Design Considerations for Nonlinear Scattering

by Christian Fazi, Frank Crowne, and Marc Ressler

ARL-TR-5684 September 2011

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Army Research Laboratory

Adelphi, MD 20783-1197

ARL-TR-5684 September 2011

Design Considerations for Nonlinear Scattering

Christian Fazi, Frank Crowne, and Marc Ressler

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution unlimited. 
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Design Considerations for Nonlinear Scattering

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Christian Fazi, Frank Crowne, and Marc Ressler

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14. ABSTRACT

Renewed interest in harmonic and multitone radar has motivated design efforts based on the standard approach found in the 

radar literature, i.e., the classical radar equation. In this report, we show that such approaches can be problematic in 

determining the correct power budget requirements for detecting a nonlinear target, and that it is necessary to examine the 

differences of a linear versus nonlinear target cross-section.

15. SUBJECT TERMS

Intermods; scattering; radar equation; radar cross-section

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Christian Fazi

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iii

Contents

List of Figures iv

1. Introduction 1

2. Linear Radar 1

3. Nonlinear Radar 2

4. Multipath Effects 6

5. Summary 8

6. References 9

Distribution List 10
iv

List of Figures

Figure 1. Multipath geometry. ........................................................................................................7

Figure 2. Range dependence of signal in a multipath environment................................................7

Figure 3. Change in target strength versus height of target for a target range of 100 m. ..............8
1

1. Introduction

There has been recent interest in the use of harmonic and multitone radar for various 

applications, from clutter suppression during tracking of animals and insects (1), to improving 

target classification when detecting radar tags (2). In designing systems based on these new 

technologies, it is incumbent on the designer to resist the temptation to use traditional

methodologies carelessly. This applies in particular to the radar equation, which must be 

modified so as to incorporate the way a nonlinear scatterer (target) responds to electromagnetic 

(EM) illumination, which (as we will see further) differs significantly from the response of an 

ordinary target.

2. Linear Radar

In order to discuss nonlinear effects in radar, we must first specify what is meant by a linear 

target. It is reasonable to identify a target as linear when, upon illumination by an EM field, it 

radiates a scattered field that is directly proportional to the incident field. For a monostatic radar, 

this makes it possible to reduce the properties of the scatterer to a single parameter, the

(backscattering) cross-section (3), which is purely a property of the scatterer and independent of 

any property of the excitation source. This definition leads to the well-established radar equation

(4). Assuming such a linear target is illuminated by an antenna with area A at a distance R, the 

antenna gain is:

4

2

A G 





,

where 



is the radar wavelength and ρ is the antenna efficiency, which we will assume to be 1.

A power P fed to this antenna creates a power flux



(watts/area) at the target of

2

4

PG

R





,

and, hence, an electric field at the target

 

 

 

1/2

1/2

1/2 4

ZPG

E Z inc

R

  



.

Here, Z is the antenna’s impedance. At the target, this field induces a current, which in turn 

radiates a scattered field back to the antenna:
2

 

 

1/2 1/2

1/2 2

4

ZPG

E E sc inc R R

 

 



,

where  is the radar cross-section. If the distance R back to the receiver is large, this scattered 

field consists of a plane wave whose amplitude has fallen off by 1/R

2

. Then, once more taking 

into account the receiver gain, the field induced at the receiver is

 

1/2 1/2

4 2

4

G ZP G

E E rec sc

R

  

        ,

and the received power is 

 

 

1 2

2 4

4

2 2

3 4

4

PGA P Z E A rec rec

R

PG

R



  





 



,

which is proportional to 1/R

4

, the classical result.

3. Nonlinear Radar

This derivation fails for a nonlinear scatterer. To illustrate why, let us assume that the

nonlinearity of the scatterer is small enough to admit a power-law expansion, a situation that 

illustrates our argument and is also simple mathematically (although other associated types of 

nonlinearities—e.g., clipping, Schottky barriers, etc.—lead to similar results). Let us illuminate 

the target with two signals, E1 and E2, close enough in frequency that the antenna can emit and 

receive both efficiently. We then assume that the response of the scatterer takes the form

1/2 2 3 I E E E sc inc inc inc       ,

where α and β are constants that are specific to the target, and Isc is the current at the target 

generated by the incident field Einc. Since this field contains two fields, E1 and E2, at two 

frequencies, ω1 and ω2, the current Isc will also contain components at these frequencies. For a 

linear target, there will be no frequencies in the return signal other than these. However, if we 

view the target as a ―device‖ in the circuit-theoretic sense, we know that the current induced by 

Einc must also contain additional frequencies. Thus, if 

cos cos     1 1 1 2 2 2 E E t E t inc         ,

the contributions to Isc from the squared term can be written
3

   

     

 

   

   

2

2

cos cos 1 1 1 2 2 2

2

2

cos 2 cos cos 1 1 1 1 2 1 1 2 2

2

2

cos 2 2 2

1 1 2 2 1 cos 2 2 1 cos 2 2 1 1 1 2 2 2 2 2

cos cos 1 2 1 2 1 2 1 2 1 2

E E t E t inc

E t E E t t

E t

E t E t

E E t t

           

          

   

                 

                             

,

which contains the frequencies 

1 2 1 2 1 2 0, 2 , 2 , , and       ,

along with ω1 and ω2. For the contributions to Isc from the cubic term, we have the following 

expression:

   

     

     

   

   

 

3

3

cos cos 1 1 1 2 2 2

3

3 2 2 cos 3 cos cos 1 1 1 1 2 1 1 2 2

2 2 3 3 3 cos cos cos 1 2 1 1 2 2 2 2 2

1 3

cos 3 3 3cos 1 1 1 1 1 4

1 3

cos 3 3 3cos 2 2 2 2 2 4

3 2

1 cos 2 2 1 2 1 1 2

E E t E t inc

E t E E t t

E E t t E t

E t t

E t t

E E t

           

          

          

           

           

       

   

   

   

   

cos 2 2

3 2

cos 1 cos 2 2 1 2 1 1 2 2 2

1 3 3 3 3 2 cos 3 3 cos 1 1 1 1 1 2 1 1 4 4 2

1 3 3 3 3 2 cos 3 3 cos 2 2 2 2 1 2 2 2 4 4 2

3 2

cos 2 2 cos 2 2 1 2 1 2 1 2 1 2 1 2 4

3

1 4

t

E E t t

E t E E E t

E t E E E t

E E t t

E

     

           

              

              

                             

    

2

cos 2 2 cos 2 2 2 2 1 2 1 2 1 2 1 E t t                             

,

which contains the frequencies 

3 , 3 , , , 2 , 2 , 2 , and 2 1 2 1 2 1 2 1 2 2 1 2 1                 .

There are two things to note about this result:

(1) the responses at the illuminating radar frequencies ω1 and ω2 are no longer linear, which 

makes the nominal cross-sections at these frequencies field dependent, and

(2) Unlike the frequencies generated by the squared term in Isc, the fact that ω1 ≈ ω2 implies 

that 2ω1 – ω2 ≈ ω1 and 2ω2 – ω1 ≈ ω2; i.e., these frequencies lie within the antenna 
4

bandwidth. In mixer theory, these frequencies are referred to as intermodulation products

(5).

Let us consider only the in-band signals returning to the antenna. Assuming the antenna gain and 

impedance are the same for frequencies ω1 and ω2, the scattered fields at the various radiated 

frequencies take the form

 

 

 

 

 

 

 

1 3 3 1/2 3 2

1 1 1 1 2 4 2

1/2

1/2 1/2 3

2

1/2 2

1 1 2 4 2

4 4

1 3 3 1/2 3 2

2 2 2 1 2 4 2

1/2

1/2 1/2 3

2

1/2 2

2 2 1 4 2

4 4

1 3 2

2

1 2 1 2 4

E E E E E sc R

ZG ZG P P P

R R

E E E E E sc R

ZG ZG P P P

R R

E E E sc R

                 

 

          

   

                 

 

          

   

      

 

 

 

 

 

3/2

3 1/2

3/2 1 2 4 4

4

3/2

1 3 3 2 1/2 2

2 1 1 2 2 1 4 4 3/2 4

4

ZG

P P

R

ZG

E E E P P sc R R



     

              . (1)

Then the fields arriving at the intermod frequencies are

   

 

   

 

1/2 3/2 2 3 1/2 2 2 1 2 1 2 1 2 4 4 2 4

4

1/2 3/2 2 3 2

2 2 2 1 2 1 2 1 4 4 2 4

4

G Z G E E P P rec sc

R

G Z G E E P P rec sc

R

               



                





,

and the powers received at the intermod frequencies are

   

 

 

   

 

 

2 2 4 1 2 2 9

2 2 1 2 1 2 1 2 16 4 8

4

2 5 2 9 2 2

5 8 1 2 16 4

2 2 4 1 2 2 9

2 2 2 1 2 1 2 1 16 4 8

4

2 5 2 9 2 2

5 8 2 1 16 4

Z G A P Z E A P P rec sc

R

Z G P P

R

Z G A P Z E A P P rec sc

R

Z G P P

R

         





 



         





 



.

Looking past the complexity of these expressions, let us focus on the R–dependence of the 

intermod terms (5). Equation 1 reveals that Prec (2ω1 – ω2) and Prec (2ω2 – ω1) both fall off as

1/R

8

, rather than the traditional 1/R

4

predicted by the radar equation. Note that this drastic 
5

attenuation of receiver power at the new frequencies will be somewhat mitigated by the 

increased antenna gain coefficient G

5

.

At first glance, it appears that we have identified a near-field component of the radiation from 

the scatterer, since there is clearly no power at infinity from a field that falls off faster than 1/R. 

To counter this assertion, let us consider the bistatic case. Let the transmit antenna have a gain Gt

and be a distance Rt from the target, while the receive antenna has a gain Gr and is at a distance 

Rr from the target. Assume that both Rt and Rr are large compared to wavelength and target 

dimensions. Then the same derivation as above leads to the following expression for the 

intermod fields at the scatterer: 

 

 

 

 

 

 

3/2

1 3 1 3 2 1/2 2

1 2 1 2 1 2 4 4 3/2 3

4

3/2

1 3 1 3 2 1/2 2

2 1 1 2 2 1 4 4 3/2 3

4

Z Gt t E E E P P sc R R r r Rt

Z Gt t E E E P P sc R R r r Rt

 

                     

 

                     

.

Then the field detected at the receiver is of the form

   

 

   

 

1/2 3/2 1/2 3/2 3 1/2 2 2 1 2 1 2 1 2 4 4 2 3

4

1/2 3/2 1/2 3/2 3 2

2 2 2 1 2 1 2 1 4 4 2 3

4

Gr

Z G G t t r E E P P rec sc

R Rr t

Gr

Z G G t t r E E P P rec sc

R Rr t

                

                





.

Now the intermod products have a factor of 

3/2 1/2

3

G G t r

R R t r

, and the powers received at the intermod 

frequencies go as

3

6 2

G Gt r

R R t r

. Clearly, the initial trip from transmitter to target satisfies the far-field 

condition by construction, while the 1/Rr

2

behavior arising from the trip from target to receive 

antenna shows prima faciae that it is also satisfied on this leg of the trip, as well. There will, in 

fact, be near-field contributions to the radiation from the scatterer, but they contain higher 

powers of 1/Rr

2

and, hence, are even more attenuated by distance. Note that in the bistatic case,

the coefficient  may depend on the angle between the receiver and transmitter boresights.

The nonlinear radar cross-section becomes very small when it involves higher order terms. For 

example, a 1 MW source would barely have 300 m range.
6

4. Multipath Effects

In situations where the radar and the target are both relatively close to the ground, multipath 

effects can further exacerbate this range-dependent attenuation. Although in rare cases multipath 

may actually enhance a target return, the most likely effect at low grazing angles is destructive 

interference caused by the ground reflection. Vertical polarization can provide some advantages 

due to pseudo-Brewster angle effects, but the Brewster angle ranges from about 13° (moist soils) 

to about 30° (arid soils), while the grazing angle from a 2-m high sensor to a point 100 m away is 

1.1°. At these shallow angles, the behavior of both horizontally and vertically polarized waves is

essentially the same, with an inversion of sign upon reflection, and with very little loss in 

amplitude. 

In general, the effect of multipath cancellation is to add another 1/R

2

to the power dependence

each way. To demonstrate where this dependence comes from, consider a radar located near the 

ground illuminating a target near the ground. To simplify the model, consider the geometry 

shown in figure 1. An antenna at a height h off the ground illuminates a point on a target a 

horizontal distance R, where the point is a height s off the ground. The signal from the antenna is 

taken to be the sum of a direct propagation from antenna to target and an indirect propagation 

involving a bounce off the ground at a horizontal distance R from the target, with  < 1. 

Assuming that the reflection causes a complete phase reversal with no loss of amplitue (again, 

this is a reasonable assumption for shallow grazing angles, or purely horizontal polarization), the 

signal at the target is of the form:

  cos cos   0 1 2 E E R D D D

c c

 

      

 

,

where 

 

1

0

E R

R



as before, and

 

 

2 2 2

1

1

2 2 2

2

2 2

D h R

D s R

D h s R





  

 

  

.

At distances 

R h s  ,

we find that

1 1 2 2 2

1 2 2 1

D D D h s hs

R

 

 

      

                 

and 
7

   

1

cos cos

2 2 2 0 1 2 1 2

1

cos 2

E E D D D D D D

c c

D

R c

    



 



.

Figure 1. Multipath geometry.

Clearly, this change in amplitude at the target corresponds to a 1/R

4

change in the power from 

antenna to target, and a 1/R

8

change for the round trip, even for a linear target. In figure 2, we plot 

the amplitude, in dB, versus the range to the target for a radar located 2 m off the ground 

illuminating a target 0.25 m off the ground. The 6 dB peaks in the signals for the 2000 and 

3000 MHz curves is due to constructive interference, but farther out in range, all of the curves 

approach a 20 dB/decade fall-off in amplitude, thus demonstrating the 1/R

2

rule. Consider the 

region between 15 and 20 m in the example of figure 2. Here the illumination near 1000 MHz is 

definitely influenced by an additional 1/R

2

term and, hence, a 1/R

8

falloff is expected. Combining 

this with the effect of the nonlinear target gives a total falloff of 1/R

16. This would mean a 

change of range from 50 to 100 m would result in a loss of signal of 2

16 or 48 dB per octave or 

160/dB per decade.

2 meter source, 0.25 meter target

-15

-10

-5

0

5

10

0 10 20 30 40 50 60 70 80 90 100

meters

dB

1000MHz

2000MHz

3000MHz

Figure 2. Range dependence of signal in a multipath environment.
8

Multipath is of particular importance when the target height above ground can vary quickly, as is 

the case when insects are tagged with non-linear transponders (1, 2). In flight, the insect can 

appear as a free-space target, but as it approaches, or crawls along the ground, its effective crosssection is seriously reduced, as can be seen in figure 3. Here we examine the effects of varying 

the target height at a fixed range of 100 m from the radar. Note that there are situations where 

additional nulls can occur (e.g., 3000 MHz at 2.5 m height, 2000 MHz at 3.75 m height), and this 

happens more frequently at smaller wavelengths and may cause a fast variation in target 

signature as the target height varies.

Variation with target height - range=100 meters

-50

-40

-30

-20

-10

0

10

0 0.5 1 1.5 2 2.5 3 3.5 4

height - meters

dB

1000MHz

2000MHz

3000MHz

Figure 3. Change in target strength versus height of target for a 

target range of 100 m.

5. Summary

We emphasize here that we are not advocating any modification of the radar equation, itself.

Rather, in this paper we are simply asserting that in nonlinear radar, the radar cross-section 

(RCS) of a target is no longer independent of distance from the source, and has to be redefined 

for each order of nonlinearity in the target response. 
9

6. References

1. O’Neal, M. E.; Landis, D. A.; Rothwell, E.; Kempel, L.; Reinhard, D. Tracking Insects with 

Harmonic Radar: a Case Study. American Entomologist, Winter 2004, pp 212–218.

2. Charvat, G. L.; Rothwell, E. J.; Kempel, L. C. Harmonic Radar Tag Measurement and 

Characterization. IEEE Antennas and Propagation Society International Symposium June 

2003, 2, 696–699.

3. Jackson, J. D. Classical Electrodynamics; 2

nd ed. John Wiley & Son: New York, 1975, Sec. 

9.6.

4. Hovanessian, S. A. Radar System Design and Analysis; Artech House Inc.: Dedham MA, 

1984), Ch. 1.

5. Maas, S. Microwave Mixers; Artech House Inc.: Norwood MA, 1993, Sec. 5.2.
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