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4/26/2012
1
Nonlinear Radar Concepts
Gregory J. Mazzaro
Electronics Engineer
U.S. Army Research Laboratory
Adelphi, MD 2012 1
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Presentation Overview
• Introduction
• Nonlinearity: AM-AM, AM-PM, Physical Sources
• Radar: Transmitted & Received Waveforms
• Prior (Published) Work
• Mathematical Modeling
• Power Series, Even- vs. Odd-Order Nonlinearities
• Responses to Waveforms: Single-Tone, Two-Tone
• Linearization Techniques
• Recent Experiments & Summary 2 f
E V
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3
Presentation Overview
• Introduction
• Nonlinearity: AM-AM, AM-PM, Physical Sources
• Radar: Transmitted & Received Waveforms
• Prior (Published) Work
• Mathematical Modeling
• Power Series, Even- vs. Odd-Order Nonlinearities
• Responses to Waveforms: Single-Tone, Two-Tone
• Linearization Techniques
• Recent Experiments & Summary 2 f
E V
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Nonlinearity vs. Linearity
For a linear system,
[1, pg 29]
Amplitude-Modulation to
Amplitude-Modulation (AM-AM)
Amplitude-Modulation to
Phase-Modulation
(AM-PM)
linear
“small-signal” “small-signal”
“large-signal”
“large-signal”
frequency conversion
For a non-linear system,
1 1 2 2 1 1 2 2
, ,i i i
a x a x a y a y
YH A
X
The system response depends on the amplitude of the input signals.
1 1 1 1 2 2
2 2 1 1 2 2
y t x t h t x y a x a x
Y X H x y a y a y
X
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Sources of Nonlinearity
• any active element operated under large-signal conditions
e.g. diodes, transistors, amplifiers, mixers
• many passive elements, even if the signals are not very large
metal-oxide-metal contacts, metal-metal contacts, dirty contacts, material defects [2, 3]
“rusty bolt” effects; usually below the system noise level
heating / “electro-thermal” [4]
above the noise level; can limit dynamic range of system
Vin R
Iout voltage applied, current flows
resistor heats up resistance
increases current decreases
resistor cools down resistance
decreases
current increases
input: DC
output: sinusoidal
nonlinear system
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Nonlinear Radar
Tx
Rx
transP
inP
outP
recP
in trans 2P P
R
rec out 2P P
R
outP
inP
target
Challenges:
• Devices require more power than is incident during normal
operation in order to drive them into non-linear behavior.
• The received waveform is generally not a linear sum
of the device response(s).
• Received responses are usually very weak compared to the
transmitted probe signals.
transmitted = { f1, f2, f3, … }
received = { f1, f2, f3, …,
fa, fb, fc, … }
Target presence/location is
indicated by receiving frequencies
that were not transmitted.
co
nve
rsio
n lo
ss
target generates fa, fb, fc, …
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Prior Work: “Harmonic Radar”
2 3
sc inc inc inc ...E E E ER
[6]
[5]
[7]
[8]
[9]
• built conformal tags for insect tracking
• successfully tracked out to 58 m
at Pt = 20 dBm, using horn antennas
with G = 22 dBi, ft = 5.9 GHz, fr = 11.8 GHz
• automotive radar, extended to Vulnerable Road Users
• simulations show detection possible > 22 m at 80 GHz,
Pt = 15 dBm with antenna G = 16 dBi
• built 2 prototypes (slot-array, horn antenna) for measuring
heartbeat and respiratory rate
• successfully measured vital signs using 12 GHz (0 dBm)
and 24 GHz (-9 dBm), d = 1 m
harmonics
intermodulation
self-generated
harmonics
radar eqn.
• metal surfaces (at normal incidence)
generate mostly odd-order nonlinear products
X
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Presentation Overview
• Introduction
• Nonlinearity: AM-AM, AM-PM, Physical Sources
• Radar: Transmitted & Received Waveforms
• Prior (Published) Work
• Mathematical Modeling
• Power Series, Even- vs. Odd-Order Nonlinearities
• Responses to Waveforms: Single-Tone, Two-Tone
• Linearization Techniques
• Recent Experiments & Summary 2 f
E V
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9
Power-Series Model
2 3
out 1 in 2 in 3 in in
1
...N
n
n
n
V aV a V a V a V
Let the device response be memoryless and approximated by
with out inV V
simple polynomial model, a1…an are complex numbers (amplitude and phase),
no hysteresis (memory) effects, otherwise use Volterra Series model [1] :
out 1 in 2 in 3 in in
1
1 2 in 1 in 2 in 1 2
...
, ,..., ... ...
N
n
n
n n n n
t
V t H V t H V t H V t H V t
H h V t V t V t d d d
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Odd-Order Nonlinearity
2 3
out 1 in 2 in 3 in in
1
...N
n
n
n
V aV a V a V a V
Let the device response be memoryless and approximated by
out in
3
in in
5 7
in in
9 11
in in
10 tanh
1010
3
4 34
3 63
124 2764...
567 31185
V V
V V
V V
V V
with out inV V
An example amplifier response [10] is:
gain = 10
saturation at ±10 V
Vin Vout
180 °symmetric around
Vin = Vout = 0 (“odd”)
n = odd terms only, because
out in out inV V V V
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2 3
out 1 in 2 in 3 in in
1
...N
n
n
n
V aV a V a V a V
Let the device response be memoryless and approximated by
with out inV V
An example rectifier response is:
Even-Order Nonlinearity
out in
2 4
in in
6 8
in in
10
in
5 4 1 sech 2
5 25
32 1536
61 1385
36864 8257536
50521...
2972712960
V V
V V
V V
V
Vin
+
_
Vo
ut
+
_
symmetric around
Vin = 0 (“even”)
n = even terms only, because
out in out inV V V V
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Typical (Odd + Even) Nonlinearity
An example nonlinear response is:
out in
in
10 tanh
151 sech 2
16
V V
V
2
out 1 in 2 in
3 4
3 in 4 in
5 6
5 in 6 in
7 8
7 in 8 in ...
V aV a V
a V a V
a V a V
a V a V
• not symmetric around Vin = Vout = 0
• not symmetric around Vin = 0
small-signal
(linear)
response
Vin Vout NL
large-signal response
large-signal
response
Typical nonlinearities have
both odd and even power-series terms.
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Single-Tone Response
2 3
out 1 in 2 in 3 in in
1
...N
n
n
n
E a E a E a E a E
Let the device response be memoryless and approximated by
Let the input waveform be a sinusoid / complex exponential:
0in 0 0 0cos
2
j t j tEE E t e e
Then the device response is
2 3
0 0 0out 1 2 3
2 30 2 2 3 3 30 0 0
1 2 3
...2 2 2
2 3 3 ...2 4 8
j t j t j t j t j t j t
j t j t j t j t j t j t j t j t j t j t
E E EE a e e a e e a e e
E E Ea e e a e e e a e e e e e
0 0
2 3
2 0 3 0out 1 0 0 0 0 0cos 1 cos 2 3cos cos 3 ...
2 4
a E a EE a E t t t t
linear even-order odd-order
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1-Tone Response: Odd-Order NL
The example amplifier response is: 3 5 7 9
out in in in in in
10 4 34 12410 ...
3 3 63 567V V V V V V
in 0 0 0
00
cos 1 V
1 MHz2
V V t V
f
Let the input be a single tone,
at 1 MHz with amplitude = 1 V:
The output is a sum of sinusoids
at 1 MHz, 3 MHz, 5 MHz, etc:
out 1 0 0
3
3 0 0
5
5 0 0
7
7 0 0
9
9 0 0
cos
cos 3
cos 5
cos 7
cos 9 ...
V V t
V t
V t
V t
V t
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input = { f }
output = { f, 3f, 5f, 7f, 9f, … }
The example amplifier response is: 3 5 7 9
out in in in in in
10 4 34 12410 ...
3 3 63 567V V V V V V
in 0 0 0
00
cos 1 V
1 MHz2
V V t V
f
Let the input be a single tone,
at 1 MHz with amplitude = 1 V:
The output is a sum of sinusoids
at 1 MHz, 3 MHz, 5 MHz, etc:
out 1 0 0
3
3 0 0
5
5 0 0
7
7 0 0
9
9 0 0
cos
cos 3
cos 5
cos 7
cos 9 ...
V V t
V t
V t
V t
V t
1-Tone Response: Odd-Order NL
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The example rectifier response is:
in 0 0 0
00
cos 1 V
1 MHz2
V V t V
f
Let the input be a single tone,
at 1 MHz with amplitude = 1 V:
The output is a sum of sinusoids
at 2 MHz, 4 MHz, 6 MHz, etc:
2
out 2 0 0
4
4 0 0
6
6 0 0
8
8 0 0
10
10 0 0
cos 2
cos 4
cos 6
cos 8
cos 10 ...
V V t
V t
V t
V t
V t
2 4 6 8
out in in in in
5 25 61 1385...
32 1536 36864 8257536V V V V V
1-Tone Response: Even-Order NL
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The example rectifier response is:
in 0 0 0
00
cos 1 V
1 MHz2
V V t V
f
Let the input be a single tone,
at 1 MHz with amplitude = 1 V:
The output is a sum of sinusoids
at 2 MHz, 4 MHz, 6 MHz, etc:
2
out 2 0 0
4
4 0 0
6
6 0 0
8
8 0 0
10
10 0 0
cos 2
cos 4
cos 6
cos 8
cos 10 ...
V V t
V t
V t
V t
V t
input = { f }
output = { 2f, 4f, 6f, 8f, 10f, … }
2 4 6 8
out in in in in
5 25 61 1385...
32 1536 36864 8257536V V V V V
1-Tone Response: Even-Order NL
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The example nonlinear response is: out in in10tanh 15 16 1 sech 2V V V
in 0 0 0
00
cos 1 V
1 MHz2
V V t V
f
Let the input be a single tone,
at 1 MHz with amplitude = 1 V:
2
out 1 0 0 2 0 0
3 4
3 0 0 4 0 0
5 6
5 0 0 6 0 0
7 8
7 0 0 8 0 0
9
9 0 0
cos cos 2
cos 3 cos 4
cos 5 cos 6
cos 7 cos 8
cos 9 ...
V V t V t
V t V t
V t V t
V t V t
V t
The output is a sum of sinusoids
at 1 MHz, 2 MHz, 3 MHz, 4 MHz, etc:
1-Tone Response: Typical NL
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The example nonlinear response is:
in 0 0 0
00
cos 1 V
1 MHz2
V V t V
f
Let the input be a single tone,
at 1 MHz with amplitude = 1 V:
2
out 1 0 0 2 0 0
3 4
3 0 0 4 0 0
5 6
5 0 0 6 0 0
7 8
7 0 0 8 0 0
9
9 0 0
cos cos 2
cos 3 cos 4
cos 5 cos 6
cos 7 cos 8
cos 9 ...
V V t V t
V t V t
V t V t
V t V t
V t
The output is a sum of sinusoids
at 1 MHz, 2 MHz, 3 MHz, 4 MHz, etc:
out in in10tanh 15 16 1 sech 2V V V
input = { f }
output = { f, 2f, 3f, 4f, 5f, 6f, … }
1-Tone Response: Typical NL
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Two-Tone Response
2 3
out 1 in 2 in 3 in ...E a E a E a E Let the device response be memoryless and approximated by
Let the input waveform be a two-tone continuous wave:
1 1 2 21 2in 1 1 1 2 2 2cos cos
2 2
j t j t j t j tE EE E t E t e e e e
Then the device response is
1 1 2 2 1 1 2 2 1 1 2 2
1 2 3
1 2 1 2 1 2out 1 2 3 ...
2 2 2 2 2 2
j t j t j t j t j t j t j t j t j t j t j t j tE E E E E EE a e e e e a e e e e a e e e e
1 2 0E E E
1 2 1 2 1 2 1 21 1 2 2 1 1 2 2
1 1 2 2 1 1 2 2
1 2
22 2 2 20 0
out 1 2
3 3 3 3
320
3
4 2 2 2 22 4
9 9
38
j t j t j t j tj t j t j t j t j t j t j t j t
j t j t j t j t j t j t j t j t
j t
E EE a e e e e a e e e e e e e e
e e e e e e e e
Ea e e
1 2 1 2 1 2
1 2 1 2 2 1 2 1
2 2 2
2 2 2 2
3 ...
3 3
j t j t j t
j t j t j t j t
e e
e e e e
2
0out 1 in 2 1 2 1 2 2 1
3
03 in 1 2 1 2 1 2 1 2 2 1
2 cos 2 cos 2 2cos 2cos2
9 cos 3 cos 3 3cos 2 3cos 2 3cos 2 3cos 2 ...4
EE a E a t t t t
Ea E t t t t t t
intermodulation (IMD) / mixing
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The example nonlinear response is: out in in10tanh 15 16 1 sech 2V V V
in 1 1 2 2cos cosV V t V t Let the input be a two-tone continuous wave,
at 99 MHz and 101 MHz with total amplitude = 2 V:
21
1 21 2 1 21V 99 MHz 101MHz
2 2V V f f
The output is a sum of distortion products:
1 1
2 2
2 1 2 1
1 2 1 2
1 2 2 1
1 2 2 1
1 2 2 1
2 198 MHz 3 297 MHz ...
2 202 MHz 3 303 MHz ...
2 MHz 2 2 4 MHz ...
200 MHz 2 2 400 MHz ...
2 97 MHz 2 103 MHz
3 2 95 MHz 3 2 105 MHz
4 3 93 MHz 4 3 107 MHz
... ...
f f
f f
f f f f
f f f f
f f f f
f f f f
f f f f
intermodulation (IMD) / mixing products
Two-Tone Response
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The example nonlinear response is:
out in in10tanh 15 16 1 sech 2V V V
intermodulation
difference frequencies
(beat frequency)
harmonics
fundamental
tones
1 21 299 MHz 101MHz
2 2f f
Two-Tone Response
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23
1 21 299 MHz 101MHz
2 2f f
3rd
-ord
er
inte
rmo
du
latio
n
fun
da
me
nta
l to
ne
5th-o
rde
r in
term
od
ula
tion
7th-o
rde
r in
term
od
ula
tion
2 12 103 MHzf f 1 22 97 MHzf f
2 13 2 105 MHzf f
2 14 3 107 MHzf f
1 23 2 95 MHzf f
1 24 3 93 MHzf f
“upper” IMD
products
“lower” IMD
products
fun
da
me
nta
l to
ne
3rd
-ord
er
inte
rmo
du
latio
n
5th-o
rde
r in
term
od
ula
tion
7th-o
rde
r in
term
od
ula
tion
Two-Tone Response
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Presentation Overview
• Introduction
• Nonlinearity: AM-AM, AM-PM, Physical Sources
• Radar: Transmitted & Received Waveforms
• Prior (Published) Work
• Mathematical Modeling
• Power Series, Even- vs. Odd-Order Nonlinearities
• Responses to Waveforms: Single-Tone, Two-Tone
• Linearization Techniques
• Recent Experiments & Summary 2 f
E V
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Linearization Techniques
A good signal generator may produce distortion
at -60 dBc (dB with respect to the carrier) at 0 dBm output.
This is likely not sufficient to detect our
targets-of-interest at standoff ranges.
f 1
f f 2
2f 1
-f2
2f 2
-f1
Pin
-60 dBc
system-generated
IMD
desired
tones
NL f
Pin
generate
distortion before
the nonlinearity
f
Pout
nonlinearity
un-distorts
the signal
predistortion [11,12]
cancellation [13]
NL f
Pin
f
+ f
out
of phase
w/ IMD
180
f
Pout
f
Pin filtering
f
Pout
25
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Presentation Overview
• Introduction
• Nonlinearity: AM-AM, AM-PM, Physical Sources
• Radar: Transmitted & Received Waveforms
• Prior (Published) Work
• Mathematical Modeling
• Power Series, Even- vs. Odd-Order Nonlinearities
• Responses to Waveforms: Single-Tone, Two-Tone
• Linearization Techniques
• Recent Experiments & Summary 2 f
E V
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Prec
Ptrans
fin
AR
50W1000
amp
Rohde & Schwarz
FSP 40 GHz analyzer
receive
filter
Tektronix
AWG7052
waveform
generator
12-ft Megaphase F130 cable
15-ft Megaphase
F130 cable
transmit
filter
target height
= 9 in
12-ft
cable
antenna height
= 39 in,
horizontal
cente
r-of-
TE
M to R
x-a
nte
nna
dis
tance =
45 in
·
y
x z
programmable
attenuators
receive antenna = A.H. Systems SAS-510-4
LPA (290 MHz to 4 GHz)
HP 33322G
HP 8494H
Recent Experiments: GTEM Cell
27
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Single-Tone Experiment
28
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Multi-Tone Experiment
29
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Summary
30
• Nonlinear radar is a technique that is well-suited for detecting electronic targets.
• Nonlinear radar has the advantage of high clutter rejection
(because most clutter reflects the same frequencies as those transmitted).
• Nonlinear targets may be (1st-order) modeled by a power series
with complex polynomial coefficients.
• A variety of waveforms may be transmitted (e.g. 1 tone, 2 simultaneous tones).
• A variety of signatures may be collected (e.g. harmonics, intermodulation).
• Some form of linearization (e.g. filtering, cancellation) must typically be used
to mitigate system-generated nonlinear distortion.
• Wireless reception of nonlinear spectral content has been demonstrated
at ARL, inside a GTEM cell, using several RF devices.
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References
[1] J. C. Pedro and N. B. Carvalho, Intermodulation Distortion in Microwave and Wireless Circuits. Boston, MA: Artech House, 2003.
[2] C. Vicente and H. L. Hartnagel, “Passive-intermodulation analysis between rough rectangular waveguide flanges,” IEEE Transactions on
Microwave Theory and Techniques, Vol. 53, No. 8, Aug. 2005, pp. 2515–2525.
[3] H. Huan and F. Wen-Bin, “On passive intermodulation at microwave frequencies,” in Proceedings of the Asia-Pacific Electromagnetic
Conference, Nov. 2003, pp. 422–425.
[4] J. R. Wilkerson, K. G. Gard, A. G. Schuchinsky, and M. B. Steer, “Electro-thermal theory of intermodulation distortion in lossy microwave
components,” IEEE Transactions on Microwave Theory and Techniques, Vol. 56, No. 12, Dec. 2008.
[5] N. Tahir and G. Brooker, “Recent developments and recommendations for improving harmonic radar tracking systems,” in Proceedings of the 5th
European Conference on Antennas and Propagation, Apr. 2011, pp. 1531–1535.
[6] D. Psychoudakis, W. Moulder, C. C. Chen, H. Zhu, and J. L. Volakis, “A portable low-power harmonic radar system and conformal tag for insect
tracking”, IEEE Antennas and Wireless Propagation Letters, Vol. 7, 2008, pp. 444–447.
[7] J. Saebboe, V. Viikari, T. Varpula, and H. Seppa, “Harmonic automotive radar for VRU classification”, in Proceedings of the International Radar
Conference: Surveillance for a Safer World, Oct. 2009, pp. 1–5.
[8] C. Fazi and F. Crowne, “Nonlinear radar signatures from metal surfaces”, in Proceedings of the International Radar Conference: Surveillance for
a Safer World, Oct. 2009, pp. 1–6.
[9] L. Chioukh, H. Boutayeb, K. Wu, and D. Deslandes, “Monitoring vital signs using remote harmonic radar concept”, in Proceedings of the 2011
European Radar Conference, Oct. 2011, pp. 381-384.
[10] K. G. Gard, L. E. Larson, and M. B. Steer, "The impact of RF front-end characteristics on the spectral regrowth of communications signals,"
IEEE Transactions on Microwave Theory and Techniques, Vol. 56, June 2005, pp. 2179-2186.
[11] W. Woo, M. D. Miller, and J. S. Kenney, “A hybrid digital/RF envelope predistortion linearization system for power amplifiers”, IEEE Transactions
on Microwave Theory and Techniques, Vol. 53, No. 1, 2005, pp. 229–237.
[12] M. Helaoui, S. Boumaiza, A. Ghazel, and F. M. Ghannouchi, “Power and efficiency enhancement of 3G multicarrier amplifiers using digital
signal processing with experimental validation”, IEEE Transactions on Microwave Theory and Techniques, Vol. 54, No. 4, 2006, pp. 1396–1404.
[13] K. J. Cho, W. J. Kim, J. H. Kim, and S. P. Stapleton, “Linearity optimization of a high power Doherty amplifier based on post-distortion
compensation”, IEEE Transactions on Microwave Theory and Techniques, Vol. 15, No. 11, 2005, pp. 748–750.
[14] G. J. Mazzaro, K. G. Gard, and M. B. Steer, “Linear amplification by time-multiplexed spectrum,” IET Circuits, Devices, & Systems, Vol. 4, No. 5,
Sept. 2010, pp. 392-402.
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