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NIST Technical Note 1522

Phase Dependence in Radar

Cross Section Measurements

Lorant Muth

QC dT

I AA Institute of Standards and Technology

^Administration, U.S. Department of Commerce .U5753

#1522

2001 c.a.

NIST CiNTENNIAL

NIST Technical Note 1522

Phase Dependence in Radar

Cross Section Measurements

Lorant A. Muth

Radio-Frequency Technology Division

Electronics and Electromagnetic Technology Division

September 2001

'^C^^-^^ J

U.S. Department of Commerce

Donald L. Evans, Secretary

National Institute of Standards and Technology

Karen H. Brown. Aclinsi Director
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept adequately.

Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the

entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Technical Note 1522

Natl. Inst. Stand. Technol. Tech. Note 1422, 16 pages (September 2001)

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PHASE DEPENDENCE IN

RADAR CROSS SECTION MEASUREMENTS

Lorant A. Muth

Radio-Frequency Technology Division

National Institute of Standards and Technology

Boulder, Colorado 80305

A new measurement and analysis technique to isolate the background

signals present in radar cross section measurements is presented. By

definition the measured RCS of a target is independent of the measured

phase, but it is not independent of the phase difference between the

theoretically correct signal and the background error signals present in

the measurements. By varying the phase of the theoretical signal and

holding the phase of the error signal constant, one can separate these

two components.

In the calibration model, where the radar cross section of the calibration

target is known, the error signals can be removed from the measurements

to obtain an accurate system calibration. When the radar cross section

of the target is unknown, only error signals with a constant phase can be

removed from the measurements. Error signals that vary in-phase with

the theoretical signal will introduce a bias that increases the uncertainty

of the measurements.

Key words: measured error fields; measured scattered fields; measured

RCS amplitude and phase; radar cross section; RCS

1. Introduction

By definition the radar cross section (RCS) of a target is the squared amphtude of the

electric field scattered by a target located at infinity and illuminated by a plane wave.

In practice, the reflected electric field is measured monostatically or at a given bistatic

angle. In either case, the distance d between a target and the transmitting and receiving

antennae is very large, kd » 27r, where k is the transmitted wavenumber. The measured

signal S is composed of a theoretically correct scattered electric field, additional scattered

fields that originate from the environment of the measurement range, and distortions due

to instrumentation nonlinearity and noise. The measured complex electric field signal S is given by

S{r,e,b,P)=re'^-^-be'^, (la)

where r and 6 are the amplitude and phase of the reflected electric field signal from the

target (that could possibly include in-phase error signals), and 6 and are the amphtude
o

d

-1.0 -0.5 0.0 0.5 1.0

Figure 1 . The normalized complex electric field r in ideal RCS measure- ments describes a circle centered on the origin as the phase varies. The

background signal 6e^^ found in real measurements S displaces the center of the ideal circle away from the origin. By removing the background

signal from the measurements, RCS calibration accuracy is improved.

and phase of the electric field signal reflected from the environment (which includes calibration and instrumentation effects and noise). The first term in eq (la) describes a circle

in the complex plane centered on the origin as 9 varies from to 27r, and the second term

is a constant that moves the center of the theoretical circle into the complex plane. Figure

1 illustrates the difference between ideal and real RCS measurements as modeled in eq

(la). Note that there are two values of the phase 9 — where the measured RCS yields

the theoretical value, that is, S = r at these points. These values can be determined easily

from the ratio b/r (see below). We can use eq (la) to formulate simple measurement- (or

data-) based definitions of r^, the theoretical RCS, and 6^, the measured background RCS.
Thus,

^27r SdO = 27r6e'^ (16)

Jo

and

/•27r /

|5|2rf^ = 27r(r2 + 62). (Ic)

Somewhat more complicated expressions result if the integrals in eqs (lb) and (Ic) are evaluated between finite angles 9i to O2 (see Appendix A). These integral equations indicate

that we can resolve the signal into a theoretical RCS r and background b component if we

obtain the measurements 5 as a function of 0. A difficulty here is that the measured phase

of S, denoted by 7, where

r sin 9 -\- b s'm P ,„. tan7 = — -, (2) r cos -\- b cos p

is a nonlinear function of the unknown amplitudes r and b and the unknown phases 9

and /3. Moreover, none of the nonlinear parameters of interest in eq (la) and (2) can be

determined independently of S.

A sfightly modified formulation can be used to obtain the background signals present in

the data. Let be^^ = {bi^bq); then eq (la) can be rewritten as

{Si-bjf + {SQ-bQ)'' = r\ (3a)

where (following accepted usage in the RCS community) the subscripts / and Q denote

the real and imaginary components of a complex quantity. The phase ^ of r is given by

tan^ = ^^—^. (36)

bi -bi

Equation (3a) is independent of 9, and is well suited to obtain r^, the RCS of a calibration or an unknown target, in the presence of a complex background signal 6. In Figure 1, the

real and imaginary components 5/ and Sq of the electric field data are constrained on

a data circle with radius r centered at {bi^bq). The measured complex electric field S is expressed with respect to the origin, and the fractional measurement error is given by S/r.

Obviously, if we can determine and remove the background signal 6 from the data, the

measurement error can be significantly reduced. Experimental realization of this approach

is, however, challenging: we need to vary 9 without significantly varying (5. Variations in (3 such that 5(3 « 69 will be incorporated into the overall uncertainty of the measurement.

In Section 4 we apply eq (3) to measurements made on a cafibration cylinder and sphere

and obtain the background signals present in the data.

Finally, we note that only the resultant of many component error signals appears in eq

(1). Most of the important sources of error signals found on RCS measurement ranges

have been discussed in reference [1], where a general framework of radar cross section

uncertainty analysis is presented. A RCS measurement error equation was developed in

reference [2]. However, these studies have not demonstrated the magnitude of each of the
d

d

d

od

.1. -.J. .•- .V. . . . .

v-y

A...

-100 100 e-p

Figure 2. Contours of the RCS measurement error k^ as function of

the phase 6 — P and the dimensionless error parameter e. See eqs (la),

(5) and (6a). Contour hnes from -1.2 dB (broken) to +1.2 dB (sohd)

in steps of 0.2 dB are shown. For a given k the phase — p must be

specified to uniquely determine e.

specific errors present in real RCS data. To characterize the individual errors (and uncertainties) listed in reference [1], we need to implement experimental procedures specifically

designed for the purpose. Measurement programs to accomplish this are in various stages

of completion at a number of RCS facilities around the country. In this study, we will

not try to determine the component uncertainties of a measurement system, but will concentrate on a new measurement technique designed to determine the background signals

present in RCS measurements. When such signals are removed from the data the accuracy

of RCS calibrations can be significantly improved.
2. Errors in Radar Cross Section Measurements

If in eqs (1) 5 denotes the calibrated electric field data, then the measured RCS of the

target is

|5|2 = rV(e,^,/?), (4)

where the error parameter e is defined as

' - ;. (5)

and the calibration factor n, (or equivalently, the measurement error) is given by

K^ = 1 + 2e cos(6' - /?) + e^ (6a)

We note that if cos(^-/?) = -|, (66)

then «; = 1 and |5| = r; that is, there is no measurement error. These points are indicated

in Figure 1, where the theoretical and shifted circles intersect.

\S\^ is independent of the measured phase 7, but is dependent on the phase difference — (5. Hence, a non-ideal RCS measurement (a measurement that is contaminated by

interactions and contributions from the environment and the instrumentation) is phase

dependent, as seen in eqs (4) and (6a). Finally, in practice one typically expects e << 1, but this is not required in the analysis.

A contour plot of the RCS measurement error k^ is presented in Figure 2. Both negative

(broken) and positive (solid) contour lines are shown. The contour lines for dB are defined by eq (6b). As the phase difference 6 — f5 varies over all angles, e is seen to vary

significantly for a given value of n. Thus, in addition to specifying k, we need to specify

the phase 6 — (3 to uniquely determine e. In current measurement practice, this phase

information is unavailable.

If in eq (la) S is uncalibrated data, then only the ratio of two uncalibrated data sets can be exhibited in a manner similar to eq (4). Thus,

Si

where the subscripts denote two independent uncalibrated measurements, and the ratio of

measurements removes the system's transfer function from the uncalibrated data set. This

equation has the same form as eq (4), if we define

si ^ ^ , (8)
and similarly for the ratios of Vi and k^, z = 1, 2. Thus,

\S2i? = 4,4,{€2,e2,P2,ei,ei,f3^). (9)

Here subscript 1 refers to a calibration artifact, and subscript 2 refers to a target. Explicitly,

l+2e2Cos(^2-/g2) + e2 l+2eicos(6'i-/?i) + €2

2 _ -^ ^^c^woy^^j fy-jj ^C2

Equation (9) has six measurement parameters. ^21 is simply the ratio of two uncalibrated

measurements S21 to the ratio of the corresponding theoretical cross sections r2i . This

ratio gives the calibrated RCS measurement error due to all sources of error identified in

reference [1]. When r^i is known, K21 can be evaluated from the data.

3. The Dual Calibration Technique

The comparison of the ratio of two uncahbrated measurements on two different targets

(see eq (8)) to the ratio of the corresponding theoretical RCS values (see eq (9)) yields

the "dual calibration error" «2i [3]. The theoretical RCSs of the two target artifacts are usually assumed to be known accurately, so that computational errors are neghgible in this

comparison.

In practice, k^i is usually evaluated using two different-sized simple cahbration artifacts

such as cyhnders or spheres. Typically, dual cahbration errors are ±0.2 dB in the range

from 2 to 18 GHz, when comparing two closely-sized cyhnders from the standard cyhnder

set [3,4]. Large errors, on the order of ±0.4 to ±0.8 dB, have been observed, especially

when cylinders and spheres were compared.

When small values (e.g., up to ±0.2 dB) of «2i are obtained, the usual interpretation is that a good calibration of the RCS measurement system has been achieved. This conclusion, however, may not be warranted, since, specifying measurement errors without phase

information will not uniquely determine the calibration error parameter ei

To graphically illustrate ^21 with its six measurement parameters, we need to make some

simplifying assumptions. First, we make the nonrestrictive and reahstic assumption that

the phase differences in the calibration and target measurements are the same. Then we

assume that £2 = Q^ei, where a is some arbitrary real constant (which can be varied to

explore the behaviour of K21). For purposes of illustration, we will examine two cases: (i) a — 1.01, and (ii) a = 0.5. Case (i) represents a simphfied model of repeatability, and

case (ii) is a typical dual calibration result. The contours of K21 for these two cases are shown in Figures 3 and 4. Qualitatively, the contour plots for the two cases are similar:

for any contour level, ei varies significantly as the phase varies. Hence, an independent

assessment of the measured phase is needed to determine the error parameters ei and 62 uniquely from a dual calibration result. We observe then that small dual calibration errors do not necessarily imply small error parameters.

The dual calibration error K21 in eq (9) is simply a known measurement error valid specifically for the two targets under consideration. Generally vahd cahbration errors cannot be
o

d

CSJ o

o

d

-100 100 e-p

Figure 3. The repeatability of RCS calibrations. Contours of the RCS

measurement error «;2i as a function of the phase — f3 (assumed the

same for both measurements) and the dimensionless error parameters

ei and €2 = l.Olei. Contour lines from -0.02 dB (broken) to +0.02 dB

(solid) in steps of 0.002 dB are shown. For a given K21 the phase —

must be specified to uniquely determine ei

.

deduced from such a comparison! The dual calibration technique simply does not provide

enough information to determine the error parameters 61,2- We want to determine the

calibration error ei, since this error is propagated into all subsequent measurements on

real targets.

The dual calibration technique does not determine the calibration error parameters 61,2

uniquely, because important phase information is not available and is not included in the

analysis. As indicated in eq (la), we must specify phase information to determine and

remove unwanted errors signals from the data.
o

COd

CMd

o

d

-100 100

Figure 4. RCS measurement errors. Contours of K21 as a function of the

phase 9 — 13 (assumed the same for both measurements) and the error parameters ei and e^ = 0.5ei. Contour hues from —1.2 dB (broken) to

+ 1.2 dB (solid) in steps of 0.2 dB are shown. For a given «2i the phase

9 — (5 must be specified to uniquely determine e\.

4. Phase-Dependent RCS Data Analysis

In current practice, a large number of RCS measurements on a stationary target are integrated to minimize noise and to reduce the I-Q circularity error in the data. In these

measurements, the phase ^ in eq (1) is constant for a stationary target, and the two parameters r and h of the received electric field signal 5, modeled in eq (la), cannot be

separated. However, the phase

^ = yrf, (11)
where A is the electromagnetic wavelength, and d is the distance of the target from the

radar, will vary from to 27r if we vary the distance d over A/2. The resulting variation

in the amplitude of the electric field can be safely neglected, since d/X » 1. Given such

phase- dependent data we can isolate the two parameters of the signal using the method of

weighted orthogonal-distance regression [5-7].

The orthogonal-distance regression-analysis technique is especially well suited to problems

in which all the model variables have significant errors. In eq (10) the quantities {Sj,Sq)

have measurement errors, and we seek to minimize the sum of squares of the orthogonal

04 -

0.2

E

O 0.0

-j

0.2 -

-04

-0.4

I HH IQ data

-0.4

"~T

-0.2 —r—

0.0 2

-0.2 _J

0.0

I

0.2

I

0.4

1 VV IQ data

I I 0.4

(m)

Figure 5. RCS measurements on a 12-inch diameter sphere. The theoretical

circles with centers at the origin are mostly above or below the data points. The

centers of the data circles (indicated by + signs) have been shifted away from the

origins by the background signals. See Figures 1 and 6.

distances between each data point and the circle described by the model equation (10).

The correlation between the variables 5/ and Sq are automatically taken into account by

the model equation, and the analysis can be used to determine the location of the center

of the circle (6/,6q) and the radius of the circle r. We consider two cases:

i. the calibration model, where the RCS of the calibration target r^ is known,

ii. the unknown-target model, where the RCS of the target is unknown.

9
In this approach, reproducibility of the results, when both models are applied to the same

calibration artifact (with known RCS), is easily checked.

Figure 5 shows (5/,5'q) data obtained using a 12-inch diameter sphere mounted on a foam column. Both the HH and VV polarizations were measured with a pulse repetition frequency of 5000 s~^ For each polarization approximately 6E+5 data points were

recorded; the distance d to the target was ^^610 m. The phase of the data varied approximately 100°, corresponding to a distance variation in the location of the sphere of roughly

A/8 « 0.41 cm at 9.24 GHz. This variation in sphere position was caused by steady and

0.4 -

0.2

E

O 0.0

-0.2 -

-0.4 -

HH IQ data

-0,4 -0.2

I. .

I L_

0.0 0.2 0.4

T

-0.4 -0.2 —I

0.0 —

r0.2 r

0.4

VV IQ data

(ml

Figure 6. RCS measurements on a 12-inch diameter sphere. The data circles

with centers indicated by + signs have been shifted away from the origins by the

background signals. The shifted theoretical circles are now embedded in the data.

See Figures 1 and 5.

gusting winds of about 20 km/hour. The theoretical RCS of a 12-inch diameter perfectly

conducting sphere is Vq ^ 0.0713 m^, or tq « 0.267 m.

In Figure 5 the theoretical circles with radii tq are centered at the coordinate origins, and

the centers of the data circles, denoted by the + signs, have been translated away from the

origins {also see Figure 1). Similar data were obtained for a cylinder 5 in in diameter and

5 in in height; the cylinder data plots are qualitatively similar to the sphere plots, and,

therefore, are not shown here.

10
We used ODRPACK [6] to obtain the centers {bi,bQ) and the radius r of the data circles

for the calibration cylinder and sphere. We assumed the initial values to be (0,0) for

the centers and the known theoretical values for the radii. In the calibration model the

theoretical r was held fixed, and in the unknown target model both the center and the radius

of the circle were allowed to vary to obtain nonlinear least-squares solutions. Uncertainty

bounds of 2a were automatically provided by the code.

Figure 6 shows the {Si,Sq) sphere measurements and the centers of theoretical circles

shifted away from the origins to minimize the nonhnear-least-squares residuals. Unlike in

0,04

002

E

O 00

-0.02

-0 04

HH IQ noise and residuals

0.04 -0.02 —I

0.0

-0.04

I

-0.02

I

0.0

I

0.02

I VV IQ noise and residuals

0.04L_

0.02 0.04

(m)

Figure 7. Nonlinear least-squares complex residuals superimposed on the measurement system noise. The residuals are for the solutions shown in Figure 6 with

the parameters given in Table 1 for the Sc artifact (calibration model).

Figure 5, the theoretical circle (the calibration model) is now embedded in the data. In

Table 1, we show the model parameters with their uncertainties for the sphere and cylinder.

Here the subscripts c and t refer to the calibration and unknown target models, respectively.

The uncertainties, indicated in parentheses, are of the order of 1 % or better; that is, all the

parameters are well determined. In the unknown target model, we find a notable difference

between the theoretical values r and the solutions. These systematic (Type B) errors indicate the presence of interactions with the environment and the instrumentation. In

Figure 7 we compare the complex system noise with the complex residuals of the nonlinear

11
TABLE 1. Parameter Solutions* with Uncertainties**

obtained with ODRPACK

Artifact Channel 7o Qo r tq

HH -1.38E-3 3.50£;-3 2.74^-1 2.67E - 1 1.37£;-2

(3.90E - 5) {A.29E - 5) (5.42^ - 5)

W 1.98E-3 3.58^;-

3

2.53E - 1 2.67£; - 1 1.62£;-2

(3.14£;-6) (6.04E-6) (3.99^;- 6)

HH 3.71£;-3 9.12£;-3 2.67£; - 1 2.67E-1 3.69E - 2

(7.83E-6) (7.81£;-6)

W -3.64£;-3 -1.05^-2 2.67E - 1 2.67E - 1 4.16£;-2

(6.10£;-6) (4.21E-6)

Ct HH 2.15E - 2 9.26£; - 3 4.83^ - 1 4.51^-1 4.85E - 2

(1.29E-3) (2.90E-4) (1.31£;-3)

W l.lOE-2 1.76E-3 4.38E-1 4.51^-1 2.54£;-2

(2.42£'-4) (4.16£;-5) (2.37£; - 4)

Cc HH -1.06£;-2 2.50£;-3 A.blE - 1 4.51E - 1 2.41E - 2

{1.59E-b) (3.95^-6)

W 2.36£;-2 4.05£;-3 4.51E - 1 4.51E - 1 b.SlE-2

{1.2AE-5) (6.37^-5)

'All quantities in the table are in meters, except for e, which is dimensionless

'* ±2a uncertainties are shown in parentheses

12
analysis. The residual amplitudes are seen to be of the same order of magnitude as the noise

amplitudes. One can conclude from these plots that the data in Figure 5 are appropriately

represented by the nonlinear model equation (10).

5. Future Efforts

In this study we explored a new measurement and analysis technique that allows us to

determine the generalized background signal in RCS measurements. This background can be subtracted from the data to improve the accuracy of the calibration or measurement.

The results of this study are very encouraging, and warrant further exploration of this

measurement and analysis techniques. We need to design and implement experiments

where the movement of the sphere (or some other target) is under careful experimental

control to demonstrate and support further the validity of the results presented in this

study. This should not be difficult, since only very small excursions, of the order of an

eighth of a wavelength, are needed. Larger excursions should improve the results, provided

we keep /? relatively stable. Currently, steps are being taken to repeat these measurements

under more controlled conditions.

The author thanks Dr. Dale Diamond and Mr. John Liles at the RCS measurement range

at NAWCWD, China Lake, for providing the data used in this study. The author also

thanks Mr. John Denson, Deputy Director, NAWCWD and Mr. Dick Dickson, Head,

Etcheron Valley Range, NAWCWD for their support and encouragement. The support

of the DoD triservice CCG is also acknowledged. The author is appreciative of Dr. Jack

C. M. Wang, Statistical Engineering Division, Information Technology Laboratory, NIST,

who provided guidance in the use of ODRPACK.

References

[1] Wittmann, R.C; Francis, M.H.; Muth, L.A.; Lewis, R.L. Proposed uncertainty analysis

for RCS measurements. Natl. Inst. Stand. Technol. NISTIR 5019, January 1994.

[2] Muth, L. A. Radar cross section calibration errors and uncertainties. Proc, Antenna

Meas. Tech. Assoc, Monterey Bay, CA, pp. 115 - 119; 4 - 8 Oct. 1999.

[3] Chizever, H.M.; Soerens, R.J.; Kent, B.M. On reducing primary cahbration errors in

radar cross section measurements. Proc, Antenna Meas. Tech. Assoc, Seattle, WA, pp.

383 - 388; 30 Sept - 3 Oct. 1996.

[4] Kent, B.M.; Wood, W. D. The squat cylinder and modified bicone primary static RCS

range cahbration standards. Proc, Antenna Meas. Tech. Assoc, Boston, MA, pp. 319 -

324; 17 - 12 Nov. 1997.

[5] Boggs, P.T.; Rogers, J.E.; Orthogonal Distance Regression. Contemporary Mathematics, 112, 1990.

[6] Boggs, P.T.; Byrd, R.H.; Donaldson, J.R.; Schnabel, R.B. Algorithm 676 - ODRPACK:

Software for Weighted Orthogonal Distance Regression. ACM Trans. Math. Software, 15; 1989.

13
[7] Boggs, P.T.; Byrd, R.H.; Rogers, J.E.; Schnabel, R.B. User's Reference Guide for

ODRPACK Version 2.01 Software for Weighted Orthogonal Distance Regression. Natl.

Inst. Stand. Technol. NISTIR 4834; June 1992.

Appendix A

The Data Integrals Over a Finite Phase Interval

In eqs (1) we evaluated the data integrals over the range of phase angles:

27r Sd9 = 27r6e^^ (16)

^2n r \2jn _ o„^^2 \S\''de = 27r{r' + b''). (Ic)

Jo

Ordinarily, data are available only over a finite range of 9. The appropriate integrals then

become:

Sd9 = be'^{92 -9i)- ir{e'^' - e'^' ) (^1)

/ \S\^d9 = {r^ + b^){92 - 9i) + 2r6lsin(^2 -^=) - sin(^i - /?)]. (^2)

These form a simultaneous set of equations that can be solved for the basic parameters r,

6, and /?, if S is known as a function of 9. We still encounter the difficulty pointed out in

the main text. Because only the phase 7 of 5 is known (see eq (2)), these equations are nonlinear in the basic parameters and cannot be solved algebraically.

14
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