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Behavioral Modeling ofNonlinear Amplifiers with
Memory
Michael Steer
with
Jie Huand Aaron Walker
1
Copyright 2009 to 2011 by M. Steer, J. Hu and A. Walker.
Not to be posted on the web or distributed electronically without permission.
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2
RALEIGH
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NC STATE UNIVERSITY
http://www.businessweek.com/lifestyle/which-is-americas-best-
city-09202011.html
September 20, 2011
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And … (for the US) Current rankings
• No. 1 Region to Find Knowledge Workers, Forbes• No. 1 Best City for Business, Forbes
• No. 1 Area for Tech Business, Silicon Valley LeadershipGroup
• No. 1 City with the Happiest Workers, Hudson Employment
Index• No. 2 Most Educated City, American Community Survey
• No. 3 Best City for Entrepreneurs, Entrepreneur.com
• No. 3 Top Metro Overall, Expansion Management Mayor’sChallenge
• No. 3 High Value Labor Market Quotient, ExpansionManagement
• No. 3 Top States for Work Force Training, ExpansionManagement
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Outline
• Background• Nonlinear Metrology
– Novel 1-channel VSA-based NVNA
– Novel Hybrid NVNA
• Behavioral Modeling – Grey-Box Model ID Methodology – ID Algorithm & Hybrid GA Optimizer – Modeling Cosite Interference
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Multi Slice Behavioral Model
• Wired Characterization Utilizing IMD PhaseInformation – High power amplifier at 450 MHz, suitable for CDMA450
standard
H 1(s)
H 1(s)
K(s)
K(s)
NL1
NL2
y(t)
f A
H 2(s)
x(t)
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Model Fit
• Single slice fit underestimates IM3 magnitude,
splits difference in phase asymmetry
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Model Fit
~3º
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Model is an Application Specific Abstraction Applications of nonlinear behavioral models
Compact RF transistor models
PA linearizationModel long-term memory
Wireless transceiver RF frontend
Top-down designDesign space exploration
Bottom-up verification
Design verification
Cosite simulation
Co-located radios
Same frequency band Similar to Near-Far problem
Encoder
Modulator Transmitter
Receiver Decoder
Demodulator
Channel
Gain, NF,
Linearity, Filter
roll off, ...
I
QDuplexer/
Switch
PA 90°
BPF
I
QDuplexer/
Switch
Duplexer/
Switch
PAPA 90°90°
BPFBPF
Parametric
Behavioral
Model
Design and test
component
Data-based
Behavioral
Model
I
QDuplexer/
Switch
PA 90
°
BPF
I
QDuplexer/
Switch
Duplexer/
Switch
PAPA 90
°
90
°
BPFBPF
Transistor, R, L, C, ...
Top-Down
DesignSpecification
Bottom-Up
Verification
FIR, IIR, Polynomial, ...
Co-located Radios
TX (Wanted)Band
Filter
TX (Interferer)Band
Filter PA
RX (Victim)Band
Filter LNAPA
TX (Interferer)Band
Filter PA
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Modeling Broadband Response (EMI)
TX2 (WCDMA @900MHz)
BandFilter
RXBandFilter
LNA
PA
TX1 (1-tone @500MHz)
BandFilter
PA2f
22f
1+f
2
2f 1-f 2f 2-f 1
f 1
f 2
2f 1
2f 2-f 1
f 2+f 1
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Model attributes Fidelity (frequency range & accuracy of the modeled response)
Development complexity (model parameter estimation)
Simulation compatibility & speed IP protection
Types of RF Nonlinear Behavioral Models
Model
examples
Formula-based:
Computationally efficient,Low fidelity (for top-down design)
Circuit simulator-based:
Computationally complex,High fidelity (for verification)
Black-box Parametric models (IP3, NF, …)
Power series
Memory Polynomial
X-parameter (Harmonic balance,
memory modeled in fundamental band)
Volterra series (modify simulator,
complex parameter estimation)
Grey-box(structure
based on
physical insight)
Weiner-Hammerstein family
Multi-slice
Weiner-Hammerstein w/lin-feedback
General RF frontend model (general
circuit simulator, complex parameter
estimation)
Grey-box model
High fidelity for verification & cosite simulation Supported in general circuit simulators
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Grey-box Model ID, Prior Work vs. Here
Model
Parameters
ExcitationOptimizer
Local: Gradient, Simplex
Global: GA, SA
Error
Function
Select
ResponseSimulator
ResponseCalculation
DUT Model
DUT
Measurement
Response
Parameterize
Model
Model ID Algorithm: a typical Parameter Estimation Step
Simple models with
easily derived expression
Case-by-case
ad-hoc ID algorithm
General ID algorithm
minimize user intervention
General models
require circuit simulation
Experiment design
ensure no under-determined optimization problem
Deterministic optimizer
(depend on initial solution
supplied, prong to trapping
in local optimums)
Hybrid-Genetic optimizer
find multiple optimums for
user selection
(typical stochastic optimizer
find only 1 global solution) 12
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• What is measured – VNA: Ratios (between excitation and response)
– NVNA: Signals (power, phase)
• Calibration (receiver & test-set frequency response)
– VNA: Relative (between port waves)
– NVNA: Relative + Absolute (within a signal)
• ‘Absolute’ phase calibration – Phase measured is relative to sampling time – Need receiver w/linear-phase
• Different spectral components in a signal• Delay by same amount
– Correct phase dispersion error• Multi-harmonic generator (maintain relative phase)
• Characterize using linear-phase RX – Nose-to-nose oscilloscope, electro-optical sampling.
• Using the phase transfer standard – 1) determine phase dispersion error – 2) serve as phase transfer mechanism
Nonlinear Metrology: Background
Amplifier
a1
b1
b2
a2
f
f
f
f
between port waves
within a signal
Amplifier
Fundamental
f 13
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NL Metrology, Prior Work vs. This Work
Broaden NVNA application Phase essential to IM cancellation in multi-stages circuits
Use available instruments (1-channel receiver, no multi-harmonic generator)
Improvements (dynamic range, tone spacing)
NVNA using broadband receivers 4-ch sub-sampling receiver & multi-harmonic generator
60dB dynamic range 1-ch VSA & switches & AWG & sampling oscilloscope 75dB dynamic range
Power/area efficient for on-chip self-characterization
NVNA using narrowband receivers
4-port VNA & multi-harmonic generator Tone spacing 1.242MHz (80dB dynamic range) (10MHz in commercial version)
4-port VNA & phase-lock CW source & sampling oscilloscope Tone spacing 200Hz (40dB dynamic range, increasing to 80dB at 200kHz)
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1-ch VSA-based NVNA
1-ch VSA-based NVNA AWG & sampling oscilloscope
Repeatability verified [Remley06] 1-ch VSA & switches
75dB dynamic range36MHz bandwidth
In-band characterization onlyMimic multiple synchronous channels
Apply to on-chip self-characterizationUbiquitous integrated receivers Power/area efficient
Sub-sampling Mixer-based NVNA [Verspecht95] Multi-harmonic generator
Determine receiver phase dispersion error 4-ch sub-sampling receiver
60dB dynamic range20GHz bandwidth
Sampling Oscilloscope,
Sub-sample Mixer,Broadband Receiver
ch1 ch2 ch3 ch4
a1 b1 b2 a2
LoadDUT
For absolute phase calibration:Connect multi-harmonic phase reference
a1 b1 b2 a2
LoadDUT
VSA
AWG
PC(VSA software,
post processing)Trigger
GPIB
Control
For absolute phase calibration:
Connect multi-tone calibration signalSampling
Oscilloscope(nose-to-nose calibrated)
AWG
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Nonlinear Metrology: 1-ch VSA-based NVNA (Chapter 3.2)
Mimic multiple synchronous channels
Repeat multi-tone excitation Hot-switch port waves
Continuous sampling during switching interval
Align sequential meas. using sampler timebaseExcitation/distortion on f-grid
Design minimum tone spacing
Concatenate wave segments of different tone-spacings(uniform time duration, equal DFT bins for noise to distribute)
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1-ch VSA-based NVNA
Verify with sub-sampling mixer-based NVNALinear TF: Mag < 0.2dB, Phase < 2deg
NLTF: Mag < 10%, Phase < 5deg
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4-port VNA-based Hybrid NVNA Hybrid NVNA
Phase-lock CW sourceRelate phase of during power sweep
Sampling oscilloscopeRelate phase of diff. spectral components
Tone spacing40dB dynamic range @200HzIncreasing to 80dB @200kHzLimited by oscilloscope memory depth
& spectral leakage (FFT window)
4-port VNA-based NVNA [Blockley05]
4-port VNA80dB dynamic range250kHz instantaneous bandwidth
Multi-harmonic generatorRelate phase of diff. spectral components20GHz bandwidth achieved in NVNA
Tone spacing1.242MHz (10MHz in commercial version)
a1 b1 b2 a2
LoadDUT
Connect for phase
calibration measurement
PC(post processing)
GPIB Control
4-port VNA,
Narrowband Receiver
A B C DR
Sampling Oscilloscope,
Broadband Receiver
ch1 ch2 ch3 ch4
Phase-Lock
CW Sources
10 MHz
AA BB
4-port VNA,Narrowband Receiver
A B C D
a1 b1 b2 a2
LoadDUT
R
Multi-HarmonicPhase
Reference
Connect forphase calibration
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4-port VNA-based Hybrid NVNA DUT measurement
2-tone excitation (200kHz tone-spacing @2GHz)
Lower
Fundamental
Lower
IM3
Phase jumps due to switching
attenuators in excitation source
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Time-Invariant Phase
‘Absolute’ phase measurement DUT port waves characterized as signals
Phase jumps in excitation is part of signal
Time-Invariant phase Previously considered only for alignment [Blockley06]
Phase is relative to sampling time
Cancel phase variation in excitation
Phase contribution only from DUT: DUT characterization
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Grey-box Model ID Methodology
Model
Parameters
ExcitationOptimizer
Local: Gradient, Simplex
Global: GA, SA
Error
Function
Select
ResponseSimulator
ResponseCalculation
DUT Model
DUT
Measurement
Response
Parameterize
Model
Model ID Algorithm: a typical Parameter Estimation Step
Simple models with
easily derived expression
Case-by-case
ad-hoc ID algorithm
General ID algorithm
minimize user intervention
General models
require circuit simulation
Experiment design
ensure no under-determined optimization problem
Deterministic optimizer
(depend on initial solution
supplied, prong to trapping
in local optimums)
Hybrid-Genetic optimizer
find multiple optimums for
user selection
(typical stochastic optimizer
find only 1 global solution)21
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Identification Algorithm
L-N-L [Boutayeb95]
N-L-N [Zhu95]
Deterministic
Optimizer 1 Solution SLin,S NL
SLin,Init S NL,Init
SLin S NL
Deterministic
SLin(0)
S NL
(held const)
SLin,Init
Deterministic
S NL,Ini tSLin
(held const)
SLin(k)
S NL(k)
Stochastic(1st pass estimate)
{SLin,Init}optional {S NL,Init}optional
{SLin} {S NL}
Select
Select
Stochastic
S NL
(held const)
{SLin,Init}
Stochastic
{S NL,Init}SLin
(held const)
{{S1Lin(k)}}
{S1Lin(k)}
Select
{{S2 NL(k)}}
{S2 NL(k)}
Select
Select
Case-by-caseDeterministic optimizer
Depend on initial solutionTrap in local optimum
L-N-L [Crama05]
L-N [Vandersteen99]
General RF frontend modelMultiple solution
Due to incomplete/noisy data
Stochastic optimizer
Find multiple local optimums
& global optimum
Minimize user intervention
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Model Optimizer
Stochastic optimizer Conventional hybrid-GA
1 global optimum
Parallel-GA [Quintero08]
& Niching-GA [Dilettoso06]Multiple sub-populations
Around local optimums
Tree Anneal [Bilbro90]& No-Revisit-GA [Yuen08]Record past searches to guide future searches
Hybrid-GA optimizerRecord past searches
Exist population around local optimums
Use as initial solution in local search
GAGen n: Cross-Over + Adap.
Mutate + Select
Gen 1
Hybrid
LocalSearch:
Gradient
Simplex
Local Search Methods:
Gradient, Simplex
Global Memory Methods:
Tabu Search, Hash Table, Tree
1 BestIndividualPopnPop1Popinit
GA
Gen 1
Hybrid
Local
Search
Gen n
Solution Selection
Compiled
Solution
Popinit Pop1 Popn
M Best
Individuals
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Model Optimizer
Hybrid-GA performance
Find global & multiple local optimums
User select based on error and physical intuition
Number of function evaluations
same to 1/7th of conventional hybrid-GA
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Model fundamental and IM3 transmission response Asymmetric phase of IM3
Due to baseband impedance (parameterize as freq. varying)
Parameterize impedance in other bands as constant
Identified baseband impedance
Reflect inductive nature of bias network
Correlates with S21 measurement
Power Amplifier w/Long-term Memory
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-4 -2 0 2 4
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15IM3
ToneSpace (Mhz)
d B m
-4 -2 0 2 4
-45
-40
-35
-30
-25
-20
-15
-10IM3
ToneSpace (Mhz)
D e g
Meas -20.3dB
Meas -19.4dB
Meas -18.4dB
Meas -17.4dB
Meas -16.5dBMeas -15.4dB
Meas -14.5dB
Meas -13.5dB
Meas -12.5dB
Meas -11.5dB
Meas -10.4dB
Meas -9.5dBm
MDL -20.3dBm
MDL -19.4dBm
MDL -18.4dBm
MDL -17.4dBm
MDL -16.5dBm
MDL -15.4dBm
MDL -14.5dBm
MDL -13.5dBm
MDL -12.5dBm
MDL -11.5dBm
MDL -10.4dBm
MDL -9.5dBm
-4 -2 0 2 4
-2
0
2
4
6
8
10f1f2
ToneSpace (Mhz)
d B m
-4 -2 0 2
171
172
173
174
175
176
177
178f1f2
ToneSpace (Mhz)
D e g
Power Amplifier w/Long-term Memory
Modeled transmission responseUp to 2-tone P
-1dB
; -20dBc IM3 (state-of-art Black-box model P-4dB)
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Cosite interferenceCo-located radios in same frequency band
Similar to Near-Far problem
Cosite simulation Formula-based BER evaluation [Isaacs91]
Using parametric models (IP3, NF),
MeasurementModulated signal, >2 port: phase can’t be measured
Amplitude measurement to be modeled
Cosite Interference
TX2 (WCDMA @900MHz)
900MHz
Filter
RX
LNA
PA
TX1 (1-tone @500MHz)
500MHz
Filter PAZHL1042J
4-port VNA A B C DR
10 MHz
Phase-Lock
CW Sources
cable
cable Anachoic
Chamber ZRL1150LN
cable
UPC1678GV
Horn
Helical
Horn
Load
AA
BB
CC
Vector Signal
Generator
Power Meter
VSA
Co-located Radios
TX (Wanted)Band
Filter
TX (Interferer)Band
Filter PA
RX (Victim)Band
Filter LNAPA
TX (Interferer)Band
Filter PA
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Model broadband EMI transmission response
Notch centered at IM frequencies 2f 2 & 2f 2-f 1Due to baseband impedance (parameterize as freq. varying)
Correlates with S21 measurements
Cosite Interference
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Summary
Nonlinear Characterization (for multi-tone excitation)
1-ch VSA-based NVNA 1-channel receiver (power/area efficient for on-chip self-characterization)
75dB dynamic Range
4-port VNA-based hybrid NVNA Phase-locked CW source as effective phase transfer mechanism when using
narrowband receivers
Tone spacing 200Hz (40dB dynamic range, increasing to 80dB at 200kHz)
Grey-box Model ID for RF Frontend General ID algorithm for minimum user intervention
Hybrid-GA optimizer to find multiple optimums
Direct calculation algorithm for Volterra circuit analysis Model PA w/long-term memory
Fidelity up to P-1dB (comparable to state of the art)
Model cosite interference Fidelity in broadband EMI response
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References
J. Hu, K. G. Gard, N. B. Carvalho, and M. B. Steer ,
“Dynamic Time-Frequency Wave-forms for VSACharacterization of PA Long-term Memory Effects," 71stAutomated RF Techniques Group Conf. Digest, June 2007.
J. Hu, K. G. Gard, N. B. Carvalho, and M. B. Steer, “Time-Frequency Characterization of Long-Term Memory in Nonlinear Power Amplifiers," 2008 IEEE MTT-SInternational Microwave Symposium Digest, Jun. 2008, pp.
269-272.
J. Hu, J. Q. Lowry, K. G. Gard, and M. B. Steer, “NonlinearRadio Frequency Model Identification Using A HybridGenetic Optimizer for Minimal User Intervention,” In Press
J. Hu, K. G. Gard, and M. B. Steer, “Calibrated NonlinearVector Network Measurement Without Using a Multi-Harmonic Generator,” In Press
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