Research on Probe Position Error Analysis and Correction in Antenna Time-domain Near-field Measurement System

Bin Cui, Zheng-hui XUE, Xiao-wen XU Electronic Engineering Department

Beijing Institute of Technology, Beijing, 100081 China zhxue@bit.edu.cn

For the antenna time-domain near-field measurement system, the probe position error is one of the most important error sources.Traditionally, many methods of its analysis and correction have been presented. However, these methods work only when the positionerror is much smaller than the AUT’s wave length. Besides, the computation of these traditional methods is always large. This paper bases on the time-domain to frequency-domain transform of the practical time-domain measurements, applies a new method of unequally spaced near-field to far-field transformation to the measurement data. The new algorithm just needs the known theory locations of sampling points and the approximate probe position errors instead of the usual restriction that sampling points be located on a plane rectangular grid. The final measurements illustrate that the algorithm can eliminate the effect caused by probe positionerror which improves the accuracy of the test.

Index Terms—Antenna time-domain near-field measurement, probe position error, unequal spaced near-field to far-field transformation

I. INTRODUCTION

With the development of the measurement techniques, people pay more and more attention to the “accuracy” of a measurement. Although there are many kinds of errors existing in the measurement systems, probe position error is more essential. Therefore, how to eliminate the position errors becomes an important task in the process of a measurement. When the probe scanned, the ideal situation is all the scan lines are fully parallel and perpendicular, but the record data is collected from an interval not an isolated point, for the reason that the position error is produced. Generally, there exists X-Y and Z axis error in a measurement system, Z axis error affects the edge flap and X-Y axis error affects the main lobe. At present, the researchers can give the quantitative magnitude of the position errors[1] and some researchers have studied the error correction methods to the probe’s position errors in the antenna near field measurement[2][3]. However, these methods are proved to be effective only when the position interval is much smaller than the wavelength, so the antenna measurement requires a higher standard accuracy of mechanical positioning. But it is hardly practical for the manufacturing techniques and the cost of the machines at present.

Traditionally, the uniformly spaced sampling fast Fourier transforms algorithm is applied in the antenna near field measurement. The fast Fourier transforms requires the measurement points locate on the same interval of a rectangular grid and the sampling interval must be smaller than 0.5� , it’s hardly practical. To overcome the difficulties mentioned above, we present the non-uniform near field to far field transform when the maintenance of positioning tolerances becomes more difficult as frequency is increased. It doesn’t require the measurement points to locate on a uniform rectangular grid, only requires using laser positioning device to obtain the accurate position data of the measurement points.

Then the AUT’s far field pattern can be computed by using the unequally spaced fast Fourier transform algorithm.

II. BASIC METHOD

The basic theory of antenna near field measurement is mode expansion method, it means that to use a probe whose characters are known to get electric field amplitude and phase of every measurement point on a plane of the AUT’s near field. According to the Kerns plane wave coupling matrix theory[4], as long as the relationship between probe response and spectral function of AUT built, the AUT’s far field pattern can be got.

According to the plane wave theory, any electromagnetic field can be extended as:

( , , ) ( , )x y

jk rx y

k k

E r b e dk k� � � � � �� � � (1)

where sin cosxk k � �� , sin sinyk k � �� , 2 2 2 2z x yk k k k� � � ,

k is the wave number, b( , )� � becomes spectral function. It is assumed that the probe is on the point(x,y,d),

considering a single plane wave, the above function may be modeled as:

(2) ( )'0 0 ' ( ) ( )x y zj k x k y k db a F t K e s K� � � �

where is the complex amplitude of reflection wave of the

receiving probe transmission line port, is the complex amplitude of incident wave of the AUT transmission line port,

is the antenna transmission coefficient,

'0b

0a

( )t K ( )s K is the receiving probe coefficient. 'F is the impedance mismatch factor.

The probe response function can be got after overlaying all the effects of plane wave:

( )' '0 0( ) ( ) ( ) x y zj k x k y k d

x yb P a F t K s K e dk k� � � ��� (3)

It is assumed that ( , ) ( , ) ( , )x y x y x yD k k s k k t k k� � , also the inverse Fourier transformation may be modeled as: ___________________________________

978-1-4244-8559-8/11/$26.00 ©2011 IEEE

'02 '

0

1( , ) ( ) exp( )4x yD k k b r jk r dxdy

a F� �� � (4)

So, (3) and (4) constitute a pair of Fourier transformation of antenna near field measurement. Usually, different

( , )x yD k k and '' ( , )x yD k k must be got in two independent near field measurements, we rotate the probe : 090

( , ) ( ) ( )ˆ ˆ ˆ ˆ ( ) ( )

( )

x y

m c m c

m m c c

D k k t k s kt m t c s m s ct s t s

�

�

�

��

'' ( , ) ( ) ( )ˆ ˆ ˆ ˆ ( ) ( )

( )

x y

m c c m

m c c m

D k k t k s kt m t c s m s ct s t s

�

� �

�� (5)

where , are AUT’s main polarization component and

cross polarization component; , are probe’s main polarization component and cross polarization component. To solve (5):

mt ct

ms cs

''

''2 2

1

m c m m

c m c c

m c

c mm c

s s t tDs s t tD

s s Ds s Ds s

� � � � �� �� � � � � �� �

� � � � � � ��� �

� �

�� �

� � � � ��� � �� �the polarizability is defined as :( ) ( ) / ( ) 1s c mk s k s k� � �

''

2

( ) ( )( )( ) ( )( )

1 ( )

s

m mm

s

k D kD ks k s kt k

k

�

�

��

�''

2

( ) ( )( )( ) ( )( )

1 ( )

s

m mc

s

k D kD ks k s kt k

k

�

�

��

� (6)

To get the AUT’s spectral function , and antenna far field pattern:

mt ct

00( ) ( , )jk rx y

j aE r e t kr

k� �� (7)

where � = = .zk cos cosk A EThen use unequally spaced fast Fourier transform algorithm

to solve the near field to far field equation, (2) is modeled as series form:

(8) 0 ( ) vu njk rvu

v ub r e� ����

where ˆ ˆ ˆvu v u vuk k x k y z�� , 2 20vu v uk k k� � � � 2

x n xL x L� � � , y nL y L� � � y

From (4) and (8), the following equation can de deduced:

2 '

0 ( , )vu v ux y

a F D k kL L

� � (9)

(8) is modeled as matrix form: b Q�� (10)

where ( )nb b r� , ,,vu njk r

n vuQ Q e�� � vu� ��The essence of the near field to far field transform

algorithm is using probe response matrix b to solve (10) and get unknown number vu� ,then calculate ( , )x yD k k from (9). Finally, the AUT’s far field pattern can be obtained.

III. SIMULATION

We began with an S-band standard gain horn with a sample surface size of 1000mm X 1000mm and an operating frequency of 3.95 GHz. The sample surface to the AUT distance is 320mm. These data consist of 121 points in x by 121 points in y on an ideal grid space.

For the first case, we used a position error is 0.16� of the form

0.1cos(0.35 ) cos(0.65 )

0.1cos(0.25 ) cos(0.15 )

0.1cos(0.15 ) cos(0.11 )

x

y

z

m n

m n

m n

�

�

�

��

� � � �� � � �� � � �� � � �� � � �

where m is the x index and n is the y index. The root mean square error is 0.07� .

-100 -80 -60 -40 -20 0 20 40 60 80 100-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

farfield/dB

elevation/deg

Theory pattern NSFFT FFT

Fig.1. H-plane farfield pattern of the horn. The solid line corresponds to the actual pattern. The solid line with stars corresponds to the pattern simulated by NSFFT. The dashed line with stars shows the pattern simulated by FFT.

From Fig. 1 we can find that when the maximal position error is 0.16� , the result simulated by traditional fast Fourier transform algorithm gets worse. Otherwise the result got by unequally spaced fast Fourier transforms algorithm is almost the same with actual pattern.

For the second case was a more severe measurement, in this test we increased the position error to 0.47� , the root mean square error is 0.25� of the form

IV. CONCLUSION

From the computer simulations it is proved that if the root mean square error is bigger than 0.17 � ,the results got by traditional fast Fourier transform algorithm are very different from actual patterns; if the root mean square error is bigger than 0.24 � , the results got by traditional fast Fourier transform algorithm have no similarities compared with original patterns. However, almost the same results compared with actual patterns can be got in a shorter time by adopting the unequally spaced fast Fourier transform algorithm. Compared with traditional error correction methods, this algorithm can deal with bigger position errors which are several times than the traditional ones. It is very important to improve the upper frequency limit of existing equipments.

0.25 cos(0.35 ) cos(0.65 )

0.25 cos(0.25 ) cos(0.15 )

cos(0.15 ) cos(0.11 )

x

y

z

m n

m n

m n

�

�

�

��

� � � �� � � �� � � �� � � �� � � �

-100 -80 -60 -40 -20 0 20 40 60 80 100-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

farfield/dB

elevation/deg

Theory pattern NSFFT FFT

REFERENCES[1] Allen C. Newell, Error Analysis Techniques for Planar Near-Field

Measurements, IEEE Trans, Vol. 36, No. 6, June 1988 [2] Ronald C Wittmann, Bradley K Alpert, and Micheal H Francis. Near-

Field Antenna Measurements Using Nonideal measurement Location. IEEE Transactions on Antennas and Propagation, 1998, 24, (5): 716-722.

[3] Lorant A Muth and Richard Lewis. A General Technique to Correct Probe Position in Near-Field Measurements to Arbitrary Accuracy. IEEE Transaction on Antennas and Propagation, 1990, 38, (12): 1925-1932

[4] D.M.Kerns, Plane-wave scattering-matrix theory of antennas and antenna-antenna interaction. NBS. Monogragh 162,1981

[5] Thorkild B. Hansen, Arthur D. Yaghjian. Formulation of probe-corrected planar

Fig.2. H-plane farfield pattern of the horn. The solid line corresponds to the actual pattern. The solid line with stars corresponds to the pattern simulated by NSFFT. The dashed line with stars shows the pattern simulated by FFT. [6] Near-field scanning in the time domain. IEEE Transactions on

Antennas and Propagation, 1995, 43(6): 569-584.

From the Fig. 2 we can find that the traditional fast Fourier transform algorithm can not deal with the data with such a big position error. Otherwise, the unequally spaced fast Fourier transforms algorithm still keeps a higher accuracy of the computation.

[7] Thorkild B. Hansen, Arthur D. Yaghjian. Planar near-field scanning in the time domain, part 2: sampling theorems and computation schemes. IEEE Transactions on Antennas and Propagation, 1994, 42(9): 1292-1300.