leus, and G. B. Giannakis, “BayesianEstimation of Clipped Gaussian Processes with Application toOFDM”, in Proc. EUSIPCO, vol.1, pp.181-184, Sep. 2002.

22/24


References II

[4] O. A. Gouba and Y. Louet “Theoretical analysis of thetrade-off between efficiency and linearity of the High PowerAmplifier in OFDM context”, in European Wireless 2012,Poznan, Poland, Apr. 2012.

[5] D. Wulich, “Definition of efficient PAPR in OFDM”, IEEECom. Letters, vol. 9, n. 9, p. 832 - 834, sept. 2005.

[6] O. A. Gouba and Y. Louet, “Joint study PAPR reduction andHPA predistortion”, URSI GASS 2011, Istanbul, Turkey, Aug.2011.

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Thank for your attention.

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Thank for your attention.

[email protected]

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