volterra series based rf power amplifier behavioral modeling

- 1 -June 11, 2006 University College Dublin

IMS-2006 Workshop

Volterra Series Based RF Power Amplifier Behavioral Modeling

Anding Zhu, and Thomas J. Brazil

RF & Microwave Research GroupUniversity College Dublin

Dublin 4, Ireland

Tel: +353-1-7161912Fax: +353-1-2830921

E-mail: [email protected]

- 2 -IMS-2006, June 11, 2006 University College Dublin

Introduction- What is Volterra Series?- Why Volterra Series?

Volterra Model Extraction- Discrete Time Domain

Simplified Model Structures- Special Cases- Direct Pruning- Reformatting the Expansion

Conclusions

Outline


Linear systemwith memory

Nonlinear systemwithout memory

1

0( ) ( ) ( )

(0) ( ) (1) ( 1)

m

iy n w i x n i

w x n w x n

−

=

= × −

= + − +

∑1

02

0 1 2

( ) [ ( )]

( ) [ ( )]

mi

ii

y n a x n

a a x n a x n

−

=

=

= + + +

∑

1 1 1

1 20 0 0

1 1 1

30 0 0

( ) ( ) ( ) ( , ) ( ) ( )

( , , ) ( ) ( ) ( )

m m m

i i j

m m m

i j k

y n h i x n i h i j x n i x n j

h i j k x n i x n j x n k

− − −

= = =

− − −

= = =

= − + − −

+ − − − +

∑ ∑∑

∑∑∑

Volterra Series Nonlinear systemwith memory

2nd-order Volterra kernel

3rd-order Volterra kernel

1st-order Volterra kernel

Volterra Series in the Discrete Time Domain


Volterra Series in the Discrete Frequency Domain

1 21 2

1 2 31 2 3

1

2 1 2 1 2, 1

3 1 2 3 1 2 3, , 1

( ) ( ) ( )

( , ) ( ) ( )

( , , ) ( ) ( ) ( )

n n

M

n nk k Mk k m

M

n n nk k k Mk k k m

Y m H m X m

H k k X k X k

H k k k X k X k X k

=− ++ =

=− ++ + =

=

+

+

+

∑

∑

Using the Discrete Fourier Transform (DFT), we could write the Volterra series in the frequency domain as

Multi-dimensional DFT of Volterra Kernels


1 2 1 3

1 2 1 3 2 4 0 5 4

1

101 1 1

*3 1 2 3 1 2 3

0 0

1 1 1 1 1* *

5 1 2 3 4 5 1 2 3 4 50

( ) ( ) ( )

( , , ) ( ) ( ) ( )

( , , , , ) ( ) ( ) ( ) ( ) ( )

m

im m m

i i i i

m m m m m

i i i i i i i i i

y n h i x n i

h i i i x n i x n i x n i

h i i i i i x n i x n i x n i x n i x n i

−

=

− − −

= = =

− − − − −

= = = = =

= −

+ − − −

+ − − − − −

+

∑

∑∑∑

∑∑∑∑∑

Low-pass Equivalent Model

input output

PA

0ω 0ω

0

( )Re[ ( ) ]j t

x tx t e ω

=0

( )Re[ ( ) ]j t

y ty t e ω

=

conjugate

In wireless communication systems, a PA is normally used to transmit modulated signals where only the envelope carries useful information and even-order nonlinear components can be omitted. Therefore, the Volterra model can be written in the low-pass equivalent format as


AdvantagesA firm mathematical foundation

-- closed form expression. Output is linear in relation to the coefficients

-- model can be extracted in a direct way by using linear optimization.Easily implemented in hardware

-- the structure is the same as for linear filters, number of coefficients can be decided in advance.

Volterra Series Based Models

Model structure too complicated-- number of coefficients increases exponentially with the degree of

nonlinearities and with memory length. (e.g., 4320 for 5th order with memory 8).Model extraction quite difficult-- involves complicated measurement procedures and complicated algorithms.Convergence problem-- cannot model discontinuities.

Problems


Introduction- What is Volterra Series- Why Voterra Series

Volterra Model Extraction- Discrete time domain

Simplified Model Structures- Special Cases- Direct Pruning- Reformat the Expansion

Conclusions

Outline


Model Extraction Methodologies

2. Arbitrary Inputs (Statistical Signals)Use arbitrary sampled inputs and outputs in the discrete time domain.Simple measurement configuration, only one measurement needed.Can formulate the model extraction problem in a vector-matrix form, and then find

the solution by employing least squares estimation.A critical issue is that the matrix could be ill-conditioned. These excitation inputs have to be relatively broadband to cover the entire range of

frequencies of interest. Has become more popular in recent years since practical test equipment is available.

1. Specialized Inputs (Deterministic Signals)Use Two-/Multi-tone or Impulsive signals. Power and frequency sweeps to cover multiple amplitude and frequency ranges. Not experimentally efficient: involve complicated measurement procedures.

Several methodologies have been used for Volterra kernel extraction, including spectral analysis in the frequency domain, least squares in the time-domain, some solutions in the mixed-domain, and so on. However, in terms of excitation signals, they can be generally clustered in two groups :


Example: Test Bench for Low-pass Model Extraction (Discrete Time Domain)

VSAPAESG

Send Test I/QSignal Back to ADS

3GPP Downlink Signal Source

ESG4438CSinkE1

RFPowOn=NOArbOn=YESScalingFactor=1FileName="ADS_3GPP_DL"SampleClk=(SamplesPerChip*3.84) MHzIQModFilter=throughARBRefFreq=10e6 HzARBRef=InternalAmplitude=SignalPower+RF_Cable_LossFrequency=RF_Freq HzStop=ESG_StopStart=ESG_StartInterfaceSelector=0Address=19Interface="gpib"

NumericSinkSource

Stop=3000Start=60ControlSimulation=YESPlot=None

CxToRectC1

VARUser_Defined_Variables

ESG_Power=SignalPower+RF_Cable_LossRF_Cable_Loss=0ESG_Stop=ESG_Start+NumSlotsMeasured*SamplesPerSlotESG_Start=(ChipsPerFilterLength/2)*SamplesPerChipSamplesPerSlot=ChipsPerSlot*SamplesPerChipRF_Freq=2140 MHzNumSlotsMeasured=30SignalPower=-5

EqnVar

TimedToCxT1

RESR1R=50 Ohm

VARSimulation_Variables

EqnVar

DFController

TimedSinkT2

ControlSimulation=YESStop=20 usecStart=0 usecPlot=None

SUB_3GPP_DL_SourceS1

FCarrier=RF_FreqSignalPower=dbmtow(SignalPower)

3GPPSource Test

Ref

3GPP Downlink Sink

VARUser_Defined_Variables

ESG_Power=SignalPower+RF_Cable_LossRF_Cable_Loss=0.44ESG_Stop=ESG_Start+NumSlotsMeasured*SamplesPerSlotESG_Start=(ChipsPerFilterLength/2)*SamplesPerChipSamplesPerSlot=ChipsPerSlot*SamplesPerChipRF_Freq=1950 MHzNumSlotsMeasured=1SignalPower=0

EqnVar

T imedSinkT5


DelayRFD1

IncludeCarrierPhaseShift=NoInterpolationMethod=noneDelay=(1/3.84/8*29) usec

VARSimulation_Variables

EqnVar

DFController

VSA89600SourceV2

RecordingFile=SetupFile="3gppdl.set"TStep=0 secRepeatData=Single passVSAT race=BPause=NOOutputType=TimedControlSimulation=NOVSAT itle="Simulation source"

VSA

TimedSinkT3


TimedToCxT4

DelayD2N=29

NumericSinkSink

Stop=3000Start=60ControlSimulation=YESPlot=None

USampleRFU1

ExcessBW=0.5InsertionPhase=0Ratio=4Type=PolyPhaseFilter

SplitterRFS1

Generate I/Q Signal in ADS

Send SimulatedI/Q Signal to ESG

Measured I/Q Signal Back in ADS

Arbitrary Signal Source, e.g., W-CDMA (3.84 MChips/Sec)


=y Xθ ( ) ( ) ( )e k d k y k= −

1ˆ ( ) ( )H Hn −=θ X X X d

21

( ) | ( ) |N Hk

J e kθ=

= =∑ e e

Parameters to be extracted:

min

Solution

1 2

1 2

(1), (1),(2), (2),

N M

x xx x

×

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

X

1

2

1M

hh

×

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

θ

then

1

(1)(2)

N

yy

×

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

y

and

Data gathered:(1), (2), ( )x x x N(1), (2), ( )d d d N

Example: Least Squares

input:output:

Formulate input and output vector:

Products of inputs

Volterra kernels


Linear System Volterra System

time-shift lost

1n nX X +→

3

3

3

3

3

3 3

2

3

2

( 1)( )

( 1)( 1)

( ) ( )( 1) ( 1)( 2)

( ) ( 1) ( ) ( 1)( 1)( 2)

( ) ( 1) (

( 1)( 1)

( 1) ( )

( 2)( 2)

( )2)

) ( 2

x nx nx nx n

x n x nx n x nx nx n x n x n x nx nx n

x n x n x

x nx n

x n x n

x nx nn x n

nx

+⎡ ⎤ ⎡⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢ −⎢ ⎥ +⎢ ⎥⎢ ⎥⎢ ⎥

− −⎢ ⎥⎢ ⎥−⎢ ⎥

+ −⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−

++

+

−−−

−⎢ ⎥⎣ ⎦ ⎣

≠

⎤⎥⎥⎥

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎦

nX1nX +

1n nX X +→

( 1)( ) ( )

( 1) ( 1)

( 2) ( 2)(

( 1)

1) (1 )

x nx n x nx n

x

x n

x n m x n mx nn m

n

x m

+⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− + − +⎢ ⎥ ⎢ ⎥

− +⎢ ⎥

+

− +⎢ ⎥⎣ ⎦ ⎣ ⎦

=

time-shift kept

nX 1nX +

Input vector stepped from time (n) to (n+1):

Time-Shift Problem in ExtractionIn some applications, e.g., predistortion, fast adaptive linear filtering algorithms are normally used to extract the nonlinear Volterra kernels, but we may encounter a time-shift problem as follows:


New Volterra Model StructureUsing V-vector Algebra

x(n) x(n-1) x(n-2) x(n-3)x3(n) x3(n-1) x3(n-2) x3(n-3)x2(n)x(n-1) x2(n-1)x(n-2) x2(n-2)x(n-3) x(n)x2(n-1) x(n-1)x2(n-2) x(n-2)x2(n-3)x2(n)x(n-2) x2(n-1)x(n-3)

x(n)x2(n-2) x(n-1)x2(n-3)

time-shift kept

A key advantage of this approach is that any fast algorithm for linear filtercoefficient estimation (kernel extraction) can be used to extract the model.

[Zhu, IMS’03]

x(n)

x3(n)

FIR

FIR

FIR

FIR

∑


Sub-filter 1

Sub-filter 2

Sub-filter kInpu

t prim

ary

sign

al

+∑

( )d n

( )y n

( )x n

( )e n

1( )y n

2 ( )y n

( )ky n

-+

Adaptive Parallel RLS Updating

1( )e n

2 ( )e n

( )ke n

2

1( ) ( )

nn k

i ik

J n e nλ −

=

=∑min

update sub-filters

update system’s coefficients

( ) ( ) ( )i i ie n d n y n= −

don’t know( )id n

( ) ( )ie n e n=

( ) ( ) ( )e n d n y n= −

But… delayed convergenceTrade-offs possible

Shorter time cycle for each iteration

Updating of Coefficients in Parallel(Real-time system)

[Zhu, IMS’03]


33.5 33.7 33.9 34.1 34.3 34.5 34.7 34.9 35.1 35.3 35.5 35.70.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (usec)

Out

put A

mpl

itude

(Vol

ts)

Measurement ResultsAM/AM AM/PM ModelNew Behavioral Model

0 10 20 30 40 50-38

-36

-34

-32

-30

-28

-26

-24

Record SetN

MSE

(dB

)

AM/AM AM/PM ModelNew Behavioral Model

Sample Experimental Results

( ) ( )( ) ( )

2 2

, , , ,1

2 2

, ,1

: 10 log

M meas mod meas modI k I k Q k Q kk

M meas measI k Q kk

y y y yNMSE

y y

=

=

⎧ ⎫⎡ ⎤− + −⎢ ⎥⎪ ⎪⎣ ⎦= ⎨ ⎬⎡ ⎤+⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭

∑∑

(Time Domain)

Model Fidelity Evaluation: Normalized Mean Square Error


2 .1 3 2 .1 3 2 2 .1 3 4 2 .1 3 6 2 .1 3 8 2 .1 4 2 .1 4 2 2 .1 4 4 2 .1 4 6 2 .1 4 8 2 .1 5 -6 0

-5 0

-4 0

-3 0

-2 0

-1 0

0

1 0

F re q u e n c y (G H z )

Am

plitu

de (d

Bm

)

M e a s u r e m e n t R e s u ltsA M /A M A M /P M M o d e lN e w B e h a v io r a l M o d e l

56.4 / 57.0

43.3 / 44.5

13.17

VolterraModel

54.6

40.0

13.16

AM/AM AM/PM Model

56.5 / 57.4ACPR (dBC) (+/-10MHz)

43.5 / 44.4ACPR (dBC) (+/-5MHz)

13.19Gain (dB)

MeasurementResultsPerformance

Sample Experimental Results(Frequency Domain)





Conclusions

Outline


Three-Box Model

Linear Filter

Memoryless Nonlinearity

AM/PM AM/AM


AM/PM AM/AM Linear Filter


AM/PM AM/AM FilterFilter

Two-Box Models

Special Cases: Separate Nonlinearity and Memory

Wiener Model

Hammerstein Model

Only measure Nonlinearity (AM/AM and AM/PM) at the center frequency. Linear filters capture the memory.Cannot predict interaction.


PolySpectral Model

[Silva, ARFTG’01]

Other Similar Models- Parallel Wiener Model [Kenney, TransMTT’02]- Parallel Hammerstein Model- … …

Single frequency measurement instead of multi-dimension characterization.

Difficult to construct for higher-orders.

Memory and nonlinearities modeled separately.Interaction can not be fully characterized. Extraction effort increased for high orders.



Volterra Model Extraction- Discrete Time Domain


Conclusions

Outline


Pruned Model (I): Memory Polynomial Model

,1 0

[ ] [ ( )]Q m

qk q

q ky n a x n k

= =

= −∑∑

x3(n)

x3(n-1)

x3(n-3)0

0x3(n-2)

Lose fidelity of the model in many cases

[Kim, EL’01]

*All off-diagonal coefficients are set to zero.

| | 0m ni i− =

Polynomial functions


Pruned Model (II): Near-Diagonality Reduction

x(n-2)x(n-3)

x(n-1)x(n-3)

0

x2(n) x(n)x(n-2) x(n)x(n-3)x(n)x(n-1)

x(n-1)x(n-2)

x2(n-2)

x2(n-3)

x2(n-1)

| |m ni i l− ≤

l

Simplify model structure Keep high fidelity Trade-off possible

[Zhu, MWCL’04]

FIR

FIR

FIR

FIR

∑

FIR

Prim

ary

Sign

al G

ener

ator

x(n)

x3(n)

x(n) y(n)

*keep some off-diagonals which are close to main diagonal.


-32.0-35.5-35.9-36.1NMSE (dB)

1254133244Number of coefficients

0123l

“Full model” “Memory polynomial model”

“Pruned model”

Comparison of the pruned models


-- Model Low-order DynamicsPruned Model (III): Deviation Reduction

( , ) ( ) ( )e t x t x tτ τ= − −Set

1

( ) ( )rS r

r

y t a x t+∞

=

=∑

( )11

( ) [ ( ), , ] [ , ]r

D r r i ii

y t g x t e t dτ τ τ τ=

= ⋅ ∏∫ ∫ …

( ) ( ) ( )S Dy t y t y t= +The Classic Volterra series can be written as

where

Extract separately Complicated procedures

Normally truncate yD(t) to first order if small enough during

( )1( ) [ ( ), ] ,B

A

T

D Ty t g x t e t dτ τ τ

−= ∫

( , )e t τ [ , ]A BT T−

[Filicori, EL’91][Mirri, TransCS’02][Ngoya, IMS’00]

Modified/Dynamic Volterra series


In the finite discrete time domain

1

11

1 1 0( ) ( ) ( ) ( ) ( ) ( , )

P P Np p

p pp p i

y n a n x n x n g i e n i−

−

= = =

= +∑ ∑ ∑

( , ) ( ) ( )e n i x n i x n= − −Re-substitute

1

11

1 1 0( ) ( ) ( ) ( ) ( ) ( )

P P Np p

p pp p i

y n h n x n x n h i x n i−

−

= = =

= + −∑ ∑ ∑

Conclusion: Dynamic Volterra model is a truncated classic Volterra model, i.e., r=1.

1 1

1 1

11 1 1 0 1

( ) ( ) ( ) { [ ( ) ( , ) ( )]}r

r r

p rP P N Np p r

p p r jp p r i i i j

y n h n x n x n h i i x n i−

− −−

= = = = = =

= + −∑ ∑ ∑ ∑ ∑ ∏…

In the same way, we can write the classic Volterra series as


New Pruning Algorithm: 1 r L≤ ≤ “Deviation Reduction”

1 1

1 1

11 1 1 0 1

( ) ( ) ( ) { [ ( ) ( , ) ( )]}r

r r

p rP P N Np p r

p p r jp p r i i i j

y n h n x n x n h i i x n i−

− −−

= = = = = =

= + −∑ ∑ ∑ ∑ ∑ ∏…

1r = 1( ) ( )px n x n i− −

2 ( ) ( ) ( )px n x n i x n j− − −2r =

AdvantageThis model can be directly extracted from discrete time-domain measured input and output data by using least-squares estimation.More flexible: enable trade-off between accuracy and complexity.

[Zhu,IMS’06]

Control order of dynamics

In many cases, the first-order truncation is not enough to capture the dynamics of the system. More terms have to be added in to improve the accuracy of the model.

*Poster Session, Wed, June 14


Index Table

h(i), h(0,0,i), h(0,0,0,0,i)

x(n-i), x2(n)x(n-i), x4(n)x(n-i), …0 0 0 0 00 0 0 0 10 0 0 0 20 0 0 1 10 0 0 1 20 0 0 2 20 0 1 1 10 0 1 1 20 0 1 2 20 0 2 2 20 1 1 1 10 1 1 1 20 1 1 2 20 1 2 2 20 2 2 2 21 1 1 1 11 1 1 1 21 1 1 2 21 1 2 2 21 2 2 2 22 2 2 2 2

0 0 00 0 10 0 20 1 10 1 20 2 21 1 11 1 21 2 22 2 2

012

Static part

r=1:

r=2:

r=3:

r=4:

r=5:

r=0:

1st-order dynamics

2nd-order dynamics

3rd-order dynamics

4th-order dynamics

5th-order dynamics

x(n)x(n-i1)x(n-i2), x3(n)x(n-i1)x(n-i2), …

x(n-i1)x(n-i2)x(n-i3), x2(n)x(n-i1)x(n-i2 )x(n-i3), …

x(n)x(n-i1)x(n-i2 )x(n-i3)x(n-i4), …

x(n-i1)x(n-i2 )x(n-i3)x(n-i4)x(n-i5), …

h(0), h(0,0,0), h(0,0,0,0,0)

h(0,i1,i2), h(0,0,0,i1,i2)

h(i1,i2,i3), h(0,0,i1,i2,i3)

h(0,i1,i2,i3,i4)

h(i1,i2,i3,i4,i5)

x(n), x3(n), x5(n), …Example:


2441784818Number of Coefficients

-39.1-38.8-38.2-32.3NMSE (dB)

4321Order of Deviation

2125 2130 2135 2140 2145 2150 2155-40

-30

-20

-10

0

10

20

30

40

Frequency (MHz)

Am

plitu

de

meas1st2nd3rd

The experimental results show that increasing the order of the deviation does not improve much in accuracy but the number of coefficients increases dramatically, which means that the low-order dynamics dominate the nonlinear distortions generated by the PA. Therefore, it is reasonable to set r to a small number, i.e., only keep low-order dynamics, so that we can reduce the model complexity but keep the model fidelity.

Experimental Test





Conclusions

Outline


-- Model Long-term Memory Effects

Classic Volterra Expansion-- Based on Finite Impulse Response;

-- Number of delays directly relies on the actual memory length;

-- Not efficient to model “slow” memory.

Reformat the Expansion: Laguerre-Volterra Model

Reformat the Expansion-- Use more efficient Orthonormal Basis Functions, e.g., Laguerre functions, to replace the Dirac Impulse Responses;

-- Avoiding “ill-condition” since it is orthogonal;

-- But “a priori” knowledge required, e.g., find the pole.


Tapped Delay Line (FIR)

Memory Dependent-- Number of delays directly relies on the actual memory length.

more efficient to model “long term” memory

Laguerre Filter

Memory Independent-- with feedback but fixed pole, so keep stable.

2 1* 1

1 1

1( , )

1 1

k

kzL z

z zλ λλλ λ

−−

− −

− ⎛ ⎞− += ⎜ ⎟− −⎝ ⎠

( 1)( ) kkH z z− −=


( ) ( ) ( ) ( )1 1 2 1 3

1 1 1 1*

1 1 1 3 1 2 3 1 2 30 0 0

( ) , , ( ) ( )M M M M

i i i i iy n h i x n i h i i i x n i x n i x n i

− − − −

= = = =

= × − + × − − − +∑ ∑∑∑

0( ) ( ) ( ) ( , ) ( )k k k

ml n m x n m L z x nϕ λ

+∞

=

= − =∑

L M<< number of coefficients reduced

1 2 3

1 2 1 3

1 1 1 1*

1 3 1 2 30 0 0

( ) ( ) ( ) ( , , ) ( ) ( ) ( )L L L L

k k k kk k k k k

y n c k l n c k k k l n l n l n− − − −

= = = =

= + +∑ ∑∑∑

Laguerre-Volterra Model

Laguerre functions

Classic Volterra Model

[Zhu, IMS’05]


Nonlinearities: up to 5th OrderNumber of Laguerre Filters: L=3Total Coefficients: 81Pole: 0.2λ =

AM - AM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

Input Amplitude

Out

put A

mpl

itude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

3.5

Input Amplitude

Phas

e Sh

ift

AM - PM

To model this power amplifier:

HBT Class A, 2.14 GHzWCDMA signal source

12,000 sampling points

Experimental Example


37 37.5 38 38.5 39 39.5 40 40.5 41-4

-3

-2

-1

0

1

2

3

4

Time(usec)

Rea

l Par

tMeasuredModeled

37 37.5 38 38.5 39 39.5 40 40.5 41-3

-2

-1

0

1

2

3

4

Time(usec)

Imag

inar

y Pa

rt

MeasuredModeled

-2 0 2

-3

-2

-1

0

1

2

3

Qua

drat

ure

In-Phase

Measured Trajactory Diagram

-2 0 2

-3

-2

-1

0

1

2

3

Qua

drat

ure

In-Phase

Modeled Trajactory Diagram


Model Fidelity Evaluation and Comparison

( ) ( )( ) ( )

2 2

, , , ,1

2 2

, ,1

: 10log

M meas mod meas modI k I k Q k Q kk

M meas measI k Q kk

y y y yNMSE

y y

=

=

⎧ ⎫⎡ ⎤− + −⎢ ⎥⎪ ⎪⎣ ⎦= ⎨ ⎬⎡ ⎤+⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭

∑∑

AM/AM AM/PM Model(9 parameters)

Classic Volterra Model(81 parameters)

Classic Volterra Model(605 parameters)

0 5 10 15 20 25 30-42

-40

-38

-36

-34

-32

-30

-28

-26

-24

-22

Record Set

NM

SE (d

B)

No. of samples in each record (M)= 1000

new model requires much smaller number of parameters to reach the same accuracy.

Laguerre-Volterra Model( 81 parameters)

Laguerre-Volterra Model( 20 parameters)


Volterra models can be used to characterize both nonlinearity and memory effects of power amplifies.The general Volterra model has to be simplified in practical applications. No “best-model” in general: model structure and pruning algorithm selection depends on the characteristics of the system and model fidelity requirements.Both the complexity of model structure and the feasibility of model extraction have to be considered. Simpler model structure sometimes means more difficult model extraction.Future work: Identify physical sources of nonlinearities and memory effects, then directly relate to the parameters of the model. Parametric Model is desirable.

Conclusions


References1. V. J. Mathews, G. L. Sicuranza, Polynomial Signal Processing, John Wiley & Sons, 2000.2. V. Z. Marmarelis, Nonlinear Dynamic Modeling of Physiological Systems, John Wiley & Sons,

2004. 3. J. C. Pedro, S. A. Maas, “A comparative overview of microwave and wireless power-amplifier

behavioral modeling approaches”, IEEE Trans. Microwave Theory Tech., Vol 53,no. 4, pp.1150-1163, April 2005.

4. C. P. Silva, et al, “Introduction to polyspectral modeling and compensation techniques for wideband communications systems”, in the 58th ARFTG Conference Digest, pp. 1-15, 2001.

5. J. Kim, et al, “Digital predistortion of wideband signals based on power amplifier model with memory”, Electronics Letters, vol. 37, no. 23, pp. 1417-1418, 2001.

6. D. Mirri, et al, “A modified Volterra series approach for nonlinear dynamic systems modeling”, IEEE Trans. Circuits and Systems I, Vol. 49, Issue 8, pp.1118 – 1128, Aug. 2002.

7. E. Ngoya, et al, “Accurate RF and microwave system level modeling of wideband nonlinear circuits”, in IEEE MTT-S Int. Microwave Symp. Dig., pp. 79 - 82 , Boston, June 2000.

8. H. Ku, et al, “Quantifying memory effects in RF power amplifiers”, IEEE Trans. Microwave Theory Tech., Vol 50,no. 12, pp. 2843-2849, Dec. 2002.

9. A. Zhu, J. Dooley, and T. J. Brazil, “Simplified Volterra Series Based Behavioral Modeling of RF Power Amplifiers Using Deviation-Reduction”, in IEEE MTT-S Int. Microwave Symp. Dig., WEPG-03, San Francisco, June 2006.

10. A. Zhu and T. J. Brazil, “RF Power Amplifiers Behavioral Modeling Using Volterra Expansion with Laguerre Functions”, in IEEE MTT-S Int. Microwave Symp. Dig., WE4D-1, Long Beach, June 2005.

11. A. Zhu and T. J. Brazil, “Behavioral Modeling of RF Power Amplifiers Based on Pruned VolterraSeries”, the IEEE Microwave and Wireless Components Letters, Vol. 14, pp. 563-565, December 2004.

12. A. Zhu, M. Wren and T. J. Brazil, “An Efficient Volterra-Based Behavioral Model for Wideband RF Power Amplifiers”, in IEEE MTT-S Int. Microwave Symp. Dig., pp. 787-790, Philadelphia, June 2003.


Acknowledgements

Science Foundation Ireland

The EU FP6 Network of ExcellenceTARGET


Vito Volterra (1860-1940)

The best material model of a cat is another, or preferably the same, cat.-- Norbert Wiener

volterra series based rf power amplifier behavioral modeling

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