nonlinear radar concepts -...

4/26/2012

1

Nonlinear Radar Concepts

Gregory J. Mazzaro

Electronics Engineer

U.S. Army Research Laboratory

Adelphi, MD 2012 1

Approved for Public Release


2

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


• Introduction









E V



4

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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5

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



6

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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7

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



8


• Introduction









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] :


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



10

Odd-Order Nonlinearity

2 3


1

...N

n

n

n

V aV a V a V a V


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



4/26/2012

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11

2 3


1

...N

n

n

n

V aV a V a V a V


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



12

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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13

Single-Tone Response

2 3


1

...N

n

n

n

E a E a E a E a E


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



14

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



4/26/2012

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15

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





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



16

The example rectifier response is:

in 0 0 0

00

cos 1 V

1 MHz2

V V t V

f





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



4/26/2012

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17

The example rectifier response is:

in 0 0 0

00

cos 1 V

1 MHz2

V V t V

f





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



18

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



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


at 1 MHz, 2 MHz, 3 MHz, 4 MHz, etc:

1-Tone Response: Typical NL



4/26/2012

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19

The example nonlinear response is:

in 0 0 0

00

cos 1 V

1 MHz2

V V t V

f



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


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



20

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



4/26/2012

11

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



22

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



4/26/2012

12

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



24


• Introduction









E V



4/26/2012

13

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



26


• Introduction









E V



4/26/2012

14

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



Single-Tone Experiment

28



4/26/2012

15

Multi-Tone Experiment

29



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.



4/26/2012

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31

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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