analog behavioral modeling for high frequency based on parameter definition (n = number of ports)...

© Copyright Agilent Technologies 2011

David E. RootResearch Fellow

Mihai MarcuSenior Applications Engineer

Agilent Technologies, Inc.

Analog Behavioral Modelingfor High Frequency Components and Systems

Frontiers in Analog Circuits

July 2011


Outline

• Introduction

• Fundamental concepts in behavioral modeling

• Functional block-models– Time domain: nonlinear time-series– Frequency domain: X-parameters– Mixed methods: dynamic X-parameters

• Conclusions


July 2011


Behavioral Models and Design Hierarchy


July 2011

Equivalent Circuit Model“Compact Model”

Device

Circuit

System

Embedding Variables

( )( ) : ( , , ..., , ..., )ny t i f v v i i

( )v t( )v t( )i t

Multivariate functions for i1, i2 Behavioral Model:

Accurate model of lower level componentfor simulation at next

highest level


When Are Behavioral Models Needed

• Complex systems– Transistor-level models become too “expensive” (resource-intensive)

• IP-protection is required– Other models reveal the IP

• Other models are simply not available– New technologies, existing HW


July 2011


Outline

• Introduction



• Conclusions


July 2011


Some of the Features Needed in Behavioral Models

• … in addition to accurately describing the behavior…

• Cascadability of nonlinear systems arbitrary topology

• Hierarchical modeling


July 2011


Some of the Features Needed in Behavioral ModelsCascadability of Models• Predict signals (magnitude and phase), including harmonics

(IMD) through chains of cascaded nonlinear components

• Linear S-parameter theory does not apply– Most previous attempts to generalize are incorrect

(power-dependent S-parameters, hot-S22)


July 2011

21

Sin(2πf0t)

Freq

Pin

ters

tage

1f0

13f0

12f0

Freq

Pin

f0Freq

Pou

t

2223f02f0f0


Some of the Features Needed in Behavioral ModelsHierarchical Modeling

• Model the cascade directly, or

• A cascade of many models reduced to one


July 2011

Dev 1 Dev 2 Dev 1 Dev 2

Mod 1 Mod 2

Mod 1 Mod 2Composite

Model(Higher Level)


Three Components of Behavioral Modeling

• Model formulation– Time-domain– Frequency-domain– Time-frequency-domain

• Experiment design– Stimulus to excite relevant dynamics

• Model identification– Procedure to determine model parameters

• Briefly reviewed using S-parameters


July 2011


S-ParametersPhysical Interpretation

• Based on the traveling waves concept

• A 2-port example (extendable to arbitrary N-port)


July 2011

Incident Transmitted

Transmitted Incident

DUT

Port 1 Port 2

1 11 1 12 2

2 21 1 22 2

b S a S ab S a S a

11 1S a1b

1a

2a

2b

12 2S a

21 1S a

22 2S a


S-Parameter Behavioral Model

• Model formulation – frequency domain

• Experiment design– One tone, low-power stimulus applied successively at each port

• Model identification:– based on parameter definition (N = number of ports)


July 2011

1 11 1 12 2

2 21 1 22 2

b S a S ab S a S a

0

, , 1,k

iij

ajk j

bS i j Na


Model Formulation

• Time-domain - nonlinear ODEs– NLTS (Non-Linear Time Series)– Transient simulation and all other simulators

• Frequency-domain - nonlinear spectral map– X-parameters– Harmonic-Balance simulation

• Time-frequency-domain – ODE-coupled envelopes– X-parameters with memory effects dynamic X-parameters– Envelope simulation

• Formulate model-equations in forms native to simulatorFrontiers in Analog Circuits

July 2011


Model FormulationTime-Domain

• Natural for strongly nonlinear low-order (lumped) systems


July 2011

( ) , , , , ,I t F V t V t V t I t

time time

StimulusResponse


Model FormulationFrequency-Domain

• Natural for low distortion (sparse spectrum) & high-frequency


July 2011

,...),,( 321 AAAFB kk

frequency frequency

A1

A2A3 A4

B1

B2B3 B4

B5B6B0

Stimulus (multiple harmonics)Response


Experiment Design

• Design the experiment such that it exercises the relevant states and dynamics of the system

• Can be achieved in both “worlds”:– Simulation– Measurement

• Conceptually the same (in measurement versus simulation), but… different challenges


July 2011


Experiment DesignSimulation

• Exercise all required states and dynamics


July 2011

System Under Test


Experiment DesignMeasurement

• Exercise the HW on all required states and dynamics

• Calibrated magnitude and phase for all harmonics/IMD


July 2011

Magnitude and Phase Data Acquisition

RFIC

A1k B1l B2m A2nReferenceplanes

New phase calibration standard

NVNA


Nonlinear Vector Network Analyzer (NVNA)combine with previous page• Based on the PNA-X platform

• Measures magnitude and phase of relevant frequency components

• Essential nonlinear measurements for nonlinear models (like NLTS and X-parameters)


July 2011

PNA-XNetwork Analyzer

+

Phase Reference

+

Meas. Science Algorithms & Software

= NVNA


Nonlinear Vector Network Analyzer (NVNA)Waveforms and Complex Spectra Measurements


July 2011

pkBpkA

Port IndexHarmonic Index

2 kA1kA

1kB 2kB

1I 2I

time time


Outline

• Introduction



• Conclusions


July 2011


Dynamical Systems and State-Space

• The dynamics of the nonlinear system can be assumed to be described by a system of nonlinear ODEs

• The sampled solution of the ODE, y(t), is a time-series

• The solution of the dynamical equations for state variables, u(t) is a time-parameterized trajectory in the State-Space


July 2011

( ) ( 1) ( )( ) ( ,... , , ,... )n n my t f y y x x x

( ) ( ), ( ) vector of state equ

( ) (

ation

), ( ) sca

s

lar outputy t h u t

u t f u t x t

x t


State-Space and Time-Series

• The multi-dimensional space spanned by the state variables is known as the state-space

• Any measureable output is a projection of this trajectory versus time and it is a time-series


July 2011


Model Identification for NLTS

• Stimulate system– Sufficiently complex stimulus

• Embed– Create auxiliary variables to represent the waveform

• Sample data– Data directly or the envelope– Difficult if multiple time-scales

• Fit– Approximate nonlinear function f(.)


July 2011


Excitation Design

• Stimulate all relevant (observable) dynamics

• Sweep power and frequency to “cover the state-space”


July 2011

‘Two-tone’

f1 f1+ff1 f1+f

f1+f

‘Three-tone’

f1+f?

‘Modulation’ (CDMA)

f1f1+f

f2

fn

‘Multi-tone’ or ‘Multi-sine’


Embedding

• Building-up state-space to define ODE


July 2011

i(t)

v(t)

A

Bi(t)

v(t)

i(t)

v(t)

A

B

i(t)

v(t)

v’(t)

B

A

i(t)

v(t)

v (t)

B

A( ) ( ( ))i t i v t ( ) ( ( ), ( ))i t i v t v t

X(t) Y(t)

( )

( )

( ) [ ( ), ( ),..., ( )]( ) [ ( ), ( ),..., ( )]

m

n

x t x t x t x ty t y t y t y t


Sample Data

• Dense enough in all dimensions


July 2011

X(t) Y(t)

( ) ( )1 1 1 1 1 1

( ) ( )2 2 2 2 2 2

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

m n

m n

m np p p p p p

x t x t x t y t y t y tx t x t x t y t y t y t

x t x t x t y t y t y t

nth order derivativeStimulus Response


Function Fit

• For nth derivative of the output

• Many function-types may be used– Polynomials– Radial basis functions– Look-up tables– …– ANNs

• Artificial neural networks (ANNs) a very good option


July 2011

( ) ( 1) ( )( ,... , , ,... )n n my f y y x x x


Function FitArtificial Neural Networks (ANNs)

• An ANN is a parallel processor made up of simple, interconnected processing units, called neurons, with weighted connections


July 2011

baxwsvxxFI

i

K

kikkiiK

1 11 ),...,(

sigmoidweights biases

x1

...

xk


Function Fit ANN General Properties

• Fit any nonlinear function of any number of variables (universal approximation theorem)

• Infinitely differentiable better distortion than naïve splines or low-order polynomials

• Easy to train (fit) using standard tools

• Easy to train on scattered data


July 2011

baxwsvxxFI

i

K

kikkiiK

1 11 ),...,(


Function FitANN Training

• wki,ak,b – obtained by training


July 2011

fANN

…

… …

,ki kw aweightsbiases

y(n-1)(t) y(n-2)(t) … y(t) x(n)(t) x(n-1)(t) … x(t)

y(n)(t)

( ) ( 1) ( 2) ( ) ( 1)( ) , ,..., , , ,...,n n n n nANNy t f y t y t y t x t x t x t


Model ImplementationAn Example

• ODE in circuit simulator


July 2011

xx x(1)

+-

x(m-1) x(m)+- f vn

+-

x(1) x(2)+-

yv2v1

+-

v2v3

+-

vn-1vn

+-

( ) ( 1) ( ), , , , ,n n my f y x x x

2

1

( )1

1 2

1

, , , , , , ,n

n n

n

mnv f v v v

v yv v

v v

x x x


NLTS – ExampleGaAs HBT MMIC

• Agilent HMMC 5200– Series-shunt amplifier– Gain: 9.5 dB @ 1.5 GHz


July 2011

Actual Circuit Detailed circuit model


NLTS Accuracy and SpeedSingle-Tone Stimulus


July 2011

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4-22 6

0

20

40

60

80

100

120

140

160

-20

180

dBm(In_model[::,1],z1[::,1])

ph

ase

(Ou

t_m

od

el[:

:,1

])

dBm(In_IC[::,1],z1ic[::,1])

ph

ase

(Ou

t_IC

[::,

1])

Fundamental Phase

Time psec0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80.0 2.0

2.0

2.5

3.0

3.5

4.0

1.5

4.5

0 200

1.5

2.0

2.5

3.0

3.5

4.0

1.0

4.5

Time psecTime Domain Waveforms

229.68 seconds

11315.67 seconds

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4-22 6

7

8

9

10

11

12

13

6

14

dbm(In_model[::,1],z1[::,1])

dB

m(O

ut_

mo

de

l[::,

1],

z2[:

:,1

])-d

Bm

(In

_m

od

el[:

:,1

],z1

[::,

1])

dbm(In_IC[::,1],z1ic[::,1])

dB

m(O

ut_

IC[:

:,1

],z2

ic[:

:,1

])-d

Bm

(In

_IC

[::,

1],

z1ic

[::,

1])

Fundamental Gain

6

14

-22 6

dBm

dBm

180

-20

1 - 19 GHz

(2) (2)1 2 1 2 1 2( ) ( , ( ), ( ), ( ), ( ), ( ), ( ))i i iI t f I V t V t V t V t V t V t

19 neurons

-22 6dBm

NLTS modelCkt model


WCDMA Lower ACLR Comparison:Circuit Co-Sim vs. NLTSA Model vs. Measured

0

10

20

30

40

50

60

70-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

Input Power (dBm)

AC

LR (d

B

Circuit Co-Sim 5MHz Lower

NLTSA Model 5 MHz Lower

Circuit Co-Sim 10 MHz Lower

NLTSA Model 10 MHz Lower

Measured Data 5 MHz Lower

Measured Data 10 MHz Lower

NLTS Accuracy and SpeedResults 3GPP WCDMA Lower ACLR


July 2011

294 sec/pt NLTS

1532 sec/pt Ckt.

Measured

Model: - 40 neurons, cascadable

- generated from sinusoidal signals only

- works in Transient, HB, Envelope

Simulation: 3 GHz (courtesy Greg Jue)Measured: 2.4 GHz


Outline

• Introduction



• Conclusions


July 2011


X-Parameters

• Similar to S-parameters, but for nonlinear behavior

• The mathematically correct superset of S-parameters

• Applicable to:– Both large- and small-signal– Both linear and nonlinear systems

• All pieces of the puzzle work together:– Measure– Model– Simulate


July 2011


X-Parameters

• Cascadable models – correctly account for mismatch

• Model identification from a simple set of measurements

• Very accurate


July 2011


X-ParametersFundamental Concepts

• Spectral map of complex large input phasors to large complex output phasors


July 2011

2A1A

1B 2B1 1 11 12 21 22

2 2 11 12 21 22

( , , , ..., , , ...)( , , , ..., , , ...)

k k

k k

B F D C A A A AB F D C A A A A

Port Index Harmonic Index


X-Parameters for Multi-Harmonic StimulusHarmonics Are Large-Signals

• Response is a nonlinear function of stimulus

• Behavior can be described by X-parameters

• Phase-coefficient comes from time-invariance:– Output of delayed input is the delayed output


July 2011

, , 11 12 21 22, , , . , ,p k p kB F D C A A A A

( )

( ), ,

1,1 1, 2,

1,1 1, 2,

, 1, , , , 2, , , 1,

, 1, , , , 2, , , 1,

1, , 1,

k kq k k

k kq k k

FDCRp p

FB kp k p k

DCR X

B

DCS q N A A P k K A P k K

DCS q N A A P k K A P k K

p N k K

X P

1,1 j phase AP e


• If all stimuli are small except for the fundamental simplify the general nonlinear spectral mapping by spectral linearization

• X(S) and X(T) terms synthetic load-pull

X-Parameters for Single-Tone StimulusHarmonics Are Small-Signals


July 2011

( )1,1 ,, ; ,

11

, 1

( ), 1,1.

( ) *1,1 ,, ; ,

11

, 1 1,1 ,

q Nl K

S k lq lp k q l

ql

q

FB kp k p k

q Nl K

T k lq lp k q

l

lql

q l

X A P AB X A P X A P A

Mismatch terms linear in Aq,l

Perfectly matched response

Mismatch terms linear in A*q,l


Practical Applications of X-Parameters

• Black-box description holds for arbitrary systems (amplifiers, mixers, receivers, transmitters, etc.)

• Supports multi-variable swept conditions:– bias– RF-power– frequency– load-pull– additional user-defined variables


July 2011


Single-Tone Amplifier


July 2011

CktXpar


Mismatch Effects in a Nonlinear Cascade


July 2011

CascadedX-parameter models

Cascaded PA circuits

Mismatch

Mismatch


Mismatch Effects in a Nonlinear CascadeResults


July 2011

Fundamental

2nd Harmonic

3rd Harmonic

Overlaid Results

Model (Red)Vs.

Circuit (Blue)


Systems with Frequency ConversionMixer (2-Tone: RF, LO)


July 2011

1.0E9

2.0E9

3.0E9

4.0E9

5.0E9

6.0E9

7.0E9

8.0E9

9.0E9

1.0E10

0.0

1.1E10

-160

-140

-120

-100

-80

-60

-40

-180

-20

freq, Hz

dB

m(v

ou

t_ck

t)

m3

freq+(0.05 GHz)

dB

m(v

ou

t_m

dl)

m4

m3indep(m3)=plot_vs(dBm(vout_ckt), freq)=-38.914

2.500E8

m4indep(m4)=plot_vs(dBm(vout_mdl), freq+(0.05 GHz))=-38.914

3.000E8

1.0E9

2.0E9

3.0E9

4.0E9

5.0E9

6.0E9

7.0E9

8.0E9

9.0E9

1.0E10

0.0

1.1E10

-150

-100

-50

0

50

100

150

-200

200

freq, Hz

ph

ase

(vo

ut_

ckt)

m1

freq+(0.05 GHz)

ph

ase

(vo

ut_

md

l)

m2

m1indep(m1)=plot_vs(phase(vout_ckt), freq)=-50.105

2.500E8

m2indep(m2)=plot_vs(phase(vout_mdl), freq+(0.05 GHz))=-50.105

3.000E8

CktXpar

v out_mdlv out_ckt

HarmonicBalanceHB1

Order[2]=5Order[1]=5Freq[2]=2 GHzFreq[1]=1.75 GHzMaxOrder=5

HARMONIC BALANCE

P_1TonePORT3

Freq=2 GHzP=polar(dbmtow(-50),0)Z=50 OhmNum=3 P_1Tone

PORT4

Freq=1.75 GHzP=polar(dbmtow(-5),0)Z=50 OhmNum=4

V_DCSRC2Vdc=5 V

X4PXNP1File="a0_Mixer.ds"

4

1 2

3 Ref

RR2R=50 Ohm

RR1R=50 Ohm

P_1TonePORT1


PORT2


V_DCSRC1Vdc=5 V

x_2_GilCellMixX1

IF_portRF_port

LO_port

Bias_port


Systems with Frequency Conversion Mixer Sub-System - Spectrum Measurement Setup

• RFpwr = -50

• LOpwr = -5 dBm


July 2011

v out_mdl

v out_ckt

X4PXNP1File="d0_MixerBPF.ds"

4

1 2

3 Ref

RR2R=50 Ohm

RR1R=50 Ohm

BPF_Cheby shevBPF1

N=15Ripple=0.5 dBBWpass=100 MHzFcenter=250 MHz

V_DCSRC2Vdc=5 V

HarmonicBalanceHB1

Order[2]=5Order[1]=5Freq[2]=2 GHzFreq[1]=1.75 GHzMaxOrder=5

HARMONIC BALANCE

P_1TonePORT1


PORT2


V_DCSRC1Vdc=5 V

x_2_GilCellMixX1

IF_portRF_port

LO_port

Bias_port

P_1TonePORT4


P_1TonePORT3

Freq=2 GHzP=polar(dbmtow(-50),0)Z=50 OhmNum=3


Systems with Frequency ConversionMixer Sub-System – Results: Output Spectrum

• small intentional display offset – to distinguish traces


July 2011

1.0E9

2.0E9

3.0E9

4.0E9

5.0E9

6.0E9

7.0E9

8.0E9

9.0E9

1.0E10

0.0

1.1E10

-800

-600

-400

-200

-1000

0

freq, Hz

dB

m(v

ou

t_ck

t)

freq+(0.05 GHz)

dB

m(v

ou

t_m

dl)

1.0E9

2.0E9

3.0E9

4.0E9

5.0E9

6.0E9

7.0E9

8.0E9

9.0E9

1.0E10

0.0

1.1E10

-100

0

100

-200

200

freq, Hz

ph

ase

(vo

ut_

ckt)

freq+(0.05 GHz)

ph

ase

(vo

ut_

md

l)

Mag

Phase

CktXpar


Amplifier Load-Pull


July 2011

Pd

el_

ckt_

con

tou

rs_

pP

de

l_m

dl_

con

tou

rs_

p

Pdel_dBm

PA

E_

ckt_

con

tou

rs_

pP

AE

_m

dl_

con

tou

rs_

p

PAE

PA

E_

ckt_

con

tou

rs_

pP

AE

_m

dl_

con

tou

rs_

p

PAE

Pd

el_

ckt_

con

tou

rs_

pP

de

l_m

dl_

con

tou

rs_

p

Pdel_dBm

CktXpar

50 Ω X-parameters

Load-dependent X-parameters

-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4-20 6

0

5

10

15

-5

20

Power in (dBm)

Pow

er o

ut (

dBm

)


Load-Dependent Waveforms27 Ω Validation of Measurement-Based 50 Ω Model


July 2011

v1

100 200 300 400 500 6000 700

-0.5

0.0

0.5

-1.0

1.0

time, psec

v2

100 200 300 400 500 6000 700

-1.0

-0.5

0.0

0.5

-1.5

1.0

time, psec

100 200 300 400 500 6000 700

-0.005

0.000

0.005

-0.010

0.010

time, psec

i1

100 200 300 400 500 6000 700

-0.02

0.00

0.02

0.04

-0.04

0.05

time, psec

i2

Measurement-Based X-parameter Model Independent NVNA Data


Rough Comparison of Methods and Applicability

NLTS• Works in TA, HB, Envelope

• Excellent for strongly nonlinear, but lumped (low-order ODE) systems

• Training non-algorithmic

• Experiment design not fully solved

• Not as robust for convergence

• Scales well with complexity

• Great gains in simulation speed

X-Parameters• Frequency-domain natural for highly

nonlinear, distributed, broad-band circuits

• Experiment design completely solved

• Highly automated model identification

• Works in HB and Envelope

• Very robust for convergence

• Always accurate if sampled densely

• Complexity increases rapidly for multiple tones


July 2011


Outline

• Introduction



• Conclusions


July 2011


Why Mixed-Methods

• Applies to systems under large-signal modulated drive

• Long-term memory effects are present

• Time-varying spectra for all inputs, outputs and state-variables

• Perfectly suited for Envelope simulations

• Well matched for data from NVNA


July 2011


Envelope Domain for Long-Term Memory

• Xh(t) – a set of time-varying complex-waveforms (amplitude and phase) at each harmonic-index, h

• Modeling problem: map input envelopes to output envelopes


July 2011

Freq. (GHz)1 2 3 4DC

Time-varying spectrum

02

0

( ) Re ( )H

j h f th

h

x t X t e

B2[1](t) – time-domain envelope


Merging Frequency and Time Domains

• The static spectral mapping becomes a differential equation in envelope domain


July 2011

11 12 21 22( , , ..., , , ...)pk pkB f A A A A

1 1 110 10 11 11 20 21 2,..., , , ,..., , ,..., ,..., ,...,n m

pk pk pk pk kB f B t B t A t A t A t A t A t A t A t

Envelope index

Order of time derivative

21 21 20 11

22011 21

,

,

B t f B t A t

dB tg A t B t

dt

Example:

Port index


Envelope Model: Amplifier with Self-Heating


July 2011

Pulsed RF signal at 1GHz: Thermal Time Const. 10usec

10 20 30 400 50

0.1

0.2

0.3

0.0

0.4

Fundamental Input

time, usec time, usec10 20 30 400 50

1

2

3

0

4

Fundamental Output

10 20 30 400 50

0.01

0.02

0.03

0.00

0.04

10

20

30

0

40

Third Harmonic Output Mag & Phase

time, usec



July 2011

Dynamic X-parameterPNA-X NVNA

Dynamic Compression Characteristic

Out

put (

Vp)

Input (Vp)

Dynamic X-Parameters - Memory Effects [12]

Power Sweep @ 19.2 kHz Tone Spacing


Conclusions

• Time, frequency and mixed domain methods:– NLTS could become practical soon– X-parameters are mature, both measurement and simulation– Dynamic X-parameters proven for memory

• Commercial availability– NVNA – based on PNA-X HW platform– X-parameters available in circuit and system simulators (ADS, Genesys,

SystemVue)

• Great opportunities for applications– Specifications of active components by X-parameters– Better understanding of nonlinear behavior new design practices– Nonlinear stability, matching, etc.


July 2011


References

[1] J. Wood, D. E. Root, N. B. Tufillaro, “A behavioral modeling approach to nonlinear model-order reduction for RF/microwave ICs and systems,” IEEE Transactions on Microwave Theory and Techniques, Vol. 52, Issue 9, Part 2, Sept. 2004 pp. 2274-2284

[2] D. E. Root, J. Wood, and N. Tufillaro, “New Techniques for Non-Linear Behavioral Modeling of Microwave/RF ICs from Simulation and Nonlinear Microwave Measurements,” in 40th ACM/IEEE Design Automation Conference Proceedings, Anaheim, CA, USA, June 2003, pp. 85-90 (videotaped)

[3] D. E. Root, “Nonlinear Analog Behavioral Modeling of Microwave Devices and Circuits,” IEEE Microwave Theory and Techniques Distinguished Microwave Lecture (now available from MTT-S Speaker’s Bureau)

[4] Agilent HMMC-5200 DC-20 GHz HBT Series-Shunt Amplifier, Data Sheet, August 2002.

[5] J. Verspecht, M. Vanden Bossche, F. Verbeyst, “Characterizing Components under Large Signal Excitation: Defining Sensible `Large Signal S-Parameters'?!,” in 49th IEEE ARFTG Conference Dig., Denver, CO, USA, June 1997, pp. 109-117.

[6] D. E. Root, J. Verspecht, D. Sharrit, J. Wood, and A. Cognata, “Broad-Band Poly-Harmonic Distortion (PHD) Behavioral Models from Fast Automated Simulations and Large-Signal Vectorial Network Measurements”, IEEE Transactions on Microwave Theory and Techniques Vol. 53. No. 11, November, 2005 pp. 3656-3664

[7] J. Wood, D. E. Root, editors, Fundamentals of Nonlinear Behavioral Modeling for RF and Microwave Design, 1st ed. Norwood, MA, USA, Artech House, 2005.

[8] D. E. Root, D. Sharrit, J. Verspecht, “Nonlinear Behavioral Models with Memory: Formulation, Identification, and Implementation,” 2006 IEEE MTT-S International Microwave Symposium Workshop (WSL) on Memory Effects in Power Amplifiers

[9] Blockley et al 2005 IEEE MTT-S International Microwave Symposium Digest, Long Beach, CA, USA, June 2005.

[10] Soury et al 2005 IEEE International Microwave Symposium Digest pp. 975-978

[11] J. Verspecht and D. E. Root, “Poly-Harmonic Distortion Modeling,” in IEEE Microwave Theory and Techniques Microwave Magazine, June, 2006.

[12] J. Verspecht, J. Horn, L. Betts, D. Gunyan, R. Pollard, C. Gillease, and D. E. Root, “Extension of X-parameters to include long-term dynamic memory effects,” IEEE MTT-S International Microwave Symposium Digest, 2009. pp 741-744, June, 2009

[13] J. Horn, L. Betts, C. Gillease, D. Gunyan, J. Xu, J. Verspecht, and D. E. Root, “X-Parameters: Extending S-parameters for Measurement, Modeling, and Simulation of Nonlinear Components,” 2008 IEEE Power Amplifier Symposium, Orlando, FL


July 2011

analog behavioral modeling for high frequency based on parameter definition (n = number of ports)...

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