An Improved Antenna Group Delay Measurement Method Using a Three-antenna Extrapolation Technique

In order to minimize the error due to multiple reflections between antennas in the conventional group delay (GD) measurement, an improved antenna GD measurement method is proposed. In this method, antenna group delay is measured as a function of distances using a three-antenna extrapolation method. The GD is determined by averaging a set of measured GD values according to a derived multiple-reflection error model. Measurement in frequency band of (1575.42 ± 16) MHz for a circularly polarized helical antenna is presented, which gives the detail measurement procedures and validates the method. The uncertainty evaluation for this measurement was carried out as well, and an expanded uncertainty of 0.20 ns (k = 2) has been achieved. One more measurement example in frequency band of (4000 ± 10) MHz for a standard gain horn antenna with an expanded uncertainty of 0.12 ns (k = 2) is also presented briefly in this paper.


Introduction
Antenna is an essential part of a transmitting or receiving system that is designed to radiate or to receive electromagnetic waves [1].Antenna performance is characterized by various parameters which can be obtained through measurements [2], [3].In the past, antenna measurements always aimed at gain, pattern, efficiency and some other main parameters [4][5][6][7][8][9][10].With the development of satellite navigation, remote sensing, modern communication, et al., antenna group delay (GD) is paid more and more attentions nowadays.Especially for global navigation satellite systems (GNSS), antenna GD measurement accuracy will set the boundaries of how well it will perform [11][12][13][14][15].The group delay of the antenna versus aspect angle is defined in the standard of Radio Technical Commission for Aeronautics (RTCA), and its limit is also specified from 0.65 ns to 2.5 ns related to the elevation angle at GPS L1 frequency (1575.42MHz) which shows the requirements for high accuracy antenna GD measurement [16].
Standard antenna method, two identical antenna method and three-antenna method are three conventional antenna GD measurement methods, and the three-antenna method is the most accurate one among them [17], [18].All these methods work at some fixed distances and measurement error caused by multiple reflections between transmitting (Tx) and receiving (Rx) antennas is hardly taken into consideration.In this paper, an improved antenna GD measurement method based on three-antenna extrapolation technique is proposed.In this method, no standard antenna is required and the measurement error caused by multiple reflections can be reduced effectively.
In this paper, the measurement methodology is described in Sec. 2, and Section 3 gives the measurement procedures and results.The uncertainty evaluation is discussed in Sec. 4, and the conclusion is given in Sec. 5.

Methodology
The group delay is defined as follows [19]: where GD(f) is group delay (in second), f is frequency, and φ(f) is phase function (in degree).For engineering purpose, group delay is often calculated using the following approximation: where f c is center frequency, f 1 and f 2 are any two different frequencies on opposite sides of f c within the required frequency range.

Conventional Three-antenna GD Measurement
Three-antenna method does not need any standard antenna, but two auxiliary antennas (marked: A and B) are required besides the antenna under test (AUT, marked: C).The measurement is divided into three groups and two antennas are contained in each group.The measuring sequence is shown in Tab. 1. Taking Group 1 for example, equation ( 3) is obtained: where φ AB is the total phase function of the insertion loss measured between Tx and Rx antennas, φ ab is the sum of the phase function of antenna A and B, φ d is the phase function associated with measuring distance, φ 0 is the phase function of the measurement system itself measured in case of a through connection, which is defined as the Tx output and Rx input transmission lines connected directly.
Taking (3) into (2), the group delay can be obtained, as shown in (4): where GD AB , GD ab , GD d and GD 0 are group delays corresponding to phase functions φ AB , φ ab , φ d and φ 0 , GD a and GD b are group delays of antenna A and B respectively, d is the distance between the apertures of Tx and Rx antennas, and c is the speed of light.
After Group 1 is finished, GD ab is calculated by solving the above equations.Similarly, the GD ac and GD bc are calculated after Group 2 and 3 are finished.Finally GD c is calculated by solving (7): .

Improved GD Measurement Method
During the process of antenna measurement, the electromagnetic wave propagates from Tx antenna to Rx antenna, but only a part of the arriving wave is received by Rx antenna and other is reflected back into the space.After the reflected wave arriving at Tx antenna, a part of it is reflected back towards Rx antenna similarly, so in this way the multiple reflections are formed between the two antennas and this process is repeated endlessly.Due to multiple reflections, φ(f) and dφ(f) are measured differently at different distances, and this causes group delay measurement errors obviously.Here, in order to simplify the analysis process, we ignore higher order reflections and only consider that the wave is reflected by Rx and Tx antennas for one time individually.The reflected wave is superposed with direct wave and then they are received together by Rx antenna.
The expression of the electric field at the aperture of Rx antenna is: where E represents electric field, E 0 is the amplitude of E, β 1 and β 2 are scattering characteristics of Rx and Tx antennas looking into the antenna aperture, β = β 1 β 2 , k = 2π/λ [20].
Using Euler's formula, equation ( 8) can be written as: In most of our practical measurements, for minimizing the antenna multiple reflections, the measurement is carried out as far as possible subject to a sufficient signalnoise ratio, which results that β/d 2 is much smaller than one, and (10) can be assumed: So equation ( 9) can be written as: Also because β/d 2 is much smaller than one, the following approximation is made: So equation ( 11) can be written as: Clearly, the phase of E is: The phase error caused by multiple reflections is: So the group delay multiple-reflection error model is: where ΔGD is group delay measurement error caused by multiple reflections.
Using (16) to calculate ΔGD and correct the measurement results is a direct way to eliminate ΔGD.In order to calculate ΔGD accurately, β 1 and β 2 should be determined in prior, and this process is very complicated and timeconsuming.
In the improved method, firstly GD cn is measured using extrapolation range where GD cn (n = 1, 2, 3, …, N) is the group delay of antenna C measured at N different distances, and then the mean value of GD cn is calculated as the measurement result as shown in (17).According to (16), ΔGD is a cosine function of d and the period is λ/2, so its positive parts and negative parts are counteracted by averaging, and ΔGD is reduced in this way.

Antenna Extrapolation Range
The measurement is performed in a newly constructed antenna extrapolation range at the National Institute of Metrology (NIM), China [2].The transmitting carriage of the system can move freely along two 10-meter-long precision rails in direction of the longitudinal axis of the range (z-axis).The rails are aligned and installed precisely on the chamber ground with maximum deviations of 0.08 mm in vertical and 0.06 mm in horizontal, respectively.The transmitting carriage is guided by a ball-screw driver with a maximum speed of 80 mm/s and positioning resolution of 1 μm.The actual relative movement of the carriage is determined by a laser interferometer with error of 0.1 μm.The photo of the chamber is shown in Fig. 1.

Measurement Procedures
In this part, practical measurement of group delay for a circularly polarized helical antenna is given.The main measurement setup involved is summarized in Tab. 2. The measurement starts at 5 meters and with an extrapolation range of 0.96 m, i.e., five times wavelength with respect to 1575.42 MHz (5 λ L1 ).
The system is set up according to Fig. 2. Cables with N type connectors are used to connect antennas and other equipment in the system.Taking Group 1 as an example, firstly antenna B is mounted on Rx tower, and then it is aligned using optical alignment system until the normal of the antenna aperture is parallel with z-axis.Then, antenna A is mounted on transmitting carriage and its attitude is adjusted using the adjustable mounts until the apertures of A and B are well aligned.First of all, a so-called "through connection" measurement is performed.Tx and Rx antennas are taken off and the cables are connected together using an adaptor.The signal generator output power is adjusted to a proper level to ensure the mixers working in a linear region, and then GD 0 is measured.The GD of the adaptor itself should be measured individually to compensate GD 0 .Throughconnection is performed before each group and after the last group, four times in total.
Then Antenna A and B are mounted on transmitting carriage and Rx towers respectively and aligned once again, and then GD AB is obtained at different distances by the insertion loss measurement.Following a similar procedure, the GD AC and GD BC are determined.
The group delay of AUT on each position GD cn is calculated according to theory described in Sec.2.2, and then GD cMean is calculated according to (17).
The mismatch between the antenna and the cable it connects can lead to GD measuring error, so mismatch correction is necessary.Considering mismatch, the following equations are obtained from (7): where ΔGD ij is GD measurement error caused by the mismatch between antenna i (i = a, b or c) and the cable it connects, and j (T or R) is the notation of transmitting or receiving cable.GD klMis is the sum of group delay of antenna k and l (k = a or b, l = b or c) considering mismatch.
By solving (18), the group delay of antenna C (GD c ) after mismatch correction can be obtained: Comparing GD c with the solution of (7), the mismatch correction factor is obtained as shown in (20) where Δφ R is the phase measuring error caused by the mismatch between Rx antenna (i.e., antenna B and C specified in Tab. 1) and the cable it connects, Δφ bT is the phase measuring error caused by the mismatch between antenna B and its cable when B is used as a transmitting antenna.Γ R and Γ B are the reflection coefficients looking into the Rx cable and antenna B feeding port respectively [19].
The final group delay measurement result of antenna C after mismatch correction is shown in (25):

Measurement Results
The GD cN values at 1575.42 MHz frequency with respect to different distances are shown in Fig. 3.A curve fitted according to (16) and GD cN is also shown in this figure and its expression is where A 1 and A 2 are frequency-dependent fitting coefficients.We can see that GD cN varies with distance periodically and the period is approximately λ L1 /2, the same as the fitted curve in accordance with (16).Although the amplitudes of GD cN at some distances are a little different from the fitted curve, this is still reasonable considering the cable flexure, the random error and other reflections from the absorbers on the walls, ceiling and floor.Generally speaking, the variation of GD cN is in accordance with ( 16) mainly caused by multiple reflections and it is proved that ΔGD can be reduced according to the theory described in Sec.2.2.In fact, the measurement process of three-antenna extrapolation method is similar as that of conventional three-antenna method when it is performed at N different distances in series.So according to Fig. 3, the measurement results vary from 2.27 ns to 2.59 ns if using conventional three-antenna method compared with that of 2.43 ns if using the proposed method, and it is proved that the proposed method is improved in measurement accuracy.
GD c at different frequencies is shown in Fig. 4, which shows the GD variation with frequency of antenna C.

Measurement for a Standard Gain Horn Antenna
One more measurement example is performed in frequency band of (4000 ± 10) MHz with frequency step of 0.5 MHz using Narda 643 SGH antenna following the same procedure with the above measurement.In this measurement, antennas are connected to waveguides directly without using waveguide-to-coaxial transducer in order to guarantee better stability of the system.
The antenna GD measurement result at 4 GHz frequency point vary from 1.38 ns to 1.45 ns with distance, and the mean value is 1.42 ns.In the frequency band of (4000 ± 10) MHz, the measurement results vary from 0.86 ns to 1.78 ns with frequency and an expanded uncertainty of 0.12 ns is achieved.

Uncertainty Evaluation
The uncertainty budget for the measurement in GPS L1 frequency band is shown in Tab. 3. The analysis according to [21] of each source will be given as follows:  u m : measurement repeatability: Because group delay measurement result GD c is the mean value of N times measurement results, u m is determined by (27): mMax m 0.1604 ns 0.0179 ns 80 where std mMax is the maximum standard deviation of N times measurement results over all the frequency points.

 u AC : antenna connection repeatability:
To evaluate u AC , the antenna is connected and reconnected to the cable 10 times and GD is measured every time when the antenna is connected.u AC is shown as follows: where std ACMax is the maximum standard deviation of the 10 times measurement results over all the frequency points.
 u CC : cable connection repeatability: The evaluation process of u CC is similar with that of u AC where Tx and Rx cables are connected and reconnected 10 times instead and the result is 0.0094 ns.
 u CF : cable flexure: During the process of extrapolation measurement, the LO cable is flexing with the movement of the transmitting carriage, and as a result GD of the measurement system itself is changing.
To evaluate the u CF , a section of LO cable on transmitting carriage is bended at different parts and GD is measured under through-connection.u CF is determined by (29): 0.0440 ns 0.0311 ns 2 2 where ΔGD SMax is the maximum deviation among the group delay values measured when the cable is bended at different parts over all the frequency points.
 u D : RF subsystem drift: During the measuring process, GD of the measurement system itself is drifting with time, and as a result, u D is caused.As described in Sec.3.2, GD 0 is measured 4 times in total from GD 01 to GD 04 , and equation ( 30) is obtained: where i = 2, 3, 4, j = i -1, ΔGD 0ij is the deviation of GD 0i and GD 0j .u D is shown in (31): 0Max D 0.0105 ns 0.0061 ns 3 3 where ΔGD 0Max is the maximum ΔGD 0ij over all the frequency points.
 u AM : antenna misalignment: Azimuth, pitch and roll are three kinds of typical error angles in antenna misalignment, and the uncertainty caused by them are u AM1 , u AM2 and u AM3 respectively.According to our practices, azimuth error angle is often less than 0.5°, so when evaluating u AM1 , GD is measured under azimuth angle ±0.5° and compared with the one measured under 0° individually.u AM1 is shown in (32): AM1Max AM1 0.0800 ns 0.0566 ns 2 2 where ΔGD AM1Max is the maximum absolute value of the above comparing results over all the frequency points.
The evaluation process of u AM2 and u AM3 is similar with that of u AM1 , where u AM2 = 0.0566 ns and u AM3 = 0.0116 ns.
 u SN : RF subsystem nonlinearity: During the process of extrapolation measurement, the received signal strength varies with d.Because of the nonlinearity of the mixers, different input power will cause different responses, and as a result u SN is caused.To evaluate u SN , the distance is set to 5.5 m and GD measured under signal generator output power 0 dBm, -10 dBm, -15 dBm and -20 dBm are compared with the one measured under -5 dBm individually.u SN is shown in (34): SNMax SN 0.0084 ns 0.0048 ns 3 3 where ΔGD SNMax is the maximum absolute value of the above comparing results over all the frequency points.
 u R : residual error: According to (16), the amplitude of ΔGD varies with d, which means the positive parts and negative parts of ΔGD cannot be counteracted totally when calculating the mean value of GD cN .The residual error is: The expressions of ΔGD R and ΔGD P1 are calculated according to (16) at 1575.42 MHz and shown in Tab. 4, where ΔGD P1 is the first peak amplitude of ΔGD.
According to Fig. 3, ΔGD P1 is not more than 0.16 ns, so ΔGD R is not more than 0.0011 ns, which is too small to be taken into consideration.ΔGD R on other frequency points is similar with this.
 u d : laser interferometer positioning error: The laser interferometer positioning error is Δd = 0.1 μm, so the group delay measurement error caused by Δd is:

Conclusion
Accurate antenna GD measurement is becoming more and more important for satellite navigation, remote sensing, modern communication and other military and civil application fields.In this paper, an improved antenna group delay measurement method based on the three antenna extrapolation technique is proposed.A practical group delay measurement for a circularly polarized helical antenna validates the derived multiple-reflection error model, and the detail procedures give an example for implementation of this method.The measurement error due to multiple reflections is reduced by 0.16 ns compared with the conventional three-antenna method at GPS L1 frequency.The uncertainty of this measurement is evaluated as well, and an expanded uncertainty of 0.20 ns has been achieved for this case.One more measurement example in frequency band of (4000 ± 10) MHz for a standard gain horn antenna is also presented briefly with an expanded uncertainty of 0.12 ns.The implementation of this method for active antenna will be the emphasis of further study.

A
vector network analyzer (VNA) together with external mixers are used in the radio frequency (RF) measurement subsystem of the antenna extrapolation range.The VNA provides local oscillation (LO) signal and receives intermediate frequency (IF) signal, which is the product of frequency mixing.LO signal goes into reference/test mixers individually after processed by LO/IF distribution

Fig. 2 .
Fig. 2. System schematic of antenna extrapolation range.unit,and then mixed with RF transmitting and receiving signal to generate IF signal.System schematic is illustrated in Fig.2.

Fig. 3 .Fig. 4 .
Fig. 3.The GD cN values and the fitted curve at GPS L1 frequency with respect to different distances.

Tab. 4 .
which means that the ΔGD d is negligible.The values and expressions of ΔGD R and ΔGD P1 .